Datasets:

TopicNet

/

NIPS

like 0

Follow

TopicNet 1

Receptive field formation in natural scene environments: comparison of single cell learning rules Brian S. Blais Brown University Physics Department Providence, Rl 02912 N.lntrator School of Mathematical Sciences Tel-Aviv University Ramat-Aviv, 69978 ISRAEL H. Shouval Institute for Brain and Neural Systems Brown University Providence, Rl 02912 Leon N Cooper Brown University Physics Department and Institute for Brain and Neural Systems Brown University Providence, Rl 02912 Abstract We study several statistically and biologically motivated learning rules using the same visual environment, one made up of natural scenes, and the same single cell neuronal architecture. This allows us to concentrate on the feature extraction and neuronal coding properties of these rules. Included in these rules are kurtosis and skewness maximization, the quadratic form of the BCM learning rule, and single cell ICA. Using a structure removal method, we demonstrate that receptive fields developed using these rules depend on a small portion of the distribution. We find that the quadratic form of the BCM rule behaves in a manner similar to a kurtosis maximization rule when the distribution contains kurtotic directions, although the BCM modification equations are computationally simpler. B. S. Blais, N. Intrator, H. Shouval and L N. Cooper 424 1 INTRODUCTION Recently several learning rules that develop simple cell-like receptive fields in a natural image environment have been proposed (Law and Cooper, 1994; Olshausen and Field, 1996; Bell and Sejnowski, 1997). The details of these rules are different as well as their computational reasoning, however they all depend on statistics of order higher than two and they all produce sparse distributions. In what follows we investigate several specific modification functions that have the. general properties of BCM synaptic modification functions (Bienenstock et al., 1982), and study their feature extraction properties in a natural scene environment. Several of the rules we consider are derived from standard statistical measures (Kendall and Stuart, 1977), such as skewness and kurtosis, based on polynomial moments. We compare these with the quadratic form of BCM (Intrator and Cooper, 1992), though one should note that this is not the only form that could be used. By subjecting all of the learning rules to the same input statistics and retina/LGN preprocessing and by studying in detail the single neuron case, we eliminate possible network/lateral interaction effects and can examine the properties of the learning rules themselves. We compare the learning rules and the receptive fields they form, and introduce a procedure for directly measuring the sparsity of the representation a neuron learns. This gives us another way to compare the learning rules, and a more quantitative measure of the concept of sparse'representations. 2 MOTIVATION We use two methods for motivating the use of the particular rules. One comes from Projection Pursuit (Friedman, 1987) and the other is Independent Component Analysis (Comon, 1994). These methods are related, as we shall see, but they provide two different approaches for the current work. 2.1 EXPLORATORY PROJECTION PURSUIT Diaconis and Freedman (1984) show that for most high-dimensional clouds (of points), most low-dimensional projections are approximately Gaussian. This finding suggests that important information in the data is conveyed in those directions whose single dimensional projected distribution is far from Gaussian. Intrator (1990) has shown that a BCM neuron can find structure in the input distribution that exhibits deviation from Gaussian distribution in the form of multimodality in the projected distributions. This type of deviation is particUlarly useful for finding clusters in high dimensional data. In the natural scene environment, however, the structure does not seem to be contained in clusters. In this work we show that the BCM neuron can still find interesting structure in non-clustered data. The most common measures for deviation from Gaussian distribution are skewness and. kurtosis which are functions of the first three and four moments of the distribution respectively. Rules based on these statistical measures satisfy the BCM conditions proposed in Bienenstock et aI. (1982), including a threshold-based stabilization. The details of these rules and some of the qualitative features of the stabilization are different, however. In addition, there are some learning rules, such as the ICA rule of Bell and Sejnowski (1997) and the sparse coding algorithm of Olshausen and Field (1995), which have been used with natural scene inputs to produce oriented receptive fields. We do not include these in our comparison be- RF Formation in Natural Scenes: Comparison of Single Cell Learning Rules 425 cause they are not single cell learning rules, and thus detract from our immediate goal of comparing rules with the same input structure and neuronal architecture. 2.2 INDEPENDENT COMPONENT ANALYSIS Recently it has been claimed that the independent components of natural scenes are the edges found in simple cells (Bell and Sejnowski, 1997). This was achieved through the maximization of the mutual entropy of a set of mixed signals. Others (Hyvarinen and Oja, 1996) have claimed that maximizing kurtosis can also lead to the separation of mixed signals into independent components. This alternate connection between kurtosis and receptive fields leads us into a discussion of ICA. Independent Component AnalYSis (ICA) is a statistical signal processing technique whose goal is to express a set of random variables as a linear mixture of statistically independent variables. The problem of ICA is then to find the transformation from the observed mixed signals to the "unmixed" independent sources. The search for independent components relies on the fact that a linear mixture of two nonGaussian distributions will become more Gaussian than either of them. Thus, by seeking projections which maximize deviations from Gaussian distribution, we recover the original (independent) signals. This explains the connection of ICA to the framework of exploratory projection pursuit. 3 SYNAPTIC MODIFICATION RULES In this section we outline the derivation for the learning rules in this study. Neural activity is assumed to be a positive quantity, so for biological plausibility we denote by c the rectified activity (T(d . m), where (T(.) is a smooth monotonic function with a positive output (a slight negative output is also allowed). (T' denotes the derivative of the sigmoidal. The rectification is required for all rules that depend on odd moments because these vanish in symmetric distributions such as natural scenes. We study the following measures(Kendall and Stuart, 1977, for review) : Skewness 1 This measures the deviation from symmetry, and is of the form: 51 = E[c3 ]j E1.5[C2]. (1) A maximization of this measure via gradient ascent gives = \.5E [c (c - E[c 3]jE[c2]) (TId] eM where em is defined as E[c 2 ]. "V51 = \ .5E eM [c (c - E[c 3]jeM) (TId] (2) Skewness 2 Another skewness measure is given by 52 = E[c3 ] - E1.5[C2]. (3) This measure requires a stabilization mechanism which we achieve by requiring that the vector of weights, denoted by m, has norm of 1. The gradient of 52 is "V52 = 3E [c2 - cJE[c2]] = 3E JeM) (TId] ,II 11= 1 (4) [c (c - m Kurtosis 1 Kurtosis measures deviation from Gaussian distribution mainly in the tails of the distribution. It has the form Kl = E[c 4 ]jE2[C2] - 3. (5) This measure has a gradient of the form 1 "VKl = -2E eM [c (c2 - E[c4]jE[c2]) (TId] 1 = -2E eM [c (c 2 - E[c4]je M) (TId]. (6) 426 B. S. Blais, N. Intrator. H. Shouval and L N. Cooper Kurtosis 2 As before, there is a similar form which requires some stabilization: K2 = E[c4 ] - 3E2[C2]. (7) This measure has a gradient of the form 'V K2 = 4E [c3 - cE[c2]] = 3E [c(c2 - eM )](1'd], II m 11= 1. (8) Kurtosis 2 and ICA It has been shown that kurtosis, defined as K2 = E [c 4 ] - 3E 2 [c 2 ] can be used for ICA(Hyvarinen and Oja, 1996). Thus, finding the extrema of kurtosis of the projections enables the estimation of the independent components. They obtain the following expression m = ~ (E- l [ddT] E [d(m? d)3] - 3m). (9) which leads to an iterative "fixed-point algorithm". Quadratic BCM The Quadratic BCM (QBCM) measure as given in (Intrator and Cooper, 1992) is of the form QBCM = !E[c3 ] 3 !E2[C2]. 4 (10) Maximizing this form using gradient ascent gives the learning rule: (11) 4 METHODS We use 13x13 circular patches from 12 images of natural scenes, presented to the neuron each iteration of the learning. The natural scenes are preprocessed either with a Difference of Gaussians (DOG) filter(Law and Cooper, 1994) or a whitening filter (Oja, 1995; Bell and Sejnowski, 1995), which eliminates the second order correlations. The moments of the output, c, are calculated iteratively, and when it is needed (Le. K2 and 8 2 ) we also normalize the weights at each iteration. For Oja's fixed-point algorithm, the learning was done in batches of 1000 patterns over which the expectation values were performed. However, the covariance matrix was calculated over the entire set of input patterns. 5 5.1 RESULTS RECEPTIVE FIELDS The resulting receptive fields (RFs) formed are shown in Figure 1 for both the DOGed and whitened images. Every learning rule developed oriented receptive fields, though some were more sensitive to the preprocessing than others. The additive versions of kurtosis and skewness, K2 and 8 2 respectively, developed RFs with a higher spatial frequency, and more orientations, in the whitened environment than in the DOGed environment. The multiplicative versions of kurtosis and skewness, Kl and 8 1 respectively, as well as QBCM, sampled from many orientations regardless of the preprocessing. 8 1 gives receptive fields with lower spatial frequencies than either QBCM or Kl. 427 RF Formation in Natural Scenes: Comparison of Single Cell Learning Rules This disappears with the whitened inputs, which implies that the spatial frequency of the RF is related to the dependence of the learning rule on the second moment. Example receptive fields using Oja's fixed-point ICA algorithm not surprisingly look qualitatively similar to those found using the stochastic maximization of K 2 ? The output distributions for all of the rules appear to be double exponential. This distribution is one which we would consider sparse, but it would be difficult to compare the sparseness of the distributions merely on the appearance of the output distribution alone. In order to determine the sparseness of the code, we introduce a method for measuring it directly. Receptive Fields from Natural Scene Input DOGed Whitened Output Distribution Output Distribution -20 0 20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 -20 0 20 o 20 fIJ ~~~I A I ?11.11 ~~~I 1\ I ~.11 i1 ~~~I A I ~.II a ~~~I A I ~t! Ii1Ii ~~~VSJ ~? ? ? ~~:I 1\ I 1\1 ?LI g Figure 1: Receptive fields using DOGed (left) and whitened (right) image input obtained from learning rules maximizing (froni top to bottom) the Quadratic BCM objective function, Kurtosis (multiplicative), Kurtosis (additive), Skewness (multiplicative), and Skewness (additive). Shown are three examples (left to right) from each learning rule as well as the log of the normalized output distribution, before the application of the rectifying sigmoid. 5.2 STRUCTURE REMOVAL: SENSITIVITY TO OUTLIERS Learning rules which are dependent on large polynomial moments, such as Quadratic BCM and kurtosis, tend to be sensitive to the tails of the distribution. In the case of a sparse code the outliers, or the rare and interesting events, are what is important. Measuring the degree to which the neurons form a sparse code can be done in a straightforward and systematic fashion. The procedure involves simply eliminating from the environment those patterns for which the neuron responds strongly. These patterns tend to be the high contrast edges, and are thus the structure found in the image. The percentage of patterns that needs to be removed in order to cause a change in the receptive field gives a direct measure of the sparsity of the coding. The results of this structure removal B. S. Blais, N. Intrator, H Shouval and L N. Cooper 428 are shown in Figure 2. For Quadratic BCM and kurtosis, one need only eliminate less than one half of a percent of the input patterns to change the receptive field significantly. To make this more precise, we define a normalized difference between two mean zero vectors cos a), where a is the angle between the two vectors. This measure as V == has a value of zero for identical vectors, and a maximum value of one for orthogonal vectors. H1 - Also shown in Figure 2 is the normalized difference as a function of the percentage eliminated, for the different learning rules. RF differences can be seen with as little as a tenth of a percent, which suggests that the neuron is coding the information in a very sparse manner. Changes of around a half a percent and above are visible as significant orientation, phase, or spatial frequency changes. Although both skewness and Quadratic BCM depend primarily on the third moment, QBCM behaves more like kurtosis with regards to sparse coding. Structure Removal for BCM, Kurtosis, and Skew 0.3 ?II ? ?I! ?1'1 II ? II rI II BCM BCM ? ". Kl S, SI rI) . , " ;' ' . [I ~ ~0.25 2l e<= ~ - - 0 C> ' 0 BCM Kurtosis 1 Skew 1 ," 0.2 CI> ~0.15 iN ~ 0.1 E 0 ZO.05 --- ----- 0 Figure 2: Example receptive fields (left), and normalized difference measure (right), resulting from structure removal using QBCM, Kl, and 8 1 , The RFs show the successive deletion of top 1% of the distribution. On the right is the normalized difference between RFs as a function of the percentage deleted in structure removal. The maximum possible value of the difference is 1. 6 DISCUSSION This study attempts to compare several learning rules which have some statistical or biological motivation, or both. For a related study discussing projection pursuit and BCM see (Press and Lee, 1996). We have used natural scenes to gain some more insight about the statistics underlying natural images. There are several outcomes from this study: ? All rules used, found kurtotic distributions. ? The single cell lCA rule we considered, which used the subtractive form of kurtosis, achieved receptive fields qualitatively similar to other rules discussed. ? The Quadratic BCM and the multiplicative version of kurtosis are less sensitive to the second moments of the distribution and produce oriented RFs even when the data is not whitened. The subtractive versions of kurtosis and skewness are sensitive and produces oriented RFs only after sphering the data (Friedman, 1987; Field, 1994). RF Fonnation in Natural Scenes: Comparison of Single Cell Learning Rules 429 ? Both Quadratic BCM and kurtosis are sensitive to the elimination of the upper 1/2% portion of the distribution. The sensitivity to small portions of the distribution represents the other side of the coin of sparse coding. ? The skew rules' sensitivity to the upper parts of the distribution is not so strong. ? Quadratic BCM learning rule, which has been advocated as a projection index for finding multi-modality in high dimensional distribution, can find projections emphasizing high kurtosis when no cluster structure is present in the data. ACKNOWLEDGMENTS This work, was supported by the Office of Naval Research, the DANA Foundation and the National Science Foundation. References Bell, A. J. and Sejnowski, T. J. (1995). An information-maximisation approach to blind separation and blind deconvolution. Neural Computation, 7(6}:1129-1159. Bell, A. J. and Sejnowski, T. J. (1997). The independent components of natural scenes are edge filters. Vision Research. in press. Bienenstock, E . L., Cooper, L. N., and Munro, P. W. (1982) . Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. Journal of Neuroscience, 2:32-48. Comon, P. (1994). Independent component analysis, a n'ew concept? Signal Processing, 36:287-314. Field, D. J. (1994). What is the goal of sensory coding. Neural Computation, 6:559-601. Friedman, J. H. (1987). Exploratory projection pursuit. Journal of the American Statistical Association, 82:249-266. Hyvarinen, A. and Oja, E. (1996). A fast fixed-point algorithm for independent component analysis. Int. Journal of Neural Systems, 7(6):671-687. Intrator, N. (1990). A neural network for feature extraction. In Touretzky, D. S. and Lippmann, R. P., editors, Advances in Neural Information Processing Systems, volume 2, pages 719-726. Morgan Kaufmann, San Mateo, CA. Intrator, N. and Cooper, L. N. (1992) . Objective function formulation of the BCM theory of visual cortical plasticity: Statistical connections, stability conditions. Neural Networks, 5:3-17. Kendall, M. and Stuart, A. (1977). The Advanced Theory of Statistics, volume 1. MacMillan Publishing, New York. Law, C. and Cooper, L. (1994). Formation of receptive fields according to the BCM theory in realistic visual environments. Proceedings National Academy of Sciences, 91:7797-7801. Oja, E. (1995). The nonlinear pca learning rule and signal separation - mathematical analysis. Technical Report A26, Helsinki University, CS and Inf. Sci. Lab. Olshausen, B. A. and Field, D. J. (1996). Emergence of simple cell receptive field properties by learning a sparse code for natural images. Nature, 381:607-609. Press, W. and Lee, C. W. (1996) . Searching for optimal visual codes: Projection pursuit analysis of the statistical structure in natural scenes. In The Neurobiology of Computation: Proceedings of the fifth annual Computation and Neural Systems conference. Plenum Publishing Corporation. 

1458 |@word version:4 eliminating:1 polynomial:2 norm:1 covariance:1 moment:8 contains:1 current:1 comparing:1 si:1 visible:1 realistic:1 additive:3 plasticity:1 enables:1 alone:1 half:2 unmixed:1 successive:1 sigmoidal:1 simpler:1 mathematical:2 c2:12 direct:1 become:1 qualitative:1 multimodality:1 introduce:2 manner:2 ica:9 themselves:1 examine:1 multi:1 brain:2 little:1 underlying:1 israel:1 what:3 skewness:12 developed:3 finding:4 transformation:1 extremum:1 corporation:1 quantitative:1 every:1 k2:5 appear:1 positive:2 tid:5 before:2 approximately:1 mateo:1 suggests:2 co:1 ramat:1 statistically:2 acknowledgment:1 maximisation:1 procedure:2 vsj:1 bell:6 significantly:1 projection:11 specificity:1 maximizing:3 straightforward:1 regardless:1 rule:43 insight:1 stability:1 searching:1 exploratory:3 v52:1 plenum:1 particularly:1 observed:1 cloud:1 bottom:1 removed:1 environment:9 depend:4 shouval:4 derivation:1 zo:1 fast:1 sejnowski:6 formation:4 outcome:1 whose:2 statistic:4 emergence:1 kurtosis:26 interaction:2 achieve:1 academy:1 normalize:1 cluster:3 double:1 produce:4 develop:1 odd:1 school:1 advocated:1 strong:1 c:1 jem:2 come:1 implies:1 involves:1 concentrate:1 direction:2 fij:1 filter:3 stochastic:1 stabilization:4 elimination:1 explains:1 clustered:1 brian:1 biological:2 around:1 considered:1 estimation:1 sensitive:5 gaussian:7 vkl:1 office:1 derived:1 naval:1 mainly:1 contrast:1 dependent:1 eliminate:2 entire:1 bienenstock:3 lgn:1 i1:1 orientation:4 denoted:1 development:1 spatial:4 mutual:1 field:24 extraction:3 eliminated:1 identical:1 represents:1 stuart:3 look:1 subjecting:1 others:2 report:1 primarily:1 retina:1 oriented:4 oja:7 diaconis:1 national:2 phase:1 friedman:3 attempt:1 investigate:1 circular:1 mixture:2 edge:3 orthogonal:1 kurtotic:2 measuring:3 maximization:5 deviation:6 rare:1 motivating:1 providence:3 sensitivity:3 systematic:1 physic:2 lee:2 nongaussian:1 american:1 derivative:1 li:1 coding:7 cje:1 int:1 satisfy:1 blind:2 performed:1 multiplicative:4 h1:1 lab:1 kendall:3 portion:3 recover:1 rectifying:1 formed:1 kaufmann:1 rectified:1 touretzky:1 synaptic:2 frequency:4 e2:2 sampled:1 gain:1 x13:1 higher:2 formulation:1 done:2 though:2 strongly:1 binocular:1 correlation:1 nonlinear:1 aviv:2 olshausen:3 effect:1 brown:4 concept:2 requiring:1 normalized:5 symmetric:1 iteratively:1 outline:1 demonstrate:1 percent:3 reasoning:1 image:7 recently:2 common:1 sigmoid:1 behaves:2 rl:3 volume:2 tail:2 slight:1 discussed:1 association:1 significant:1 ai:1 cortex:1 whitening:1 inf:1 claimed:2 selectivity:1 discussing:1 seen:1 morgan:1 determine:1 maximize:1 signal:7 ii:7 fonnation:1 smooth:1 technical:1 plausibility:1 e1:2 whitened:6 vision:1 expectation:1 iteration:2 achieved:2 cell:11 addition:1 source:1 modality:1 eliminates:1 ascent:2 tend:2 seem:1 architecture:2 motivated:1 expression:1 pca:1 munro:1 york:1 cause:2 useful:1 percentage:3 neuroscience:1 ddt:1 shall:1 express:1 four:1 threshold:1 deleted:1 preprocessed:1 ce:1 tenth:1 merely:1 angle:1 separation:3 patch:1 quadratic:12 annual:1 activity:2 scene:16 ri:2 helsinki:1 leon:1 sphering:1 department:2 lca:1 alternate:1 according:1 em:6 biologically:1 modification:4 comon:2 outlier:2 computationally:1 equation:1 rectification:1 skew:3 mechanism:1 needed:1 studying:1 pursuit:6 gaussians:1 intrator:8 batch:1 coin:1 original:1 denotes:1 top:2 include:1 publishing:2 a26:1 seeking:1 objective:2 quantity:1 receptive:19 dependence:1 responds:1 exhibit:1 gradient:5 lateral:1 sci:1 code:5 index:1 difficult:1 negative:1 upper:2 neuron:9 immediate:1 neurobiology:1 blais:4 precise:1 dog:1 required:1 kl:5 c3:4 connection:3 c4:3 bcm:24 deletion:1 pattern:6 sparsity:2 rf:11 including:1 event:1 natural:19 advanced:1 disappears:1 review:1 removal:6 law:3 mixed:3 interesting:2 dana:1 foundation:2 degree:1 conveyed:1 editor:1 subtractive:2 surprisingly:1 supported:1 side:1 institute:2 fifth:1 sparse:10 regard:1 calculated:2 cortical:1 sensory:1 made:1 qualitatively:2 preprocessing:3 projected:2 san:1 far:1 hyvarinen:3 lippmann:1 assumed:1 search:1 iterative:1 nature:1 ca:1 tel:1 symmetry:1 motivation:2 freedman:1 allowed:1 neuronal:3 je:3 fashion:1 cooper:11 exponential:1 vanish:1 third:1 learns:1 emphasizing:1 specific:1 deconvolution:1 ci:1 sparseness:2 entropy:1 simply:1 appearance:1 visual:5 contained:1 macmillan:1 monotonic:1 relies:1 goal:3 ii1ii:1 lntrator:1 change:4 included:1 ew:1

Intrusion Detection with Neural Networks Jake Ryan* Department of Computer Sciences The University of Texas at Austin Austin, TX 78712 Department of Electrical and Computer Engineering The University of Texas at Austin Austin, TX 78712 raven@cs.utexas.edu mj@orac.ece . utexas.edu Meng-Jang Lin Risto Miikkulainen Department of Computer Sciences The University of Texas at Austin Austin, TX 78712 risto@cs.utexas.edu Abstract With the rapid expansion of computer networks during the past few years, security has become a crucial issue for modern computer systems. A good way to detect illegitimate use is through monitoring unusual user activity. Methods of intrusion detection based on hand-coded rule sets or predicting commands on-line are laborous to build or not very reliable. This paper proposes a new way of applying neural networks to detect intrusions. We believe that a user leaves a 'print' when using the system; a neural network can be used to learn this print and identify each user much like detectives use thumbprints to place people at crime scenes. If a user's behavior does not match hislher print, the system administrator can be alerted of a possible security breech. A backpropagation neural network called NNID (Neural Network Intrusion Detector) was trained in the identification task and tested experimentally on a system of 10 users. The system was 96% accurate in detecting unusual activity, with 7% false alarm rate. These results suggest that learning user profiles is an effective way for detecting intrusions. 1 INTRODUCTION Intrusion detection schemes can be classified into two categories: misuse and anomaly intrusion detection. Misuse refers to known attacks that exploit the known vulnerabilities of the system. Anomaly means unusual activity in general that could indicate an intrusion. ?Currently: MCI Communications Corp., 9001 N. IH 35, Austin, TX 78753; jake.ryan@mci.com. 944 1. Ryan, M-J. Lin and R. Miikkulainen If the observed activity of a user deviates from the expected behavior, an anomaly is said to occur. Misuse detection can be very powerful on those attacks that have been programmed in to the detection system. However, it is not possible to anticipate all the different attacks that could occur, and even the attempt is laborous. Some kind of anomaly detection is ultimately necessary. One problem with anomaly detection is that it is likely to raise many false alarms. Unusual but legitimate use may sometimes be considered anomalous. The challenge is to develop a model of legitimate behavior that would accept novel legitimate use. It is difficult to build such a model for the same reason that it is hard to build a comprehensive misuse detection system: it is not possible to anticipate aU possible variations of such behavior. The task can be made tractable in three ways: (1) Instead of general legitimate use, the behavior of individual users in a particular system can be modeled. The task of characterizing regular patterns in the behavior of an individual user is an easier task than trying to do it for aU users simultaneously. (2) The patterns of behavior can be learned for examples of legitimate use, instead of having to describe them by hand-COding possible behaviors. (3) Detecting an intrusion real-time, as the user is typing commands, is very difficult because the order of commands can vary a lot. In many cases it is enough to recognize that the distribution of commands over the entire login session, or even the entire day, differs from the usual. The system presented in this paper, NNID (Neural Network Intrusion Detector), is based on these three ideas. NNID is a backpropagation neural network trained to identify users based on what commands they use during a day. The system administrator runs NNID at the end of each day to see if the users' sessions match their normal pattern. If not, an investigation can be launched. The NNID model is implemented in a UNIX environment and consists of keeping logs of the commands executed, forming command histograms for each user, and learning the users' profiles from these histograms. NNID provides an elegant solution to off-line monitoring utilizing these user profiles. In a system of 10 users, NNID was 96% accurate in detecting anomalous behavior (i.e. random usage patterns), with a false alarm rate of 7%. These results show that a learning offline monitoring system such as NNID can achieve better performance than systems that attempt to detect anomalies on-line in the command sequences, and with computationally much less effort. The rest of the paper outlines other approaches to intrusion detection and motivates the NNID approach in more detail (sections 2 and 3), presents the implementation and an evaluation on a real-world computer system (sections 4 and 5), and outlines some open issues and avenues for future work (section 6). 2 INTRUSION DETECTION SYSTEMS Many misuse and anomaly intrusion detection systems (lDSs) are based on the general model proposed by Denning (1987). This model is independent of the platform, system vulnerability, and type of intrusion. It maintains a set of historical profiles for users, matches an audit record with the appropriate profile, updates the profile whenever necessary, and reports any anomalies detected. Another component, a rule set, is used for detecting misuse. Actual systems implement the general model with different techniques (see Frank 1994; Mukherjee et al. 1994, for an overview). Often statistical methods are used to measure how anomalous the behavior is, that is, how different e.g. the commands used are from normal behavior. Such approaches require that the distribution of subjects' behavior is known. The behavior can be represented as a rule-based model (Garvey and Lunt 1991), in terms of predictive pattern generation (Teng et al. 1990), or using state transition analysis (Porras Intrusion Detection with Neural Networks 945 et al. 1995). Pattern matching techniques are then used to detennine whether the sequence of events is part of normal behavior, constitutes an anomaly, or fits the description of a known attack. IDSs also differ in whether they are on-line or off-line. Off-line IDSs are run periodically and they detect intrusions after-the-fact based on system logs. On-line systems are designed to detect intrusions while they are happening, thereby allowing for quicker intervention. On-line IDSs are computationally very expensive because they require continuous monitoring. Decisions need to be made quickly with less data and therefore they are not as reliable. Several IDSs that employ neural networks for on-line intrusion detection have been proposed (Debar et al. 1992; Fox et al. 1990). These systems learn to predict the next command based on a sequence of previous commands by a specific user. Through a shifting window, the network receives the w most recent commands as its input. The network is recurrent, that is, part of the output is fed back as the input for the next step; thus, the network is constantly observing the new trend and "forgets" old behavior over time. The size of the window is an important parameter: If w is too small, there will be many false positives; if it is too big, the network may not generalize well to novel sequences. The most recent of such systems (Debar et al. 1992) can predict the next command correctly around 80% of the time, and accept a command as predictable (among the three most likely next commands) 90% of the time. One problem with the on-line approach is that most of the effort goes into predicting the order of commands. In many cases, the order does not matter much, but the distribution of commands that are used is revealing. A possibly effective approach could therefore be to collect statistics about the users' command usage over a period of time, such as a day, and try to recognize the distribution of commands as legitimate or anomalous off-line. This is the idea behind the NNID system. 3 THE NNID SYSTEM The NNID anomaly intrusion detection system is based on identifying a legitimate user based on the distribution of commands she or he executes. This is justifiable because different users tend to exhibit different behavior, depending on their needs of the system. Some use the system to send and receive e-mail only, and do not require services such as programming and compilation. Some engage in all kinds of activities including editing, programming, e-mail, Web browsing, and so on. However, even two users that do the same thing may not use the same application program. For example, some may prefer the "vi" editor to "emacs", favor "pine" over "elm" as their mail utility program, or use "gcc" more often than "cc" to compile C programs. Also, the frequency with which a command is used varies from user to user. The set of commands used and their frequency, therefore, constitutes a 'print' of the user, reflecting the task performed and the choice of application programs, and it should be possible to identify the user based on this information. It should be noted that this approach works even if some users have aliases set up as shorthands for long commands they use frequently, because the audit log records the actual commands executed by the system. Users' privacy is not violated, since the arguments to a command do not need to be recorded. That is, we may know that a user sends e-mail five times a day, but we do not need to know to whom the mail is addressed. Building NNID for a particular computer system consists of the following three phases: 1. Collecting training data: Obtain the audit logs for each user for a period of several days. For each day and user, form a vector that represents how often the user executed each command. 946 1 Ryan, M-J. Un and R. Miikkulainen as cut expr ghostview Id man netstat rm tcsh vi awk cvs fgrep gmake fess mesg nm rsh tee virtex be date filter grep look metamail objdump sed test w 61btex df find gs Ipq rillCdir perl sendmail tgif wc calendar diff finger gzip Ipr more pgp sh top whereis cat du fmt hostname Iprm movemail ping sort tput xbiff++ chmOd dvips from id Is mpage ps strip tr xca1c comsat egrep ftp ifConfig machine mt pwd stty tty xdvi cp elm gcc Ispell mail mv rcp tail uname xhost cpp emacs gdb fast make netscape resize tar vacation xterm Table 1: The 100 commands used to describe user behavior. The number of times the user executed each of these commands during the day was recorded, mapped into a nonlinear scale of 11 intervals, and concatenated into a l00-dimensional input vector, representing the usage pattern for that user for that day. 2. Training: Train the neural network to identify the user based on these command distribution vectors. 3. Perfonnance: Let the network identify the user for each new command distribution vector. If the network's suggestion is different from the actual user, or if the network does not have a clear suggestion, signal an anomaly. The particular implementation of NNID and the environment where it was tested is described in the next section. 4 EXPERIMENTS The NNID system was built and tested on a machine that serves a particular research group at the Department of Electrical and Computer Engineering at the University of Texas at Austin. This machine has 10 total users; some are regular users, with several other users logging in intennittently. This platfonn was chosen for three reasons: 1. The operating system (NetBSD) provides audit trail logging for accounting purposes and this option had been enabled on this system. 2. The number of users and the total number of commands executed per day are on an order of magnitude that is manageable. Thus, the feasibility of the approach could be tested with real-world data without getting into scalability issues. 3. The system is relatively unknown to outsiders and the users are all known to us, so that it is likely that the data collected on it consists of nonnal user behavior (free of intrusions). Data was collected on this system for 12 days, resulting in 89 user-days. Instead of trying to optimize the selection of features (commands) for the input, we decided to simply use a set of 100 most common commands in the logs (listed in Table 1), and let the network figure out what infonnation was important and what superfluous. Intelligent selection of features might improve the results some but the current approach is easy to implement and proves the point. In order to introduce more overlap between input vectors, and therefore better generalization, the number of times a command was used was divided into intervals. There were 11 intervals, non-linearly spaced, so that the representation is more accurate at lower frequencies where it is most important. The first interval meant the command was never used; the second that it was used once or twice, and so on until the last interval where the command was used more than 500 times. The intervals were represented by values from 0.0 to 1.0 in 0.1 increments. These values, one for each command, were then concatenated into a 100-dimensional command distribution vector (also called user vector below) to be used as input to the neural network. Intrusion Detection with Neural Networks 947 The standard three-layer backpropagation architecture was chosen for the neural network. The idea was to get results on the most standard and general architecture so that the feasibility of the approach could be demonstrated and the results would be easily replicable. More sophisticated architectures could be used and they would probably lead to slightly better results. The input layer consisted of 100 units, representing the user vector; the hidden layer had 30 units and the output layer 10 units, one for each user. The network was implemented in the PlaNet Neural Network simulator (Miyata 1991). 5 RESULTS To avoid overtraining, several training sessions were run prior to the actual experiments to see how many training cycles would give the highest performance. The network was trained on 8 randomly chosen days of data (65 user vectors), and its performance was tested on the remaining 4 days (24 vectors) after epochs 30, 50, 100,200, and 300, of which 100 gave the best performance. Four splits of the data into training and testing sets were created by randomly picking 8 days for training. The reSUlting four networks were tested in two tasks: 1. Identifying the user vectors of the remaining 4 days. If the activation of the output unit representing the correct user was higher than those of all other units, and also higher than 0.5, the identification was counted as correct. Otherwise, a false positive was counted. 2. Identifying 100 randomly-generated user vectors. If all output units had an activation less than 0.5, the network was taken to correctly identify the vector as an anomaly (i.e. not any of the known users in the system). Otherwise, the most highly active output unit identifies the network's suggestion. Since all intrusions occur under one of the 10 user accounts, there is a 111 0 chance that the suggestion would accidentally match the compromised user account and the intrusion would not be detected. Therefore, 1/10 of all such cases were counted as false negatives. The second test is a suggestive measure of the accuracy of the system. It is not possible to come up with vectors that would represent a good sampling of actual intrusions; the idea here was to generate vectors where the values for each command were randomly drawn from the distribution of values for that command in the entire data set. In other words, the random test vectors had the same first-order statistics as the legitimate user vectors, but had no higher-order correlations. Therefore they constitute a neutral but realistic sample of unusual behavior. All four splits led to similar results. On average, the networks rejected 63% of the random user vectors, leading to an anomaly detection rate of 96%. They correctly identified the legitimate user vectors 93% of the time, giving a false alarm rate of 7%. Figure 1 shows the output of the network for one of the splits. Out of 24 legitimate user vectors, the network identified 22. Most of the time the correct output unit is very highly activated, indicating high certainty of identification. However, the activation of the highest unit was below 0.5 for two of the inputs, resulting in a false alarm. Interestingly, in all false alarms in all splits, the falsely-accused user was always the same. A closer look at the data set revealed that there were only 3 days of data on this user. He used the system very infrequently, and the network could not learn a proper profile for him. While it would be easy to fix this problem by collecting more data in this case, we believe this is a problem that would be difficult to rule out in general. No matter how much data one collects, there may still not be enough for some extremely infrequent user. Therefore, we believe the results obtained in this rather small data set give a realistic picture of the performance of the NNID system. 948 1. Ryan, M-l. lin and R. Miikkulainen O~t.,ut;': .> 4 ? ) 2 3 ... ? :l ~t? - 4 , ? "" 6 ~ 0 L Z ? ~ D.itrut . . ~ .. . .. , ~ u " ~~- n..,.!. - ~"":;l e ~~t ~t<'4'" . ? . . . ~ .. - o ? ~ ? -. " ... _ ? - ~ , t " e 9 ? 0 1 2 3 00. ....... ? 5 6 ? u ~ ~< ~ , ? . ~t ~ 0 1 2 Oo.t"", ~ , " o::.+~" Eo 6 9 .. " 5 6 7 8 9 < , Out.~, ~ " ~ 0 . ? ? g ? ? . " t ? ? ?? ~ ~~,.,,, ? - ?? D~t~2~'56 " 5 6 7 B 9 ? " tI . ?? .? . ... D.np;;.. '" ~ ? ?? D~~L tI ? 5 6 7 B D.~t '" ? ? . ' . ? O.lt('t-rt .. ... ?? ? , xr~ct .. ~ . ~ Out.~, ~"O' ? ~ ~Zl.'5 . ti " Ou<.~. , , ~ ~ L ?" ~ t " o::...~"'.'~ ? s Figure 1: User identification with the NNID Network. The output layer of NNID is shown for each of the 24 test vectors in one of the 4 splits tested. The output units are lined up from left to right, and their activations are represented by the size of the squares. In this split there were two false alarms: one is displayed in the top right with activation 0.01, and one in the second row from the bottom, second column from the left with 0.35. All the other test vectors are identified correctly with activation higher than 0.5. 6 DISCUSSION AND FUTURE WORK An important question is, how well does the performance of NNID scale with the number of users? Although there are many computer systems that have no more than a dozen users, most intrusions occur in larger systems with hundreds of users. With more users, the network would have to make finer distinctions, and it would be difficult to maintain the same low level of false alarms. However, the rate of detecting anomalies may not change much, as long as the network can learn the user patterns well. Any activity that differs from the user's normal behavior would still be detected as an anomaly. Training the network to represent many more users may take longer and require a larger network, but it should be possible because the user profiles share a lot of common structure, and neural networks in general are good at learning such data. Optimizing the set of commands included in the user vector, and the size of the value intervals, might also have a large impact on performance. It would be interesting to determine the curve of performance Intrusion Detection with Neural Networks 949 versus the number of users, and also see how the size of the input vector and the granularity of the value intervals affect that curve. This is the most important direction of future work. Another important issue is, how much does a user's behavior change over time? If behavior changes dramatically, NNID must be recalibrated often or the number of false positives would increase. Fortunately such retraining is easy to do. Since NNID parses daily activity of each user into a user-vector, the user profile can be updated daily. NNID could then be retrained periodically. In the current system it takes only about 90 seconds and would not be a great burden on the system. 7 CONCLUSION Experimental evaluation on real-world data shows that NNID can learn to identify users simply by what commands they use and how often, and such an identification can be used to detect intrusions in a network computer system. The order of commands does not need to be taken into account. NNID is easy to train and inexpensive to run because it operates off-line on daily logs. As long as real-time detection is not required, NNID constitutes a promising, practical approach to anomaly intrusion detection. Acknowledgements Special thanks to Mike Dahlin and Tom Ziaja for feedback on an earlier version of this paper, and to Jim Bednar for help with the PlaNet simulator. This research was supported in part by DOD-ARPA contract F30602-96-1-0313, NSF grant IRI-9504317, and the Texas Higher Education Coordinating board grant ARP-444. References Debar, H., Becker, M., and Siboni, D. (1992). A neural network component for an intrusion detection system. In Proceedings of the 1992 IEEE Computer Society Symposium on Research in Computer Security and Privacy, 240-250. Denning, D. E. (1987). An intrusion detection model. IEEE Transactions on Software Engineering, SE-13:222-232. Fox, K. L., Henning, R. R., Reed, J. H., and Simonian, R. (1990). A neural network approach towards intrusion detection. In Proceedings of the 13th National Computer Security Conference, 125-134. Frank, J. (1994). Artificial intelligence and intrusion detection: Current and future directions. In Proceedings of the National 17th Computer Security Conference. Garvey, T. D., and Lunt, T. F. (1991). Model-based intrusion detection. In Proceedings of the 14th National Computer Security Conference. Miyata, Y. (1991). A User's Guide to PlaNet Version 5.6 -A Toolfor Constructing, Running, and Looking in to a PDP Network. Computer Science Department, University of Colorado, Boulder, Boulder, CO. Mukherjee, B., Heberlein, L. T., and Levitt, K. N. (1994). Network intrusion detection. IEEE Network, 26-41. Porras, P. A., IIgun, K., and Kemmerer, R. A. (1995). State transition analysis: A rulebased intrusion detection approach. IEEE Transactions on Software Engineering, SE21 : 181-199. Teng, H. S., Chen, K., and Lu, S. C. (1990). Adaptive real-time anomaly detection using inductively generated sequential patterns. In Proceedings of the 1990 IEEE Symposium on Research in Computer Security and Privacy, 278-284. 

1459 |@word version:2 manageable:1 retraining:1 risto:2 open:1 accounting:1 detective:1 thereby:1 tr:1 interestingly:1 past:1 current:3 com:1 activation:6 must:1 planet:3 realistic:2 periodically:2 designed:1 update:1 intelligence:1 leaf:1 record:2 detecting:6 provides:2 attack:4 five:1 become:1 symposium:2 consists:3 shorthand:1 introduce:1 privacy:3 falsely:1 expected:1 rapid:1 behavior:22 ldss:1 frequently:1 simulator:2 porras:2 actual:5 window:2 what:4 kind:2 elm:2 certainty:1 rcp:1 collecting:2 ti:3 rm:1 zl:1 unit:10 grant:2 intervention:1 positive:3 service:1 engineering:4 id:6 meng:1 ipr:1 might:2 twice:1 au:2 collect:2 compile:1 co:1 programmed:1 grep:1 decided:1 practical:1 testing:1 implement:2 differs:2 backpropagation:3 xr:1 revealing:1 matching:1 word:1 refers:1 regular:2 suggest:1 get:1 selection:2 applying:1 optimize:1 misuse:6 demonstrated:1 send:1 go:1 iri:1 identifying:3 legitimate:10 rule:4 utilizing:1 enabled:1 variation:1 increment:1 updated:1 infrequent:1 user:80 anomaly:17 programming:2 engage:1 trail:1 colorado:1 trend:1 infrequently:1 expensive:1 mukherjee:2 cut:1 observed:1 bottom:1 mike:1 quicker:1 electrical:2 awk:1 cycle:1 highest:2 environment:2 predictable:1 inductively:1 ultimately:1 trained:3 raise:1 predictive:1 logging:2 easily:1 represented:3 tx:4 finger:1 cat:1 train:2 fast:1 effective:2 describe:2 detected:3 artificial:1 larger:2 otherwise:2 calendar:1 favor:1 statistic:2 sequence:4 tput:1 date:1 achieve:1 description:1 scalability:1 getting:1 p:1 bednar:1 ftp:1 depending:1 develop:1 recurrent:1 oo:1 help:1 implemented:2 c:2 indicate:1 come:1 differ:1 direction:2 correct:3 filter:1 education:1 require:4 fix:1 generalization:1 investigation:1 anticipate:2 ryan:5 around:1 considered:1 normal:4 great:1 predict:2 pine:1 vary:1 purpose:1 currently:1 infonnation:1 utexas:3 vulnerability:2 him:1 always:1 rather:1 avoid:1 tar:1 command:43 she:1 intrusion:35 detect:6 entire:3 accept:2 hidden:1 nonnal:1 issue:4 among:1 proposes:1 platform:1 special:1 fes:1 once:1 never:1 having:1 sampling:1 represents:1 look:2 constitutes:3 future:4 report:1 np:1 intelligent:1 few:1 employ:1 modern:1 randomly:4 simultaneously:1 recognize:2 comprehensive:1 individual:2 national:3 phase:1 maintain:1 attempt:2 detection:28 highly:2 evaluation:2 sh:1 behind:1 superfluous:1 compilation:1 activated:1 accurate:3 closer:1 necessary:2 daily:3 perfonnance:1 fox:2 old:1 arpa:1 column:1 earlier:1 neutral:1 hundred:1 dod:1 too:2 varies:1 thanks:1 contract:1 off:5 picking:1 quickly:1 recorded:2 nm:1 possibly:1 administrator:2 leading:1 account:3 coding:1 matter:2 mv:1 vi:2 performed:1 try:1 lot:2 observing:1 garvey:2 sort:1 maintains:1 option:1 sed:1 square:1 accuracy:1 spaced:1 identify:7 generalize:1 identification:5 lu:1 monitoring:4 justifiable:1 cc:1 finer:1 executes:1 classified:1 overtraining:1 detector:2 ping:1 whenever:1 strip:1 rsh:1 inexpensive:1 frequency:3 ut:1 ou:1 sophisticated:1 back:1 reflecting:1 higher:5 day:17 tom:1 editing:1 rejected:1 until:1 correlation:1 hand:2 receives:1 web:1 nonlinear:1 gdb:1 believe:3 building:1 usage:3 consisted:1 during:3 noted:1 trying:2 outline:2 gzip:1 orac:1 cp:1 novel:2 common:2 mt:1 overview:1 tail:1 he:2 cv:1 session:3 had:5 longer:1 operating:1 recent:2 optimizing:1 corp:1 tee:1 fortunately:1 eo:1 determine:1 period:2 signal:1 match:4 long:3 lin:3 divided:1 coded:1 feasibility:2 impact:1 anomalous:4 df:1 histogram:2 sometimes:1 pwd:1 represent:2 receive:1 addressed:1 interval:8 sends:1 crucial:1 launched:1 rest:1 probably:1 subject:1 tend:1 elegant:1 henning:1 thing:1 granularity:1 revealed:1 split:6 enough:2 easy:4 affect:1 fit:1 gave:1 architecture:3 identified:3 idea:4 avenue:1 illegitimate:1 texas:5 whether:2 utility:1 becker:1 effort:2 constitute:1 dramatically:1 clear:1 listed:1 se:1 category:1 generate:1 nsf:1 coordinating:1 correctly:4 per:1 group:1 four:3 drawn:1 year:1 run:4 unix:1 powerful:1 place:1 decision:1 prefer:1 resize:1 layer:5 ct:1 g:1 activity:7 occur:4 rulebased:1 scene:1 software:2 wc:1 argument:1 extremely:1 relatively:1 department:5 slightly:1 boulder:2 taken:2 computationally:2 lined:1 know:2 tractable:1 fed:1 end:1 unusual:5 serf:1 detennine:1 appropriate:1 jang:1 vacation:1 top:2 remaining:2 running:1 fmt:1 exploit:1 giving:1 concatenated:2 f30602:1 build:3 prof:1 jake:2 expr:1 society:1 print:4 question:1 rt:1 usual:1 said:1 exhibit:1 mapped:1 mail:6 whom:1 collected:2 reason:2 modeled:1 reed:1 difficult:4 executed:5 frank:2 negative:1 gcc:2 implementation:2 motivates:1 proper:1 unknown:1 allowing:1 displayed:1 communication:1 looking:1 jim:1 pdp:1 tty:1 retrained:1 required:1 security:7 crime:1 learned:1 distinction:1 below:2 pattern:9 challenge:1 program:4 perl:1 built:1 reliable:2 including:1 shifting:1 event:1 overlap:1 typing:1 predicting:2 representing:3 scheme:1 improve:1 alias:1 identifies:1 picture:1 created:1 ipq:1 deviate:1 prior:1 epoch:1 acknowledgement:1 recalibrated:1 par:1 generation:1 suggestion:4 interesting:1 versus:1 editor:1 share:1 austin:8 row:1 supported:1 last:1 keeping:1 free:1 accidentally:1 offline:1 guide:1 characterizing:1 curve:2 feedback:1 login:1 world:3 transition:2 replicable:1 made:2 adaptive:1 alerted:1 historical:1 counted:3 miikkulainen:4 transaction:2 l00:1 emacs:2 suggestive:1 active:1 continuous:1 un:1 compromised:1 table:2 promising:1 mj:1 learn:5 miyata:2 expansion:1 du:1 constructing:1 linearly:1 big:1 alarm:8 profile:9 levitt:1 board:1 forgets:1 hislher:1 pgp:1 dozen:1 specific:1 lunt:2 raven:1 burden:1 ih:1 false:12 sequential:1 magnitude:1 browsing:1 chen:1 easier:1 led:1 lt:1 simply:2 likely:3 forming:1 happening:1 chance:1 constantly:1 towards:1 man:1 experimentally:1 hard:1 change:3 included:1 diff:1 operates:1 called:2 teng:2 cpp:1 ece:1 mci:2 total:2 experimental:1 arp:1 indicating:1 audit:4 people:1 meant:1 violated:1 tested:7

281 PERFORMANCE OF SYNTHETIC NEURAL NETWORK CLASSIFICATION OF NOISY RADAR SIGNALS S. C. Ahalt F. D. Garber I. Jouny A. K . Krishnamurthy Department of Electrical Engineering The Ohio State University, Columbus, Ohio 43210 ABSTRACT This study evaluates the performance of the multilayer-perceptron and the frequency-sensitive competitive learning network in identifying five commercial aircraft from radar backscatter measurements. The performance of the neural network classifiers is compared with that of the nearest-neighbor and maximum-likelihood classifiers. Our results indicate that for this problem, the neural network classifiers are relatively insensitive to changes in the network topology, and to the noise level in the training data. While, for this problem, the traditional algorithms outperform these simple neural classifiers, we feel that neural networks show the potential for improved performance. INTRODUCTION The design of systems that identify objects based on measurements of their radar backscatter signals has traditionally been predicated upon decision-theoretic methods of pattern recognition [1]. While it is true that these methods are characterized by a well-defined sense of optimality, they depend on the availability of accurate models for the statistical properties of the radar measurements. Synthetic neural networks are an attractive alternative to this problem, since they can learn to perform the classification from labeled training data, and do not require knowledge of statistical models [2]. The primary objectives of this investigation are; to establish the feasibility of using synthetic neural networks for the identification of radar objects, and to characterize the trade-oft's between neural network and decision-theoretic methodologies for the design of radar object identification systems. The present study is focused on the performance evaluation of systems operating on the received radar backscatter signals of five commercial aircraft; the Boeing 707, 727, 747, the DC-lO, and the Concord. In particular, we present results for the multi-layer perceptron and the frequency-sensitive competitive learning (FSCL) synthetic network models [2,3] and compare these with results for the nearestneighbor and maximum-likelihood classification algorithms. In this paper, the performance of the classification algorithms is evaluated by means 282 Ahalt, Garber, Jouny and Krishnamurthy of computer simulation studies; the results are compared for a number of conditions concerning the radar environment and receiver models. The sensitivity of the neural network classifiers, with respect to the number of layers and the number of hidden units, is investigated. In each case, the results obtained using the synthetic neural network classifiers are compared with those obtained using an (optimal) maximumlikelihood classifier and a (minimum-distance) nearest-neighbor classifier. PROBLEM DESCRIPTION The radar system is modeled as a stepped-frequency system measuring radar backscatter at 8, 11, 17, and 28 MHz. The 8-28 MHz band of frequencies was chosen to correspond to the "resonant region" of the aircraft, i.e., frequencies with wavelengths approximately equal to the length of the object. The four specific frequencies employed for this study were pre-selected from the database maintained at The Ohio State University ElectroScience Laboratory compact radar range as the optimal features among the available measurements in this band [4] . Performance results are presented below for systems modeled as having in-phase and quadrature measurement capability (coherent systems) and for systems modeled as having only signal magnitude measurement capability (non coherent systems). For coherent systems, the observation vector X = [(xI, x~), (x~, x~), (x~, x~), (xt x~)] T represents the in-phase and quadrature components of the noisy backscatter measurements of an unknown target. The elements of X correspond to the complex scattering coefficient whose magnitude is the square root of the measured cross section (in units of square meters, m 2 ), and whose complex phase is that of the measured signal at that frequency. For noncoherent systems, the observation vector X = [aI, a2, a3, a4]T consists of components which are the magnitudes of the noisy backscatter measurements corresponding to the square root of the measured cross section. For the simulation experiments, it is assumed that the received signal is the result of a superposition of the backscatter signal vector S and noise vector W which is modeled as samples from an additive white Gaussian process. COHERENT MEASUREMENTS In the case of a coherent radar system, the kth frequency component of the obser- vation vector is given by: xL = (s{ + wi), (1) where sL and s~ are the in-phase and quadrature components of the backscatter signal, and wi and W~ are the in-phase and quadrature components of the sample of the additive white Gaussian noise process at that frequency. Each of these components is modeled as a zero-mean Gaussian random variable with variance u 2 /2 Performance of Synthetic Neural Network Classification so that the total additive noise contribution at each frequency is complex-valued Gaussian with zero mean and variance 0'2. During operation, the neural network classifier is presented with the observation vector, of dimension eight, consisting of the in-phase and quadrature components of each of the four frequency measurements; (2) Typically, the neural net is trained using 200 samples of the observation vector X for each of the five commercial aircraft discussed above. The desired output vectors are of the form (3) = where di,j 1 for the desired aircraft and is 0 otherwise. Thus, for example, the output vector di for the second aircraft is 0,1,0,0,0, with a 1 appearing in the second position. The structure of the neural nets used can be represented by [8, nl, ... , nh, 5], where there are 8 input neurons, ni hidden layer neurons in the h hidden layers, and 5 output neurons. The first experiment tested the perceptron nets of varying architectures, as shown in Figures 1, and 2. As can be seen, there was little change in performance between the various nets. The effects of the signal-to-noise ratio of the data observed during the training phase on the performance of the perceptron was also investigated. The results are presented in Figure 3. The network showed little change in performance until a training data SNR of 20 dB was reached. We repeated this basic experiment using a winner-take-all network, the FSCL net [3]. Figure 4 shows that the performance of this network is also effected minimally by changes in network architecture. When the FSCL net is trained with noisy data, as shown in Fig. 5, the performance decreases as the SNR of the training data increases, however, the overall performance is still very close to the performance of the multi-layer perceptron. Our final coherent-data experiment compared the performance of the multi-layer perceptron, the FSCL net, a max-likelihood classifier and the nearest neighbor classifier. The results are shown in Figure 6. For this experiment, the training data had no superimposed noise. These results show that the max-likelihood classifier is superior, but requires full knowledge of the noise distribution. On average, the FSCL net performs better than the perceptron, but the nearest neighbor classifier performs better than either of the neural network models. 283 284 Ahalt, Garber, Jouny and Krishnamurthy 100 90 E- 80 E- - 8x5x5 ----- , 8x 10x 5 ------ .... ~ 8x2Ox 5 8x3Ox5 .......... 8x40x 5 ~ 70 ....... 60 l ~ ~ \~ \ 50 ~ \ t: ? 40 ~ 30 " 20 ~, . 10 ~ '\r:::.. 0 -30 -25 ?20 -15 ?10 o ?5 5 10 15 20 SNR Idbl Figure 1: Performance of the perceptron with different number of hidden units. 100 90 c- 80 - ----_. -----~ 70 I\~, 80 l eai 8x1Ox5.200 8x1Ox10x5.18OO 8X1Ox10x10x5.18OO ~ 50 ~ \ 40 \ 30 \ 20 \ 10 ~ '\ 0 ?30 ?25 ?20 ?15 -10 ?5 o 5 10 15 20 SNR Idbl Figure 2: Performance of the perceptron with 1, 2 and 3 hidden layers. Perfonnance of Synthetic Neural Network Classification 100 90 r- 80 r- ----_. ------ - .......... 70 0 ___ - t-- 80 l 15 ~ NoIse Free -5 db Odb 6db 12db 20 db , i\ 50 \ '\ 40 ~ 30 \\ \\ 20 " \ ~\ 10 .~ 0 -3b -25 -20 -15 -10 "~ o -5 - ... ....... .... . 5 10 15 20 SNRldbl Figure 3: Performance of the perceptron for different SNR of the training data. 100 90 ,.- 80 r- ----_. 8 x 10 x5 8x2Ox5 8x30x5 8 x40x5 8x50x5 ------ .......... 70 ~ 80 l ~ e 50 ~ ,~ \ ~ 40 30 \ \ 20 10 \ ~ .'\k. 0 -30 -25 -20 -15 -10 -5 o 5 10 15 20 SNRldbl Figure 4: Performance of FSCL with varying no. of hidden units. 285 286 Ahalt, Garber, Jouny and Krishnamurthy 100 90 80 :-- ----_. ------ :- - ?????? 0 ? ?? ..... 70 ,~ 60 l 50 ? 40 e Noise Free -5 db Odb 6db 12db \ , \ ...... ' .. .~ 30 ~.... \~'.'. \ \ 20 ~~ 10 '.~ 0 -30 -25 -20 -15 -10 -5 ~ o .. ...... 5 10 15 20 SNR Idbl Figure 5: Performance of the FSCL network for different SNR of the training data. 100 90 80 :-- ----_. ------ :-- ..."..~ 60 ? perceptron 8x1Ox5 max. likelihood nearest neighbor ~ 70 tg FSCL 8x1Ox5 ~ ~~ 50 1\, 40 ~ ,, 30 ~\ ~\ ~~ 20 10 0 -30 -25 -20 -15 -10 -5 o 5 10 15 20 SNR Idbl Figure 6: Comparison of all four classifiers for the coherent data case. Performance of Synthetic Neural Network Classification NONCOHERENT MEASUREMENTS For the case of a noncoherent radar system model, the the observation vector is given by: kth frequency component of (4) where, as before, s{ and s~ are the in-phase and quadrature components of the backscatter signal, and wI and w~ are the in-phase and quadrature components of the additive white Gaussian noise. Hence, while the underlying noise process is additive Gaussian, the resultant distribution of the observation components is Rician for the non coherent system model. For the case of non coherent measurements, the neural network classifier is presented with a four-dimensional observation vector whose components are the magnitudes of the noisy measurements at each of the four frequencies; (5) As in the coherent case, the neural net is typically trained with 200 samples for each of the five aircraft using exemplars of the form discussed above. The structure of the neural nets in this experiment was [4, nl, ... ,nh, 5] and the same training and testing procedure as in the coherent case was followed. Figure 7 shows a comparison of the performance of the neural net classifiers with both the maximum likelihood and nearest neighbor classifiers. As before, the max-likelihood out-performs the other classifiers, with the nearestneighbor classifier is second in performance, and the neural network classifiers perform roughly the same. CONCLUSIONS These experiments lead us to conclude that neural networks are good candidates for radar classification applications. Both of the neural network learning methods we tested have a similar performance and they are both relatively insensitive to changes in network architecture, network topology, and to the noise level of the training data. Because the methods used to implement the neural networks classifiers were relatively simple, we feel that the level of performance of the neural classifiers is quite impressive. Our ongoing research is concentrating on improving neural classifier performance by introducing more sophisticated learning algorithms such as the LVQ algorithm proposed by Kohonen [5]. We are also investigating methods of improving the performance of the perceptron, for example, by increasing training time. 287 288 Ahalt, Garber, Jouny and Krishnamurthy 100 90 :- 80 :-- ----_. ------ - FSCL4x20x5 perceptron 4X2Ox5 max-Okellhood near~, ralghbor-O db --- 70 I~ \\ '\ \ 60 l !5 50 \\ \\, I: ? 40 30 .~ 20 ~\, '\\ , , 10 ,~ ~ ,, . ' 0 -30 -25 -20 -15 -10 -5 0 5 10 15 20 SNR rdbl Figure 7: Comparison of all four classifiers for the non coherent data case. References [1] B. Bhanu, "Automatic target recognition: State of the art survey," IEEE Transactions on Aerospace and Electronic Systems, vol. AES-22, no. 4, pp. 364-379, July 1986. [2] R. R. Lippmann, "An Introduction to Computing with Neural Nets," IEEE ASSP Magazine, vol. 4, no. 2, pp. 4-22, April 1987. [3] S. C. Ahalt, A. K. Krishnamurthy, P. Chen, and D. E. Melton, "A new competitive learning algorithm for vector quantization using neural networks," Neural Networks, 1989. (submitted). [4] F. D. Garber, N. F. Chamberlain, and O. Snorrason, "Time-domain and frequency-domain feature selection for reliable radar target identification," in Proceedings of the IEEE 1988 National Itadar Conference, pp . 79-84, Ann Arbor, MI, April 20-21, 1988. [5] T . Kohonen, Self-Organization and Associative Memory, 2nd Ed. Springer-Veralg, 1988. Berlin: 

146 |@word aircraft:7 effect:1 establish:1 true:1 indicate:1 hence:1 nd:1 objective:1 laboratory:1 simulation:2 primary:1 white:3 attractive:1 traditional:1 x5:1 during:2 self:1 rician:1 kth:2 maintained:1 distance:1 require:1 berlin:1 stepped:1 investigation:1 theoretic:2 bhanu:1 performs:3 aes:1 measured:3 length:1 modeled:5 ratio:1 ohio:3 x5x5:1 superior:1 additive:5 a2:1 winner:1 insensitive:2 nh:2 boeing:1 discussed:2 design:2 selected:1 unknown:1 perform:2 superposition:1 measurement:13 observation:7 sensitive:2 neuron:3 ai:1 automatic:1 assp:1 gaussian:6 had:1 odb:2 obser:1 noncoherent:3 dc:1 five:4 impressive:1 operating:1 varying:2 consists:1 showed:1 superimposed:1 likelihood:7 aerospace:1 coherent:12 backscatter:9 sense:1 roughly:1 multi:3 seen:1 minimum:1 below:1 typically:2 pattern:1 employed:1 little:2 hidden:6 oft:1 increasing:1 signal:10 july:1 underlying:1 full:1 overall:1 classification:8 among:1 max:5 reliable:1 memory:1 characterized:1 art:1 cross:2 equal:1 concerning:1 having:2 feasibility:1 represents:1 basic:1 multilayer:1 classifier:24 unit:4 columbus:1 meter:1 before:2 national:1 engineering:1 phase:9 consisting:1 ahalt:6 organization:1 approximately:1 db:9 minimally:1 nearestneighbor:2 evaluation:1 concord:1 nl:2 near:1 range:1 lo:1 accurate:1 testing:1 architecture:3 topology:2 free:2 implement:1 perceptron:13 perfonnance:1 procedure:1 neighbor:6 desired:2 dimension:1 pre:1 mhz:2 close:1 selection:1 measuring:1 tg:1 transaction:1 introducing:1 eai:1 compact:1 lippmann:1 snr:9 band:2 investigating:1 characterize:1 focused:1 survey:1 vation:1 receiver:1 identifying:1 outperform:1 sl:1 assumed:1 synthetic:8 conclude:1 xi:1 sensitivity:1 learn:1 traditionally:1 krishnamurthy:6 vol:2 feel:2 chamberlain:1 target:3 commercial:3 four:6 magazine:1 improving:2 investigated:2 complex:3 domain:2 element:1 recognition:2 noise:12 melton:1 labeled:1 database:1 observed:1 potential:1 repeated:1 quadrature:7 fig:1 electrical:1 availability:1 coefficient:1 region:1 resonant:1 electronic:1 decrease:1 trade:1 root:2 decision:2 position:1 xl:1 candidate:1 environment:1 reached:1 competitive:3 effected:1 layer:7 capability:2 followed:1 radar:15 trained:3 depend:1 contribution:1 square:3 ni:1 specific:1 xt:1 variance:2 upon:1 correspond:2 identify:1 a3:1 quantization:1 identification:3 represented:1 various:1 optimality:1 magnitude:4 relatively:3 department:1 submitted:1 chen:1 ed:1 wavelength:1 whose:3 garber:6 quite:1 valued:1 evaluates:1 wi:3 frequency:14 otherwise:1 pp:3 resultant:1 di:2 mi:1 springer:1 noisy:5 final:1 associative:1 concentrating:1 knowledge:2 net:12 lvq:1 ann:1 sophisticated:1 kohonen:2 scattering:1 change:5 available:1 operation:1 methodology:1 improved:1 april:2 eight:1 evaluated:1 description:1 total:1 appearing:1 arbor:1 predicated:1 alternative:1 until:1 maximumlikelihood:1 object:4 a4:1 oo:2 ongoing:1 exemplar:1 nearest:6 received:2 tested:2

Self-similarity properties of natural images ANTONIO TURIEL; GERMAN MATOt NESTOR PARGA t Departamento de Fisica Te6rica. Universidad AutOnoma de Madrid Cantoblanco, 28049 Madrid, Spain and JEAN-PIERRE N ADAL? Laboratoire de Physique Statistique de I'E. N.S. , Ecole Normale Superieure 24, rue Lhomond, F-75231 Paris Cedex OS, France Abstract Scale invariance is a fundamental property of ensembles of natural images [1]. Their non Gaussian properties [15, 16] are less well understood, but they indicate the existence of a rich statistical structure. In this work we present a detailed study of the marginal statistics of a variable related to the edges in the images. A numerical analysis shows that it exhibits extended self-similarity [3, 4, 5]. This is a scaling property stronger than self-similarity: all its moments can be expressed as a power of any given moment. More interesting, all the exponents can be predicted in terms of a multiplicative log-Poisson process. This is the very same model that was used very recently to predict the correct exponents of the structure functions of turbulent flows [6]. These results allow us to study the underlying multifractal singularities. In particular we find that the most singular structures are one-dimensional: the most singular manifold consists of sharp edges. Category: Visual Processing. 1 Introduction An important motivation for studying the statistics of natural images is its relevance for the modeling of the visual system. In particular, the epigenetic development ? e-mail: amturiel@delta.ft.uam.es t e-mail: matog@cab.cnea.edu.ar +To whom correspondence should be addressed. e-mail: parga@delta.ft.uam.es ?e-mail: nadal@lps.ens.fr 'Laboratoire associe au C.N.R.S. (U.R.A. 1306), a l'ENS, et aux Universites Paris VI et Paris VII. 837 Self-similarity PropeT1ies of Natural Images could lead to the adaptation of visual processing to the statistical regularities in the visual scenes [8, 9, 10, 11, 12, 13]. Most of these predictions on the development of receptive fields have been obtained using a gaussian description of the environment contrast statistics. However non Gaussian properties like the ones found by [15, 16] could be important. To gain further insight into non Gaussian aspects of natural scenes we investigate the self similarity properties of an edge type variable [14]. Scale invariance in natural images is a well-established property. In particular it appears as a power law behaviour of the power spectrum of luminosity contrast: S(f) ex: IfIL'! (the parameter 1] depends on the particular images that has been included in the dataset). A more detailed analysis of the scaling properties of the luminosity contrast was done by [15, 16]. These authors noted the possible analogy between the statistics of natural images and turbulent flows. There is however no model to explain the scaling behaviour that they observed. On the other hand, a large amount of effort has been put to understand the statistics of turbulent flows and to develop predictable models (see e.g. [17]). Qualitative and quantitative theories of fully developed turbulence elaborate on the original argument of Kolmogorov [2]. The cascade of energy from one scale to another is described in terms of local energy dissipation per unit mass within a box of linear size r. This quantity, fr, is given by: (1) where Vi(X) is the ith component of the velocity at point x. This variable has Sel/Similarity (SS) properties that is, there is a range of scales r (called the inertial range) where: (2) here < f~ > denotes the pth moment of the energy dissipation marginal distribution. A more general scaling relation, called Extended Self-Similarity (ESS) has been found to be valid in a much larger scale domain. This relation reads (3) where p(p, q) is the ESS exponent of the pth moment with respect to the qth moment. Let us notice that if SS holds then Tp = Tqp(p, q). In the following we will refer all the moments to < f; >. 2 The Local Edge Variance For images the basic field is the contrast c(x), that we define as the difference between the luminosity and its average. By analogy with the definition in eq. (1) we will consider a variable that accumulates the value of the variation of the contrast. We choose to study two variables, defined at position x and at scale r. The variable fh,r(X) takes contributions from edges transverse to a horizontal segment of size r: fh,r(X) l1 = r xl +r Xl /))2 (ac(x -ay dy (4) X'={y,X2} A vertical variable fv,r(X) is defined similarly integrating along the vertical direction. We will refer to the value of the derivative of the contrast along a given direction as an edge transverse to that direction. This is justified in the sense that in the presence of borders this derivative will take a great value, and it will almost vanish A. Turiel, G. Mato, N. Parga and l-P. Nadal 838 if evaluated inside an almost-uniformly illuminated surface. Sharp edges will be the maxima ofthis derivative. According to its definition, ?/,r(x) ( 1 = h, v) is the local linear edge variance along the direction 1 at scale r. Let us remark that edges are well known to be important in characterizing images. A recent numerical analysis suggests that natural images are composed of statistically independent edges [18]. We have analyzed the scaling properties of the local linear edge variances in a set of 45 images taken into a forest, of 256 x 256 pixels each (the images have been provided to us by D. Ruderman; see [16] for technical details concerning them). An analysis of the image resolution and of finite size effects indicates the existence of upper and lower cut-offs. These are approximately r = 64 and r = 8, respectively. First we show that SS holds in a range of scales r with exponents Th,p and Tv,p. This is illustrated in Fig. (1) where the logarithm of two moments of horizontal and vertical local edge variances are plotted as a function of In r; we see that SS holds, but not in the whole range. ESS holds in the whole considered range; two representative graphs are shown in Fig. (2). The linear dependence of In < ?f,r > vs In < ?f,r > is observed in both the horizontal (l = h) and the vertical (l = v) directions. This is similar to what is found in turbulence, where this property has been used to obtain a more accurate estimation of the exponents of the structure functions (see e.g. [17] and references therein) . The exponents Ph(p, 2) and Pv(p,2), estimated with a least squares regression, are shown in Fig. (3) as a function of p . The error bars refer to the statistical dispersion. From figs. (1-3) one sees that the horizontal and vertical directions have similar statistical properties. The SS exponents differ, as can be seen in Fig(I); but, surprisingly, ESS not only holds in both directions, but it does it with the same ESS exponents, i.e. Ph(P,2) '" Pv(p, 2). 3 ESS and multiplicative processes Let us now consider scaling models to predict the Jrdependence of the ESS exponents Pl(p, 2). (Since ESS holds, the SS exponents Tl ,p can be obtained from the Pl(p, 2)' s by measuring 72,2). The simplest scaling hypothesis is that, for a random variable ?r(x) observed at the scale r (such as ?/,r(x)), its probability distribution Pr(?r(x) = ?) can be obtained from any other scale L by Pr(?) = a(r~ L) PL (a(r~ L)) (5) From this one derives easily that a(r, L) = [~:~~P/p (for any p) and p(p, 2) ex: p; if SS holds, Tp ex: p: for turbulent flows this corresponds to the Kolmogorov prediction for the SS exponents [2] . Fig (3) shows that this naive scaling is violated. This discrepancy becomes more dramatic if eq. (5) is expressed in terms of a normalized variable. Taking ?~ = limp -+ oo < ?~+l > / < ?~ > ( that can be shown to be the maximum value of ?r, which in fact is finite) the new variable is defined as ir = ?r/?~ ; 0 < ir < 1. If Pr(J) is the distribution of ir, the scaling relation eq.(5) reads Pr(J) = PL(J) ; this identity does not hold as can be seen in Fig. (4). A way to generalize this scaling hypothesis is to say that a is no longer a constant as in eq. (5), but an stochastic variable. Thus, one has for Pr(J) : (6) This scaling relation has been first introduced in the context of turbulent flows [6, 19, 7]. Eq. (6) is an integral representation of ESS with general (not necessarily 839 Self-similarity Properties of Natural Images linear) exponents: once the kernel G rL is chosen, the p(p, 2)'s can be predicted. It can also be phrased in terms of multiplicative processes [20, 21] : now ir = aiL, where the factor a itself becomes a stochastic variable determined by the kernel G rL (1na). Since the scale L is arbitrary (scale r can be reached from any other scale L') the kernel must obey a composition law, GrLI ?G L' L = GrL. Consequently ir can be obtained through a cascade of infinitesimal processes G6 == G r ,r+6r' Specific choices of G6 define different models of ESS. The She-Leveque (SL) [6] model corresponds to a simple process such that a is 1 with probability 1 - sand is a constant f3 with probability s. One can see that s = ll,lF In( <~tl;? and that this stochastic process yields a log-Poisson distribution for a [22]. It also gives ESS with exponents p(p, q) that is expressed in terms of the parameter f3 as follows [6]: p(p,q) 1 - f3P - (1 - (3)p (1- (3)q = 1- f3 q - (7) We can now test this models with the ESS exponents obtained with the image data set. The resulting fit for the SL model is shown in Fig. (3). Both the vertical and horizontal ESS exponents can be fitted with {3 = 0.50 ? 0.03. The integral representation of ESS can also be directly tested on the probability distributions evaluated from the data. In Fig. (4) we show the prediction for Pr (f) obtained from PL(f) using eq. (6) , compared with the actual Pr(f). The parameter f3 allows us to predict all the ESS exponents p(p,2). To obtain the SS exponents 7p we need another parameter. This can be chosen e.g. as 72 or as the asymptotic exponent ~, given by f~ ex: r-t::., r ? 1; we prefer~. As 7p = 72 p(P, 2), then from the definition of f~ one can see that ~ = -1":!13' A least square fit of 7p was used to determine ~, obtaining ~h = 0.4 ? 0.2 for the horizontal variable and ~v = 0.5 ? 0.2. for the vertical one. 4 Multifractal analysis Let us now partition the image in sets of pixels with the same singularity exponent h of the local edge variance: fr ex: rh. This defines a multifractal with dimensions D(h) given by the Legendre transform of 7p (see e.g. [17]): D(h) = inip{ph+d-7p}, where d = 2 is the dimension of the images. We are interested in the most singular of these manifolds; let us call Doo its dimension and h min its singularity exponent. Since f~ is the maximum value of the variable fr, the most singular manifold is given by the set of points where fr = f~, so h min = -~. Using again that 7p = -~ (1- {3) p(P, 2) with p(P, 2) given by the SL model, one has Doo = d- (1~13)' From our data we obtain Doo ,h = 1.3 ? 0.3 and Doo ,v = 1.1 ? 0.3. As a result we can say that Doo,h "" Doo,v "" 1: the most singular structures are almost onedimensional. This reflects the fact that the most singular manifold consists of sharp edges. 5 Conclusions We insist on the main result of this work, which is the existence of non trivial scaling properties for the local edge variances. This property appears very similar to the one observed in turbulence for the local energy dissipation. In fact, we have seen that the SL model predicts all the relevant exponents and that, in particular, it describes the scaling behaviour of the sharpest edges in the image ensemble. It would also be interesting to have a simple generative model of images which - apart 840 A. Turiel, G. Mato, N. Parga and J-P. Nadal from having the correct power spectrum as in [23] - would reproduce the self-similar properties found in this work. Acknowledgements We are grateful to Dan Ruderman for giving us his image data base. We warmly thank Bernard Castaing for very stimulating discussions and Zhen-Su She for a discussion on the link between the scaling exponents and the dimension of the most singular structure. We thank Roland Baddeley and Patrick Tabeling for fruitful discussions. We also acknowledge Nicolas BruneI for his collaboration during the early stages of this work. This work has been partly supported by the FrenchSpanish program "Picasso" and an E.V. grant CHRX-CT92-0063. References [1] Field D. J., 1. Opt. Soc. Am. 4 2379-2394 (1987). [2] Kolmogorov, Dokl. Akad. Nauk. SSSR 30, 301-305 (1941). [3] Benzi R., Ciliberto S., Baudet C., Ruiz Chavarria G. and Tripiccione C., Europhys. Lett. 24 275-279 (1993) [4] Benzi, Ciliberto, Tripiccione, Baudet, Massaioli, and Succi, Phys. Rev. E 48, R29 (1993) [5] Benzi, Ciliberto, Baudet and Chavarria Physica D 80 385-398 (1995) [6] She and Leveque, Phys. Rev. Lett. 72,336-339 (1994). [7] Castaing, 1. Physique II, France 6, 105-114 (1996) [8] Barlow H. B., in Sensory Communication (ed. Rosenblith W.) pp. 217. (M.I.T. Press, Cambridge MA, 1961). [9] Laughlin S. B., Z. Naturf. 36 910-912 (1981). [10] van Hateren J.H. 1. Compo Physiology A 171157-170,1992. [11] Atick J. J. Network 3 213-251, 1992. [12] Olshausen B.A. and Field D. J., Nature 381, 607-609 (1996). [13] Baddeley R., Cognitive Science, in press (1997). [14] Turiel A., Mato G., Parga N. and Nadal J.-P., to appear in Phys. Rev. Lett., 1998. [15] Ruderman D. and Bialek, Phys. Rev. Lett. 73,814 (1994) [16] Ruderman D., Network 5,517-548 (1994) [17] Frisch V., Turbulence, Cambridge Vniv. Press (1995). [18] Bell and Sejnowski, Vision Research 37 3327-3338 (1997). [19] Dubrulle B., Phys. Rev. Lett. 73 959-962 (1994) [20] Novikov, Phys. Rev. E 50, R3303 (1994) [21] Benzi, Biferale, Crisanti, Paladin, Vergassola and Vulpiani, Physica D 65, 352-358 (1993). [22] She and Waymire, Phys. Rev. Lett. 74, 262-265 (1995). [23] Ruderman D., Vision Research 37 3385-3398 (1997). Self-similarity Properties of Natural Images In < ?~ > 841 b a .. ." .. " Inr Inr Figure 1: Test of SS. We plot In < ?f r > vs. In r for p = 3 and 5; r from 8 to 64 pixels. a) horizontal direction, l = h. b) vertical direction, l = v. In < ?~ > b a , // ? . / ." / / .-/ ' ""....?' .... / '" ,/ , "./'" In < ?~ >~ ? ? ..", /' / / ././ , // ......./...-' . ,,, ,,-- . , ....... ...",/ // ./ . ~. , .. . // -' .... ' ." .".... .-/ .,..,/ /./ ./ ./ // In ?f.r < ?~ > Figure 2: Test of ESS. We plot In < > vs. In < ?~, r > for p=3, 5; r from 8 to r = 64 pixels. a) horizontal direction, l = h. b) vertical direction, l = v. A. Turiel, G. Mato, N. Parga and J-P. Nadal 842 p(p, 2) b a 12 1. p p Figure 3: ESS exponents p(p, 2), for the vertical and horizontal variables. a) horizontal direction, Ph (P, 2) . b) vertical direction, pv (p, 2). The solid line represents the fit with the SL model. The best fit is obtained with (3v '" (3h '" 0.50. P 18 16 14 12 10 8 ++ + + + 6 + 4 + 2 ++++ + + + + 0 0 0.05 0.1 0.15 0.2 f Figure 4: Verification of the validity of the integral representation of ESS, eq.(6) with a log-Poisson kernel, for horizontal local edge variance. The largest scale is L = 64. Starting from the histogram Pdf) (denoted with crosses), and using a log-Poisson distribution with parameter (3 = 0.50 for the kernel GrL , eq.(6) gives a prediction for the distribution at the scale r = 16 (squares). This has to be compared with the direct evaluation of Pr (I) (diamonds). Similar results hold for other pairs of scales. Although not shown in the figure, the test for vertical case is as good as for horizontal variable. 

1460 |@word stronger:1 dramatic:1 solid:1 moment:7 ecole:1 qth:1 must:1 numerical:2 partition:1 limp:1 plot:2 v:3 generative:1 adal:1 es:18 ith:1 compo:1 along:3 direct:1 qualitative:1 consists:2 dan:1 inside:1 insist:1 actual:1 becomes:2 spain:1 provided:1 underlying:1 mass:1 what:1 nadal:5 ail:1 developed:1 quantitative:1 unit:1 grant:1 appear:1 understood:1 local:9 accumulates:1 approximately:1 au:1 therein:1 suggests:1 range:5 statistically:1 lf:1 bell:1 cascade:2 physiology:1 statistique:1 integrating:1 turbulence:4 put:1 context:1 fruitful:1 starting:1 resolution:1 insight:1 his:2 variation:1 grl:2 hypothesis:2 velocity:1 cut:1 predicts:1 observed:4 ft:2 environment:1 predictable:1 grateful:1 segment:1 easily:1 kolmogorov:3 sejnowski:1 picasso:1 europhys:1 jean:1 larger:1 say:2 s:10 statistic:5 transform:1 itself:1 fr:5 adaptation:1 relevant:1 superieure:1 nauk:1 description:1 regularity:1 oo:1 develop:1 ac:1 novikov:1 eq:8 soc:1 predicted:2 indicate:1 differ:1 direction:13 sssr:1 correct:2 stochastic:3 sand:1 behaviour:3 opt:1 singularity:3 pl:5 physica:2 hold:9 considered:1 aux:1 great:1 predict:3 early:1 fh:2 estimation:1 largest:1 reflects:1 offs:1 gaussian:4 normale:1 sel:1 she:4 indicates:1 contrast:6 sense:1 am:1 relation:4 reproduce:1 france:2 interested:1 pixel:4 denoted:1 exponent:23 development:2 autonoma:1 marginal:2 field:4 once:1 f3:4 having:1 represents:1 discrepancy:1 composed:1 nestor:1 ciliberto:3 epigenetic:1 brunei:1 investigate:1 inr:2 evaluation:1 physique:2 analyzed:1 accurate:1 edge:17 integral:3 logarithm:1 plotted:1 fitted:1 modeling:1 ar:1 tp:2 measuring:1 crisanti:1 fundamental:1 universidad:1 na:1 again:1 choose:1 cognitive:1 derivative:3 tqp:1 de:4 vi:2 depends:1 multiplicative:3 reached:1 contribution:1 square:3 ir:5 variance:7 ensemble:2 yield:1 castaing:2 generalize:1 sharpest:1 parga:6 mato:4 explain:1 phys:7 rosenblith:1 ed:1 definition:3 infinitesimal:1 energy:4 pp:1 gain:1 dataset:1 inertial:1 appears:2 done:1 box:1 evaluated:2 stage:1 atick:1 hand:1 horizontal:12 ruderman:5 su:1 o:1 defines:1 olshausen:1 effect:1 validity:1 normalized:1 barlow:1 read:2 illustrated:1 ll:1 during:1 self:9 noted:1 benzi:4 pdf:1 ay:1 dissipation:3 l1:1 image:22 recently:1 rl:2 onedimensional:1 refer:3 composition:1 cambridge:2 turiel:5 similarly:1 similarity:9 surface:1 longer:1 base:1 patrick:1 recent:1 apart:1 multifractal:3 seen:3 determine:1 f3p:1 ii:1 technical:1 cross:1 concerning:1 roland:1 prediction:4 basic:1 regression:1 vision:2 poisson:4 histogram:1 kernel:5 justified:1 doo:6 addressed:1 laboratoire:2 singular:7 cedex:1 flow:5 call:1 presence:1 fit:4 effort:1 remark:1 antonio:1 detailed:2 amount:1 ph:4 category:1 simplest:1 sl:5 notice:1 delta:2 estimated:1 per:1 graph:1 almost:3 dy:1 scaling:14 prefer:1 illuminated:1 correspondence:1 scene:2 x2:1 phrased:1 aspect:1 argument:1 min:2 tv:1 according:1 legendre:1 describes:1 lp:1 rev:7 pr:8 g6:2 taken:1 german:1 turbulent:5 studying:1 uam:2 obey:1 pierre:1 existence:3 original:1 denotes:1 giving:1 quantity:1 receptive:1 dependence:1 bialek:1 exhibit:1 thank:2 link:1 manifold:4 mail:4 whom:1 trivial:1 akad:1 diamond:1 upper:1 vertical:12 dispersion:1 finite:2 acknowledge:1 luminosity:3 extended:2 communication:1 sharp:3 arbitrary:1 transverse:2 introduced:1 pair:1 paris:3 fv:1 established:1 bar:1 dokl:1 program:1 power:4 natural:10 zhen:1 naive:1 r29:1 acknowledgement:1 asymptotic:1 law:2 fully:1 interesting:2 analogy:2 verification:1 collaboration:1 surprisingly:1 supported:1 allow:1 understand:1 laughlin:1 characterizing:1 taking:1 van:1 dimension:4 lett:6 valid:1 rich:1 sensory:1 author:1 pth:2 spectrum:2 nature:1 nicolas:1 obtaining:1 forest:1 necessarily:1 rue:1 domain:1 main:1 rh:1 motivation:1 border:1 whole:2 fig:9 representative:1 madrid:2 en:2 elaborate:1 tl:2 lhomond:1 position:1 pv:3 baudet:3 xl:2 vanish:1 ruiz:1 specific:1 departamento:1 derives:1 ofthis:1 cab:1 vii:1 visual:4 expressed:3 corresponds:2 ma:1 stimulating:1 identity:1 consequently:1 included:1 determined:1 uniformly:1 called:2 bernard:1 invariance:2 e:2 partly:1 relevance:1 violated:1 hateren:1 baddeley:2 tested:1 ex:5

Refractoriness and Neural Precision Michael J. Berry n and Markus Meister Molecular and Cellular Biology Department Harvard University Cambridge, MA 02138 Abstract The relationship between a neuron's refractory period and the precision of its response to identical stimuli was investigated. We constructed a model of a spiking neuron that combines probabilistic firing with a refractory period. For realistic refractoriness, the model closely reproduced both the average firing rate and the response precision of a retinal ganglion cell. The model is based on a "free" firing rate, which exists in the absence of refractoriness. This function may be a better description of a spiking neuron's response than the peri-stimulus time histogram. 1 INTRODUCTION The response of neurons to repeated stimuli is intrinsically noisy. In order to take this trial-to-trial variability into account, the response of a spiking neuron is often described by an instantaneous probability for generating an action potential. The response variability of such a model is determined by Poisson counting statistics; in particular, the variance in the spike count is equal to the mean spike count for any time bin (Rieke, 1997). However, recent experiments have found far greater precision in the vertebrate retina (Berry, 1997) and the HI interneuron in the fly visual system (de Ruyter, 1997). In both cases, the neurons exhibited sharp transitions between silence and nearly maximal firing. When a neuron is firing near its maximum rate, refractoriness causes spikes to become more regularly spaced than for a Poisson process with the same firing rate. Thus, we asked the question: does the refractory period play an important role in a neuron's response precision under these stimulus conditions? 2 FIRING EVENTS IN RETINAL GANGLION CELLS We addressed the role of refractoriness in the precision of light responses for retinal ganglion cells. 2.1 RECORDING AND STIMULATION Experiments were performed on the larval tiger salamander. The retina was isolated from the eye and superfused with oxygenated Ringer's solution. Action potentials from retinal Refractoriness and Neural Precision 111 ganglion cells were recorded extracellularly with a multi-electrode array, and their spike times measured relative to the beginning of each stimulus repeat (Meister, 1994). Spatially uniform white light was projected from a computer monitor onto the photoreceptor layer. The intensity was flickered by choosing a new value at random from a Gaussian distribution (mean J, standard deviation oJ) every 30 ms. The mean light level (J= 4'10-3 W/m2) corresponded to photopic (daylight) vision. Contrast C is defined here as the temporal standard deviation of the light intensity divided by the mean, C = 01/ I. Recordings extended over 60 repeats of a 60-sec segment of random flicker. The qualitative features of ganglion cell responses to random flicker stimulation at 35 % contrast are seen in Fig. 1. First, spike trains had extensive periods in which no spikes were seen in 60 repeated trials. Many spike trains were sparse, in that the silent periods covered a large fraction of the total stimulus time. Second, during periods of firing, the peri-stimulus time histogram (PSTH) rose from zero to the maximum firing rate (-200 Hz) on a time scale comparable to the time interval between spikes (-10 ms). We have argued that these responses are better viewed as a set of discrete firing "events" than as a continuously varying firing rate (Berry, 1997). In general, the firing events were bursts containing more than one spike (Fig. IB). Identifiable firing events were seen across cell types; similar results were also found in the rabbit retina (Berry, 1997). >. '..-wc Q) ..c 2 A 1 0 60 ~ ?c 40 ~' If: N I , I 'I~~ ~ . ?' .I 1 \ ~ ,.~. \>.~ ~-(- /..'1 '~*': >f. 0 ! : ) ..0: ~~,{ 20 .'J: I'r '~ .~' :~ ~~ I- .- 300 I' W? 8 ,It: C Q) ni a: 0 43.4 43.5 43.6 43.7 43.8 Time (5) Figure 1: Response of a salamander ganglion cell to random flicker stimulation. (A) Stimulus intensity in units of the mean for a O.5-s segment, (B) spike rasters from 60 trials, and (C) the firing rate r(t). 2.2 FIRING EVENT PRECISION Discrete episodes of ganglion cell firing were recognized from the PSTH as a contiguous period of firing bounded by periods of complete silence. To provide a consistent demarcation of firing events, we drew the boundaries of a firing event at minima v in the PSTH that were significantly lower than neighboring maxima PI and P2' such that ~ PIP2 ~ ? with 95 % confidence (Berry, 1997). With these boundaries defined, every spike in each trial was assigned to exactly one firing event. Iv M. J Berry and M. Meister 112 Measurements of both timing and number precision can be obtained if the spike train is parsed into such firing events. For each firing event i, we accumulated the distribution of spike times across trials and calculated several statistics: the average time Tj of the first spike in the event and its standard deviation OTj across trials, which quantified the temporal jitter of the first spike; similarly, the average number N j of spikes in the event and its variance ONj 2 across trials, which quantified the precision of spike number. In trials that contained zero spikes for event i, no contribution was made to Tj or OTj , while a value of zero was included in the calculation of Nj and ONj 2 . For the ganglion cell shown in Fig. 1, the temporal jitter oT of the first spike in an event was very small (1 to 10 ms). Thus, repeated trials of the same stimulus typically elicit action potentials with a timing uncertainty of a few milliseconds. The temporal jitter of all firing events was distilled into a single number Tby taking the median o"er all events. The variance ON 2 in the spike count was remarkably low as well: it often approached the lower bound imposed by the fact that individual trials necessarily produce integer spike counts. Because ON 2 ? N for all events, ganglion cell spike trains cannot be completely characterized by their firing rate (Berry, 1997). The spike number precision of a cell was average variance over events and dividing by the average assessed by comp.utin,fo. spike count: F =(ON- Jj(N). This quantity, also known as the Fano factor, has a value of one for a Poisson process with no refractoriness. tHe 3 PROBABILISTIC MODELS OF A SPIKE TRAIN We start by reviewing one of the simplest probabilistic models of a spike train, the inhomogeneous Poisson model. Here, the measured spike times {t j } are used to estimate the instantaneous rate r(t) of spike generation during a time Lit . This can be written fonnallyas where M is the number of repeated stimulus trials and e( x) is the Heaviside function x~O} 1 () ex= o . x<O We can randomly generate a sequence of spike trains from a set of random numbers between zero and one: {aj } with a j E (0,1]. If there is a spike at time tj , then the next spike time tj+1 is found by numerically solving the equation 1,+, -Ina;+! = Jr(t)dt . t, 3.1 INCLUDING AN ABSOLUTE REFRACTORY PERIOD In order to add refractoriness to the Poisson spike-generator, we expressed the firing rate as the product of a "free" firing rate q(t) , which obtains when the neuron is not refractory, and a recovery function w(t), which describes how the neuron recovers from refractoriness (Johnson, 1983; Miller, 1985). When the recovery function is zero, spiking is not possible; and when it is one, spiking is not affected. The modified rule for selecting spikes then becomes I,... , -lna j +1 = Jq(t)w(t-t;)dt . " For an absolute refractory period of time J1, the weight function is zero for times between o and J1 and one otherwise Refractoriness and Neural Precision 113 w{t;,u) = 1- B{t )B(,u - t) Because the refractory period may exclude spiking in a given time bin, the probability of firing a spike when not prevented by the refractory period is higher than predicted by r( t). This free firing rate q( t ; ,u) can be estimated by excluding trials where the neuron is unable to fire due to refractoriness The sum is restricted to spike times ti nearest to the time bin on a given trial. This restriction follows from the assumption that the recovery function only depends on the time since the last action potential. Notice that this new probability obeys the inequality q( t ) ~ p( t) and also that it depends upon the refractory period ,u. -N I 4.5 Q) ~ a: t - + - -i- -1- -f - -f" - ~- - .. - -. - - - -- ? 4.3 OJ c ?c u:: ? 4.1 1.00 lJ... .... 0.75 0 ( ,) ctS 0.50 0 c ctS u.. 0.25 ? ? ? ? ----------------------.--.-~ 0.00 - - 5 .,.. 4 =: 3 e! 2 (J) E .... Q) ::s 8. 1 ~ 0 E ?_____________ ? ? ? ? _?___? __ S __ ? o 1 2 3 4 __ ,.. __? 5 Refractory Period (ms) Figure 2: Results for model spike trains with an absolute refractory period. (A) Mean firing rate averaged over a 60-s segment (circles), (B) Fano factor F, a measure of spike number precision in an event (triangles), and (C) temporal jitter 't'(diamonds) plotted versus the absolute refractory period ,u. Shown in dotted in each panel is the value for the real data. With this definition of the free firing rate, we can now generate spike trains with the same first order statistics (i.e., the average firing rate) for a range of values of the refractory period ,u. For each value of ,u, we can then compare the second order statistics (i.e., the precision) of the model spike trains to the real data. To this end, the free rate q( t) was M. 1 Berry and M. Meister 114 calculated for a 60-s segment of the response to random flicker of the salamander ganglion cell shown in Fig. 1. Then, q(t) was used to generate 60 spike trains. Firing events were identified in the set of model spike trains, and their precision was calculated. Finally, this procedure was repeated 10 times for each value ofthe refractory period. Figure 2A plots the firing rate (circles) generated by the model, averaged over the entire 60-s segment of random flicker with error bars equal to the standard deviation of the rate among the 10 repeated sets. The firing rate of the model matches the actual firing rate for the real ganglion cell (dashed) up to refractory periods of J1 :: 4 ms, although the deviation for larger refractory periods is still quite small. For large enough values of the absolute refractory period, there will be inter-spike intervals in the real data that are shorter than J1. In this case, the free firing rate q( t) cannot be enhanced enough to match the observed firing rate. While the mean firing rate is approximately constant for refractory periods up to 5 ms, the precision changes dramatically. Figure 2B shows that the Fano factor F (triangles) has the expected value of 1 for no refractory period, but drops to ~ 0.2 for the largest refractory period. In Fig. 2e, the temporal jitter 'l" (diamonds) also decreases as refractoriness is added, although the effect is not as large as for the precision of spike number. The sharpening of temporal precision is due to the fact that the probability q( t) rises more steeply than r(t) (see Fig. 4), so that the first spike occurs over a narrower range oftimes. The number precision of the model matches the real data for J1 = 4 to 4.5 ms and the timing precision matches for ~ :: 4 ms. Therefore, a probabilistic spike generator with an absolute refractory period can match both the average firing rate and the precision of a retinal ganglion cell's spike train with roughly the same value of one free parameter. 3.2 USING A RELATIVE REFRACTORY PERIOD Salamander ganglion cells typically have a relative refractory period that lasts beyond their absolute refractory period. This can be seen in Fig. 3A from the distribution of interspike intervals P{L1) for the ganglion cell shown above - the absolute refractory period lasts for only 2 ms, while relative refractoriness extends to ~ 5 ms. We can include the effects of relative refractoriness by using weight values in w( t) that are between zero and one. Figure 3 illustrates a parameter-free method for determining this weight function. If there were no refractoriness and a neuron had a constant firing rate q, then the inter-spike interval distribution would drop exponentially. This behavior is seen from the curve fit in Fig. 3A for intervals in the range 5 to 10 ms. The recovery function w(t) can then be found from the inter-spike interval distribution (Berry, 1998) Notice in Fig. 3B, that the recovery function w(t) is zero out to 3 ms, rises almost linearly between 3 and 5 ms, and then reaches unity beyond 5 ms. Using the weight function shown in Fig. 3B, the free firing rate q(t) was calculated and 10 sets of 60 spike trains were generated. The results, summarized in Table 1, give very close agreement with the real data: Table 1: Results for a Relative Refractory Period QUANTITY REAL DATA MODEL STD. DEV. Firing Rate Timing Precision 'l" Number Precision F 4.43 Hz 3.20ms 0.250 4.44 Hz 2.95 ms 0.266 0.017 Hz 0.09ms 0.004 Refractoriness and Neural Precision 115 Thus, a Poisson spike generator with a relative refractory period reproduces the measured precision. A similar test, performed over a population of ganglion cells, also yielded close agreement (Berry, 1998). 1000 >u c: A 0 0 0000 0 100 0 Q) :::J 0 CT .... Q) u. 10 ~OOO 0 0 000 0 0 1 0 2 4 6 8 10 12 10 12 Inter-Spike Interval (ms) c: 0 U c: 1.0 B :::J U. ~ 0.5 Q) > 0 u Q) 0.0 a:: 0 2 4 6 8 Time (ms) Figure 3: Determination of the relative refractory period. (A) The interspike interval distribution (diamonds) is fit by an exponential curve (solid), resulting in (B) the recovery function. Not only is the average firing rate well-matched by the model, but the firing rate in each time bin is also very similar. Figure 4A compares the firing rate for the real neuron to that generated by the model. The mean-squared error between the two is 4 %, while the counting noise, estimated as the variance of the standard error divided by the variance of r{t) , is also 4 %. Thus, the agreement is limited by the finite number of repeated trials. Figure 4B compares the free firing rate q( t) to the observed rate firing r{ t). q( t) is equal to r(t) at the beginning of a firing event, but becomes much larger after several spikes have occurred. In addition, q(t) is generally smoother than r(t), because there is a greater enhancement in q(t) at times following a peak in r(t). In summary, the free firing rate q( t) can be calculated from the raw spike train with no more computational difficulty than r{t), and thus can be used for any spiking neuron. Furthermore, q(t) has some advantages over r(t): 1) in conjunction with a refractory spike-generator, it produces the correct response precision; 2) it does not saturate at high firing rates, so that it can continue to distinguish gradations in the neuron's response. Thus, q( t) may prove useful for constructing models of the input-output relationship of a spiking neuron (Berry, 1998). Acknowledgments We would like to thank Mike DeWeese for many useful conversations. One of us, MJB, acknowledges the support of the National Eye Institute. The other, MM, acknowledges the support of the National Science Foundation. Figure 4: Illustration of the free fIring rate. (A) The observed fIring rate r(t) for real data (solid) is compared to that from the model (dotted). (B) The free rate q(t) (thick) is shown on the same scale as r(t) (thin). All rates used a time bin of 0.25 ms and boxcar smoothing 'over 9 bins. References Berry, M. J., D. K. Warland, and M. Meister, The Structure and Precision ofRetinal Spike Trains. PNAS, USA, 1997.94: pp. 5411-5416. Berry II, M. J. and Markus Meister, Refractoriness and Neural Precision. 1. Neurosci., 1998. in press. De Ruyter van Steveninck, R. R., G. D. Lewen, S. P. Strong, R. Koberle, and W. Bialek, Reliability and Variability in Neural Spike Trains. Science, 1997.275: pp. 1805-1808. Johnson, D. H. and A. Swami, The Transmission of Signals by Auditory-Nerve Fiber Discharge Patterns. J. Acoust. Soc. Am., 1983. 74: pp. 493-501. Meister, M., J. Pine, and D. A. Baylor, Multi-Neuronal Signals from the Retina: Acquisition and Analysis. 1. Neurosci. Methods, 1994.51: pp. 95-106. Miller, M . 1. Algorithms for Removing Recovery-Related Distortion gtom Auditory-Nerve Discharge Patterns. J. Acoust. Soc. Am., 1985.77: pp. 1452-1464. Rieke, F., D. K. Warland, R. R. de Ruyter van Steveninck, and W. Bialek, Spikes: Exploring the Neural Code . 1997, Cambridge, MA: MIT Press. 

1461 |@word trial:15 solid:2 selecting:1 written:1 realistic:1 j1:5 interspike:2 plot:1 drop:2 beginning:2 psth:3 burst:1 constructed:1 become:1 qualitative:1 prove:1 combine:1 inter:4 expected:1 roughly:1 behavior:1 multi:2 actual:1 vertebrate:1 becomes:2 bounded:1 matched:1 panel:1 acoust:2 sharpening:1 nj:1 temporal:7 every:2 ti:1 exactly:1 unit:1 timing:4 firing:52 approximately:1 quantified:2 limited:1 range:3 obeys:1 averaged:2 steveninck:2 acknowledgment:1 procedure:1 elicit:1 significantly:1 confidence:1 onto:1 cannot:2 close:2 gradation:1 restriction:1 imposed:1 rabbit:1 recovery:7 m2:1 rule:1 array:1 population:1 rieke:2 discharge:2 enhanced:1 play:1 agreement:3 harvard:1 std:1 observed:3 role:2 mike:1 fly:1 episode:1 decrease:1 rose:1 asked:1 reviewing:1 segment:5 solving:1 swami:1 upon:1 completely:1 triangle:2 fiber:1 train:17 approached:1 corresponded:1 choosing:1 quite:1 larger:2 distortion:1 otherwise:1 statistic:4 noisy:1 reproduced:1 sequence:1 advantage:1 maximal:1 product:1 neighboring:1 description:1 electrode:1 enhancement:1 transmission:1 produce:2 generating:1 measured:3 nearest:1 strong:1 dividing:1 lna:1 predicted:1 soc:2 p2:1 inhomogeneous:1 closely:1 correct:1 thick:1 bin:6 argued:1 larval:1 exploring:1 mm:1 pine:1 largest:1 onj:2 mit:1 gaussian:1 modified:1 varying:1 conjunction:1 salamander:4 steeply:1 contrast:2 demarcation:1 am:2 accumulated:1 typically:2 lj:1 entire:1 jq:1 among:1 smoothing:1 equal:3 distilled:1 biology:1 identical:1 lit:1 nearly:1 thin:1 stimulus:10 few:1 retina:4 randomly:1 national:2 individual:1 fire:1 light:4 tj:4 shorter:1 iv:1 circle:2 plotted:1 isolated:1 dev:1 contiguous:1 deviation:5 uniform:1 johnson:2 peri:2 peak:1 probabilistic:4 michael:1 continuously:1 squared:1 recorded:1 containing:1 account:1 potential:4 exclude:1 de:3 retinal:5 sec:1 summarized:1 depends:2 performed:2 extracellularly:1 start:1 contribution:1 ni:1 variance:6 miller:2 spaced:1 ofthe:1 raw:1 comp:1 fo:1 reach:1 definition:1 raster:1 acquisition:1 pp:5 recovers:1 auditory:2 intrinsically:1 conversation:1 oftimes:1 nerve:2 higher:1 dt:2 response:14 ooo:1 refractoriness:17 furthermore:1 aj:1 usa:1 effect:2 assigned:1 spatially:1 white:1 during:2 m:20 complete:1 l1:1 instantaneous:2 stimulation:3 spiking:8 refractory:29 exponentially:1 occurred:1 numerically:1 measurement:1 cambridge:2 boxcar:1 similarly:1 fano:3 had:2 reliability:1 add:1 recent:1 inequality:1 continue:1 seen:5 minimum:1 greater:2 recognized:1 period:32 dashed:1 ii:1 smoother:1 signal:2 pnas:1 match:5 characterized:1 calculation:1 determination:1 divided:2 molecular:1 prevented:1 vision:1 poisson:6 histogram:2 cell:17 addition:1 remarkably:1 addressed:1 interval:8 median:1 ot:1 exhibited:1 recording:2 hz:4 regularly:1 integer:1 near:1 counting:2 enough:2 fit:2 identified:1 silent:1 otj:2 cause:1 jj:1 action:4 dramatically:1 generally:1 useful:2 covered:1 simplest:1 generate:3 flicker:5 millisecond:1 notice:2 dotted:2 estimated:2 discrete:2 affected:1 monitor:1 deweese:1 fraction:1 sum:1 ringer:1 jitter:5 uncertainty:1 extends:1 almost:1 comparable:1 layer:1 hi:1 bound:1 ct:3 distinguish:1 identifiable:1 yielded:1 markus:2 wc:1 department:1 jr:1 across:4 describes:1 unity:1 restricted:1 equation:1 count:5 end:1 meister:7 include:1 warland:2 parsed:1 mjb:1 question:1 quantity:2 spike:57 occurs:1 added:1 bialek:2 unable:1 thank:1 cellular:1 code:1 relationship:2 illustration:1 baylor:1 daylight:1 rise:2 diamond:3 neuron:16 finite:1 extended:1 variability:3 excluding:1 sharp:1 intensity:3 extensive:1 beyond:2 bar:1 pattern:2 oj:2 including:1 event:21 difficulty:1 eye:2 acknowledges:2 koberle:1 lewen:1 berry:13 determining:1 relative:8 ina:1 generation:1 versus:1 generator:4 foundation:1 consistent:1 pi:1 summary:1 repeat:2 last:3 free:13 silence:2 institute:1 taking:1 absolute:8 sparse:1 van:2 boundary:2 calculated:5 curve:2 transition:1 made:1 projected:1 far:1 obtains:1 reproduces:1 table:2 ruyter:3 investigated:1 necessarily:1 constructing:1 linearly:1 neurosci:2 noise:1 repeated:7 neuronal:1 fig:10 precision:28 exponential:1 ib:1 saturate:1 removing:1 er:1 exists:1 drew:1 illustrates:1 interneuron:1 ganglion:15 visual:1 expressed:1 contained:1 pip2:1 ma:2 viewed:1 narrower:1 absence:1 tiger:1 change:1 included:1 determined:1 total:1 photoreceptor:1 tby:1 support:2 assessed:1 heaviside:1 ex:1

Characterizing Neurons in the Primary Auditory Cortex of the Awake Primate U sing Reverse Correlation R. Christopher deC harms decharms@phy.ucsf.edu Michael M . Merzenich merz@phy.ucsf.edu w. M. Keck Center for Integrative Neuroscience University of California, San Francisco CA 94143 Abstract While the understanding of the functional role of different classes of neurons in the awake primary visual cortex has been extensively studied since the time of Hubel and Wiesel (Hubel and Wiesel, 1962), our understanding of the feature selectivity and functional role of neurons in the primary auditory cortex is much farther from complete. Moving bars have long been recognized as an optimal stimulus for many visual cortical neurons, and this finding has recently been confirmed and extended in detail using reverse correlation methods (Jones and Palmer, 1987; Reid and Alonso, 1995; Reid et al., 1991; llingach et al., 1997). In this study, we recorded from neurons in the primary auditory cortex of the awake primate, and used a novel reverse correlation technique to compute receptive fields (or preferred stimuli), encompassing both multiple frequency components and ongoing time. These spectrotemporal receptive fields make clear that neurons in the primary auditory cortex, as in the primary visual cortex, typically show considerable structure in their feature processing properties, often including multiple excitatory and inhibitory regions in their receptive fields. These neurons can be sensitive to stimulus edges in frequency composition or in time, and sensitive to stimulus transitions such as changes in frequency. These neurons also show strong responses and selectivity to continuous frequency modulated stimuli analogous to visual drifting gratings. 1 Introduction It is known that auditory neurons are tuned for a number of independent feature parameters of simple stimuli including frequency (Merzenich et al., 1973), intensity (Sutter and Schreiner, 1995), amplitude modulation (Schreiner and Urbas, 1988), and Characterizing Auditory Cortical Neurons Using Reverse Correlation 125 others. In addition, auditory cortical responses to multiple stimuli can enhance or suppress one another in a time dependent fashion (Brosch and Schreiner, 1997; Phillips and Cynader, 1985; Shamma and Symmes, 1985), and auditory cortical neurons can be highly selective for species-specific vocalizations (Wang et al., 1995; Wollberg and Newman, 1972), suggesting complex acoustic processing by these cells. It is not yet known if these many independent selectivities of auditory cortical neurons reflect a discernible underlying pattern of feature decomposition, as has often been suggested (Merzenich et al., 1985; Schreiner and Mendelson, 1990; Wang et al., 1995). Further, since sustained firing rate responses in the auditory cortex to tonal stimuli are typically much lower than visual responses to drifting bars (deCharms and Merzenich, 1996b), it has been suggested that the preferred type of auditory stimulus may still not be known (Nelken et al., 1994). We sought to develop an unbiased method for determining the full feature selectivity of auditory cortical neurons, whatever it might be, in frequency and time based upon reverse correlation. 2 Methods Recordings were made from a chronic array of up to 49 individually placed ultrafine extracellular Iridium microelectrodes, placed in the primary auditory cortex of the adult owl monkey. The electrodes had tip lengths of 10-25microns, which yield impedance values of .5-SMOhm and good isolation of signals from individual neurons or clusters of nearby neurons. We electrochemically activated these tips to add an ultramicroscopic coating of Iridium Oxide, which leaves the tip geometry unchanged, but decreases the tip impedance by more than an order of magnitude, resulting in substantially improved recording signals. These signals are filtered from .3-8kHz, sampled at 20kHz, digitized, and sorted. The stimuli used were a variant of random VlsuII Cortn: Reveree Correlltlon U.lng 2?D VI.nl Pltternl In Time Auditory Cortex: Rever.e Correlltlon U.lng 1?D Auditory Pltternl (Chordl) In Tim. t -Om.ec t- 20m.ec t-40msec t - 40m.ec x x II ! Spltlotemporal Receptive Field II SplkeT .. ln. Spectrotempoul Receptive Field Figure 1: Schematic of stimuli used for reverse correlation. white noise which was designed to allow us to characterize the responses of neurons in time and in frequency. As shown in figure 1, these stimuli are directly analogous to stimuli that have been used previously to characterize the response properties of neurons in the primary visual cortex (Jones and Palmer, 1987; Reid and Alonso, 1995; Reid et al., 1991). In the visual case, stimuli consist of spatial checkerboards that span some portion of the two-dimensional visual field and change pattern with a short sampling interval. In the auditory case, which we have studied here, the stimuli chosen were randomly selected chords, which approximately evenly span a R C. deChanns and M M. Merzenich 126 portion of the one-dimensional receptor surface of the cochlea. These stimuli consist of combinations of pure tones, all with identical phase and all with 5 msec cosineshaped ramps in amplitude when they individually turn on or off. Each chord was created by randomly selecting frequency values from 84 possible values which span 7 octaves from 110Hz to 14080Hz in even semitone steps. The density of tones in each stimulus was 1 tone per octave on average, or 7 tones per chord, but the stimuli were selected stochastically so a given chord could be composed of a variable number of tones of randomly selected frequencies. We have used sampling rates of 10-100 chords/second, and the data here are from stimuli with 50 chords/second. Stimuli with random, asynchronous onset times of each tone produce similar results. These stimuli were presented in the open sound field within an acoustical isolation chamber at 44. 1kHz sampling rate directly from audio compact disk, while the animal sat passively in the sound field or actively performed an auditory discrimination task, receiving occasional juice rewards. The complete characterization set lasted for ten minutes, thereby including 30,000 individual chords. Spike trains were collected from mUltiple sites in the cortex simultaneously during the presentation of our characterization stimulus set, and individually reverse correlated with the times of onset of each of the tonal stimuli. The reverse correlation method computes the number of spikes from a neuron that were detected, on average, during a given time preceding, during, or following a particular tonal stimulus component from our set of chords. These values are presented in spikes/s for all of the tones in the stimulus set, and for some range of time shifts. This method is somewhat analogous in intention to a method developed earlier for deriving spectrotemporal receptive fields for auditory midbrain neurons (Eggermont et al., 1983), but previous methods have not been effective in the auditory cortex. 3 Results Figure 2 shows the spectrotemporal responses of neurons from four locations in the primary auditory cortex. In each panel, the time in milliseconds between the onset of a particular stimulus component and a neuronal spike is shown along the horizontal axis. Progressively greater negative time shifts indicate progressively longer latencies from the onset of a stimulus component until the neuronal spikes. The frequency of the stimulus component is shown along the vertical axis, in octave spacing from a 110Hz standard, with twelve steps per octave. The brightness corresponds to the average rate of the neuron, in spk/s, driven by a particular stimulus component. The reverse-correlogram is thus presented as a stimulus triggered spike rate average, analogous to a standard peristimulus time histogram but reversed in time, and is identical to the spectrogram of the estimated optimal stimulus for the cell (a spike triggered stimulus average which would be in units of mean stimulus denSity). A minority of neurons in the primary auditory cortex have spectrotemporal receptive fields that show only a single region of increased rate, which corresponds to the traditional characteristic frequency of the neuron, and no inhibitory region. We have found that cells of this type (less than 10%, not shown) are less common than cells with multimodal receptive field structure. More commonly, neurons have regions of both increased and decreased firing rate relative to their mean rate within their receptive fields. For terminological convemence, these will be referred to as excitatory and inhibitory regions, though these changes in rate are not diagnostic of an underlying mechanism. Neurons with receptive fields of this type can serve as detectors of stimulus edges in both frequency space, and in time. The neuron shown in figure 2a has a receptive field structure indicative of lateral inhibition in frequency space. This cell prefers a very narrow range of frequencies, and decreases its firing rate for nearby frequencies, giving the characteristic of a sharply-tuned bandpass filter. This Characterizing Auditory Cortical Neurons Using Reverse Correlation a) ... > 127 b) 3 3.5 2.5 3 40 30 !: 2.5 2 g 1.5 !!! 3 2 8 20 1.5 0.5 -100 10 -50 0 msec d) c) N 3 J: 0 .,> g 1.5 .. 0 15 3.5 2.5 -SO msec )! 2 3 10 10;::: 2.5 5 .8 '"> 2 0 5 ., 1.5 < ~ U -5 -10 CD -100 -50 msec 0 0 ~O -40 -20 msec Figure 2: Spectrotemporal receptive fields of neurons in the primary auditory cortex of the awake primate. These receptive fields are computed as described in methods. Receptive field structures read from left to right correspond to a preferred stimulus for the neuron, with light shading indicating more probable stimulus components to evoke a spike, and dark shading indicating less probable components. Receptive fields read from right to left indicate the response of the neuron in time to a particular stimulus component. The colorbars correspond to the average firing rates of the neurons in Hz at a given time preceding, during, or following a particular stimulus component. type of response is the auditory analog of a visual or tactile edge detector with lateral inhibition. Simple cells in the primary visual cortex typically show similar patterns of center excitation along a short linear segment, surrounded by inhibition (Jones and Palmer, 1987;?Reid and Alonso, 1995; Reid et al., 1991). The neuron shown in figure 2b shows a decrease in firing rate caused by a stimulus frequency which at a later time causes an increase in rate. This receptive field structure is ideally suited to detect stimulus transients; and can be thought of as a detector of temporal edges. Neurons in the auditory cortex typically prefer this type of stimulus, which is initially soft or silent and later loud. This corresponds to a neuronal response which shows an increase followed by a decrease in firing rate. This is again analogous to neuronal responses in the primary visual cortex, which also typically show a firing rate pattern to an optimal stimulus of excitation followed by inhibition, and preference for stimulus transients such as when a stimulus is first off and then comes on. The neuron shown in figures 2c shows an example which has complex receptive field structure, with multiple regions. Cells of this type would be indicative of selectivity for feature conjunctions or quite complex stimuli, perhaps related to sounds in the animal's learned environment. Cells with complex receptive field structures are common in the awake auditory cortex, and we are in the process of quantifying the percentages of cells that fit within these different categories. Neurons were observed which respond with increased rate to one frequency range at one time, and a different frequency range at a later time, indicative of selectivity for frequency modulations(Suga, 1965). Regions of decreased firing rate can show similar patterns. The neuron shown in figure 2d is an example of this type. This pattern is strongly analogous to motion energy detectors in the visual system (Adelson and Bergen, 1985), which detect stimuli moving in space, and these cells are selective for changes in frequency. R. C. deCharms and M M. Merzenich 128 2 oct/sec 6 oct/sec 10 oct/sec 14 oct/sec 30 oct/sec 100 oct/sec ?2 oct/sec ?6 oct/sec ? 10 oct/sec ?14 oct/sec ?30 oct/sec ?100 oct/sec Figure 3: Parametric stimulus set used to explore neuronal responses to continuously changing stimulus frequency. Images axe spectrograms of stimuli from left to right in time, and spanning seven octaves of frequency from bottom to top. Each stimulus is one second. Numbers indicate the sweep rate of the stimuli in octaves per second. Based on the responses shown, we wondered whether we could find a more optimal class of stimuli for these neuron, analogous to the use of drifting bars or gratings in the primary visual cortex. We have created auditory stimuli which correspond exactly to the preferred stimulus computed for a paxticulax cell from the cell's spectrotemporal receptive field (manuscript in prepaxation), and we have also designed a paxametric class of stimuli which are designed to be particularly effective for neurons selective for stimuli of changing amplitude or frequency, which are presented here. The stimuli shown in figure 3 are auditory analogous of visual drifting grating stimuli. The stimuli axe shown as spectrograms, where time is along the horizontal axis, frequency content on an octave scale is along the vertical axis, and brightness corresponds to the intensity of the signal. These stimuli contain frequencies that change in time along an octave frequency scale so that they repeatedly pass approximately linearly through a neurons receptive field, just as a drifting grating would pass repeatedly through the receptive field of a visual neuron. These stimuli axe somewhat analogous to drifting ripple stimuli which have recently been used by Kowalski, et.al. to characterize the linearity of responses of neurons in the anesthetized ferret auditory cortex (Kowalski et al., 1996a; Kowalski et al., 1996b). Neurons in the auditory cortex typically respond to tonal stimuli with a brisk onset response at the stimulus transient, but show sustained rates that axe far smaller than found in the visual or somatosensory systems (deCharms and Merzenich, 1996a). We have found neurons in the awake animal that respond with high firing rates and significant selectivity to the class of moving stimuli shown in figure 3. An outstanding example of this is shown in figure 4. The neuron in this example showed a very high sustained firing rate to the optimal drifting stimulus, as high as 60 Hz?for one second. The neuron shown in this example also showed considerable selectivity for stimulus velocity, as well as some selectivity for stimulus direction. 4 Conclusions These stimuli enable us to efficiently quantify the response characteristics of neurons in the awake primaxy auditory cortex, as well as producing optimal stimuli for particular neurons. The data that we have gathered thus far extend our knowledge about the complex receptive field structure of cells in the primary auditory cortex, Cha racterizing Auditory Cortical Neurons Using Reverse Correlation 129 2 oct/sec 6 oct/sec 10 octIsec 14 oct/sec 30 oct/sec 100 oct/sec -2 oct/sec -6 oct/sec -10 oct/sec -14 oct/sec -30 oct/sec -100 oct/sec Figure 4: Responses of a neuron in the primary auditory cortex of the awake primate to example stimuli take form our characterization set, as shown in figure 3. In each panel, the average response rate histogram in spikes per second is shown below rastergrams showing the individual action potentials elicited on,each of twenty trials. and show some considerable analogy with neurons in the primary visual cortex. In addition, they indicate that it is possible to drive auditory cortical cells to high rates of sustained firing, as in the visual cortex. This method will allow a number of future questions to be addressed. Since we have recorded many neurons simultaneously, we are interested in the interactions among large populations of neurons and how these relate to stimuli. We are also recording responses to these stimuli while monkeys are performing cognitive tasks involving attention and learning, and we hope that this will give us insight into the effects on cell selectivity of the context provided by other stimuli, the animal's behavioral state or awareness of the stimuli, and the animal's prior learning of stimulus sets. 5 References Adelson EH, Bergen JR (1985) Spatiotemporal energy models for the perception of motion. J. Opt. Soc. Am. A, 2, 284-299. Brosch M, Schreiner CE (1997) Time course of forward masking tuning curves in cat primary auditory cortex. J Neurophysiol, 77, 923-43. deCharms Re, Merzenich MM (1996a) Primary cortical representation of sounds by the coordination of action-potential timing. Nature, 381, 610-3. deCharms RC, Merzenich MM (1996b) Primary cortical representation of sounds by the coordination of action-potential timing. Nature, 381, 610-613. EggeI1I).ont JJ, Aertsen AM, Johannesma PI (1983) Quantitative characterisation procedure for auditory neurons based on the spectro-temporal receptive field. Hear Res, 10, 167-90. Hubel DH, Wiesel TN (1962) Receptive fields, binocular interaction and functional archtecture in the cat's visual cortex. J. Physiol., 160, 106-154. Jones JP, Palmer LA (1987) The two-dimensional spatial structure of simple receptive 130 R. C. deCharms and M M. Merzenich fields in cat striate cortex. J Neurophysiol, 58, 1187-211. Kowalski N, Depireux DA, Shamma SA (1996a) Analysis of dynamic spectra in ferret primary auditory cortex. I. Characteristics of single-unit responses to moving ripple spectra. J Neurophysiol, 76, 3503-23. Kowalski N, Depireux DA, Shamma SA (1996b) Analysis of dynamic spectra in ferret primary auditory cortex. II. Prediction of unit responses to arbitrary dynamic spectra. J Neurophysiol, 76, 3524-34. Merzenich MM, Jenkins WM, Middlebrooks JC (1985) Observations and hypotheses on special organizational features of the central auditory nervous system. In: Dynamic Aspects of Neocortical Function. Edited by E. G. a. W. M. C. G. Edelman. New York: Wiley, pp. 397-423. Merzenich MM, Knight PL, Roth GL (1973) Cochleotopic organization of primary auditory cortex in the cat. Brain Res, 63, 343-6. Nelken I, Prut Y, Vaadia E, Abeles M (1994) In search of the best stimulus: an optimization procedure for finding efficient stimuli in the? cat auditory cortex. Hear Res, 72, 237-53. Phillips DP, Cynader MS (1985) Some neural mechanisms in the cat's auditory cortex underlying sensitivity to combined tone and wide-spectrum noise stimuli. Hear Res, 18, 87-102. Reid RC, Alonso JM (1995) Specificity of monosynaptic connections from thalamus to visual cortex. Nature, 378,281-4. Reid RC, Soodak RE, Shapley RM (1991) Directional selectivity and spatiotemporal structure of receptive fields of simple cells in cat striate cortex. J Neurophysiol, 66, 505-29. Ringach DL, Hawken MJ, Shapley R (1997) Dynamics of orientation tuning in macaque primary visual cortex. Nature, 387, 281-4. Schreiner CE, Mendelson JR (1990) Functional topography of cat primary auditory cortex: distribution of integrated excitation. J Neurophysiol, 64, 1442-59. Schreiner CE, Urbas JV (1988) Representation of amplitude in the auditory cortex of the cat. II. Comparison between cortical fields. Hear. Res., 32, 49-64. Shamma SA, Symmes D (1985) Patterns of inhibition in auditory cortical cells in awake squirrel monkeys. Hear Res, 19, 1-13. Suga N (1965) Responses of cortical auditory neurones to frequency modulated sounds in echo-locating bats. Nature, 206, 890-l. Sutter ML, Schreiner CE (1995) Topography of intensity tuning in cat primary auditory cortex: single-neuron versus multiple-neuron recordings. J Neurophysiol, 73, 190-204. Wang X, Merzenich MM, Beitel R, Schreiner CE (1995) Representation of a speciesspecific vocalization in the primary auditory cortex of the common marmoset: temporal and spectral characteristics. J Neurophysiol, 74, 2685-706. Wollberg Z, Newman JD (1972) Auditory cortex of squirrel monkey: response patterns of single cells to species-specific vocalizations. Science, 175, 212-214. 

1462 |@word trial:1 wiesel:3 disk:1 open:1 cha:1 integrative:1 decomposition:1 brightness:2 thereby:1 shading:2 phy:2 selecting:1 tuned:2 yet:1 physiol:1 discernible:1 designed:3 progressively:2 discrimination:1 leaf:1 selected:3 nervous:1 tone:8 indicative:3 sutter:2 short:2 farther:1 filtered:1 characterization:3 location:1 preference:1 rc:3 along:6 edelman:1 sustained:4 shapley:2 behavioral:1 brain:1 ont:1 jm:1 provided:1 monosynaptic:1 underlying:3 linearity:1 panel:2 substantially:1 monkey:4 developed:1 finding:2 temporal:3 quantitative:1 exactly:1 rm:1 middlebrooks:1 whatever:1 unit:3 producing:1 reid:8 timing:2 receptor:1 firing:11 modulation:2 approximately:2 might:1 studied:2 shamma:4 palmer:4 range:4 suga:2 bat:1 procedure:2 johannesma:1 thought:1 intention:1 specificity:1 context:1 center:2 chronic:1 roth:1 attention:1 wondered:1 pure:1 schreiner:9 insight:1 array:1 deriving:1 population:1 analogous:9 hypothesis:1 velocity:1 particularly:1 observed:1 role:2 bottom:1 wang:3 region:7 decrease:4 chord:8 knight:1 edited:1 environment:1 reward:1 ideally:1 dynamic:5 segment:1 serve:1 cynader:2 upon:1 neurophysiol:8 spk:1 multimodal:1 cat:10 train:1 effective:2 detected:1 newman:2 quite:1 ramp:1 echo:1 vocalization:3 triggered:2 vaadia:1 interaction:2 electrode:1 keck:1 cluster:1 ripple:2 produce:1 tim:1 develop:1 sa:3 strong:1 soc:1 grating:4 indicate:4 come:1 somatosensory:1 quantify:1 direction:1 coating:1 filter:1 transient:3 enable:1 owl:1 opt:1 probable:2 pl:1 mm:5 squirrel:2 sought:1 spectrotemporal:6 coordination:2 sensitive:2 individually:3 hope:1 depireux:2 conjunction:1 lasted:1 detect:2 am:2 dependent:1 bergen:2 typically:6 integrated:1 initially:1 selective:3 interested:1 among:1 orientation:1 animal:5 spatial:2 special:1 field:30 sampling:3 identical:2 jones:4 adelson:2 future:1 others:1 stimulus:79 randomly:3 microelectrodes:1 composed:1 simultaneously:2 individual:3 geometry:1 phase:1 organization:1 highly:1 nl:1 light:1 activated:1 edge:4 re:8 increased:3 earlier:1 soft:1 organizational:1 characterize:3 spatiotemporal:2 abele:1 combined:1 density:2 twelve:1 sensitivity:1 off:2 receiving:1 michael:1 enhance:1 tip:4 continuously:1 again:1 reflect:1 recorded:2 central:1 stochastically:1 oxide:1 cognitive:1 checkerboard:1 actively:1 suggesting:1 potential:3 sec:23 jc:1 caused:1 vi:1 onset:5 performed:1 later:3 portion:2 wm:1 elicited:1 masking:1 om:1 prut:1 characteristic:5 efficiently:1 yield:1 correspond:3 kowalski:5 gathered:1 directional:1 confirmed:1 drive:1 detector:4 energy:2 frequency:27 pp:1 sampled:1 auditory:50 knowledge:1 amplitude:4 manuscript:1 response:23 improved:1 though:1 strongly:1 just:1 binocular:1 correlation:9 until:1 horizontal:2 christopher:1 perhaps:1 effect:1 contain:1 unbiased:1 merzenich:13 read:2 ringach:1 white:1 semitone:1 during:4 excitation:3 m:1 octave:8 complete:2 neocortical:1 tn:1 motion:2 image:1 novel:1 recently:2 common:3 juice:1 functional:4 khz:3 jp:1 axe:4 analog:1 extend:1 significant:1 composition:1 phillips:2 tuning:3 had:1 moving:4 cortex:43 surface:1 longer:1 inhibition:5 add:1 showed:2 driven:1 reverse:11 selectivity:11 greater:1 somewhat:2 preceding:2 spectrogram:3 recognized:1 signal:4 ii:4 multiple:6 full:1 sound:6 thalamus:1 long:1 schematic:1 prediction:1 variant:1 involving:1 histogram:2 cochlea:1 dec:1 cell:18 addition:2 spacing:1 addressed:1 interval:1 decreased:2 ferret:3 recording:4 hz:5 isolation:2 fit:1 silent:1 shift:2 whether:1 tonal:4 tactile:1 locating:1 york:1 cause:1 jj:1 prefers:1 repeatedly:2 action:3 neurones:1 latency:1 clear:1 dark:1 extensively:1 ten:1 category:1 percentage:1 inhibitory:3 millisecond:1 neuroscience:1 estimated:1 per:5 diagnostic:1 urbas:2 four:1 characterisation:1 changing:2 jv:1 ce:5 micron:1 respond:3 lng:2 hawken:1 prefer:1 followed:2 sharply:1 awake:9 nearby:2 aspect:1 span:3 passively:1 performing:1 extracellular:1 combination:1 jr:2 smaller:1 primate:4 marmoset:1 midbrain:1 ln:1 brosch:2 previously:1 turn:1 mechanism:2 jenkins:1 occasional:1 spectral:1 chamber:1 drifting:7 jd:1 top:1 giving:1 eggermont:1 unchanged:1 sweep:1 question:1 spike:9 receptive:26 primary:27 parametric:1 striate:2 traditional:1 aertsen:1 dp:1 reversed:1 lateral:2 alonso:4 evenly:1 acoustical:1 seven:1 collected:1 spanning:1 minority:1 length:1 relate:1 decharms:7 negative:1 suppress:1 twenty:1 vertical:2 neuron:55 observation:1 sing:1 extended:1 digitized:1 arbitrary:1 intensity:3 connection:1 california:1 acoustic:1 learned:1 narrow:1 macaque:1 adult:1 bar:3 suggested:2 below:1 pattern:8 perception:1 hear:5 including:3 eh:1 axis:4 created:2 prior:1 understanding:2 determining:1 relative:1 encompassing:1 topography:2 analogy:1 versus:1 awareness:1 surrounded:1 cd:1 pi:1 excitatory:2 course:1 placed:2 gl:1 asynchronous:1 allow:2 wide:1 characterizing:3 anesthetized:1 curve:1 cortical:14 transition:1 computes:1 nelken:2 forward:1 made:1 commonly:1 san:1 ec:3 far:2 compact:1 spectro:1 preferred:4 evoke:1 ml:1 hubel:3 sat:1 harm:1 francisco:1 spectrum:5 continuous:1 search:1 impedance:2 nature:5 mj:1 ca:1 brisk:1 complex:5 da:2 linearly:1 noise:2 neuronal:5 site:1 referred:1 fashion:1 wiley:1 msec:6 bandpass:1 minute:1 specific:2 showing:1 peristimulus:1 iridium:2 dl:1 mendelson:2 consist:2 magnitude:1 suited:1 explore:1 visual:21 correlogram:1 corresponds:4 loud:1 dh:1 oct:23 sorted:1 presentation:1 quantifying:1 considerable:3 change:5 content:1 specie:2 pas:2 merz:1 la:1 indicating:2 soodak:1 modulated:2 ucsf:2 outstanding:1 ongoing:1 audio:1 correlated:1

Statistical Models of Conditioning Peter Dayan* Brain & Cognitive Sciences E25-2IDMIT Cambridge, MA 02139 Theresa Long 123 Hunting Cove Williamsburg, VA 23185 Abstract Conditioning experiments probe the ways that animals make predictions about rewards and punishments and use those predictions to control their behavior. One standard model of conditioning paradigms which involve many conditioned stimuli suggests that individual predictions should be added together. Various key results show that this model fails in some circumstances, and motivate an alternative model, in which there is attentional selection between different available stimuli. The new model is a form of mixture of experts, has a close relationship with some other existing psychological suggestions, and is statistically well-founded. 1 Introduction Classical and instrumental conditioning experiments study the way that animals learn about the causal texture of the world (Dickinson, 1980) and use this information to their advantage. Although it reached a high level of behavioral sophistication, conditioning has long since gone out of fashion as a paradigm for studying learning in animals, partly because of the philosophical stance of many practitioners, that the neurobiological implementation of learning is essentially irrelevant. However, more recently it has become possible to study how conditioning phenomena are affected by particular lesions or pharmacological treatments to the brain (eg Gallagher & Holland, 1994), and how particular systems, during simple learning tasks, report information that is consistent with models of conditioning (Gluck & Thompson, 1987; Gabriel & Moore, 1989). In particular, we have studied the involvement of the dopamine (DA) system in the ventral tegmental area of vertebrates in reward based learning (Montague et aI, 1996; Schultz et aI, 1997). The activity of these cells is consistent with a model in which they report a temporal difference (TO) based prediction error for reward "This work was funded by the Surdna Foundation. 118 P. Dayan and T. Long (Sutton & Barto, 1981; 1989). This prediction error signal can be used to learn correct predictions and also to learn appropriate actions (Barto, Sutton & Anderson, 1983). The DA system is important since it is crucially involved in normal reward learning, and also in the effects of drugs of addiction, self stimulation, and various neural diseases. The TO model is consistent with a whole body of experiments, and has even correctly anticipated new experimental findings. However, like the Rescorla-Wagner (RW; 1972) or delta rule, it embodies a particular additive model for the net prediction made when there are multiple stimuli. Various sophisticated conditioning experiments have challenged this model and found it wanting. The results support competitive rather than additive models. Although ad hoc suggestions have been made to repair the model, none has a sound basis in appropriate prediction. There is a well established statistical theory for competitive models, and it is this that we adopt. In this paper we review existing evidence and theories, show what constraints a new theory must satisfy, and suggest and demonstrate a credible candidate. Although it is based on behavioral data, it also has direct implications for our neural theory. 2 Data and Existing Models Table 1 describes some of the key paradigms in conditioning (Dickinson, 1980; Mackintosh, 1983). Although the collection of experiments may seem rather arcane (the standard notation is even more so), in fact it shows exactly the basis behind the key capacity of animals in the world to predict events of consequence. We will extract further biological constraints implied by these and other experiments in the discussion. In the table,l (light) and s (tone) are potential predictors (called conditioned stimuli or CSs), of a consequence, r, such a~ the delivery of a reward (called an unconditioned stimulus or US). Even though we use TO rules in practice, we discuss some of the abstract learning rules without much reference to the detailed time course of trials. The same considerations apply to TD. In Pavlovian conditioning, the light acquires a positive association with the reward in a way that can be reasonably well modeled by: ,6.Wl(t) = al(t)(r(t) - wl(t?l(t), (1) where let) E {O, I} represents the presence of the light in trial t (s(t) will similarly represent the presence of a tone), Wl(t) (we will often drop the index t) represents the strength of the expectation about the delivery of reward ret) in trial t if the light is also delivered, and al(t) is the learning rate. This is just the delta rule. It also captures well the probabilistic contingent nature of conditioning - for binary ret) E {O, I}, animals seem to assess il = P[r(t)ll(t) = 1J - P[r(t)ll(t) = OJ, and then only expect reward following the light (in the model, have WI > 0) if il > O. Pavlovian conditioning is easy to explain under a whole wealth of rules. The trouble comes in extending equation 1 to the case of multiple predictors (in this paper we consider just two). The other paradigms in table 1 probe different aspects of this. The one that is most puzzling is (perversely) called downwards unblocking (Holland, 1988). In a first set of trials, an association is established between the light and two presentations of reward separated by a few (u) seconds. In a second set, a tone is included with the light, but the second reward is dropped. The animal amasses less reward in conjunction with the tone. However, when presented with the tone Statistical Models of Conditioning 119 1 Name Pavlovian 2 Overshadowing l+s-tr 3 Inhibitory l-tr } { Z+s-t? 4 Blocking Upwards unblocking Downwards unblockinK 5 .6 Set 1 I l-tr l-tr I -t rflur Set 2 -t r l+s-tr 1+ s -t rflur l+s-tr Test r l~ {I ~ r~ s~ 1 r'i } s~f s~? s~r s~ ?~ Table 1: Paradigms. Sets 1 and 2 are separate sets of learning trials, which are continued until convergence. Symbols land s indicate presentation of lights and tones as potential predictors. The 't+ in the test set indicates that the associations of the predictors are tested, prodUCing the listed results. In overshadowing, association with the reward can be divided between the light and the sound, indicated by r!. In some cases overshadowing favours one stimulus at the complete expense of the other; and at the end of very prolonged training, all effects of overshadowing can disappear. In blocking, the tone makes no prediction of r. In set 2 of inhibitory conditioning, the two types of trials are interleaved and the outcome is that the tone predicts the absence of reward. In upwards and downwards unblocking, the 6" indicates that the delivery of two rewards is separated by time u. For downwards unblocking, if u is small, then s is associated with the absence of r; if u is large, then s is associated with the presence of r. alone, the animal expects the presence rather than the absence of reward. On the face of it, this seems an insurmountable challenge to prediction-based theories. First we describe the existing theories, then we formalise some potential replacements. One theory (called a US-processing theory) is due to Rescorla & Wagner (RW; 1972), and, as pointed out by Sutton & Barto (1981), is just the delta rule. For RW, the animal constructs a net prediction: V(t) = wi(t)l(t) + ws{t)s{t) (2) for r(t), and then changes flWi(t) = Cti(t)(r(t) - V(t?l(t) (and similarly for ws(t? using the prediction error r(t) - V(t). Its foundation in the delta rule makes it computationally appropriate (Marr, 1982) as a method of making predictions. TD uses the same additive model in equation 2, but uses r(t) + V(t + 1) - V(t) as the prediction error. RW explains overshadowing, inhibitory conditioning, blocking, and upwards unblocking, but not downwards unblocking. In overshadowing, the terminal association between I and r is weaker if I and s are simultaneously trained - this is expected under RW since learning stops when V(t) = r{t), and W, and Ws will share the prediction. In inhibitory conditioning, the sound comes to predict the absence of r. The explanation of inhibitory conditioning is actually quite complicated (Konorski, 1967; Mackintosh, 1983); however RW provides the simple account that WI = r for the I -t r trials, forcing Ws = -r for the 1+ s -t . trials. In blocking, the prior association between I and r means that Wi = r in the second set of trials, leading to no learning for the tone (since V(t) - r(t) = 0). In upwards unblocking, Wi = r at the start of set 2. Therefore, r(t) - WI = r > 0, allowing Ws to share in the prediction. As described above, downwards unblocking is the key thorn in the side of RW. Since the TD rule combines the predictions from different stimuli in a similar way, P. Dayan and T. Long 120 it also fails to account properly for downwards unblocking. This is one reason why it is incorrect as a model of reward learning. The class of theories (called CS-processing theories) that is alternative to RW does not construct a net prediction V(t), but instead uses equation 1 for all the stimuli, only changing the learning rates O!l(t) and O!s(t) as a function of the conditioning history of the stimuli (eg Mackintosh, 1975; Pearce & Hall, 1980; Grossberg, 1982). A standard notion is that there is a competition between different stimuli for a limited capacity learning processor (Broadbent, 1958; Mackintosh, 1975; Pearce & Hall, 1980), translating into competition between the learning rates. In blocking, nothing unexpected happens in the second set of trials and equally, the tone does not predict anything novel. In either case as is set to '" 0 and so no learning happens. In these models, downwards unblocking now makes qualitative sense: the surprising consequences in set 2 can be enough to set as ?0, but then learning according to equation 1 can make Ws > O. Whereas Mackintosh's (1975) and Pearce and Hall's (1980) models only consider competition between the stimuli for learning, Grossberg's (1982) model incorporates competition during representation, so the net prediction on a trial is affected by competitive interactions between the stimuli. In essence, our model provides a statistical formalisation of this insight. 3 New Models From the previous section, it would seem that we have to abandon the computational basis of the RW and TD models in terms of making collective predictions about the reward. The CS-processing models do not construct a net prediction of the reward, or say anything about how possibly conflicting information based on different stimuli should be integrated. This is a key flaw - doing anything other than well-founded prediction is likely to be maladaptive. Even quite successful pre-synaptic models, such as Grossberg (1982), do not justify their predictions. We now show that we can take a different, but still statistically-minded approach to combination in which we specify a parameterised probability distribution P[r(t)ls(t), l(t)] and perform a form of maximum likelihood (ML) inference, updating the parameters to maximise this probability over the samples. Consider three natural models of P[r(t)/s(t), l(t)]: Pa[r(t)ls(t),l(t)] P M[r(t)/s(t), l(t)] P J[r(t)/s(t), l(t)] N[w1l(t) + wss(t), (72] 7l"1 (t)N[Wl' (72] N[WI7l"I(t)l(t) + 7l"s(t)N[ws, (72] + 1i'(t).,v[w, r2] + wsnAt)s(t), (72] (3) (4) (5) where N[J.L, (72] is a normal distribution, with mean J.L and variance (72. In the latter two cases, 0 ::; 7l"1 (t) + 7l" s (t) ::; I, implementing a form of competition between the stimuli, and 7l".(t) = 0 if stimulus * is not presented. In equation 4, N[w, r2] captures the background expectation if neither the light nor the tone wins, and 1i'(t) = 1 - 7l"1(t) - n"s{t). We will show that the data argue against the first two and support the third of these models. Rescorla-Wagner: Pa[r(t)/s(t), l(t)] The RW rule is derived as ML inference based on equation 3. The only difference is the presence of the variance, (72. This is useful for capturing the partial reinforcement effect (see Mackintosh, 1983), in which if r(t) is corrupted by substantial noise (ie (72 ?0), then learning to r is demonstrably slower. As we discussed above, 121 Statistical Models of Conditioning downwards unblocking suggests that animals are not using P G [r( t) Is(t), I (t)] as the basis for their predictions. Competitive mixture of experts: P M[r(t)ls(t), l(t)] PM[r(t)ls(t),l(t)] is recognisable as the generative distribution in a mixture of Gaussians model (Nowlan, 1991; Jacobs et ai, 1991b). Key in this model are the mixing proportions 7r1(t) and 7r s (t). Online variants of the E phase of the EM algorithm (Dempster et ai, 1977) compute posterior responsibilities as ql(t) + qs(t) + q(t) = 1, where ql(t) <X 7r1(t)e-(r(t)-w1I(t)),2/2(T2 (and similarly for the others), and then perform a partial M phase as L\wl(t) <X (r(t) - WI (t?ql(t) L\ws(t) <X (r(t) - ws(t?qs(t) (6) which has just the same character as the presynaptic rules (depending on how 7r1 (t) is calculated). As in the mixture of experts model, each expert (each stimulus here) that seeks to predict r(t) (ie each stimulus * for which q. (t) f; 0) has to predict the whole of r(t) by itself. This means that the model can capture downwards unblocking in the following way. The absence of the second r in the second set of trials forces 7r s (t) > 0, and, through equation 6, this in turn means that the tone will come to predict the presence of the first r. The time u between the rewards can be important because of temporal discounting. This means that there are sufficiently large values of u for which the inhibitory effect of the absence of the second reward will be dominated. Note also that the expected reward based on l(t) and s(t) is the sum (7) Although the net prediction given in equation 7 is indeed based on all the stimuli, it does not directly affect the course of learning. This means that the model has difficulty with inhibitory conditioning. The trouble with inhibitory conditioning is that the model cannot use Ws < 0 to counterbalance WI > 0 - it can at best set Ws =0, which is experimentally inaccurate. Note, however, this form of competition bears some interesting similarities with comparator models of conditioning (see Miller & Matzel, 1989). It also has some problems in explaining overshadowing, for similar reasons. Cooperative mixture of experts: P J[r(t)ls(t), l(t)] The final model P J[r(t)ls(t), l(t)] is just like the mixture model that Jacobs et al (1991a) suggested (see also Bordley, 1982). One statistical formulation of this model considers that, independently, where Pl(t) and Ps(t) are inverse variances. This makes (72 = (Pl(t) + Ps(t?-l 7r1(t) = PI(t)(72 7r s (t) = Ps(t)(72. Normative learning rules should emerge from a statistical model of uncertainty in the world. Short of such a model, we used: 7r1 (t) L\WI = o:w-(-) 6(t) PI t where 6(t) = r(t) - 7r1(t)Wl (t) - 7rs (t)ws (t) is the prediction error; the 1/ Pl(t) term in changing WI makes learning slower if WI is more certainly related to r (ie if PI (t) is greater); the 0.1 substitutes for background noise; if 62 (t) is too large, then PI + Ps P. Dayan and T. Long 122 t'reolctrVe vanances; I:SloCklng I:SIOCKlng ana unDloCklng ......... ... r-u"gj;\'--'--' 8 , t'reolctove vanances: UnDlocKlng i 6 ;:_1:1 /~~ .... -"- 4 r . "" ~~L-~1~ 0 --~2~ 0 --~M ~~ ~ Tim. to 2nd I8warn 00 200 400 600 Trial 800 1000 00 200 400 600 800 1000 Trial Figure 1: Blocking and downwards unblocking with 5 steps to the first reward; and a variable number to the second. Here, the discount factor "y = 0.9, and O:w = 0.5, O:p = 0.02, f.L = 0.75. For blocking, the second reward remains; for unblocking it is removed after 500 trials. a) The terminal weight for the sound after learning - for blocking it is always small and positive; for downwards unblocking, it changes from negative at small ~u to positive at large ~ u. b,c) Predictive variances Pl(t) and P.. (t). In blocking, although there is a small change when the sound is introduced because of additivity of the variances, learning to the sound is substantially prevented. In downwards unblocking, the surprise omission of the second reward makes the sound associable and unblocks learning to it. pr is shared out in proportion of to capture the insight that there can be dramatic changes to variabilities; and the variabilities are bottom-limited. Figure 1 shows the end point and course of learning in blocking and downwards unblocking. Figure 1a confirms that the model captures downwards unblocking, making the terminal value of Ws negative for short separations between the rewards and positive for long separations. By comparison, in the blocking condition, for which both rewards are always presented, W s is always small and positive. Figures 1b,c show the basis behind this behaviour in terms of Pl(t) and Ps{t). In particular, the heightened associability of the sound in unblocking following the prediction error when the second reward is removed accounts for the behavior. As for the mixture of experts model (and also for comparator models), the presence of 11'j(t) and nAt) makes the explanation of inhibitory conditioning and overshadowing a little complicated. For instance, if the sound is associable (Ps(t) ? 0), then it can seem to act as a conditioned inhibitor even if Ws = O. Nevertheless, unlike the mixture of experts model, the fact that learning is based on the joint prediction makes true inhibitory conditioning possible. 4 Discussion Downwards unblocking may seem like an extremely abstruse paradigm with which to refute an otherwise successful and computationally sound model. However, it is just the tip of a conditioning iceberg that would otherwise sink TD. Even in other reinforcement learning applications of TO, there is no a priori reason why predictions should be made according to equation 2 - the other statistical models in equations 4 and 5 could also be used. Indeed, it is easy to generate circumstances in which these more competitive models will perform better. For the neurobiology, experiments on the behavior of the DA system in these conditioning tasks will help specify the models further. The model is incomplete in various important ways. First, it makes no distinction between preparatory and consumatory conditioning (Konorski, 1967). There is evidence that the predictions a CS makes about the affective value of USs fall in a different class from the predictions it makes about the actual USs that appear. Statistical Models of Conditioning 123 For instance, an inhibitory stimulus reporting the absence of expected delivery of food can block learning to the delivery of shock, implying that aversive events form a single class. The affective value forms the preparatory aspect, is likely what is reported by the DA cells, and perhaps controls orienting behavior, the characteristic reaction of animals to the conditioned stimuli that may provide an experimental handle on the attention they are paid. Second, the model does not use opponency (Konorski, 1967; Solomon & Corbit, 1974; Grossberg, 1982) to handle inhibitory conditioning. This is particularly important, since the dynamics of the interaction between the opponent systems may well be responsible for the importance of the delay u in downwards unblocking. Serotonin is an obvious candidate as an opponent system to DA (Montague et a11996). We also have not specified a substrate for the associabilities or the attentional competition - the DA system itself may well be involved. Finally, we have not specified an overall model of how the animal might expect the contingency of the world to change over time - which is key to the statistical justification of appropriate learning rules. References [1] Barto, AG, Sutton, RS & Anderson, CW (1983). IEEE Transactions on Systems, Man, and Cybernetics, 13, pp 834-846. [2] Bordley, RF (1982). Journal of the Operational Research Society, 33, 171-174. [3] Broadbent, DE (1958). Perception and Communication. London: Pergamon. [4] Buhusi, CV & Schmajuk, NA. Hippocampus, 6, 621-642. [5] Dempster, AP, Laird, NM & Rubin, DB (1977). Proceedings of the Royal Statistical Society, B-39,1-38. [6] Dickinson, A (1980). Contemporary Animal Learning Theory. Cambridge, England: Cambridge University Press. [7] Gabriel, M & Moore, J, editors (1989). Learning and Computational Neuroscience. Cambridge, MA: MIT Press. [8] Gallagher, M & Holland, PC (1994). PNAS, 91, 11771-6. [9] Gluck, MA & Thompson, RF (1987). Psychological Reviews, 94, 176-191. [10] Grossberg, S (1982). Psychological Review, 89,529-572. [11] Holland, PC (1988). Journal of Experimental Psychology: Animal Behavior Processes, 14, 261-279. [12] Jacobs, RA, Jordan, MI & Barto, AG (1991). Cognitive Science, 15, 219-250. [13] Jacobs, RA, Jordan, MI, Nowlan, SJ & Hinton, GE (1991). Neural Computation, 3, 79-87. [14] Konorski, J (1967). Integrative Activity of the Brain. Chicago, 11: Chicago University Press. [15] Mackintosh, NJ (1975). Psychological Review, 82,276-298. [16] Mackintosh, NJ (1983). Conditioning and Associative Learning. Oxford, UK: Oxford University Press. [17] Marr, 0 (1982). Vision. New York, NY: Freeman. [18] Miller, RR & Matzel, LD (1989). In SB Klein & RR Mowrer, editors, Contemporary Learning Theories: Pavlovian Conditioning and the Status of Traditional Theory. Hillsdale, NJ: Lawrence Erlbaum. [19] Montague, PR, Dayan, P & Sejnowski, TK (1996). Journal of Neuroscience, 16, 1936-1947. [20] Nowlan, SJ (1991). Soft Competitive Adaptation: Neural Network Learning Algorithms Based on Fitting Statistical Mixtures. PhD Thesis, Department of Computer Science, CamegieMellon University. [21] Pearce, JM & Hall, G (1980). Psychological Review, 87,532-552. [22] Rescorla, RA & Wagner, AR (1972). In AH Black & WF Prokasy, editors, Classical Conditioning II: Current Research and Theory, pp 64-69. New York, NY: Appleton-CenturyCrofts. [23] Schultz, W, Dayan, P & Montague, PR (1997). Science, 275, 1593-1599. [24] Solomon, RL & Corbit, JD (1974). Psychological Review, 81, 119-145. [25] Sutton, RS & Barto, AG (1981). Psychological Review, 882, pp 135-170. [26] Sutton, RS & Barto, AG (1989). In Gabriel & Moore (1989). 

1463 |@word trial:15 proportion:2 instrumental:1 nd:1 seems:1 hippocampus:1 integrative:1 confirms:1 seek:1 crucially:1 r:4 jacob:4 paid:1 dramatic:1 tr:6 ld:1 hunting:1 existing:4 reaction:1 current:1 surprising:1 nowlan:3 must:1 additive:3 chicago:2 drop:1 alone:1 generative:1 implying:1 tone:12 short:2 provides:2 direct:1 become:1 addiction:1 incorrect:1 qualitative:1 combine:1 fitting:1 behavioral:2 affective:2 indeed:2 ra:3 preparatory:2 expected:3 nor:1 behavior:5 brain:3 terminal:3 freeman:1 td:5 prolonged:1 little:1 actual:1 food:1 jm:1 vertebrate:1 notation:1 what:2 substantially:1 ret:2 finding:1 ag:4 nj:3 temporal:2 act:1 exactly:1 uk:1 control:2 appear:1 producing:1 positive:5 maximise:1 dropped:1 consequence:3 sutton:6 oxford:2 ap:1 might:1 black:1 studied:1 suggests:2 limited:2 gone:1 statistically:2 grossberg:5 responsible:1 practice:1 block:1 area:1 drug:1 pre:1 suggest:1 cannot:1 close:1 selection:1 attention:1 l:6 thompson:2 independently:1 rule:12 continued:1 insight:2 q:2 marr:2 handle:2 notion:1 justification:1 cs:1 heightened:1 dickinson:3 substrate:1 us:5 pa:2 mowrer:1 particularly:1 updating:1 predicts:1 maladaptive:1 blocking:11 cooperative:1 bottom:1 capture:5 removed:2 contemporary:2 disease:1 substantial:1 dempster:2 reward:27 dynamic:1 motivate:1 trained:1 predictive:1 basis:5 sink:1 joint:1 montague:4 various:4 additivity:1 separated:2 describe:1 london:1 sejnowski:1 outcome:1 quite:2 say:1 otherwise:2 serotonin:1 itself:2 laird:1 abandon:1 unconditioned:1 delivered:1 online:1 hoc:1 advantage:1 final:1 associative:1 net:6 rr:2 rescorla:4 interaction:2 adaptation:1 mixing:1 competition:7 convergence:1 p:6 extending:1 r1:6 tk:1 tim:1 depending:1 help:1 insurmountable:1 c:3 come:3 indicate:1 correct:1 translating:1 ana:1 implementing:1 hillsdale:1 explains:1 behaviour:1 biological:1 refute:1 pl:5 sufficiently:1 hall:4 normal:2 lawrence:1 predict:6 ventral:1 adopt:1 wl:6 minded:1 mit:1 tegmental:1 always:3 inhibitor:1 rather:3 barto:7 conjunction:1 derived:1 properly:1 indicates:2 likelihood:1 sense:1 wf:1 inference:2 flaw:1 dayan:6 sb:1 integrated:1 inaccurate:1 w:15 overall:1 priori:1 animal:13 construct:3 represents:2 anticipated:1 report:2 stimulus:20 t2:1 others:1 few:1 simultaneously:1 individual:1 phase:2 replacement:1 certainly:1 mixture:9 light:10 behind:2 pc:2 implication:1 partial:2 incomplete:1 opponency:1 causal:1 formalise:1 psychological:7 instance:2 soft:1 ar:1 overshadowing:8 w1i:1 challenged:1 expects:1 predictor:4 delay:1 successful:2 erlbaum:1 too:1 reported:1 corrupted:1 punishment:1 ie:3 probabilistic:1 tip:1 together:1 e25:1 na:1 thesis:1 nm:1 solomon:2 possibly:1 cognitive:2 expert:7 leading:1 account:3 potential:3 de:1 satisfy:1 ad:1 responsibility:1 doing:1 reached:1 competitive:6 start:1 complicated:2 ass:1 il:2 matzel:2 variance:5 characteristic:1 miller:2 none:1 cybernetics:1 processor:1 history:1 ah:1 unblocking:21 explain:1 synaptic:1 against:1 pp:3 involved:2 obvious:1 associated:2 mi:2 stop:1 treatment:1 credible:1 sophisticated:1 actually:1 specify:2 formulation:1 though:1 anderson:2 just:6 parameterised:1 until:1 broadbent:2 perhaps:1 indicated:1 orienting:1 name:1 effect:4 true:1 discounting:1 stance:1 moore:3 eg:2 pharmacological:1 ll:2 self:1 during:2 acquires:1 essence:1 anything:3 complete:1 demonstrate:1 upwards:4 consideration:1 novel:1 recently:1 iceberg:1 stimulation:1 rl:1 conditioning:32 association:6 discussed:1 cambridge:4 appleton:1 ai:4 cv:1 pm:1 similarly:3 pointed:1 funded:1 similarity:1 gj:1 cove:1 posterior:1 involvement:1 irrelevant:1 forcing:1 binary:1 contingent:1 greater:1 paradigm:6 signal:1 ii:1 multiple:2 sound:10 pnas:1 england:1 long:6 divided:1 equally:1 prevented:1 va:1 prediction:31 variant:1 circumstance:2 expectation:2 essentially:1 dopamine:1 vision:1 represent:1 cell:2 whereas:1 background:2 wealth:1 unlike:1 db:1 incorporates:1 seem:5 jordan:2 practitioner:1 presence:7 easy:2 enough:1 affect:1 psychology:1 favour:1 peter:1 york:2 action:1 gabriel:3 useful:1 detailed:1 involve:1 listed:1 discount:1 demonstrably:1 rw:10 generate:1 inhibitory:12 delta:4 neuroscience:2 correctly:1 klein:1 affected:2 key:7 nevertheless:1 changing:2 neither:1 shock:1 sum:1 inverse:1 uncertainty:1 thorn:1 reporting:1 separation:2 delivery:5 interleaved:1 capturing:1 konorski:4 activity:2 strength:1 constraint:2 dominated:1 aspect:2 extremely:1 pavlovian:4 department:1 according:2 combination:1 describes:1 em:1 character:1 wi:11 making:3 happens:2 repair:1 pr:3 computationally:2 equation:10 remains:1 discus:1 turn:1 ge:1 end:2 studying:1 available:1 gaussians:1 opponent:2 probe:2 apply:1 appropriate:4 alternative:2 slower:2 jd:1 substitute:1 trouble:2 embodies:1 disappear:1 classical:2 society:2 implied:1 pergamon:1 added:1 traditional:1 win:1 cw:1 attentional:2 separate:1 capacity:2 argue:1 presynaptic:1 considers:1 reason:3 modeled:1 relationship:1 index:1 ql:3 expense:1 negative:2 aversive:1 implementation:1 collective:1 perform:3 allowing:1 pearce:4 neurobiology:1 variability:2 communication:1 hinton:1 omission:1 introduced:1 specified:2 philosophical:1 distinction:1 conflicting:1 established:2 suggested:1 corbit:2 perception:1 challenge:1 rf:2 oj:1 royal:1 explanation:2 event:2 natural:1 force:1 difficulty:1 wanting:1 counterbalance:1 extract:1 review:7 prior:1 expect:2 bear:1 a11996:1 suggestion:2 interesting:1 foundation:2 contingency:1 consistent:3 rubin:1 editor:3 share:2 land:1 pi:4 course:3 side:1 weaker:1 explaining:1 fall:1 face:1 wagner:4 emerge:1 schmajuk:1 calculated:1 world:4 made:3 collection:1 reinforcement:2 schultz:2 founded:2 transaction:1 prokasy:1 sj:2 neurobiological:1 status:1 ml:2 why:2 table:4 learn:3 reasonably:1 nature:1 operational:1 da:6 whole:3 noise:2 nothing:1 lesion:1 body:1 fashion:1 downwards:17 bordley:2 ny:2 formalisation:1 fails:2 candidate:2 third:1 normative:1 symbol:1 r2:2 evidence:2 importance:1 texture:1 gallagher:2 nat:1 phd:1 conditioned:4 gluck:2 surprise:1 sophistication:1 likely:2 unexpected:1 holland:4 ma:3 cti:1 comparator:2 presentation:2 shared:1 absence:7 man:1 change:5 experimentally:1 included:1 justify:1 called:5 partly:1 experimental:3 puzzling:1 support:2 latter:1 theresa:1 tested:1 phenomenon:1 mackintosh:8

Regularisation in Sequential Learning Algorithms J oao FG de Freitas Cambridge University Engineering Department Cambridge CB2 IPZ England jfgf@eng.cam.ac.uk [Corresponding author] Mahesan Niranjan Cambridge University Engineering Department Cambridge CB2 IPZ England niranjan@eng.cam.ac.uk Andrew H Gee Cambridge University Engineering Department Cambridge CB2 IPZ England ahg@eng.cam.ac.uk Abstract In this paper, we discuss regularisation in online/sequential learning algorithms. In environments where data arrives sequentially, techniques such as cross-validation to achieve regularisation or model selection are not possible. Further, bootstrapping to determine a confidence level is not practical. To surmount these problems, a minimum variance estimation approach that makes use of the extended Kalman algorithm for training multi-layer perceptrons is employed. The novel contribution of this paper is to show the theoretical links between extended Kalman filtering, Sutton's variable learning rate algorithms and Mackay's Bayesian estimation framework. In doing so, we propose algorithms to overcome the need for heuristic choices of the initial conditions and noise covariance matrices in the Kalman approach. 1 INTRODUCTION Model estimation involves building mathematical representations of physical processes using measured data. This problem is often referred to as system identification, time-series modelling or machine learning. In many occasions, the system being modelled varies with time. Under this circumstance, the estimator needs to be Regularisation in Sequential Learning Algorithms 459 updated sequentially. Online or sequential learning has many applications in tracking and surveillance, control systems, fault detection, communications, econometric systems, operations research, navigation and other areas where data sequences are often non-stationary and difficult to obtain before the actual estimation process. To achieve acceptable generalisation, the complexity of the estimator needs to be judiciously controlled. Although there are various reliable schemes for controlling model complexity when training en bloc (batch processing), the same cannot be said about sequential learning. Conventional regularisation techniques cannot be applied simply because there is no data to cross-validate. Consequently, there is ample scope for the design of sequential methods of controlling model complexity. 2 NONLINEAR ESTIMATION A dynamical system may be described by the following discrete, stochastic state space representation: (1) Wk +dk (2) g(Wk, tk) + Vk where it has been assumed that the model parameters (Wk E R<J) constitute the states of the system, which in our case represent the weights of a multi-layer perceptron (MLP). g is a nonlinear vector function that may change at each estimation step k, tk denotes the time at the k-th estimation step and dk and Vk represent zero mean white noise with covariances given by Qk and Rk respectively. The noise terms are often called the process noise (dk) and the measurement noise (Vk). The system measurements are encoded in the output vector Yk E Rm. The estimation problem may be reformulated as having to compute an estimate Wk of the states Wk using the set of measurements Yk = {Yl, Y2, "', Yk}. The estimate Wk can be used to predict future values of the output y. We want Wk to be an unbiased, minimum variance and consistent estimate (Gelb 1984). A minimum variance (unbiased) estimate is one that has its variance less than or equal to that of any other unbiased estimator. Since the variance of the output Y depends directly on the variance of the parameter estimates (Astrom 1970), the minimum variance framework constitutes a regularisation scheme for sequential learning. The conditional probability density function of Wk given Yk (p(wkIYk)) constitutes the complete solution of the estimation problem (Bar-Shalom and Li 1993, Ho and Lee 1964, Jazwinski 1970). This is simply because p(wkIYk) embodies all the statistical information about Wk given the measurements Yk and the initial condition Woo This is essentially the Bayesian approach to estimation, where instead of describing a model by a single set of parameters, it is expressed in terms of the conditional probability p(wkIYk) (Jaynes 1986, Jazwinski 1970). The estimate Wk can be computed from p(wklY k) according to several criteria, namely MAP estimation (peak of the posterior), minimum variance estimation (centroid of the posterior) and minimax estimation (median of the posterior). The Bayesian solution to the optimal estimation problem is (Ho and Lee 1964): P(Wk+1,Yk+I IYk) p(Yk+1I Yk) J p(Yk+1IYk, Wk+l )p(wk+1lwk)P(Wk IYk)dwk J J p(Yk+lIYk, Wk+1 )p(Wk+llwk)p(Wk IYk)dwk+l dWk (3) where the integrals run over the parameter space. This functional integral difference equation governing the evolution of the posterior density function is not suitable 1. R G. d. Freitas, M. Niranjan andA. H. Gee 460 for practical implementation (Bar-Shalom and Li 1993, Jazwinski 1970). It involves propagating a quantity (the posterior density function) that cannot be described by a finite number of parameters. The situation in the linear case is vastly simpler. There the mean and covariance are sufficient statistics for describing the Gaussian posterior density function. In view of the above statements, it would be desirable to have a framework for nonlinear estimation similar to the one for linear-Gaussian estimation. The extended Kalman filter (EKF) constitutes an attempt in this direction (Bar-Shalom and Li 1993, Gelb 1984). The EKF is a minimum variance estimator based on a Taylor series expansion of the nonlinear function g(w) around the previous estimate. The EKF equations for a linear expansion are given by: + Qk)Gk+l [Rk + Gk+1 (Pk + Qk)Gk+1]-1 Wk + Kk+l(Yk+l - Gk+l Wk) Pk + Qk - Kk+lGk+l (Pk + Qk) (Pk (4) (5) (6) where Pk denotes the covariance of the weights. In the general multiple input, multiple output (MIMO) case, g E ~m is a vector function and G represents the Jacobian of the network outputs with respect to the weights. The EKF provides a minimum variance Gaussian approximation to the posterior probability density function. In many cases, p(wkIYk) is a multi-modal (several peaks) function. In this scenario, it is possible to use a committee of Kalman filters, where each individual filter approximates a particular mode, to produce a more accurate approximation (Bar-Shalom and Li 1993, Kadirkamanathan and Kadirkamanathan 1995). The parameter covariances of the individual estimators may be used to determine the contribution of each estimator to the committee. In addition, the parameter covariances serve the purpose of placing confidence intervals on the output prediction. 3 TRAINING MLPs WITH THE EKF One of the earliest implementations of EKF trained MLPs is due to Singhal and Wu (Singhal and Wu 1988). In their method, the network weights are grouped into a single vector w that is updated in accordance with the EKF equations. The entries of the Jacobian matrix are calculated by back-propagating the m output values through the network. The algorithm proposed by Singhal and Wu requires a considerable computational effort. The complexity is of the order mq2 multiplications per estimation step. Shah, Palmieri and Datum (1992) and Puskorius and Feldkamp (1991) have proposed strategies for decoupling the global EKF estimation algorithm into local EKF estimation sub-problems, thereby reducing the computational time. The EKF is an improvement over conventional MLP estimation techniques, such as back-propagation, in that it makes use of second order statistics (covariances). These statistics are essential for placing error bars on the predictions and for combining separate networks into committees of networks. Further, it has been proven elsewhere that the back-propagation algorithm is simply a degenerate of the EKF algorithm (Ruck, Rogers, Kabrisky, Maybeck and Oxley 1992). However, the EKF algorithm for training MLPs suffers from serious difficulties, namely choosing the initial conditions (wo, Po) and the noise covariance matrices Rand Q. In this work, we propose the use of maximum likelihood techniques, such as back-propagation computed over a small set of initial data, to initialise the Regularisation in Sequential Learning Algorithm~ 461 EKF-MLP estimator. The following two subsections? describe ways of overcoming the difficulty of choosing R and Q. 3.1 ELIMINATING Q BY UPDATING P WITH BACK-PROPAGATION . To circumvent the problem of choosing the process noise covariance Q, while at the same time increasing computational efficiency, it is possible to extend an algorithm proposed by Sutton (Sutton 1992) to the nonlinear case. In doing so, the weights covariance is approximated by a diagonal matrix with entries given by pqq = exp(,8q), where,8 is updated by error back-propagation (de Freitas, Niranjan and Gee 1997). The Kalman gain K k and the weights estimate Wk are updated using a variation of the Kalman equations, where the Kalman gain and weights update equations are independent of Q (Gelb 1984), while the weights covariance P is updated by backpropagation. This algorithm lessens the burden of choosing the matrix Q by only having to choose the learning rate scalar 1]. The performance of the EKF algorithm with P updated by back-propagation will be analysed in Section 4. 3.2 KALMAN FILTERING AND BAYESIAN TECHNIQUES A further improvement on the EKF algorithm for training MLPs would be to update Rand Q automatically each estimation step. This can be done by borrowing some ideas from the Bayesian estimation field. In particular, we shall attempt to link Mackay's work (Mackay 1992, Mackay 1994) on Bayesian estimation for neural networks with the EKF estimation framework. This theoretical link should serve to enhance both methods. Mackay expresses the prior, likelihood and posterior density functions in terms of the following Gaussian approximations: 1 (a 2 (7) p(w) = (27r)q/2 a -q/2 exp - "2llwll ) 1 ,8~ 2 p(Yklw) = (27r)n/2,8-n/2 exp ( - "2 L.,..(Yk - fn,q(w, <Pk)) ) A (8) k=l p(wIYk ) 1 1 T = (27r)q/2IAI- 1 / 2 exp ( - 2(w - WMP) A(w - WMP)) (9) where in,q(w, <Pk) represents the estimator and the hyper-parameters a and ,8 control the variance of the prior distribution of weights and the variance of the measurement noise. a also plays the role of the regularisation coefficient. The posterior is obtained by approximating it with a Gaussian function, whose mean wMP is given by a minimum of the following regularised error function: a S(w) = "2llwl12 ,8~ + "2 L.,..(Yk - A fn,q(w, <Pk)) 2 (10) k==l The posterior covariance A is the Hessian of the above error function. In Mackay's estimation framework, also known as the evidence framework, the parameters ware obtained by minimising equation (10), while the hyper-parameters a and ,8 are obtained by maximising the evidence p(Yk la,,8) after approximating the posterior density function by a Gaussian function. In doing so, the following recursive formulas for a and ,8 are obtained: '1 n-'1 ak+1 = L: q 2 and ,8k+1 = n 2 i=l Wi L::k=l (Yk - in,q(Wk, <Pk)) A J. F. G. d. Freitas, M. Niranjan andA. H. Gee 462 The quantity 'Y represents the effective number of parameters 'Y = 2J~=1 >.:~a' where the Ai correspond to the eigenvalues of the Hessian of the error function without the regularisation term . Instead of adopting Mackay's evidence framework, it is possible to maximise the posterior density function by performing integrations over the hyper-parameters analytically (Buntine and Weigend 1991, Mackay 1994). The latter approach is known as the MAP framework for 0 and {3. The hyper-parameters computed by the MAP framework differ from the ones computed by the evidence framework in that the former makes use of the total number of parameters and not only the effective number of parameters. That is, 0 and {3 are updated according to: q n Ok+l = ",q 2 and {3k+1 = n 2 L.,..i=l Wi l:k=l (Yk - /n,q(Wk , <Pk)) A By comparing the equations for the prior, likelihood and posterior density functions in the Kalman filtering framework (Ho and Lee 1964) with equations (7), (8) and (9) we can establish the following relations: P=A- 1 Q=o-IIq_A- 1 , and R={3-1Im where Iq and 1m represent identity matrices of sizes q and m respectively. Therefore, it is possible to update Q and R sequentially by expressing them in terms of the sequential updates of 0 and {3. -60~--:':: 10---::2'=""0- ::'::30- --=4'=""0-::'::5o----:6O ':--7=o----:8o~-90 :'::---,-' 100 I -~-8~10~82~0-~ ~M~0--~ ~860~-8~ 70--880 ~-8~90-~900 I Figure 1: Prediction using the conventional EKF algorithm for a network with 20 hidden neurons. Actual output [. . .J and estimated output [-J. 4 RESULTS To compare the performance of the conventional EKF algorithm, the EKF algorithm with P updated by back-propagation, and the EKF algorithm with Rand Q updated sequentially according to the Bayesian MAP framework, noisy data was generated from the following nonlinear, non-stationary, multivariate process: Y (t) - { Xl (t) + X2 (t) + v(t) 4sin(xdt)) + X2(t) sin(0.03(t - 200)) + v(t} 200 1 ::; t ::; 200 < t ::; 1000 Regularisation in Sequential Learning Algorithms 463 where the inputs Xi are uniformly distributed random sequences with variance equal . to 1 and v(t) corresponds to uniformly distributed noise with variance equal to 0.1. Figure 1 shows the prediction obtained using the conventional EKF algorithm. To \ " ," , ,, 3.S ~ I , I , I , I , I , 3 .? ri s I 2 ~ 1.5 a: 05 10 Tnal 12 14 16 18 20 Figure 2: Output error for the conventional EKF algorithm [. .. ], the EKF algorithm with P updated by back-propagation [- . -], the EKF algorithm with Rand Q updated sequentially according to the Bayesian MAP framework [-], and the EKF algorithm with the Bayesian evidence framework [- - -] . compare the four estimation frameworks, an MLP with 20 neurons in the hidden layer was selected. The initial conditions were obtained by using back-propagation on the first 100 samples and assigning to P a diagonal matrix with diagonal elements equal to 10. The matrices R and Q in the conventional EKF algorithm were chosen, by trial and error, to be identity matrices. In the EKF algorithm with P updated by back-propagation, R was chosen to be equal to the identity matrix, while the learning rate was set to 0.01. Finally, in the EKF algorithm with Rand Q updated sequentially, the initial Rand Q matrices were chosen to be identity matrices. The prediction errors obtained for each method with random input data are shown in Figure 2. It is difficult to make a fair comparison between the four nonlinear estimation methods because their parameters were optimised independently. However, the results suggest that the prediction obtained with the conventional EKF training outperforms the predictions of the other methods. This may be attributed to the facts that, firstly, in this simple problem it is possible to guess the optimal values for Rand Q and, secondly, the algorithms to update the noise covariances may affect the regularisation performance of the EKF algorithm. This issue, and possible solutions, is explored in depth by the authors in (de Freitas et al. 1997). 5 Conclusions In this paper, we point out the links between Kalman filtering, gradient descent algorithms with variable learning rates and Bayesian estimation. This results in two algorithms for eliminating the problem of choosing the initial conditions and the noise covariance matrices in the training of MLPs with the EKF. These algorithms are illustrated on a toy problem here, but more extensive experiments have been reported in (de Freitas et al. 1997). Improved estimates may be readily obtained by combining the estimators into com- 464 J. R G. d. Freitas, M. Niranjan and A. H. Gee mit tees or extending the training methods to recurrent networks. Finally, the computational time may be reduced by decoupling the network weights. Acknowledgements Joao FG de Freitas is financially supported by two University of the Witwatersrand Merit Scholarships, a Foundation for Research Development Scholarship (South Africa) and a Trinity College External Studentship (Cambridge). References Astrom, K. J. (1970). Introduction to Stochastic Control Theory, Academic Press. Bar-Shalom, Y. and Li, X. R. (1993). Estimation and Tracking: Principles, Techniques and Software, Artech House, Boston. Buntine, W. L. and Weigend, A. S. (1991). Bayesian back-propagation, Complex Systems 5: 603-643. de Freitas, J., Niranjan, M. and Gee, A. (1997). Hierarchichal BayesianKalman models for regularisation and ARD in sequential learning, Technical Report CUED/F-INFENG/TR 307, Cambridge University, http:j jsvrwww .eng.cam.ac.ukj-jfgf. Gelb, A. (ed.) (1984). Applied Optimal Estimation, MIT Press. Ho, Y. C. and Lee, R. C. K. (1964). A Bayesian approach to problems in stochastic estimation and control, IEEE Transactions on Automatic Control AC-9: 333339. Jaynes, E. T. (1986). Bayesian methods: General background, in J. H. Justice (ed.), Maximum Entropy and Bayesian Methods in Applied Statistics, Cambridge University Press, pp. 1-25. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory, Academic Press. Kadirkamanathan, V. and Kadirkamanathan, M. (1995). Recursive estimation of dynamic modular RBF networks, in D. S. Touretzky, M. C. Mozer and M. E. Hasselmo (eds), Advances in Neural Information Processing Systems 8, pp. 239-245. Mackay, D. J. C. (1992). Bayesian interpolation, Neural Computation 4(3): 415-447. Mackay, D. J. C. (1994). Hyperparameters: Optimise or integrate out?, in G. Heidbreder (ed.), Maximum Entropy and Bayesian Methods. Puskorius, G. V. and Feldkamp, 1. A. (1991). Decoupled extended Kalman filter training of feedforward layered networks, International Joint Conference on Neural Networks, Seattle, pp. 307-312. Ruck, D. W., Rogers, S. K., Kabrisky, M., Maybeck, P. S. and Oxley, M. E. (1992). Comparative analysis of backpropagation and the extended Kalman filter for training multilayer perceptrons, IEEE Transactions on Pattern Analysis and Machine Intelligence 14(6): 686-690. Shah, S., Palmieri, F. and Datum, M. (1992). Optimal filtering algorithms for fast learning in feedforward neural networks, Neural Networks 5: 779-787. Singhal, S. and Wu, 1. (1988). Training multilayer perceptrons with the extended Kalman algorithm, in D. S. Touretzky (ed.), Advances in Neural Information Processing Systems, Vol. 1, San Mateo, CA, pp. 133-140. Sutton, R. S. (1992). Gain adaptation beats least squares?, Proceedings of the Seventh Yale Workshop on Adaptive Learning Systems, pp. 161-166. 

1464 |@word trial:1 eliminating:2 justice:1 lwk:1 eng:4 covariance:14 thereby:1 tr:1 initial:7 series:2 outperforms:1 freitas:9 africa:1 comparing:1 jaynes:2 com:1 analysed:1 assigning:1 readily:1 fn:2 update:5 stationary:2 intelligence:1 selected:1 guess:1 provides:1 firstly:1 simpler:1 mathematical:1 multi:3 feldkamp:2 automatically:1 actual:2 increasing:1 heidbreder:1 joao:1 bootstrapping:1 rm:1 uk:3 control:5 before:1 maximise:1 engineering:3 accordance:1 local:1 sutton:4 ak:1 ware:1 optimised:1 interpolation:1 mateo:1 practical:2 recursive:2 witwatersrand:1 backpropagation:2 cb2:3 area:1 confidence:2 suggest:1 cannot:3 selection:1 layered:1 conventional:8 map:5 independently:1 estimator:9 initialise:1 variation:1 updated:13 controlling:2 play:1 regularised:1 element:1 approximated:1 updating:1 role:1 yk:16 mozer:1 environment:1 complexity:4 cam:4 dynamic:1 trained:1 serve:2 efficiency:1 po:1 joint:1 various:1 fast:1 describe:1 effective:2 hyper:4 choosing:5 whose:1 heuristic:1 encoded:1 modular:1 statistic:4 noisy:1 online:2 sequence:2 eigenvalue:1 propose:2 adaptation:1 combining:2 degenerate:1 achieve:2 validate:1 seattle:1 extending:1 produce:1 comparative:1 tk:2 cued:1 iq:1 andrew:1 ac:5 propagating:2 recurrent:1 measured:1 ard:1 involves:2 differ:1 direction:1 filter:5 stochastic:4 rogers:2 trinity:1 secondly:1 im:1 around:1 exp:4 scope:1 predict:1 purpose:1 estimation:32 lessens:1 oxley:2 grouped:1 hasselmo:1 mit:2 gaussian:6 ekf:31 surveillance:1 earliest:1 vk:3 improvement:2 modelling:1 likelihood:3 centroid:1 jazwinski:4 borrowing:1 relation:1 hidden:2 issue:1 development:1 integration:1 mackay:10 equal:5 field:1 having:2 represents:3 ipz:3 placing:2 constitutes:3 future:1 report:1 serious:1 individual:2 attempt:2 detection:1 mlp:4 navigation:1 arrives:1 puskorius:2 accurate:1 integral:2 decoupled:1 taylor:1 theoretical:2 mahesan:1 singhal:4 entry:2 mimo:1 seventh:1 lgk:1 buntine:2 reported:1 varies:1 density:9 peak:2 international:1 lee:4 yl:1 bloc:1 enhance:1 vastly:1 choose:1 external:1 li:5 toy:1 de:6 wk:22 coefficient:1 depends:1 ahg:1 view:1 doing:3 contribution:2 mlps:5 square:1 variance:14 qk:5 correspond:1 modelled:1 bayesian:16 identification:1 suffers:1 touretzky:2 ed:5 pp:5 attributed:1 gain:3 palmieri:2 subsection:1 back:12 ok:1 modal:1 improved:1 rand:7 iai:1 done:1 governing:1 wmp:3 nonlinear:7 propagation:11 mode:1 building:1 y2:1 unbiased:3 evolution:1 analytically:1 former:1 illustrated:1 white:1 sin:2 criterion:1 occasion:1 complete:1 oao:1 novel:1 functional:1 physical:1 extend:1 approximates:1 measurement:5 expressing:1 cambridge:9 ai:1 automatic:1 posterior:13 multivariate:1 shalom:5 scenario:1 tee:1 fault:1 minimum:8 employed:1 determine:2 artech:1 multiple:2 desirable:1 technical:1 england:3 jfgf:2 cross:2 minimising:1 academic:2 niranjan:7 controlled:1 prediction:7 infeng:1 multilayer:2 essentially:1 circumstance:1 represent:3 adopting:1 addition:1 want:1 background:1 interval:1 median:1 south:1 ample:1 feedforward:2 affect:1 idea:1 judiciously:1 effort:1 wo:1 reformulated:1 hessian:2 constitute:1 maybeck:2 reduced:1 http:1 estimated:1 per:1 discrete:1 shall:1 vol:1 express:1 four:2 wkiyk:4 econometric:1 surmount:1 weigend:2 run:1 wu:4 acceptable:1 layer:3 datum:2 yale:1 x2:2 ri:1 software:1 performing:1 department:3 according:4 llwll:1 wi:2 equation:8 discus:1 describing:2 committee:3 merit:1 operation:1 batch:1 ho:4 shah:2 denotes:2 xdt:1 embodies:1 scholarship:2 establish:1 approximating:2 quantity:2 kadirkamanathan:4 strategy:1 diagonal:3 said:1 financially:1 gradient:1 link:4 separate:1 maximising:1 kalman:14 kk:2 difficult:2 statement:1 gk:4 design:1 implementation:2 neuron:2 finite:1 descent:1 gelb:4 beat:1 situation:1 extended:6 communication:1 overcoming:1 namely:2 extensive:1 bar:6 dynamical:1 pattern:1 reliable:1 optimise:1 suitable:1 difficulty:2 circumvent:1 minimax:1 scheme:2 iyk:4 woo:1 prior:3 acknowledgement:1 multiplication:1 regularisation:12 filtering:6 proven:1 validation:1 foundation:1 integrate:1 sufficient:1 consistent:1 principle:1 elsewhere:1 supported:1 gee:6 perceptron:1 fg:2 distributed:2 overcome:1 calculated:1 depth:1 studentship:1 author:2 adaptive:1 san:1 transaction:2 global:1 sequentially:6 anda:2 assumed:1 xi:1 ca:1 decoupling:2 expansion:2 complex:1 ruck:2 pk:10 noise:11 hyperparameters:1 fair:1 astrom:2 referred:1 en:1 sub:1 xl:1 house:1 jacobian:2 rk:2 formula:1 explored:1 dk:3 evidence:5 essential:1 burden:1 workshop:1 sequential:11 boston:1 entropy:2 simply:3 expressed:1 llwl12:1 tracking:2 scalar:1 corresponds:1 ukj:1 conditional:2 identity:4 consequently:1 rbf:1 considerable:1 change:1 generalisation:1 reducing:1 uniformly:2 dwk:3 called:1 total:1 la:1 perceptrons:3 college:1 latter:1

Adaptive choice of grid and time reinforcement learning ? In Stephan Pareigis stp@numerik.uni-kiel.de Lehrstuhl Praktische Mathematik Christian-Albrechts-Uni versitiit Kiel Kiel, Germany Abstract We propose local error estimates together with algorithms for adaptive a-posteriori grid and time refinement in reinforcement learning. We consider a deterministic system with continuous state and time with infinite horizon discounted cost functional. For grid refinement we follow the procedure of numerical methods for the Bellman-equation. For time refinement we propose a new criterion, based on consistency estimates of discrete solutions of the Bellmanequation. We demonstrate, that an optimal ratio of time to space discretization is crucial for optimal learning rates and accuracy of the approximate optimal value function. 1 Introduction Reinforcement learning can be performed for fully continuous problems by discretizing state space and time, and then performing a discrete algorithm like Q-Iearning or RTDP (e.g. [5]). Consistency problems arise if the discretization needs to be refined, e.g. for more accuracy, application of multi-grid iteration or better starting values for the iteration of the approximate optimal value function. In [7] it was shown, that for diffusion dominated problems, a state to time discretization ratio k/ h of Ch'r, I > 0 has to hold, to achieve consistency (i.e. k = o(h)). It can be shown, that for deterministic problems, this ratio must only be k / h = C, C a constant, to get consistent approximations of the optimal value function. The choice of the constant C is crucial for fast learning rates, optimal use of computer memory resources and accuracy of the approximation. We suggest a procedure involving local a-posteriori error estimation for grid refinement, similar to the one used in numerical schemes for the Bellman-equation (see [4]). For the adaptive time discretization we use a combination from step size con- Adaptive Choice of Grid and Time in Reinforcement Learning 1037 trol for ordinary differential equations and calculations for the rates of convergence of fully discrete solutions of the Bellman-equation (see [3]). We explain how both methods can be combined and applied to Q-Iearning. A simple numerical example shows the effects of suboptimal state space to time discretization ratio, and provides an insight in the problems of coupling both schemes. 2 Error estimation for adaptive choice of grid We want to approximate the optimal value function V : of the following problem: Minimize n C IRd J(x, u(.)) := 1 00 n -+ IR in a state space u(.): IR+ -+ A measurable, e- pr g(Yx,u( .)(r), u(r))dr, (1) where 9 : n X A -+ IR+ is the cost function, and Yx,u( .)(.) is the solution of the differential equation (2) y(t) f(y(t), u(t)), y(O) x. = = As a trial space for the approximation of the optimal value function (or Q-function) we use locally linear elements on simplizes Si, i = 1, ... , N s which form a triangulation of the state space, N s the number of simplizes. The vertices shall be called Xi, i 1, . .. , N, N the dimension of the trial space 1 . This approach has been used in numerical schemes for the Bellman-equation ([2], [4]). We will first assume, that the grid is fixed and has a discretization parameter = k = maxdiam{Si}. i Other than in the numerical case, where the updates are performed in the vertices of the triangulation, in reinforcement learning only observed information is available. We will assume, that in one time step of size h > 0, we obtain the following information: ? the current state Yn E n, ? an action an E A, ? the subsequent state Yn+1 := YYn,a n(h) ? the local cost rn = r(Yn, an) = Joh e-PTg(YYn,an(r),an(r))dr. The state Yn, in which an update is to be made, may be any state in finite, and an locally constant . n. A shall be The new value of the fully discrete Q-function Qi (Yn, an) should be set to shall be where rn ( ) + e -phTTk v h Yn+l , V; (Yn+d = minaQi(Yn+l,a). We call the right side the update function (3) We will update Qi in the vertices {Xd~l of the triangulation in one ofthe following two ways: 1 When an adaptive grid is used, then N s and N depend on the refinement. Kaczmarz-update. Let nates, such that >.7 (AI, .. . , AN) be the vector of barycentric coordi- N Yn = 2: Aixi, O:SAi:Sl, foralli=I, ... ,N. i=1 Then update (4) Kronecker-update. Let 53 Yn and x be the vertex of 5, closest to Yn (if there is a draw, then the update can be performed in all winners). Then update Q~ only in x according to (5) Each method has some assets and drawbacks . In our computer simulations the Kaczmarz-update seemed to be more stable over the Kronecker-update (see [6]) . However , examples may be constructed where a (Holder-) continuous bounded optimal value function V is to be approximated, and the Kaczmarz-update produces an approximation with arbitrarily high "."sup-norm (place a vertex x of the triangulation in a point where V is infinity, and use as update states the vertex x in turn with an arbitrarily close state x) . d: Kronecker-update will provide a bounded approximation if V is bounded. Let be the fully-discrete optimal value function Vhk (xd = min{r(xi' a) + e-PhVhk (Yxi,a(h)), a Vhk i = 1, . . . , N . Then it can be shown, that an approximationyerformed by Kronecker-update will (with respect to the "."sup-norm), eventually be caught in an c-neighborhood of if the data points Yo, Yl, Y2, . . . are sufficiently dense. Under regularity conditions on V, c may be bounded by2 VI: (6) As a criterion for grid refinement we choose a form of a local a posteriori error estimate as defined in [4] . Let (x) = mina Q~ (x, a) be the current iterate of the optimal value function. Let ax E U be the minimizing control ax = argmina Q~ (x, a). Then we define vI: (7) vI:, If Vhk is in the c-neighborhood of then it can be shown, that (for every x E and simplex Sx with x E Sx, ax as above) n O:S e(x) :S sup P(z , az , Vhk) - inf P( z , az , Vhk). z ESz: z ES:t If Vhk is Lipschitz-continuous, then an estimate using only Gronwall's inequality bounds the right side and therefore e(x) by C where C depends on the Lipschitzconstants of and the cost g . vl' p\' 2With respect to the results in [3] we assume, that also E: ~ C(h + 7;:) can be shown. Adaptive Choice of Grid and Time in Reinforcement Learning 1039 The value ej := maXxesj eh(x) defines a function, which is locally constant on every simplex. We use ej, j = 1, ... , N as an indicator function for grid refinement. The (global) tolerance value tolk for ej shall be set to Ns tolk = C * (L; edlNs, i=l where we have chosen 1 :::; C :::; 2. We approximate the function e on the simplizes in the following way, starting in some Yn E Sj: 1. 2. 3. 4. apply a control a E U constantly on [T, T + h] receive value rn and subsequent state Yn+l calculate the update value Ph(x, a, Vf) if (IPh(x,a, vt) - Vt(x)l ~ ej) then ej := IPh(x,a, Vhk) - Vt(x)1 It is advisable to make grid refinements in one sweep. We also store (different to the described algorithm) several past values of ej in every simplex, to be able to distinguish between large ej due to few visits in that simplex and the large ej due to space discretization error. For grid refinement we use a method described in ([1]). 3 A local criterion for time refinement Why not take the smallest possible sampling rate? There are two arguments for adaptive time discretization. First, a bigger time step h naturally improves (decreases) the contraction rate of the iteration, which is e- ph . The new information is conveyed from a point further away (in the future) for big h, without the need to store intermediate states along the trajectory. It is therefore reasonable to start with a big h and refine where needed. The second argument is, that the grid and time discretization k and h stand in a certain relation. In [3] the estimate lV(x) - vt(x)1 :::; C(h + k ..Jh)' for all x En, C a constant is proven (or similar estimates, depending on the regularity of V). For obvious reasons, it is desirable to start with a coarse grid (storage, speed), i.e. k large. Having a too small h in this case will make the approximation error large. Also here, it is reasonable to start with a big h and refine where needed. What can serve as a refinement criterion for the time step h? In numerical schemes for ordinary differential equations, adaptive step size control is performed by estimating the local truncation error of the Taylor series by inserting intermediate points. In reinforcement learning, however, suppose the system has a large truncation error (i.e. it is difficult to control) in a certain region using large h and locally constant control functions. If the optimal value function is nearly constant in this region, we will not have to refine h. The criterion must be, that at an intermediate point, e.g. at time h12, the optimal value function assumes a value considerably smaller (better) than at time h . However, if this better value is due to error in the state discretization, then do not refine the time step. ? We define a function H on the simplices of the triangulation. H(S) > holds the time-step which will be used when in simplex S. Starting at a state Yn E n, Yn E Sn at time T > 0, with the current iterate of the Q-function Q~ (Vhk respectively) the following is performed: s. Pareigis 1040 1. apply a control a E U constantly on [T, T + h] 2. take a sample at the intermediate state z = YYn,a(h/2) 3. if (H(Sn) < C*vdiam{Sn}) then end. else: 4. compute Vl(z) = millb Q~(z, b) 5. compute Ph/2(Yn, a, Vt) = rh/2(Yn, a) + e- ph / 2Vt(z) 6. compute Ph(Yn, a, Vt) = rh(Yn, a) +e-phVl(Yn+d 7. if (Ph/ 2(Yn, a, Vhk) S Ph(Yn , a, Vhk)-tol) update H(Sn) = H(Sn)/2 The value C is currently set to C 2 = C(Yn, a) = -lrh/2(Yn, a) p rh(Yn, a)/, whereby a local value of MI~gh2 is approximated, MJ (x) approximation of l\7g(x, a)1 (if 9 is sufficiently regular). = maxa If(x, a)l, Lg an tol depends on the local value of Vhk and is set to tOl(x) = 0.1 * vt(x). How can a Q-function Q:~:~(x, a), with state dependent time and space discretisation be approximated and stored? We have stored the time discretisation function H locally constant on every simplex. This implies (if H is not constant on 0), that there will be vertices Xj, such that adjacent triangles hold different values of H . The Q-function, which is stored in the vertices, then has different choices of H(xj). We solved this problem, by updating a function Q'H(Xj, a) with Kaczmarz-update and the update value PH(Yn) (Yn , a, Vt), Yn in an to Xj adjacent simplex, regardless of the different H-values in Xj. Q'H(Xj, a) therefore has an ambiguous semantic: it is the value if a is applied for 'some time ', and optimal from there on. 'some time'depends here on the value of H in the current simplex. It can be shown, that IQ~(Xj)/2(xj,a) - Q'H(Xj)(xj,a)1 is less than the space discretization error. 4 A simple numerical example We demonstrate the effects of suboptimal values for space and time discretisation with the following problem. Let the system equation be iJ = f(y, u) := (~1 ~) (y - v), v =( .375 ) .375 ' yEO = [0,1] x [0,1] (8) The stationary point of the uncontrolled system is v. The eigenvalues of the system are {u + i, U - i}, u E [-c, cJ. The system is reflected at the boundary. The goal of the optimal control shall be steer the solution along a given trajectory in state space (see figure 1), minimizing the integral over the distance from the current state to the given trajectory. The reinforcement or cost function is therefore chosen to be g(y) dist(L, y)t, (9) = where L denotes the set of points in the given trajectory. The cost functional takes the form ( 10) Adaptive Choice of Grid and Time in Reinforcement Learning 1041 IL 0.5 ~ o~ o ______ ~ ______ ~ 0.5 Figure 1: The left picture depicts the L-form of the given trajectory. The stationary point of the system is at (.375, .375) (depicted as a big dot). The optimal value function computed by numerical schemes on a fine fixed grid is depicted with too large time discretization (middle) and small time discretization (right) (rotated by about 100 degrees for better viewing). The waves in the middle picture show the effect of too large time steps in regions where 9 varies considerably. In the learning problem, the adaptive grid mechanism tries to resolve the waves (figure 1, middle picture) which come from the large time discretization. This is depicted in figure 2. We used only three different time step sizes (h 0.1, 0.05 and 0.025) and started globally with the coarsest step size 0.1. = Figure 2: The adaptive grid mechanism refines correctly. However, in the left picture, unnecessary effort is spended in resolving regions, in which the time step should be refined urgently. The right picture shows the result, if adaptive time is also used. Regions outside the L-form are refined in the early stages of learning while h was still large. An additional coarsening should be considered in future work. We used a high rate of random jumps in the process and locally a certainty equivalence controller to produce these pictures. 1042 5 S. Pareigis Discussion of the methods and conclusions We described a time and space adaptive method for reinforcement learning with discounted cost functional. The ultimate goal would be, to find a self tuning algorithm which locally adjusted the time and space discretization automatically to the optimal ratio. The methods worked fine in the problems we investigated, e.g. nonlinearities in the system showed no problems. Nevertheless, the results depended on the choice of the tolerance values C, tol and tolk' We used only three time discretization steps to prevent adjacent triangles holding time discretization values too far apart. The smallest state space resolution in the example is therefore too fine for the finest time resolution. A solution can be, to eventually use controls that are of higher order (in terms of approximation of control functions) than constant (e.g. linear, polynomial, or locally constant on subintervals of the finest time interval). This corresponds to locally open loop controls. The optimality of the discretization ratio time/space could not be proven. Some discontinuous value functions 9 gave problems, and we had problems handling stiff systems, too. The learning period was considerably shorter (about factor 100 depending on the requested accuracy and initial data) in the adaptive cases as opposed to fixed grid and time with the same accuracy. From our experience, it is difficult in numerical analysis to combine adaptive time and space discretization methods. To our knowledge this concept has not yet been applied to the Bellman-equation. Theoretical work is still to be done. We are aware, that triangulation of the state space yields difficulties in implementation in high dimensions. In future work we will be using rectangular grids. We will also make some comparisons with other algorithms like Parti-game ([5]). To us, a challenge is seen in handling discontinuous systems and cost functions as they appear in models with dry friction for example, as well as algebro-differential systems as they appear in robotics. References [1] E. Bansch. Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Engrg. 3, Vol. 3:181-191, 1991. [2] M. Falcone. A numerical approach to the infinite horizon problem of deterministic control theory. Appl Math Optim 15:1-13, 1987. [3] R. Gonzalez and M. Tidball. On the rates of convergence of fully discrete solutions of Hamilton-Jacobi equations. INRIA, Rapports de Recherche, No 1376, Programme 5, 1991. [4] L. Griine. An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation. Numerische Mathematik, Vol. 75, No. 3:319-337, 1997. [5] A. W. Moore and C. G. Atkeson. The parti-game algorithm for variable resolution reinforcement learning in multidimensional state-spaces. Machine Learning, Volume 21, 1995. [6] S. Pareigis. Lernen der Losung der Bellman-Gleichung durch Beobachtung von kontinuierlichen Prozepen. PhD thesis, Universitat Kiel, 1996. [7] S. Pareigis. Multi-grid methods for reinforcement learning in controlled diffusion processes. In D. S. Touretzky, M. C . Mozer, and M. E. Hasselmo, editors, Advances in Neural Information Processing Systems, volume 9. The MIT Press, Cambridge, 1997. 

1465 |@word trial:2 middle:3 joh:1 polynomial:1 norm:2 open:1 simulation:1 contraction:1 initial:1 series:1 past:1 current:5 discretization:19 optim:1 urgently:1 si:2 yet:1 must:2 finest:2 refines:1 subsequent:2 numerical:10 mesh:1 christian:1 update:19 stationary:2 recherche:1 provides:1 coarse:1 math:1 kiel:4 albrechts:1 along:2 constructed:1 differential:4 combine:1 dist:1 multi:2 bellman:7 discounted:2 globally:1 automatically:1 resolve:1 estimating:1 bounded:4 what:1 rtdp:1 maxa:1 certainty:1 sai:1 every:4 multidimensional:1 xd:2 iearning:2 control:11 yn:28 appear:2 hamilton:2 local:9 depended:1 inria:1 equivalence:1 appl:1 kaczmarz:4 procedure:2 foralli:1 regular:1 suggest:1 get:1 close:1 storage:1 measurable:1 deterministic:3 regardless:1 starting:3 caught:1 rectangular:1 resolution:3 numerische:1 insight:1 parti:2 suppose:1 aixi:1 element:1 approximated:3 praktische:1 updating:1 observed:1 solved:1 calculate:1 region:5 decrease:1 mozer:1 trol:1 depend:1 serve:1 triangle:2 fast:1 neighborhood:2 refined:3 outside:1 eigenvalue:1 propose:2 inserting:1 loop:1 achieve:1 az:2 convergence:2 regularity:2 produce:2 yxi:1 rotated:1 coupling:1 advisable:1 depending:2 iq:1 ij:1 implies:1 come:1 drawback:1 discontinuous:2 viewing:1 adjusted:1 hold:3 sufficiently:2 considered:1 early:1 smallest:2 estimation:2 currently:1 by2:1 esz:1 hasselmo:1 mit:1 ej:8 ax:3 yo:1 posteriori:3 dependent:1 vl:2 relation:1 germany:1 stp:1 aware:1 having:1 sampling:1 nearly:1 future:3 simplex:8 few:1 integral:1 experience:1 shorter:1 discretisation:3 taylor:1 theoretical:1 steer:1 ordinary:2 cost:8 vertex:8 too:6 universitat:1 stored:3 varies:1 considerably:3 combined:1 yl:1 together:1 von:1 thesis:1 opposed:1 choose:1 dr:2 yeo:1 nonlinearities:1 de:2 rapport:1 vi:3 depends:3 performed:5 try:1 sup:3 start:3 wave:2 minimize:1 il:1 ir:3 accuracy:5 holder:1 yield:1 ofthe:1 dry:1 trajectory:5 asset:1 explain:1 touretzky:1 obvious:1 naturally:1 mi:1 jacobi:2 con:1 pareigis:5 knowledge:1 improves:1 cj:1 higher:1 follow:1 reflected:1 lehrstuhl:1 done:1 stage:1 defines:1 effect:3 concept:1 y2:1 moore:1 semantic:1 adjacent:3 game:2 self:1 ambiguous:1 whereby:1 criterion:5 mina:1 demonstrate:2 functional:3 winner:1 volume:2 cambridge:1 ai:1 tuning:1 grid:24 consistency:3 engrg:1 had:1 dot:1 stable:1 argmina:1 closest:1 showed:1 triangulation:6 stiff:1 inf:1 apart:1 store:2 certain:2 inequality:1 discretizing:1 arbitrarily:2 durch:1 vt:9 der:2 seen:1 additional:1 period:1 resolving:1 desirable:1 calculation:1 visit:1 bigger:1 controlled:1 qi:2 impact:1 involving:1 controller:1 iteration:3 robotics:1 receive:1 want:1 fine:3 interval:1 else:1 crucial:2 coarsening:1 call:1 intermediate:4 stephan:1 iterate:2 xj:10 gave:1 suboptimal:2 ultimate:1 effort:1 ird:1 action:1 tol:4 locally:9 ph:8 sl:1 correctly:1 discrete:7 shall:5 vol:2 nevertheless:1 prevent:1 diffusion:2 place:1 reasonable:2 h12:1 draw:1 gonzalez:1 vf:1 bound:1 uncontrolled:1 distinguish:1 refine:4 kronecker:4 infinity:1 worked:1 dominated:1 speed:1 argument:2 min:1 optimality:1 friction:1 performing:1 coarsest:1 according:1 coordi:1 combination:1 smaller:1 pr:1 equation:11 resource:1 mathematik:2 turn:1 eventually:2 mechanism:2 needed:2 end:1 available:1 apply:2 away:1 ptg:1 assumes:1 denotes:1 yx:2 sweep:1 distance:1 sci:1 reason:1 ratio:6 minimizing:2 difficult:2 lg:1 holding:1 implementation:1 finite:1 rn:3 barycentric:1 able:1 challenge:1 memory:1 difficulty:1 eh:1 indicator:1 scheme:6 picture:6 started:1 sn:5 fully:5 proven:2 lv:1 degree:1 conveyed:1 consistent:1 editor:1 truncation:2 side:2 jh:1 lrh:1 tolerance:2 boundary:1 dimension:3 stand:1 seemed:1 made:1 adaptive:17 reinforcement:12 refinement:12 jump:1 programme:1 far:1 atkeson:1 sj:1 approximate:4 uni:2 global:1 unnecessary:1 xi:2 continuous:4 why:1 mj:1 subintervals:1 requested:1 investigated:1 dense:1 rh:3 big:4 arise:1 en:1 depicts:1 simplices:1 n:1 comput:1 nates:1 phd:1 sx:2 horizon:2 depicted:3 ch:1 iph:2 corresponds:1 constantly:2 goal:2 lipschitz:1 infinite:2 called:1 e:1 handling:2

Independent Component Analysis for identification of artifacts in Magnetoencephalographic recordings Ricardo Vigario 1 ; Veikko J ousmiiki2 , Matti Hiimiiliiinen2, Riitta Hari2, and Erkki Oja 1 1 Lab. of Computer & Info. Science Helsinki University of Technology P.O. Box 2200, FIN-02015 HUT, Finland {Ricardo.Vigario, Erkki.Oja}@hut.fi 2 Brain Research Unit, Low Temperature Lab. Helsinki University of Technology P.O. Box 2200, FIN-02015 HUT, Finland {veikko, msh, hari}@neuro.hut.fi Abstract We have studied the application of an independent component analysis (ICA) approach to the identification and possible removal of artifacts from a magnetoencephalographic (MEG) recording. This statistical technique separates components according to the kurtosis of their amplitude distributions over time, thus distinguishing between strictly periodical signals, and regularly and irregularly occurring signals. Many artifacts belong to the last category. In order to assess the effectiveness of the method, controlled artifacts were produced, which included saccadic eye movements and blinks, increased muscular tension due to biting and the presence of a digital watch inside the magnetically shielded room. The results demonstrate the capability of the method to identify and clearly isolate the produced artifacts. 1 Introduction When using a magnetoencephalographic (MEG) record, as a research or clinical tool, the investigator may face a problem of extracting the essential features of the neuromagnetic ? Corresponding author R. Vigario, 230 v. Jousmiiki, M. Hiimiiliiinen, R. Hari and E. Oja signals in the presence of artifacts. The amplitude of the disturbance may be higher than that of the brain signals, and the artifacts may resemble pathological signals in shape. For example, the heart's electrical activity, captured by the lowest sensors of a whole-scalp magnetometer array, may resemble epileptic spikes and slow waves (Jousmili and Hari 1996). The identification and eventual removal of artifacts is a common problem in electroencephalography (EEG), but has been very infrequently discussed in context to MEG (Hari 1993; Berg and Scherg 1994). The simplest and eventually most commonly used artifact correction method is rejection, based on discarding portions of MEG that coincide with those artifacts. Other methods tend to restrict the subject from producing the artifacts (e.g. by asking the subject to fix the eyes on a target to avoid eye-related artifacts, or to relax to avoid muscular artifacts). The effectiveness of those methods can be questionable in studies of neurological patients, or other non-co-operative subjects. In eye artifact canceling, other methods are available and have recently been reviewed by Vigario (I 997b) whose method is close to the one presented here, and in Jung et aI. (1998). This paper introduces a new method to separate brain activity from artifacts, based on the assumption that the brain activity and the artifacts are anatomically and physiologically separate processes, and that their independence is reflected in the statistical relation between the magnetic signals generated by those processes. The remaining of the paper will include an introduction to the independent component analysis, with a presentation of the algorithm employed and some justification of this approach. Experimental data are used to illustrate the feasibility of the technique, followed by a discussion on the results. 2 Independent Component Analysis Independent component analysis is a useful extension of the principal component analysis (PC A). It has been developed some years ago in context with blind source separation applications (Jutten and Herault 1991; Comon 1994). In PCA. the eigenvectors of the signal covariance matrix C = E{xx T } give the directions oflargest variance on the input data x. The principal components found by projecting x onto those perpendicular basis vectors are uncorrelated, and their directions orthogonal. However, standard PCA is not suited for dealing with non-Gaussian data. Several authors, from the signal processing to the artificial neural network communities, have shown that information obtained from a second-order method such as PCA is not enough and higher-order statistics are needed when dealing with the more demanding restriction of independence (Jutten and Herault 1991; Comon 1994). A good tutorial on neural ICA implementations is available by Karhunen et al. (1997). The particular algorithm used in this study was presented and derived by Hyvarinen and Oja (1997a. 1997b). 2.1 The model In blind source separation, the original independent sources are assumed to be unknown, and we only have access to their weighted sum. In this model, the signals recorded in an MEG study are noted as xk(i) (i ranging from 1 to L, the number of sensors used, and k denoting discrete time); see Fig. 1. Each xk(i) is expressed as the weighted sum of M ICAfor Identification of Artifacts in MEG Recordings 231 independent signals Sk(j), following the vector expression: M Xk = La(j)sdj) = ASk, (1) j=l where Xk = [xk(1), ... , xk(L)]T is an L-dimensional data vector, made up of the L mixtures at discrete time k. The sk(1), ... , sk(M) are the M zero mean independent source signals, and A = [a(1), . .. , a(M)] is a mixing matrix independent of time whose elements ail are th.e unknown coefficients of the mixtures. In order to perform ICA, it is necessary to have at least as many mixtures as there are independent sources (L ~ M). When this relation is not fully guaranteed, and the dimensionality of the problem is high enough, we should expect the first independent components to present clearly the most strongly independent signals, while the last components still consist of mixtures of the remaining signals. In our study, we did expect that the artifacts, being clearly independent from the brain activity, should come out in the first independent components. The remaining of the brain activity (e.g. a and J-L rhythms) may need some further processing. The mixing matrix A is a function of the geometry of the sources and the electrical conductivities of the brain, cerebrospinal fluid, skull and scalp. Although this matrix is unknown. we assume it to be constant, or slowly changing (to preserve some local constancy). The problem is now to estimate the independent signals Sk (j) from their mixtures, or the equivalent problem of finding the separating matrix B that satisfies (see Eq. 1) (2) In our algorithm, the solution uses the statistical definition of fourth-order cumulant or kurtosis that, for the ith source signal, is defined as kurt(s(i)) = E{s(i)4} - 3[E{s(i)2}]2, where E( s) denotes the mathematical expectation of s. 2.2 The algorithm The initial step in source separation, using the method described in this article, is whitening, or sphering. This projection of the data is used to achieve the uncorrelation between the solutions found, which is a prerequisite of statistical independence (Hyvarinen and Oja 1997a). The whitening can as well be seen to ease the separation of the independent signals (Karhunen et al. 1997). It may be accomplished by PCA projection: v = V x, with E{ vv T } = I. The whitening matrix V is given by -=T , V -- A- 1 / 2 ..... where A = diag[-\(1), ... , -\(M)] is a diagonal matrix with the eigenvalues of the data covariance matrix E{xxT}, and 8 a matrix with the corresponding eigenvectors as its columns. Consider a linear combination y = w T v of a sphered data vector v, with Ilwll = 1. Then E{y2} = .1 andkurt(y) = E{y4}-3, whose gradientwithrespecttow is 4E{v(wTv)3} . Based on this, Hyvarinen and Oja (1997a) introduced a simple and efficient fixed-point algorithm for computing ICA, calculated over sphered zero-mean vectors v, that is able to find one of the rows of the separating matrix B (noted w) and so identify one independent source at a time - the corresponding independent source can then be found using Eq. 2. This algorithm, a gradient descent over the kurtosis, is defined for a particular k as 1. Take a random initial vector Wo of unit norm. Let l = 1. 232 R. Vigario, v. Jousmiiki, M. Hiimiiliiinen, R. Hari and E. Oja 2. Let Wi = E{V(W[.1 v)3} - 3Wl-I. The expectation can be estimated using a large sample OfVk vectors (say, 1,000 vectors). 3. Divide Wi by its norm (e.g. the Euclidean norm 4. Ilwll = JLi wI J. lflwT wi-II is not close enough to 1, let I = 1+1 andgo back to step 2. Otherwise, output the vector Wi. In order to estimate more than one solution, and up to a maximum of lvI, the algorithm may be run as many times as required. It is, nevertheless, necessary to remove the infonnation contained in the solutions already found, to estimate each time a different independent component. This can be achieved, after the fourth step of the algorithm, by simply subtracting the estimated solution s = w T v from the unsphered data Xk . As the solution is defined up to a multiplying constant, the subtracted vector must be multiplied by a vector containing the regression coefficients over each vector component of Xk. 3 Methods The MEG signals were recorded in a magnetically shielded room with a 122-channel whole-scalp Neuromag-122 neuromagnetometer. This device collects data at 61 locations over the scalp, using orthogonal double-loop pick-up coils that couple strongly to a local source just underneath, thus making the measurement "near-sighted" (HamaHi.inen et al. 1993). One of the authors served as the subject and was seated under the magnetometer. He kept his head immobile during the measurement. He was asked to blink and make horizontal saccades, in order to produce typical ocular artifacts. Moreover, to produce myographic artifacts, the subject was asked to bite his teeth for as long as 20 seconds. Yet another artifact was created by placing a digital watch one meter away from the helmet into the shieded room. Finally, to produce breathing artifacts, a piece of metal was placed next to the navel. Vertical and horizontal electro-oculograms (VEOG and HEOG) and electrocardiogram (ECG) between both wrists were recorded simultaneously with the MEG, in order to guide and ease the identification of the independent components. The bandpassfiltered MEG (0.03-90 Hz), VEOG, HEOG, and ECG (0.1-100 Hz) signals were digitized at 297 Hz, and further digitally low-pass filtered, with a cutoff frequency of 45 Hz and downsampled by a factor of 2. The total length of the recording was 2 minutes. A second set of recordings was perfonned, to assess the reproducibility of the results. Figure 1 presents a subset of 12 spontaneous MEG signals from the frontal, temporal and occipital areas. Due to the dimension of the data (122 magnetic signals were recorded), it is impractical to plot all MEG signals (the complete set is available on the internet - see reference list for the adress (Vigario 1997a?. Also both EOG channels and the electrocardiogram are presented. 4 Results Figure 2 shows sections of9 independent components (IC's) found from the recorded data, corresponding to a I min period, starting 1 min after the beginning of the measurements. The first two IC's, with a broad band spectrum, are clearly due to the musclular activity originated from the biting. Their separation into two components seems to correspond, on the basis of the field patterns, to two different sets of muscles that were activated during the process. IC3 and IC5 are, respectively showing the horizontal eye movements and the eye blinks, respectively. IC4 represents cardiac artifact that is very clearly extracted. In agreement with Jousmaki and Hari (1996), the magnetic field pattern of IC4 shows some predominance on the left. ICA/or Identification 0/ Artifacts in MEG Recordings 233 MEG [ 1000 fTlcm I-- ---l saccades I-- ---l blinking EOG [ 500 IlV ECG [ 500 IlV I-- biting ---l MEG ~=::::::::::::::=:: ~?'104~ rJ. ......... M ,J.\.......1iIIiM~.. :: t... :;::::::;:::~= ~::::::::;::= ~ ?? ",~Jrt ..,. t .... ~,.~ . ? .J.. . .../\""$""~I 2 t :; :; 4 ~ 5 t ., ... ...., ,'fIJ'\, ..........-. ,..,d ,LIlt ... ., I? .,............. ................. " ....,..,."........ . .... Dei ..... " .'''IIb'''*. rt -1I\JY. ? --- I p", . . . . , . . . . . . . . . . . . at ...'.... I; rp .. ,P.... , .,...............' tMn':M.U ... , , ..... ' U\..,.--II..------'-__ ooII..Jl,,- ".'tIItS 5 ~ 6 t VEOG It ... 11.1. HEOG ~UijuJJJ.LU Wl Uij.lJU.LllU.UUUllUUij,UU~ijJJJ ECG 10 s Figure 1: Samples of MEG signals, showing artifacts produced by blinking, saccades, biting and cardiac cycle. For each of the 6 positions shown, the two orthogonal directions of the sensors are plotted. The breathing artifact was visible in several independent components, e.g. IC6 and IC7. It is possible that, in each breathing the relative position and orientation of the metallic piece with respect to the magnetometer has changed. Therefore, the breathing artifact would be associated with more than one column of the mixing matrix A, or to a time varying mixing vector. To make the analysis less sensible to the breathing artifact, and to find the remaining artifacts, the data were high-pass filtered, with cutoff frequency at 1 Hz. Next, the independent component IC8 was found. It shows clearly the artifact originated at the digital watch, located to the right side of the magnetometer. The last independent component shown, relating to the first minute of the measurement, shows an independent component that is related to a sensor presenting higher RMS (root mean squared) noise than the others. 5 Discussion The present paper introduces a new approach to artifact identification from MEG recordings, based on the statistical technique of Independent Component Analysis. Using this method, we were able to isolate both eye movement and eye blinking artifacts, as well as R. Vigario, 234 v. Jousmiiki, M HtJmlJliiinen, R. Hari and E. Oja cardiac, myographic, and respiratory artifacts. The basic asswnption made upon the data used in the study is that of independence between brain and artifact waveforms. In most cases this independence can be verified by the known differences in physiological origins of those signals. Nevertheless, in some eventrelated potential (ERP) studies (e.g. when using infrequent or painful stimuli), both the cerebral and ocular signals can be similarly time-locked to the stimulus. This local time dependence could in principle affect these particular ICA studies. However, as the independence between two signals is a measure of the similarity between their joint amplitude distribution and the product of each signal's distribution (calculated throughout the entire signal, and not only close to the stimulus applied), it can be expected that the very local relation between those two signals, during stimulation, will not affect their global statistical relation. 6 Acknowledgment Supported by a grant from Junta Nacional de Investiga~ao Cientifica e Tecnologica, under its 'Programa PRAXIS XXI' (R.Y.) and the Academy of Finland (R.H.). References Berg, P. and M. Scherg (1994). A multiple source approach to the correction of eye artifacts. Electroenceph. clin. Neurophysiol. 90, 229-241. Comon, P. (1994). Independent component analysis - a new concept? Signal Processing 36,287-314. Hamalainen, M., R. Hari, R. Ilmoniemi, 1. Knuutila, and O. Y. Lounasmaa (1993, April). Magnetoencephalography-theory, instrumentation, and applications to noninvasive studies of the working human brain. Reviews o/Modern Physics 65(2), 413-497. Hari, R. (1993). Magnetoencephalography as a tool of clinical neurophysiology. In E. Niedermeyer and F. L. da Silva (Eds.), Electroencephalography. Basic principles, clinical applications, and relatedjields, pp. 1035-1061 . Baltimore: Williams & Wilkins. Hyvarinen, A. and E. Oja (l997a). A fast fixed-point algorithm for independent component analysis. Neural Computation (9), 1483-1492. Hyvarinen, A. and E. Oja (1997b). One-unit learning rules for independent component analysis. In Neural Information Processing Systems 9 (Proc. NIPS '96). MIT Press. Jousmiiki, Y. and R. Hari (1996). Cardiac artifacts in magnetoencephalogram. Journal o/Clinical Neurophysiology 13(2), 172-176. Jung, T.-P., C. Hwnphries, T.-W. Lee, S. Makeig, M. J. McKeown, Y. lragui, and T. Sejnowski (1998). Extended ica removes artifacts from electroencephalographic recordings. In Neural Information Processing Systems 10 (Proc. NIPS '97). MIT Press. Jutten, C. and 1. Herault (1991). Blind separation of sources, part i: an adaptive algorithm based on neuromimetic architecture. Signal Processing 24, 1-10. Karhunen, J., E. Oja, L. Wang, R. Vigmo, and J. Joutsensalo (1997). A class of neural networks for independent component analysis. IEEE Trans. Neural Networks 8(3), 1-19. Vigmo, R. (1997a). WWW adress for the MEG data: http://nuc1eus.hut.firrvigarioINIPS97_data.html. Vigmo, R. (1997b). Extraction of ocular artifacts from eeg using independent component analysis. To appear in Electroenceph. c/in. Neurophysiol. ICAfor Identification ofArtifacts in MEG Recordings 235 ~~~ IC1 ------,--y~-------------------------.-.------~~.. ,.. ~ U ... IC2 IC3 .", ''' ... '' .. ' <> . ).\~ .\ C:> ? \ ~~~}a ~~-" ____I4-_. _ . . . ._.---_._. . . .-.__ . """"""?t;_-"'' '.... ~ . . . . .-......,..... ~_1 IC4 IC5 IC6 ~ ...W" .... "1011 ...~"_f.... ..".,.""'_ /tJ'IfII/'h I' ...... d1b .. ~*W,.'tJ ...... r' .. ns... IC7 ICB ICg ~._-~.,. . . . . .t . . Wt:n:ePWt.~..,.~I'NJ'~~ I 10 s I Figure 2: Nine independent components found from the MEG data. For each component the left, back and right views of the field patterns generated by these components are shown full line stands for magnetic flux coming out from the head, and dotted line the flux inwards. 

1466 |@word neurophysiology:2 norm:3 seems:1 riitta:1 covariance:2 pick:1 initial:2 denoting:1 kurt:1 yet:1 must:1 visible:1 shape:1 remove:2 plot:1 device:1 xk:8 beginning:1 ith:1 record:1 junta:1 filtered:2 location:1 mathematical:1 magnetoencephalographic:3 inside:1 expected:1 ica:7 brain:9 electroencephalography:2 xx:1 moreover:1 lowest:1 ail:1 developed:1 finding:1 impractical:1 nj:1 temporal:1 questionable:1 makeig:1 unit:3 conductivity:1 grant:1 appear:1 producing:1 local:4 studied:1 ecg:4 scherg:2 collect:1 co:1 ease:2 perpendicular:1 locked:1 acknowledgment:1 wrist:1 area:1 projection:2 downsampled:1 tmn:1 shielded:2 onto:1 close:3 periodical:1 context:2 restriction:1 equivalent:1 www:1 williams:1 occipital:1 starting:1 painful:1 rule:1 array:1 his:2 jli:1 justification:1 target:1 spontaneous:1 infrequent:1 distinguishing:1 us:1 origin:1 agreement:1 element:1 infrequently:1 iib:1 located:1 constancy:1 electrical:2 wang:1 cycle:1 movement:3 digitally:1 asked:2 neuromagnetic:1 upon:1 basis:2 neurophysiol:2 joint:1 xxt:1 fast:1 sejnowski:1 artificial:1 whose:3 say:1 relax:1 otherwise:1 statistic:1 eigenvalue:1 kurtosis:3 subtracting:1 product:1 coming:1 biting:4 loop:1 mixing:4 reproducibility:1 achieve:1 academy:1 double:1 produce:3 mckeown:1 unsphered:1 illustrate:1 eq:2 ic1:1 resemble:2 come:1 uu:1 direction:3 waveform:1 fij:1 human:1 fix:1 ao:1 of9:1 strictly:1 extension:1 correction:2 hut:5 ic:2 finland:3 ilv:2 proc:2 infonnation:1 helmet:1 predominance:1 wl:2 tool:2 weighted:2 neuromag:1 mit:2 clearly:6 sensor:4 gaussian:1 avoid:2 varying:1 eventrelated:1 derived:1 electroencephalographic:1 underneath:1 entire:1 relation:4 uij:1 orientation:1 html:1 herault:3 icb:1 magnetically:2 field:3 ilmoniemi:1 extraction:1 placing:1 broad:1 represents:1 others:1 stimulus:3 modern:1 pathological:1 oja:11 praxis:1 preserve:1 simultaneously:1 geometry:1 jrt:1 introduces:2 mixture:5 pc:1 activated:1 tj:2 necessary:2 orthogonal:3 divide:1 euclidean:1 plotted:1 increased:1 column:2 asking:1 subset:1 nacional:1 lee:1 physic:1 squared:1 recorded:5 containing:1 slowly:1 ricardo:2 potential:1 de:1 electrocardiogram:2 coefficient:2 blind:3 piece:2 root:1 view:1 lab:2 portion:1 wave:1 capability:1 ass:2 variance:1 correspond:1 identify:2 blink:3 identification:8 produced:3 lu:1 multiplying:1 served:1 ago:1 icg:1 canceling:1 ed:1 definition:1 frequency:2 ocular:3 pp:1 hwnphries:1 associated:1 couple:1 lounasmaa:1 ask:1 dimensionality:1 blinking:3 amplitude:3 back:2 higher:3 tension:1 reflected:1 april:1 box:2 strongly:2 just:1 working:1 horizontal:3 jutten:3 artifact:38 concept:1 y2:1 during:3 noted:2 rhythm:1 presenting:1 complete:1 demonstrate:1 temperature:1 silva:1 ranging:1 epwt:1 fi:2 recently:1 common:1 stimulation:1 immobile:1 cerebral:1 jl:1 belong:1 discussed:1 he:2 relating:1 measurement:4 ai:1 bandpassfiltered:1 similarly:1 access:1 similarity:1 whitening:3 l997a:1 lju:1 instrumentation:1 accomplished:1 muscle:1 captured:1 seen:1 employed:1 period:1 signal:30 ii:2 multiple:1 full:1 rj:1 clinical:4 long:1 magnetometer:4 sdj:1 jy:1 controlled:1 feasibility:1 neuro:1 regression:1 basic:2 patient:1 expectation:2 vigario:7 achieved:1 xxi:1 operative:1 baltimore:1 source:13 recording:9 isolate:2 tend:1 subject:5 electro:1 hz:5 regularly:1 effectiveness:2 extracting:1 near:1 presence:2 enough:3 independence:6 affect:2 architecture:1 restrict:1 expression:1 pca:4 epileptic:1 rms:1 wo:1 nine:1 useful:1 eigenvectors:2 band:1 category:1 simplest:1 http:1 tutorial:1 dotted:1 estimated:2 discrete:2 tecnologica:1 nevertheless:2 changing:1 erp:1 cutoff:2 verified:1 kept:1 year:1 sum:2 run:1 fourth:2 throughout:1 separation:6 lilt:1 internet:1 followed:1 guaranteed:1 activity:6 scalp:4 i4:1 helsinki:2 erkki:2 min:2 sphering:1 according:1 combination:1 cardiac:4 wi:5 skull:1 making:1 cerebrospinal:1 comon:3 anatomically:1 projecting:1 heart:1 eventually:1 needed:1 irregularly:1 neuromimetic:1 available:3 prerequisite:1 multiplied:1 away:1 magnetic:4 subtracted:1 rp:1 original:1 denotes:1 remaining:4 include:1 clin:1 sighted:1 already:1 spike:1 saccadic:1 rt:1 dependence:1 diagonal:1 hamalainen:1 gradient:1 separate:3 separating:2 sensible:1 meg:19 length:1 y4:1 msh:1 inwards:1 info:1 fluid:1 implementation:1 ic6:2 unknown:3 perform:1 vertical:1 fin:2 descent:1 extended:1 head:2 digitized:1 jousmaki:1 community:1 introduced:1 required:1 nip:2 ic4:3 trans:1 able:2 pattern:3 breathing:5 bite:1 perfonned:1 demanding:1 disturbance:1 technology:2 dei:1 eye:9 created:1 eog:2 review:1 removal:2 meter:1 relative:1 fully:1 expect:2 digital:3 teeth:1 metal:1 article:1 principle:2 uncorrelated:1 seated:1 row:1 changed:1 jung:2 placed:1 last:3 supported:1 guide:1 side:1 vv:1 face:1 calculated:2 dimension:1 noninvasive:1 stand:1 electroenceph:2 author:3 commonly:1 made:2 coincide:1 adaptive:1 hyvarinen:5 flux:2 dealing:2 global:1 hari:10 assumed:1 inen:1 navel:1 spectrum:1 physiologically:1 sk:4 reviewed:1 channel:2 matti:1 eeg:2 metallic:1 joutsensalo:1 ic2:1 diag:1 lvi:1 did:1 da:1 whole:2 noise:1 adress:2 respiratory:1 fig:1 slow:1 n:1 position:2 originated:2 minute:2 discarding:1 showing:2 list:1 physiological:1 essential:1 consist:1 ilwll:2 knuutila:1 occurring:1 karhunen:3 rejection:1 suited:1 simply:1 expressed:1 contained:1 neurological:1 watch:3 saccade:3 veikko:2 satisfies:1 extracted:1 coil:1 presentation:1 magnetoencephalography:2 eventual:1 room:3 included:1 muscular:2 typical:1 wt:1 principal:2 total:1 pas:2 experimental:1 la:1 wilkins:1 berg:2 sphered:2 cumulant:1 frontal:1 investigator:1

A Revolution: Belief Propagation Graphs With Cycles ? In Brendan J. Frey? http://wvw.cs.utoronto.ca/-frey Department of Computer Science University of Toronto David J. C. MacKay http://vol.ra.phy.cam.ac.uk/mackay Department of Physics, Cavendish Laboratory Cambridge University Abstract Until recently, artificial intelligence researchers have frowned upon the application of probability propagation in Bayesian belief networks that have cycles. The probability propagation algorithm is only exact in networks that are cycle-free. However, it has recently been discovered that the two best error-correcting decoding algorithms are actually performing probability propagation in belief networks with cycles. 1 Communicating over a noisy channel Our increasingly wired world demands efficient methods for communicating bits of information over physical channels that introduce errors. Examples of real-world channels include twisted-pair telephone wires, shielded cable-TV wire, fiber-optic cable, deep-space radio, terrestrial radio, and indoor radio. Engineers attempt to correct the errors introduced by the noise in these channels through the use of channel coding which adds protection to the information source, so that some channel errors can be corrected. A popular model of a physical channel is shown in Fig. 1. A vector of K information bits u = (Ut, ... ,UK), Uk E {O, I} is encoded, and a vector of N codeword bits x = (Xl! ... ,XN) is transmitted into the channel. Independent Gaussian noise with variance (12 is then added to each codeword bit, .. Brendan Frey is currently a Beckman Fellow at the Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign. B. J Frey and D. J. C. MacKay 480 Gaussian noise with variance (J2 --U--'~~I~__ E __ nc_o_d_e_r__ ~1 ~ .I~___ x y D_e_c_od_e_r__ ~--U~.~ Figure 1: A communication system with a channel that adds Gaussian noise to the transmitted discrete-time sequence. producing the real-valued channel output vector y = (Y!, ... ,YN). The decoder must then use this received vector to make a guess U at the original information vector. The probability P" (e) of bit error is minimized by choosing the Uk that maximizes P(ukly) for k = 1, ... , K. The rate K/N of a code is the number of information bits communicated per codeword bit. We will consider rate ~ 1/2 systems in this paper, where N == 2K. The simplest rate 1/2 encoder duplicates each information hit: X2k-l = X2k = Uk, k = 1, ... , K. The optimal decoder for this repetition code simply averages together pairs of noisy channel outputs and then applies a threshold: Uk = 1 if (Y2k-l + Y2k)/2 > 0.5, 0 otherwise. (1) Clearly, this procedure has the effect of reducing the noise variance by a factor of 1/2. The resulting probability p,,(e) that an information bit will be erroneously decoded is given by the area under the tail of the noise Gaussian: p,,(e) -0.5) = 4> ( (J2/2 ' (2) where 4>0 is the cumulative standard normal distribution. A plot of p,,(e) versus (J for this repetition code is shown in Fig. 2, along with a thumbnail picture that shows the distribution of noisy received signals at the noise level where the repetition code gives p,,(e) == 10- 5 ? More sophisticated channel encoders and decoders can be used to increase the tolerable noise level without increasing the probability of a bit error. This approach can in principle improve performance up to a bound determined by Shannon (1948). For a given probability of bit error P,,(e), this limit gives the maximum noise level that can be tolerated, no matter what channel code is used. Shannon's proof was nonconstructive, meaning that he showed that there exist channel codes that achieve his limit, but did not present practical encoders and decoders. The curve for Shannon's limit is also shown in Fig. 2. The two curves described above define the region of interest for practical channel coding systems. For a given P,,(e), if a system requires a lower noise level than the repetition code, then it is not very interesting. At the other extreme, it is impossible for a system to tolerate a higher noise level than Shannon's limit. 2 Decoding Hamming codes by probability propagation One way to detect errors in a string of bits is to add a parity-check bit that is chosen so that the sum modulo 2 of all the bits is O. If the channel flips one bit, the receiver will find that the sum modulo 2 is 1, and can detect than an error occurred. In a simple Hamming code, the codeword x consists of the original vector A Revolution: Belief Propagation in Graphs with Cycles 481 le-l Shannon limit Concatenated Code le-5 '-----I~_----'c..........L..L_ 0.2 _ _ _ _....L.__ _ _..L____L_......u._~LI__...J.........J 0.4 0.5 0.6 Standard deviation of Gaussian noise, U 0.3 0.7 0.8 Figure 2: Probability of bit error Ph (e) versus noise level u for several codes with rates near 1/2, using 0/1 signalling. It is impossible to obtain a Ph(e) below Shannon's limit (shown on the far right for rate 1/2). "H-PP" = Hamming code (rate 4/7) decoded by probability propagation (5 iterations); "H-Exact" = Hamming code decoded exactly; "LDPCC-PP" = low-density parity-check coded decoded by probability propagation; "TC-PP" = turbo code decoded by probability propagation. The thumbnail pictures show the distribution of noisy received signals at the noise levels where the repetition code and the Shannon limit give Ph (e) = 10- 5 . u in addition to several parity-check bits, each of which depends on a different subset of the information bits. In this way, the Hamming code can not only detect errors but also correct them. The code can be cast in the form of the conditional probabilities that specify a Bayesian network. The Bayesian network for a K = 4, N = 7 rate 4/7 Hamming code is shown in Fig. 3a. Assuming the information bits are uniformly random, we have P(Uk) = 0.5, Uk E {0,1}, k = 1,2,3,4. Codeword bits 1 to 4 are direct copies of the information bits: P(xkluk) = 6(Xk,Uk), k = 1,2,3,4, where 6(a, b) = 1 if a = b and 0 otherwise. Codeword bits 5 to 7 are parity-check bits: P(XSIUI,U2,U3) = 6(X5,Ul EB U2 EB U3), P(XaIU.,U2,U4) = 6(Xa,Ul EB U2 EB U4), P(x7Iu2,U3,U4) = 6(X7,u2EBU3EBu4), where EB indicates addition modulo 2 (XOR). Finally, the conditional channel probability densities are (3) for n = 1, ... , 7. The probabilities P(ukly) can be computed exactly in this belief network, using Lauritzen and Spiegelhalter's algorithm (1988) or just brute force computation. However, for the more powerful codes discussed below, exact computations are intractable. Instead, one way the decoder can approximate the probabilities P( Uk Iy) is by applying the probability propagation algorithm (Pearl 1988) to the Bayesian network. Probability propagation is only approximate in this case because the B. 1. Frey and D. J. C. MacKay 482 (a) (b) (d) (c) Figure 3: (a) The Bayesian network for a K = 4, N = 7 Hamming code. (b) The Bayesian network for a K = 4, N = 8 low-density parity-check code. (c) A block diagram for the turbocode linear feedback shift register. (d) The Bayesian network for a K = 6, N = 12 turbocode. network contains cycles (ignoring edge directions), e.g., UI-XS-U2-X6-UI. Once a channel output vector y is observed, propagation begins by sending a message from Yn to Xn for n = 1, ... ,7. Then, a message is sent from Xk to Uk for k = 1,2,3,4. An iteration now begins by sending messages from the information variables 'Ill, U2, U3, U4 to the parity-check variables Xs, X6, X7 in parallel. The iteration finishes by sending messages from the parity-check variables back to the information variables in parallel. Each time an iteration is completed, new estimates of P( Uk Iy) for k = 1,2,3,4 are obtained. The Pb (e) - (j curve for optimal decoding and the curve for the probability propagation decoder (5 iterations) are shown in Fig. 2. Quite surprisingly, the performance of the iterative decoder is quite close to that of the optimal decoder. Our expectation was that short cycles would confound the probability propagation decoder. However, it seems that good performance can be obtained even when there are short cycles in the code network. For this simple Hamming code, the complexities of the probability propagation decoder and the exact decoder are comparable. However, the similarity in performance between these two decoders prompts the question: "Can probability propagation decoders give performances comparable to exact decoding in cases where exact decoding is computationally intractable?" A Revolution: Belief Propagation in Graphs with Cycles 3 483 A leap towards the limit: Low-density parity-check codes Recently, there has been an explosion of interest in the channel coding community in two new coding systems that have brought us a leap closer to Shannon's limit. Both of these codes can be described by Bayesian networks with cycles, and it turns out that the corresponding iterative decoders are performing probability propagation in these networks. Fig. 3b shows the Bayesian network for a simple low-density parity-check code (Gallager 1963). In this network, the information bits are not represented explicitly. Instead, the network defines a set of allowed configurations for the codewords. Each parity-check vertex qi requires that the codeword bits {Xn}nEQ; to which qi is connected have even parity: P(qil{xn}nEQ;) = 8(qi' EB xn), (4) nEQi where q is clamped to 0 to ensure even parity. Here, Qi is the set of indices of the codeword bits to which parity-check vertex qi is connected. The conditional probability densities for the channel outputs are the same as in Eq. 3. One way to view the above code is as N binary codeword variables along with a set of linear (modulo 2) equations. If in the end we want there to be K degrees of freedom, then the number of linearly independent parity-check equations should be N - K. In the above example, there are N = 8 codeword bits and 4 paritychecks, leaving K = 8 - 4 = 4 degrees of freedom. It is these degrees of freedom that we use to represent the information vector u. Because the code is linear, a K -dimensional vector u can be mapped to a valid x simply by multiplying by an N x K matrix (using modulo 2 addition). This is how an encoder can produce a low-density parity-check codeword for an input vector. Once a channel output vector y is observed, the iterative probability propagation decoder begins by sending messages from y to x. An iteration now begins by sending messages from the codeword variables x to the parity-check constraint variables q. The iteration finishes by sending messages from the parity-check constraint variables back to the codeword variables. Each time an iteration is completed, new estimates of P(xnly) for n = 1, . . . , N are obtained. After a valid (but not necessarily correct) codeword has been found, or a prespecified limit on the number of iterations has been reached, decoding stops. The estimate of the codeword is then mapped back to an estimate ii of the information vector. Fig. 2 shows the performance of a K = 32,621, N = 65,389 low-density paritycheck code when decoded as described above. (See MacKay and Neal (1996) for details.) It is impressively close to Shannon's limit - significantly closer than the "Concatenated Code" (described in Lin and Costello (1983? which was considered the best practical code until recently. 4 Another leap: Turbocodes The codeword for a turbocode (Berrou et al. 1996) consists of the original information vector, plus two sets of bits used to protect the information. Each of these two sets is produced by feeding the information bits into a linear feedback shift register (LFSR), which is a type of finite state machine. The two sets differ in that one set is produced by a permuted set of information bits; i.e., the order of the bits is scrambled in a fixed way before the bits are fed into the LFSR. Fig. 3c shows a block diagram (not a Bayesian network) for the LFSR that was used in our experiments. B. 1. Frey and D. J. C. MacKay 484 Each box represents a delay (memory) element, and each circle performs addition modulo 2. When the kth information bit arrives, the machine has a state Sk which can be written as a binary string of state bits b4b3b2blbo as shown in the figure. bo of the state Sk is determined by the current input Uk and the previous state Sk-l' Bits b1 to b4 are just shifted versions of the bits in the previous state. Fig. 3d shows the Bayesian network for a simple turbocode. Notice that each state variable in the two constituent chains depends on the previous state and an information bit. In each chain, every second LFSR output is not transmitted. In this way, the overall rate of the code is 1/2, since there are K = 6 information bits and N = 6 + 3 + 3 = 12 codeword bits. The conditional probabilities for the states of the non permuted chain are P(sllsl-I' Uk) =1 if state sl follows Sk-l for input Uk, 0 otherwise. (5) The conditional probabilities for the states in the other chain are similar, except that the inputs are permuted. The probabilities for the information bits are uniform, and the conditional probability densities for the channel outputs are the same as in Eq.3. Decoding proceeds with messages being passed from the channel output variables to the constituent chains and the information bits. Next, messages are passed from the information variables to the first constituent chain, SI. Messages are passed forward and then backward along this chain, in the manner of the forwardbackward algorithm (Smyth et al. 1997). After messages are passed from the first chain to the second chain s2, the second chain is processed using the forwardbackward algorithm. To complete the iteration, messages are passed from S2 to the information bits. Fig. 2 shows the performance of a K = 65,536, N = 131,072 turbocode when decoded as described above, using a fixed number (18) of iterations. (See Frey (1998) for details.) Its performance is significantly closer to Shannon's limit than the performances of both the low-density parity-check code and the textbook standard "Concatenated Code" . 5 Open questions We are certainly not claiming that the NP-hard problem (Cooper 1990) of probabilistic inference in general Bayesian networks can be solved in polynomial time by probability propagation. However, the results presented in this paper do show that there are practical problems which can be solved using approximate inference in graphs with cycles. Iterative decoding algorithms are using probability propagation in graphs with cycles, and it is still not well understood why these decoders work so well. Compared to other approximate inference techniques such as variational methods, probability propagation in graphs with cycles is unprincipled. How well do more principled decoders work? In (MacKay and Neal 1995), a variational decoder that maximized a lower bound on n~=1 P(ukly) was presented for low-density parity-check codes. However, it was found that the performance of the variational decoder was not as good as the performance of the probability propagation decoder. It is not difficult to design small Bayesian networks with cycles for which probability propagation is unstable. Is there a way to easily distinguish between those graphs for which propagation will work and those graphs for which propagation is unstable? A belief that is not uncommon in the graphical models community is that short cycles are particularly apt to lead probability propagation astray. Although it is possible to design networks where this is so, there seems to be a variety of interesting networks A Revolution: Belief Propagation in Graphs with Cycles 485 (such as the Hamming code network described above) for which propagation works well, despite short cycles. The probability distributions that we deal with in decoding are very special distributions: the true posterior probability mass is actually concentrated in one microstate in a space of size 2M where M is large (e.g., 10,000). The decoding problem is to find this most probable microstate, and it may be that iterative probability propagation decoders work because the true probability distribution is concentrated in this microstate. We believe that there are many interesting and contentious issues in this area that remain to be resolved. Acknowledgements We thank Frank Kschischang, Bob McEliece, and Radford Neal for discussions related to this work, and Zoubin Ghahramani for comments on a draft of this paper. This research was supported in part by grants from the Gatsby foundation, the Information Technology Research Council, and the Natural Sciences and Engineering Research Council. References C. Berrou and A. Glavieux 1996. Near optimum error correcting coding and decoding: Turbo-codes. IEEE 7hmsactions on Communications 44, 1261-1271. G. F. Cooper 1990. The computational complexity of probabilistic inference using Bayesian belief networks. Artificial Intelligence 42, 393-405. B. J. Frey 1998. Graphical Models for Machine Learning and Digital Communication, MIT Press, Cambridge, MA. See http://vwv . cs. utoronto. cal -frey. R. G. Gallager 1963. Low-Density Parity-Check Codes, MIT Press, Cambridge, MA. S. Lin and D. J. Costello, Jr. 1983. Error Control Coding: Fundamentals and Applications, Prentice-Hall Inc., Englewood Cliffs, NJ. S. L. Lauritzen and D. J. Spiegelhalter 1988. Local computations with probabilities on graphical structures and their application to expert systems. Journal of the Royal Statistical Society B 50, 157-224. D. J. C. MacKay and R. M. Neal 1995. Good codes based on very sparse matrices. In Cryptography and Coding. 5th IMA Conference, number 1025 in Lecture Notes in Computer Science, 100-111, Springer, Berlin Germany. D. J. C. MacKay and R. M. Neal 1996. Near Shannon limit performance of low density parity check codes. Electronics Letters 32, 1645-1646. Due to editing errors, reprinted in Electronics Letters 33, 457-458. J. Pearl 1988. Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, San Mateo, CA. C. E. Shannon 1948. A mathematical theory of communication. Bell System Technical Journal 27, 379-423, 623-656. P. Smyth, D. Heckerman, and M. I. Jordan 1997. Probabilistic independence networks for hidden Markov probability models. Neural Computation 9, 227-270. 

1467 |@word version:1 polynomial:1 seems:2 open:1 electronics:2 configuration:1 contains:1 phy:1 current:1 protection:1 si:1 must:1 written:1 neq:2 plot:1 intelligence:2 guess:1 signalling:1 xk:2 short:4 prespecified:1 draft:1 toronto:1 mathematical:1 along:3 direct:1 consists:2 vwv:1 manner:1 introduce:1 ra:1 increasing:1 begin:4 twisted:1 maximizes:1 mass:1 what:1 string:2 textbook:1 lfsr:4 nj:1 fellow:1 every:1 exactly:2 hit:1 uk:15 brute:1 control:1 grant:1 yn:2 producing:1 before:1 understood:1 frey:9 engineering:1 local:1 limit:13 despite:1 cliff:1 plus:1 eb:6 mateo:1 practical:4 block:2 communicated:1 procedure:1 area:2 bell:1 significantly:2 zoubin:1 shielded:1 close:2 cal:1 prentice:1 impossible:2 applying:1 correcting:2 communicating:2 his:1 cavendish:1 modulo:6 smyth:2 exact:6 element:1 particularly:1 u4:4 qil:1 observed:2 solved:2 region:1 cycle:17 connected:2 forwardbackward:2 principled:1 ui:2 complexity:2 turbocodes:1 cam:1 upon:1 easily:1 resolved:1 represented:1 fiber:1 artificial:2 choosing:1 quite:2 encoded:1 valued:1 otherwise:3 encoder:2 noisy:4 sequence:1 j2:2 achieve:1 wvw:1 constituent:3 optimum:1 wired:1 produce:1 ac:1 lauritzen:2 received:3 eq:2 c:2 differ:1 direction:1 correct:3 feeding:1 probable:1 considered:1 hall:1 normal:1 u3:4 beckman:2 radio:3 currently:1 leap:3 council:2 repetition:5 brought:1 clearly:1 mit:2 gaussian:5 check:19 indicates:1 brendan:2 detect:3 inference:4 hidden:1 germany:1 overall:1 issue:1 ill:1 special:1 mackay:9 once:2 represents:1 minimized:1 np:1 intelligent:1 duplicate:1 nonconstructive:1 ima:1 attempt:1 freedom:3 interest:2 message:12 englewood:1 certainly:1 uncommon:1 extreme:1 arrives:1 chain:10 edge:1 closer:3 explosion:1 circle:1 y2k:2 deviation:1 subset:1 vertex:2 uniform:1 delay:1 encoders:2 tolerated:1 density:13 fundamental:1 probabilistic:4 physic:1 decoding:11 together:1 iy:2 expert:1 coding:7 matter:1 inc:1 register:2 explicitly:1 depends:2 view:1 reached:1 parallel:2 xor:1 variance:3 kaufmann:1 maximized:1 bayesian:14 produced:2 multiplying:1 researcher:1 bob:1 pp:3 proof:1 hamming:9 stop:1 popular:1 ut:1 sophisticated:1 actually:2 back:3 tolerate:1 higher:1 x6:2 specify:1 editing:1 box:1 xa:1 just:2 until:2 mceliece:1 propagation:30 defines:1 believe:1 effect:1 true:2 laboratory:1 neal:5 deal:1 x5:1 complete:1 performs:1 reasoning:1 meaning:1 variational:3 recently:4 permuted:3 physical:2 b4:1 tail:1 he:1 occurred:1 discussed:1 cambridge:3 illinois:1 similarity:1 add:3 posterior:1 showed:1 codeword:17 binary:2 transmitted:3 morgan:1 berrou:2 signal:2 ii:1 champaign:1 technical:1 lin:2 coded:1 qi:5 expectation:1 iteration:11 represent:1 addition:4 want:1 diagram:2 source:1 leaving:1 comment:1 sent:1 jordan:1 near:3 variety:1 independence:1 finish:2 reprinted:1 shift:2 passed:5 ul:2 deep:1 ph:3 concentrated:2 processed:1 simplest:1 http:3 sl:1 exist:1 shifted:1 notice:1 thumbnail:2 per:1 discrete:1 vol:1 threshold:1 pb:1 contentious:1 backward:1 graph:9 sum:2 letter:2 powerful:1 comparable:2 bit:41 x2k:2 bound:2 distinguish:1 turbo:2 optic:1 constraint:2 erroneously:1 x7:2 performing:2 department:2 tv:1 jr:1 remain:1 heckerman:1 increasingly:1 cable:2 costello:2 confound:1 computationally:1 equation:2 turn:1 flip:1 fed:1 end:1 sending:6 tolerable:1 apt:1 original:3 include:1 ensure:1 completed:2 graphical:3 concatenated:3 ghahramani:1 society:1 added:1 question:2 codewords:1 kth:1 thank:1 mapped:2 berlin:1 decoder:21 astray:1 terrestrial:1 unstable:2 assuming:1 code:40 index:1 difficult:1 frank:1 claiming:1 design:2 wire:2 markov:1 urbana:1 finite:1 communication:4 discovered:1 community:2 prompt:1 david:1 introduced:1 pair:2 cast:1 protect:1 pearl:2 proceeds:1 below:2 indoor:1 royal:1 memory:1 belief:9 natural:1 force:1 advanced:1 improve:1 technology:2 spiegelhalter:2 picture:2 acknowledgement:1 lecture:1 interesting:3 impressively:1 versus:2 digital:1 foundation:1 degree:3 principle:1 surprisingly:1 parity:21 free:1 copy:1 supported:1 l_:1 institute:1 sparse:1 curve:4 feedback:2 xn:5 world:2 cumulative:1 valid:2 forward:1 san:1 far:1 unprincipled:1 approximate:4 receiver:1 b1:1 scrambled:1 iterative:5 sk:4 why:1 channel:23 ca:2 kschischang:1 ignoring:1 necessarily:1 did:1 linearly:1 s2:2 noise:14 allowed:1 cryptography:1 fig:10 gatsby:1 cooper:2 decoded:7 xl:1 clamped:1 utoronto:2 revolution:4 x:2 intractable:2 demand:1 tc:1 simply:2 gallager:2 bo:1 u2:6 applies:1 radford:1 springer:1 ma:2 conditional:6 glavieux:1 towards:1 hard:1 telephone:1 determined:2 corrected:1 reducing:1 uniformly:1 except:1 engineer:1 shannon:12

Learning Human-like Knowledge by Singular Value Decomposition: A Progress Report Thomas K. Landauer Darrell Laham Department of Psychology & Institute of Cognitive Science University of Colorado at Boulder Boulder, CO 80309-0345 {landauer, dlaham}@psych.colorado.edu Peter Foltz Department of Psychology New Mexico State University Las Cruces, NM 88003-8001 pfoltz@crl.nmsu.edu Abstract Singular value decomposition (SVD) can be viewed as a method for unsupervised training of a network that associates two classes of events reciprocally by linear connections through a single hidden layer. SVD was used to learn and represent relations among very large numbers of words (20k-60k) and very large numbers of natural text passages (lk70k) in which they occurred. The result was 100-350 dimensional "semantic spaces" in which any trained or newly aibl word or passage could be represented as a vector, and similarities were measured by the cosine of the contained angle between vectors. Good accmacy in simulating human judgments and behaviors has been demonstrated by performance on multiple-choice vocabulary and domain knowledge tests, emulation of expert essay evaluations, and in several other ways. Examples are also given of how the kind of knowledge extracted by this method can be applied. 1 INTRODUCTION Traditionally, imbuing machines with human-like knowledge has relied primarily on explicit coding of symbolic facts into computer data structures and algorithms. A serious limitation of this approach is people's inability to access and express the vast reaches of unconscious knowledge on which they rely, knowledge based on masses of implicit inference and irreversibly melded data. A more important deficiency of this state of affairs is that by coding the knowledge ourselves, (as we also do when we assign subjectively hypothesized rather than objectively identified features to input or output nodes in a neural net) we beg important questions of how humans acquire and represent the cOOed knowledge in the fIrSt place. T. K. Landauer; D. Laham and P. Foltz 46 Thus. from both engineering and scientific perspectives. there are reasons to try to design learning machines that can ocquire human-like quantities of human-like knowledge from the same sources as humans. The success of such techniques would not prove that the same mechanisms are used by humans. but because we presently do not know how the problem can be solved in principle, successful simulation may offer theoretical insights as well as practical applications. In the work reported here we have found a way to induce significant amounts of knowledge about the meanings of passages and of their constituent vocabularies of words by training on large bodies of natural text. In general terms, the method simultaneously extracts the similarity between words (the likelihood of being used in passages that convey similar ideas) and the similarity of passages (the likelihood of containing words of similar meaning). The conjoint estimation of similarity is accomplished by a fundamentally simple representational technique that exploits mutual constraints implicit in the occurrences of very many words in very many contexts. We view the resultant system both as a means for automatically learning much of the semantic content of words and passages. and as a potential computational model for the process that underlies the corresponding human ability. While the method starts with data about the natural contextual co-occurrences of words. it uses them in a different manner than has previously been applied. A long -standing objection to co-occurrence statistics as a source of linguistic knowledge (Chomsky's 1957) is that many grammatically acceptable expressions. for example sentences with potentially unlimited embedding structures. cannot be produced by a finite Markov process whose elements are transition probabilities from word to word. If word-word probabilities are insufficient to generate language. then. it is argued, acquiring estimates of such probabilities cannot be a way that language can be learned. However, our approach to statistical knowledge learning differs from those considered in the past in two ways. First. the basic associational data from which knowledge is induced are not transition frequencies between successive individual words or phrases. but rather the frequencies with which particular words appear as components of relatively large natural passages, utterances of the kind that humans use to convey complete ideas. The result of this move is that the statistical regularities reflected are relations among unitary expressions of meaning. rather than syntactic constraints on word order that may serve additional purposes such as output and input processing efficiencies. error protection. or esthetic style. Second, the mutual constraints inherent in a multitude of such local ~ occurrence relations are jointly satisfied by being forced into a global representation of lower dimensionality. This constraint satisfaction. a form of induction. was accomplished by singular value decomposition. a linear factorization technique that produces a representational structure equivalent to a three layer neural network model with linear activation functions. 2 THE TEXT ANALYSIS MODEL AND METHOD The text analysis process that we have explored is called Latent Semantic Analysis (LSA) (Deerwester et al .? 1990; Landauer and Dumais. 1997). It comprises four steps: (1) A large body of text is represented as a matrix [ij], in which rows stand for individual word types. columns for meaning-bearing passages such as sentences or paragraphs. mel cells contain the frequency with which a word occurs in a passage. (2) Cell entries (freqi) are transformed to: log(freqi, + I) L(( fref/,] {freq. ]'l -1-, ~fr:~v *10 ~fre~v ) a measure of the first order association of a word and its context. Learning Human Knowledge by Singular Value Decomposition: A Progress Report 47 (3) The matrix is then subjected to singular value decomposition (Berry, 1992): [ij] = [ik] [kk] Uk]' in which [ik] and Uk] have orthonormal columns, [kk] is a diagonal matrix of singular values, and k <= max (ij). (4) Finally, all but the d largest singular values are set to zero. Pre-multiplication of the right-hand matrices produces a least-squares best approximation to the original matrix given the number of dimensions, d, (hidden units in a corresponding neural net model representation) that are retained. The SVD with dimension reduction constitutes a constraint-satisfaction induction process in that it predicts the original observations on the basis of linear relations among the abstracted representations of the data that it has retained. By hypothesis, the analysis induces human-like relationships among passages and words because humans also make inferences about semantic relationships from abstracted representations based on limited data, and do so by an analogous process. In the result, each word and passage is represented as a vector of length d. Performance depends strongly on the choice of number of dimensions. The optimal number is typically around 300. The similarity of any two words, any two text passages, or any word and any text passage, are computed by measures on their vectors. We have most often used the cosine (of the contained angle between the vectors in semantic d-space) which we interpret as the degree of qualitative similarity of meaning. The length of vectors is also useful and interpretable. 3 TESTS OF LSA'S PERFORMANCE LSA's ability to simulate human knowledge and meaning relations has been tested in a variety of ways. Here we describe two relatively direct sources of evidence and briefly list several others. 3.1 VOCABULARY & DOMAIN KNOWLEDGE TESTS In all cases, LSA was ftrst trained on a large text corpus intended to be representative of the text from which humans gain most of the semantic knowledge to be simulated. In a previously reported test (Landauer and Dumais, 1997), LSA was trained on approximately five million words of text sampled from a high-school level encyclopedia, then tested on multiple choice items from the Educational Testing Service Test of English as a Foreign Language (TOEFL). These test questions present a target word or short phrase and ask the student to choose the one of four alternative words or phrases that is most similar in meaning. LSA's answer was determined by computing the cosine between the derived vector for the target word or phrase and each of the alternatives and choosing the largest. LSA was correct on 64% of the 80 items, identical to the average of a large sample of students from non-English speaking countries who had applied for admission to U. S. colleges. When in error, LSA made choices positively correlated (product-moment r = .44) with those preferred by students. We have recently replicated this result with training on a similar sized sample from the Associated Press newswire In a new set of tests, LSA was trained on a popular introductory psychology textbook (Myers, 1995) and tested with the same four-alternative multiple choice tests used for students in two large classes. In these experiments, LSA's score was about 6O%-lower than the class averages but above passing level, and far above guessing probability. Its errors again resembled those of students; it got right about half as many of questions rated difftcult by the test constructors as ones rated easy (Landauer, Foltz and Laham, 1997). 3.2 ESSA Y TESTS 48 T. K. Landauer, D. Laham and P. Foltz Word-wad meaning similarities are a good test of knowledge-indeed, vocabulary tests are the best single measure of human intelligence. However, they are not sufficient to assess the correspondence of LSA and human knowledge because people usually express knowledge via larger verbal strings, such as sentences, paragraphs and articles. Thus, just as multiple choice tests of student knowledge are often supplemented by essay tests whose content is then judged by humans, we wished to evaluate the adequacy of LSA' s representation of knowledge in complete passages of text. We could not have LSA write essays because it has no means for producing sentences. However, we were able to assess the accum::y with which LSA could extract and represent the knowledge expressed in essays written by students by simulating judgments about their content that were made by human readers (Landauer, Laham, Rehder, & Schreiner, in press). In these tests, students were asked to write short essays to cover an assigned topic or to answer a posed question. In various experiments, the topics included anatomy axI function of the heart, phenomena from introductory psychology, the history of the Panama Canal, and tolerance of diversity in America. In each case, LSA was first trained either on a large sample of instructional text from the same domain or, in the latter case, on combined text from the very large number of essays themselves, to produce a highdimensional (100-350 dimensions in the various tests) semantic space. We then represented each essay simply as the vector average of the vectors for the words it contained. Two properties of these average vectors were then used to measure the quality and quantity of knowledge conveyed by an essay: (1) the similarity (measured as the cosine of the angle between vectors) of the student essay and one or more standard essays, and (2) the total amount of domain specific content, measured as the vector length. In each case, two human experts independently rated the overall quality of each essay on a five or ten point scale. The judges were either university course instructors or professional exam readers from Educational Testing Service. The LSA measures were calibrated with respect to the judges' rating scale in several different ways, but because they gave nearly the same results only one will be described here. In this method, each student essay was compared to a large (90-200) set of essays previously scored by experts, and the ten most similar (by cosine) identified. The target essay was then assigned a "quality" score component consisting of the cosine-weighted average of the ten. A second, "relevant quantity", score component was the vector length of the student essay. Finally, regression on expert scores was used to weight the quality and quantity scores (However, the weights in all cases were so close to equal that merely adding them would have given comparable results). Calibration was performed on data independent of that used to evaluate the relation between LSA and expert ratings. The correlation between the LSA score for an essay and that assigned by the average of the human readers was .80, .64, .XX and .84 for the four sets of exams. The comparable correlation between one reader and the other was .83, .65, .XX and .82, respectively. In the heart topic case, each student had also taken a carefully constructed "objective" test over the same material (a short answer test with near perfect scoring agreement). The correlation between the LSA essay score and the objective test was .81, the average correlation for the two expert readers .74. A striking aspect of these results is that the LSA representations were based on analyses of the essays that took no account of word order, each essay was treated as a "bag of words". In extracting meaning from a text, human readers presumably rely on syntax as well as the mere combination of words it contains, yet they were no better at agreeing on an essay's quality or in assigning a score that predicted a performance on a separate test of knowledge. Apparently, either the relevant information conveyed by word order in sentences is redundant with the information that can be inferred from the combination of words in the essay, or the processes used by LSA and humans extract different but compensatingly useful information. Learning HumanKnowledgc by Singular Value Decomposition: A Progress Report 3.3 49 OTHER EVIDENCE LSA has been compared with human knowledge in several additional ways, some confrrming the correspondence, others indicating limitations. Here are some examples, all based on encyclopedia corpus training. (1) Overall LSA similarity between antonyms (mean cos = .18) was equivalent to that between synonyms (mean cos .17) in triplets sampled from an antonym/synonym dictionary (W & R Chambers, 1989), both of which significantly exceeded that for unrelated pairs (mean cos = .01; ps < .0001). However for antonym (but not for synonym) pairs a dominant dimension of difference could easily be extracted by computing a one dimensional unfolding using the LSA cosines from a set of words listed in Rogel's (1992) thesaurus as related respectively to the two members of the pair. = (2) Anglin (1970) asked children and adults to sort words varying in concept relations ad parts of speech. LSA wont-word similarity correlates .50 with children and .32 with adults for the number of times they sorted two words together. Conceptual structure is reflected, but grammatical classification, strong in the adult data, is not. (3) When people are asked to decide that a letter string is a wont, they do so faster if they have just read a sentence that does not contain the word but implies a related concept (e.g. Till, Mross & Kintsch, 1988). LSA mirrors this result with high similarities between the same sentences and words. (Landauer & Dumais, 1997). (4) People frequently make a logical error, called the conjunction error by Tversky ad Kahneman (1974), in which they estimate that the probability that an object is a member of a class is greater than that it is a member of a superset class when the description of the object is "similar" to the description of the subset. For example, when told that "Linda is a young woman who is single, outspoken ...deeply concerned with issues of discrimination and social justice," over 80% of even statistically sophisticated subjects rate it more likely that Linda is a feminist bank teller than that Linda is a bank teller (Tversky & Kahneman, 1980). LSA similarities between descriptions of people ad occupations of this kind taken from Shafir, Smith and Osherson (1990) were computed as the cosine between the vector averages of words in the paired person-occupation descriptions. Conjunction error statements were more similar to the subset than superset statement in 12 out of 14 cases (p<.01), showing that LSA's representation of sentential meaning reflected similarity relations of the sort that have been hypothesized to underlie the conjunction fallacy in human judgment. (5) A semantic subspace was constructed for words from natural kind and artifact categories whose differential preservation is characteristic of agnosias due to local damage from herpes simplex encephalitis (Warrington & Shallice, 1984). Principal components analysis of the similarities among these words as represented by LSA revealed that categories that tend to be lost contain words that are more highly inter-related than those in preserved categories (Laham, in press). Of course, LSA does not capture all of the human knowledge conveyed by text. Some of the shortfall is probably due merely to the use of training corpora that are still imperfectly representative of the language experience of a typical person, and to lack of knowledge from non-textual sources. For example, in all these studies, less total text was used for LSA training than even a single educated adult would have read However, a more fundamental restriction is that the analysis does not reflect order relations between words, and therefore cannot extract infonnation that depends on syntax. Because the analysis discovers and represents only unsigned continuous similarities, it can be used to induce only certain classes of structural relations, not including ones that express Boolean, causal or other non-commutative logical relations. As we have seen, this lack does not prevent accurate simulation of human cognition in many cases, possibly because humans also 50 T. K Landauer, D. Laham and P. Foltz frequently rely on similarity rather than syntax-based. discrete logic (fversky axl Kahneman, 1983); however, it does limit the utility of the results for populating the symbolic data structures commonly used to represent knowledge in traditional AI. On the other hand, as examples in the next section show, continuous-valued similarity relations can be fruitfully applied if appropriate computational use is made of them. 4 SAMPLE APPLICATIONS LSA has been used successfully in a variety of experimental applications, including the essay scoring techniques described earlier. Here are some additional examples: (1) The technique has been used to improve automatic information retrieval by 20-30% over otherwise identical methods by allowing users' queries to match documents with the desired conceptual meaning but expressed in different words (Dumais, 1991, 1994). (2) By training on corpora of translated text in which the words of corresponding paragraphs in the two languages are combined in the "bags of words", LSA has been able to provide at least as good retrieval when queries and documents are in a different language as when in the same language (Landauer and Littman, 1990). (3) LSA-based measures of the similarity of student essays on a topic to instructional texts can predict how much an individual student will learn from a particular text (Wolfe et al., in press; Rehder et al., in press). To do this, the full set of student essays and the texts in question are aligned along a single dimension that best accommodates the LSA similarities among them. Estimates from one such experiment showed that using LSA to choose the optimal one of four texts for each student (a text that is slightly more sophisticated than the student) rather than assigning all students the overall best text (which LSA also picked correctly) increased the average amount learned by over 40%. (4) LSA-based measures of conceptual similarity between successive sentences accurately predict differences in judged coherence and measured comprehensibility of text (Foltz, Kintsch and Landauer, in press). 5 SUMMARY SVD-based learning of the structure underlying the use of words in meaningful contexts has been found capable of deriving and representing the similarity of meaning of words and text passages in a manner that accurately simulates corresponding similarity relations as reflected in several sorts of human judgments and behavior. The validity of the resulting representation of meaning similarities has been established in a variety of ways, and the utility of its knowledge representation illustrated by several educational axl cognitive psychological research applications. It is obviously too early to assess whether the particular computational model is a true analog of the process used by the human brain to accomplish the same things. However, the basic process, the representation of myriad local associative relations between components and larger contexts of experience in a joint space of lower dimensionality, offers, for the first time, a candidate for such a mechanism that has been shown sufficient to approximate human knowledge acquisition from natural sources at natural scale. Acknowledgments We thank members of the LSA research group at the University of Colorado for valuable collaboration and advice: Walter Kintsch, Bob Rehder, Mike Wolfe, & M. E. Shreiner. We especially acknowledge two participants from the Spring 1997 LSA seminar at CU whose unpublished work is described: Alan Sanfey (3.3.4) and Michael Emerson (3.3.1). Thanks also to Susan T. Dumais of Bellcore. Learning Human Knowledge by Singular Value Decomposition: A Progress Repol1 51 References Anglin, J. M. (1970). The growth of word meaning. Cambridge, MA: MIT. Berry, M. W. (1992). Large scale singular value computations. International Journal of Supercomputer Applications, 6, 13-49. Deerwester, S., Dumais, S. T., Furnas, G. W., Landauer, T. K., & Harshman, R. (1990). Indexing By Latent Semantic Analysis. Journal of the American Society For Information Science. 41, 391-407. Dumais, S. T. (1991). Improving the retrieval of infonnation from external sources. Behavior Research Methods, Instruments and Computers. 23,229-236. Dumais, S. T. (1994). Latent semantic indexing (LSI) and TREC-2. In D. Harman (Ed.), National Institute of Standards and Technology Text Retrieval Conference. NIST special publication. Foltz, P. W., Kintsch, W., & Landauer, T. K. (in press). Analysis of text coherence using Latent Semantic Analysis. Discourse Processes. Laham, D. (in press). Latent Semantic Analysis approaches to categorization. Proceedings of the Cognitive Science Society. 1997. Landauer, T. K., & Dumais, S. T. (1997). A solution to Plato's problem: The Latent Semantic Analysis theory of the acquisition, induction, and representation of knowledge. Psychological Review, 104,211-240. Landauer, T. K., Foltz, P. W., & Laham, D. (1997). Latent Semantic Analysis passes the test: knowledge representation and multiple-choice testing. Manuscript in preparation. Landauer, T. K., Laham, D., Rehder, B. & Schreiner, M .E. (in press). How well can passage meaning be derived without using word order: A comparison of Latent Semantic Analysis and humans. Proceedings of the Cognitive Science Society, 1997. Landauer, T. K., & Littman, M. L. (1990). Fully automatic cross-language document retrieval using latent semantic indexing. In Proceedings of the Sixth Annual Conference of the UW Centre for the New Oxford English Dictionary and Text Research (pp. 31-38). Waterloo, Ontario: UW Centre for the New OED. Myers, D. G. (1995). Psychology. Fourth Edition. NY, NY: Worth. Rehder, B., Schreiner, M. E., Wolfe, B. W., Laham, D., Landauer, T. K., & Kintsch, W. (in press). Using Latent Semantic Analysis to assess knowledge: Some technical considerations. Discourse Processes. ShafIf, E., Smith, E. E., & Osherson, D. N. (1990). Typicality and reasoning judgments. Memory & Cognition, 3, 229-239. Till, R. E., Mross, E. F., & Kintsch. W. (1988). Time course of priming for associate and inference words in discourse context. Memory and Cognition, 16, 283-299. Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 18S, 1124-1131. Tversky, A., & Kahneman, D. (1980). Judgments of and by representativeness. In D. Kahneman, P. Slovic, & A. Tversky (Eds.), Judgment under uncertainty: Heuristics and biases. New York: Cambridge University Press. Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90,293-315. Warrington, E. K., & Shallice, T. (1984). Category-specific semantic impairments. Brain, 107, 829-853. Wolfe, M. B., Schreiner, M. E., Rehder, B., Laham, D., Foltz, P. W., Kintsch, W., & Landauer, T. K. (in press). Learning from text: Matching readers and text by Latent Semantic Analysis. Discourse Processes. 

1468 |@word kintsch:7 briefly:1 cu:1 justice:1 essay:24 simulation:2 decomposition:7 moment:1 reduction:1 contains:1 score:8 document:3 past:1 contextual:1 protection:1 activation:1 yet:1 assigning:2 written:1 extensional:1 interpretable:1 discrimination:1 half:1 intelligence:1 item:2 affair:1 rehder:6 smith:2 short:3 node:1 successive:2 five:2 admission:1 along:1 constructed:2 direct:1 differential:1 ik:2 qualitative:1 prove:1 introductory:2 paragraph:3 manner:2 inter:1 indeed:1 behavior:3 themselves:1 frequently:2 brain:2 automatically:1 xx:2 unrelated:1 underlying:1 mass:1 linda:3 kind:4 psych:1 string:2 textbook:1 axl:2 growth:1 uk:2 shafir:1 unit:1 underlie:1 lsa:40 appear:1 producing:1 harshman:1 educated:1 service:2 engineering:1 local:3 limit:1 oxford:1 approximately:1 co:6 factorization:1 limited:1 statistically:1 practical:1 acknowledgment:1 testing:3 lost:1 differs:1 emerson:1 got:1 significantly:1 matching:1 word:52 induce:2 pre:1 instructor:1 chomsky:1 symbolic:2 cannot:3 close:1 wad:1 judged:2 unsigned:1 context:5 restriction:1 equivalent:2 demonstrated:1 educational:3 independently:1 typicality:1 schreiner:4 insight:1 orthonormal:1 deriving:1 embedding:1 traditionally:1 constructor:1 laham:12 analogous:1 unconscious:1 target:3 colorado:3 user:1 us:1 hypothesis:1 agreement:1 associate:2 element:1 wolfe:4 predicts:1 mike:1 solved:1 capture:1 susan:1 valuable:1 deeply:1 asked:3 littman:2 oed:1 tversky:6 trained:5 myriad:1 serve:1 efficiency:1 basis:1 translated:1 kahneman:7 easily:1 joint:1 osherson:2 represented:5 various:2 america:1 walter:1 forced:1 describe:1 query:2 herpes:1 choosing:1 whose:4 heuristic:2 larger:2 posed:1 valued:1 otherwise:1 objectively:1 ability:2 statistic:1 wont:2 syntactic:1 jointly:1 associative:1 obviously:1 myers:2 net:2 essa:1 took:1 product:1 fr:1 relevant:2 aligned:1 till:2 ontario:1 representational:2 description:4 intuitive:1 constituent:1 regularity:1 darrell:1 p:1 produce:3 categorization:1 perfect:1 object:2 exam:2 measured:4 ij:3 school:1 wished:1 progress:4 strong:1 predicted:1 judge:2 beg:1 implies:1 anatomy:1 emulation:1 correct:1 human:33 material:1 argued:1 crux:1 assign:1 anglin:2 around:1 considered:1 presumably:1 cognition:3 predict:2 dictionary:2 early:1 purpose:1 estimation:1 bag:2 infonnation:2 waterloo:1 largest:2 successfully:1 weighted:1 unfolding:1 mit:1 rather:5 varying:1 publication:1 conjunction:4 linguistic:1 derived:2 likelihood:2 inference:3 foreign:1 typically:1 hidden:2 relation:14 transformed:1 overall:3 among:6 classification:1 issue:1 bellcore:1 special:1 mutual:2 equal:1 identical:2 represents:1 unsupervised:1 constitutes:1 nearly:1 simplex:1 report:3 others:2 fundamentally:1 inherent:1 serious:1 primarily:1 simultaneously:1 national:1 individual:3 intended:1 ourselves:1 consisting:1 highly:1 evaluation:1 accurate:1 capable:1 experience:2 desired:1 causal:1 theoretical:1 psychological:3 increased:1 column:2 earlier:1 boolean:1 cover:1 phrase:4 entry:1 subset:2 imperfectly:1 successful:1 fruitfully:1 too:1 reported:2 answer:3 accomplish:1 dumais:9 combined:2 calibrated:1 slovic:1 person:2 fundamental:1 thanks:1 international:1 shortfall:1 standing:1 told:1 michael:1 together:1 again:1 reflect:1 nm:1 satisfied:1 containing:1 choose:2 possibly:1 woman:1 sentential:1 cognitive:4 external:1 expert:6 american:1 style:1 account:1 potential:1 diversity:1 coding:2 student:18 representativeness:1 depends:2 ad:3 performed:1 try:1 view:1 picked:1 apparently:1 start:1 relied:1 sort:3 participant:1 ass:4 square:1 who:2 characteristic:1 judgment:9 accurately:2 produced:1 populating:1 mere:1 worth:1 bob:1 history:1 reach:1 ed:2 sixth:1 acquisition:2 frequency:3 pp:1 resultant:1 associated:1 gain:1 newly:1 fre:1 sampled:2 popular:1 ask:1 logical:2 knowledge:35 dimensionality:2 carefully:1 sophisticated:2 manuscript:1 exceeded:1 reflected:4 strongly:1 just:2 implicit:2 correlation:4 hand:2 warrington:2 lack:2 quality:5 artifact:1 scientific:1 hypothesized:2 contain:3 concept:2 validity:1 true:1 harman:1 assigned:3 read:2 semantic:19 freq:1 illustrated:1 mel:1 cosine:8 syntax:3 complete:2 passage:16 reasoning:2 meaning:15 consideration:1 discovers:1 recently:1 million:1 association:1 occurred:1 analog:1 interpret:1 significant:1 cambridge:2 ai:1 automatic:2 newswire:1 centre:2 language:8 had:2 access:1 calibration:1 similarity:23 subjectively:1 dominant:1 showed:1 perspective:1 certain:1 success:1 accomplished:2 scoring:2 seen:1 additional:3 greater:1 redundant:1 preservation:1 multiple:5 full:1 alan:1 technical:1 faster:1 match:1 offer:2 long:1 retrieval:5 cross:1 paired:1 underlies:1 basic:2 regression:1 represent:4 cell:2 preserved:1 objection:1 singular:10 source:6 country:1 comprehensibility:1 probably:1 pass:1 induced:1 subject:1 tend:1 simulates:1 thing:1 member:4 plato:1 grammatically:1 adequacy:1 extracting:1 unitary:1 near:1 structural:1 revealed:1 easy:1 superset:2 concerned:1 variety:3 psychology:5 gave:1 identified:2 idea:2 whether:1 expression:2 utility:2 peter:1 speech:1 speaking:1 passing:1 york:1 impairment:1 useful:2 listed:1 amount:3 encyclopedia:2 ten:3 induces:1 category:4 generate:1 lsi:1 canal:1 correctly:1 write:2 discrete:1 express:3 group:1 four:5 prevent:1 uw:2 vast:1 merely:2 deerwester:2 angle:3 letter:1 fourth:1 uncertainty:2 striking:1 place:1 reader:7 decide:1 thesaurus:1 coherence:2 acceptable:1 comparable:2 layer:2 correspondence:2 annual:1 constraint:5 deficiency:1 unlimited:1 aspect:1 simulate:1 spring:1 relatively:2 department:2 combination:2 slightly:1 agreeing:1 presently:1 indexing:3 boulder:2 instructional:2 heart:2 taken:2 previously:3 mechanism:2 know:1 subjected:1 instrument:1 appropriate:1 chamber:1 simulating:2 occurrence:4 alternative:3 professional:1 supercomputer:1 thomas:1 original:2 exploit:1 especially:1 society:3 move:1 objective:2 question:5 quantity:4 occurs:1 damage:1 diagonal:1 guessing:1 traditional:1 subspace:1 separate:1 thank:1 simulated:1 accommodates:1 topic:4 reason:1 induction:3 length:4 retained:2 relationship:2 insufficient:1 kk:2 acquire:1 mexico:1 ftrst:1 potentially:1 statement:2 design:1 shallice:2 allowing:1 observation:1 markov:1 finite:1 melded:1 acknowledge:1 nist:1 trec:1 inferred:1 rating:2 pair:3 unpublished:1 connection:1 sentence:8 learned:2 textual:1 established:1 adult:4 able:2 usually:1 panama:1 max:1 including:2 reciprocally:1 memory:2 event:1 satisfaction:2 natural:7 rely:3 treated:1 representing:1 improve:1 rated:3 technology:1 extract:4 utterance:1 text:30 review:2 teller:2 berry:2 multiplication:1 occupation:2 fully:1 limitation:2 versus:1 conjoint:1 degree:1 conveyed:3 sufficient:2 article:1 principle:1 bank:2 collaboration:1 row:1 course:3 summary:1 english:3 verbal:1 bias:2 foltz:9 institute:2 tolerance:1 grammatical:1 dimension:6 vocabulary:4 transition:2 stand:1 axi:1 made:3 commonly:1 replicated:1 far:1 social:1 correlate:1 approximate:1 preferred:1 logic:1 abstracted:2 global:1 corpus:4 conceptual:3 landauer:20 accum:1 continuous:2 latent:11 fallacy:2 triplet:1 learn:2 improving:1 bearing:1 priming:1 domain:4 synonym:3 scored:1 edition:1 child:2 convey:2 body:2 positively:1 advice:1 representative:2 ny:2 furnas:1 seminar:1 comprises:1 explicit:1 candidate:1 young:1 resembled:1 specific:2 showing:1 supplemented:1 agnosia:1 explored:1 list:1 multitude:1 evidence:2 adding:1 mirror:1 associational:1 antonym:3 commutative:1 simply:1 likely:1 expressed:2 contained:3 acquiring:1 discourse:4 extracted:2 ma:1 viewed:1 sized:1 sorted:1 crl:1 content:4 included:1 determined:1 typical:1 principal:1 called:2 total:2 svd:4 la:1 experimental:1 meaningful:1 indicating:1 college:1 highdimensional:1 people:5 latter:1 inability:1 preparation:1 evaluate:2 tested:3 phenomenon:1 correlated:1

A Superadditive-Impairment Theory of Optic Aphasia Michael C. Mozer Dept. of Computer Science University of Colorado Boulder; CO 80309-0430 Mark Sitton Dept. of Computer Science University of Colorado Boulder; CO 80309-0430 Martha Farah Dept. of Psychology University of Pennsylvania Phila., PA 19104-6196 Abstract Accounts of neurological disorders often posit damage to a specific functional pathway of the brain. Farah (1990) has proposed an alternative class of explanations involving partial damage to multiple pathways. We explore this explanation for optic aphasia, a disorder in which severe perfonnance deficits are observed when patients are asked to name visually presented objects, but surprisingly, performance is relatively nonnal on naming objects from auditory cues and on gesturing the appropriate use of visually presented objects. We model this highly specific deficit through partial damage to two pathways-one that maps visual input to semantics, and the other that maps semantics to naming responses. The effect of this damage is superadditive, meaning that tasks which require one pathway or the other show little or no performance deficit, but the damage is manifested when a task requires both pathways (i.e., naming visually presented objects). Our model explains other phenomena associated with optic aphasia, and makes testable experimental predictions. Neuropsychology is the study of disrupted cognition resulting from damage to functional systems in the brain. Generally, accounts of neuropsychological disorders posit damage to a particular functional system or a disconnection between systems. Farah (1990) suggested an alternative class of explanations for neuropsychological disorders: partial damage to multiple systems, which is manifested through interactions among the loci of damage. We explore this explanation for the neuropsychological disorder of optic aphasia. Optic aphasia, arising from unilateral left posterior lesions, including occipital cortex and the splenium of the corpus callosum (Schnider, Benson, & Scharre, 1994), is marked by a deficit in naming visually presented objects, hereafter referred to as visual naming (Farah, 1990). However, patients can demonstrate recognition of visually presented objects nonverbally, for example, by gesturing the appropriate use of an object or sorting visual items into their proper superordinate categories (hereafter, visual gesturing). Patients can also name objects by nonvisual cues such as a verbal definition or typical sounds made by the objects (hereafter, auditory naming). The highly specific nature of the deficit rules out an explanation in terms of damage to a single pathway in a standard model of visual naming (Figure 1), suggesting that a more complex model is required, involving A Superadditive-Impainnent Theory of Optic Aphasia FIGURE 1. A standard box-and-arrow model of visual naming. The boxes denote levels of representation, and the arrows denote pathways mapping from one level of representation to another. Although optic aphasia cannot be explained by damage to the vision-to-semantics pathway or the semantics-ta-naming pathway, Farah (1990) proposed an explanation in terms of partial damage to both pathways (the X's). 67 visual auditory mUltiple semantic systems or multiple pathways to visual naming. However, a mere parsimonious account is suggested by Farah (1990): Optic aphasia might arise from partial lesions to two pathways in the standard model-those connecting visual input to semantics, and semantics to naming-and the effect of damage to these pathways is superadditive, meaning that tasks which require only one of these pathways (e.g., visual gesturing, or auditory naming) will be relatively unimpaired, whereas tasks requiring both pathways (e.g., visual naming) will show a significant deficit. 1 A MODEL OF SUPERADDITIVE IMPAIRMENTS We present a computational model of the superadditive-impairment theory of optic aphasia by elaborating the architecture of Figure 1. The architecture has four pathways: visual input to semantics (V~S), auditory input to semantics (A~S), semantics to naming (S~N), and semantics to gesturing (S~G). Each pathway acts as an associative memory. The critical property of a pathway that is required to explain optic aphasia is a speed-accuracy trade off: The initial output of a pathway appears rapidly, but it may be inaccurate. This "quick and dirty" guess is refined over time, and the pathway output asymptotically converges on the best interpretation of the input. We implement a pathway using the architecture suggested by Mathis and Mozer (1996). In this architecture, inputs are mapped to their best interpretations by means of a two-stage process (Figure 2). First, a quick, one-shot mapping is performed by a multilayer feedforward connectionist network to transform the input directly to its corresponding output. This is followed by a slower iterative clean-up process carried out by a recurrent attractor network. This architecture shows a speed-accuracy trade off by virtue of the .assumption that the feed forward mapping network does not have the capacity to produce exactly the right output to every input, especially when the inputs are corrupted by noise or are otherwise incomplete. Consequently, the clean up stage is required to produce a sensible interpretation of the noisy output of the mapping network. Fully distributed attractor networks have been used for similar purposes (e.g., Plaut & Shallice, 1993). For simplicity, we adopt a localist-attractor network with a layer of state units and a layer of radial basis function (RBF) units, one RBF unit per attractor. Each RBF or attractor unit measures the distance of the current state to the attractor that it represents. The activity of attractor unit i, aj, is: FIGURE 2. Connectionist implementation of a processing pathway. The pathway consists of feedforward mapping network followed by a recurrent cleanup or attractor network. Circles denote connectionist processing units and arrows denote connections between units or between layers of units. clean up network mapping network M. C. Mozer, M. Sitton and M. Farah 68 (1) (2) where set) is the state unit activity vector at time t, ~i is the vector denoting the location of attractor i, and ~i is the strength of the attractor. The strength detennines the region of the state space over which an attractor will exert its pull, and also the rate at which the state will converge to the attractor. The state units receive input from the mapping network and from the attractor units and are updated as follows: s;(t) = d;(t)e;(t) + (I - d;(t? La/t -I )~j; (3) j where sift) is the activity of state unit i at time t, e; is the ith output of the mapping net, is the ith element of attractor j, and d; is given by d .(t) I = h[I __ e;(_t-..,.-_l)] ej(t) (4) ej(t) where h[.] is a linear threshold function that bounds activity between -1 and +1, weighted time average of the ith output of the mapping net, = cxej(t) + (I -CX)e;(t-I) ~j; ej IS a (5) In all simulations, cx = .02. The activity of the state units are governed by two forces-the external input from the feedforward net (first tenn in Equation 3) and the attractor unit activities (second tenn). The parameter d; acts as a kind of attentional mechanism that modulates the relative influence of these two forces. The basic idea is that when the input coming from the mapping net is changing, the system should be responsive to the input and should not yet be concerned with interpreting the input. In this case, the input is copied straight through to the state units and hence dj should have a value close to I. When the input begins to stabilize, however, the focus shifts to interpreting the input and following the dynamics of the attractor network. This shift corresponds to d j being lowered to zero. The weighted time average in the update rule for d j is what allows for the smooth transition of the function to its new value. For certain constructions of the function d, Zemel and Mozer (in preparation) have proven convergence of the algorithm to an attractor. Apart from speed-accuracy trade off, these dynamics have another important consequence for the present model, particularly with respect to cascading pathways. If pathway A feeds into pathway S, such as V~S feeding into S~N, then the state unit activities of A act as the input to S. Because these activities change over time as the state approaches a well-fonned state, the dynamics of pathway S can be quite complex as it is forced to deal with an unstable input. This property is important in explaining several phenomena associated with optic aphasia. 1.1 PATTERN GENERATION Patterns were constructed for each of the five representational spaces: visual and auditory input, semantic, name and gesture responses. Each representational space was arbitrarily made to be 200 dimensional. We generated 200 binary-valued (-1,+1) patterns in each space, which were meant to correspond to known entities of that representational domain. For the visual, auditory, and semantic spaces, patterns were partitioned into 50 similarity clusters with 4 ?siblings per cluster. Patterns were chosen randomly subject to two constraints: patterns in different clusters had to be at least 80? apart, and siblings had to be between 25? and 50? apart. Because similarity of patterns in the name and gesture spaces was irrelevant to our modeling, we did not impose a similarity structure on these spaces. A Superadditive-Impainnent Theory of Optic Aphasia 69 Instead, we generated patterns in these spaces at random subject to the constraint that every pattern had to be at least 60? from every other. After generating patterns in each of the representational spaces, we established arbitrary correspondences among the patterns such that visual pattern n. auditory pattern n, semantic pattern n, name pattern n, and gesture pattern n all represented the same concept. That is, the appropriate response in a visual-naming task to visual pattern n would be semantic pattern n and name pattern n. 1.2 TRAINING PROCEDURE The feedforward networks in the four pathways (V-"7S, A-"7S, S-"7N, and S-"70) were independently trained on all 200 associations using back propagation. Each of these networks contained a single hidden layer of 150 units, and all units in the network used the symmetric activation function to give activities in the range [-1,+1]. The amount of training was chosen such that performance on the training examples was not perfect; usually several elements in the output would be erroneous-i.e., have the wrong sign-and others would not be exactly correct-i.e., -lor + 1. This was done to embody the architectural assumption that the feedforward net does not have the capacity to map every input to exactly the right output, and hence, the clean-up process is required. Training was not required for the clean-up network. Due to the localist representation of attractors in the clean-up network, it was trivial to hand wire each clean-up net with the 200 attractors for its domain, along with one rest-state attractor. All attractor strengths were initialized to the same value, ~= 15, except the rest-state attractor, for which P=5. The rest-state attractor required a lower strength so that even a weak external input would be sufficient to kick the attractor network out of the rest state. 1.3 SIMULATION METHODOLOGY After each pathway had been trained, the model was damaged by "lesioning" or removing a fraction y of the connections in the V -"7S and S-"7N mapping networks. The lesioned connections were chosen at random and an equal fraction was removed from the two pathways. The clean-up nets were not damaged. The architecture was damaged a total of 30 different times, creating 30 simulated patients who were tested on each of the four tasks and on all 200 input patterns for a task. The results we report come from averaging across simulated patients and input patterns. Responses were detennined after the system had been given sufficient time to relax into a name or gesture attractor, which was taken to be the response. Each response was classified as one of the following mutually exclusive response types: correct, perseveration (response is the same as that produced on any of the three immediately preceding trials), visual error (the visual pattern corresponding to the incorrect response is a sibling of the visual pattern corresponding to the correct response), semantic error, visual+semantic error, or other error. 1.4 PRIMING MECHANISM Priming-the increased availability of recently experienced stimuli-has been found across a wide variety of tasks in normal subjects. We included priming in our model as a strengthening (increasing the Pi parameter) of recently visited attractors (see Mathis & Mozer 1996, for details, and Becker, Behrmann, & Moscovitch, 1993, for a related approach). In the damaged model, this mechanism often gave rise to perseverations. 2 RESULTS We have examined the model's behavior as we varied the amount of damage, quantified by the parameter y. However, we report on the perfonnance of simulated patients with y = .30. This intermediate amount of damage yielded no floor or ceiling effects, and also produced error rates for the visual-naming task in the range of 30-40%, roughly the median performance of patients in the literature. 70 TABLE 1. Error rate of the damaged model on various tasks. M. C. Mozer, M. Sitton and M. Farah task aUOltory gesturing auditory naming visual gesturing visual naming error rate U.U"'/o 0.5% 8.7% 36.8% Table 1 presents the error rates of the model on four tasks. The pattern of errors shows a qualitative fit to human patient data. The model produced no errors on the auditory gesturing task because the two component pathways (A~S and S~G) were undamaged. Relatively few errors were made on the auditory-naming and visual-gesturing tasks, each of which involved one damaged pathway, because the clean-up nets were able to compensate for the damage. However, the error rate for the visual-naming task was quite large, due t'O damage on both of its component pathways (V~S and S~N). The error rate for visual naming cannot be accounted for by summing the effects of the damage to the two component pathways because the sum of the error rates for auditory naming and visual gesturing, each of which involves one of the two partially damaged pathways, is nearly four times smaller. Rather, the effects of damage on these pathways interact, and their interaction leads to superadditive impairments. When a visual pattern is presented to the model, it is mapped by the damaged V~S pathway into a corrupted semantic representation which is then cleaned up. While the corruption is sufficiently minor that clean up will eventually succeed, the clean up process is slowed considerably by the corruption. During the period of time in which the semantic clean-up network is searching for the correct attractor, the corrupted semantic representation is nonetheless fed into the damaged S~N pathway. The combined effect of the (initially) noisy semantic representation serving as input to a damaged pathway leads to corruption of the naming representation past the point where it can be cleaned-up properly. Interactions in the architecture are inevitable, and are not merely a consequence of some arbitrary assumption that is built into our model. To argue this point, we consider two modifications to the architecture that might eliminate the interaction in the damaged model. First, if we allowed the V~S pathway to relax into a well-formed state before feeding its output into the S~N pathway, there would be little interaction-the effects of the damage would be additive. However, cortical pathways do not operate sequentially, one stage finishing its computation and then turning on the next stage. Moreover, in the undamaged brain, such a processing strategy is unadaptive, as cascading partial results from one pathway to the next can speed processing without the introduction of errors (McClelland, 1979). Second, the interaction might be eliminated by making the S~N pathway continually responsive to changes in the output of the V ~S pathway. Then, the rate of convergence of the V ~S pathway would be irrelevant to determining the eventual output of the S~N pathway. However, because the output of the S~N pathway depends not only on its input but its internal state (the state of the clean-up net), one cannot design a pathway that is continually responsive to changes in the input and is also able to clean up noisy responses. Thus, the two modifications one might consider to eliminate the interactions in the damaged model seriously weaken the computational power of the undamaged model. We therefore conclude that the framework of our model makes it difficult to avoid an interaction of damage in two pathways. A subtle yet significant aspect of the model's performance is that the error rate on the visual-gesturing task was reliably higher than the error rate on the auditory-naming task, despite the fact that each task made use of one damaged pathway, and the pathways were damaged to the same degree. The difference in performance is due to the fact that the damaged pathway for the visual-gesturing task is the first in a cascade of two, while the damaged pathway for the auditory-naming task is the second. The initially noisy response from a damaged pathway early in the system propagates to subsequent pathways, and A Superadditive-Impairment Theory of Optic Aphasia 71 although the damaged pathway will eventually produce the correct response, this is not sufficient to ensure that subsequent pathways will do so as well. 2.1 DISTRIBUTION OF ERRORS FOR VISUAL OBJECT NAMING Figure 2 presents the model's error distribution for the visual-naming task. Consistent with the patient data (Farah, 1990), the model produces many more semantic and perseveration errors than by chance. The chance error proportions were computed by assuming that if the correct response was not made, then all other responses had an equal probability of being chosen. To understand the predominance of semantic errors, consider the effect of damage to the V ~S pathway. For relatively small amounts of damage, the mapping produced will be close to the correct mapping. "Close" here means that the Euclidean distance in the semantic output space between the correct and perturbed mapping is small. Most of the time, minor perturbation of the mapping will be compensated for by the clean-up net. Occasionally, the perturbation will land the model in a different attractor basin, and a different response will be made. However, when the wrong attractor is selected, it will be one "close" to the correct attractor, i.e., it will likely be a sibling in the same pattern cluster as the correct attractor. In the case of the V ~S pathway, the siblings of the correct attract or are by definition semantically related. A semantic error will be produced by the model when a sibling semantic attractor is chosen, and then this pattern is correctly mapped to a naming response in the S~N pathway. In addition to semantic errors, the other frequent error type in visual naming is perseverations. The priming mechanism is responsible for the significant number of perseverations, although in the unlesioned model, it facilitates processing of repeated stimuli without producing perseverations. Just as important as the presence of perseverative and semantic errors is the absence of visual errors, a feature of optic aphasia that contrasts sharply with visual agnosia (Farah, 1990). The same mechanisms explain why the rate of visual errors is close to its chance value and why visual+semantic errors are above chance. Visual-naming errors occur because there is an error either in the V~S or S~N mappings, or both. Since the erroneous outputs of these pathways show a strong tendency to be similar to the correct output, and because semantic and name similarity does not imply visual similarity (the patterns were paired randomly), visual errors should only occur by chance. When a visual error does occur, though, there is a high probability that the error is also semantic because of the strong bias that already exists toward producing semantic errors. This is the reason why more visual+semantic errors occur than by chance and why the proportion of these ? actual ? cI1ance FIGURE 3. Distribution of error types made by model on the V~N task (black bars) relative to chance (grey bars). visual vis+sem Error type pe~Yendwe 72 M. C. Mozer, M. Sitton and M Farah errors is only slightly less than the proportion of visual errors. Plaut and Shallice (1993) have proposed a connectionist model to account for the distribution of errors made by optic aphasics. Although their model was not designed to account for any of the other phenomena associated with the disorder, it has much in common with the model we are proposing. Unlike our model, however, theirs requires the assumption that visually similar objects also share semantic similarity. This assumption might be questioned, especially because our model does not require this assumption to produce the correct distribution of error responses. 3 DISCUSSION In demonstrating superadditive effects of damage, we have offered an account of optic aphasia that explains the primary phenomenon: severe impairments in visual naming in conjunction with relatively spared performance on naming from verbal description or gesturing the appropriate use of a visually presented object. The model also explains the distribution of errors on visual naming. Although we did not have the space in this brief report to elaborate, the model accounts for several other distinct characteristics of optic aphasia, including the tendency of patients to "home in" on the correct name for a visually presented object when given sufficient time, and a positive correlation between the error rates on naming and gesturing responses to a visual object (Sitton, Mozer, & Farah, 1998). Further, the model makes several strong predictions which have yet to be tested experimentally. One such prediction, which was apparent in the results presented earlier, is that a higher error rate should be observed on visual gesturing than on auditory naming when the tasks are equated for difficulty, as our simulation does. More generally, we have strengthened the plausibility of Farah's (1990) hypothesis that partial damage to two processing pathways may result in close-to-normal performance on tasks involving one pathway or the other while yielding a severe performance deficit on tasks involving both damaged pathways. The superadditive-impairment theory thus may provide a more parsimonious account of various disorders that were previously believed to require more complex architectures or explanations. 4 ACKNOWLEDGMENTS This research was supported by grant 97-18 from the McDonnell-Pew Program in Cognitive Neuroscience. 5 REFERENCES Becker, S., Behrmann, M., & Moscovitch, K. (1993). Word priming in attractor networks. Proceedings of the Fifteenth Annual Conference of the Cognitive Science Society (pp. 231-236). Hillsdale, NJ: Erlbaum. Farah, M. J. (1990). Visual agnosia. Cambridge, MA: MIT PresslBradford Books. Mathis, D. W., & 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). Hillsdale, NJ: Erlbaum. McClelland, J. L. (1979). On the time relations of mental processes: An examination of systems of processes in cascade. Psychological Review, 86, 287-330. Plaut, D., & Shallice, T. (1993). Perseverative and semantic influences on visual object naming errors in optic aphasia: A connectionist approach. Journal of Cognitive Neuroscience, 5,89-112. Schnider, A., Benson, D. E, and Scharre, D. W. (1994). Visual agnosia and optic aphasia: Are they anatomically distinct? Cortex, 30, 445-457. Sitton, M., Mozer, M. C., & Farah, M. (1998). Diffuse lesions in a modular connectionisl architecture: An account of optic aphasia. Manuscript submitted for publication. 

1469 |@word trial:1 proportion:3 grey:1 simulation:3 shot:1 initial:1 hereafter:3 denoting:1 seriously:1 elaborating:1 past:1 current:1 activation:1 yet:3 cottrell:1 additive:1 subsequent:2 designed:1 update:1 cue:2 tenn:2 guess:1 item:1 selected:1 ith:3 mental:1 plaut:3 location:1 five:1 lor:1 along:1 constructed:1 incorrect:1 consists:1 qualitative:1 pathway:65 roughly:1 behavior:1 embody:1 brain:3 little:2 actual:1 increasing:1 begin:1 moreover:1 what:1 kind:1 proposing:1 nj:2 every:4 act:3 exactly:3 wrong:2 unit:18 grant:1 producing:2 continually:2 before:1 positive:1 consequence:2 despite:1 might:5 black:1 exert:1 examined:1 quantified:1 co:2 range:2 neuropsychological:3 acknowledgment:1 responsible:1 implement:1 procedure:1 cascade:2 word:1 radial:1 lesioning:1 cannot:3 close:6 influence:2 map:3 quick:2 compensated:1 eighteenth:1 occipital:1 independently:1 simplicity:1 disorder:7 immediately:1 rule:2 undamaged:3 cascading:2 pull:1 searching:1 updated:1 construction:1 unconscious:1 colorado:2 damaged:19 hypothesis:1 pa:1 element:2 recognition:1 particularly:1 observed:2 region:1 trade:3 removed:1 neuropsychology:1 mozer:10 asked:1 lesioned:1 dynamic:3 trained:2 basis:1 represented:1 various:2 forced:1 distinct:2 zemel:1 refined:1 quite:2 apparent:1 modular:1 valued:1 relax:2 otherwise:1 transform:1 noisy:4 associative:1 net:10 interaction:8 coming:1 strengthening:1 frequent:1 rapidly:1 detennined:1 representational:4 description:1 convergence:2 cluster:4 produce:5 generating:1 perfect:1 converges:1 object:15 recurrent:2 minor:2 strong:3 involves:1 come:1 posit:2 detennines:1 correct:14 human:1 hillsdale:2 explains:3 require:4 feeding:2 sufficiently:1 normal:2 visually:8 cognition:1 mapping:16 adopt:1 early:1 purpose:1 visited:1 predominance:1 callosum:1 weighted:2 mit:1 rather:1 avoid:1 ej:3 publication:1 conjunction:1 focus:1 finishing:1 properly:1 contrast:1 spared:1 attract:1 inaccurate:1 eliminate:2 initially:2 hidden:1 relation:1 semantics:10 nonnal:1 among:2 equal:2 eliminated:1 represents:1 nearly:1 inevitable:1 connectionist:5 others:1 report:3 stimulus:2 few:1 randomly:2 attractor:33 highly:2 superadditive:11 severe:3 yielding:1 partial:7 perfonnance:2 incomplete:1 euclidean:1 initialized:1 circle:1 weaken:1 psychological:1 increased:1 modeling:1 earlier:1 localist:2 erlbaum:2 perturbed:1 corrupted:3 considerably:1 combined:1 disrupted:1 off:3 michael:1 connecting:1 external:2 creating:1 cognitive:4 book:1 account:9 suggesting:1 stabilize:1 availability:1 depends:1 vi:1 performed:1 farah:15 formed:1 accuracy:3 who:1 characteristic:1 correspond:1 weak:1 produced:5 mere:1 perseveration:6 corruption:3 straight:1 classified:1 submitted:1 explain:2 ed:1 definition:2 nonetheless:1 pp:2 involved:1 associated:3 auditory:15 subtle:1 back:1 appears:1 feed:2 manuscript:1 ta:1 higher:2 methodology:1 response:19 done:1 box:2 though:1 just:1 stage:4 correlation:1 hand:1 propagation:1 aj:1 name:9 effect:9 requiring:1 concept:1 hence:2 symmetric:1 semantic:25 deal:1 during:1 demonstrate:1 interpreting:2 meaning:2 recently:2 common:1 functional:3 association:1 interpretation:3 theirs:1 significant:3 cambridge:1 pew:1 dj:1 unimpaired:1 had:6 lowered:1 cortex:2 similarity:6 posterior:1 irrelevant:2 apart:3 occasionally:1 certain:1 manifested:2 binary:1 arbitrarily:1 impose:1 preceding:1 floor:1 converge:1 period:1 multiple:4 sound:1 smooth:1 gesture:4 plausibility:1 compensate:1 believed:1 naming:36 paired:1 prediction:3 involving:4 basic:1 multilayer:1 vision:1 patient:10 fifteenth:1 receive:1 whereas:1 addition:1 median:1 rest:4 operate:1 unlike:1 subject:3 facilitates:1 kick:1 presence:1 feedforward:5 intermediate:1 concerned:1 variety:1 fit:1 psychology:1 gave:1 pennsylvania:1 architecture:10 idea:1 sibling:6 shift:2 becker:2 unilateral:1 questioned:1 impairment:7 generally:2 amount:4 conscious:1 category:1 mcclelland:2 sign:1 neuroscience:2 arising:1 per:2 correctly:1 serving:1 four:5 threshold:1 demonstrating:1 changing:1 clean:15 asymptotically:1 merely:1 fraction:2 sum:1 architectural:1 parsimonious:2 home:1 bound:1 layer:4 followed:2 perseverative:2 copied:1 correspondence:1 presslbradford:1 yielded:1 activity:9 annual:2 strength:4 occur:4 optic:20 constraint:2 sharply:1 diffuse:1 aspect:1 speed:4 relatively:5 mcdonnell:1 across:2 smaller:1 slightly:1 partitioned:1 modification:2 making:1 benson:2 explained:1 slowed:1 anatomically:1 boulder:2 taken:1 ceiling:1 equation:1 mutually:1 previously:1 eventually:2 mechanism:5 aphasia:19 locus:1 fed:1 appropriate:4 responsive:3 alternative:2 slower:1 dirty:1 ensure:1 testable:1 especially:2 society:2 already:1 damage:25 strategy:1 exclusive:1 primary:1 distance:2 deficit:7 mapped:3 attentional:1 capacity:2 entity:1 sensible:1 simulated:3 argue:1 unstable:1 trivial:1 toward:1 reason:1 assuming:1 difficult:1 rise:1 implementation:1 design:1 proper:1 shallice:3 reliably:1 wire:1 varied:1 perturbation:2 arbitrary:2 required:6 cleaned:2 connection:3 established:1 able:2 suggested:3 superordinate:1 usually:1 pattern:28 bar:2 perception:1 program:1 built:1 including:2 memory:1 explanation:7 power:1 critical:1 difficulty:1 force:2 examination:1 turning:1 brief:1 imply:1 carried:1 review:1 literature:1 determining:1 relative:2 fully:1 generation:1 unadaptive:1 proven:1 degree:1 offered:1 sufficient:4 consistent:1 basin:1 propagates:1 pi:1 land:1 share:1 accounted:1 surprisingly:1 supported:1 verbal:2 bias:1 disconnection:1 understand:1 explaining:1 wide:1 distributed:1 cortical:1 transition:1 forward:1 made:8 fonned:1 equated:1 sequentially:1 corpus:1 summing:1 conclude:1 iterative:1 why:4 table:2 nature:1 sem:1 interact:1 complex:3 mathis:3 priming:5 domain:2 did:2 arrow:3 noise:1 arise:1 lesion:3 allowed:1 repeated:1 referred:1 elaborate:1 strengthened:1 experienced:1 governed:1 pe:1 behrmann:2 removing:1 erroneous:2 specific:3 sift:1 agnosia:3 virtue:1 exists:1 unlesioned:1 modulates:1 sorting:1 cx:2 explore:2 likely:1 visual:52 contained:1 neurological:1 partially:1 corresponds:1 chance:7 ma:1 succeed:1 marked:1 consequently:1 rbf:3 eventual:1 absence:1 martha:1 change:3 included:1 typical:1 except:1 experimentally:1 semantically:1 averaging:1 total:1 experimental:1 la:1 tendency:2 internal:1 mark:1 moscovitch:2 cleanup:1 meant:1 preparation:1 dept:3 tested:2 phenomenon:4

297 A NETWORK FOR IMAGE SEGMENTATION USING COLOR Anya Hurlbert and Tomaso Poggio Center for Biological Information Processing at Whitaker College Department of Brain and Cognitive Science and the MIT AI Laboratory Cambridge, MA 02139 (hur lbert@wheaties.ai.mit.edu) ABSTRACT We propose a parallel network of simple processors to find color boundaries irrespective of spatial changes in illumination, and to spread uniform colors within marked reglOns. . INTRODUCTION To rely on color as a cue in recognizing objects, a visual system must have at least approximate color constancy. Otherwise it might ascribe different characteristics to the same object under different lights. But the first step in using color for recognition, segmenting the scene into regions of different colors, does not require color constancy. In this crucial step color serves simply as a means of distinguishing one object from another in a given scene. Color differences, which mark material boundaries, are essential, while absolute color values are not. The goal of segmentation algorithms is to achieve this first step toward object recognition by finding discontinuities in the image irradiance that mark material boundaries. The problems that segmentation algorithms must solve is how to choose color labels, how to distinguish material boundaries from other changes in the image that give rise to color edges, and how to fill in uniform regions with the appropriate color labels. (Ideally, the color labels should remain constant under changes in the illumination or scene composition and color edges should occur only at material boundaries.) Rubin and Richards (1984 ) show that algorithms can solve the second problem under some conditions by comparing the image irradiance signal in distinct spectral channels on either side of an edge. The goal of the segmentation algorithms we discuss here is to find boundaries between regions of different surface spectral reflectances and to spread uniform colors within them, without explicitly requiring the colors to be constant under changes in illumination. The color labels we use are analogous to the CIE chromaticity coordinates x and y. Under the single source assumption, they change across space 298 Hurlbert and Poggio only when the surface spectral reflectance changes, except when strong specularities are present. (The algorithms therefore require help at a later stage to identify between color label changes due to specularities, which we have not yet explicitly incorporated.) The color edges themselves are localised with the help of luminance edges, by analogy with psychophysics of segmentation and filling-in. The Koftka Ring illusion, for example, indicates that color is attributed to surfaces by an interaction between an edge-finding operator and a filling-in operator. 1 The interaction is justified by the fact that in the real world changes in surface spectral reflectance are almost always accompanied by changes in brightness. Color Labels We assume that surfaces reflect light according to the neutral-interface-reflection model. In this model (Lee, 1986 , Shaefer, 1984 [3]) the image irradiance I(X,y,A) is the sum of two components, the surface reflection and the body reflection: I(x, y, A) = L(r(x, y), A)[a(r, A)g(6(r)) + bh(6(r))], where A labels wavelength and r( x, y) is the point on the 3D surface to which the image coordinates (x, y) correspond. L(r(x, y), A) is the illumination on the surface. a(r, A) is the spectral reflectance factor of the body reflection component and g(6(r)) its magnitude, which depends on the viewing geometry parameters lumped together in 6(r). The spectral reflectance factor of the specular, or surface reflection, component b is assumed to be constant with respect to A, as is true for inhomogeneous materials such as paints and plastics. For most materials, the magnitude of the specular component h depends strongly on the viewing geometry. Using the single source assumption, we may factor the illumination L into separate spatial and spectral components (L(r, A) L(r)c(A)). Multiplying I by the spectral sensitivities of the color sensors i = 1,2,3 and integrating over wavelength yields the triplet of color values (R, G, B), where and so forth and where the a i and bi are the reflectance factors in the spectral channels defined by the sensor spectral sensitivities. We define the hues u and v as R u= - -__- - R+G+B and 1 Note that Land's original retinex algorithm, which thresholds and swns the differences in image irradiance between adjacent points along many paths, accounts for the contribution of edges to color, without introducing a separate luminance edge detector. A Network for Image Segmentation Using Color G v=----- R+G+B at each pixel. In Lambertian reflection, the specular reflectance factor b is zero. In this case, u and v are piecewise constant: they change in the image only when the ai(x,y) change. Thus u or v mark discontinuities in the surface spectral reflectance function, e.g they mark material boundaries. Conversely, image regions of constant u correspond to regions of constant surface color. Synthetic images generated with standard computer graphics algorithms (using, for example, the Phong reflectance model) behave in this way: u is constant across the visible surface of a shaded sphere. Across specularities, u in general changes but often not much. Thus one approach to the segmentation problem is to find regions of "constant" u and their boundaries . The difficulty with this approach is that real u data are noisy and unreliable: u is the quotient of numbers that are not only noisy themselves but also, at least for biological photosensor spectral sensitivities, very close to one another. The goals of segmentation algorithms are therefore to enhance discontinuities in u and, within the regions marked by the discontinuities, to smoothe over the noise and fill in the data where they are unreliable. We have explored several methods of meeting these goals. Segmentation Algorithms One method is to regularize - to eliminate the noise and fill in the data, while preserving the discontinuities. Using an algorithm based on Markov Random Field techniques, we have obtained encouraging results on real images (see Poggio et al., 1988) . The MRF technique exploits the constraint that u should be piecewise constant within the discontinuity contours and uses image brightness edges as guides in finding the contours. An alternative to the MRF approach is a cooperative network that fills in data and filters out noise while enforcing the constraint of piecewise constancy. The network, a type of Hopfield net, is similar to the cooperative stereo network of Marr and Poggio (1976). Another approach consists of a one-pass winner-take-all scheme. Both algorithms involve loading the initial hue values into discrete bins, an undesirable and biologically unlikely feature . Although they produce good results on noisy synthetic images and can be improved by modification (see Hurlbert, 1989), another class of algorithms which we now describe are simple and effective, especially on parallel computers such as the Connection Machine. Averaging Network One way to avoid small step changes in hue across a uniform surface resulting from initial loading into discrete bins is to relax the local requirement for piecewise 299 300 Hurlbert and Poggio b. 41 97 " 74 Figure 1: (a) Image of a Mondrian-textured sphere - the red channel. (b) Vertical slice through the specularity in a 75 x 75 pixel region of the three-channel image (R + G + B) of the same sphere. constancy and instead require only that hue be smooth within regions delineated by the edge input. We will see that this local smoothness requirement actually yields an iterative algorithm that provides asymptotically piecewise constant hue regions. To implement the local smoothness criterion we use an averaging scheme that simply replaces the value of each pixel in the hue image with the average of its local surround, iterating many times over the whole image. The algorithm takes as input the hue image (either the u-image or the v-image) and one or two edge images, either luminance edges alone, or luminance edges plus u or v edges, or u edges plus v edges. The edge images are obtained by performing Canny edge detection or by using a thresholded directional first derivative. On each iteration, the value at each pixel in the hue image is replaced by the average of its value and those in its contributing neighborhood. A neighboring pixel is allowed to contribute if (i) it is one of the four pixels sharing a full border with the central pixel (ii) it shares the same edge label with the central pixel in all input edge images (iii) its value is non-zero and (iv) its value is within a fixed range of the central pixel value. The last requirement simply reinforces the edge label requirement when a hue image serves as an input edge image - the edge label requirement allows only those pixels that lie on the same side of an edge to be averaged, while the other insures that only those pixels with similar hues are averaged. More formally A Network for Image Segmentation Using Color where Cn(hf,j) is the set of N(C n ) pixels among the next neighbors of i,j that differ from h~. less than a specified amount and are not crossed by an edge in the edge map(s) (on the assumption that the pixel (i,j) does not belong to an edge). The iteration of this operator is similar to nonlinear diffusion and to discontinuous regularization of the type discussed by Blake and Zisserman (1987), Geman and Geman (1984) and Marroquin (9]. The iterative scheme of the above equation can be derived from minimization via gradient descent of the energy function E = L:Ei,j with = where V(x, y) V(x - y) is a quadratic potential around 0 and constant for above a certain value. Ix - yl The local averaging smoothes noise in the hue values and spreads uniform hues across regions marked by the edge inputs. On images with shading but without strong specularities the algorithm performs a clean segmentation into regions of different hues. Conclusions The averaging scheme finds constant hue regions under the assumptions of a single source and no strong specularities. A strong highlight may originate an edge that could then "break" the averaging operation. In our limited experience most specularities seem to average out and disappear from the smoothed hue map, largely because even strong specularities in the image are much reduced in the initial hue image. The iterative averaging scheme completely eliminates the remaining gradients in hue. It is possible that more powerful discrimination of specularities will require specialized routines and higher-level knowledge (Hurlbert, 1989). Yet this simple network alone is sufficient to reproduce some psychophysical phenomena. In particular, the interaction between brightness and color edges enables the network to mimic such visual "illusions" as the Koftka Ring. We replicate the illusion in the following way. A black-and-white Koft'ka Ring (a uniform grey annulus against a rectangular bipartite background, one side black and the other white) (Hurlbert and Poggio, 1988b) is filtered through the lightness filter estimated in 301 302 Hurlbert and Poggio h. a. c. 9.49679872 9.1989 9 72 9.1122449 9 299 Figure 2: (a) A 75x75 pixel region of the u image, including the specularity. (b) The image obtained after 500 iterations of the averaging network on (a), using as edge input the Canny edges of the luminance image. A threshold on differences in the v image allows only similar v values to be averaged. (c) Vertical slice through center of (a). (d) Vertical slice at same coordinates through (b) (note different scales of (c) and (d?. A Network for Image Segmentation Using Color the way described elsewhere (Hurlbert and Poggio, 1988a). (For black-and-white images this step replaces the operation of obtaining u and v: in both cases the goal is to eliminate spatial gradients of in the effective illumination.) The filtered Koffka Ring is then fed to the averaging network together with the brightness edges. When in the input image the boundary between the two parts of the background continues across the annulus, in the output image (after 2000 iterations of the averaging network) the annulus splits into two semi-annuli of different colors in the output image, dark grey against the white half, light grey against the black half (Hurlbert, 1989). When the boundary does not continue across the annulus, the annulus remains a uniform grey. These results agree with human perception. Acknowledgements This report describes research done within the Center for Biological Information Processing, in the Department of Brain and Cognitive Sciences, and at the Artificial Intelligence Laboratory. This research is sponsored by a grant from the Office of Naval Research (ONR), Cognitive and Neural Sciences Division; by the Artificial Intelligence Center of Hughes Aircraft Corporation; by the Alfred P. Sloan Foundation; by the National Science Foundation; by the Artificial Intelligence Center of Hughes Aircraft Corporation (SI-801534-2); and by the NATO Scientific Affairs Division (0403/87). Support for the A. I. Laboratory's artificial intelligence research is provided by the Advanced Research Projects Agency of the Department of Defense under Army contract DACA76-85-C-001O, and in part by ONR contract NOOOI485-K-0124. Tomaso Foggio is supported by the Uncas and Helen Whitaker Chair at the Massachusetts Institute of Technology, Whitaker College. References John Rubin and Whitman Richards. Colour VISIon: representing material categories. Artificial Intelligence Laboratory Memo 764, Massachusetts Institute of Technology, 1984. Hsien-Che Lee. Method for computing the scene-illuminant chromaticity from specular highlights. Journal of the Optical Society of America, 3:1694-1699, 1986. Steven A. Shafer. Using color to separate reflection components. Color Research and Applications, 10(4):210-218, 1985. Tomaso Poggio, J. Little, E. Gamble, W. Gillett, D. Geiger, D. Weinshall, M. Villalba, N. Larson, T. Cass, H. Biilthoff, M. Drumheller, P. Oppenheimer, W. Yang, and A. Hurlbert. The MIT Vision Machine. In Proceedings Image Understanding Workshop, Cambridge, MA, April 1988. Morgan Kaufmann, San Mateo, CA. David Marr and Tomaso Poggio. Cooperative computation of stereo disparity. Science, 194:283-287, 1976. Anya C. Hurlbert. The Computation of Color. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, 1989. Jose L. Marroquin. Probabilistic Solution of Inverse Problems. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, 1985. 303 304 Hurlbert and Poggio Andrew Blake and Andrew Zisserman. Visual Reconstruction. MIT Press, Cambridge, Mass, 1987. Stuart Geman and Don Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern A nalysis and Machine Intelligence, PAMI-6:721-741, 1984. Anya C. Hurlbert and Tomaso A. Poggio. Learning a color algorithm from examples. In Dana Z. Anderson, editor, Neural Information Processing Systems. American Institute of Physics, 1988. A. C. Hurlbert and T. A. Poggio. Synthesizing a color algorithm from examples. Science, 239:482-485, 1988. 

147 |@word aircraft:2 loading:2 replicate:1 grey:4 brightness:4 shading:1 initial:3 disparity:1 ka:1 comparing:1 si:1 yet:2 must:2 john:1 visible:1 enables:1 sponsored:1 discrimination:1 alone:2 cue:1 half:2 intelligence:6 affair:1 filtered:2 provides:1 contribute:1 along:1 consists:1 tomaso:5 themselves:2 brain:2 encouraging:1 little:1 provided:1 project:1 mass:1 weinshall:1 finding:3 corporation:2 grant:1 segmenting:1 local:5 path:1 pami:1 might:1 plus:2 black:4 mateo:1 conversely:1 shaded:1 limited:1 bi:1 range:1 averaged:3 hughes:2 implement:1 illusion:3 integrating:1 close:1 undesirable:1 operator:3 bh:1 map:2 center:5 helen:1 rectangular:1 fill:4 regularize:1 marr:2 coordinate:3 analogous:1 distinguishing:1 us:1 recognition:2 continues:1 richards:2 geman:4 cooperative:3 constancy:4 steven:1 region:14 agency:1 ideally:1 mondrian:1 bipartite:1 division:2 textured:1 completely:1 whitman:1 hopfield:1 america:1 distinct:1 describe:1 effective:2 artificial:5 neighborhood:1 solve:2 relax:1 otherwise:1 noisy:3 net:1 propose:1 reconstruction:1 interaction:3 canny:2 neighboring:1 achieve:1 forth:1 requirement:5 produce:1 ring:4 object:4 help:2 andrew:2 strong:5 quotient:1 differ:1 inhomogeneous:1 discontinuous:1 filter:2 stochastic:1 human:1 viewing:2 material:8 bin:2 require:4 biological:3 daca76:1 around:1 blake:2 label:10 x75:1 minimization:1 mit:4 sensor:2 always:1 avoid:1 office:1 derived:1 naval:1 indicates:1 eliminate:2 unlikely:1 reproduce:1 pixel:14 among:1 spatial:3 psychophysics:1 field:1 stuart:1 filling:2 mimic:1 report:1 piecewise:5 koffka:1 national:1 replaced:1 geometry:2 detection:1 light:3 edge:33 experience:1 poggio:13 iv:1 reglons:1 restoration:1 introducing:1 neutral:1 uniform:7 recognizing:1 graphic:1 synthetic:2 sensitivity:3 lee:2 yl:1 contract:2 probabilistic:1 physic:1 enhance:1 together:2 thesis:2 reflect:1 central:3 choose:1 cognitive:3 american:1 derivative:1 account:1 potential:1 accompanied:1 explicitly:2 sloan:1 depends:2 crossed:1 later:1 break:1 red:1 hf:1 parallel:2 contribution:1 kaufmann:1 characteristic:1 largely:1 correspond:2 identify:1 yield:2 directional:1 bayesian:1 plastic:1 annulus:6 multiplying:1 processor:1 detector:1 sharing:1 against:3 energy:1 hurlbert:14 attributed:1 massachusetts:4 hur:1 color:37 knowledge:1 segmentation:12 routine:1 marroquin:2 irradiance:4 actually:1 higher:1 zisserman:2 improved:1 april:1 done:1 strongly:1 anderson:1 stage:1 ei:1 nonlinear:1 ascribe:1 scientific:1 requiring:1 true:1 regularization:1 laboratory:4 white:4 chromaticity:2 adjacent:1 lumped:1 larson:1 criterion:1 performs:1 interface:1 reflection:7 image:42 specialized:1 winner:1 phong:1 belong:1 discussed:1 composition:1 cambridge:5 surround:1 ai:3 gibbs:1 smoothness:2 surface:13 certain:1 onr:2 continue:1 meeting:1 preserving:1 morgan:1 signal:1 ii:1 semi:1 full:1 smooth:1 sphere:3 mrf:2 vision:2 iteration:4 cie:1 justified:1 background:2 source:3 crucial:1 eliminates:1 photosensor:1 seem:1 yang:1 iii:1 split:1 specular:4 cn:1 defense:1 colour:1 stereo:2 iterating:1 involve:1 amount:1 dark:1 hue:17 category:1 reduced:1 estimated:1 reinforces:1 smoothe:1 discrete:2 alfred:1 four:1 threshold:2 clean:1 thresholded:1 diffusion:1 nalysis:1 luminance:5 asymptotically:1 relaxation:1 sum:1 jose:1 inverse:1 powerful:1 almost:1 smoothes:1 geiger:1 distinguish:1 replaces:2 quadratic:1 occur:1 constraint:2 scene:4 chair:1 performing:1 optical:1 department:3 according:1 remain:1 across:7 describes:1 delineated:1 biologically:1 modification:1 equation:1 agree:1 remains:1 discus:1 fed:1 serf:2 operation:2 lambertian:1 appropriate:1 spectral:12 alternative:1 original:1 remaining:1 whitaker:3 exploit:1 reflectance:9 especially:1 disappear:1 society:1 psychophysical:1 paint:1 che:1 gradient:3 separate:3 originate:1 toward:1 enforcing:1 localised:1 memo:1 rise:1 synthesizing:1 vertical:3 markov:1 descent:1 behave:1 hsien:1 incorporated:1 smoothed:1 david:1 specified:1 connection:1 discontinuity:6 perception:1 pattern:1 including:1 difficulty:1 rely:1 advanced:1 representing:1 scheme:5 technology:4 lightness:1 irrespective:1 understanding:1 acknowledgement:1 contributing:1 anya:3 highlight:2 specularity:2 analogy:1 dana:1 foundation:2 sufficient:1 rubin:2 editor:1 share:1 land:1 elsewhere:1 koftka:2 supported:1 last:1 side:3 guide:1 institute:5 neighbor:1 absolute:1 slice:3 boundary:10 world:1 contour:2 san:1 transaction:1 approximate:1 biilthoff:1 nato:1 unreliable:2 assumed:1 don:1 iterative:3 triplet:1 channel:4 ca:2 obtaining:1 spread:3 whole:1 noise:4 border:1 shafer:1 allowed:1 body:2 lie:1 ix:1 swns:1 explored:1 essential:1 workshop:1 phd:2 magnitude:2 illumination:6 specularities:8 simply:3 wavelength:2 insures:1 army:1 visual:3 ma:4 marked:3 goal:5 change:13 except:1 averaging:9 pas:1 gamble:1 formally:1 college:2 mark:4 retinex:1 support:1 illuminant:1 phenomenon:1

Bach in a Box - Real-Time Harmony Randall R. Spangler and Rodney M. Goodman* Computation and Neural Systems California Institute of Technology, 136-93 Pasadena, CA 91125 Jim Hawkins t 88B Milton Grove Stoke Newington, London N16 8QY, UK Abstract We describe a system for learning J. S. Bach's rules of musical harmony. These rules are learned from examples and are expressed as rule-based neural networks. The rules are then applied in realtime to generate new accompanying harmony for a live performer. Real-time functionality imposes constraints on the learning and harmonizing processes, including limitations on the types of information the system can use as input and the amount of processing the system can perform. We demonstrate algorithms for generating and refining musical rules from examples which meet these constraints. We describe a method for including a priori knowledge into the rules which yields significant performance gains. We then describe techniques for applying these rules to generate new music in real-time. We conclude the paper with an analysis of experimental results. 1 Introduction The goal of this research is the development of a system to learn musical rules from examples of J.S. Bach's music, and then to apply those rules in real-time to generate new music in a similar style. These algorithms would take as input a melody such *rspangle@micro.caltech.edu, rogo@micro.caltech.edu tjhawkins@cix.compulink.co.uk R. R. Spangler; R. M. Goodman and J Hawkins 958 I~II- JIJ Figure 1: Melody for Chorale #1 "Aus meines Herzens Grunde" Figure 2: J. S. Bach's Harmony For Chorale #1 as Figure 1 and produce a complete harmony such as Figure 2. Performance of this harmonization in real-time is a challenging problem. It also provides insight into the nature of composing music. We briefly review the representation of input data and the process of rule base generation. Then we focus on methods of increasing the performance of rule-based systems. Finally we present our data on learning the style of Bach. 1.1 Constraints Imposed by Real-Time Functionality A program which is to provide real-time harmony to accompany musicians at live performances faces two major constraints. First, the algorithms must be fast enough to generate accompaniment without detectable delay between the musician playing the melody and the algorithm generating the corresponding harmony. For musical instrument sounds with sharp attacks (plucked and percussive instruments, such as the harp or piano), delays of even a few tens of milliseconds between the start of the melody note and the start of the harmony notes are noticeable and distracting. This limits the complexity of the algorithm and the amount of information it can process for each timestep. Second, the algorithms must base their output only on information from previous timesteps. This differentiates our system from HARMONET (Hild, Feulnzer and Menzel, 1992) which required knowledge of the next note in the future before generating harmony for the current note. 1.2 Advantages of a Rule-Based Algorithm A rule-based neural network algorithm was chosen over a recurrent network or a non-linear feed-forward network. Neural networks have been previously used for harmonizing music with some success (Mozer, 1991)(Todd, 1989). However, rulebased algorithms have several advantages when dealing with music. Almost all music has some sort of rhythm and is tonal, meaning both pitch and duration of individual notes are quantized. This presents problems in the use of continuous networks, which must be overtrained to reasonably approximate discrete behavior. 959 Bach in a Box-Real-Time Harmony Rule-based systems are inherently discrete, and do not have this problem. Furthermore it is very difficult to determine why a non-linear multi-layer network makes a given decision or to extract the knowledge contained in such a network. However, it is straightforward to determine why a rule-based network produced a given result by examining the rules which fired. This aids development of the algorithm, since it is easier to determine where mistakes are being made. It allows comparison of the results to existing knowledge of music theory as shown below, and may provide insight into the theory of musical composition beyond that currently available. Rule-based neural networks can also be modified via segmentation to take advantage of additional a priori knowledge. 2 Background 2.1 Representation of Input Data The choice of input representation greatly affects the ability of a learning algorithm to generate meaningful rules. The learning and inferencing algorithms presented here speak an extended form of the classical figured bass representation common in Bach's time. Paired with a melody, figured bass provides a sufficient amount of information to reconstruct the harmonic content of a piece of music. Figured bass has several characteristics which make it well-disposed to learning rules. It is a symbolic format which uses a relatively small alphabet of symbols. It is also hierarchical - it specifies first the chord function that is to be played at the current note/timestep, then the scale step to be played by the bass voice, then additional information as needed to specify the alto and tenor scale steps. This allows our algorithm to fire sets of rules sequentially, to first determine the chord function which should be associated with a new melody note, and then to use that chord function as an input attribute to subsequent rulebases which determine the bass, alto, and tenor scale steps. In this way we can build up the final chord from simpler pieces, each governed by a specialized rulebase. 2.2 Generation of Rulebases Our algorithm was trained on a set of 100 harmonized Bach chorales. These were translated from MIDI format into our figured bass format by a preprocessing program which segmented them into chords at points where any voice changed pitch. Chord function was determined by simple table lookup in a table of 120 common Bach chords based on the scale steps played by each voice in the chord. The algorithm was given information on the current timestep (MelO-TeO), and the previous two timesteps (Mell-Func2). This produced a set of 7630 training examples, a subset of which are shown below: MelO D E F G FuncO V 17 IV V 800 82 81 80 80 BaO Bl B3 Bl BO AIO A2 AO A2 Al TeO TO T2 Tl T2 Mell E D E F Funcl I V 17 IV 801 81 82 81 80 Bal BO Bl B3 Bl All AO A2 AO A2 Tel T2 TO T2 Tl Me12 C E D E Func2 I I V 17 R. R. Spangler; R. M. Goodman and 1. Hawkins 960 A rulebase is a collection of rules which predict the same right hand side (RHS) attribute (for example, FunctionO). All rules have the form IF Y=y... THEN X=x. A rule's order is the number of terms on its left hand side (LHS). Rules are generated from examples using a modified version of the ITRULE algorithm. (Goodman et al., 1992) All possible rules are considered and ranked by a measure of the information contained in each rule defined as J(X; Y = y) = p(y) [P(x1Y)log (p;~~~)) + (I - p(xly))log (11-!;~~~)) ] (1) This measure trades off the amount of information a rule contains against the probability of being able to use the rule. Rules are less valuable if they contains little information. Thus, the J-measure is low when p{xly) is not much higher than p(x) . Rules are also less valuable if they fire only rarely (p(y) is small) since those rules are unlikely to be useful in generalizing to new data. A rulebase generated to predict the current chord's function might start with the following rules: 1. IF HelodyO 2. IF Function1 AND Helody1 AND HelodyO 3. IF Function1 AND HelodyO 2.3 p(corr) J-meas 0.621 0.095 E THEN FunctionO I V THEN FunctionO V7 0.624 0.051 THEN FunctionO V7 0.662 0.049 D D V D Inferencing Using Rulebases Rule based nets are a form of probabilistic graph model. When a rulebase is used to infer a value, each rule in the rule base is checked in order of decreasing rule J-measure. A rule can fire if it has not been inhibited and all the clauses on its LHS are true. When a rule fires, its weight is added to the weight of the value which it predicts, After all rules have had a chance to fire, the result is an array of weights for all predicted values. 2.4 Process of Harmonizing a Melody Input is received a note at a time as a musician plays a melody on a MIDI keyboard. The algorithm initially knows the current melody note and the data for the last two timesteps. The system first uses a rule base to determine the chord function which should be played for the current melody note. For example, given the melody note "e" , "it might playa chord function "IV", corresponding to an F -Major chord. The program then uses additional rulebases to specify how the chord will be voiced. In the example, the bass, alto, and tenor notes might be set to "BO", "AI", and "T2" , corresponding to the notes "F", "A", and "e". The harmony notes are then converted to MIDI data and sent to a synthesizer, which plays them in real-time to accompany the melody. Bach in a Box-Real-Time Harmony 3 961 Improvement of Rulebases The J-measure is a good measure for determining the information-theoretic worth of rules. However, it is unable to take into account any additional a priori knowledge about the nature of the problem - for example, that harmony rules which use the current melody note as input are more desirable because they avoid dissonance between the melody and harmony. 3.1 Segmentation A priori knowledge of this nature is incorporated by segmenting rulebases into moreand less-desirable rules based on the presence or absence of a desired LHS attribute such as the current melody note (MelodyO). Rules lacking the attribute are removed from the primary set of rules and placed in a second "fallback" set. Only in the event that no primary rules are able to fire is the secondary set allowed to fire. This gives greater impact to the primary rules (since they are used first) without the loss of domain size (since the less desirable rules are not actually deleted). Rulebase segmentation provides substantial improvements in the speed of the algorithm in addition to improving its inferencing ability. When an unsegmented rule base is fired, the algorithm has to compare the current input data with the LHS of every rule in the rulebase. However, processing for a segmented rulebase stops after the first segment which fires a rule on the input data. The algorithm does not need to spend time examining rules in lower-priority segments of that rulebase. This increase in efficiency allows segmented rule bases to contain more rules without impacting performance. The greater number of rules provides a richer and more robust knowledge base for generating harmony. 3.2 Realtime Dependency Pruning When rules are used to infer a value, the rules weights are summed to generate probabilities. This requires that all rules which are allowed to fire must be independent of one another. Otherwise, one good rule could be overwhelmed by the combined weight of twenty mediocre but virtually identical rules. To prevent this problem, each segment of a rulebase is analyzed to determine which rules are dependent with other rules in the same segment. Two rules are considered dependent if they fire together on more than half the training examples where either rule fires. For each rule, the algorithm maintains a list of lower rank rules which are dependent with the rule. This list is used in real-time dependency pruning. Whenever a rule fires on a given input, all rules dependent on it are inhibited for the duration of the input. This ensures that all rules which are able to fire for an input are independent. 3.3 Conflict Resolution When multiple rules fire and predict different values, an algorithm must be used to resolve the conflict. Simply picking the value with the highest weight, while most likely to be correct, leads to monotonous music since a given melody then always produces the same harmony. To provide a more varied harmony, our system exponentiates the accumulated rule R. R. Spangler, R. M Goodman and J Hawkins 962 Table 1: Rulebase Segments RHS FunctionO SopranoO BassO AltoO TenorO REQUIRED LHS FOR SEGMENT MelodyO, Functionl, Function2 MelodyO,Functionl MelodyO MelodyO, FunctionO FunctionO, SopranoO (none) SopranoO, BassO (none) SopranoO, BassO, AltoO. FunctionO SopranoO, Bas80, AltoO (none) RULES llO 380 346 74 125 182 267 533 52 164 115 Table 2: Rulebase Performance RHS FunctionO SopranoO Bas80 AltoO TenorO RULEBASE un8egmented segmented unsegmented # 2 un8egmented unsegmented 8egmented unsegmented #2 un8egmented segmented unsegmented #2 un8egmented segmented unsegmented #2 RULES 1825 816 428 74 307 307 162 800 800 275 331 331 180 AVG EVAL 1825 428 428 74 307 162 162 800 275 275 331 180 180 CORRECT 55% 56% 50% 95% 70% 70% 65% 63% 63% 59% 73% 74% 67% weights for the possible outcomes to produce probabilities for each value, and the final outcome is chosen randomly based on those probabilities. It is because we use the accumulated rule weights to determine these probabilities that all rules which are allowed to fire must be independent of each other. If no rules at all fire, the system uses a first-order Bayes classifier to determine the RlIS value based on the current melody note. This ensures that the system will always return an outcome compatible with the melody. 4 Results Rulebases were generated for each attribute. Up to 2048 rules were kept in each rule base. Rules were retained if they were correct at least 30% of the time they fired, and had a J-measure greater than 0.001. The rulebases were then segmented. These rulebases were tested on 742 examples derived from 27 chorales not used in the training set. The number of examples correctly inferenced is shown for each rule base before and after segmentation. Also shown is the average number of rules evaluated per test example; the speed of inferencing is proportional to this number. To determine whether segmentation was in effect only removing lower J-measure rules, we removed low-order rules from the unsegmented rule bases until they had the same average number of rules evaluated as the segmented rule bases. In all cases, segmenting the rulebases reduced the average rules fired per example without lowering the accuracy of the rule bases (in some cases, segmentation even increased accuracy). Speed gains from segmentation ranged from 80% for TenorO up to 320% for FunctionO. In comparison, simply reducing the size of the unsegmented 963 Bach in a Box-Real-Time Harmony rulebase to match the speed of the segmented rulebase reduced the number of correctly inferred examples by 4% to 6%. The generated rules for harmony have a great deal of similarity to accepted harmonic transitions (Ottman, 1989). For example, high-priority rules specify common chord transitions such as V-V7-I (a classic way to end a piece of music). 5 Remarks The system described in this paper meets the basic objectives described in Section 1. It learns harmony rules from examples of the music of J.S. Bach. The system is then able to harmonize melodies in real-time. The generated harmonies are sometimes surprising (such as the diminished 7th chord near the end of "Happy Birthday"), yet are consistent with Bach harmony. I 1\ .. I I I Figure 3: Algorithm's Bach-Like Harmony for "Happy Birthday" Rulebase segmentation is an effective method for incorporating a priori knowledge into learned rulebases. It can provides significant speed increases over unsegmented rule bases with no loss of accuracy. Acknowledgements Randall R. Spangler is supported in part by an NSF fellowship. References J. Bach (Ed.: A. Riemenschneider) (1941) 371 Harmonized Chorales and 96 Chorale Melodies. Milwaukee, WI: G. Schirmer. H. Hild, J. Feulner & W. Menzel. (1992) HARMONET: A Neural Net for Harmonizing Chorales in the Style of J. S. Bach. In J. Moody (ed.), Advances in Neural Information Processing Systems 4,267-274. San Mateo, CA: Morgan Kaufmann. M. Mozer, T. Soukup. {1991} Connectionist Music Composition Based on Melodic and Stylistic Constraints. In R. Lippmann (ed.), Advances in Neural Information Processing Systems 3. San Mateo, CA: Morgan Kaufmann. P. Todd. (1989) A Connectionist Approach to Algorithmic Composition. Computer Music Joumal13(4}:27-43. R. Goodman, P. Smyth, C. Higgins, J. Miller. {1992} Rule-Based Neural Networks for Classification and Probability Estimation. Neural Computation 4(6}:781-804. R. Ottman. (1989) Elementary Harmony. Englewood Cliffs, NJ: Prentice Hall. 

1470 |@word version:1 briefly:1 llo:1 contains:2 accompaniment:1 existing:1 feulner:1 current:10 surprising:1 synthesizer:1 yet:1 must:6 subsequent:1 half:1 rulebase:15 provides:5 quantized:1 attack:1 simpler:1 harmonize:1 behavior:1 multi:1 decreasing:1 resolve:1 little:1 increasing:1 alto:3 riemenschneider:1 nj:1 every:1 classifier:1 uk:2 segmenting:2 before:2 todd:2 limit:1 mistake:1 cliff:1 meet:2 birthday:2 might:3 au:1 mateo:2 challenging:1 co:1 melodic:1 symbolic:1 mediocre:1 prentice:1 live:2 applying:1 function2:1 imposed:1 musician:3 straightforward:1 duration:2 resolution:1 rule:94 insight:2 array:1 higgins:1 stoke:1 classic:1 play:2 speak:1 smyth:1 us:4 predicts:1 ensures:2 bass:7 trade:1 removed:2 chord:15 valuable:2 highest:1 substantial:1 mozer:2 complexity:1 trained:1 segment:6 efficiency:1 translated:1 alphabet:1 fast:1 describe:3 london:1 effective:1 outcome:3 richer:1 spend:1 reconstruct:1 otherwise:1 ability:2 final:2 advantage:3 net:2 jij:1 basso:3 fired:4 bao:1 produce:3 generating:4 recurrent:1 inferencing:4 received:1 noticeable:1 predicted:1 xly:2 correct:3 functionality:2 attribute:5 func2:2 melody:20 ao:3 itrule:1 harmonization:1 elementary:1 mell:2 accompanying:1 hild:2 hawkins:4 considered:2 hall:1 great:1 algorithmic:1 predict:3 major:2 a2:4 estimation:1 harmony:24 currently:1 teo:2 always:2 modified:2 harmonizing:4 avoid:1 derived:1 focus:1 refining:1 improvement:2 rank:1 greatly:1 dependent:4 accumulated:2 unlikely:1 initially:1 pasadena:1 classification:1 priori:5 impacting:1 development:2 summed:1 harmonet:2 dissonance:1 identical:1 future:1 t2:5 connectionist:2 micro:2 few:1 inhibited:2 randomly:1 individual:1 fire:16 spangler:5 englewood:1 eval:1 analyzed:1 grove:1 lh:5 iv:3 desired:1 monotonous:1 increased:1 aio:1 subset:1 delay:2 examining:2 menzel:2 dependency:2 combined:1 probabilistic:1 off:1 picking:1 together:1 moody:1 priority:2 v7:3 style:3 return:1 account:1 converted:1 lookup:1 piece:3 start:3 sort:1 maintains:1 bayes:1 rodney:1 voiced:1 accuracy:3 musical:5 characteristic:1 kaufmann:2 miller:1 yield:1 produced:2 none:3 worth:1 whenever:1 checked:1 ed:3 against:1 associated:1 gain:2 stop:1 knowledge:9 segmentation:8 actually:1 feed:1 higher:1 specify:3 evaluated:2 box:4 furthermore:1 until:1 hand:2 unsegmented:9 fallback:1 disposed:1 b3:2 effect:1 contain:1 true:1 ranged:1 deal:1 rhythm:1 bal:1 distracting:1 complete:1 demonstrate:1 theoretic:1 meaning:1 harmonic:2 common:3 specialized:1 clause:1 function1:2 significant:2 cix:1 composition:3 ai:1 had:3 similarity:1 base:13 playa:1 milwaukee:1 keyboard:1 success:1 caltech:2 morgan:2 additional:4 greater:3 performer:1 determine:10 ii:1 multiple:1 sound:1 desirable:3 infer:2 segmented:9 match:1 bach:16 melo:2 paired:1 impact:1 pitch:2 basic:1 sometimes:1 qy:1 background:1 addition:1 fellowship:1 goodman:6 accompany:2 virtually:1 sent:1 near:1 presence:1 enough:1 affect:1 timesteps:3 whether:1 tonal:1 remark:1 useful:1 amount:4 ten:1 reduced:2 generate:6 specifies:1 nsf:1 millisecond:1 correctly:2 per:2 discrete:2 harp:1 deleted:1 prevent:1 figured:4 kept:1 lowering:1 timestep:3 graph:1 almost:1 stylistic:1 realtime:2 decision:1 layer:1 played:4 newington:1 constraint:5 rulebased:1 speed:5 format:3 relatively:1 percussive:1 wi:1 randall:2 previously:1 detectable:1 differentiates:1 needed:1 know:1 milton:1 instrument:2 end:2 available:1 apply:1 hierarchical:1 voice:3 overtrained:1 music:14 soukup:1 build:1 classical:1 bl:4 objective:1 added:1 primary:3 unable:1 retained:1 happy:2 difficult:1 twenty:1 perform:1 extended:1 incorporated:1 jim:1 varied:1 sharp:1 inferred:1 required:2 conflict:2 california:1 learned:2 beyond:1 able:4 below:2 program:3 including:2 event:1 ranked:1 chorale:7 technology:1 extract:1 review:1 piano:1 acknowledgement:1 determining:1 lacking:1 loss:2 generation:2 limitation:1 proportional:1 sufficient:1 consistent:1 imposes:1 playing:1 compatible:1 changed:1 placed:1 last:1 supported:1 side:2 institute:1 face:1 transition:2 forward:1 made:1 collection:1 preprocessing:1 avg:1 san:2 approximate:1 pruning:2 lippmann:1 midi:3 dealing:1 sequentially:1 conclude:1 tenor:3 continuous:1 why:2 table:4 learn:1 nature:3 robust:1 ca:3 composing:1 reasonably:1 inherently:1 tel:1 funco:1 improving:1 domain:1 rh:3 allowed:3 tl:2 aid:1 governed:1 learns:1 removing:1 symbol:1 meas:1 list:2 incorporating:1 corr:1 overwhelmed:1 easier:1 generalizing:1 simply:2 likely:1 expressed:1 contained:2 bo:3 chance:1 goal:1 harmonized:2 absence:1 content:1 diminished:1 determined:1 reducing:1 secondary:1 accepted:1 experimental:1 meaningful:1 rarely:1 tested:1

A Model of Early Visual Processing Laurent Itti, Jochen Braun, Dale K. Lee and Christof Koch {itti, achim, jjwen, koch}Gklab.caltech.edu Computation & Neural Systems, MSC 139-74 California Institute of Technology, Pasadena, CA 91125, U.S.A. Abstract We propose a model for early visual processing in primates. The model consists of a population of linear spatial filters which interact through non-linear excitatory and inhibitory pooling. Statistical estimation theory is then used to derive human psychophysical thresholds from the responses of the entire population of units. The model is able to reproduce human thresholds for contrast and orientation discrimination tasks, and to predict contrast thresholds in the presence of masks of varying orientation and spatial frequency. 1 INTRODUCTION A remarkably wide range of human visual thresholds for spatial patterns appears to be determined by the earliest stages of visual processing, namely, orientation- and spatial frequency-tuned visual filters and their interactions [18, 19, 3, 22, 9]. Here we consider the possibility of quantitatively relating arbitrary spatial vision thresholds to a single computational model. The success of such a unified account should reveal the extent to which human spatial vision indeed reflects one particular stage of processing. Another motivation for this work is the controversy over the neural circuits that generate orientation and spatial frequency tuning in striate cortical neurons (13, 8, 2]. We think it is likely that behaviorally defined visual filters and their interactions reveal at least some of the characteristics of the underlying neural circuitry. Two specific problems are addressed: (i) what is the minimal set of model components necessary to account for human spatial vision, (ii) is there a general decision strategy which relates model responses to behavioral thresholds and which obviates case-by-case assumptions about the decision strategy in different behavioral situations. To investigate these questions, we propose a computational model articulated around three main stages: first, a population of bandpass linear filters extracts visual features from a stimulus; second, linear filters interact through non-linear excitatory and inhibitory pooling; third, a noise model and decision strategy are assumed in order to relate the model's output to psychophysical data. 174 2 L Itti, 1. Braun, D. K. Lee and C. Koch MODEL We assume spatial visual filters tuned for a variety of orientations e E e and spatial periods A E A. The filters have overlapping receptive fields in visual space. Quadrature filter pairs, p{~(r and F{~d, are used to compute a phase-independent linear energy response, E>.,6, to a visual stimulus S. A small constant background activity, f, is added to the linear energy responses: E>. ,6 = \I . /(peven >' ,6 * S)2 + (podd * S)2 + f >.,6 Filters have separable Gaussian tuning curves in orientation and spatial frequency. Their corresponding shape in visual space is close to that of Gabor filters, although not separable along spatial dimensions. 2.1 Pooling: self excitation and divisive inhibition A model based on linear filters alone would not correctly account for the non-linear response characteristics to stimulus contrast which have been observed psychophysically [19]. Several models have consequently introduced a non-linear transducer stage following each linear unit [19]. A more appealing possibility is to assume a non-linear pooling stage [6, 21, 3, 22]. In this study, we propose a pooling strategy inspired by Heeger's model for gain control in cat area VI [5, 6]. The pooled response R>.,6 of a unit tuned for (A, 0) is computed from the linear energy responses of the entire population: E'Y R>. >',6 + 1] (1) ,6 - So + L>'I,61W>.,6(N, OI)E~/,61 where the sum is taken over the entire population and W>.,6 is a two-dimensional Gaussian weighting function centered around (A,O), and 1] a background activity. The numerator in Eq. 1 represents a non-linear self-excitation term. The denominator represents a divisive inhibitory term which depends not only on the activity of the unit (A,O) of interest, but also on the responses of other units . We shall see in Section 3 that, in contrast to Heeger's model for electrophysiological data in which all units contribute equally to the pool, it is necessary to assume that only a subpopulation of units with tuning close to (A, 0) contribute to the pool in order to account for psychophysical data. Also, we assume, > 15 to obtain a power law for high contrasts [7], as opposed to Heeger's physiological model in which, = 15 = 2 to account for neuronal response saturation at high contrasts. Several interesting properties result from this pooling model. First, a sigmoidal transducer function - in agreement with contrast discrimination psychophysics - is naturally obtained through pooling and thus need not be introduced post-hoc. The transducer slope for high contrasts is determined by ,-15, the location of its inflexion point by 5, and the slope at this point by the absolute value of, (and 15). Second, the tuning curves of the pooled units for orientation and spatial period do not depend of stimulus contrast, in agreement with physiological and psychophysical evidence [14]. In comparison, a model which assumes a non-linear transducer but no pooling exhibits sharper tuning curves for lower contrasts. Full contrast independence of the tuning is achieved only when all units participate in the inhibitory pool; when only sub-populations participate in the pool, some contrast dependence remains. 2.2 Noise model: Poisson lX It is necessary to assume the presence of noise in the system in order to be able to derive psychophysical performance from the responses of the population of pooled A Model of Early Visual Processing 175 units. The deterministic response of each unit then represents the mean of a randomly distributed "neuronal" response which varies from trial to trial in a simulated psychophysical experiment . Existing models usually assume constant noise variance in order to simplify the subsequent decision stage [18]. Using the decision strategy presented below, it is however possible to derive psychophysical performance with a noise model whose variance increases with mean activity, in agreement with electrophysiology [16]. In what follows, Poisson cx noise will be assumed and approximated by a Gaussian random variable with variance mean cx (0' is a constant close to unity). = 2.3 Decision strategy We use tools from statistical estimation theory to compute the system's behavioral response based on the responses of the population of pooled units. Similar tools have been used by Seung and Sompolinsky [12] under the simplifying assumption of purely Poisson noise and for the particular task of orientation discrimination in the limit of an infinite population of oriented units. Here, we extend this framework to the more general case in which any stimulus attribute may differ between the two stimulus presentations to be discriminated by the model. Let's assume that we want to estimate psychophysical performance at discriminating between two stimuli which differ by the value of a stimulus parameter ((e.g . contrast, orientation, spatial period). The central assumption of our decision strategy is that the brain implements an unbiased efficient statistic T(R; (), which is an estimator of the parameter ( based {R).,I/; A E A, () E 0}. The efficient statistic is on the population response R the one which, among all possible estimators of (, has the property of minimum variance in the estimated value of ( . Although we are not suggesting any putative neuronal correlate for T, it is important to note that the assumption of efficient statistic does not require T to be prohibitively complex; for instance, a maximum likelihood estimator proposed in the decision stage of several existing models is asymptotically (with respect to the number of observations) a efficient statistic. = Because T is efficient, it achieves the Cramer-Rao bound [1]. Consequently, when the number of observations (i .e. simulated psychophysical trials) is large, E[T] = ( and var[T] = 1/3(() where E[.] is the mean over all observations, var[.] the variance, and 3(() is the Fisher information. The Fisher information can be computed using the noise model assumption and tuning properties of the pooled units: for a random variable X with probability density f(x; (), it is given by [1]: J(() = E [:c In/(X;()r For our Poisson cx noise model and assuming that different pooled units are independent [15], this translates into: One unit R). ,I/: All independent units: The Fisher information computed for each pooled unit and three types of stimulus parameters ( is shown in Figure 1. This figure demonstrates the importance of using information from all units in the population rather than from only one unit optimally tuned for the stimulus: although the unit carrying the most information about contrast is the one optimally tuned to the stimulus pattern, more information L. lui, 1 Braun, D. K. Lee and C. Koch 176 about orientation or spatial frequency is carried by units which are tuned to flanking orientations and spatial periods and whose tuning curves have maximum slope for the stimulus rather than maximum absolute sensitivity. In our implementation, the derivatives of pooled responses used in the expression of Fisher information are computed numerically. orientation spatial frequency Figure 1: Fisher information computed for contrast, orientation and spatial frequency. Each node in the tridimensional meshes represents the Fisher information for the corresponding pooled unit (A, B) in a model with 30 orientations and 4 scales. Arrows indicate the unit (A, B) optimally tuned to the stimulus. The total Fisher information in the population is the sum of the information for all units. Using the estimate of ( and its variance from the Fisher information, it is possible to derive psychophysical performance for a discrimination task between two stimuli with parameters (1 ~ (2 using standard ideal observer signal discrimination techniques [4] . For such discrimination, we use the Central Limit Theorem (in the limit of large number of trials) to model the noisy responses of the system as two Gaussians with means (1 and (2, and variances lTi 1/:1((d and lTi 1/:1((2) respectively. A decision criterion D is chosen to minimize the overall probability of error; since in our case lT1 =f. lT2 in general, we derive a slightly more complicated expression for performance P at a Yes/No (one alternative forced choice) task than what is commonly used with models assuming constant noise [18]: = D = (2 lT i - (llT~ - = lT1lT2J((1 - (2)2 + 2(lTr - lTi) log(lT!/lT2) 2 2 lT1 - lT2 P= ~+~erf((2-D) 2 4 lT2..J2 + ~erf(D-(l) 4 lT1..J2 where erf is the Normal error function. The expression for D extends by continuity to D = ((2 - (1)/2 when lT1 = lT2 . This decision strategy provides a unified, taskindependent framework for the computation of psychophysical performance from the deterministic responses of the pooled units. This strategy can easily be extended to allow the model to perform discrimination tasks with respect to additional stimulus parameters, under exactly the same theoretical assumptions. 3 3.1 RESULTS Model calibration The parameters of the model were automatically adjusted to fit human psychophysical thresholds measured in our laboratory [17] for contrast and orientation discrimination tasks (Figure 2). The model used in this experiment consisted of 60 orientations evenly distributed between 0 and 180deg. One spatial scale at 4 cycles per degree (cpd) was sufficient to account for the data. A multidimensional simplex method with simulated annealing overhead was used to determine the best fit of the model to the data [10]. The free parameters adjusted during the automatic A Model of Early VlSUal Processing 177 fits were: the noise level a, the pooling exponents 'Y and &, the inhibitory pooling constant 5, and the background firing rates, E and rJ. The error function minimized by the fitting algorithm was a weighted average of three constraints: 1) least-square error with the contrast discrimination data in Figure 2.a; 2) least-square error with the orientation discrimination data in Figure 2.h; 3) because the data was sparse in the "dip-shaped" region of the curve in Figure 2.a, and unreliable due to the limited contrast resolution of the display used for the psychophysics, we added an additional constraint favoring a more pronounced "dip", as has been observed by several other groups [11, 19, 22] . Data fits used for model calibration: a . -_ _ _ _ _---..:a:::.., iii ~u ~ ~ 0- ~ ~ 10-2 c:Ch Q)~ E :5 10-3 L..-_ _ _ _ _ _--...J oc: Q) 10 .- ?2 10 0 0.2 0.4 stimulus contrast mask contrast Transducer function: - ~50.----________c~ Q) C o a. ~ ~ ~ 0.5 Q) > uQ) 8. d Q) Ch o a. Ch c: (5 I; Orientation tuning: 0.5 stimulus contrast ~ ~ O~~----~----=-~ -100 0 100 stimulUS orientation (deg) Figure 2: The model (solid lines) was calibrated using data from two psychophysical experiments: (a) discrimination between a pedestal contrast (a.a) and the same pedestal plus an increment contrast (a.{3); (b) discrimination between two orientations near vertical (b.a and b.{3). After calibration, the transducer function of each pooled unit (c) correctly exhibits an accelerating non-linearity near threshold (contrast ~ 1%) and compressive non-linearity for high contrasts (Weber's law). We can see in (d) that pooling among units with similar tuning properties sharpens their tuning curves. Model parameters were: a ~ 0.75,,), ~ 4,?5 ~ 3.5,E ~ 1%, '1 ~ 1.7Hz,S such that transducer inflexion point is at 4x detection threshold contrast, orientation tuning FWHM=68deg (full width at half maximum), orientation pooling FWHM=40deg. Two remaining parameters are the orientation tuning width, (7'8, of the filters and the width, (7'We, of the pool. It was not possible from the data in Figure 2 alone to unambiguously determine these parameters. However, for any given (7'8, (7'W8 is uniquely determined by the following two qualitative constraints: first, a small pool size is not desirable because it yields contrast-dependent orientation tuning; it however appears from the data in Figure 2.h that this tuning should not vary much over a wide range of contrasts. The second constraint is qualitatively derived from Figure 3.a: for large pool sizes, the model predicted significant interference between mask and test patterns even for large orientation differences. Such inter- 178 L Itti, 1. Braun, D. K. Lee and C. Koch ference was not observed in the data for orientation differences larger than 45deg . It consequently seems that a partial inhibitory pool, composed only of a fraction of the population of oriented filters with tuning similar to the central excitatory unit, accounts best for the psychophysical data. Finally, (76 was fixed so as to yield a correct qualitative curve shape for Figure 3.a. 3.2 Predictions We used complex stimuli from masking experiments to test the predictive value of the model (Figure 3). Although it was necessary to use some of the qualitative properties of the data seen in Figure 3.a to calibrate the model as detailed above, the calibrated model correctly produced a quantitative fit of this data. The calibrated model also correctly predicted the complex data of Figure 3.h. a c::10 0 a ~ > Q) ~ > Q) CD 5 Q) ~ "C (5 .r; (/J Q) ~ .r; ...... c:: 0 no mask orientation (deg) mask 0 30 60 90 "C (5 b 6 4 2 .r; (/J Q) ~ .r; ...... 2 4 8 mask spatial freq. (cpd) Figure 3: Prediction of psychophysical contrast thresholds in the presence of an oblique mask. The mask was a 50%-contrast stochastic oriented pattern (a). and the superimposed test pattern was a sixth-derivative of Gaussian bar (j3). In (a), threshold elevation (i.e. ratio of threshold in the presence of mask to threshold in the absence of mask) was measured for varying mask orientation, for mask and test patterns at 4 cycles per degree (cpd). In (b), orientation difference between test and mask was fixed to 15deg, and threshold elevation was measured as a function of mask spatial frequency. Solid lines represent model predictions, and dashed lines represent unity threshold elevation. 4 DISCUSSION AND CONCLUSION We have developed a model of early visual processing in humans which accounts for a wide range of measured spatial vision thresholds and which predicts behavioral thresholds for a potentially unlimited number of spatial discriminations. In addition to orientation- and spatial-frequency-tuned units, we have found it necessary to assume two types of interactions between such units: (i) non-linear self-excitation of each unit and (ii) divisive normalization of each unit response relative to the responses of similarly tuned units. All model parameters are constrained by psychophysical data and an automatic fitting procedure consistently converged to the same parameter set regardless of the initial position in parameter space. Our two main contributions are the small number of model components and the un i.fied, task-independent decision strategy. Rather than making different assumptions about the decision strategy in different behavioral tasks, we combine the information contained in the responses of all model units in a manner that is optimal for any behavioral task. We suggest that human observers adopt a similarly optimal decision procedure as they become familiar with a particular task (" task set"). Although here we apply this decision strategy only to the discrimination of stimulus contrast, orientation, and spatial frequency, it can readily be generalized to arbitrary discriminations such as, for example, the discrimination of vernier targets. A Model of Early Vzsual Processing 179 So far we have considered only situations in which the same decision strategy is optimal for every stimulus presentation. We are now studying situations in which the optimal decision strategy varies unpredictably from trial to trial (" decision uncertainty"). For example, situations in which the observer attempts to detect an increase in either the spatial frequency or the contrast of stimulus. In this way, we hope to learn the extent to which our model reflects the decision strategy adopted by human observers in an even wider range of situations. We have also assumed that the model's units were independent, which is not strictly true in biological systems (although the main source of correlation between neurons is the overlap between their respective tuning curves, which is accounted for in the model). The mathematical developments necessary to account for fixed or variable covariance between units are currently under study. In contrast to other models of early visual processing [5, 6], we find that the psychophysical data is consistent only with interactions between similarly tuned units (e.g., "near-orientation inhibition")' not with interactions between units of very different tuning (e.g., "cross-orientation inhibition") . Although such partial pooling does not render tuning functions completely contrast-independent, an additional degree of contrast-independence could be provided by pooling across different spatial locations. This issue is currently under investigation. In conclusion, we have developed a model based on self-excitation of each unit, divisive normalization [5, 6] between similarly tuned units, and an ideal observer decision strategy. It was able to reproduce a wide range of human visual thresholds. The fact that such a simple and idealized model can account quantitatively for a wide range of psychophysical observations greatly strengthens the notion that spatial vision thresholds reflect processing at one particular neuroanatomical level. Acknowledgments: This work was supported by NSF-Engineering Research Center (ERC), NIMH, ONR, and the Sloan Center for Theoretical Neurobiology. References [1] Cover TM, Thomas JA. Elem Info Theo, Wiley & Sons, 1991 [2] Ferster D, Chung S, Wheat H. Nature 1996;380(6571):249-52 [3] Foley JM. J Opt Soc A 1994;11(6):1710-9 [4] Green DM, Swets JA. Signal Detectability and Psychophys. Wiley & Sons, 1966. [5] Heeger DJ. Comput Models of Vis Processing, MIT Press, 1991 [6] Heeger DJ . Vis Neurosci 1992;9:181-97 [7] Nachmias J, Sansbury RV. Vis Res 1974;14:1039-42 [8] Nelson S, Toth L, Sheth B, Sur M. Science 1994;265(5173):774-77 [9] Perona P, Malik J. J Opt Soc A 1990;7(5):923-32 [10] Press WH, Teukolsky SA, et al. Num Rec in C. Cambridge University Press, 1992 [ll] Ross J, Speed HD. Proc R Soc B 1991;246:61-9 [12] Seung HS, Sompolinksy H. Proc Natl Acad Sci USA 1993;90:10749-53. [13] Sillito AM. Progr Brain Res 1992;90:349-84 [14] Skottun BC, Bradley A, Sclar G et al. J Neurophys 1987;57(3):773-86 [15] Snippe HP, Koenderink JJ. Bioi Cybern 1992;67:183-90 [16] Teich MC, Thrcott RG, Siegel RM. IEEE Eng Med Bioi 1996;Sept-Oct,79-87 [17] Wen J, Koch C , Braun J. Proc ARVO 1997;5457 [18] Wilson HR, Bergen JR. Vis Res 1979; 19: 19-32 [19] Wilson HR. Bioi Cybern 1980;38: 171-8 [20] Wilson HR, McFarlane DK, Phillips GC. Vis Res 1983;23;873-82. [21] Wilson HR, Humanski R. Vis Res 1993;33(8):1133-50 [22] Zenger B, Sagi D. Vis Res 1996;36(16):2497-2513. 

1471 |@word h:1 trial:6 sharpens:1 seems:1 teich:1 simplifying:1 covariance:1 eng:1 solid:2 initial:1 tuned:11 bc:1 existing:2 bradley:1 neurophys:1 readily:1 mesh:1 subsequent:1 shape:2 discrimination:16 alone:2 half:1 oblique:1 num:1 provides:1 contribute:2 location:2 lx:1 node:1 sigmoidal:1 mathematical:1 along:1 become:1 transducer:7 consists:1 qualitative:3 overhead:1 fitting:2 behavioral:6 combine:1 manner:1 swets:1 inter:1 mask:14 indeed:1 brain:2 inspired:1 automatically:1 jm:1 provided:1 zenger:1 underlying:1 linearity:2 circuit:1 what:3 developed:2 compressive:1 unified:2 w8:1 quantitative:1 multidimensional:1 every:1 braun:5 exactly:1 prohibitively:1 demonstrates:1 rm:1 control:1 unit:41 christof:1 engineering:1 sagi:1 limit:3 acad:1 laurent:1 firing:1 plus:1 limited:1 range:6 fwhm:2 acknowledgment:1 implement:1 procedure:2 area:1 gabor:1 subpopulation:1 suggest:1 close:3 cybern:2 deterministic:2 center:2 regardless:1 resolution:1 estimator:3 hd:1 population:13 notion:1 increment:1 target:1 vzsual:1 agreement:3 approximated:1 strengthens:1 rec:1 predicts:1 observed:3 region:1 wheat:1 cycle:2 sompolinsky:1 nimh:1 seung:2 controversy:1 depend:1 carrying:1 predictive:1 purely:1 completely:1 easily:1 cat:1 articulated:1 forced:1 whose:2 larger:1 arvo:1 statistic:4 erf:3 think:1 noisy:1 hoc:1 propose:3 interaction:5 j2:2 pronounced:1 wider:1 derive:5 measured:4 sa:1 eq:1 soc:3 predicted:2 indicate:1 differ:2 correct:1 attribute:1 filter:13 stochastic:1 snippe:1 centered:1 human:10 require:1 ja:2 investigation:1 elevation:3 opt:2 biological:1 adjusted:2 strictly:1 koch:6 around:2 cramer:1 normal:1 considered:1 predict:1 circuitry:1 achieves:1 early:7 vary:1 adopt:1 estimation:2 proc:3 currently:2 ross:1 tool:2 reflects:2 weighted:1 hope:1 mit:1 behaviorally:1 gaussian:4 rather:3 varying:2 wilson:4 earliest:1 derived:1 consistently:1 likelihood:1 superimposed:1 greatly:1 contrast:35 detect:1 am:1 dependent:1 bergen:1 entire:3 achim:1 pasadena:1 perona:1 favoring:1 reproduce:2 overall:1 among:2 orientation:33 issue:1 sheth:1 exponent:1 development:1 ference:1 spatial:28 constrained:1 psychophysics:2 field:1 shaped:1 represents:4 jochen:1 simplex:1 minimized:1 stimulus:22 simplify:1 quantitatively:2 cpd:3 wen:1 randomly:1 oriented:3 composed:1 inflexion:2 familiar:1 phase:1 attempt:1 detection:1 interest:1 possibility:2 investigate:1 natl:1 partial:2 necessary:6 respective:1 re:6 theoretical:2 minimal:1 instance:1 rao:1 cover:1 calibrate:1 optimally:3 varies:2 psychophysically:1 calibrated:3 density:1 sensitivity:1 discriminating:1 lee:4 pool:8 central:3 reflect:1 opposed:1 derivative:2 itti:4 chung:1 koenderink:1 account:10 suggesting:1 pedestal:2 pooled:11 sloan:1 vi:8 depends:1 idealized:1 observer:5 complicated:1 masking:1 slope:3 contribution:1 minimize:1 oi:1 square:2 variance:7 characteristic:2 yield:2 yes:1 produced:1 mc:1 converged:1 llt:1 sixth:1 energy:3 frequency:11 dm:1 naturally:1 gain:1 wh:1 electrophysiological:1 appears:2 unambiguously:1 response:21 stage:7 msc:1 correlation:1 overlapping:1 continuity:1 reveal:2 usa:1 consisted:1 unbiased:1 true:1 progr:1 laboratory:1 freq:1 ll:1 numerator:1 self:4 during:1 width:3 uniquely:1 excitation:4 oc:1 criterion:1 generalized:1 weber:1 discriminated:1 extend:1 relating:1 numerically:1 elem:1 significant:1 cambridge:1 phillips:1 tuning:19 automatic:2 hp:1 similarly:4 erc:1 dj:2 calibration:3 inhibition:3 onr:1 success:1 caltech:1 seen:1 minimum:1 additional:3 determine:2 period:4 signal:2 ii:2 relates:1 full:2 desirable:1 rj:1 dashed:1 rv:1 cross:1 post:1 equally:1 prediction:3 j3:1 denominator:1 vision:5 poisson:4 represent:2 normalization:2 achieved:1 background:3 remarkably:1 want:1 addition:1 addressed:1 annealing:1 source:1 pooling:14 hz:1 med:1 near:3 presence:4 ideal:2 iii:1 variety:1 independence:2 fit:5 tm:1 translates:1 expression:3 accelerating:1 render:1 jj:1 detailed:1 generate:1 nsf:1 inhibitory:6 estimated:1 correctly:4 per:2 detectability:1 shall:1 group:1 threshold:19 lti:3 asymptotically:1 fraction:1 sum:2 uncertainty:1 extends:1 putative:1 decision:19 bound:1 display:1 activity:4 constraint:4 unlimited:1 speed:1 separable:2 nachmias:1 jr:1 across:1 slightly:1 son:2 unity:2 appealing:1 primate:1 making:1 lt2:5 flanking:1 interference:1 taken:1 remains:1 studying:1 adopted:1 gaussians:1 apply:1 uq:1 alternative:1 thomas:1 obviates:1 assumes:1 remaining:1 neuroanatomical:1 humanski:1 psychophysical:18 malik:1 question:1 added:2 strategy:16 receptive:1 dependence:1 striate:1 exhibit:2 simulated:3 sci:1 participate:2 evenly:1 nelson:1 extent:2 assuming:2 sur:1 ratio:1 sharper:1 relate:1 potentially:1 vernier:1 info:1 implementation:1 perform:1 vertical:1 neuron:2 observation:4 situation:5 extended:1 neurobiology:1 gc:1 arbitrary:2 introduced:2 namely:1 pair:1 california:1 able:3 bar:1 usually:1 pattern:6 below:1 psychophys:1 mcfarlane:1 saturation:1 green:1 power:1 overlap:1 hr:4 technology:1 carried:1 extract:1 foley:1 sept:1 relative:1 law:2 interesting:1 var:2 degree:3 ltr:1 sufficient:1 consistent:1 cd:1 excitatory:3 accounted:1 supported:1 free:1 theo:1 allow:1 institute:1 wide:5 absolute:2 sparse:1 distributed:2 curve:8 dimension:1 cortical:1 dip:2 dale:1 commonly:1 qualitatively:1 far:1 correlate:1 unreliable:1 deg:7 assumed:3 un:1 sillito:1 learn:1 nature:1 toth:1 ca:1 interact:2 complex:3 main:3 neurosci:1 arrow:1 motivation:1 noise:11 quadrature:1 fied:1 neuronal:3 siegel:1 lt1:4 wiley:2 sub:1 position:1 heeger:5 bandpass:1 comput:1 third:1 weighting:1 theorem:1 specific:1 dk:1 physiological:2 evidence:1 importance:1 rg:1 cx:3 electrophysiology:1 lt:2 likely:1 visual:15 contained:1 sclar:1 ch:3 teukolsky:1 oct:1 bioi:3 presentation:2 consequently:3 ferster:1 fisher:8 absence:1 determined:3 infinite:1 lui:1 unpredictably:1 total:1 divisive:4

Hierarchical Non-linear Factor Analysis and Topographic Maps Zoubin Ghahramani and Geoffrey E. Hinton Dept. of Computer Science, University of Toronto Toronto, Ontario, M5S 3H5, Canada http://www.cs.toronto.edu/neuron/ {zoubin,hinton}Ocs.toronto.edu Abstract We first describe a hierarchical, generative model that can be viewed as a non-linear generalisation of factor analysis and can be implemented in a neural network. The model performs perceptual inference in a probabilistically consistent manner by using top-down, bottom-up and lateral connections. These connections can be learned using simple rules that require only locally available information. We then show how to incorporate lateral connections into the generative model. The model extracts a sparse, distributed, hierarchical representation of depth from simplified random-dot stereograms and the localised disparity detectors in the first hidden layer form a topographic map. When presented with image patches from natural scenes, the model develops topographically organised local feature detectors. 1 Introduction Factor analysis is a probabilistic model for real-valued data which assumes that the data is a linear combination of real-valued uncorrelated Gaussian sources (the factors). After the linear combination, each component of the data vector is also assumed to be corrupted by additional Gaussian noise. A major advantage of this generative model is that, given a data vector, the probability distribution in the space of factors is a multivariate Gaussian whose mean is a linear function of the data. It is therefore tractable to compute the posterior distribution exactly and to use it when learning the parameters of the model (the linear combination matrix and noise variances). A major disadvantage is that factor analysis is a linear model that is insensitive to higher order statistical structure of the observed data vectors. One way to make factor analysis non-linear is to use a mixture of factor analyser modules, each of which captures a different linear regime in the data [3]. We can view the factors of all of the modules as a large set of basis functions for describing the data and the process of selecting one module then corresponds to selecting an appropriate subset of the basis functions. Since the number of subsets under consideration is only linear in the number of modules, it is still tractable to compute Hierarchical Non-linear Factor Analysis and Topographic Maps 487 the full posterior distribution when given a data point. Unfortunately, this mixture model is often inadequate. Consider, for example, a typical image that contains multiple objects. To represent the pose and deformation of each object we want a componential representation of the object's parameters which could be obtained from an appropriate factor analyser. But to represent the multiple objects we need several of these componential representations at once, so the pure mixture idea is not tenable. A more powerful non-linear generalisation of factor analysis iF to have a large set of factors and to allow any subset of the factors to be selected. This can be achieved by using a generative model in which there is a high probability of generating factor activations of exactly zero. 2 Rectified Gaussian Belief Nets The Rectified Gaussian Belief Net (RGBN) uses multiple layers of units with states that are either positive real values or zero [5]. Its main disadvantage is that computing the posterior distribution over the factors given a data vector involves Gibbs sampling. In general, Gibbs sampling can be very time consuming, but in practice 10 to 20 samples per unit have proved adequate and there are theoretical reasons for believing that learning can work well even when the Gibbs sampling fails to reach equilibrium [10]. We first describe the RGBN without considering neural plausibility. Then we show how lateral interactions within a layer can be used to perform probabilistic inference correctly using locally available information. This makes the RGBN far more plausible as a neural model than a sigmoid belief net [9, 8] because it means that Gibbs sampling can be performed without requiring units in one layer to see the total top-down input to units in the layer below. The generative model for RGBN's consists of multiple layers of units each of which has a real-valued unrectified state, Yj, and a rectified state, [Yj]+, which is zero if Yj is negative and equal to Yj otherwise. This rectification is the only non-linearity in the network. 1 The value of Yj is Gaussian distributed with a standard deviation (Jj and a mean, ih that is determined by the generative bias, gOj, and the combined effects of the rectified states of units, k, in the layer above: Yj = gOj + Lgkj[Yk]+ (1) k The rectified state [Yj]+ therefore has a Gaussian distribution above zero, but all of the mass of the Gaussian that falls below zero is concentrated in an infinitely dense spike at zero as shown in Fig. la. This infinite density creates problems if we attempt to use Gibbs sampling over the rectified states, so, following a suggestion by Radford Neal, we perform Gibbs sampling on the unrectified states. Consider a unit, j, in some intermediate layer of a multilayer RGBN. Suppose that we fix the unrectified states of all the other units in the net. To perform Gibbs sampling, we need to stochastically select a value for Yj according to its distribution given the unrectified states of all the other units. If we think in terms of energy functions, which are equal to negative log probabilities (up to a constant), the rectified states of the units in the layer above contribute a quadratic energy term by determining Yj. The unrectified states of units, i, in the layer below contribute a constant if [Yj]+ is 0, and if [Yj]+ is positive they each contribute a quadratic term 1 The key arguments presented in this paper hold for general nonlinear belief networks as long as the noise is Gaussian; they are not specific to the rectification nonlinearity. Z Ghahramani and G. E. Hinton 488 a c b I I I W / ~----..J,' , Top-down , '-_ .. " -3-2-1 0 1 2 3 -3-2-1 0 1 2 3 Y because of the effect of [Yj] + on Y Figure 1: a) Probability density in which all the mass of a Gaussian below zero has been replaced by an infinitely dense spike at zero. b) Schematic of the density of a unit's unrectified state. c) Bottomup and top-down energy functions corresponding to b. Yi. (2) where h is an index over all the units in the same layer as j including j itself. Terms that do not depend on Yj have been omitted from Eq. 2. For values of Yj below zero there is a quadratic energy function which leads to a Gaussian distribution. The same is true for values of Yj above zero, but it is a different quadratic (Fig . Ic) . The Gaussian distributions corresponding to the two quadratics must agree at Yj 0 (Fig. Ib). Because this distribution is piecewise Gaussian it is possible to perform Gibbs sampling exactly. = Given samples from the posterior, the generative weights of a RGBN can be learned by using the online delta rule to maximise the log probability of the data. 2 (3) The variance of the local Gaussian noise of each unit, o}, can also be learned by an online rule, D-.o} f [(Yj - Yj)2 - o}]. Alternatively, o} can be fixed at I for all hidden units and the effective local noise level can be controlled by scaling the generative weights. = 3 The Role of Lateral Connections in Perceptual Inference In RGBNs and other layered belief networks, fixing the value of a unit in one layer causes correlations between the parents of that unit in the layer above. One of the main reasons why purely bottom-up approaches to perceptual inference have proven inadequate for learning in layered belief networks is that they fail to take into account this phenomenon, which is known as "explaining away." Lee and Seung (1997) introduced a clever way of using lateral connections to handle explaining away effects during perceptual inference. Consider the network shown in Fig. 2. One contribution, Ebelow, to the energy of the state of the network is the squared difference between the unrectified states of the units in one layer, Yj, a.nd the top-down expectations generated by the states of units in the layer above. Assuming the local noise models for the lower layer units all have unit variance, and 2 If Gibbs sampling has not been run long enough to reach equilibrium, the delta rule follows the gradient of the penalized log probability of the data [10]. The penalty term is the Kullback-Liebler divergence between the equilibrium distribution and the distribution produced by Gibbs sampling. Other things being equal, the delta rule therefore adjusts the parameters that determine the equilibrium distribution to reduce this penalty, thus favouring models for which Gibbs sampling works quickly. Hierarchical Non-linear Factor Analysis and Topographic Maps 489 ignoring biases and constant terms that are unaffected by the states of the units Ebe\ow = ~ l:)Yj - = ~ I)Yj - Yj)2 j = gkj and mkl = - Lj gkjglj Rearranging this expression and setting rjk Ebe\ow = ~ LyJ j 2:k[Yk]+9kj)2. (4) j L[Yk]+ LYjrjk k ~ L[Yk]+ L[y!l+mkl . k j we get (5) I This energy function can be exactly implemented in a network with recognition weights, rjk, and symmetric lateral interactions, mkl. The lateral and recognition connections allow a unit, k, to compute how Ebe\ow for the layer below depends on its own state and therefore they allow it to follow the gradient of E or to perform Gibbs sampling in E . Figure 2: A small segment of a network, showing the generative weights (dashed) and the recognition and lateral weights (solid) which implement perceptual inference and correctly handle explaining away effects. Seung's trick can be used in an RGBN and it eliminates the most neurally implausible aspect of this model which is that a unit in one layer appears to need to send both its state Y and the top-down prediction of its state Y to units in the layer above. Using the lateral connections, the units in the layer above can, in effect, compute all they need to know about the top-down predictions. In computer simulations, we can simply set each lateral connection mkl to be the dot product - 2: j gkjglj. It is also possible to learn these lateral connections in a more biologically plausible way by driving units in the layer below with unit-variance independent Gaussian noise and using a simple anti-Hebbian learning rule. Similarly, a purely local learning rule can learn recognition weights equal to the generative weights . .If units at one layer are driven by unit-variance, independent Gaussian noise, and these in turn drive units in the layer below using the generative weights, then Hebbian learning between the two layers will learn the correct recognition weights [5]. 4 Lateral Connections in the Generative Model When the generative model contains only top-down connections, lateral connections make it possible to do perceptual inference using locally available information. But it is also possible, and often desirable, to have lateral connections in the generative model. Such connections can cause nearby units in a layer to have a priori correlated activities, which in turn can lead to the formation of redundant codes and, as we will see, topographic maps. Symmetric lateral interactions between the unrectified states of units within a layer have the effect of adding a quadratic term to the energy function EMRF = ~ L: L k Mkl YkYI, (6) I which corresponds to a Gaussian Markov Random Field (MRF). During sampling, this term is simply added to the top-down energy contribution. Learning is more difficult. The difficulty sterns from the need to know the derivatives of the partition function of the MRF for each data vector. This partition function depends on the 490 Z Ghahramani and G. E. Hinton top-down inputs to a layer so it varies from one data vector to the next, even if the lateral connections themselves are non-adaptive . Fortunately, since both the MRF and the top-down prediction define Gaussians over the states of the units in a layer, these derivatives can be easily calculated. Assuming unit variances, tlYj; = , ([Yj]+(Y; - ii;) + [Yj]+ ~ [M(I + M)-ll;. ii.) (7) where M is the MRF matrix for the layer including units i and k, and I is the identity matrix. The first term is the delta rule (Eq. 3); the second term is the derivative of the partition function which unfortunately involves a matrix inversion. Since the partition function for a multivariate Gaussian is analytical it is also possible to learn the lateral connections in the MRF. Lateral interactions between the rectified states of units add the quadratic term ~ Lk Ll Mkl [Yk]+[YzJ+? The partition function is no longer analytical, so computing the gradient of the likelihood involves a two-phase Boltzmann-like procedure: !19ji = f ([Yj]+Yi) * - ([Yj]+Yi r) , (8) where 0* averages with respect to the posterior distribution of Yi and Yj, and 0averages with respect to the posterior distribution of Yj and the prior of Yi given units in the same layer as j. This learning rule suffers from all the problems of the Boltzmann machine, namely it is slow and requires two-phases. However, there is an approximation which results in the familiar one-phase delta rule that can be described in three equivalent ways: (1) it treats the lateral connections in the generative model as if they were additional lateral connections in the recognition model; (2) instead of lateral connections in the generative model it assumes some fictitious children with clamped values which affect inference but whose likelihood is not maximised during learning; (3) it maximises a penalized likelihood of the model without the lateral connections in the generative model. 5 Discovering depth in simplified stereograms Consider the following generative process for stereo pairs. Random dots of uniformly distributed intensities are scattered sparsely on a one-dimensional surface, and the image is blurred with a Gaussian filter. This surface is then randomly placed at one of two different depths, giving rise to two possible left-to-right disparities between the images seen by each eye. Separate Gaussian noise is then added to the image seen by each eye. Some images generated in this manner are shown in Fig. 3a. Figure 3: a) Sample data from the stereo disparity problem. The left and right column of each 2 x 32 image are the inputs to the left and right eye, respectively. Periodic boundary conditions were used. The value of a pixel is represented by the size of the square, with white being positive and black being negative. Notice that pixel noise makes it difficult to infer the disparity, i.e. the vertical shift between the left and right columns, in some images. b) Sample images generated by the model after learning. We trained a three-layer RGBN consisting of 64 visible units, 64 units in the first hidden layer and 1 unit in the second hidden layer on the 32-pixel wide stereo Hierarchical Non-linear Factor Analysis and Topographic Maps 491 disparity problem. Each of the hidden units in the first hidden layer was connected to the entire array of visible units, i.e. it had inputs from both eyes. The hidden units in this layer were also laterally connected in an MRF over the unrectified units. Nearby units excited each other and more distant units inhibited each other, with the net pattern of excitation/inhibition being a difference of two Gaussians. This MRF was initialised with large weights which decayed exponentially to zero over the course of training. The network was trained for 30 passes through a data set of 2000 images. For each image we used 16 iterations of Gibbs sampling to approximate the posterior distribution over hidden states. Each iteration consisted of sampling every hidden unit once in a random order. The states after the fourth iteration of Gibbs sampling were used for learning, with a learning rate of 0.05 and a weight decay parameter of 0.001. Since the top level of the generative process makes a discrete decision between left and right global disparity we used a trivial extension of the RGBN in which the top level unit saturates both at 0 and 1. a __ IEI[I:I_1II_-=-_.-:rr::JI...___I:IIUI::JI-L1D-.--:tIl::Jl-=-::l .-'-' _______ OW''--o--.,u'-'-''__=_-..._.-'-"._ ._--="TI:~?:I=-[J b c Figure 4: Generative weights of a three-layered RGBN after being trained on the stereo disparity problem. a) Weights from the top layer hidden unit to the 64 middle-layer hidden units. b) Biases of the middle-layer hidden units, and c) weights from the hidden units to the 2 x 32 visible array. Thirty-two of the hidden units learned to become local left-disparity detectors, while the other 32 became local right-disparity detectors (Fig. 4c). The unit in the second hidden layer learned positive weights to the left-disparity detectors in the layer below, and negative weights to the right detectors (Fig. 4a). In fact, the activity of this top unit discriminated the true global disparity of the input images with 99% accuracy. A random sample of images generated by the model after learning is shown in Fig. 3b. In addition to forming a hierarchical distributed representation of disparity, units in the hidden layer self-organised into a topographic map. The MRF caused high correlations between nearby units early in learning, which in turn resulted in nearby units learning similar weight vectors. The emergence of topography depended on the strength of the MRF and on the speed with which it decayed. Results were relatively insensitive to other parametric changes. We also presented image patches taken from natural images [1] to a network with units in the first hidden layer arranged in laterally-connected 2D grid. The network developed local feature detectors, with nearby units responding to similar features (Fig. 5). Not all units were used, but the unused units all clustered into one area. 6 Discussion Classical models of topography formation such as Kohonen's self-organising map [6] and the elastic net [2, 4] can be thought of as variations on mixture models where additional constraints have been placed to encourage neighboring hidden units to have similar generative weights . The problem with a mixture model is that it cannot handle images in which there are several things going on at once. In contrast, we 492 Z. Ghahramani and G. E. Hinton Figure 5: Generative weights of an RGBN trained on 12 x 12 natural image patches: weights from each of the 100 hidden units which were arranged in a 10 x 10 sheet with toroidal boundary conclitions. have shown that topography can arise in much richer hierarchical and componential generative models by inducing correlations between neighboring units. There is a sense in which topography is a necessary consequence of the lateral connection trick used for perceptual inference. It is infeasible to interconnect all pairs of units in a cortical area. If we assume that direct lateral interactions (or interactions mediated by interneurons) are primarily local, then widely separated units will not have the apparatus required for explaining away. Consequently the computation of the posterior distribution will be incorrect unless the generative weight vectors of widely separated units are orthogonal. If the generative weights are constrained to be positive, the only way two vectors can be orthogonal is for each to have zeros wherever the other has non-zeros. Since the redundancies that the hidden units are trying to model are typically spatially localised, it follows that widely separated units must attend to different parts of the image and units can only attend to overlapping patches if they are laterally interconnected. The lateral connections in the generative model assist in the formation of the topography required for correct perceptual inference. Acknowledgements. We thank P. Dayan, B. Frey, G. Goodhill, D. MacKay, R. Neal and M. Revow. The research was funded by NSERC and ITRC. GEH is the Nesbitt-Burns fellow of CIAR. References [1] A. Bell & T. J. Sejnowski. The 'Independent components' of natural scenes are edge filters. Vision Research, In Press . [2] R. Durbin & D. Willshaw. An analogue approach to the travelling salesman problem using an elastic net method. Nature, 326(16):689-691, 1987. [3] Z. Ghahramani & G. E. Hinton. The EM algorithm for mixtures of factor analyzers. Univ. Toronto Technical Report CRG-TR-96-1, 1996. [4] G. J . Goodhill & D. J . Willshaw. Application of the elatic net algorithm to the formation of ocular dominance stripes. Network: Compo in Neur. Sys ., 1:41-59, 1990. [5] G. E. Hinton & Z. Ghahramani. Generative models for cliscovering sparse clistributed representations. Philos. Trans. Roy. Soc . B, 352:1177-1190, 1997. [6] T. Kohonen. Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43:59-69, 1982. [7] D. D. Lee & H. S. Seung. Unsupervised learning by convex and conic cocling. In M. Mozer, M. Jordan, & T. Petsche, eds., NIPS 9. MIT Press, Cambridge, MA, 1997. [8] M. S. Lewicki & T. J. Sejnowski. Bayesian unsupervised learning of higher order structure. In NIPS 9. MIT Press, Cambridge, MA, 1997. [9] R. M. Neal. Connectionist learning of belief networks. Arti/. Intell., 56:71-113, 1992. [10] R. M. Neal & G. E. Hinton. A new view of the EM algorithm that justifies incremental and other variants. Unpublished Manuscript, 1993. 

1472 |@word middle:2 inversion:1 nd:1 simulation:1 excited:1 arti:1 tr:1 solid:1 contains:2 disparity:12 selecting:2 favouring:1 activation:1 must:2 distant:1 visible:3 partition:5 generative:27 selected:1 discovering:1 maximised:1 sys:1 compo:1 contribute:3 toronto:5 organising:1 direct:1 become:1 incorrect:1 consists:1 manner:2 themselves:1 considering:1 ocs:1 linearity:1 mass:2 developed:1 fellow:1 every:1 ti:1 exactly:4 laterally:3 toroidal:1 willshaw:2 unit:72 positive:5 maximise:1 attend:2 local:9 treat:1 apparatus:1 depended:1 consequence:1 frey:1 black:1 burn:1 thirty:1 yj:28 practice:1 implement:1 procedure:1 area:2 bell:1 thought:1 zoubin:2 get:1 cannot:1 clever:1 layered:3 sheet:1 www:1 equivalent:1 map:9 send:1 convex:1 pure:1 rule:10 adjusts:1 array:2 handle:3 variation:1 suppose:1 us:1 trick:2 roy:1 recognition:6 stripe:1 sparsely:1 bottom:2 observed:1 module:4 role:1 capture:1 connected:3 yk:5 mozer:1 stereograms:2 seung:3 trained:4 depend:1 segment:1 topographically:1 purely:2 creates:1 basis:2 easily:1 represented:1 separated:3 univ:1 describe:2 effective:1 sejnowski:2 analyser:2 formation:5 whose:2 richer:1 widely:3 valued:3 plausible:2 otherwise:1 topographic:7 think:1 emergence:1 itself:1 online:2 advantage:1 rr:1 net:8 analytical:2 interaction:6 product:1 interconnected:1 kohonen:2 neighboring:2 ontario:1 inducing:1 parent:1 generating:1 incremental:1 object:4 fixing:1 pose:1 eq:2 soc:1 implemented:2 c:1 involves:3 correct:3 filter:2 require:1 fix:1 clustered:1 biological:1 crg:1 extension:1 hold:1 ic:1 equilibrium:4 driving:1 major:2 early:1 omitted:1 mit:2 gaussian:20 probabilistically:1 believing:1 likelihood:3 contrast:1 sense:1 inference:10 itrc:1 dayan:1 interconnect:1 lj:1 entire:1 typically:1 hidden:20 going:1 pixel:3 priori:1 constrained:1 mackay:1 equal:4 once:3 field:1 sampling:16 unsupervised:2 report:1 connectionist:1 develops:1 piecewise:1 inhibited:1 primarily:1 randomly:1 divergence:1 intell:1 resulted:1 familiar:1 replaced:1 phase:3 consisting:1 attempt:1 interneurons:1 mixture:6 edge:1 encourage:1 necessary:1 orthogonal:2 unless:1 deformation:1 theoretical:1 column:2 disadvantage:2 deviation:1 subset:3 inadequate:2 varies:1 corrupted:1 periodic:1 combined:1 density:3 decayed:2 probabilistic:2 lee:2 quickly:1 squared:1 stochastically:1 derivative:3 til:1 account:1 iei:1 blurred:1 caused:1 depends:2 performed:1 view:2 contribution:2 square:1 accuracy:1 became:1 variance:6 bayesian:1 l1d:1 produced:1 rectified:8 cybernetics:1 drive:1 m5s:1 liebler:1 unaffected:1 detector:7 implausible:1 reach:2 suffers:1 ed:1 energy:8 initialised:1 ocular:1 proved:1 organized:1 appears:1 manuscript:1 higher:2 follow:1 arranged:2 correlation:3 nonlinear:1 overlapping:1 mkl:6 effect:6 requiring:1 true:2 consisted:1 spatially:1 symmetric:2 neal:4 white:1 ll:2 during:3 self:3 excitation:1 trying:1 performs:1 image:18 consideration:1 sigmoid:1 ji:3 discriminated:1 insensitive:2 exponentially:1 jl:1 componential:3 cambridge:2 gibbs:14 philos:1 grid:1 similarly:1 nonlinearity:1 analyzer:1 had:1 dot:3 funded:1 longer:1 surface:2 inhibition:1 add:1 multivariate:2 posterior:8 own:1 driven:1 yi:5 seen:2 additional:3 fortunately:1 determine:1 redundant:1 dashed:1 ii:2 full:1 multiple:4 neurally:1 desirable:1 infer:1 hebbian:2 technical:1 plausibility:1 ebe:3 long:2 dept:1 controlled:1 schematic:1 prediction:3 mrf:9 variant:1 multilayer:1 vision:1 expectation:1 iteration:3 represent:2 achieved:1 addition:1 want:1 source:1 eliminates:1 pass:1 thing:2 jordan:1 unused:1 intermediate:1 enough:1 affect:1 reduce:1 idea:1 shift:1 ciar:1 expression:1 assist:1 penalty:2 stereo:4 cause:2 jj:1 adequate:1 locally:3 concentrated:1 http:1 goj:2 notice:1 delta:5 per:1 correctly:2 discrete:1 geh:1 dominance:1 key:1 redundancy:1 tenable:1 run:1 powerful:1 fourth:1 topologically:1 patch:4 decision:1 scaling:1 layer:42 quadratic:7 durbin:1 activity:2 strength:1 constraint:1 scene:2 nearby:5 aspect:1 speed:1 argument:1 relatively:1 according:1 neur:1 combination:3 em:2 biologically:1 wherever:1 taken:1 rectification:2 agree:1 describing:1 turn:3 fail:1 know:2 tractable:2 travelling:1 salesman:1 available:3 gaussians:2 hierarchical:8 away:4 appropriate:2 petsche:1 top:15 assumes:2 responding:1 giving:1 ghahramani:6 classical:1 added:2 spike:2 parametric:1 gradient:3 ow:4 separate:1 thank:1 lateral:25 trivial:1 reason:2 rjk:2 assuming:2 code:1 index:1 difficult:2 unfortunately:2 localised:2 negative:4 rise:1 stern:1 boltzmann:2 perform:5 maximises:1 vertical:1 neuron:1 markov:1 unrectified:9 anti:1 hinton:8 saturates:1 canada:1 intensity:1 introduced:1 namely:1 pair:2 required:2 unpublished:1 connection:22 learned:5 nip:2 trans:1 below:9 pattern:1 goodhill:2 regime:1 including:2 belief:7 analogue:1 natural:4 difficulty:1 eye:4 conic:1 lk:1 mediated:1 extract:1 kj:1 prior:1 acknowledgement:1 determining:1 topography:5 suggestion:1 organised:2 fictitious:1 proven:1 geoffrey:1 consistent:1 uncorrelated:1 course:1 penalized:2 placed:2 infeasible:1 bias:3 allow:3 fall:1 explaining:4 wide:1 sparse:2 distributed:4 boundary:2 depth:3 calculated:1 cortical:1 adaptive:1 simplified:2 far:1 approximate:1 kullback:1 global:2 assumed:1 consuming:1 alternatively:1 bottomup:1 why:1 learn:4 nature:1 rearranging:1 elastic:2 ignoring:1 main:2 dense:2 noise:10 arise:1 child:1 fig:9 scattered:1 slow:1 fails:1 lyj:1 clamped:1 perceptual:8 ib:1 down:11 specific:1 showing:1 decay:1 ih:1 adding:1 justifies:1 simply:2 infinitely:2 forming:1 nserc:1 lewicki:1 radford:1 corresponds:2 ma:2 viewed:1 identity:1 consequently:1 revow:1 change:1 infinite:1 generalisation:2 typical:1 determined:1 uniformly:1 total:1 la:1 select:1 incorporate:1 h5:1 phenomenon:1 correlated:1

Linear concepts and hidden variables: An empirical study Adam J. Grove Dan Rothe NEC Research Institute 4 Independence Way Princeton NJ 08540 grove@research.nj.nec.com Department of Computer Science University of Illinois at Urbana-Champaign 1304 W. Springfield Ave. Urbana 61801 danr@cs.uiuc.edu Abstract Some learning techniques for classification tasks work indirectly, by first trying to fit a full probabilistic model to the observed data. Whether this is a good idea or not depends on the robustness with respect to deviations from the postulated model. We study this question experimentally in a restricted, yet non-trivial and interesting case: we consider a conditionally independent attribute (CIA) model which postulates a single binary-valued hidden variable z on which all other attributes (i.e., the target and the observables) depend. In this model, finding the most likely value of anyone variable (given known values for the others) reduces to testing a linear function of the observed values. We learn CIA with two techniques: the standard EM algorithm, and a new algorithm we develop based on covariances. We compare these, in a controlled fashion, against an algorithm (a version of Winnow) that attempts to find a good linear classifier directly. Our conclusions help delimit the fragility of using the CIA model for classification: once the data departs from this model, performance quickly degrades and drops below that of the directly-learned linear classifier. 1 Introduction We consider the classic task of predicting a binary (0/1) target variable zo, based on the values of some n other binary variables ZI ??? Zft,. We can distinguish between two styles of learning approach for such tasks. Parametric algorithms postulate some form of probabilistic model underlying the data, and try to fit the model's parameters. To classify an example we can compute the conditional probability distribution for Zo given the values of the known variables, and then predict the most probable value. Non-parametric algorithms do not assume that the training data has a particular form. They instead search directly in the space of possible classification functions, attempting to find one with small error on the training set of examples. An important advantage of parametric approaches is that the induced model can be used to support a wide range of inferences, aside from the specified classification task. On the other hand, to postulate a particular form of probabilistic model can be a very strong assumption. "Partly supported by ONR grant NOOOI4-96-1-0550 while visiting Harvard University. Linear Concepts and Hidden Variables: An Empirical Study 501 So it is important to understand how robust such methods are when the real world deviates from the assumed model. In this paper, we report on some experiments that test this issue. We consider the specific case of n + 1 conditionally independent attributes Zi together with a single unobserved variable z, also assumed to be binary valued, on which the Zi depend (henceforth, the binary CIA model); see Section 2. In fact, such models are plausible in many domains (for instance, in some language interpretation tasks; see [GR96]). We fit the parameters of the CIA model using the well-known expectation-maximization (EM) technique [DLR77], and also with a new algorithm we have developed based on estimating covariances; see Section 4. In the nonparametric case, we simply search for a good linear separator. This is because the optimal predictors for the binary CIA model (i.e., for predicting one variable given known values for the rest) are also linear. This means that our comparison is "fair" in the sense that neither strategy can choose from classifiers with more expressive power than the other. As a representative of the non-parametric class of algorithms, we use the Winnow algorithm of [Lit881, with some modifications (see Section 6). Winnow works directly to find a "good" linear separator. It is guaranteed to find a perfect separator if one exists, and empirically seems to be fairly successful even when there is no perfect separator [GR96, Blu9?]. It is also very fast. Our experimental methodology is to first generate synthetic data from a true CIA model and test performance; we then study various deviations from the model. There are various interesting issues involved in constructing good experiments, including the desirability of controlling the inherent "difficulty" of learning a model. Since we cannot characterize the entire space, we consider here only deviations in which the data is drawn from a CIA model in which the hidden variable can take more than two values. (Note that the optimal classifier given Zo is generally not linear in this case.) Our observations are not qualitatively surprising. CIA does well when the assumed model is correct, but performance degrades when the world departs from the model. But as we discuss, we found it surprising how fragile this model can sometimes be, when compared against algorithms such as Winnow. This is even though the data is not linearly separable either, and so one might expect the direct learning techniques to degrade in performance as well. But it seems that Wmnow and related approaches are far less fragile. Thus the main contribution of this work is that our results shed light on the specific tradeoff between fitting parameters to a probabilistic model, versus direct search for a good classifier. Specifically, they illustrate the dangers of predicting using a model that is even "slightly" simpler than the distribution actually generating the data, vs. the relative robustness of directly searching for a good predictor. This would seem to be an important practical issue, and highlights the need for some better theoretical understanding of the notion of "robustness". 2 Conditionally Independent Attributes Throughout we assume that each example is a binary vector z E {O, 1}n+l, and that each example is generated independently at random according to some unknown distribution on {O, 1}n+l. We use Xi to denote the i'th attribute, considered as a random variable, and Zi to denote a value for Xi. In the conditionally independent attribute (CIA) model, examples are generated as follows. We postulate a "hidden" variable Z with Ie values, which takes values z for 0 $ z < Ie with probability a. ~ O. Since we must have E::~ a. 1 there are Ie - 1 independent parameters. Having randomly chosen a value z for the hidden variable, we choose the value Zi for each observable Xi: the value is 1 with probability p~.}, and 0 otherwise. Here p~.} E [0,1). The attributes' values are chosen independently of each other, although z remains fixed. Note that there are thus (n + 1)1e probability parameters p~.). In the following, let l' denote the set of all (n + 1)1e + Ie - 1 parameters in the model. From this point, and until Section 7, we always assume that Ie = 2 and in this case, to simplify notation, we write al as a, ao (= 1 - a) as ai, p! as Pi and p~ as qi. = A. 1. Grove and D. Roth 502 3 The Expectation-Maximization algorithm (EM) One traditional unsupervised approach to learning the parameters of this model is to find the maximum-likelihood parameters of the distribution given the data. That is, we attempt to find the set of parameters that maximizes the probability of the data observed. Finding the maximum likelihood parameterization analytically appears to be a difficult problem, even in this rather simple setting. However, a practical approach is to use the wellknown Expectation-Maximization algorithm (EM) [DLR77], which is an iterative approach that always converges to a local maximum of the likelihood function. In our setting, the procedure is as follows. We simply begin with a randomly chosen parameterization p, and then we iterate until (apparent) convergence: 1 Expectation: For all zi, compute Ui = p-p(zi 1\ Z = 1) and Vi = p-p(zi 1\ Z = 0). = = Maximization: Reestimate P as follows (writing U Ei Ui and V Ei Vi): a f- E:=I Ui/(U + V) P; f- E{i::i~=I} u;./U qj f- E{i::i~=I} Vi/V. After convergence has been detected all we kno'w is that we are near a [ocdi minima of the likelihood function. Thus it is prudent to repeat the process with many different restarts. (All our experiments were extremely conservative concerning the stopping criteria at each iteration, and in the number of iterations we tried.) But in practice, we are never sure that the true optimum has been located. 4 Covariances-Based approach Partly in response to concern just expressed, we also developed another heuristic technique for learning P. The algorithm, which we call COY, is based on measuring the covariance between pairs of attributes. Since we do not see Z, attributes will appear to be correlated. In fact, if the CIA model is correct, it is easy to show that covariance between Xi and X j (defined as Yi,; = ~,; - ~I-'; where~, 1-';, ~,; are the expectations of Xi, Xj, (Xi and Xj), respectively), will be Yi,j aa'did; where di denotes Pi - qi. We also know that the expected value of Xi is ~ = aPi + a'qi. Furthermore, we will be able to get very accurate estimates of ~ just by observing the proportion of samples in which Zi is 1. Thus, if we could estimate both a and di it would be trivial to solve for estimates of Pi and qi. To estimate di, suppose we have computed all the pairwise covariances using the data; we use fli,; to denote our estimate of Yi,j' For any distinct j, Ie i= i we clearly have aa l = IV'rd.r"/o1 so we could estimate d; using this equation. A better estimate would be "/o to consider all pairs j, Ie and average the individual estimates. However, not all individual estimates are equally good. It can be shown that the smaller Y;,II is, the less reliable we should expect the estimate to be (and in the limit, where X; and XII are perfectly uncorrelated, we get no valid estimate at all). This suggests that we use a weighted average, with the weights proportional to Yj,II. Using these weights leads us to the next equation for determining 5i , which, after simplification, is: = 5; E j ,II:j;t1l;ti IYi,jYi,II I (E;:#i IYi,; 1)2 - E;:;;ti if,; E;,II:;;tll;ti IY;,II I E;,II:;;tll IY;,II I - 2 Ej:j;ti IYj,i I dl. By substituting the estimates 'Oi,; we get an estimate for aa' This estimate can be computed in linear time except for the determination of Ej,II:j;tll IYj,II I which, although quadratic, does not depend on i and so can be computed once and for all. Thus it takes O(n2) time in total to estimate aa'd; for all i. It remains only to estimate a and the signs of the di'S. Briefly, to determine the signs we first stipulate that do is positive. (Because we never see z, one sign can be chosen at random.) IThe maximization phase works as though we were estimating parameters by taking averages based on weighted labeled data (Le., in which we see z). If i i is a sample point, these fictional data 1) with weight Ui/U and (ii, z 0) with weight Vi/V. points are (ii,Z = = Linear Concepts and Hidden Variables: An Empirical Study 503 In principle, then, the sign of 0; will then be equal to the sign of Yo,;, which we have an estimate for. In practice, this can statistically unreliable for small sample sizes and so we use a more involved ''voting'' procedure (details omitted here). Finally we estimate Q. We have found no better method of doing this than to simply search for the optimal value, using likelihood as the search criterion. However, this is only a I-dimensional search and it turns out to be quite efficient in practice. 5 Linear Separators and CIA Given a fully parameterized CIA model, we may be interested in predicting the value of one variable, say Xo, given known values for the remaining variables. One can show that in fact the optimal prediction region is given by a linear separator in the other variables, although we omit details of this derivationhere. 2 This suggest an obvious learning strategy: simply try to find the line which minimizes this loss on the training set. Unfortunately, in general the task of finding a linear separator that minimizes disagreements on a collection of examples is known to be NP-hard [HS92]. So instead we use an algorithm called Winnow that is known to produce good results when a linear separator exists, as well as under certain more relaxed assumptions [Lit9I], and appears to be quite effective in practice. 6 Learning using a Winnow-based algorithm The basic version of the Winnow algorithm [Lit88] keeps an n-dimensional vector w = (1011" .1On ) of positive weights (Le., is the weight associated with the ith feature), which it updates whenever a mistake is made. Initially, the weight vector is typically set to assign equal positive weight to all features. The algorithm has 3 parameters, a promotion parameter Q > I, a demotion parameter 0 < f3 < 1 and a threshold 8. For a > 8. If the given instance (:1:1, ? "1 :l: n ) the algorithm predicts that :1:0 = 1 iff E~l algorithm predicts 0 and the label (Le., :1:0) is 1 (positive example) then the weights which correspond to active attributes (:1:, = 1) are promoted-the weight 10, is replaced by a larger weight Q ? Wi. Conversely, if algorithm predicts 1 and the received label is 0, then the weights which correspond to active features are demoted by factor {3. We allow for negative weights as follows. Given an example (:1:1" "1 :l: n ), we rewrite it as an example over 2n variables (Y1, 'Y21 ?.. I 'Y2n) where y, = :1:, and Yn+, 1 - :1:,. We then apply Winnow just as above to learn 2n (positive) weights. If wi is the weight associated with :1:, and wi is the weight associated with :l:n+i (Le., 1 - :1:,), then the prediction rule is simply to compare E~=l(wi:l:, + wi(1 - :1:,)) with the threshold. w, W,:I:, = In the experiments described here we have made two significant modifications to the basic algorithm. To reduce variance, our final classifier is a weighted average of several classifiers; each is trained using a subs ample from the training set, and its weight is based based on how well it was doing on that sample. Second, we biased the algorithm so as to look for "thick" classifiers. To understand this, consider the case in which the data is perfectly linearly separable. Then there will generally be many linear concepts that separate the training data we actually see. Among these, it seems plausible that we have a better chance of doing well on the unseen test data if we choose a linear concept that separates the positive and negative training examples as "widely" as possible. The idea of having a wide separation is less clear when there is no perfect separator, but we can still appeal to the basic intuition. To bias the search towards "thick" separators, we change Wmnow's training rule somewhat. We now have a new margin parameter T. As before, we always update when our current hypothesis makes a mistake, but now we also update if I E~=l Wi:l:, - 8 I is less than T, even if the prediction is correct. In our experiments, we found that performance when using this version of Winnow is better than that of the basic algorithm, so in this paper we present results for the former. 2A derivation for the slightly different case, for predicting z, can be found in [MP69J. A. J Grove and D. Roth 504 7 Experimental Methodology Aside from the choice of algorithm used, the number of attributes n, and the sample size 8, our experiments also differed in two other dimensions. These are the type of process generating the data (we will be interested in various deviations from CIA), and the "difficulty" of the problem. These features are determined by the data model we use (i.e., the distribution over {O, I} ft used to generate data sets). Our first experiments consider the case where the data really is drawn from a binary CIA distribution. We associated with any such distribution a "difficulty" parameter B, which is the accuracy with which one could predict the value of Z if one actually knew the correct model. (Of course, even with knowledge of the correct model we should not expect 100% accuracy.) The ability to control B allows us to select and study models with different qualitative characteristics. In particular, this has allowed us concentrated most of our experiments on fairly "hard" instances 3 , and to more meaningfully compare trials with differing numbers of attributes. We denote by CIA(n, 2, b) the class of all data models which are binary CIA distributions over n variables with difficulty b. 4 The next family of data models we used are also CIA models, but now using more than two values for the hidden variable. We denote the family using Ie values as CIA(n, Ie, b) where n and b are as before. When Ie > 2 there are more complex correlation patterns between the Xi than when Ie 2. Furthermore, the optimal predictor is not necessarily linear. The specific results we discuss in the next section have concentrated on this case. Given any set of parameters, including a particular class of data models, our experiments are designed with the goal of good statistical accuracy. We repeatedly (typically 100 to 300 times) choose a data model at random from the chosen class, choose a sample of the appropriate size from this model, and then run all our algorithms. Each algorithm produces a (linear) hypothesis. We measure the success rate Salg (i.e., the proportion of times a hypothesis makes the correct prediction of :1:0) by drawing yet more random samples from the data model being used. In the test phase we always draw enough new samples so that the confidence interval for Salg, for the results on a single model, has width at most ? 1%. We use the Salg values to construct a normalized measure of performance (denoted T) as follows. Let Sbest be the best possible accuracy attainable for predicting:l:o (i.e., the accuracy achieved by the actual model generating the data). Let Sconst denote the performance of the best possible constant prediction rule (i.e., the rule that predicts the most likely a priori value for :1:0). Note that Sconst and Sbest can vary from model to model. For each model we compute :alg--;onst ,and our normalized statistic T is the average of these values. It can be best- const thought of as measuring the percentage of the possible predictive power, over a plausible baseline, that an algorithm achieves. = 8 Results We only report on a small, but representative, selection of our experiments in any detail. For instance, although we have considered many values of n ranging from 10 to 500, here we show six graphs giving the learning curves for CIA(n, Ie, 0.90) for n = 10,75, and for Ie = 2,3,5; as noted, we display the T statistic. The error bars show the standard error,s providing a rough indication of accuracy. Not surprisingly, when the data model is binary 3Note that if one simply chooses parameters of a CIA model independently at random, without examining the difficulty of the model or adjusting for n, one will get many trivial problems, in which it is easy to predict Z with nearly 100% accuracy, and thus predict optimally for Xo. 41t is nontrivial to efficiently select random models from this class. Briefly, our scheme is to choose each parameter in a CIA model independently from a symmetric beta distribution. Thus, the model parameters will have expected value 0.5. We choose the parameter of the beta distribution (which determines concentration about 0.5) so that the average B value, of the models thus generated, equals b. Finally, we use rejection sampling to find CIA models with B values that are exactly b ? 1%. 5Computed as the observed standard deviation, divided by the square root of the number of trials. Linear Concepts and Hidden Variables: An Empirical Study 505 CIA, the EM algorithm does extremely well, learning significantly (if not overwhelmingly) faster than Winnow. But as we depart from the binary CIA assumption, the performance of EM quickly degrades. CI"(10.2.0.1IO) ,"" Cl"(78.2.0.1IO) '00 --_..... -. .. '00 loo Joo I ?? z t 40 J: .. m - - oa>I ......, .. ,0 ... '00 .oIT~~ , ,1,1 40 , -EM ...... 00 lOO ~i'.~/ j: ~ _~~! ~.!-,-4? )......, -.... - - oa>I 00 '0 }oo m - E.. - - oa>I .. '00 '000 ... CIA(78,',O.1O) - - oa>I I"" J. . . t 0 ......, - '000 Figure 3: CIA(1O,3,0.9) ,0 . ... '00 .01 Tralring ~ '000 Figure 4: CIA(75,3,0.9) CIA(7????o..a) CIAC10.a,O.1O) 10 . ........ 40 l loo '-----' , 140 too I ... -EM 00 l40 .dT"INng~ '00 .oIT,.......~ ? ?" m 00 l ,0 .. '000 . '00 ...... ? "m Figure 2: CIA(75,2,0.9) OA,(10 ?? ,O.IO) I"" of ,.?f .... -EM Figure 1: CIA(10,2,0.9) j40 .- ,, j 20 a -r' )- 0 " -1-- ......, -20 .. ......,~,o~----~~~,~~~-----=*~~,_ .oIT'. . . . ExemPM - _---t--r I m ... - E.. - - oa>I '0 eo '00 .oIT~~ 'ODD Figure 5: CIA(10,5,0.9) Figure 6: CIA(75,5,0.9) When Ie = 3 performances is, on the whole, very similar for Winnow and EM. But when Ie = 5 Winnow is already superior to EM; significantly and uniformly so for n = 10. For fixed Ie the difference seems to become somewhat less dramatic as n increases; in Figure 6 (for n = 75) Winnow is less obviously dominant, and in fact is not better than EM until the sample size has reached 100. (But when 8 ~ n, meaning that we have fewer samples than attributes, the performance is unifonnly dismal anyway.) Should we attribute this degradation to the binary CIA assumption, or to the EM itself? This question is our reason for also considering the covariance algorithm. We see that the results for COY are generally similar to EM's, supporting our belief that the phenomena we see are properties inherent to the model rather than to the specific algorithm being used. Similarly (the results are omitted) we have tried several other algorithms that try to find good linear separators directly, including the classic Perceptron algorithm [MP69); our version of Winnow was the best on the experiments we tried and thus we conjecture that its performance is (somewhat) indicative of what is possible for any such approach. As the comparison between n 10 and n = 75 illustrates, there is little qualitative differ- = 506 A. J. Grove and D. Roth ence between the phenomena observed as the number of attributes increases. Nevertheless, as n grows it does seem that Winnow needs more examples before its performance surpasses that of the other algorithms (for any fixed k). As already noted, this may be due simply to the very "noisy" nature of the region 8 $ n. We also have reasons to believe that this is partially an artifact of way we select models. As previously noted, we also experimented with varying "difficulty" (B) levels. Although we omit the corresponding figures we mentioned that the main difference is that Winnow is a little faster in surpassing EM when the data deviates from the assumed model, but when the data model really is binary CIA, and EM converge even faster to an optimal performance. These patterns were confinned when we tried to compare the approaches on real data. We have used data that originates from a problem in which assuming a hidden "context" variable seems somewhat plausible. The data is taken from the context-sensitive spelling correction domain. We used one data set from those that were used in [GR96]. For example, given sentences in which the word passed or past appear, the task is to determine, for each such occurrence, which of the two it should be. This task may be modeled by thinking of the "context" as a hidden variable in our sense. Yet when we tried to learn in this case under the CIA model, with a binary valued hidden variable, the results were no better than just predicting the most likely classification (around 70%). Winnow, in contrast, performed extremely well and exceeds 95% on this task. We hesitate to read much into our limited real-data experiments, other than to note that so far they are consistent witli the more careful experiments on synthetic data. 9 Conclusion By restricting to a binary hidden variable, we have been able to consider a "fair" comparison between probabilistic model construction, and more traditional algorithms that directly learn a classification-at least in the sense that both have the same expressive power. Our conclusions concerning the fragility of the former should not be surprising but we believe that given the importance of the problem it is valuable to have some idea of the true significance of the effect. As we have indicated, in many real-world cases, where a model of the sort we have considered here seems plausible, it is impossible to nail down more specific characterizations of the probabilistic model. Our results exhibit how important it is to use the correct model and how sensitive are the results to deviations from it, when attempting to learn using model construction. The purpose of this paper is not to advocate that in practice one should use either Winnow or binary CIA in exactly the form considered here. A richer probabilistic model should be used along with a model selection phase. However, studying the problem in a restricted and controlled environment in crucial so as to understand the nature and significance of this fundamental problem. References [Blu97] A. Blum. Empirical support for winnow and weighted majority based algorithms: results on a calendar scheduling domain. Machine Learning, 26: 1-19, 1997. [DLR77] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Royal Statistical SOCiety B, 39: 1-38, 1977. [GR96] A. R. Golding and D. Roth. Applying winnow to context-sensitive spelling correcton. In Proc. 13th International Conference on Machine Learning (ML' 96), pages 182-190, 1996. [HS92] K. HOffgen and H. Simon. Robust trainability of single neurons. In Proc. 5th Annu. Workshop on Comput. Learning Theory, pages 428-439, New York, New York, 1992. ACM Press. [Lit88] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2:285-318,1988. [Lit91] N. Littlestone. Redundant noisy attributes, attribute errors, and linear threshold learning using Winnow. In Proc. 4th Annu. Workshop on Comput. Learning Theory, pages 147-156, San Mateo, CA, 1991. Morgan Kaufmann. [MP69] M. L. Minsky and S. A. Papert. Perceptrons. MIT Press, Cambridge, MA, 1969. 

1473 |@word trial:2 briefly:2 version:4 hoffgen:1 seems:6 proportion:2 tried:5 covariance:7 attainable:1 dramatic:1 past:1 current:1 com:1 surprising:3 yet:3 must:1 drop:1 designed:1 update:3 aside:2 v:1 fewer:1 parameterization:2 indicative:1 ith:1 characterization:1 simpler:1 demoted:1 along:1 direct:2 beta:2 become:1 qualitative:2 dan:1 fitting:1 advocate:1 pairwise:1 expected:2 uiuc:1 actual:1 little:2 considering:1 abound:1 begin:1 estimating:2 underlying:1 notation:1 maximizes:1 what:1 minimizes:2 developed:2 differing:1 finding:3 unobserved:1 nj:2 ti:4 voting:1 shed:1 exactly:2 classifier:8 control:1 originates:1 grant:1 omit:2 appear:2 yn:1 positive:6 before:3 local:1 limit:1 mistake:2 api:1 io:3 might:1 mateo:1 suggests:1 conversely:1 confinned:1 limited:1 range:1 statistically:1 practical:2 testing:1 yj:1 practice:5 procedure:2 danger:1 empirical:5 thought:1 significantly:2 confidence:1 word:1 suggest:1 get:4 cannot:1 selection:2 scheduling:1 context:4 impossible:1 writing:1 applying:1 roth:4 delimit:1 independently:4 rule:4 classic:2 searching:1 notion:1 anyway:1 target:2 controlling:1 suppose:1 construction:2 hypothesis:3 harvard:1 located:1 predicts:4 labeled:1 observed:5 ft:1 region:2 valuable:1 mentioned:1 intuition:1 environment:1 dempster:1 ui:4 trained:1 depend:3 rewrite:1 ithe:1 predictive:1 observables:1 various:3 derivation:1 zo:3 distinct:1 fast:1 effective:1 detected:1 apparent:1 heuristic:1 quite:2 valued:3 plausible:5 solve:1 say:1 otherwise:1 larger:1 widely:1 ability:1 drawing:1 statistic:2 unseen:1 calendar:1 itself:1 noisy:2 final:1 laird:1 obviously:1 advantage:1 indication:1 stipulate:1 iff:1 convergence:2 optimum:1 produce:2 generating:3 adam:1 perfect:3 converges:1 help:1 illustrate:1 develop:1 oo:1 odd:1 received:1 strong:1 c:1 differ:1 thick:2 correct:7 attribute:18 kno:1 assign:1 ao:1 really:2 probable:1 correction:1 around:1 considered:4 predict:4 substituting:1 vary:1 achieves:1 omitted:2 purpose:1 proc:3 label:2 sensitive:3 weighted:4 mit:1 promotion:1 clearly:1 rough:1 always:4 desirability:1 rather:2 ej:2 varying:1 tll:3 overwhelmingly:1 yo:1 likelihood:6 contrast:1 ave:1 baseline:1 sense:3 inference:1 stopping:1 entire:1 typically:2 initially:1 hidden:13 springfield:1 interested:2 issue:3 classification:6 among:1 y2n:1 prudent:1 denoted:1 priori:1 fairly:2 equal:3 once:2 never:2 having:2 f3:1 construct:1 sampling:1 look:1 unsupervised:1 nearly:1 thinking:1 others:1 report:2 simplify:1 inherent:2 np:1 randomly:2 individual:2 replaced:1 phase:3 minsky:1 attempt:2 danr:1 light:1 accurate:1 grove:5 unifonnly:1 iv:1 incomplete:1 littlestone:2 theoretical:1 instance:4 classify:1 ence:1 measuring:2 maximization:5 deviation:6 surpasses:1 predictor:3 successful:1 examining:1 too:1 loo:3 characterize:1 optimally:1 synthetic:2 chooses:1 fundamental:1 international:1 ie:16 probabilistic:7 together:1 quickly:3 iy:2 postulate:4 choose:7 henceforth:1 iyj:2 style:1 postulated:1 lit91:1 depends:1 vi:4 performed:1 try:3 root:1 observing:1 doing:3 reached:1 sort:1 simon:1 contribution:1 square:1 oi:1 accuracy:7 variance:1 characteristic:1 efficiently:1 kaufmann:1 correspond:2 reestimate:1 whenever:1 against:2 involved:2 obvious:1 associated:4 di:4 adjusting:1 noooi4:1 knowledge:1 actually:3 appears:2 dt:1 restarts:1 methodology:2 response:1 though:2 furthermore:2 just:4 until:3 correlation:1 hand:1 ei:2 expressive:2 artifact:1 indicated:1 grows:1 believe:2 effect:1 concept:6 true:3 normalized:2 former:2 analytically:1 sbest:2 read:1 symmetric:1 conditionally:4 width:1 noted:3 criterion:2 trying:1 ranging:1 meaning:1 superior:1 empirically:1 interpretation:1 dismal:1 surpassing:1 significant:1 cambridge:1 ai:1 rd:1 similarly:1 illinois:1 language:1 joo:1 fictional:1 iyi:2 dominant:1 winnow:21 irrelevant:1 wellknown:1 certain:1 binary:16 onr:1 success:1 yi:3 morgan:1 minimum:1 relaxed:1 promoted:1 somewhat:4 eo:1 determine:2 converge:1 redundant:1 ii:12 full:1 reduces:1 champaign:1 exceeds:1 faster:3 determination:1 divided:1 concerning:2 equally:1 controlled:2 qi:4 prediction:5 basic:4 y21:1 expectation:5 iteration:2 sometimes:1 achieved:1 hesitate:1 interval:1 crucial:1 biased:1 rest:1 jyi:1 sure:1 induced:1 ample:1 meaningfully:1 seem:2 call:1 near:1 easy:2 enough:1 iterate:1 independence:1 fit:3 zi:9 xj:2 perfectly:2 reduce:1 idea:3 tradeoff:1 oit:4 fragile:2 qj:1 golding:1 whether:1 six:1 passed:1 york:2 repeatedly:1 generally:3 clear:1 nonparametric:1 concentrated:2 generate:2 percentage:1 sign:5 xii:1 write:1 demotion:1 threshold:4 nevertheless:1 blum:1 drawn:2 neither:1 graph:1 run:1 parameterized:1 throughout:1 family:2 nail:1 separation:1 draw:1 guaranteed:1 distinguish:1 simplification:1 display:1 quadratic:1 nontrivial:1 anyone:1 extremely:3 attempting:2 separable:2 conjecture:1 department:1 according:1 smaller:1 slightly:2 em:17 wi:6 modification:2 restricted:2 xo:2 taken:1 equation:2 remains:2 previously:1 discus:2 turn:1 know:1 studying:1 apply:1 indirectly:1 disagreement:1 appropriate:1 cia:37 occurrence:1 robustness:3 denotes:1 remaining:1 onst:1 const:1 giving:1 society:1 question:2 already:2 depart:1 degrades:3 parametric:4 strategy:2 concentration:1 traditional:2 spelling:2 visiting:1 exhibit:1 t1l:1 separate:2 oa:6 majority:1 degrade:1 trivial:3 reason:2 assuming:1 o1:1 modeled:1 providing:1 difficult:1 unfortunately:1 negative:2 unknown:1 observation:1 neuron:1 urbana:2 supporting:1 y1:1 pair:2 specified:1 dlr77:3 sentence:1 learned:1 able:2 bar:1 below:1 pattern:2 including:3 reliable:1 royal:1 belief:1 power:3 difficulty:6 predicting:7 scheme:1 deviate:2 understanding:1 determining:1 relative:1 fully:1 expect:3 highlight:1 loss:1 interesting:2 proportional:1 versus:1 lit88:2 rothe:1 consistent:1 rubin:1 principle:1 uncorrelated:1 pi:3 course:1 supported:1 repeat:1 surprisingly:1 bias:1 allow:1 understand:3 perceptron:1 institute:1 wide:2 taking:1 curve:1 dimension:1 world:3 valid:1 qualitatively:1 collection:1 made:2 san:1 far:2 observable:1 unreliable:1 keep:1 ml:1 active:2 assumed:4 knew:1 xi:8 search:7 iterative:1 learn:5 nature:2 fragility:2 robust:2 ca:1 alg:1 cl:1 complex:1 separator:11 constructing:1 domain:3 fli:1 did:1 necessarily:1 main:2 significance:2 linearly:2 whole:1 n2:1 fair:2 allowed:1 representative:2 fashion:1 differed:1 sub:1 papert:1 richer:1 comput:2 down:1 departs:2 annu:2 specific:5 appeal:1 experimented:1 concern:1 dl:1 exists:2 workshop:2 restricting:1 importance:1 ci:1 nec:2 illustrates:1 margin:1 rejection:1 simply:7 likely:3 expressed:1 gr96:4 partially:1 aa:4 chance:1 determines:1 acm:1 ma:1 conditional:1 goal:1 careful:1 towards:1 experimentally:1 hard:2 change:1 specifically:1 except:1 determined:1 uniformly:1 degradation:1 conservative:1 total:1 called:1 partly:2 experimental:2 trainability:1 perceptrons:1 select:3 support:2 princeton:1 phenomenon:2 correlated:1

Multiresolution Tangent Distance for Affine-invariant Classification Nuno Vasconcelos Andrew Lippman MIT Media Laboratory, 20 Ames St, E15-320M, Cambridge, MA 02139, {nuno,lip }@media.mit.edu Abstract The ability to rely on similarity metrics invariant to image transformations is an important issue for image classification tasks such as face or character recognition. We analyze an invariant metric that has performed well for the latter - the tangent distance - and study its limitations when applied to regular images, showing that the most significant among these (convergence to local minima) can be drastically reduced by computing the distance in a multiresolution setting. This leads to the multi resolution tangent distance, which exhibits significantly higher invariance to image transformations, and can be easily combined with robust estimation procedures. 1 Introduction Image classification algorithms often rely on distance metrics which are too sensitive to variations in the imaging environment or set up (e.g. the Euclidean and Hamming distances), or on metrics which, even though less sensitive to these variations, are application specific or too expensive from a computational point of view (e.g. deformable templates). A solution to this problem, combining invariance to image transformations with computational simplicity and general purpose applicability was introduced by Simard et al in [7]. The key idea is that, when subject to spatial transformations, images describe manifolds in a high dimensional space, and an invariant metric should measure the distance between those manifolds instead of the distance between other properties of (or features extracted from) the images themselves. Because these manifolds are complex, minimizing the distance between them is a difficult optimization problem which can, nevertheless, be made tractable by considering the minimization of the distance between the tangents to the manifolds -the tangent distance (TO) - instead of that between the manifolds themselves. While it has led to impressive results for the problem of character recognition [8] , the linear approximation inherent to the TO is too stringent for regular images, leading to invariance over only a very narrow range of transformations. 844 N. Vasconcelos and A. Lippman In this work we embed the distance computation in a multi resolution framework [3], leading to the multiresolution tangent distance (MRTD). Multiresolution decompositions are common in the vision literature and have been known to improve the performance of image registration algorithms by extending the range over which linear approximations hold [5, 1]. In particular, the MRTD has several appealing properties: 1) maintains the general purpose nature of the TD; 2) can be easily combined with robust estimation procedures, exhibiting invariance to moderate non-linear image variations (such as caused by slight variations in shape or occlusions); 3) is amenable to computationally efficient screening techniques where bad matches are discarded at low resolutions; and 4) can be combined with several types of classifiers. Face recognition experiments show that the MRTD exhibits a significantly extended invariance to image transformations, originating improvements in recognition accuracy as high as 38%, for the hardest problems considered. 2 The tangent distance Consider the manifold described by all the possible linear transformations that a pattern lex) may be subject to (1) Tp [lex)] = 1('ljJ(x, p)), where x are the spatial coordinates over which the pattern is defined, p is the set of parameters which define the transformation, and 'ljJ is a function typically linear on p, but not necessarily linear on x. Given two patterns M(x) and N(x), the distance between the associated manifolds - manifold distance (MD) - is T(M, N) = min IITq[M(x)] - Tp[N(x)]W. p,q (2) For simplicity, we consider a version of the distance in which only one of the patterns is subject to a transformation, i.e. T(M, N) = min IIM(x) p Tp[N(x)]lf, (3) but all results can be extended to the two-sided distance. Using the fact that \7 p Tp[N(x)] = \7pN('ljJ(x, p)) = \7 p '?(x, p)\7xN('?(x, p)), (4) where \7pTp is the gradient of Tp with respect to p, Tp[N(x)] can, for small p, be approximated by a first order Taylor expansion around the identity transformation Tp[N(x)] = N(x) + (p - If\7p 'ljJ(x,p)\7 x N(x). This is equivalent to approximating the manifold by a tangent hyper-plane, and leads to the TD. Substituting this expression in equation 3, setting the gradient with respect to p to zero, and solving for p leads to p ~ [~'VP;6(X' P ) 'Vx N(x) 'V); N(X)'V~;6(x, P)]-' ~ D(x)'Vp;6(x, P l'VxN(x) + I, (5) where D(x) = M(x) - N(x). Given this optimal p, the TD between the two patterns is computed using equations I and 3. The main limitation of this formulation is that it relies on a first-order Taylor series approximation, which is valid only over a small range of variation in the parameter vector p . 2.1 Manifold distance via Newton's method The minimization of the MD of equation 3 can also be performed through Newton's method, which consists of the iteration pn+1 = pn _ 0: [\7~ T/p=pn] -I \7 p Tlp=pn (6) 845 Multiresolution Tangent Distancefor Affine-invariant Classification where \7 p / and \7~ / are, respectively, the gradient and Hessian of the cost function of equation 3 with respect to the parameter p, \7p/ = 2 L [M(x) - Tp[N(x)]) V'pTp[N(x)] x V'~ / = 2 L [-V'pTp[N(x)] \7~Tp[N(x)] + [M(x) - N(x)] V'~Tp[N(x)]] . x Disregarding the term which contains second-order derivatives (V'~Tp[N(x)]), choosing pO I and Q: 1, using 4, and substituting in 6 leads to equation 5. I.e. the TO corresponds to a single iteration of the minimization of the MD by a simplified version of Newton's method, where sec!ond-orderderivatives are disregarded. This reduces the rate of convergence of Newton's method, and a single iteration may not be enough to achieve the local minimum, even for simple functions. It is, therefore, possible to achieve improvement if the iteration described by equation 6 is repeated until convergence. = = 3 The multiresolution tangent distance The iterative minimization of equation 6 suffers from two major drawbacks [2]: 1) it may require a significant number of iterations for convergence and 2), it can easily get trapped in local minima. Both these limitations can be, at least partially, avoided by embedding the computation of the MD in a multiresolution framework, leading to the multiresolution manifold distance (MRMD). For its computation, the patterns to classify are first subject to a multiresolution decomposition, and the MD is then iteratively computed for each layer, using the estimate obtained from the layer above as a starting point, where, Dl(x) = M(x) - Tp~ [N(x)]. If only one iteration is allowed at each imageresolution, the MRMD becomes the multiresolution extension of the TO, i.e. the multi resolution tangent distance (MRTO). To illustrate the benefits of minimization over different scales consider the signal J(t) = E{;=1 sin(wkt ), and the manifold generated by all its possible translations J'(t,d) = J(t + d). Figure 1 depicts the multiresolution Gaussian decomposition of J(t), together with the Euclidean distance to the points on the manifold as a function of the translation associated with each of them (d). Notice that as the resolution increases, the distance function has more local minima, and the range of translations over which an initial guess is guaranteed to lead to convergence to the global minimum (at d = 0) is smaller. I.e., at higher resolutions, a better initial estimate is necessary to obtain the same performance from the minimization algorithm. Notice also that, since the function to minimize is very smooth at the lowest resolutions, the minimization will require few iterations at these resolutions if a procedure such as Newton's method is employed. Furthermore, since the minimum at one resolution is a good guess for the minimum at the next resolution, the computational effort required to reach that minimum will also be small. Finally, since a minimum at low resolutions is based on coarse, or global, information about the function or patterns to be classified, it is likely to be the global minimum of at least a significant region of the parameter space, if not the true global minimum. 846 N. Vasconcelos and A. Lippman ?R B .~5ISa {\Z\Z\] -UJj -F \lJ -t;: Ll -I~ ..::.. .. .. .... . . . . . . ... .. ~ .. . .... . ..:.. ?. ?? ?? ???? ..I~ ?? ?? --"' . _ .:.. . . .. ... . . .. Figure 1: Top: Three scales of the multiresolution decomposition of J(t) . Bottom: Euclidean distance VS. translation for each scale. Resolution decreases from left to right. 4 Affine-invariant classification There are many linear transformations which can be used in equation 1. In this work, we consider manifolds generated by affine transformations 1jJ(x,p) =[ X 0 y 0 1000] 0 x yIP = ~(x)p, (8) where P is the vector of parameters which characterize the transformation. Taking the gradient of equation 8 with respect to p. V'p1jJ(x,p) = ~(x)T. using equation 4. and substituting in equation 7. p~+1 = pr +" [ ~ 4> (x) TV x N ' (x) viN' (x) 4> (xl ] -I L D'(x)~(x)TV'xN'(x), (9) x PI?' where N'(x) = N(1jJ(x, and D'(x) = M(x) - N'(x). For a given levell of the multiresolution decomposition, the iterative process of equation 9 can be summarized as follows. 1. Compute N'(x) by warping the pattern to classify N(x) according to the best current estimate of p, and compute its spatial gradient V'xN'(x). 2. Update the estimate of PI according to equation 9. 3. Stop if convergence, otherwise go to 1. Once the final PI is obtained, it is passed to the multiresolution level below (by doubling the translation parameters), where it is used as initial estimate. Given the values of Pi which minimize the MD between a pattern to classify and a set of prototypes in the database, a K-nearest neighbor classifier is used to find the pattern's class. 5 Robust classifiers One issue of importance for pattern recognition systems is that of robustness to outliers, i.e errors which occur with low probability, but which can have large magnitude. Examples are errors due to variation of facial features (e.g. faces shot with or without glasses) in face recognition, errors due to undesired blobs of ink or uneven line thickness in character recognition, or errors due to partial occlusions (such as a hand in front of a face) or partially Multiresolution Tangent Distance/or Affine-invariant Classification 847 missing patterns (such as an undoted i). It is well known that a few (maybe even one) outliers of high leverage are sufficient to throw mean squared error estimators completely off-track [6] . Several robust estimators have been proposed in the statistics literature to avoid this problem. In this work we consider M-estimators [4] which can be very easily incorporated in the MD classification framework. M-estimators are an extension of least squares estimators where the square function is substituted by a functional p(x) which weighs large errors less heavily. The robust-estimator version of the tangent distance then becomes to minimize the cost function T(M, N) = min p(M(x) - Tp[N{x)]) , (10) p I: x and it is straightforward to show that the "robust" equivalent to equation 9 is p~+' ~ pr +" [~P"[D(X))oI>(X)TI7XN'(X)I7;;:N'(X)oI>(X)T]-' x [~P'[D(X))oI>(X)Tl7xN' (X)] , (11) where D(x) = M(x) - N'(x) and p'(x) and p"(x) are, respectively, the first and second derivatives of the function p( x) with respect to its argument. 6 Experimental results In this section, we report on experiments carried out to evaluate the performance of the MD classifier. The first set of experiments was designed to test the invariance of the TD to affine transformations of the input. The second set was designed to evaluate the improvement obtained under the multiresolution framework. 6.1 Affine invariance of the tangent distance Starting from a single view of a reference face, we created an artificial dataset composed by 441 affine transformations of it. These transformations consisted of combinations of all rotations in the range from - 30 to 30 degrees with increments of 3 degrees, with all scaling transformations in the range from 70% to 130% with increments of 3%. The faces associated with the extremes of the scaling/rotation space are represented on the left portion of figure 2. On the right of figure 2 are the distance surfaces obtained by measuring the distance associated with several metrics at each of the points in the scaling/rotation space. Five metrics were considered in this experiment: the Euclidean distance (ED), the TD, the MD computed through Newton's method, the MRMD, and the MRTD. While the TD exhibits some invariance to rotation and scaling, this invariance is restricted to a small range of the parameter space and performance only slightly better than the obtained with the ED. The performance of the MD computed through Newton's method is dramatically superior, but still inferior to those achieved with the MRTD (which is very close to zero over the entire parameter space considered in this experiment), and the MRMD. The performance of the MRTD is in fact impressive given that it involves a computational increase of less than 50% with respect to the TD, while each iteration of Newton's method requires an increase of 100%, and several iterations are typically necessary to attain the minimum MD. N. Vasconcelos and A. Lippman 848 -30 !i ~ -0 a: 0 1.3 0.7 Scaling Figure 2: Invariance of the tangent distance. In the right, the surfaces shown correspond to ED, TO, MO through Newton's method, MRTO, and MRMO. This ordering corresponds to that of the nesting of the surfaces, i.e. the ED is the cup-shaped surface in the center, while the MRMO is the flat surface which is approximately zero everywhere. 6.2 Face recognition To evaluate the performance of the multiresolution tangent distance on a real classification task, we conducted a series of face recognition experiments, using the Olivetti Research Laboratories (ORL) face database. This database is composed by 400 images of 40 subjects, 10 images per subject, and contains variations in pose, light conditions, expressions and facial features, but small variability in terms of scaling, rotation, or translation. To correct this limitation we created three artificial datasets by applying to each image three random affine transformations drawn from three multivariate normal distributions centered on the identity transformation with different covariances. A small sample of the faces in the database is presented in figure 3, together with its transformed version under the set of transformations of higher variability. Figure 3: Left: sample of the ORL face database. Right: transformed version. We next designed three experiments with increasing degree of difficulty. In the first, we selected the first view of each subject as the test set, using the remaining nine views as training data. In the second, the first five faces were used as test data while the remaining five were used for training. Finally, in the third experiment, we reverted the roles of the datasets used in the first. The recognition accuracy for each of these experiments and each of the datasets is reported on figure 4 for the ED, the TO, the MRTD, and a robust version of this distance (RMRTO) with p(x) = 1x2 if x::; aT and p(x) = ~2 otherwise, where T is a threshold (set to 2.0 in our experiments), and a a robust version of the error standard deviation defined as a = median lei - median (ei )1 /0.6745. Several conclusions can be taken from this figure. First, it can be seen that the MRTD provides a significantly higher invariance to linear transformations than the ED or the TO, MultiresolUlion Tangent Distance for Affine-invariant Classification 849 increasing the recognition accuracy by as much as 37.8% in the hardest datasets. In fact, for the easier tasks of experiments one and two, the performance of the multiresolution classifier is almost constant and always above the level of 90% accuracy. It is only for the harder experiment that the invariance of the MRTO classifier starts to break down. But even in this case, the degradation is graceful- the recognition accuracy only drops below 75% for considerable values of rotation and scaling (dataset D3). On the other hand, the ED and the single resolution TO break down even for the easier tasks, and fail dramatically when the hardest task is performed on the more difficult datasets. Furthermore, their performance does not degrade gracefully, they seem to be more invariant when the training set has five views than when it is composed by nine faces of each subject in the database. I 'iJ .... '~--------~,~-=-=~~ ~ , ' , ___ _ ...l _ _ _ _ _ __ _ l... , - - --1- - , , , , - - -- - - --r- --- - - - - -t- - . .. _ __ _____ _ L _ _ _ _ ___ , ,, _ _ _ _ ______ L- __ _ __ __ ,, ,, am . ---- - -- - -- - -r-- ---- - -t - - - - --- - t, ,- ,, ...1 ___ __ _ __ l- I , __ .... -------;----...---......... :0 __ ~ L . _ _ _ ....1_ _ _ I .OII IIL. r ? iii"" ----- - - - j- ------ - -t - - - - - - - - r -TD"""" _ I . _ _ ______ IL _____ __ ...1! _ _ _ ?_ _ _ _ .l- 11041 10>00_ OO~ _ JOQI _ ..,CV _ ,, _ ____ __ ,LI __ _ ____ ,, ,, ,, _ __ ___ _ _ ,, ,, ,L, _ ,, -10 _ _ ____ __ ... _ _ _ __ ____ j l'W'> _ _ _ _ _ _ _ _ ~ __ __ _ _ ~ -- - ---- ~ -- --- -- - ~ - _ ~ _ _ _ _ _ _ _ ... _ _ _ _ _ _ _ _ ... , I , , , I I , I I I 1Mb IIRm - ~?~?~:~j?~~~~~~~}~~~~~~~~~ : : '~ -- - - -- -- t-- -- -- -- -~ - -- ----t" _ _ _ _ _ _ _ _ 1- ! I _ , - -- - -~ - , , ,, ,, ...1_ ______ , _ ,, , ,, , "".(1,1 ____ __ _ _ _ ~- - ---- --+ ---- --- -~, , ,, , , , '!O.(U. ., . ---- - -I-- ------ - - ------ - r -m- ,GI;IIIIII_ 110 m ? ? _ :1111,1,1 . __ _ _ _ _ ___ L ___ _ _ __ ...l ___ ___ ___ _ ?>.011 _ _ _ __ ___ I I - , _ _ ~ , , I____ _ ___ I ? +____ _____ t _ , , _____ _ _ I _ _ _ _ _ ___ ~ I __ __ ___ L _____ ____ _ , , ,, , ,;. Figure 4: Recognition accuracy. From left to right: results from the first, second, and third experiments. Oatasets are ordered by degree of variability: 00 is the ORL database 03 is subject to the affine transfonnations of greater amplitude. Acknowledgments We would like to thank Federico Girosi for first bringing the tangent distance to our attention, and for several stimulating discussions on the topic. References [1J P. Anandan, J. Bergen, K. Hanna, and R. Hingorani. Hierarchical Model-Based Motion Estimation. In M. Sezan and R. Lagendijk, editors, Motion Analysis and Image Sequence Processing, chapter 1. Kluwer Academic Press, 1993. [2J D. Bertsekas. Nonlinear Programming. Athena Scientific, 1995. [3J P. Burt and E. Adelson. The Laplacian Pyramid as a Compact Image Code. IEEE Trans. on Communications, Vol. 31:532-540,1983. [4] P. Huber. Robust Statistics. John Wiley, 1981 . [5] B. Lucas and T. Kanade. An Iterative Image Registration Technique with an Application to Stereo Vision. In Proc. DARPA Image Understanding Workshop, 198 I. [6J P. Rousseeuw and A. Leroy. Robust Regression and Outlier Detection. John Wiley, 1987. [7] P. Simard, Y. Le Cun, and J. Denker. Efficient Pattern Recognition Using a New Transformation Distance. In Proc. Neurallnfonnation Proc. Systems, Denver, USA, 1994. [8] P. Simard, Y. Le Cun, and 1. Denker. Memory-based Character Recognition Using a Transformation Invariant Metric. In Int. Conference on Pattern Recognition, Jerusalem, Israel, 1994. 

1474 |@word version:7 covariance:1 decomposition:5 harder:1 shot:1 initial:3 series:2 contains:2 current:1 john:2 girosi:1 shape:1 tlp:1 designed:3 update:1 drop:1 v:1 selected:1 guess:2 plane:1 coarse:1 provides:1 ames:1 five:4 consists:1 huber:1 themselves:2 multi:3 td:8 considering:1 increasing:2 becomes:2 medium:2 lowest:1 israel:1 transformation:23 transfonnations:1 classifier:6 bertsekas:1 local:4 approximately:1 range:7 acknowledgment:1 ond:1 lf:1 i____:1 lippman:4 procedure:3 significantly:3 attain:1 regular:2 get:1 close:1 applying:1 equivalent:2 missing:1 center:1 go:1 straightforward:1 starting:2 attention:1 jerusalem:1 resolution:13 simplicity:2 estimator:6 nesting:1 embedding:1 variation:7 coordinate:1 increment:2 heavily:1 programming:1 recognition:16 expensive:1 approximated:1 database:7 bottom:1 role:1 region:1 vxn:1 ordering:1 decrease:1 environment:1 solving:1 completely:1 easily:4 po:1 darpa:1 represented:1 chapter:1 describe:1 artificial:2 hyper:1 choosing:1 otherwise:2 federico:1 ability:1 statistic:2 gi:1 final:1 blob:1 sequence:1 mb:1 combining:1 multiresolution:18 deformable:1 achieve:2 convergence:6 extending:1 illustrate:1 andrew:1 oo:1 pose:1 ij:1 nearest:1 throw:1 involves:1 exhibiting:1 drawback:1 correct:1 centered:1 vx:1 stringent:1 require:2 extension:2 hold:1 around:1 considered:3 normal:1 iil:1 mo:1 substituting:3 major:1 purpose:2 estimation:3 proc:3 sensitive:2 minimization:7 mit:2 gaussian:1 always:1 pn:5 avoid:1 improvement:3 am:1 glass:1 bergen:1 typically:2 lj:1 entire:1 originating:1 transformed:2 issue:2 classification:9 among:1 oii:1 lucas:1 spatial:3 yip:1 once:1 vasconcelos:4 shaped:1 hardest:3 adelson:1 report:1 inherent:1 few:2 composed:3 occlusion:2 detection:1 screening:1 extreme:1 light:1 amenable:1 partial:1 necessary:2 facial:2 euclidean:4 taylor:2 weighs:1 classify:3 tp:13 measuring:1 applicability:1 cost:2 deviation:1 conducted:1 too:3 front:1 characterize:1 reported:1 thickness:1 combined:3 st:1 off:1 together:2 squared:1 simard:3 leading:3 derivative:2 li:1 sec:1 summarized:1 int:1 caused:1 performed:3 view:5 break:2 analyze:1 portion:1 start:1 maintains:1 vin:1 minimize:3 square:2 oi:3 accuracy:6 il:1 correspond:1 vp:2 classified:1 reach:1 suffers:1 ptp:3 ed:7 nuno:2 associated:4 hamming:1 stop:1 dataset:2 amplitude:1 higher:4 formulation:1 though:1 furthermore:2 until:1 hand:2 ei:1 nonlinear:1 ujj:1 scientific:1 lei:1 usa:1 consisted:1 true:1 laboratory:2 iteratively:1 undesired:1 sin:1 ll:1 inferior:1 motion:2 image:19 reverted:1 common:1 rotation:6 superior:1 functional:1 denver:1 slight:1 kluwer:1 significant:3 cambridge:1 cup:1 cv:1 similarity:1 impressive:2 surface:5 neurallnfonnation:1 multivariate:1 olivetti:1 moderate:1 seen:1 minimum:12 greater:1 anandan:1 employed:1 signal:1 isa:1 reduces:1 smooth:1 levell:1 match:1 academic:1 laplacian:1 regression:1 vision:2 metric:8 iteration:9 pyramid:1 achieved:1 median:2 bringing:1 wkt:1 subject:9 seem:1 leverage:1 iii:1 enough:1 idea:1 prototype:1 i7:1 expression:2 passed:1 effort:1 stereo:1 hessian:1 nine:2 jj:2 dramatically:2 maybe:1 rousseeuw:1 reduced:1 notice:2 trapped:1 track:1 per:1 vol:1 key:1 nevertheless:1 threshold:1 drawn:1 d3:1 registration:2 imaging:1 everywhere:1 almost:1 scaling:7 orl:3 layer:2 guaranteed:1 leroy:1 occur:1 x2:1 flat:1 argument:1 min:3 graceful:1 tv:2 according:2 combination:1 smaller:1 slightly:1 character:4 appealing:1 cun:2 outlier:3 invariant:10 pr:2 restricted:1 sided:1 taken:1 computationally:1 equation:14 fail:1 tractable:1 denker:2 hierarchical:1 hingorani:1 robustness:1 top:1 remaining:2 newton:9 approximating:1 ink:1 warping:1 lex:2 md:11 exhibit:3 gradient:5 distance:37 thank:1 athena:1 gracefully:1 degrade:1 topic:1 manifold:14 code:1 minimizing:1 difficult:2 datasets:5 discarded:1 extended:2 incorporated:1 variability:3 communication:1 burt:1 introduced:1 required:1 narrow:1 trans:1 below:2 pattern:14 memory:1 difficulty:1 rely:2 e15:1 improve:1 mrmd:4 carried:1 created:2 ljj:4 literature:2 understanding:1 tangent:18 limitation:4 degree:4 affine:11 sufficient:1 editor:1 pi:4 translation:6 drastically:1 iim:1 neighbor:1 template:1 face:14 taking:1 benefit:1 xn:3 valid:1 made:1 simplified:1 avoided:1 compact:1 global:4 iterative:3 lip:1 kanade:1 nature:1 robust:10 hanna:1 expansion:1 complex:1 necessarily:1 substituted:1 main:1 repeated:1 allowed:1 depicts:1 wiley:2 xl:1 third:2 down:2 embed:1 bad:1 specific:1 showing:1 disregarding:1 dl:1 workshop:1 importance:1 magnitude:1 disregarded:1 easier:2 led:1 likely:1 ordered:1 partially:2 doubling:1 corresponds:2 relies:1 extracted:1 ma:1 stimulating:1 identity:2 considerable:1 degradation:1 invariance:12 experimental:1 uneven:1 latter:1 evaluate:3

Structural Risk Minimization for Nonparametric Time Series Prediction Ron Meir* Department of Electrical Engineering Technion, Haifa 32000, Israel rmeir@dumbo.technion.ac.il Abstract The problem of time series prediction is studied within the uniform convergence framework of Vapnik and Chervonenkis. The dependence inherent in the temporal structure is incorporated into the analysis, thereby generalizing the available theory for memoryless processes. Finite sample bounds are calculated in terms of covering numbers of the approximating class, and the tradeoff between approximation and estimation is discussed. A complexity regularization approach is outlined, based on Vapnik's method of Structural Risk Minimization, and shown to be applicable in the context of mixing stochastic processes. 1 Time Series Prediction and Mixing Processes A great deal of effort has been expended in recent years on the problem of deriving robust distribution-free error bounds for learning, mainly in the context of memory less processes (e.g. [9]). On the other hand, an extensive amount of work has been devoted by statisticians and econometricians to the study of parametric (often linear) models of time series, where the dependence inherent in the sample, precludes straightforward application of many of the standard results form the theory of memoryless processes. In this work we propose an extension of the framework pioneered by Vapnik and Chervonenkis to the problem of time series prediction. Some of the more elementary proofs are sketched, while the main technical results will be proved in detail in the full version of the paper. Consider a stationary stochastic process X = { ... ,X-1, X 0, X 1, ... }, where Xi is a random variable defined over a compact domain in R and such that IXil ::; B with probability 1, for some positive constant B. The problem of one-step prediction, in the mean square sense, can then be phrased as that of finding a function f (.) of the infinite past, such that E IXo - f(X=~) 12 is minimal, where we use the notation xf = (Xi, Xi ti, ... ,Xj ), ?This work was supported in part by the a grant from the Israel Science Foundation Structural Risk Minimization/or Nonparametric Time Series Prediction 309 j ~ i. It is well known that the optimal predictor in this case is given by the conditional mean, E[XoIX:!J While this solution, in principle, settles the issue of optimal prediction, it does not settle the issue of actually computing the optimal predictor. First of all, note that ~o compute the conditional mean, the probabilistic law generating the stochastic process X must be known. Furthermore, the requirement of knowing the full past, X=-~, is of course rather stringent. In this work we consider the more practical situation, where a finite sub-sequence Xi" = (Xl, X 2,??? ,XN ) is observed, and an optimal prediction is needed, conditioned on this data. Moreover, for each finite sample size N we allow the pre.dictors to be based only on a finite lag vector of size d. Ultimately, in order to achieve full generality one may let d -+ 00 when N -+ 00 in order to obtain the optimal predictor. We first consider the problem of selecting an empirical estimator from a class of functions Fd,n : Rd -+ R, where n is a complexity index of the class (for example, the number of computational nodes in a feedforward neural network with a single hidden layer), and If I ::; B for f E Fd,n. Consider then an empirical predictor fd,n,N(Xi=~), i > N, for Xi based on the finite data set Xi" and depending on the d-dimensional lag vector Xi=~, where fd,n,N E Fd,n. It is possible to split the error incurred by this predictor into three terms, each possessing a rather intuitive meaning. It is the competition between these terms which determines the optimal solution, for a fixed amount of data. First, define the IX loss of a functional predictor f : Rd -+ R as L(f) = E i - f(xi=~) 12 , and let fd,n be the optimal function in Fd,n minimizing this loss. Furthermore, denote the optimal lag d predictor by fd' and its associated loss by L'd. We are then able to split the loss of the empirical predictor fd,n,N into three basic components, L(fd,n,N) = (Ld,n,N - L'd,n) + (L'd,n - L'd) + L'd, (I) = where Ld,n,N L(fd,n,N). The third term, L'd, is related to the error incurred in using a finite memory model (of lag size d) to predict a process with potentially infinite memory. We do not at present have any useful upper bounds for this term, which is related to the rate of convergence in the martingale convergence theorem, which to the best of our knowledge is unknown for the type of mixing processes we study in this work. The second term in (1) , is related to the so-called approximation error, given by Elfei (X:=-~) - fel,n (Xf=~) 12 to which it can be immediately related through the inequality IIal P- IblPI ::; pia - bll max( a, b) Ip-l . This term measures the excess error incurred by selecting a function f from a class of limited complexity Fd,n, while the optimal lag d predictor fei may be arbitrarily complex. Of course, in order to bound this term we will have to make some regularity assumptions about the latter function. Finally, the first term in (1) r~resents the so called estimation error, and is the only term which depends on the data Xl . Similarly to the problem of regression for i.i.d. data, we expect that the approximation and estimation terms lead to conflicting demands on the choice of the the complexity, n, of the functional class Fd,n. Clearly, in order to minimize the approximation error the complexity should be made as large as possible. However, doing this will cause the estimation error to increase, because of the larger freedom in choosing a specific function in Fd,n to fit the data. However, in the case of time series there is an additional complication resulting from the fact that the misspecification error L'd is minimized by choosing d to be as large as possible, while this has the effect of increasing both the approximation as well as the estimation errors. We thus expect that sOrhe optimal values of d and n exist for each sample size N. Up to this point we have not specified how to select the empirical estimator f d,n,N. In this work we follow the ideas of Vapnik [8], which have been studied extensively in the context of i.i.d observations, and restrict our selection to that hypothesis which minimizes the IX empirical error, given by LN(f) = N~d 2::~d+l i - f(x:=~)12 . For this function it is easy to establish (see for example [8]) that (Ld,n,N - L'd,n) ::; 2 sUP!E.rd,n IL(f) - LN(f)I? The main distinction from the i.i.d case, of course, is that random variables appearing in R. Meir 310 the empirical error, LN(f), are no longer independent. It is clear at this point that some assumptions are needed regarding the stochastic process X, in order that a law of large numbers may be established. In any event, it is obvious that the standard approach of using randomization and symmetrization as in the i.i.d case [3] will not work here. To circumvent this problem, two approaches have been proposed. The first makes use of the so-called method of sieves together with extensions of the Bernstein inequality to dependent data [6]. The second approach, to be pursued here, is based on mapping the problem onto one characterized by an i.i.d process [10], and the utilization of the standard results for the latter case. In order to have some control of the estimation error discussed above, we will restrict ourselves in this work to the class of so-called mixing processes. These are processes for which the 'future' depends only weakly on the 'past', in a sense that will now be made precise. Following the definitions and notation of Yu [10], which will be utilized in the sequel, let (7t = (7(Xf) and (7:+m = (7(Xt~m)' be the sigma-algebras of events generated by the random variables Xf = (X 1 ,X2 , ??? ,Xt) and Xi1.m = (X1+m ,Xl+ m+1 , ?? . ), respectively. We then define 13m, the coefficient of absolute regularity, as 13m = SUPt>l Esup {IP(BI(7I) - P(B)I : BE (7:+m} , where the expectation is taken with respect-to (71 = (7(XD. A stochastic process is said to be 13-mixing if (3m -t 0 as m -t 00. We note that there exist many other definitions of mixing (see [2] for details). The motivation for using the 13-mixing coefficient is that it is the weakest form of mixing for which uniform laws of large numbers can be established. In this work we consider two type of processes for which this coefficient decays to zero, namely algebraically decaying processes for which 13m ~ /3m- r , /3, r > 0, and exponentially mixing processes for which 13m ~ /3 exp{ -bm K } , jJ, b, I\, > O. Note that for Markov processes mixing implies exponential mixing, so that at least in this case, there is no loss of generality in assuming that the process is exponentially mixing. Note also that the usual i.i.d process may be obtained from either the exponentially or algebraically mixing process, by taking the limit I\, -t 00 or r -t 00, respectively. In this section we follow the approach taken by Yu [10] in deriving uniform laws of large numbers for mixing processes, extending her mainly asymptotic results to finite sample behavior, and somewhat broadening the class of processes considered by her. The basic idea in [10], as in many related approaches, involves the construction of an independentblock sequence, which is shown to be 'close' to the original process in a well-defined probabilistic sense. We briefly recapitulate the construction, slightly modifying the notation in [10] to fit in with the present paper. Divide the sequence xi' into 2J-lN blocks, each of size aN; we assume for simplicity that N = 2J-lNaN. The blocks are then numbered according to their order in the block-sequence. For 1 ~ j ~ J-lN define H j {i : 2(j - l)aN + 1 ~ i ~ (2j - l)aN} and Tj = {i : (2j - l)aN + 1 ~ i ~ (2j)aN}. Denote the random variables corresponding to the H j and Tj indices as X(j) = {Xi : i E H j } and X' (j) = {Xi : i E T j }. The sequence of H-blocks is then denoted by X aN = {X(j)}j:l. Now, construct a sequence of independent and identically distributed (i.i.d.) blocks {3(j) )}j:l' where 3(j) = {~i : i E H j }, such that the sequence is independent of Xi" and each block has the same distribution as the block X(j) from the original sequence. Because the process is stationary, the blocks 3(j) are not only independent but also identically distributed. The basic idea in the construction of the independent block sequence is that it is 'close', in a well-defined sense to the original blocked sequence X aN . Moreover, by appropriately selecting the number of blocks, J-lN, depending on the mixing nature of the sequence, one may relate properties of the original sequence X f", to those of the independent block sequence 3 aN (see Lemma 4.1 in [10)). = Let F be a class of bounded functions, such that 0 ~ f ~ B for any f E F. In order to Structural Risk Minimizationfor Nonparametric Time Series Prediction 311 relate the uniform deviations (with respect to F) of the original sequence Xi' to those of the independent-block sequence BaN' use is made of Lemma 4.1 from [10]. We also utilize Lemma 4.2 from [10] and modify it so that it holds for finite sample size. Consider the block-independent sequence BaN and define EJ.LN = J.L1N 'E~:1 f(B(j)) where f(=.(j?) = 'EiEH j f(~i)' j = 1,2, ... , J-lN, is a sequence of independent random variables such that 1 111 ~ aNB. In the remainder of the paper we use variables with a tilde above them to denote quantities related to the transformed block sequence. Finally, we use the symbol EN to denote the empirical average with respect to the original sequence, namely EN f = (N - d)-1 'E~d+1 f(Xi). The following result can be proved by a simple extension of Lemma 4.2 in [10] . Lemma 1.1 Suppose F is a permissible class of boundedfunctions, If I ~ B for f E :F. Then p {sup lEN f - Efl > ~ 2P {sup IEJ.LN 1 - Ell> aNt:} + 2J-lNf3aN' fE';: fE';: t:} (2) The main merit of Lemma 1.1 is in the transformation of the problem from the domain of dependent processes, implicit in the quantity lEN f - Efl, to one characterized by independent processes, implicit in the term EJ.LN Ell corresponding to the independent blocks. The price paid for this transformation is the extra term 2J-lN f3aN which appears on the r.h.s of the inequality appearing in Lemma 1.1 . 1- 2 Error Bounds The development in Section 1 was concerned with a scalar stochastic process X. In order to use the results in the context of time series, we first define a new vector-valued pro........ . . . . . . . . d+l . For cess X' = { ... ,X- 1 ,XO ,X 1 , ... } where Xi = (Xi,X i - 1, . :.... ,Xi-d) E ~ this sequence the f3-mixing coefficients obey the inequality f3m(X ' ) ~ f3m-d(X). Let F be a space of functions mapping Rd -7 R, and for each f E F let the loss function be given by ff(Xf-d) = IXi - f(X:~~W? The loss space is given by L,;: = {ff: f E F} . It is well known in the theory of empirical processes (see [7] for example), that in order to obtain upper bounds on uniform deviations of i.i.d sequences, use must be made of the so-called covering number of the function class F, with respect to the empirical it,N norm, given by it ,N(f, g) = N- 1 'E~1 If(Xd - g(Xi)l? Similarly, we denote the empirical norm with respect to the independent block sequence by [1 ,J.LN' where [1,J.LN(f,g) J-l,'/ 'E~:1 11(x(j)) - g(X(j) I, and where f(X(j?) 'EiEH j Xi and similarly for g. Following common practice we denote the t:-covering number of the functional space F using the metric p by N(t:, F, p). = = Definition 1 Let L';: be a class of real-valued functions from RD --t R, D = d + 1. For eachff E L,;:andx = (Xl,X2, .. . ,XaN ), Xi E R D , let if (x) = 'E~:lff(Xi)' Then define ?,;: = {if: if E L';:} ,where if : RaND -7 R+. In order to obtain results in terms of the covering numbers of the space L';: rather than ?,;:, which corresponds to the transformed sequence, we need the following lemma, which is not hard to prove. Lemma 2.1 For any t: > 0 N (t:, ?,;:, [1 ,J.LN) ~ N (t:jaN, L';:, h,N). R. Meir 312 PROOF The result follows by sequence of simple inequalities, showing that ll.J1.N (j, g) ~ aNh,N(f, g). I We now present the main result of this section, namely an upper bound for the uniform deviations of mixing processes, which in turn yield upper bounds on the error incurred by the empirically optimal predictor fd ,n.N. Theorem 2.1 Let X = { . .. ,Xl' X o, Xl, ... } be a bounded stationary (3-mixing stochastic process, with IXil ~ B, and let F be a class of bo unded functions, f : Rd ~ [0, B]. For each sample size N, let f~ be the function in :F which minimizes the empirical error, and 1* is the function in F minimizing the true error L(f). Then, where c' = c/128B. PROOF The theorem is established by making use of Lemma 1.1, and the basic results from the theory of uniform convergence for i.i.d. processes, together with Lemma 2.1 relating the covering numbers of the spaces iF and LF. The covering numbers of LF and Fare easily related using N(c, LF, Ll (P)) ~ N(c/2B, F, Ll (P)) . I Up to this point we have not specified J..tN and aN, and the result is therefore quite general. In order to obtain weak consistency we require that that the r.h.s. of (3) converge to zero for each c > O. This immediately yields the following conditions on J..tN (and thus also on aN through the condition 2aNJ..tN = N). Corollary 2.1 Under the conditions of Theorem 2.1, and the added requirements that d = o(aN) and N(c, F, h,N) < 00, the following choices of J..tN are sufficient to guarantee the weak consistency of the empirical predictor f N: J..tN ,..", N/t/(1+/t) (exponential mixing), (4) J..tN""" N s/{1+s), 0 < s < r (algebraic mixing), (5) where the notation aN ,..", bN implies that O(bN) ~ aN ~ O(b N ). PROOF Consider first the case of exponential mixing. In this case the r.h.s. of (3) clearly converges to zero because of the finiteness of the covering number. The fastest rate of convergence is achieved by balancing the two terms in the equation, leading to the choice J..tN '" N/t/(1+/t). In the case of algebraic mixing, the second term on the r.h.s. of (3) is of the order O(J..tNa"i/) where we have used d = o(aN). Since J..tNaN '" N, a sufficient condition to guarantee that this term converge to zero is that J..tN ,..", Ns/(1+s), 0 < s < r, as was claimed. I In order to derive bounds on the expected error, we need to make an assumption concerning the covering number of the space F. In particular, we know from the work Haussler [4J that the covering number is upper bounded as follows N(c , F, L 1 (P)) ~ e(Pdim(F) + 1) 2B) Pdim(F) ' (-7- for any measure P. Thus, assuming the finiteness of the pseudo-dimension of F guarantees a finite covering number. Structural Risk Minimization/or Nonparametric Time Series Prediction 313 3 Structural Risk Minimization The results in Section 2 provide error bounds for estimators formed by minimizing the empirical error over a fixed class of d-dimensional functions. It is clear that the complexity of the class of functions plays a crucial role in the procedure. If the class is too rich, manifested by very large covering numbers, clearly the estimation error term will be very large. On the other hand, biasing the class of functions by restricting its complexity, leads to poor approximation rates. A well-known strategy for overcoming this dilemma is obtained by considering a hierarchy of functional classes with increasing complexity. For any given sample size, the optimal trade-off between estimation and approximation can then be determined by balancing the two terms. Such a procedure was developed in the late seventies by Vapnik [8], and termed by him structural risk minimization (SRM). Other more recent approaches, collectively termed complexity regularization, have been extensively studied in recent years (e.g. [1]). It should be borne in mind, however, that in the context of time series there is an added complexity, that does not exist in the case of regression. Recall that the results derived in Section 2 assumed some fixed lag vector d. In general the optimal value of d is unknown, and could in fact be infinite. In order to achieve optimal performance in a nonparametric setting, it is crucial that the size of the lag be chosen adaptively as well. This added complexity needs to be incorporated into the SRM framework, if optimal performance in the face of unknown memory size is to be achieved. Let Fd,n, d, n E Fd ,n let N be a sequence of functions, and define F = U~l U~=l Fd,n ' For any which from [4] is upper bounded by cc-Pdim(Fd.n). We observe in passing that Lugosi and Nobel [5] have recently considered situations where the pseudo-dimension Pdim(Fd,n) is unknown, and the covering number is estimated empirically from the data. Although this line of thought is potentially very useful, we do not pursue it here, but rather assume that upper bounds on the pseudo-dimensions of Fd,n are known, as is the case for many classes of functions used in practice (see for example [9]). In line with the standard approach in [8] we introduce a new empirical function, which takes into account both the empirical error as well as the complexity costs penalizing overly complex models (large complexity index n and lag size d). Let (6) where LN(f) is the empirical error of the predictor f and the complexity penalties given by IogN1 (c, Fd,n) + Cn J-lN /64(2B)4 /-LN /64(2B)4 . ~ are (7) (8) The specific form and constants in these definitions are chosen with hindsight, so as to achieve the optimal rates of convergence in Theorem 3.1 below. The constants Cn and Cd are positive constants obeying l:~=1 e- Cn :::; 1 and similarly for Cd . A possible choice is Cn = 210g n + 1 and Cd = 210g d + 1. The value of J-lN can be chosen in accordance with Corollary 2.1. Let id,n,N minimize the empirical error LN(f) within the class of functions Fd ,n' ",!e assume that the classes Fd,n are compact, so that such a minimizer exists. Further, let IN R. Meir 314 be the function in F minimizing the complexity penalized loss (6), namely Ld n , , N(1~) = min min Ld n N(1~ n N) d2: 1 n2: 1 " " (9) The following basic result establishes the consistency of the structural risk minimization approach, and yields upper bounds on its performance. Theorem 3.1 Let Fd,n, d, n E N be sequence offunctional classes, where 1 E Fd,n is a mapping from Rd to R The expected loss of the function iN, selected according to the SRM principle, is upper bounded by EL(iN) ::; min {inf L(J) d,n d,n + Cl The main merit of Theorem 3.1 is the demonstration that the SRM procedure achieves an optimal balance between approximation and estimation, while retaining its non parametric attributes. In particular, if the optimal lag d predictor 1J belongs to Fd,no for some no, the SRM predictor would converge to it at the same rate as if no were known in advance. The same type of adaptivity is obtained with respect to the lag size d. The non parametric rates of convergence of the SRM predictor will be discussed in the full paper. References [1] A. Barron. Complexity Regularization with Application to Artificial Neural Networks. In G. Roussas, editor, Nonparametric Functional Estimation and Related Topics, pages 561-576. Kluwer Academic Press, 1991. [2] L. Gyorfi, W. HardIe, P. Sarda, and P. Vieu. Nonparametric Curve Estimation from Time Series. Springer Verlag, New York, 1989. [3] D. Haussler. Decision Theoretic Generalizations of the PAC Model for Neural Net and Other Learning Applications. Information and Computation, 100:78-150, 1992. [4] D. Haussler. Sphere Packing Numbers for Subsets of the Boolean n-Cube with Bounded Vapnik-Chervonenkis Dimesnion. J. Combinatorial Theory, Series A 69:217-232,1995. [5] G. Lugosi and A. Nobel. Adaptive Model Selection Using Empirical Complexities. Submitted to Annals Statis., 1996. [6] D. Modha and E. Masry. Memory Universal Prediction of Stationary Random Processes. IEEE Trans. Inj. Th., January, 1998. [7] D. Pollard. Convergence of Empirical Processes. Springer Verlag, New York, 1984. [8] V. N. Vapnik. Estimation of Dependences Based on Empirical Data. Springer Verlag, New York, 1992. [9] M. Vidyasagar. A Theory of Learning and Generalization. Springer Verlag, New York,1996. [10] B. Yu. Rates of convergence for empirical processes of stationary mixing sequences. Annals of Probability, 22:94-116, 1984. 

1475 |@word briefly:1 version:1 norm:2 d2:1 bn:2 recapitulate:1 paid:1 thereby:1 ld:5 series:13 selecting:3 chervonenkis:3 past:3 must:2 ixil:2 j1:1 statis:1 stationary:5 pursued:1 selected:1 node:1 ron:1 complication:1 prove:1 introduce:1 expected:2 behavior:1 considering:1 increasing:2 notation:4 moreover:2 bounded:6 anh:1 israel:2 anj:1 minimizes:2 pursue:1 developed:1 finding:1 transformation:2 hindsight:1 guarantee:3 temporal:1 pseudo:3 ti:1 xd:2 utilization:1 control:1 grant:1 f3m:2 masry:1 positive:2 engineering:1 accordance:1 modify:1 limit:1 id:1 modha:1 lugosi:2 studied:3 vieu:1 fastest:1 limited:1 bi:1 gyorfi:1 practical:1 practice:2 block:16 lf:3 procedure:3 jan:1 universal:1 empirical:21 thought:1 pre:1 numbered:1 onto:1 close:2 selection:2 risk:8 context:5 straightforward:1 simplicity:1 immediately:2 estimator:3 haussler:3 deriving:2 annals:2 construction:3 suppose:1 play:1 pioneered:1 hierarchy:1 ixi:1 hypothesis:1 utilized:1 observed:1 role:1 electrical:1 trade:1 complexity:17 ultimately:1 weakly:1 algebra:1 dilemma:1 packing:1 easily:1 artificial:1 choosing:2 quite:1 lag:10 larger:1 valued:2 precludes:1 ip:2 sequence:27 net:1 propose:1 remainder:1 mixing:23 achieve:3 intuitive:1 fel:1 competition:1 convergence:9 regularity:2 requirement:2 extending:1 generating:1 converges:1 depending:2 derive:1 ac:1 involves:1 implies:2 attribute:1 modifying:1 stochastic:7 stringent:1 settle:2 require:1 generalization:2 randomization:1 elementary:1 extension:3 hold:1 considered:2 exp:1 great:1 mapping:3 predict:1 achieves:1 estimation:12 applicable:1 combinatorial:1 symmetrization:1 him:1 establishes:1 minimization:7 clearly:3 supt:1 rather:4 ej:2 corollary:2 derived:1 mainly:2 sense:4 dependent:2 el:1 hidden:1 her:2 transformed:2 sketched:1 issue:2 denoted:1 retaining:1 development:1 ell:2 cube:1 construct:1 f3:1 seventy:1 yu:3 dumbo:1 minimized:1 future:1 inherent:2 ourselves:1 statistician:1 freedom:1 fd:27 tj:2 devoted:1 divide:1 haifa:1 minimal:1 boolean:1 pia:1 cost:1 deviation:3 subset:1 uniform:7 technion:2 predictor:15 srm:6 too:1 adaptively:1 sequel:1 probabilistic:2 xi1:1 off:1 together:2 borne:1 leading:1 expended:1 account:1 coefficient:4 depends:2 doing:1 sup:3 decaying:1 len:2 minimize:2 il:2 square:1 formed:1 yield:3 ant:1 weak:2 cc:1 submitted:1 definition:4 obvious:1 proof:4 associated:1 proved:2 xan:1 recall:1 knowledge:1 actually:1 appears:1 follow:2 rand:1 generality:2 furthermore:2 implicit:2 hand:2 hardie:1 roussas:1 effect:1 true:1 regularization:3 sieve:1 memoryless:2 deal:1 ll:3 covering:12 theoretic:1 tn:8 pro:1 meaning:1 possessing:1 recently:1 common:1 functional:5 empirically:2 exponentially:3 discussed:3 fare:1 relating:1 kluwer:1 blocked:1 rd:7 outlined:1 consistency:3 similarly:4 longer:1 recent:3 inf:1 belongs:1 termed:2 claimed:1 verlag:4 manifested:1 inequality:5 arbitrarily:1 additional:1 somewhat:1 algebraically:2 converge:3 full:4 technical:1 xf:5 academic:1 characterized:2 sphere:1 concerning:1 prediction:11 basic:5 regression:2 expectation:1 metric:1 achieved:2 finiteness:2 crucial:2 appropriately:1 permissible:1 extra:1 structural:8 feedforward:1 split:2 easy:1 concerned:1 bernstein:1 sarda:1 identically:2 xj:1 fit:2 restrict:2 idea:3 regarding:1 knowing:1 tradeoff:1 pdim:4 cn:4 effort:1 penalty:1 algebraic:2 pollard:1 passing:1 cause:1 jj:1 york:4 useful:2 clear:2 amount:2 nonparametric:7 extensively:2 meir:4 exist:3 estimated:1 overly:1 rmeir:1 penalizing:1 ce:1 utilize:1 efl:2 year:2 decision:1 bound:12 layer:1 fei:1 x2:2 phrased:1 min:3 department:1 according:2 poor:1 slightly:1 making:1 xo:1 taken:2 ln:19 equation:1 turn:1 needed:2 know:1 merit:2 mind:1 resents:1 available:1 obey:1 observe:1 barron:1 appearing:2 original:6 l1n:1 establish:1 approximating:1 added:3 quantity:2 parametric:3 strategy:1 dependence:3 usual:1 said:1 topic:1 nobel:2 assuming:2 index:3 minimizing:4 demonstration:1 balance:1 fe:2 potentially:2 relate:2 sigma:1 unknown:4 upper:9 observation:1 markov:1 finite:9 january:1 ixo:1 situation:2 tilde:1 incorporated:2 bll:1 misspecification:1 precise:1 anb:1 overcoming:1 namely:4 specified:2 extensive:1 distinction:1 conflicting:1 established:3 trans:1 able:1 below:1 biasing:1 max:1 memory:5 vidyasagar:1 event:2 circumvent:1 asymptotic:1 law:4 loss:9 expect:2 adaptivity:1 foundation:1 incurred:4 sufficient:2 principle:2 editor:1 balancing:2 cd:3 course:3 penalized:1 ban:2 supported:1 free:1 iej:1 allow:1 taking:1 face:1 absolute:1 distributed:2 curve:1 calculated:1 xn:1 dimension:3 rich:1 made:4 adaptive:1 bm:1 excess:1 compact:2 assumed:1 xi:22 nature:1 robust:1 broadening:1 complex:2 cl:1 domain:2 main:5 motivation:1 n2:1 x1:1 en:2 ff:2 martingale:1 n:1 sub:1 tna:1 obeying:1 exponential:3 xl:6 third:1 late:1 ix:2 theorem:7 specific:2 xt:2 showing:1 pac:1 symbol:1 lff:1 decay:1 weakest:1 exists:1 vapnik:7 restricting:1 conditioned:1 demand:1 generalizing:1 scalar:1 bo:1 collectively:1 springer:4 corresponds:1 minimizer:1 determines:1 conditional:2 price:1 hard:1 infinite:3 determined:1 lemma:11 called:5 inj:1 select:1 latter:2

Nonparametric Model-Based Reinforcement Learning Christopher G. Atkeson College of Computing, Georgia Institute of Technology, Atlanta, GA 30332-0280, USA ATR Human Information Processing, 2-2 Hikaridai, Seiko-cho, Soraku-gun, 619-02 Kyoto, Japan cga@cc.gatech.edu http://www .cc.gatech.edu/fac/Chris.Atkeson/ Abstract This paper describes some of the interactions of model learning algorithms and planning algorithms we have found in exploring model-based reinforcement learning. The paper focuses on how local trajectory optimizers can be used effectively with learned nonparametric models. We find that trajectory planners that are fully consistent with the learned model often have difficulty finding reasonable plans in the early stages of learning. Trajectory planners that balance obeying the learned model with minimizing cost (or maximizing reward) often do better, even if the plan is not fully consistent with the learned model. 1 INTRODUCTION We are exploring the use of nonparametric models in robot learning (Atkeson et al., 1997b; Atkeson and Schaal , 1997). This paper describes the interaction of model learning algorithms and planning algorithms, focusing on how local trajectory optimization can be used effectively with nonparametric models in reinforcement learning. We find that trajectory optimizers that are fully consistent with the learned model often have difficulty finding reasonable plans in the early stages of learning . The message of this paper is that a planner should not be entirely consistent with the learned model during model-based reinforcement learning. Trajectory optimizers that balance obeying the learned model with minimizing cost (or maximizing reward) often do better, even if the plan is not fully consistent with the learned model: 1009 Nonparametric Model-Based Reinforcement Learning '~~ A - 1" - - .--/ - ~ ~ ---e L V 2\ \ '\ V V '\ If \ [iV II '\\ l\ '\ \ ./ () \ \i\ \ \ \ \ I"" l"" / + ) . /V \ \ !"" ~ ."" \~ \ / I'---" / / ~ I" " ~ ~ ~ it ~ V Figure 1: A: Planning in terms of trajectory segments. B: Planning in terms of trajectories all the way to a goal point . Two kinds of reinforcement learning algorithms are direct (non-model-based) and indirect (model-based) . Direct reinforcement learning algorithms learn a policy or value function without explicitly representing a model of the controlled system (Sutton et al. , 1992) . Model-based approaches learn an explicit model of the system simultaneously with a value function and policy (Sutton, 1990 , 1991a,b; Barto et al. , 1995; Kaelbling et al. , 1996) . We will focus on model-based reinforcement learning , in which the learner uses a planner to derive a policy from a learned model and an optimization criterion . 2 CONSISTENT LOCAL PLANNING An efficient approach to dynamic programming, a form of global planning, is to use local trajectory optimizers (Atkeson, 1994) . These local planners find a plan for each starting point in a grid in the state space. Figure 1 compares the output of a traditional cell based dynamic programming process with the output of a planner based on integrating local plans. Traditional dynamic programming generates trajectory segments from each cell to neighboring cells, while the planner we use generates entire trajectories. These locally optimal trajectories have local policies and local models of the value function along the trajectories (Dyer and McReynolds, 1970; Jacobson and Mayne, 1970). The locally optimal trajectories are made consistent with their neighbors by using the local value function to predict the value of a neighboring trajectory. If all the local value functions are consistent with their neighbors the aggregate value function is a unique solution to the Bellman equation and the corresponding trajectories and policy are globally optimal. We would like any local planning algorithm to produce a local model of the value function so we can perform this type of consistency checking. We would also like a local policy from the local planner, so we can respond to disturbances and modeling errors. Differential dynamic programming is a local planner that has these characteristics (Dyer and McReynolds. 1970; Jacobson and Mayne, 1970). Differential dynamic programming maintains a local quadratic model of the value function along the current best trajectory x* (t): V (x,t) = Vo(t) + Vx(t)(x - x*(t))T + 0.5(x - x*(t))TVxx(t)(x - x*(t)) (1) C. G. Atkeson 1010 as well as a local linear model of the corresponding policy: U(X,t) = u*(t) + K(t)(x - x*(t)) (2) u(x, t) is the local policy at time t, the control signal u as a function of state x. u * (t) is the model's estimate of the control signal necessary to follow the current best trajectory x*(t). K(t) are the feedback gains that alter the control signals in response to deviations from the current best trajectory. These gains are also the first derivative of the policy along the current best trajectory. The first phase of each optimization iteration is to apply the current local policy to the learned model, integrating the modeled dynamics forward in time and seeing where the simulated trajectory goes. The second phase of the differential dynamic programming approach is to calculate the components of the local quadratic model of the value function at each point along the trajectory: the constant term Vo (t), the gradient Vx (t), and the Hessian Vxx (t). These terms are constructed by integrating backwards in time along the trajectory. The value function is used to produce a new policy, which is represented using a new x*(t), u*(t), and K(t). The availability of a local value function and policy is an attractive feature of differential dynamic programming. However, we have found several problems when applying this method to model-based reinforcement learning with nonparametric models: 1. Methods that enforce consistency with the learned model need an initial trajectory that obeys that model, which is often difficult to produce. 2. The integration of the learned model forward in time often blows up when the learned model is inaccurate or when the plant is unstable and the current policy fails to stabilize it. 3. The backward integration to produce the value function and a corresponding policy uses derivatives of the learned model, which are often quite inaccurate in the early stages of learning, producing inaccurate value function estimates and ineffective policies. 3 INCONSISTENT LOCAL PLANNING To avoid the problems of consistent local planners, we developed a trajectory optimization approach that does not integrate the learned model and does not require full consistency with the learned model. Unfortunately, the price of these modifications is that the method does not produce a value function or a policy, just a trajectory (x(t), u(t)). To allow inconsistency with the learned model, we represent the state history x(t) and the control history u(t) separately, rather than calculate x(t) from the learned model and u(t). We also modify the original optimization criterion C = Lk C(Xk, Uk) by changing the hard constraint that Xk+1 = f(Xk' Uk) on each time step into a soft constraint: Cnew = L [C(Xk' Uk) +~IXk+1-f(Xk,Uk)12] (3) k C(Xk' Uk) is the one step cost in the original optimization criterion. ~ is the penalty on the trajectory being inconsistent with the learned model Xk+1 = f(Xk' Uk). IXk +1 - f (Xk' Uk) I is the magnitude of the mismatch of the trajectory and the model prediction at time step k in the trajectory. ~ provides a way to control the amount of inconsistency. A small ~ reflects lack of confidence in the model, and allows Nonparametric Model-Based Reinforcement Learning //If\<'' ,,,? " ~j 1011 Figure 2: The SARCOS robot arm with a pendulum gripped in the hand. The pendulum aXIS is aligned with the fingers and with the forearm in this arm configuration. the optimized trajectory to be inconsistent with the model in favor of reducing A large). reflects confidence in the model, and forces the optimized trajectory to be more consistent with the model. ). can increase with time or with the number of learning trials . If we use a model that estimates the confidence level of a prediction, we can vary). for each lookup based on Xk and Uk. Locally weighted learning techniques provide exactly this type of local confidence estimate (Atkeson et al., 1997a) . C(Xk, Uk)' Now that we are not integrating the trajectory we can use more compact representations of the trajectory, such as splines (Cohen , 1992) or wavelets (Liu et al., 1994). We no longer require that Xk+l = f(Xk, Uk), which is a condition difficult to fulfill without having x and u represented as independent values on each time step. We can now parameterize the trajectory using the spline knot points, for example. In this work we used B splines (Cohen, 1992) to represent the trajectory. Other choices for spline basis functions would probably work just as well. We can use any nonlinear programming or function optimization method to minimize the criterion in Eq. 3. In this work we used Powell's method (Press et al., 1988) to optimize the knot points, a method which is convenient to use but not particularly efficient. 4 IMPLEMENTATION ON AN ACTUAL ROBOT Both local planning methods work well with learned parametric models. However , differential dynamic programming did not work at all with learned nonparametric models, for reasons already discussed. This section describes how the inconsistent local planning method was used in an application of model-based reinforcement learning: robot learning from demonstration using a pendulum swing up task (Atkeson and Schaal, 1997). The pendulum swing up task is a more complex version of the pole or broom balancing task (Spong, 1995) . The hand holds the axis of the pendulum, and the pendulum rotates about this hinge in an angular movement (Figure 2). Instead of starting with the pendulum vertical and above its rotational joint, the pendulum is hanging down from the hand, and the goal of the swing up task is to move the hand so that the pendulum swings up and is then balanced in the inverted position . The swing up task was chosen for study because it is a difficult dynamic maneuver and requires practice for humans to learn, but it is easy to tell if the task is successfully executed (at the end of the task the pendulum is balanced upright and does not fall down) . We implemented learning from demonstration on a hydraulic seven degree of free- 1012 . C. G. Atkeson c: 1.0 'tI 0.0 .S! ~ CD human demonstration 1st trial (imitation) 2nd trial 3rd trial -1.0 "S. -2.0 c: III ? -3.0 'S -4.0 b i ~ .! CD .?. c: .2 ?1:: -5.0 0.0 0.2 0.4 0.5 0.4 0.3 0.2 0.1 ......... -0.0 . .................. ~ -0.1 ~ -0.2 ] -0.3 III 0.0 0.2 ~ --_-. 0.6 0.8 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.8 2.0 1.8 2.0 .~.~ '., /. --,.,.,."..~ ?-.----? seconds Figure 3: The hand and pendulum motion during robot learning from demonstration using a nonparametric model. dom anthropomorphic robot arm (SARCOS Dextrous Arm located at ATR, Figure 2). The robot observed its own performance with the same stereo vision system that was used to observe the human demonstrations. The robot observed a human swinging up a pendulum using a horizontal hand movement (dotted line in Figure 3) . The most obvious approach to learning from demonstration is to have the robot imitate the human motion, by following the human hand trajectory. The dashed lines in Figures 3 show the robot hand motion as it attempts to follow the human demonstration of the swing up task, and the corresponding pendulum angles. Because of differences in the task dynamics for the human and for the robot, this direct imitation failed to swing the pendulum up, as the pendulum did not get even halfway up to the vertical position, and then oscillated about the hanging down position. The approach we used was to apply a planner to finding a swing up trajectory that worked for the robot, based on learning both a model and a reward function and using the human demonstration to initialize the planning process. The data collected during the initial imitation trial and subsequent trials was used to build a model. Nonparametric models were constructed using locally weighted learning as described in (Atkeson et al., 1997a) . These models did not use knowledge of the model structure but instead assumed a general relationship: (4) where () is the pendulum angle and x is the hand position. Training data from the demonstrations was stored in a database, and a local model was constructed to answer each query. Meta-parameters such as distance metrics were tuned using cross validation on the training set. For example, cross validation was able to quickly establish that hand position and velocity (x and x) played an insignificant role in predicting future pendulum angular velocities. The planner used a cost function that penalizes deviations from the demonstration trajectory sampled at 60H z: C(Xk, Uk) = (Xk - X~)T(Xk - X~) + uluk (5) Nonparametric Model-Based Reinforcement Learning 1013 where the state is x = ((J, il, x , x), x d is the demonstrated motion, k is the sample index, and the control is u = (x). Equation 3 was optimized using B splines to represent x and u. The knot points for x and u were initially separately optimized to minimize (6) and (7) The tolerated inconsistency, ). was kept constant during a set of trials and set at values ranging from 100 to 100000. The exact value of ). did not make much difference. Learning failed when). was set to zero , as there was no way for the learned model to affect the plan. The planning process failed when ). was set too high , enforcing the learned model too strongly. The next attempt got the pendulum up a little more. Adding this new data to the database and replanning resul ted in a movement that succeeded (trial 3 in Figure 3). The behavior shown in Figure 3 is quite repeatable. The balancing behavior at the end of the trial is learned separately and continues for several minutes, at which point the trial is automatically terminated (Schaal, 1997). 5 DISCUSSION AND CONCLUSION We applied locally weighted regression (Atkeson et aI. , 1997a) in an attempt to avoid the structural modeling errors of idealized parametric models during model-based reinforcement learning, and also to see if a priori knowledge of the structure of the task dynamics was necessary. In an exploration of the swingup task, we found that these nonparametric models required a planner that ignored the learned model to some extent. The fundamental reason for this is that planners amplify modeling error. Mechanisms for this amplification include: ? The planners take advantage of any modeling error to reduce the cost of the planned trajectory, so the planning process seeks out modeling error that reduces apparent cost . ? Some planners use derivatives of the model, which amplifies any noise in the model. Models that support fast learning will have errors and noise. For example , in order to learn a model of the complexity necessary to accurately model the full robot dynamics between the commanded and actual hand accelerations a large amount of data is required, independent of modeling technique. The input would be 21 dimensional (robot state and command) ignoring actuator dynamics. Because there are few robot trials during learning, there is not enough data to make such a model even just in the vicinity of a successful trajectory. If it was required that enough data is collected during learning to make an accurate model. robot learning would be greatly slowed down. One solution to this error amplification is to bias the nonparametric modeling tools to oversmooth the data. This reduces the benefit of nonparametric modeling, and also ignores the true learned model to some degree. Our solution to this problem is to introduce a controlled amount of inconsistency with the learned model into the planning process. The control parameter). is explicit and can be changed as a function of time, amount of data, or as a function of confidence in the model at the query point. 1014 C. G. Atkeson References Atkeson, C. G. (1994). Using local trajectory optimizers to speed up global optimization in dynamic programming. In Cowan, J. D., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6, pages 663-670. Morgan Kaufmann, San Mateo, CA. Atkeson, C . G., Moore, A. W., and Schaal, S. (1997a). Locally weighted learning. Artificial Intelligence Review, 11:11-73. Atkeson, C. G., Moore, A. W., and Schaal, S. (1997b). Locally weighted learning for control. Artificial Intelligence Review, 11:75-113. Atkeson, C. G. and Schaal, S. (1997) . Robot learning from demonstration. In Proceedings of the 1997 International Conference on Machine Learning. Barto, A. G ., Bradtke, S. J., and Singh, S. P. (1995). Learning to act using real-time dynamic programming. Artificial Intelligence, 72(1):81-138. Cohen, M. F . (1992). Interactive spacetime control for animation. Computer Graphics, 26(2):293-302. Dyer, P. and McReynolds, S. (1970). The Computational Theory of Optimal Control. Academic, NY. Jacobson, D. and Mayne, D. (1970). Differential Dynamic Programming. Elsevier, NY. Kaelbling, L. P., Littman, M. L., and Moore, A. W. (1996). Reinforcement learning: A survey. lournal of Artificial Intelligence Research, 4:237-285 . Liu, Z., Gortler, S. J., and Cohen, M. F. (1994). Hierarchical spacetime control. Computer Graphics (SIGGRAPH '94 Proceedings), pages 35-42. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1988). Numerical Recipes in C. Cambridge University Press, New York, NY. Schaal, S. (1997). Learning from demonstration . In Mozer, M. C., Jordan, M., and Petsche, T ., editors, Advances in Neural Information Processing Systems 9, pages 1040-1046. MIT Press, Cambridge, MA. Spong, M. W. (1995). The swing up control problem for the acrobot. IEEE Control Systems Magazine, 15(1):49-55. Sutton, R. S. (1990). Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. In Seventh International Machine Learning Workshop, pages 216-224. Morgan Kaufmann, San Mateo, CA. http://envy.cs.umass.edu/People/sutton/publications.html. Sutton, R. S. (1991a). Dyna, an integrated architecture for learning, planning and reacting. http://envy.cs.umass.edu/People/sutton/publications.html, Working Notes of the 1991 AAAI Spring Symposium on Integrated Intelligent Architectures pp. 151-155 and SIGART Bulletin 2, pp. 160-163. Sutton, R . S. (1991b). Planning by incremental dynamic programming. In Eighth International Machine Learning Workshop, pages 353-357. Morgan Kaufmann, San Mateo, CA. http://envy.cs.umass.edu/People/sutton/publications.html. Sutton, R. S., Barto, A. G., and Williams, R. J. (1992). Reinforcement learning is direct adaptive optimal control. IEEE Control Systems Magazine, 12:19-22. 

1476 |@word trial:11 version:1 nd:1 seek:1 initial:2 configuration:1 liu:2 uma:3 tuned:1 current:6 numerical:1 subsequent:1 intelligence:4 imitate:1 xk:16 sarcos:2 provides:1 along:5 constructed:3 direct:4 differential:6 symposium:1 introduce:1 alspector:1 behavior:2 planning:17 bellman:1 globally:1 automatically:1 actual:2 little:1 kind:1 developed:1 finding:3 ti:1 act:1 interactive:1 exactly:1 uk:11 control:15 producing:1 maneuver:1 gortler:1 local:28 modify:1 sutton:9 reacting:2 mateo:3 commanded:1 obeys:1 unique:1 practice:1 optimizers:5 powell:1 got:1 convenient:1 confidence:5 integrating:4 seeing:1 get:1 amplify:1 ga:1 applying:1 www:1 optimize:1 demonstrated:1 maximizing:2 go:1 williams:1 starting:2 survey:1 swinging:1 oscillated:1 magazine:2 exact:1 programming:14 us:2 velocity:2 particularly:1 located:1 continues:1 database:2 observed:2 role:1 parameterize:1 calculate:2 movement:3 balanced:2 mozer:1 complexity:1 reward:3 littman:1 dynamic:19 dom:1 singh:1 segment:2 learner:1 basis:1 joint:1 indirect:1 siggraph:1 represented:2 finger:1 hydraulic:1 vxx:1 fac:1 fast:1 query:2 artificial:4 tell:1 aggregate:1 quite:2 apparent:1 favor:1 advantage:1 interaction:2 neighboring:2 aligned:1 mayne:3 amplification:2 amplifies:1 recipe:1 produce:5 incremental:1 derive:1 eq:1 implemented:1 c:3 mcreynolds:3 exploration:1 human:10 vx:2 require:2 anthropomorphic:1 exploring:2 hold:1 predict:1 vary:1 early:3 replanning:1 successfully:1 tool:1 reflects:2 weighted:5 mit:1 rather:1 fulfill:1 avoid:2 barto:3 gatech:2 command:1 publication:3 focus:2 schaal:7 greatly:1 elsevier:1 inaccurate:3 entire:1 vetterling:1 integrated:3 initially:1 html:3 priori:1 plan:7 integration:2 initialize:1 having:1 ted:1 alter:1 future:1 seiko:1 spline:5 intelligent:1 few:1 simultaneously:1 phase:2 attempt:3 atlanta:1 message:1 jacobson:3 accurate:1 succeeded:1 necessary:3 iv:1 penalizes:1 modeling:8 soft:1 planned:1 cost:6 kaelbling:2 deviation:2 pole:1 successful:1 seventh:1 too:2 graphic:2 stored:1 answer:1 tolerated:1 cho:1 st:1 fundamental:1 international:3 quickly:1 aaai:1 derivative:3 japan:1 blow:1 lookup:1 stabilize:1 availability:1 explicitly:1 idealized:1 ixk:2 pendulum:18 maintains:1 minimize:2 il:1 kaufmann:3 characteristic:1 accurately:1 knot:3 trajectory:41 cc:2 history:2 pp:2 obvious:1 gain:2 sampled:1 knowledge:2 focusing:1 follow:2 response:1 strongly:1 just:3 stage:3 angular:2 hand:11 working:1 horizontal:1 christopher:1 nonlinear:1 lack:1 cnew:1 usa:1 true:1 swing:9 vicinity:1 moore:3 attractive:1 during:7 criterion:4 vo:2 motion:4 bradtke:1 ranging:1 cohen:4 discussed:1 cambridge:2 ai:1 rd:1 grid:1 consistency:3 robot:17 longer:1 own:1 tesauro:1 meta:1 inconsistency:4 inverted:1 morgan:3 signal:3 ii:1 dashed:1 full:2 kyoto:1 reduces:2 academic:1 cross:2 controlled:2 prediction:2 regression:1 spong:2 vision:1 metric:1 iteration:1 represent:3 cell:3 separately:3 resul:1 ineffective:1 probably:1 cowan:1 inconsistent:4 jordan:1 structural:1 backwards:1 iii:2 easy:1 enough:2 affect:1 architecture:3 reduce:1 penalty:1 stereo:1 soraku:1 hessian:1 york:1 ignored:1 cga:1 amount:4 nonparametric:14 locally:7 lournal:1 http:4 dotted:1 changing:1 kept:1 backward:1 halfway:1 angle:2 respond:1 planner:16 reasonable:2 entirely:1 played:1 spacetime:2 quadratic:2 constraint:2 worked:1 generates:2 speed:1 spring:1 hanging:2 describes:3 modification:1 slowed:1 equation:2 mechanism:1 dyna:1 dyer:3 end:2 apply:2 observe:1 actuator:1 hierarchical:1 enforce:1 petsche:1 original:2 include:1 hinge:1 build:1 hikaridai:1 establish:1 approximating:1 move:1 already:1 parametric:2 traditional:2 gradient:1 distance:1 rotates:1 atr:2 simulated:1 gun:1 chris:1 seven:1 collected:2 unstable:1 broom:1 reason:2 dextrous:1 enforcing:1 extent:1 modeled:1 relationship:1 index:1 rotational:1 balance:2 minimizing:2 demonstration:12 difficult:3 unfortunately:1 executed:1 sigart:1 implementation:1 policy:16 perform:1 vertical:2 forearm:1 required:3 optimized:4 learned:27 able:1 mismatch:1 eighth:1 difficulty:2 force:1 disturbance:1 predicting:1 arm:4 representing:1 technology:1 lk:1 axis:2 review:2 swingup:1 checking:1 fully:4 plant:1 validation:2 integrate:1 degree:2 consistent:10 editor:2 balancing:2 cd:2 oversmooth:1 changed:1 free:1 bias:1 allow:1 institute:1 neighbor:2 fall:1 bulletin:1 benefit:1 feedback:1 ignores:1 forward:2 made:1 reinforcement:15 san:3 adaptive:1 atkeson:16 compact:1 global:2 assumed:1 imitation:3 learn:4 ca:3 ignoring:1 complex:1 did:4 terminated:1 noise:2 animation:1 envy:3 georgia:1 ny:3 fails:1 position:5 explicit:2 obeying:2 wavelet:1 down:4 minute:1 repeatable:1 insignificant:1 workshop:2 adding:1 effectively:2 magnitude:1 acrobot:1 flannery:1 failed:3 teukolsky:1 ma:1 goal:2 acceleration:1 price:1 hard:1 upright:1 reducing:1 college:1 support:1 people:3

Synaptic Transmission: An Information-Theoretic Perspective Amit Manwani and Christof Koch Computation and Neural Systems Program California Institute of Technology Pasadena, CA 91125 email: quixote@klab.caltech.edu koch@klab.caltech.edu Abstract Here we analyze synaptic transmission from an infonnation-theoretic perspective. We derive c1osed-fonn expressions for the lower-bounds on the capacity of a simple model of a cortical synapse under two explicit coding paradigms. Under the "signal estimation" paradigm, we assume the signal to be encoded in the mean firing rate of a Poisson neuron. The perfonnance of an optimal linear estimator of the signal then provides a lower bound on the capacity for signal estimation. Under the "signal detection" paradigm, the presence or absence of the signal has to be detected. Perfonnance of the optimal spike detector allows us to compute a lower bound on the capacity for signal detection. We find that single synapses (for empirically measured parameter values) transmit infonnation poorly but significant improvement can be achieved with a small amount of redundancy. 1 Introduction Tools from estimation and infonnation theory have recently been applied by researchers (Bialek et. ai, 1991) to quantify how well neurons transmit infonnation about their random inputs in their spike outputs. In these approaches, the neuron is treated like a black-box, characterized empirically by a set of input-output records. This ignores the specific nature of neuronal processing in tenns of its known biophysical properties. However, a systematic study of processing at various stages in a biophysically faithful model of a single neuron should be able to identify the role of each stage in infonnation transfer in tenns of the parameters relating to the neuron's dendritic structure, its spiking mechanism, etc. Employing this reductionist approach, we focus on a important component of neural processing, the synapse, and analyze a simple model of a cortical synapse under two different representa?tional paradigms. Under the "signal estimation" paradigm, we assume that the input signal 202 A. Manwani and C. Koch is linearly encoded in the mean firing rate of a Poisson neuron and the mean-square error in the reconstruction of the signal from the post-synaptic voltage quantifies system performance. From the performance of the optimal linear estimator of the signal, a lower bound on the capacity for signal estimation can be computed. Under the "signal detection" paradigm, we assume that information is encoded in an all-or-none format and the error in deciding whether or not a presynaptic spike occurred by observing the post-synaptic voltage quantifies system performance. This is similar to the conventional absentipresent(Yes-No) decision paradigm used in psychophysics. Performance of the optimal spike detector in this case allows us to compute a lower bound on the capacity for signal detection. 0x- 0 NoSpI<o Poisson Encoding !'-. NoR &, Release 1 Spoke 1 Optimal Estimator 1- Stimulys h{l) Stochastic Variable EPSC Vesicle Release Amplitude EPSP Shape Optimal Detector Spike I No Spike Spike I No Spike Encodlna Synaptic Channel Decoding Figure 1: Schematic block diagram for the signal detection and estimation tasks. The synapse is modeled as a binary channel followed by a filter h(t) = at exp( -tits). where a is a random variable with probability density, P(a) = a (aa)k- 1 exp( -aa)/(k - 1)!. The binary channel, (inset, EO = Pr[spontaneous release], E1 = Pr [release failure]) models probabilistic vesicle release and h(t) models the variable epsp size observed for cortical synapses. n( t) denotes additive post-synaptic voltage noise and is assumed to be Gaussian and white over a bandwidth En. Performance of the optimal linear estimator (Wiener Filter) and the optimal spike detector (Matched Filter) quantify synaptic efficacy for signal estimation and detection respectively. 2 The Synaptic Channel Synaptic transmission in cortical neurons is known to be highly random though the role of this variability in neural computation and coding is still unclear. In central synapses, each synaptic bouton contains only a single active release zone, as opposed to the hundreds or thousands found at the much more reliable neuromuscular junction. Thus, in response to an action potential in the presynaptic terminal at most one vesicle is released (Kom and Faber, 1991). Moreover, the probability of vesicle release p is known to be generally low (0.1 to 0.4) from in vitro studies in some vertebrate and invertebrate systems (Stevens, 1994). This unreliability is further compounded by the trial-to-trial variability in the amplitude of the post-synaptic response to a vesicular release (Bekkers et. ai, 1990). In some cases, the variance in the size of EPSP is as large as the mean. The empirically measured distribution of amplitudes is usually skewed to the right (possibly biased due the inability of measuring very small events) and can be modeled by a Gamma distribution. In light of the above, we model the synapse as a binary channel cascaded by a random amplitude filter (Fig. 1). The binary channel accounts for the probabilistic vesicle release. EO Synaptic Trarumission: An Information-Theoretic Perspective 203 and ?l denote the probabilities of spontaneous vesicle release and failure respectively. We follow the binary channel convention used in digital communications (? 1 = 1-p), whereas, p is more commonly used in neurobiology. The filter h(t) is chosen to correspond to the epsp profile of a fast AMPA-like synapse. The amplitude of the filter a is modeled as random variable with density Pea), mean J.la and standard deviation aa. The CV (standard deviation/mean) of the distribution is denoted by eVa. We also assume that additive Gaussian voltage noise net) at the post-synaptic site further corrupts the epsp response. net) is assumed to white with variance a~ and a bandwidth En corresponding to the membrane time constant T. One can define an effective signal-to-noise ratio, SN R = Ea/No? given by the ratio of the energy in the epsp pulse, Eh = 00 h2 (t) dt to the noise power spectral density, No = a;/ En. The performance of the synapse depends on the SN R and not on the absolute values of Eh or an. In the above model, by regarding synaptic parameters as constants, we have tacitly ignored history dependent effects like paired-pulse facilitation, vesicle depletion, calcium buffering. etc, which endow the synapse with the nature of a sophisticated nonlinear filter (Markram and Tsodyks, 1997). 10 a) m(t) '" '~t ",!t) b) N.:" V..:~ Spike n(t) ~ SpIke X=l Y=l l?P. Effective Continuous Estimation Channel 3 . Effective Elinm. Detection Channel Figure 2: (a) Effective channel model for signal estimation. met), met), net) denote the stimulus, the best linear estimate, and the reconstruction noise respectively. (b) Effective channel model for signal detection. X and Y denote the binary variables corresponding to the input and the decision respectively. Pi and Pm are the effective error probabilities. Signal Estimation Let us assume that the spike train of the presynaptic neuron can be modeled as a doubly stochastic Poisson process with a rate A(t) = k(t) * met) given as a convolution between the stimulus met) and a filter k(t). The stimulus is drawn from a probability distribution which we assume to be Gaussian. k(t) = exp( -tiT) is a low-pass filter which models the phenomenological relationship between a neuron's firing rate and its input current. T is chosen to correspond to the membrane time constant. The exact form of k(t) is not crucial and the above form is assumed primarily for analytical tractability. The objective is to find the optimal estimator ofm(t) from the post-synaptic voltage v(t), where optimality is in a least-mean square sense. The optimal mean-square estimator is, in general, nonlinear and reduces to a linear filter only when all the signals and noises are Gaussian. However, instead of making this assumption, we restrict ourselves to the analysis of the optimal linear estimator, met) = get) * vet), i.e. the filter get) which minimizes the mean-square error E = (m(t) - m(t))2) where (.) denotes an ensemble average. The overall estimation system shown in Fig. 1 can be characterized by an effective continuous channel (Fig. 2a) where net) = met) - met) denotes the effective reconstruction noise. System performance can be quantified by E, the lower E, the better the synapse at signal transmission. The expression for the optimal filter (Wiener filter) in the frequency domain is g(w) = Smv( -w)/Svv(w) where Smv(w) is the cross-spectral density (Fourier transform of the cross-correlation Rmv) ofm(t) and set) and Svv(w) is the power spectral density of vet). The minimum mean-square error is given by, E = a~ I Smv(w) 12 / Svv(w) dw. The set S = {w 1 Svv (w) =J. O} is called the support of Svv (w). - Is A. Manwani and C. Koch 204 Another measure of system performance is the mutual information rate I (m; v) between m(t) and v(t), defined as the rate of information transmitted by v(t) about s(t). By the Data Processing inequality (Cover 1991), l(m, v) ~ l(m, m). A lower bound of l(m, m) and thus of l(m; v) is given by the simple expression lib = ~ log2[~::/w/l dw (units of bits/sec). The lower bound is achieved when n(t) is Gaussian and is independent of m(t). Since the spike train s(t) = L 6(t - ti) is a POiSSOl!process with rate k(t) * m(t), its power spectrum is given by the expression, Sss(w) = >'+ 1 K(w) 12 Smm(w) where ). is the mean firing rate. We assume that the mean (J..Lm) and variance (CT~) of m(t) are chosen such that the probability that >.(t) < 0 is negligible 1 The vesicle release process is the spike train gated by the binary channel and so it is also a Poisson process with rate (1 - E1 )>.(t). Since v(t) = L aih(t - ti) + n(t) is a filtered P~isson process, its power spectral density is given by Svv (w) =1 H(w) 12 {(J..L~+CT~)(1-E1)>'+J..L~(1-E1)21 K(w) 12 Smm(w)} + Snn{w). The cross-spectral density is given by the expression Svm(w) = (1 - Et)J..LaSmm(w)H(w)K(w). This allows us to write the mean-square error as, Is Thus, the power spectral density ofn(t) is given by Snn = >'eff(w) + Self(w). Notice that if K (w) ---+ 00, E ---+ 0 i. e. perfect reconstruction takes place in the limit of high firing rates. For the parameter values chosen, SefJ{w) ? >'e//(w), and can be ignored. Consequently, signal estimation is shot noise limited and synaptic variability increases shot noise by a factor N syn = (1 + eVa2 ) / (1 - E1)' For eVa = 0.6 and E1 = 0.6, N syn = 3.4, and for eVa = 1 and E1 = 0.6, N syn = 5. If m(t) is chosen to be white, band-limited to Bm Hz, closed-form expressions for E and lib can be obtained. The expression for lib is tedious and provides little insight and so we present only the expression for E below. 2 ,1 -1 BT E(r,BT ) = CTm [1- ~-B tan (~)l 1+, T +, E is a monotonic function of, (decreasing) and BT (increasing). ,can be considered as the effective number of spikes available per unit signal bandwidth and BT is the ratio of the signal bandwidth and the neuron bandwidth. Plots of normalized reconstruction error Er = E/CT~ and llb versus mean firing rate ().) for different values of signal bandwidth Bm are shown in Fig. 3a and Fig. 3b respectively. Observe that lib (bits/sec) is insensitive to Bm for firing rates upto 200Hz because the decrease in quality of estimation (E increases with Bm) is compensated by an increase in the number of independent samples (2Bm) available per second. This phenomenon is characteristic of systems operating in the low SNR regime. lib has the generic form, llb = B log(1 + S/(N B)), where B, S and N denote signal bandwidth, signal power and noise power respectively. For low SNR, I ~ B S / (N B) = S / N, is independent of B. So one can argue that, for our choice of parameters, a single synapse is a low SNR system. The analysis generalizes very easily to the case of multiple synapses where all are driven by the same signal s (t). (Manwani and Koch, in preparation). However, instead of presenting the rigorous analysis, we appeal to the intuition gained from the single synapse case. Since a single synapse can be regarded as a shot noise source, n parallel synapses can be treated as n parallel noise sources. Let us make the plausible lWe choose pm and O'm so that X= 30'). (std of ,X) so that Prob['x(t) ~ 0] < 0.01. Synaptic Transmission: An Information-Theoretic Perspective 205 assumption that these noises are uncorrelated. If optimal estimation is carried out separately for each synapse and the estimates are combined optimally, the effective noise variance is given by the harmonic mean of the individual variances i.e. l/u~eff = Li l/u~i. However, if the noises are added first and optimal estimation is carried out with respect to the sum, the effective noise variance is given by the arithmetic mean of the individual variances, i.e. u~ef f :::: Li u~dn2. If we assume that all synapses are similar so that U~i = u 2, u~ef f = u 2In. Plots of Er and Jib for the case of 5 identical synapses are shown in Fig. 3c and Fig. 3d respectively. Notice that Jib increases with Bm suggesting that the system is no longer in the low SNR regime. Thus, though a single synapse has very low capacity, a small amount of redundancy causes a considerable increase in performance. This is consistent with the fact the in the low S N R regime, J increases linearly with S N R , consequently, linearly with n, the number of synapses. a) b) x x x x 0 0 x x 0 o.a ~ ~ + ~ .. .. + + + + 000 000 0 0 0 0 X X X + x x x X ~ e x o W 0.7 B m = 10Hz - 0.8 E - m m - - B m=75Hz - Bm: 100Hz Bm= 100 Hz 0.5 20 40 ~ x o.a 0 0 l1li l1li 100 120 140 1l1li 200 180 l1li 80 100 l1li 80 ~ 120 140 180 180 200 ~ ~ ~ - . .. +"-: .... __ .. + + .. 1. + + + g .. ------ .. + o 0 + + .. '- 0.8 o W "0 0.7 :- 12 UQ) 0 o I IO (/) 0 UiS ."t::: .0 Q) .~ (Q o.s - Q) E o Z B B=25Hz Bm= 50Hz B m=75HZ - ~ =10Hz x o Bm= 50 Hz .~ Z 12 X Bm- 25Hz Q)m o X o.s "0 1U 1. x x x S ~. 0.5 .E2 .5 0 .? 20 Firing Rate (Hz) 40 ~ Firing Rate (Hz) Figure 3: Er and!,b vs. mean firing rate (X) for n = I [(a) and (b)] and n =5 [(c) and (d)] identical synapses respectively (different values of Em) for signal estimation. Parameter values are 101 = 0.6, 100 = 0, eVa = 0.6, ts = 0.5 msec, T = I Omsec, (7n = 0.1 mY, En = 100 Hz. 4 Signal Detection The goal in signal detection is to decide which member from a finite set of signals was generated by a source, on the basis of measurements related to the output only in a statistical sense. Our example corresponds to its simplest case, that of binary detection. The objective is to derive an optimal spike detector based on the post-synaptic voltage in a given time interval. The criterion of optimality is minimum probability of error (Pe ). A false alarm A. Manwani and C. Koch 206 (FA) error occurs when a spike is falsely detected even when no presynaptic spike occurs and a miss error (M) occurs when a spike fails to be detected. The probabilities of the errors are denoted by P, and Pm respectively. Thus, Pe = (1- Po) P, +Po Pm where Po denotes the a priori probability of a spike occurrence. Let X and Y be binary variables denoting spike occurrence and the decision respectively. Thus, X = 1 if a spike occurred else X = O. Similarly, Y = 1 expresses the decision that a spike occurred. The posterior likelihood ratio is defined as ?(v) = Pr(v I X = l)/Pr(v I X = 0) and the prior likelihood as ?0 = (1 - Po)/Po. The optimal spike detector employs the well-known likelihood ratio test, "If?(v) ~ ?0 Y=lelseY=O". When X = 1,v(t) = ah(t)+n(t) elsev(t) = n(t). Since a is a random variable, ?(v) = (f Pr(v I X = 1; a) P(a) da)/ Pr(v I X = 0). If the noise n( t) is Gaussian and white, it can be shown that the optimal decision rule reduces to a matchedfilte?, i.e. if the correlation, r between v(t) and h(t) exceeds a particular threshold (denoted by TJ), Y = 1 else Y = O. The overall decision system shown in Fig. 1 can be treated as effective binary channel (Fig. 2b). The system perfonnance can be quantified either by Pe or J (X; Y), the mutual infonnation between the binary random variables, X and Y. Note that even when n(t) = 0 (SN R = 00), Pe =j:. 0 due to the unreliability of vesicular release. Let Pe* denote the probability of error when S N R = 00. If EO = 0, Pe* = Po El is the minimum possible detection error. Let PJ and P~ denote FA and M errors when the release is ideal (El = 0, EO = 0). It can be shown that Pe = Pe* + P~[Po(1- Ed - + PJ[(l - Po)(l P~ + El (1 - P~ + PI) (1 - Po)EO] P, = PJ ' Pm = EO) - PoEl] Both PJ and P~ depend on TJ. The optimal value ofT) is chosen such that Pe is minimized. In general, PJ and P~ can not be expressed in closed-fonn and the optimal 'f} is found using the graphical ROC analysis procedure. Ifwe normalize a such that /-La = 1, PJ and P~ can be parametrically expressed in tenns ofa nonnalized threshold 'f}*, PJ = 0.5[1- Er f('f}*)], = P~ 0.5[1+ Iooo Erf(TJ* - JSNRa) P(a) da]. J(X;Y) can be computed using the fonnula for the mutual infonnation for a binary channel, J = 1i (Po (1 - Pm) + (1 Po) P,) - Po 1i(Pm ) - (1- Po)1i(P, ) where 1i(x) -x log2 (x) - (1- x) log2(1- x) is the binary entropy function. The analysis can be generalized to the case of n syna!Jses but the expressions involve n-dimensional integrals which need to be evaluated numerically. The Central Limit Theorem can be used to simplify the case of very large n. Plots of Pe and J(X; Y) versus n for different values of SNR (1,10,00) for the case of identical synapses are shown in Fig. 4a and Fig. 4b respectively. Yet again, we observe the poor perfonnance of a single synapse and the substantial improvement due to redundancy. The linear increase of J with n is similar to the result obtained for signal estimation. = 5 Conclusions We find that a single synapse is rather ineffective as a communication device but with a little redundancy neuronal communication can be made much more robust. Infact, a single synapse can be considered as a low SNR device, while 5 independent synapses in parallel approach a high SNR system. This is consistently echoed in the results for signal estimation and signal detection. The values of infonnation rates we obtain are very small compared to numbers obtained from some peripheral sensory neurons (Rieke et. ai, 1996). This could be due to an over-conservative choice of parameter values on our part or could argue for the preponderance of redundancy in neural systems. What we have presented above are preliminary results of work in progress and so the path ahead is much 2 For deterministic a, the result is well-known, but even if a is a one-sided random variable, the matched filter can be shown to be optimal. Synaptic Tranrmission: An lnformation-Theoretic Perspective a) b) SNR = In!. ..... SNR=10 --SNR=1 -4-- ~ e w 207 i' ~ 0 ?? :0 0. ."r;:====~"'-'-------::::::::~ 0.7 ~ ... -4-- SNR = In!. ..... SNR=10 --SNR= 1 * ex: 0.4 .~ 0.:1 c: E 00.2 0.' ? 0 ... o~ , --~~2~~--~3--~--~'--~~ Number of Synapses (n) .., ~~~--~2~----~3------~'----~ Number of Synapses (n) Pe (a) and l,b (b) vs. the number of synapses, n, (different values of SN R) for signal detection. SNR = Inf. corresponds to no post-synaptic voltage noise. All the synapses are assumed to be identical. Parameter values are po = 0.5, 101 0.6, 100 0, eVa = 0.6, ts =0.5 msec, T = 10 msec, an =0.1 mY, Bn = 100 Hz. = = longer than the distance we have covered so far. To the best of our knowledge, analysis of distinct individual components of a neuron from an communications standpoint has not been carried out before. Acknowledgements This research was supported by NSF, NIMH and the Sloan Center for Theoretical Neuroscience. We thank Fabrizio Gabbiani for illuminating discussions. References Bekkers, J.M., Richerson, G.B. and Stevens, C.F. (1990) "Origin of variability in quantal size in cultured hippocampal neurons and hippocampal slices," Proc. Natl. Acad. Sci. USA 87: 5359-5362. Bialek, W. Rieke, F. van Steveninck, R.D.R. and Warland, D. (1991) "Reading a neural code," Science 252: 1854-1857. Cover, T.M., and Thomas, lA. (1991) Elements ofInformation Theory. New York: Wiley. Kom, H. and Faber, D.S. (1991) "Quantal analysis and synaptic efficacy in the CNS," Trends Neurosci. 14: 439-445. Markram, H. and Tsodyks, T. (1996) "Redistibution of synaptic efficacy between neocortical pyramidal neurons," Nature 382: 807-810. Rieke, F. Warland, D. van Steveninck, R.D.R. and Bialek, W. (1996) Spikes: Exploring the Neural Code. Cambridge: MIT Press. Stevens, C.F. (1994) "What form should a cortical theory take," In: Large-Scale Neuronal Theories ofthe Brain, Koch, C. and Davis, J.L., eds., pp. 239-256. Cambridge: MIT Press. 

1477 |@word trial:2 tedious:1 pulse:2 bn:1 fonn:2 shot:3 contains:1 efficacy:3 denoting:1 current:1 yet:1 additive:2 shape:1 plot:3 v:2 device:2 record:1 filtered:1 provides:2 doubly:1 falsely:1 nor:1 brain:1 terminal:1 decreasing:1 snn:2 little:2 vertebrate:1 lib:5 increasing:1 matched:2 moreover:1 what:2 minimizes:1 ti:2 ofa:1 unit:2 christof:1 unreliability:2 before:1 negligible:1 limit:2 io:1 acad:1 encoding:1 firing:10 path:1 black:1 quantified:2 limited:2 steveninck:2 faithful:1 ofinformation:1 block:1 procedure:1 faber:2 get:2 fonnula:1 conventional:1 deterministic:1 compensated:1 center:1 estimator:7 insight:1 rule:1 regarded:1 facilitation:1 dw:2 rieke:3 transmit:2 spontaneous:2 tan:1 cultured:1 exact:1 origin:1 element:1 trend:1 std:1 observed:1 role:2 epsc:1 thousand:1 tsodyks:2 eva:5 decrease:1 substantial:1 intuition:1 nimh:1 tacitly:1 depend:1 tit:2 vesicle:8 basis:1 easily:1 po:14 various:1 train:3 distinct:1 fast:1 effective:12 detected:3 svv:6 encoded:3 kom:2 plausible:1 erf:1 transform:1 biophysical:1 net:4 analytical:1 reconstruction:5 epsp:6 poorly:1 normalize:1 transmission:5 perfect:1 derive:2 measured:2 progress:1 quantify:2 convention:1 met:7 stevens:3 filter:14 stochastic:2 pea:1 reductionist:1 eff:2 preliminary:1 dendritic:1 exploring:1 koch:7 klab:2 considered:2 deciding:1 exp:3 echoed:1 lm:1 released:1 ctm:1 estimation:18 proc:1 infonnation:8 gabbiani:1 tool:1 smv:3 mit:2 gaussian:6 vesicular:2 representa:1 rather:1 voltage:7 endow:1 release:13 focus:1 improvement:2 consistently:1 likelihood:3 rigorous:1 sense:2 tional:1 dependent:1 el:3 bt:4 pasadena:1 corrupts:1 overall:2 denoted:3 priori:1 psychophysics:1 mutual:3 identical:4 buffering:1 minimized:1 stimulus:3 simplify:1 primarily:1 employ:1 gamma:1 individual:3 ourselves:1 cns:1 detection:14 highly:1 light:1 tj:3 natl:1 integral:1 perfonnance:4 theoretical:1 lwe:1 cover:2 jib:2 measuring:1 tractability:1 deviation:2 parametrically:1 snr:14 hundred:1 optimally:1 my:2 combined:1 density:8 systematic:1 probabilistic:2 decoding:1 again:1 central:2 opposed:1 choose:1 possibly:1 li:2 account:1 potential:1 suggesting:1 coding:2 sec:2 sloan:1 depends:1 closed:2 analyze:2 observing:1 parallel:3 square:6 wiener:2 variance:7 characteristic:1 ensemble:1 correspond:2 identify:1 ofthe:1 yes:1 biophysically:1 none:1 researcher:1 history:1 ah:1 detector:6 synapsis:15 synaptic:22 email:1 ed:2 failure:2 quixote:1 energy:1 frequency:1 pp:1 e2:1 knowledge:1 amplitude:5 syn:3 sophisticated:1 ea:1 dt:1 follow:1 response:3 synapse:17 evaluated:1 box:1 though:2 stage:2 correlation:2 aih:1 nonlinear:2 quality:1 usa:1 effect:1 iooo:1 normalized:1 preponderance:1 manwani:5 white:4 skewed:1 self:1 davis:1 criterion:1 generalized:1 hippocampal:2 presenting:1 theoretic:5 neocortical:1 harmonic:1 ef:2 recently:1 spiking:1 empirically:3 vitro:1 insensitive:1 occurred:3 relating:1 l1li:5 numerically:1 significant:1 measurement:1 cambridge:2 ai:3 cv:1 pm:7 similarly:1 bekkers:2 phenomenological:1 longer:2 operating:1 etc:2 j:1 posterior:1 perspective:5 inf:1 driven:1 inequality:1 binary:13 tenns:3 caltech:2 transmitted:1 minimum:3 eo:6 paradigm:7 signal:37 rmv:1 arithmetic:1 multiple:1 reduces:2 compounded:1 exceeds:1 characterized:2 cross:3 post:8 e1:7 paired:1 schematic:1 poisson:5 llb:2 achieved:2 whereas:1 separately:1 interval:1 diagram:1 else:2 neuromuscular:1 source:3 crucial:1 standpoint:1 biased:1 pyramidal:1 ineffective:1 hz:16 member:1 presence:1 ideal:1 bandwidth:7 restrict:1 regarding:1 whether:1 expression:9 york:1 cause:1 action:1 ignored:2 generally:1 covered:1 involve:1 amount:2 band:1 simplest:1 nsf:1 notice:2 neuroscience:1 fabrizio:1 per:2 write:1 express:1 redundancy:5 threshold:2 drawn:1 pj:7 uis:1 spoke:1 sum:1 prob:1 place:1 decide:1 ofm:2 decision:6 bit:2 bound:7 ct:3 followed:1 syna:1 ahead:1 invertebrate:1 fourier:1 optimality:2 format:1 peripheral:1 poor:1 membrane:2 em:1 making:1 pr:6 sided:1 depletion:1 mechanism:1 junction:1 available:2 generalizes:1 observe:2 spectral:6 upto:1 generic:1 occurrence:2 uq:1 thomas:1 denotes:4 graphical:1 log2:3 warland:2 amit:1 objective:2 added:1 spike:25 occurs:3 fa:2 bialek:3 unclear:1 distance:1 thank:1 sci:1 capacity:6 presynaptic:4 argue:2 code:2 modeled:4 relationship:1 quantal:2 ratio:5 calcium:1 gated:1 neuron:14 convolution:1 finite:1 t:2 neurobiology:1 variability:4 communication:4 nonnalized:1 california:1 able:1 usually:1 below:1 regime:3 oft:1 reading:1 program:1 reliable:1 power:7 event:1 treated:3 eh:2 cascaded:1 technology:1 carried:3 ss:1 sn:4 prior:1 acknowledgement:1 versus:2 digital:1 h2:1 illuminating:1 consistent:1 uncorrelated:1 pi:2 supported:1 institute:1 markram:2 absolute:1 van:2 slice:1 cortical:5 dn2:1 ignores:1 sensory:1 commonly:1 made:1 bm:11 employing:1 far:1 smm:2 active:1 assumed:4 spectrum:1 infact:1 continuous:2 vet:2 quantifies:2 nature:3 transfer:1 channel:15 ca:1 robust:1 ampa:1 domain:1 da:2 linearly:3 neurosci:1 noise:18 alarm:1 profile:1 neuronal:3 fig:11 site:1 en:4 roc:1 wiley:1 fails:1 explicit:1 msec:3 pe:11 theorem:1 ofn:1 specific:1 inset:1 er:4 appeal:1 svm:1 false:1 gained:1 entropy:1 expressed:2 monotonic:1 aa:3 corresponds:2 goal:1 consequently:2 absence:1 considerable:1 miss:1 conservative:1 called:1 pas:1 la:3 zone:1 support:1 inability:1 preparation:1 bouton:1 phenomenon:1 ex:1

Relative Loss Bounds for Multidimensional Regression Problems Jyrki Kivinen Department of Computer Science P.O. Box 26 (Teollisuuskatu 23) FIN-00014 University of Helsinki, Finland Manfred K. Warmuth Department of Computer Science University of California, Santa Cruz Santa Cruz, CA 95064, USA Abstract We study on-line generalized linear regression with multidimensional outputs, i.e., neural networks with multiple output nodes but no hidden nodes. We allow at the final layer transfer functions such as the softmax function that need to consider the linear activations to all the output neurons. We use distance functions of a certain kind in two completely independent roles in deriving and analyzing on-line learning algorithms for such tasks. We use one distance function to define a matching loss function for the (possibly multidimensional) transfer function, which allows us to generalize earlier results from one-dimensional to multidimensional outputs. We use another distance function as a tool for measuring progress made by the on-line updates. This shows how previously studied algorithms such as gradient descent and exponentiated gradient fit into a common framework. We evaluate the performance of the algorithms using relative loss bounds that compare the loss of the on-line algoritm to the best off-line predictor from the relevant model class, thus completely eliminating probabilistic assumptions about the data. 1 INTRODUCTION In a regression problem, we have a sequence of n-dimensional real valued inputs Zt E R n , t 1, ... ,f, and for each input Zt a k-dimensional real-valued desired output Yt E R". Our goal is to find a mapping that at least approximately models the dependency between Zt and Yt. Here we consider the parametric case Yt f (w; Zt) where the actual output Yt corresponding to the input Zt is determined by a parameter vector w E Rm (e.g., weights in a neural network) through a given fixed model f (e.g., a neural network architecture). = = 1. Kivinen and M. K Wannuth 288 Thus, we wish to obtain parameters w such that, in some sense, I(w;:z:t} ~ Yt for all t. The most basic model 1 to consider is the linear one, which in the one-dimensional case k 1 means that I(w;:z:t) w . :Z:t for w E Rfl. In the multidimensional case O:Z:t. The we actually have a whole matrix 0 E Rkxfl of parameters and 1(0;:z:t} goodness of the fit is quantitatively measured in terms of a loss function; the square loss given by Lt,j (Yt,j - ilt,j)2 /2 is a popular choice. = = = In generalized linear regression [MN89] we fix a transfer function 4> and apply it on top of a ?(w?:z:t). Here linear model. Thus, in the one-dimensional case we would have I(w;:z:t) ? is usually a continuous increasing function from R to R, such as the logistic function that maps z to 1/(1 + e- Z ). It is still possible to use the square loss, but this can lead to problems. In particular, when we apply the logistic transfer function and try to find a weight vector w that minimizes the total square loss over f examples (:Z:t, Yt), we may have up to ?fl local minima [AHW95, Bud93]. Hence, some other choice of loss function might be more convenient. In the one-dimensional case it can be shown that any continuous strictly increasing transfer function ? has a specific matching loss function LtP such that, among other useful properties, Lt LtP(Yt, ?(w . :z:t}) is always convex in w, so local minima are not a problem [AHW95]. For example, the matching loss function for the logistic transfer function is the relative entropy (a generalization of the logarithmic loss for continuousvalued outcomes). The square loss is the matching loss function for the identity transfer function (i.e., linear regression). = The main theme of the present paper is the application of a particular kind of distance functions to analyzing learning algorithms in (possibly multidimensional) generalized linear regression problems. We consider a particular manner in which a mapping 4>: Rk -+ Rk can be used to define a distance function D.4> : Rk x Rk -+ R; the assumption we must make here is that 4> has a convex potential function. The matching loss function LtP mentioned above for a transfer function ? in the one-dimensional case is given in terms of the distance function D.tP as LtP(?(a), ?(ii)) = D.tP(ii, a). Here, as whenever we use the matching loss LtP (y, iI), we assume that Y and iI are in the range of ?, so we can write Y = ?(a) and iI ?(ii) for some a and ii. Notice that for k 1, any strictly increasing continuous function has a convex potential (i.e., integral) function. In the more interesting case k > 1, we can consider transfer functions such as the softmax function, which is commonly used to transfer arbitrary vectors a E Rk into probability vectors y (i.e., vectors such that iii ~ 0 for all i and Li iii 1). The matching loss function for the softmax function defined analogously with the one-dimensional case turns out to be the relative entropy (or Kul1back-Leibler divergence), which indeed is a commonly used measure of distance between probability vectors. For the identity transfer function, the matching loss function is the squared Euclidean distance. = = = The first result we get from this observation connecting matching losses to a general notion of distance is that certain previous results on generalized linear regression with matching loss on one-dimensional outputs [HKW95] directly generalize to multidimensional outputs. From a more general point of view, a much more interesting feature of these distance functions is how they allow us to view certain previously known learning algorithms, and introduce new ones, in a simple unified framework. To briefly explain this framework without unnecessary complications, we restrict the foUowing discussion to the case k = 1, i.e., f(w;:z:) = ?(w . :z:) E R with w E Rfl. We consider on-line learning algorithms, by which we here mean an algorithm that processes the training examples one by one, the pair (:Z:t, Yt) being processed at time t. Based Relative Loss Bounds for Multidimensional Regression Problems 289 on the training examples the algorithm produces a whole sequence of weight vectors Wt, = 1, ... ,f.. At each time t the old weight vector Wt is updated into WtH based on Zt and Yt. The best-known such algorithm is on-line gradient descent. To see some alternatives, consider first a distance function ll.,p defined on R n by some function ,p: Rn ~ Rn. (Thus, we assume that,p has a convex potential.) We represent the update somewhat indirectly by introducing a new parameter vector 6 t ERn from which the actual weights Wt are obtained by the mapping Wt ,p{6t ). The new parameters are updated by t = (1) where TJ > 0 is a learning rate. We call this algorithm the general additive algorithm with parameterization function ,p. Notice that here 6 is updated by the gradient with respect to w, so this is not just a gradient descent with reparameterization [JW98]. However, we obtain the usual on-line gradient descent when ,p is the identity function. When,p is the softmax function, we get the so-called exponentiated gradient (EG) algorithm [KW97 , HKW95]. The connection of the distance function ll.,p to the update (1) is two-fold. First, (1) can be motivated as an approximate solution to a minimization problem in which the distance ll.,p (6 t , 6 tH ) is used as a kind of penalty term to prevent too drastic an update based on a single example. Second, the distance function ll.,p can be used as a potential function in analyzing the performance of the resulting algorithm. The same distance functions have been used previously for exactly the same purposes [KW97, HKW95] in important special cases (the gradient descent and EG algorithms) but without realizing the full generality of the method. It should be noted that the choice of the parameterization function ,p is left completely free, as long as ,p has a convex potential function. (In contrast, the choice of the transfer function ? depends on what kind of a regression problem we wish to solve.) Earlier work suggests that the softmax parameterization function (Le., the EG algorithm) is particularly suited for situations in which some sparse weight vector W gives a good match to the data [HKW95, KW97]. (Because softmax normalizes the weight vector and makes the components positive, a simple transformation of the input data is typically added to realize positive and negative weights with arbitrary norm.) In work parallel to this, the analogue of the general additive update (1) in the context of linear classification, i.e., with a threshold transfer function, has recently been developed and analyzed by Grove et al. [GLS97] with methods and results very similar to ours. CesaBianchi [CB97] has used somewhat different methods to obtain bounds also in cases in which the loss function does not match the transfer function. Jagota and Warmuth [JW98] view (1) as an Euler discretization of a system of partial differential equations and investigate the performance of the algorithm as the discretization parameter approaches zero. The distance functions we use here have previously been applied in the context of exponential families by Amari [Ama85] and others. Here we only need some basic technical properties of the distance functions that can easily be derived from the definitions. For a discussion of our line of work in a statistical context see Azoury and Warmuth [AW97]. In Section 2 we review the definition of a matching loss function and give examples. Section 3 discusses the general additive algorithm in more detail. The actual relative on-line loss bounds we have for the general additive algorithm are explained in Section 4. 290 J. Kivinen and M. K. Warmuth 2 DISTANCE FUNCTIONS AND MATCIllNG LOSSES Let 4>: R k -t R k be a function that has a convex potential function P4> (i.e., 4> = V' P4> for some convex P 4>: Rk -4 R). We first define a distance/unction A4> for 4> by (2) Thus, the distance A4>(a, a) is the error we make if we approximate P4>(a) by its firstorder Taylor polynomial around a. Convexity of P 4> implies that A4> is convex in its first argument. Further, A4>(a, a) is nonnegative, and zero if and only if 4>(a) = 4>( a). I: ~ We can alternatively write (2) as A4>(a, a) = (4)(r) - 4>(a)) . dr where the integral is a path integral the value of which must be independent of the actual path chosen between a and a. In the one-dimensional case, the integral is a simple definite integral, and ? has a convex potential (i.e., integral) function if it is strictly increasing and continuous [AHW95, HKW95]. Let now 4> have range V4> ~ Rk and distance function function L4>: V4> x V4> -4 R such that L4>(4)(a) , 4>(a? a, we say that L4> is the matching loss function for 4>. A4>' = Assuming that there is a A4>(a, a) holds for all a and Example 1 Let 4> be a linear function given by 4>(a) = Aa where A E R kxk is symmetrical and positive definite. Then 4> has the convex potential function P4> (a) = aT Aa /2, and (2) gives A4>(a, a) = Ha - a)T A(a - a). Hence, L4>(Y' y) = t(y - y)T A-l(y - y) forally,YERk. 0 exp(a;)/E7=1 exp(aj), be the softmax function. It has a potential function given by PO'(a) = In E7=1 exp(aj). To see that PO' is convex, notice that the Hessian n 2PO' is given by D2PO'(a);j = dijO'i(a) - O'da)O'j(a). Given Example2 Let 0': Rk -4 Rk, O'i(a) = a vector z E Rk, let now X be a random variable that has probability O';{a) of taking E7=1 O'i{a)xl- E7=1 E7=1 0'; (a)xiO'j(a)xj the value Xi? We have zTDO'(a)z 2 EX - (EX)2 VarX ~ O. Straightforward algebra now gives the relative entropy LO'(y, y) = E;=l Yj In(Yj/Yj) as the matching loss function. (To allow Yj 0 or Yj 0, we adopt the standard convention that OlnO Oln(O/O) 0 and yln(y/O) 00 for y> 0.) 0 = = = = = = = = In the relative loss bound proofs we use the basic property [JW98, Ama85] This shows that our distances do not satisfy the triangle inequality. Usually they are not symmetrical, either. 3 THE GENERAL ADDITIVE ALGORITHM We consider on-line learning algorithms that at time t -first receive an input Zt E R n , then produce an output Yt E R k, and finally receive as feedback the desired output Yt E Rk. To define the general additive algorithm. assume we are given a transfer function Relative Loss Bounds for Multidimensional Regression Problems 291 l/J: Rk ~ Rk that has a convex potential function. (We wi11later use the matching loss as a performance measure.) We also require that all the desired outputs Y t are in the range of l/J. The algorithm's predictions are now given by Yt = l/J(Ot:et) where Ot E Rkxn is the algorithm's weight matrix at time t. To see how the weight matrix is updated, assume further we have a parameterization function ..p: R n ~ R n with a distance D....p. The algorithm maintains kn real-valued parameters. We denote by 8 t the k x n matrix of the values ofthese parameters immediately before trial t. Futher, we denote by 8 t ,i the jth row of 8t. and by ..p(8t} the matrix with ..p(8t ,i) as its jth row. Given initial parameter values 8 1 and a learning rate 1] > 0, we now define the general additive (GA) algorithm as the algorithm that repeats at each trial t the following prediction and update steps. Prediction: Upon recieving the instance :et, give the prediction Yt = l/J(..p(8t):et). Update: For j := 1, ... , k, set 8t+l,i = 8t,i - fJ(yt,i - Yt ,i ):et . Note that (2) implies \7aD..l/J(a, a)) = l/J(a) -l/J(a), so this update indeed turns out to be the same as (1) when we recall that Ll/J(Yt, Yt) D..l/J(Ot:et, at} where Yt l/J(at). = = The update can be motivated by an optimization problem given in terms of t~e loss and distance. Consider updating an old parameter matrix 8 into a new matrix 8 based on a single input :e and desired output y. A natural goal would be to minimize the loss L l/J (y, l/J( ,p (8):e ) ). However, the algorithm must avoid losing too much of the information it has gained during the previous trials and stored in the form of the old parameter matrix 8 . We thus set as the algorithm's goal to minimize the sum D.."p(8, 8) + fJLl/J(Y' l/J(,p(8):e)) where fJ > 0 is a parameter regulating how fast the algorithm is willing to move its parameters. Under certain regularity assumptions, the update rule of the GA algorithm can be shown to approximately solve this minimization problem. For more discussion and examples in the special case of linear regression, see [KW97]. An interesting related idea is using all the previous examples in the update instead of just the last one. For work along these lines in the linear case see Vovk [Vov97] and Foster [Fos91]. 4 RELATIVE LOSS BOUNDS Consider a sequence S := ((:el,yd, . .. ,(:el,Yl)) of training examples, and let Lossl/J(GA, S) 2:!=1 Ll/J(Yt, Yt) be the loss incurred by the general additive algorithm on this sequence when it always uses its current weights Ot for making the tth prediction Yt? Similarly, let Lossl/J(O, S) = 2:!=1 Ll/J(Yt, l/J(O:ed) be the loss of a fixed predictor O. Basically, our goal is to show that if some 0 achieves a small loss, then the algorithm is not doing much worse, regardless of how the sequence S was generated. Making additional probabilistic assumptions allows such on-line loss bounds to be converted into more traditional results about generalization errors [KW97]. To give the bounds for Lossl/J(GA, S) in terms of Lossl/J(O, S) we need some additional parameters. The first one is the distance D....p(8 1 ,8) where 0 = "p(8) and 8 1 is the initial parameter matrix of the GA algorithm (which can be arbitrary). The second one is defined by = bx,,,p = sup {:e T n..p(9):e 19 E Rn,:e EX} where X := {:el, ... ,:el} is the set of input vectors and n..p(9) is the Jacobian with (D,p(9))ij 81Pi(9)/80j . The value bx,,p can be interpreted as the maximum norm of = J. Kivinen and M. K. Wannuth 292 any input vector in a norm defined by the parameterization function ..p. In Example 3 below we show how bx,..p can easily be evaluated when 1/J is a linear function or the softmax function. The third parameter ctP ' defined as relates the matching loss function for the transfer function tP to the square loss. In Example 4 we evaluate this constant for linear functions, the softmax function, and the onedimensional case. Example 3 Consider bounding the value :I: TDO'( 0):1: where 0' is the softmax function. As we saw in Example 2, this value is a variance of a random variable with the range {Xl, ... ,Xn }. Hence, we have bx,O' ~ maXzex(maxixi - minixd 2 /4 ~ maXzex 11:l:11~ where 11:1:1100 = maXi IXil? If 1/J is a linear function with 1/J( 8) = A8 for a symmetrical positive definite A, we clearly 0 have bx,..p ~ Amax max:l:ex :1: 2 where Amax is the largest eigenvalue of A. Example 4 For the softmax function 0' the matching loss function LO' is the relative entropy (see Example 2), for which it is well known that LO'(y, y) 2: 2(y - y)2. Hence, we have ctP ~ 1/4. If tP is a linear function given by a symmetrical positive semidefinite matrix A, we see from Example 1 that CtP is the largest eigenvalue of A. Finally, in the special case k = 1, with ?: R -7 R differentiable and strictly increasing, we can show ctP :::; Z if Z is a bound such that 0 < ?'(z) :::; Z holds for all z. 0 Assume now we are given constants b 2: bx ,1/J and C for any parameter matrix 8 we have 2: ctP . Our first loss bound states that when the learning rate is chosen as '1 = 1/(2bc). (Proofs are omitted from this extended abstract.) The advantage of this bound is that with a fixed learning rate it holds for any 8, so we need no advance knowledge about a good 8. The drawback is the factor 2 in front of Loss tP (..p (8), S), which suggests that asymptotically the algorithm might not ever achieve the performance of the best fixed predictor. A tighter bound can be achieved by more careful tuning. Thus, iven constants K 2: 0 and R > 0, if we choose the learning rate as '1 (bcR)2 + KbcR - bcR)/(Kbc) (with '1 1/(2bc) if K 0) we obtain =( = = for any 8 that satisfies Loss tP (..p (8) , S) :::; K and d..p (8 1 , 8) :::; R. This shows that if we restrict our comparison to parameter matrices within a given distance R of the initial matrix of the algorithm, and we have a reasonably good guess K as to the loss of the best fixed predictor within this distance, this knowledge allows the algorithm to asymptotically match the performance of this best fixed predictor. Relative Loss Bounds for Multidimensional Regression Problems 293 Acknowledgments The authors thank Katy Azoury, Chris Bishop, Nicolo Cesa-Bianchi, David Helmbold, and Nick Littlestone for helpful discussions. Jyrki Kivinen was supported by the Academy of Finland and the ESPRIT project NeuroCOLT. Manfred Warmuth was supported by the NSF grant CCR 9700201. References [Ama85] S. Amari. Differential Geometrical Methods in Statistics. Springer Verlag, Berlin, 1985. [AHW95] P. Auer, M. Herbster, and M. K. Warmuth. Exponentially many local minima for single neurons. In Proc. 1995 Neural Information Processing Conference, pages 316-317. MIT Press, Cambridge, MA, November 1995. [AW97] K. Azoury and M. K. Warmuth. Relative loss bounds and the exponential family of distributions. Unpublished manuscript, 1997. [Bud93] M. Budinich. Some notes on perceptron learning. 1. Phys. A.: Math. Gen., 26:4237-4247, 1993 . [CB97] N. Cesa-Bianchi. Analysis of two gradient-based algorithms for on-line regression. In Proc. 10th Annu. Con! on Comput. Learning Theory, pages 163-170. ACM,1997. [Fos91] D. P. Foster. Prediction in the worst case. The Annals of Statistics, 19(2):10841090, 1991. [GLS97] A. J. Grove, N. Littlestone, and D. Schuurmans. General convergence results for linear discriminant updates. In Proc. 10th Annu. Con! on Comput. Learning Theory, pages 171-183. ACM, 1997. [HKW95] D. P. Helmbold, J. Kivinen, and M. K. Warmuth. Worst-case loss bounds for sigmoided linear neurons. In Proc. Neural Information Processing Systems 1995, pages 309-315. MIT Press, Cambridge, MA, November 1995. [JW98] A. K. Jagota and M. K. Warmuth. Continuous versus discrete-time nonlinear gradient descent: Relative loss bounds and convergence. Presented at Fifth Symposium on Artificial Intelligence and Mathematics, Ft. Lauderdale, FL, 1998. [KW97] J. Kivinen and M. K. Warmuth. Additive versus exponentiated gradient updates for linear prediction. Information and Computation, 132( 1): 1-64, January 1997. [MN89] P. McCullagh and J. A. NeIder. Generalized Linear Models. Chapman & Hall, New York, 1989. [Vov97] V. Vovk. Competitive on-line linear regression. In Proc. Neural Information Processing Systems 1997. MIT Press, Cambridge, MA, 1998. 

1478 |@word trial:3 briefly:1 eliminating:1 polynomial:1 norm:3 willing:1 initial:3 ours:1 bc:2 current:1 discretization:2 unction:1 varx:1 activation:1 must:3 realize:1 ixil:1 cruz:2 additive:9 update:13 intelligence:1 guess:1 warmuth:10 parameterization:5 realizing:1 manfred:2 wth:1 math:1 node:2 complication:1 along:1 differential:2 symposium:1 introduce:1 manner:1 indeed:2 continuousvalued:1 actual:4 increasing:5 project:1 what:1 kind:4 interpreted:1 minimizes:1 iven:1 developed:1 unified:1 transformation:1 multidimensional:10 firstorder:1 exactly:1 esprit:1 rm:1 gls97:2 grant:1 positive:5 before:1 local:3 analyzing:3 path:2 approximately:2 yd:1 might:2 studied:1 suggests:2 range:4 acknowledgment:1 yj:5 definite:3 matching:16 convenient:1 get:2 ga:5 context:3 map:1 yt:23 straightforward:1 regardless:1 convex:12 immediately:1 helmbold:2 rule:1 amax:2 deriving:1 reparameterization:1 notion:1 updated:4 annals:1 losing:1 us:1 particularly:1 updating:1 role:1 ft:1 worst:2 oln:1 mentioned:1 convexity:1 xio:1 algebra:1 upon:1 completely:3 triangle:1 easily:2 po:3 fast:1 artificial:1 outcome:1 valued:3 solve:2 say:1 amari:2 statistic:2 final:1 sequence:5 eigenvalue:2 differentiable:1 advantage:1 p4:4 relevant:1 gen:1 achieve:1 academy:1 convergence:2 regularity:1 produce:2 measured:1 ij:1 progress:1 implies:2 convention:1 drawback:1 mn89:2 require:1 fix:1 generalization:2 tighter:1 strictly:4 hold:3 around:1 hall:1 exp:3 mapping:3 finland:2 adopt:1 achieves:1 omitted:1 purpose:1 proc:5 saw:1 largest:2 tool:1 minimization:2 mit:3 clearly:1 always:2 e7:5 avoid:1 hkw95:6 derived:1 contrast:1 sense:1 helpful:1 el:4 typically:1 hidden:1 algoritm:1 among:1 classification:1 softmax:11 special:3 chapman:1 others:1 quantitatively:1 divergence:1 regulating:1 investigate:1 analyzed:1 semidefinite:1 tj:1 grove:2 integral:6 partial:1 euclidean:1 old:3 taylor:1 desired:4 littlestone:2 instance:1 earlier:2 tp:6 measuring:1 goodness:1 introducing:1 euler:1 predictor:5 too:2 front:1 stored:1 dependency:1 kn:1 herbster:1 probabilistic:2 off:1 v4:3 yl:1 lauderdale:1 analogously:1 connecting:1 squared:1 cesa:2 choose:1 possibly:2 dr:1 worse:1 bx:6 li:1 potential:10 converted:1 satisfy:1 depends:1 ad:1 try:1 view:3 neider:1 doing:1 sup:1 competitive:1 maintains:1 parallel:1 minimize:2 square:5 variance:1 generalize:2 basically:1 explain:1 phys:1 teollisuuskatu:1 whenever:1 ed:1 definition:2 proof:2 con:2 popular:1 recall:1 knowledge:2 actually:1 auer:1 manuscript:1 evaluated:1 box:1 generality:1 just:2 nonlinear:1 logistic:3 aj:2 usa:1 hence:4 leibler:1 eg:3 ll:7 during:1 noted:1 generalized:5 fj:2 geometrical:1 recently:1 common:1 sigmoided:1 exponentially:1 onedimensional:1 cambridge:3 tuning:1 mathematics:1 similarly:1 nicolo:1 certain:4 verlag:1 inequality:1 jagota:2 minimum:3 additional:2 somewhat:2 cesabianchi:1 ii:7 relates:1 multiple:1 full:1 technical:1 match:3 long:1 prediction:7 regression:14 basic:3 represent:1 achieved:1 receive:2 ot:4 ltp:5 call:1 iii:2 xj:1 fit:2 architecture:1 restrict:2 idea:1 motivated:2 penalty:1 hessian:1 york:1 useful:1 santa:2 bcr:2 processed:1 tth:1 nsf:1 notice:3 ccr:1 discrete:1 write:2 threshold:1 prevent:1 asymptotically:2 sum:1 ilt:1 family:2 wannuth:2 bound:18 layer:1 fl:2 fold:1 nonnegative:1 helsinki:1 argument:1 ern:1 department:2 kw97:6 ofthese:1 making:2 explained:1 equation:1 previously:4 turn:2 discus:1 drastic:1 apply:2 indirectly:1 alternative:1 top:1 a4:8 move:1 added:1 parametric:1 usual:1 traditional:1 gradient:11 distance:27 thank:1 neurocolt:1 berlin:1 chris:1 discriminant:1 assuming:1 negative:1 tdo:1 zt:7 bianchi:2 neuron:3 observation:1 fin:1 descent:6 yln:1 november:2 january:1 situation:1 rfl:2 extended:1 ever:1 rn:3 arbitrary:3 david:1 pair:1 unpublished:1 connection:1 nick:1 california:1 usually:2 below:1 max:1 analogue:1 natural:1 kivinen:7 review:1 relative:14 loss:48 interesting:3 versus:2 incurred:1 foster:2 pi:1 normalizes:1 lo:3 row:2 repeat:1 last:1 free:1 supported:2 jth:2 allow:3 exponentiated:3 perceptron:1 taking:1 fifth:1 sparse:1 feedback:1 xn:1 author:1 made:1 commonly:2 approximate:2 symmetrical:4 unnecessary:1 xi:1 alternatively:1 continuous:5 ctp:5 transfer:16 reasonably:1 ca:1 schuurmans:1 da:1 main:1 azoury:3 whole:2 bounding:1 theme:1 wish:2 exponential:2 xl:2 comput:2 kbc:1 jacobian:1 third:1 rk:13 annu:2 example2:1 specific:1 bishop:1 maxi:1 gained:1 suited:1 entropy:4 lt:2 logarithmic:1 kxk:1 katy:1 springer:1 aa:2 futher:1 a8:1 satisfies:1 acm:2 ma:3 goal:4 jyrki:2 identity:3 careful:1 mccullagh:1 determined:1 vovk:2 wt:4 total:1 called:1 l4:4 evaluate:2 ex:4

Unsupervised On-Line Learning of Decision Trees for Hierarchical Data Analysis Marcus Held and Joachim M. Buhmann Rheinische Friedrich-Wilhelms-U niversitat Institut fUr Informatik III, ROmerstraBe 164 D-53117 Bonn, Germany email: {held.jb}.cs.uni-bonn.de WWW: http://www-dbv.cs.uni-bonn.de Abstract An adaptive on-line algorithm is proposed to estimate hierarchical data structures for non-stationary data sources. The approach is based on the principle of minimum cross entropy to derive a decision tree for data clustering and it employs a metalearning idea (learning to learn) to adapt to changes in data characteristics. Its efficiency is demonstrated by grouping non-stationary artifical data and by hierarchical segmentation of LANDSAT images. 1 Introduction Unsupervised learning addresses the problem to detect structure inherent in unlabeled and unclassified data. The simplest, but not necessarily the best approach for extracting a grouping structure is to represent a set of data samples = {Xi E Rdli = 1, ... by a set of prototypes y = {Ya E Rdlo = 1, .. . K ? N. The encoding usually is represented by an assignment matrix M = (Mia), where Mia = 1 if and only if Xi belongs to cluster 0, and Mia = 0 otherwise. According to this encoding scheme, the cost function 1i (M, Y) = ~ L:~1 MiaV (Xi, Ya) measures the quality of a data partition, Le., optimal assignments and prototypes (M,y)OPt = argminM,y1i (M,Y) minimize the inhomogeneity of clusters w.r.t. a given distance measure V. For reasons of simplicity we restrict the presentation to the ' sum-of-squared-error criterion V(x, y) = !Ix - YI12 in this paper. To facilitate this minimization a deterministic annealing approach was proposed in [5] which maps the discrete optimization problem, i.e. how to determine the data assignments, via the Maximum Entropy Principle [2] to a continuous parameter es- X ,N} ,K}, Unsupervised On-line Learning of Decision Trees for Data Analysis 515 timation problem. Deterministic annealing introduces a Lagrange multiplier {3 to control the approximation of 11. (M, Y) in a probabilistic sense. Equivalently to maximize the entropy at fixed expected K-means costs we minimize the free energy :F = ~ 2:f::1ln (2::=1 exp (-{3V (Xi, Ya:))) w.r.t. the prototypes Ya:. The assignments Mia: are treated as random variables yielding a fuzzy centroid rule N N Ya: = L i=l (Mia:)xdLi=l (Mia:) , where the expected assignments (Mia:) are given by Gibbs distributions (Mia:) = :x p (-{3V (Xi,Ya:)) (1) (2) . 2:/l=1 exp ( -{3V (Xi, Ya:)) For a more detailed discussion of the DA approach to data clustering cf. [1, 3, 5]. In addition to assigning data to clusters (1,2), hierarchical clustering provides the partitioning of data space with a tree structure. Each data sample X is sequentially assigned to a nested structure of partitions which hierarchically cover the data space Rd. This sequence of special decisions is encoded by decision rules which are attached to nodes along a path in the tree (see also fig. 1). Therefore, learning a decision tree requires to determine a tree topology, the accompanying assignments, the inner node labels S and the prototypes y at the leaves. The search of such a hierarchical partition of the data space should be guided by an optimization criterion, i.e., minimal distortion costs. This problem is solvable by a two-stage approach, which on the one hand minimizes the distortion costs at the leaves given the tree structure and on the other hand optimizes the tree structure given the leaf induced partition of Rd. This approach, due to Miller & Rose [3], is summarized in section 2. The extensions for adaptive online learning and experimental results are described in sections 3 and 4, respectively. x S ~ S S /4\ j'\ a /'\ S b c d e partition ofdata space ? a f Figure 1: Right: Topology of a decision tree. Left: Induced partitioning of the data space (positions of the letters also indicate the positions of the prototypes). Decisions are made according to the nearest neighbor rule. 2 Unsupervised Learning of Decision Trees Deterministic annealing of hierarchical clustering treats the assignments of data to inner nodes of the tree in a probabilistic way analogous to the expected assignments of data to leaf prototypes. Based on the maximum entropy principle, the probability ~~j that data point Xi reaches inner node Sj is recursively defined by (see [3]): ~~root:= 1, ~~j = ~~parent(j)1ri,j, 1ri,j = exp (-,V(Xi,Sj)) 2: exp(-,V(Xi,Sk)) , kEsiblings(j) (3) M. Held and 1. M. Buhmann 516 where the Lagrange multiplier, controls the fuzziness of all the transitions 1fi,j' On the other hand, given the tree topology and the prototypes at the leaves, the maximum entropy principle naturally recommends an ideal probability cpLl< at leaf Yet, resp. at an inner node sj> cp~ = exp(-j1V(Xi,Yet)) exp(-j1V(Xi'Y/L)) ',et and cpl. L: :E = kEdescendants(j) l,J (4) CP!k' , /LEY We apply the principle of minimum cross entropy for the calculation of the prototypes at the leaves given a priori the probabilities for the parents of the leaves. Minimization of the cross entropy with fixed expected costs (HXi) = L:et (Miet)V (Xi, Yet) for the data point Xi yields the expression m~n I({(Miet)}II{Cp~parent(et)/K}) = {(M.e?} min Let (Miet) In {(M.e>)} cpJMiet) , i,parent(et) (5) where I denotes the Kullback-Leibler divergence and K defines the degree of the inner nodes. The tilted distribution Cp~parent(et) exp (-j1V (Xi, Yet)) ( (6) Miet) = H . L: /L cp i,parent(/L) exp ( - j1V (Xi, Y/L)) generalizes the probabilistic assignments (2). In the case of Euclidian distances we again obtain the centroid formula (1) as the minimum of the free energy h F = - L::l ln [L:etEY cprparent(et) exp (-j1V (Xi, Yet))]. Constraints induced by the tree structure are incorporated in the assignments (6). For the optimization of the hierarchy, Miller and Rose in a second step propose the minimization of the distance between the hierarchical probabilities CP~. and the ideal probabilities Cp~ ., the distance being measured by the Kullback-Leibler divergence ' ~T L , BjEparent(Y) I ({ Cp~,j }II{ Cp~j}) == ~,W L cp~ . N :E cpL BjEparent(Y)i=l In cp~J . (7) t,J Equation (7) describes the minimization of the sum of cross entropies between the probability densities CP~. and CP~. over the parents of the leaves. Calculating the gradients for the inner ;'odes S j ~d the Lagrange multiplier, we receive N -2, L (Xi - Sj) N {cpL - cp!,parent(j)1fi,j} := -2,:E ~1 (Xi, Sj), (8) i=l i=l N L:E V (Xi, Sj) i=l jES N {cpL - cpLparent(j)1fi,j} := L:E ~2 (Xi, Sj). (9) i=l jES The first gradient is a weighted average of the difference vectors (Xi - Sj), where the weights measure the mismatch between the probability CPtj and the probability induced by the transition 1fi,j' The second gradient (9) measures the scale - V (Xi, Sj) - on which the transition probabilities are defined, and weights them with the mismatch between the ideal probabilities. This procedure yields an algorithm which starts at a small value j1 with a complete tree and identical test vectors attached to all nodes. The prototypes at the leaves are optimized according to (6) and the centroid rule (1), and the hierarchy is optimized by (8) and (9). After convergence one increases j1 and optimizes the hierarchy and the prototypes at the leaves again. The increment of j1leads to phase transitions where test vectors separate from each other and the formerly completely degenerated tree evolves its structure. For a detailed description of this algorithm see [3]. Unsupervised On-line Learning of Decision Trees for Data Analysis 3 517 On-Line Learning of Decision Trees Learning of decision trees is refined in this paper to deal with unbalanced trees and on-line learning of trees. Updating identical nodes according to the gradients (9) with assignments (6) weighs parameters of unbalanced tree structures in an unsatisfactory way. A detailed analysis reveals that degenerated test vectors, i.e., test vectors with identical components, still contribute to the assignments and to the evolution of /. This artefact is overcome by using dynamic tree topologies instead of a predefined topology with indistinguishable test vectors. On the other hand, the development of an on-line algorithm makes it possible to process huge data sets and non-stationary data. For this setting there exists the need of on-line learning rules for the prototypes at the leaves, the test vectors at the inner nodes and the parameters / and (3. Unbalanced trees also require rules for splitting and merging nodes. Following Buhmann and Kuhnel [1] we use an expansion of order O(I/n) of (1) to estimate the prototypes for the Nth datapoint N '" N-l Ya '" Ya + 'TJa (M:;;I) N-1M POI ( N-I) XN - Ya , (10) where P~ ~ p~-1 +1/M ((M:;;I) - p~-l) denotes the probability of the occurence of class o. The parameters M and'TJa are introduced in order to take the possible non-stationarity of the data source into account. M denotes the size of the data window, and 'TJa is a node specific learning rate. Adaptation of the inner nodes and of the parameter / is performed by stochastic approximation using the gradients (8) and (9) (11) (12) For an appropriate choice of the learning rates 'TJ, the learning to learn approach of Murata et al. [4] suggests the learning algorithm (13) The flow 1 in parameter space determines the change of w N -1 given a new datapoint XN. Murata et al. derive the following update scheme for the learning rate: rN 'TJN (1 - 8)r N - 1 _ 'TJ N - I + 81 (XN, W N - 1 ) + Vl",N-l , (v2I1rNII- 'TJN-l) , (14) (15) where VI, v2 and 8 are control parameters to balance the tradeoff between accuracy and convergence rate. r N denotes the leaky average of the flow at time N. The adaptation of (3 has to observe the necessary condition for a phase transition (3 > (3erit == 1/28rnax , 8rnax being the largest eigenvalue of the covariance matrix [3] M ~a =L i=l M (Xi - Ya) (Xi - Ya)t (Mia)/L(Mia ). (16) i=l Rules for splitting and merging nodes of the tree are introduced to deal with unbalanced trees and non-stationary data. Simple rules measure the distortion costs at the prototypes of the leaves. According to these costs the leaf with highest M Held and 1. M Buhmann 518 distortion costs is split. The merging criterion combines neighboring leaves with minimal distance in a greedy fashion. The parameter M (10), the typical time scale for changes in the data distribution is used to fix the time between splitting resp. merging nodes and the update of (3. Therefore, M controls the time scale for changes of the tree topology. The learning parameters for the learning to learn rules (13)-(15) are chosen empirically and are kept constant for all experiments . 4 Experiments The first experiment demonstrates how a drifting two dimensional data source can be tracked. This data source is generated by a fixed tree augmented with transition probabilities at the edges and with Gaussians at the leaves. By descending the tree structure this ~enerates an Li.d. random variable X E R2, which is rotated around the origin of R to obtain a random variable T(N) = R(w, N)X . R is an orthogonal matrix, N denotes the number of the actual data point and w denotes the angular velocity, M = 500. Figure 2 shows 45 degree snapshots of the learning of this nonstationary data source. We start to take these snapshots after the algorithm has developed its final tree topology (after ~ 8000 datapoints). Apart from fluctuations of the test vectors at the leaves, the whole tree structure is stable while tracking the rotating data source. Additional experiments with higher dimensional data sources confirm the robustness of the algorithm w.r.t. the dimension of the data space, i. e. similiar tracking performances for different dimensions are observed, where differences are explained as differences in the data sources (figure 3) . This performance is measured by the variance of the mean of the distances between the data source trajectory and the trajectories of the test vectors at the nodes of the tree. 7) 8) Figure 2: 45 degree snapshots of the learning of a data source which rotates with a velocity w = 271"/30000 (360 degree per 30000 data samples:. A second experiment demonstrates the learning of a switching data source. The results confirm a good performance concerning the restructuring of the tree (see figure 4). In this experiment the algorithm learns a given data source and after 10000 data points we switch to a different source. As a real-world example of on-line learning of huge data sources the algorithm is applied to the hierarchical clustering of 6- dimensional LANDSAT data. The heat 519 Unsupervised On-line Learning of Decision Trees for Data Analysis -0.5 ?1 ' 1.5 2dim4dim ~ .-- 12dim ..... 18dim I~ . ~ . -3.5 -4 -4.5 ?5 ?5.5 ~ 10000 0 20000 30000 40000 50000 60000 70000 80000 90000 N Figure 3: Tracking performance for different dimensions. As data sources we use d-dimensional Gaussians which are attached to a unit sphere. To the components of every random sample X we add sin(wN) in order to introduce non stationarity. The first 8000 samples are used for the development of the tree topology. ~, A) \ a \ I k 1\ "'-, b B) . \ . \ _,_____. '" ._________ \ . .>,\" \L _.-" -'" -".(----?/l <' I:' e C J J i / ,/ / ' "\" '\ \ \ c \\ 1- 1 ""., . \ 1 ~, ,-- ..____- __}-;-''-.--~.~ Ig - j \ -----b~~<\ :' ' \ / / .? ' ( ? -\--<I!..~ \ \' . k .9" _ ", h \ v' .'6> h\,m \\Q\ \ ..__ - >I--."/_-J--; #-\01f;;<1 \ ;r" " n'/ , l/ " ~"" \ 1 1:1'" tf '\' ~ f ____ e '- I \" \ \ II \1 \ ~ a b m J k 1 def g nohi Figure 4: Learning a switching data source: top: a) the partition of the data space after 10000 data samples given the first source, b) the restructured partition after additional 2500 samples. Below: accompanying tree topologies. channel has been discarded because of its reduced resolution. In a preprocessing step all channels are rescaled to unit variance, which alternatively could be established by using a Mahalanobis distance. Note that the decision tree which clusters this data supplies us with a hierarchical segmentation of the corresponding LANDSAT image. A tree of 16 leaves has been learned on a training set of 128 x 128 data samples, and it has been applied to a test set of 128 x 128 LANDSAT pixels. The training is established by 15 sequential runs through the test set, where after each M = 16384 run a split of one node is carried out. The resulting empirical errors (0.49 training distortion and 0.55 test distortion) differ only slightly from the errors obtained by the LBG algorithm applied to the whole training set (0.42 training distortion and 0.52 test distortion). This difference is due to the fact that not every data point is assigned to the nearest leaf prototype by a decision tree induced partition. The segmentation of the test image is depicted in figure 5. 5 Conclusion This paper presents a method for unsupervised on-line learning of decision trees. We overcome the shortcomings of the original decision tree approach and extend 520 M Held and J. M. Buhmann Figure 5: Hierarchical segmentation of the test image. The root represents the original image, i.e., the gray scale version of the three color channels. it to the realm of on-line learning of huge data sets and of adaptive learning of non-stationary data. Our experiments demonstrate that the approach is capable of tracking gradually changing Or switching environments. Furthermore, the method has been successfully applied to the hierarchical segmentation of LANDSAT images. Future work will address active data selection issues to significantly reduce the uncertainty of the most likely tree parameters and the learning questions related to different tree topologies. Acknowledgement: This work has been supported by the German Israel Foundation for Science and Research Development (GIF) under grant #1-0403-001.06/95 and by the Federal Ministry for Education, Science and Technology (BMBF #01 M 3021 A/4). References [1] J. M. Buhmann and H. Kuhnel. Vector quantization with complexity costs. IEEE Transactions on Information Theory, 39(4):1133-1145, July 1993. [2] T.M. Cover and J. Thomas. Elements of Information Theory. Wiley & Sons, 1991. [3] D. Miller and K Rose. Hierarchical unsupervised learning with growing via phase transitions. Neural Computation, 8:425-450, February 1996. [4] N. Murata, K-R. Muller, A. Ziehe, and S. Amari . Adaptive on-line learning in changing environments. In M.C. Mozer, M.I. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems, number 9, pages 599-605. MIT Press, 1997. [5] K Rose, E. Gurewitz, and G.C. Fox. A deterministic annealing approach to clustering. Pattern Recognition Letters, 11(9):589-594, September 1990. 

1479 |@word version:1 covariance:1 euclidian:1 recursively:1 assigning:1 yet:5 tilted:1 partition:8 j1:2 update:2 stationary:5 greedy:1 leaf:19 provides:1 contribute:1 node:16 along:1 supply:1 combine:1 introduce:1 expected:4 growing:1 actual:1 window:1 israel:1 minimizes:1 gif:1 fuzzy:1 developed:1 every:2 demonstrates:2 control:4 partitioning:2 unit:2 grant:1 treat:1 switching:3 encoding:2 path:1 fluctuation:1 cpl:4 suggests:1 procedure:1 empirical:1 significantly:1 unlabeled:1 selection:1 descending:1 www:2 argminm:1 map:1 deterministic:4 demonstrated:1 resolution:1 simplicity:1 splitting:3 rule:9 datapoints:1 increment:1 analogous:1 resp:2 hierarchy:3 origin:1 velocity:2 element:1 recognition:1 updating:1 observed:1 highest:1 rescaled:1 rose:4 mozer:1 environment:2 complexity:1 dynamic:1 efficiency:1 completely:1 represented:1 heat:1 shortcoming:1 refined:1 encoded:1 distortion:8 otherwise:1 amari:1 inhomogeneity:1 final:1 online:1 sequence:1 eigenvalue:1 propose:1 adaptation:2 neighboring:1 description:1 parent:8 cluster:4 convergence:2 rotated:1 derive:2 measured:2 nearest:2 c:2 indicate:1 differ:1 artefact:1 guided:1 ley:1 stochastic:1 education:1 require:1 fix:1 opt:1 extension:1 accompanying:2 around:1 exp:9 label:1 largest:1 tf:1 successfully:1 weighted:1 minimization:4 federal:1 mit:1 poi:1 joachim:1 rheinische:1 fur:1 unsatisfactory:1 centroid:3 detect:1 sense:1 dim:2 landsat:5 vl:1 germany:1 pixel:1 issue:1 priori:1 development:3 special:1 identical:3 represents:1 unsupervised:8 future:1 jb:1 inherent:1 employ:1 divergence:2 phase:3 stationarity:2 huge:3 introduces:1 yielding:1 tj:2 held:5 predefined:1 edge:1 capable:1 niversitat:1 necessary:1 institut:1 orthogonal:1 tree:43 fox:1 rotating:1 weighs:1 minimal:2 cover:2 assignment:12 cost:9 density:1 probabilistic:3 squared:1 again:2 tjn:2 kuhnel:2 li:1 account:1 de:2 summarized:1 vi:1 performed:1 root:2 start:2 minimize:2 accuracy:1 variance:2 characteristic:1 wilhelms:1 miller:3 yield:2 murata:3 informatik:1 trajectory:2 mia:10 datapoint:2 metalearning:1 reach:1 email:1 energy:2 naturally:1 color:1 realm:1 segmentation:5 jes:2 higher:1 furthermore:1 angular:1 stage:1 hand:4 defines:1 quality:1 gray:1 facilitate:1 multiplier:3 evolution:1 assigned:2 leibler:2 deal:2 mahalanobis:1 indistinguishable:1 sin:1 criterion:3 complete:1 demonstrate:1 cp:14 image:6 fi:4 empirically:1 tracked:1 attached:3 extend:1 gibbs:1 rd:2 hxi:1 stable:1 add:1 belongs:1 optimizes:2 apart:1 muller:1 minimum:3 additional:2 ministry:1 determine:2 maximize:1 july:1 ii:3 adapt:1 calculation:1 cross:4 sphere:1 concerning:1 represent:1 receive:1 addition:1 ode:1 annealing:4 source:17 induced:5 flow:2 jordan:1 extracting:1 nonstationary:1 ideal:3 iii:1 recommends:1 split:2 wn:1 switch:1 topology:10 restrict:1 inner:8 idea:1 prototype:14 reduce:1 tradeoff:1 expression:1 detailed:3 simplest:1 reduced:1 http:1 per:1 discrete:1 changing:2 kept:1 sum:2 run:2 unclassified:1 letter:2 uncertainty:1 decision:17 def:1 constraint:1 ri:2 y1i:1 bonn:3 min:1 according:5 describes:1 slightly:1 son:1 evolves:1 explained:1 gradually:1 ln:2 equation:1 german:1 generalizes:1 gaussians:2 apply:1 observe:1 hierarchical:12 v2:1 appropriate:1 petsche:1 robustness:1 drifting:1 original:2 thomas:1 denotes:6 clustering:6 cf:1 top:1 calculating:1 february:1 question:1 september:1 gradient:5 distance:7 separate:1 rotates:1 reason:1 marcus:1 degenerated:2 balance:1 equivalently:1 snapshot:3 discarded:1 similiar:1 incorporated:1 rn:1 introduced:2 friedrich:1 optimized:2 learned:1 established:2 address:2 usually:1 etey:1 mismatch:2 below:1 pattern:1 rnax:2 yi12:1 treated:1 buhmann:6 solvable:1 nth:1 scheme:2 technology:1 carried:1 occurence:1 gurewitz:1 formerly:1 acknowledgement:1 foundation:1 degree:4 principle:5 editor:1 supported:1 free:2 neighbor:1 leaky:1 overcome:2 dimension:3 xn:3 transition:7 world:1 made:1 adaptive:4 preprocessing:1 ig:1 transaction:1 sj:9 uni:2 kullback:2 confirm:2 sequentially:1 reveals:1 active:1 xi:24 alternatively:1 continuous:1 search:1 timation:1 sk:1 tja:3 learn:3 channel:3 expansion:1 necessarily:1 da:1 hierarchically:1 whole:2 augmented:1 fig:1 fashion:1 wiley:1 bmbf:1 position:2 lbg:1 ix:1 learns:1 formula:1 specific:1 r2:1 grouping:2 exists:1 quantization:1 merging:4 sequential:1 entropy:8 depicted:1 likely:1 lagrange:3 restructuring:1 tracking:4 nested:1 determines:1 presentation:1 fuzziness:1 change:4 typical:1 e:1 ya:12 experimental:1 ziehe:1 romerstrabe:1 unbalanced:4 artifical:1

124 ADAPTIVE NEURAL NET PREPROCESSING FOR SIGNAL DETECTION IN NON-GAUSSIAN NOISE1 Richard P. Lippmann and Paul Beckman MIT Lincoln Laboratory Lexington, MA 02173 ABSTRACT A nonlinearity is required before matched filtering in mInimum error receivers when additive noise is present which is impulsive and highly non-Gaussian. Experiments were performed to determine whether the correct clipping nonlinearity could be provided by a single-input singleoutput multi-layer perceptron trained with back propagation. It was found that a multi-layer perceptron with one input and output node, 20 nodes in the first hidden layer, and 5 nodes in the second hidden layer could be trained to provide a clipping nonlinearity with fewer than 5,000 presentations of noiseless and corrupted waveform samples. A network trained at a relatively high signal-to-noise (SIN) ratio and then used as a front end for a linear matched filter detector greatly reduced the probability of error. The clipping nonlinearity formed by this network was similar to that used in current receivers designed for impulsive noise and provided similar substantial improvements in performance. INTRODUCTION The most widely used neural net, the adaptive linear combiner (ALe). is a singlelayer perceptron with linear input and output nodes. It is typically trained using the LMS algorithm and forms one of the most common components of adaptive filters. ALes are used in high-speed modems to construct equalization filters, in telephone links as echo cancelers, and in many other signal processing applications where linear filtering is required [9]. The purpose of this study was to determine whether multilayer perceptrons with linear input and output nodes but with sigmoidal hidden nodes could be as effective for adaptive nonlinear filtering as ALes are for linear filtering. 1 This work wa.s sponsored by the Defense Advanced Research Projects Agency and the Department of the Air Force . The views expressed are those of the authors and do not reflect the policy or position of the U . S. Government. Adaptive Neural Net Preprocessing for Signal Detection The task explored in this paper is signal detection with impulsive noise where an adaptive nonlinearity is required for optimal performance. Impulsive noise occurs in underwater acoustics and in extremely low frequency communications channels where impulses caused by lightning strikes propagate many thousands of miles [2]. This task was selected because a nonlinearity is required in the optimal receiver, the structure of the optimal receiver is known, and the resulting signal detection error rate provides an objective measure of performance. The only other previous studies of the use of multi-layer perceptrons for adaptive nonlinear filtering that we are aware of [6,8] appear promising but provide no objective performance comparisons. In the following we first present examples which illustrate that multi-layer perceptrons trained with back-propagation can rapidly form clipping and other nonlinearities useful for signal processing with deterministic training. The signal detection task is then described and theory is presented which illustrates the need for nOlllinear processing with non-Gaussian noise. Nonlinearities formed when the input to a net is a corrupted signal and the desired output is the uncorrupted signal are then presented for no noise, impulsive noise, and Gaussian noise. Finally, signal detection performance results are presented that demonstrate large improvements in performance with an adaptive nonlinearity and impulsive noise. FORMING DETERMINISTIC NONLINEARITIES A theorem proven by Kohnogorov and described in [5] demonstrates that singleinput single-output continuous nonlinearities can be formed by a multi-layer perceptron with two layers of hidden nodes. This proof, however, requires complex nonlinear functions in the hidden nodes that are very sensitive to the desired input/output function and may be difficult to realize. "More recently, Lapedes [4] presented an intuitive description of how multi-layer perceptrons with sigmoidal nonlinearities could produce continuous nonlinear mappings. A careful mathematical proof was recently developed by Cybenko [1] which demonstrated that continuous nonlinear mappings can be formed using sigmoidal nonlinearities and a multi-layer perceptron with one layer of hidden nodes. This proof, however, is not constructive and does not indicate how many nodes are required in the hidden layer. The purpose of our study was to determine whether multi-layer perceptrons with sigmoidal nonlinearities and trained using back-propagation could adaptively and rapidly form clipping nonlinearities. Initial experiments were performed to determine the difficulty of learning complex mappings using multi-layer perceptrons trained using back-propagation. Networks with 1 and 2 hidden layers and from 1 to 50 hidden nodes per layer were evaluated. Input and output nodes were linear and all other nodes included sigmoidal nonlinearities. Best overall performance was provjded by the three-layer perceptron shown in Fig. 1. It has 20 nodes in the first and 5 nodes in the second hidden layer. This network could form a wide variety of mappings and required only slightly more training than other networks. It was used in all experiments. 125 126 Lippmann and Beckman y - OUTPUT (LInear Sum) (20 Nodes) x - INPUT Figure 1: The multi-layer perceptron with linear input and output nodes that was used in all experiments. The three-layer network shown in Fig. 1 was used to form clipping and other deterministic nonlinearities. Results in Fig. 2 demonstrate that a clipping nonlinearity ('auld be formed with fewer than 1,000 input samples. Input/output point pairs were determined by selecting the input at random over the range plotted and using tlte deterministic clipping function shown as a solid line in Fig. 2. Back-propagation training [7] was used with the gain term (11) equal to 0.1 and the momentum term (0') equal to 0.5. These values provide good convergence rates for the clipping function and all other functions tested. Initial connection weights were set to small random values. The multi-layer percept ron from Fig. 1 was also used to form the four nonlinear functions shown in Fig. 3. The "Hole Punch" is useful in nonlinear signal processing. It performs much the same function as the clipper but completely eliminates amplitudes above a certain threshold le\'el. Accurate approximation of this function required more than 50,000 input samples. The "Step" has one sharp edge and could be roughly approximated after 2,000 input samples. The "Double Pulse" requires approximation of two close "pulses" and is the nonlinear function analogy of the disjoint region problem studied in [3]. In this examplf>, back-propagation training approximated the rightmost pulse first after 5,000 input samples. Both pulses were then approximated fairly well after 50,000 input samples. The "Gaussian Pulse" is a smooth curve that could be approximated well after only 2,000 input samples. These results demonstrate that back-propagation training with sigmoidal 1I0nlinearities can form many different nonlinear functions. Qualitative results on training times are similar to those reported in [.1]. In this previous study it was de mOll- Adaptive Neural Net Preprocessing for Signal Detection .... .!: 1 1000 TRIALS 40 TRIALS BEFORE TRAINING DESIRED ACTUAL ????????. ~ 0 ...... ~ 0- 1 -2_ 2 -1 2 0 -2 t -1 0 INPUT (I() 2 -2 -1 0 2 RMS ERROR O. II: 0 II: II: u.I (f) :IE ex: 200 400 606 TRIALS 800 1000 Figure 2: Clipping nonlinearities formed using back-propagation training and the multi-layer perceptron from Fig. 1 (top) and the rms error produced by these Ilonlinearities versus training time (bottom). strated that simple half-plane decision regions could be formed for classification problems with little training while complex disjoint decision regions required long training times. These new results suggest that complex nonlinearities with many sharp discontinuities require much more training time than simple smooth curves. THE SIGNAL DETECTION TASK The signal detection task was to discriminate between two equally likely input signals as shown in Fig. 4. One signal (so(t)) corresponds to no input and the other signal (Sl(t)) was a sinewa\'c pulse with fixed duration and known amplitude, frequency, and phase. Noise was added to these inputs, the resultant signal was passed through a memoryless nonlinearity, and a matched filter was then used to select hypothesis Ho corresponding to no input or HI corresponding to the sinewave pulse. The matched filter multiplied the output of the nonlinearity by the known timealigned signal waveform, integrated this product over time, and decided HI if the result was greater than a threshold and Ho otherwise. The threshold was selected to provide a minimum overall error rate. The optimum nonlinearity Ilsed in the detector depends on the noise distribu tion. If the signal levels are small relati\'e to the noise levels, then the optimum nonlinearity is approximated by f (J') = t;~ In{ (In (J')). where r .. (x) is the instantaneous probability density function of the noise (2]- This function is linear for Gaussian noise but has a clipping shape for impulsi\'e noise. 127 128 Lippmann and Beckman HOLE PUNCH STEP 2r---~---r--~---. I -, f- ., ~ ?2 I:::l no I::l 0 N.5.ooo ----Nco 50.000 ?2 -, Nco 2.000 ---_. 0 .- l?/ f/ I .......... I ????~?:?~5 ............... i.~j .............. N.5oo . ----- - "-. 0 I ., 1 2 ?2 DOUBLE PULSE 1 -1--' o 2 GAUSSIAN PULSE 2 ?2 ?1 o 2 INPUT (xl Figure 3: Four deterministic nonlinearities formed using the multi-layer perceptron from Fig. 1. Desired functions are plotted as solid lines while functions formed using back-propagation with different numbers of input samples are plotted using dots and dashes. Examples of the signal, impulsive noise and Gaussian noise are presented in Fig. 5. The signal had a fixed duration of 250 samples and peak amplitude of 1.0. The impulsive noise was defined by its amplitude distribution and inter-arrival time. Amplit udes had a zero mean, Laplacian distribution with a standard de\'iation (IJ) of 14.1 in all experiments. The standard deviation was reduced to 2.8 in Fig. 5 for illustrative purposes. Inter-arrival times (L\T) between noise impulses had a Poisson distribution. The mean inter-arrival time was varied in experiments to obtain different SIN ratios after adding noise. For example varying inter-arrival times from 500 to 2 samples results in SIN ratios that vary from roughly 1 dB to - 24 dB. Additive Gaussian noise had zero mean and a standard oeviation (IJ) of 0.1 in all experiments. ADAPTIVE TRAINING WITH NOISE The three-layer perceptron was traineq as shown in Fig. 6 using the signal plus Iloist> as the input and the uncorrupted signal as the desired output. Network weights were adapted after every sample input using back-propagation training. Adaptive nonlinearitics formed during training are shown in Fig. 7. These are similar to those Adaptive Neural Net Preprocessing for Signal Detection MEMORYlESS NONLINEARITY SO(II--- S,II)~ MATCHED FILTER DETECTOR ~-...o{ NOISE 'I ? I(x) N(tl Figure 4: The signal detection task was to discriminate between a sinewa\?e pulse and a no-input condition with additive impulsive noise. UNCORRUPTED SIGNAL IMPULSE NOISE ct = 2.8 .H~ o 50 100 150 200 250 0 12 50 100 150 200 2500 GAUSSIAN NOISE maO (J ? O. t 50 100 150 200 250 SAMPLES Figure 5: The input to the nonlinearity with no noise, additive impulsive noise, and additive Gaussian noise. required by theory. No noise results in nonlinearity that is linear over the range of the input sinewave (-1 to + 1) after fewer than 3,000 input samples. Impulsive noise at a high SIN ratio (6.T 125 or SIN -5 dB) results in a nonlinearity that clips above the signal level after roughly 5,000 input samples and then slowly forms a "Hole Punch" nonlinearity as the number of training samples increases. Gaussian noise noise results in a nonlinearity that is roughly linear over the range of the input sinewave after fewer than 5,000 input samples. = = SIGNAL DETECTION PERFORMANCE Signal detection performance was measured using a matched filter detector and the nonlinearity shown in the center of Fig. 7 for 10,000 input training samples. The error rate with a minimum-error matched filter is plotted in Fig. 8 for impulsive lIoise at SIN ratios ranging from roughly 5 dB to -24 dB. This error rate was estimated from 2,000 signal detection trials. Signal detection performance always improved with the nonlinearity and sometimes the improvement was dramatic. The error rate provided with the adaptively-formed nonlinearity is essentially identical 129 130 Lippmann and Beckman DESIRED OUTPUT 5(1) X MULTI-LAYER ) - -....--1 PERCEPTRON + Y - I. E NOISE BACK-PROPAGATION 1 - _..... ALGORITHM Figure 6: The procedure used for adaptive training. NO NOISE IMPULSE NOISE GAUSSIAN NOISE N. '00,000 N.5,000 2 ~ ... ::;) ... Q. ::;) 0 0 N.1.000 -, ?2 -2 N . 2.ooo N.3.ooo -, 0 2 ?2 ., 0 2 ?2 ., 0 2 INPUT (x) Figure 7: Nonlinearities formed with adaptive training with no additive noise, with additive impulsive noise at a SIN level of -5 dB, and with additive Gaussian noise. to that provided by a clipping nonlinearity that clips above the signal level. This error rate is roughly zero down to - 24 dB and then rises rapidly with higher levels of impulsive noise. This rapid increase in error rate below -24 dB is not shown in Fig. 8. The error rate with linear processing rises slowly as the SIN ratio drops and reaches roughly 36% when the SIN ratio is -24 dB. Further exploratory experiments demonstrated that the nonlinearity formed by back-propagation was not robust to the SIN ratio used during training. A clipping nonlinearity is only formed when the number of samples of uncorrupted sinewave input is high enough to form the linear function of the curve and the number of samples of noise pulses is low , but sufficient to form the non~ill('ar clipping section of the nonlinearity. At high noise levels the resulting nonlinearity is not linear Over the range of the input signal. It instead resembles a curve that interpolates between a flat horizontal input-output curve and the desired clipping curve. SUMMARY AND DISCUSSION In summary, it was first demonstrated that multi-layer perccptrons with linear Adaptive Neural Net Preprocessing for Signal Detection 40 w ~ < a: a: oa: 20 ~ a: w LINEAR PROCESSING 10 ADAPTIVE NONLINEAR PROCESSING o ??? ...,?????? .I......I???.? ~.~____-~. -10 -5 o 5 -20 -15 -Z5 ' SIN RATIO Figure 8: The signal detection error rate with impulsive noise when the SIN ratio after adding the noise ranges from 5 dB to - 24 dB. input and output nodes could approximate prespecified clipping nonlinearities required for signal detection with impulsive noise with fewer than 1,000 trials of back-propagation training. More complex nonlinearities could also be formed but required longer training times. Clipping nonunearities were also formed adaptively using a multi-layer perceptron with the corrupted signal as the input and the noisefree signal as the desired output. Nonlinearities learned using this approach at high S / N ratios were similar to those required by theory and improved signal detection performance dramatically at low SIN ratios. Further work is necessary to further explore the utility of this technique for forming adaptive nonlinearities. This work should explore the robustness of the nonlinearity formed to variations in the input S / N ratio. It should also explore the use of multi-layer perccptrons and backpropagation training for other adaptive nonlinear signal processing tasks such as system identification, noise removal, and channel modeling. 131 132 Lippmann and Beckman References [1] G. Cybenko. Approximation by superpositions of a sigmoidal function. Research note, Department of Computer Science, Tufts University, October 1988. [2] J. E. Evans and A. S Griffiths. Design of a sanguine noise processor based upon world-wide extremely low frequency (elf) recordings. IEEE Transactions on Communications, COM-22:528-539, April 1974. [3] W. M. Huang and R. P. Lippmann. Neural net and traditional classifiers. In D. Anderson, editor, Neural Information Processing Systems, pages 387-396, New York, 1988. American Institute of Physics. [4] A. Lapedes and R. Farber. How neural nets work. In D. Anderson, editor, Neural Information Processing Systems, pages 442-456, New York, 1988. American Institute of Physics. [5] G. G. Lorentz. The 13th problem of Hilbert. In F. E. Browder, editor, Afathematical Developments Arising from Hilbert Problems. American Mathematical Society, Providence, R.I., 1976. [6] D. Palmer and D. DeSieno. Removing random noise from ekg signals using a back propagation network, 1987. HNC Inc., San Diego, CA. [7] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland, editors, Parallel Distributed Processing, volume 1: Foundations, chapter 8. MIT Press, Cambridge, MA, 1986. [8] S. Tamura and A. Wailbel. Noise reduction using connectionist models. In Proceedings IEEE International Conference on Acoustics, Speech and Signal Processing, volume 1: Speech Processing, pages 553-556, April 1988. [9] B. Widrow and S. D. Stearns. Adaptive Signal Processing. Prentice-Hall, NJ, 1985. 

148 |@word trial:5 pulse:11 propagate:1 dramatic:1 solid:2 reduction:1 initial:2 selecting:1 lapedes:2 rightmost:1 current:1 com:1 lorentz:1 realize:1 evans:1 additive:8 shape:1 designed:1 sponsored:1 drop:1 half:1 fewer:5 selected:2 plane:1 prespecified:1 provides:1 node:18 ron:1 sigmoidal:7 mathematical:2 qualitative:1 inter:4 rapid:1 roughly:7 udes:1 multi:17 actual:1 little:1 provided:4 project:1 matched:7 developed:1 lexington:1 nj:1 every:1 demonstrates:1 classifier:1 appear:1 before:2 plus:1 studied:1 resembles:1 palmer:1 range:5 decided:1 backpropagation:1 procedure:1 griffith:1 suggest:1 close:1 prentice:1 equalization:1 cancelers:1 deterministic:5 demonstrated:3 center:1 williams:1 duration:2 exploratory:1 underwater:1 variation:1 diego:1 hypothesis:1 rumelhart:2 approximated:5 bottom:1 thousand:1 region:3 substantial:1 agency:1 trained:7 upon:1 completely:1 chapter:1 effective:1 widely:1 otherwise:1 echo:1 net:9 product:1 rapidly:3 lincoln:1 intuitive:1 description:1 convergence:1 double:2 optimum:2 produce:1 illustrate:1 oo:1 widrow:1 measured:1 ij:2 indicate:1 waveform:2 clipper:1 correct:1 farber:1 filter:8 require:1 government:1 cybenko:2 hall:1 mapping:4 lm:1 vary:1 purpose:3 beckman:5 superposition:1 sensitive:1 mit:2 gaussian:14 always:1 varying:1 combiner:1 improvement:3 greatly:1 el:1 nco:2 typically:1 integrated:1 hidden:10 overall:2 classification:1 ill:1 development:1 fairly:1 equal:2 construct:1 aware:1 noisefree:1 identical:1 elf:1 connectionist:1 richard:1 phase:1 detection:19 highly:1 accurate:1 edge:1 necessary:1 desired:8 plotted:4 modeling:1 impulsive:16 ar:1 clipping:17 deviation:1 front:1 reported:1 providence:1 corrupted:3 adaptively:3 density:1 peak:1 international:1 ie:1 physic:2 reflect:1 huang:1 slowly:2 american:3 nonlinearities:18 de:2 inc:1 caused:1 depends:1 performed:2 view:1 tion:1 parallel:1 desieno:1 formed:17 air:1 percept:1 identification:1 produced:1 processor:1 detector:4 reach:1 frequency:3 resultant:1 proof:3 gain:1 hilbert:2 amplitude:4 back:14 higher:1 improved:2 april:2 ooo:3 singleoutput:1 evaluated:1 anderson:2 horizontal:1 nonlinear:11 propagation:15 impulse:4 memoryless:2 laboratory:1 mile:1 sin:13 during:2 illustrative:1 demonstrate:3 performs:1 ranging:1 instantaneous:1 recently:2 common:1 volume:2 cambridge:1 nonlinearity:27 lightning:1 dot:1 had:4 longer:1 certain:1 uncorrupted:4 minimum:3 greater:1 determine:4 strike:1 signal:45 ale:3 ii:5 smooth:2 long:1 equally:1 laplacian:1 z5:1 multilayer:1 noiseless:1 essentially:1 poisson:1 sometimes:1 tamura:1 eliminates:1 recording:1 db:11 enough:1 variety:1 moll:1 whether:3 rms:2 defense:1 utility:1 passed:1 interpolates:1 york:2 speech:2 dramatically:1 useful:2 clip:2 mcclelland:1 reduced:2 sl:1 stearns:1 punch:3 estimated:1 disjoint:2 per:1 arising:1 four:2 threshold:3 sum:1 decision:2 layer:28 noise1:1 hi:2 dash:1 ct:1 adapted:1 flat:1 speed:1 extremely:2 relatively:1 department:2 slightly:1 end:1 multiplied:1 tuft:1 robustness:1 ho:2 top:1 relati:1 society:1 objective:2 added:1 occurs:1 traditional:1 link:1 oa:1 ratio:13 difficult:1 october:1 rise:2 design:1 policy:1 modem:1 ekg:1 hinton:1 communication:2 varied:1 sharp:2 pair:1 required:12 connection:1 acoustic:2 learned:1 discontinuity:1 below:1 difficulty:1 force:1 advanced:1 hnc:1 removal:1 singlelayer:1 filtering:5 proven:1 analogy:1 versus:1 foundation:1 sufficient:1 editor:4 summary:2 distribu:1 perceptron:12 institute:2 wide:2 distributed:1 curve:6 world:1 author:1 adaptive:19 preprocessing:5 san:1 transaction:1 approximate:1 lippmann:6 receiver:4 continuous:3 promising:1 channel:2 robust:1 ca:1 complex:5 noise:51 paul:1 arrival:4 fig:16 tl:1 position:1 momentum:1 mao:1 xl:1 theorem:1 down:1 removing:1 tlte:1 sinewave:4 explored:1 adding:2 illustrates:1 hole:3 likely:1 explore:3 forming:2 expressed:1 corresponds:1 ma:2 presentation:1 careful:1 included:1 telephone:1 determined:1 discriminate:2 perceptrons:6 select:1 internal:1 constructive:1 tested:1 ex:1

Ensemble Learning for Multi-Layer Networks David Barber? Christopher M. Bishopt Neural Computing Research Group Department of Applied Mathematics and Computer Science Aston University, Birmingham B4 7ET, U.K. http://www.ncrg.aston.ac.uk/ Abstract Bayesian treatments of learning in neural networks are typically based either on local Gaussian approximations to a mode of the posterior weight distribution, or on Markov chain Monte Carlo simulations. A third approach, called ensemble learning, was introduced by Hinton and van Camp (1993). It aims to approximate the posterior distribution by minimizing the Kullback-Leibler divergence between the true posterior and a parametric approximating distribution. However, the derivation of a deterministic algorithm relied on the use of a Gaussian approximating distribution with a diagonal covariance matrix and so was unable to capture the posterior correlations between parameters. In this paper, we show how the ensemble learning approach can be extended to fullcovariance Gaussian distributions while remaining computationally tractable. We also extend the framework to deal with hyperparameters, leading to a simple re-estimation procedure. Initial results from a standard benchmark problem are encouraging. 1 Introduction Bayesian techniques have been successfully applied to neural networks in the context of both regression and classification problems (MacKay 1992; Neal 1996). In contrast to the maximum likelihood approach which finds only a single estimate for the regression parameters, the Bayesian approach yields a distribution of weight parameters, p(wID), conditional on the training data D, and predictions are ex?Present address: SNN, University of Nijmegen, Geert Grooteplein 21, Nijmegen, The Netherlands. http://wvw.mbfys.kun . n1/snn/ email: davidbbbfys.kun.n1 tpresent address: Microsoft Research Limited, St George House, Cambridge CB2 3NH, UK. http://vvv.research.microsoft . com email: cmbishopbicrosoft.com D. Barber and C. M. Bishop 396 pressed in terms of expectations with respect to the posterior distribution (Bishop 1995). However, the corresponding integrals over weight space are analytically intractable. One well-established procedure for approximating these integrals, known as Laplace's method, is to approximate the posterior distribution by a Gaussian, centred at a mode of p(wID), in which the covariance of the Gaussian is determined by the local curvature of the posterior distribution (MacKay 1995). The required integrations can then be performed analytically. More recent approaches involve Markov chain Monte Carlo simulations to generate samples from the posterior (Neal 1996}. However, such techniques can be computationally expensive, and they also suffer from the lack of a suitable convergence criterion. A third approach, called ensemble learning, was introduced by Hinton and van Camp (1993) and again involves finding a simple, analytically tractable, approximation to the true posterior distribution. Unlike Laplace's method, however, the approximating distribution is fitted globally, rather than locally, by minimizing a Kullback-Leibler divergence. Hinton and van Camp (1993) showed that, in the case of a Gaussian approximating distribution with a diagonal covariance, a deterministic learning algorithm could be derived. Although the approximating distribution is no longer constrained to coincide with a mode of the posterior, the assumption of a diagonal covariance prevents the model from capturing the (often very strong) posterior correlations between the parameters. MacKay (1995) suggested a modification to the algorithm by including linear preprocessing of the inputs to achieve a somewhat richer class of approximating distributions, although this was not implemented. In this paper we show that the ensemble learning approach can be extended to allow a Gaussian approximating distribution with an general covariance matrix, while still leading to a tractable algorithm. 1.1 The Network Model We consider a two-layer feed-forward network having a single output whose value is given by H (1) /(x,w) = LViU(Ui'X) i=1 where w is a k-dimensional vector representing all of the adaptive parameters in the model, x is the input vector, {ud, i = 1, ... , H are the input-to-hidden weights, and {Vi}, i = 1, ... ,H are the hidden-to-output weights. The extension to multiple outputs is straightforward. For reasons of analytic tractability, we choose the sigmoidal hidden-unit activation function u(a) to be given by the error function u(a} = f! loa exp (-8 2/2) d8 (2) which (when appropriately scaled) is quantitatively very similar to the standard logistic sigmoid. Hidden unit biases are accounted for by appending the input vector with a node that is always unity. In the current implementation there are no output biases (and the output data is shifted to give zero mean), although the formalism is easily extended to include adaptive output biases (Barber and Bishop 1997) . . The data set consists of N pairs of input vectors and corresponding target output values D = {x~, t~} ,It = 1, ... , N. We make the standard assumption of Gaussian noise on the target values, with variance (3-1. The likelihood of the training data is then proportional to exp(-(3ED ), where the training error ED is ED{w) = ", 21~ (J{x~, w) ~ t~) 2 . (3) Ensemble Leamingfor Multi-Layer Networks 397 The prior distribution over weights is chosen to be a Gaussian of the form p(w) exp (-Ew(w)) (X (4) where Ew(w) = !wT Aw, and A is a matrix of hyper parameters. The treatment of (3 and A is dealt with in Section 2.1. From Bayes' theorem, the posterior distribution over weights can then be written p(wID) = z1 exp (-(3ED(w) - (5) Ew(w)) where Z is a normalizing constant. Network predictions on a novel example are given by the posterior average of the network output (f(x)) = J (6) f(x, w)p(wID) dw. This represents an integration over a high-dimensional space, weighted by a posterior distribution p(wID) which is exponentially small except in narrow regions whose locations are unknown a-priori. The accurate evaluation of such integrals is thus very difficult. 2 .Ensemble Learning Integrals of the form (6) may be tackled by approximating p(wID) by a simpler distribution Q(w). In this paper we choose this approximating distribution to be a Gaussian with mean wand covariance C. We determine the values of w and C by minimizing the Kullback-Leibler divergence between the network posterior and approximating Gaussian, given by F [Q] { J J = Q(w) } Q(w) In p(wID) dw (7) J (8) Q(w) In Q(w)dw - Q(w) Inp(wID) dw. The first term in (8) is the negative entropy of a Gaussian distribution, and is easily evaluated to give ! In det (C) + const. From (5) we see that the posterior dependent term in (8) contains two parts that depend on the prior and likelihood J Q(w)Ew(w)dw + J (9) Q(w)ED(w)dw. Note that the normalization coefficient Z-l in (5) gives rise to a constant additive term in the KL divergence and so can be neglected. The prior term Ew (w) is quadratic in w, and integrates to give Tr(CA) + ~wT Aw. This leaves the data dependent term in (9) which we write as = L J (3N Q(W)ED(W)dw = "2 I: l(xl!, tl!) (10) I!=l where l(x, t) = J Q(w) (J(x, W))2 dw - 2t J Q(w)f(x, w) dw + t2? (11) D. Barber and C. M. BisJwp 398 For clarity, we concentrate only on the first term in (11), as the calculation of the term linear in I(x, w) is similar, though simpler. Writing the Gaussian integral over Q as an average, ( ), the first term of (11) becomes H ((I(x, w?2) = L (vivju(uTx)u(uJx?). (12) i,j=I To simplify the notation, we denote the set of input-to-hidden weights (Ul' ... , UH) by u and the set of hidden-to-output weights, (VI' ... ' V H) by v. Similarly, we partition the covariance matrix C into blocks, C uu , C vu , C vv , and C vu = C~v. As the components of v do not enter the non-linear sigmoid functions, we can directly integrate over v, so that each term in the summation (12) gives ((Oij + (u - IT)T \Il ij (u - IT) + n~ (u - IT?) u (uTxi) u (uTxj)) (13) where (C vv - Oij \Il ij nij - CvuC uu -lCuV)ij + "hvj (14) C uu -IC u,v=:i C lI=:j,uC uu -1, (15) 2C uu -ICu,lI=:jVi. (16) Although the remaining integration in (13) over u is not analytically tractable, we can make use of the following result to reduce it to a one-dimensional integration (u (z?a + ao) u (z ?b + bo?)z = zaTb + bolal )) T 2 Vl a l (1 + Ib 1 ) - (a b)2 z (17) where a and b are vectors and 0.0, bo are scalar offsets. The avera~e on the left of (17) is over an isotropic multi-dimensional Gaussian, p(z) ex: exp( -z z/2), while the average on the right is over the one-dimensional Gaussian p(z) ex: exp( -z2 /2). This result follows from the fact that the vector z only occurs through the scalar product with a and b, and so we can choose a coordinate system in which the first two components of z lie in the plane spanned by a and b. All orthogonal components do not appear elsewhere in the integrand, and therefore integrate to unity. (u (zlal + 0.0) u ( 2 The integral we desire, (13) is only a little more complicated than (17) and can be evaluated by first transforming the coordinate system to an isotopic basis z, and then differentiating with respect to elements of the covariance matrix to 'pull down' the required linear and quadratic terms in the u-independent pre-factor of (13). These derivatives can then be reduced to a form which requires only the numerical evaluation of (17). We have therefore succeeded in reducing the calculation of the KL divergence to analytic terms together with a single one-dimensional numerical integration of the form (17), which we compute using Gaussian quadrature 1 . Similar techniques can be used to evaluate the derivatives of the KL divergence with respect to the mean and covariance matrix (Barber and Bishop 1997). Together with the KL divergence, these derivatives are then used in a scaled conjugate gradient optimizer to find the parameters w and C that represent the best Gaussian fit. The number of parameters in the covariance matrix scales quadratically with the number of weight parameters. We therefore have also implemented a version with 1 Although (17) appears to depend on 4 parameters, it can be expressed in terms of 3 independent parameters. An alternative to performing quadrature during training would therefore be to compute a 3-dimensionallook-up table in advance. Ensemble Learning for Multi-Layer Networks Posterior laplace fit 399 Minimum KLD fit Minimum KL fit Figure 1: Laplace and minimum Kullback-Leibler Gaussian fits to the posterior. The Laplace method underestimates the local posterior mass by basing the covariance matrix on the mode alone, and has KL value 41. The minimum KullbackLeibler Gaussian fit with a diagonal covariance matrix (KLD) gives a KL value of 4.6, while the minimum Kullback-Leibler Gaussian with full covariance matrix achieves a value of 3.9. a constrained covariance matrix s C = diag(di,? ? ?, d~) + L sisT (18) i=l which is the form of covariance used in factor analysis (Bishop 1997). This reduces the number offree parameters in the covariance matrix from k(k + 1)/2 to k(s + 1) (representing k(s + 1) - s(s - 1)/2 independent degrees of freedom) which is now linear in k. Thus, the number of parameters can be controlled by changing sand, unlike a diagonal covariance matrix, this model can still capture the strongest of the posterior correlations. The value of s should be as large as possible, subject only to computational cost limitations. There is no 'over-fitting' as s is increased since more flexible distributions Q(w) simply better approximate the true posterior. We illustrate the optimization of the KL divergence using a toy problem involving the posterior distribution for a two-parameter regression problem. Figure 1 shows the true posterior together with approximations obtained from Laplace's method, ensemble learning with a diagonal covariance Gaussian, and ensemble learning using an unconstrained Gaussian. 2.1 Hyperparameter Adaptation So far, we have treated the hyperparameters as fixed. We now extend the ensemble learning formalism to include hyperparameters within the Bayesian framework. For simplicity, we consider a standard isotropic prior covariance matrix of the form A = aI, and introduce hyperpriors given by Gamma distributions lnp (a) lnp (f3) = In {aa-l exp ( -~) } + const (19) In {f3 C- 1 exp ( -~) } + const (20) D. Barber and C. M. BisJwp 400 where a, b, c, d are constants. The joint posterior distribution of the weights and hyperparameters is given by p (w, a, ,BID) <X P (Dlw, j3) p (wla) p (a) p (,B) (21) in which lnp (Dlw,,B) - ,BED N + "2 In,B + const k -alwl 2 + '2 In a lnp (wla) (22) + const (23) We follow MacKay (1995) by modelling the joint posterior p (w, a, ,BID) by a factorized approximating distribution of the form (24) Q(w)R(a)S(,B) where Q(w) is a Gaussian distribution as before, and the functional forms of Rand S are left unspecified. We then minimize the KL divergence F[Q,R,S] = J Q(w)R(a)S(,B) In { Q(W)R(a)S(,B) } p(w,a,,BID) dwdad,B. (25) Consider first the dependence of (25) on Q(w) F [QJ J - -J Q(w)R(a)S(j3) { -,BED(W) - - Q(w) { -73ED(W) - ~lwl2 -In Q(w) } + const ~lwl2 -In Q(W)} + const (26) (27) where a = J R(a)ada and 73 = J S(,B)j3d,B. We see that (27) has the form of (8), except that the fixed hyperparameters are now replaced with their average values. To calculate these averages, consider the dependence of the functional F on R(a) F[R] JQ(W)R(a)S(j3){-~lwI2+~lna+(a-1)lna-i} -J + + - R(a) { ; (r - 1) Ina -In R(a)} da const dwdad,B (28) where r = ~ +a and lis = ~lwl2 + ~TrC+ lib. We recognise (28) as the KullbackLeibler divergence between R(a) and a Gamma distribution. Thus the optimum R(a) is also Gamma distributed R(a) We therefore obtain <X a r - 1 exp (-;) . (29) a = rs. A similar procedure for S(,B) gives 73 = uv, where u = ~ + c and 11v = (ED) + lid, in which (ED) has already been calculated during the optimization of Q(w) . This defines an iterative procedure in which we start by initializing the hyperparameters (using the mean of the hyperprior distributions) and then alternately optimize the KL divergence over Q(w) and re-estimate a and 73. 3 Results and Discussion As a preliminary test of our method on a standard benchmark problem, we applied the minimum KL procedure to the Boston Housing dataset. This is a one Ensemble Learning for Multi-Layer Networks I Method Ensemble (s == 1) Ensemble (diagonal) Laplace 401 Test Error 0.22 0.28 0.33 Table 1: Comparison of ensemble learning with Laplace's method. The test error is defined to be the mean squared error over the test set of 378 examples. dimensional regression problem, with 13 inputs, in which the data for 128 training examples was obtained from the DELVE archive 2 ? We trained a network of four hidden units, with covariance matrix given by (18) with s = 1, and specified broad hyperpriors on a and (3 (a = 0.25, b = 400, c = 0.05, and d = 2000). Predictions are made by evaluating the integral in (6). This integration can be done analytically as a consequence of the form of the sigmoid function given in (2). We compared the performance of the KL method against the Laplace framework of MacKay (1995) which also treats hyperparameters through a re-estimation procedure. In addition we also evaluated the performance of the ensemble method using a diagonal covariance matrix. Our results are summarized in Table 1. Acknowledgements We would like to thank Chris Williams for helpful discussions. Supported by EPSRC grant GR/J75425: Novel Developments in Learning Theory for Neural Networks. References Barber, D. and C. M. Bishop (1997). On computing the KL divergence for Bayesian neural networks. Technical report, Neural Computing Research Group, Aston University, Birmingham, {;.K. Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press. Bishop, C. M. (1997). Latent variables, mixture distributions and topographic mappings. Technical report, Aston University. To appear in Proceedings of the NATO Advanced Study Institute on Learning in Graphical Models, Erice. Hinton, G. E. and D. van Camp (1993). Keeping neural networks simple by minimizing the description length of the weights. In Proceedings of the Sixth Annual Conference on Computational Learning Theory, pp. 5-13. MacKay, D. J. C. (1992). A practical Bayesian framework for back-propagation networks. Neural Computation 4 (3) , 448-472. MacKay, D. J. C. (1995). Developments in probabilistic modelling with neural networks--ensemble learning. In Neural Networks: Artificial Intelligence and Industrial Applications. Proceedings of the 3rd Annual Symposium on Neural Networks, Nijmegen, Netherlands, 14-15 September 1995, Berlin, pp. 191-198. Springer. MacKay, D. J. C. (1995). Probable networks and plausible predictions - a review of practical Bayesian methods for supervised neural networks. Network: Computation in Neural Systems 6(3), 469-505. Neal, R. M. (1996). Bayesian Learning for Neural Networks. Springer. Lecture Notes in Statistics 118. 2See http://wvv . cs. utoronto. cal "-'delve I 

1480 |@word version:1 wla:2 grooteplein:1 simulation:2 r:1 covariance:21 pressed:1 tr:1 initial:1 contains:1 current:1 com:2 z2:1 activation:1 trc:1 written:1 numerical:2 additive:1 partition:1 analytic:2 alone:1 intelligence:1 leaf:1 plane:1 isotropic:2 tpresent:1 node:1 location:1 sigmoidal:1 simpler:2 symposium:1 consists:1 fitting:1 introduce:1 mbfys:1 multi:5 globally:1 encouraging:1 snn:2 little:1 lib:1 becomes:1 notation:1 mass:1 factorized:1 sist:1 unspecified:1 finding:1 scaled:2 uk:2 unit:3 grant:1 appear:2 before:1 local:3 treat:1 consequence:1 oxford:1 delve:2 limited:1 practical:2 vu:2 block:1 cb2:1 procedure:6 pre:1 inp:1 cal:1 context:1 writing:1 kld:2 www:1 optimize:1 deterministic:2 straightforward:1 williams:1 simplicity:1 wvv:1 spanned:1 pull:1 dw:9 geert:1 coordinate:2 laplace:9 target:2 element:1 expensive:1 recognition:1 epsrc:1 initializing:1 capture:2 calculate:1 region:1 transforming:1 ui:1 neglected:1 trained:1 depend:2 basis:1 uh:1 easily:2 joint:2 derivation:1 monte:2 artificial:1 hyper:1 whose:2 richer:1 plausible:1 statistic:1 topographic:1 housing:1 product:1 adaptation:1 achieve:1 bed:2 description:1 wvw:1 convergence:1 optimum:1 illustrate:1 ac:1 ij:3 strong:1 lna:2 implemented:2 c:1 involves:1 uu:5 concentrate:1 j75425:1 wid:8 sand:1 ao:1 preliminary:1 probable:1 summation:1 extension:1 ic:1 exp:9 mapping:1 optimizer:1 achieves:1 estimation:2 birmingham:2 integrates:1 basing:1 successfully:1 weighted:1 gaussian:23 always:1 aim:1 rather:1 derived:1 modelling:2 likelihood:3 contrast:1 industrial:1 camp:4 helpful:1 dependent:2 vl:1 typically:1 hidden:7 jq:1 classification:1 flexible:1 priori:1 development:2 constrained:2 integration:6 mackay:8 uc:1 f3:2 having:1 represents:1 broad:1 t2:1 report:2 quantitatively:1 simplify:1 gamma:3 divergence:12 hvj:1 replaced:1 n1:2 microsoft:2 freedom:1 evaluation:2 mixture:1 chain:2 accurate:1 integral:7 succeeded:1 orthogonal:1 hyperprior:1 re:3 nij:1 fitted:1 increased:1 formalism:2 ada:1 tractability:1 cost:1 gr:1 kullbackleibler:2 aw:2 st:1 probabilistic:1 together:3 again:1 squared:1 choose:3 d8:1 derivative:3 leading:2 li:3 toy:1 centred:1 summarized:1 coefficient:1 vi:2 performed:1 start:1 bayes:1 relied:1 complicated:1 utx:1 minimize:1 il:2 variance:1 ensemble:17 yield:1 dealt:1 bayesian:8 carlo:2 strongest:1 ed:9 email:2 sixth:1 against:1 underestimate:1 pp:2 di:1 dataset:1 treatment:2 dlw:2 back:1 appears:1 feed:1 jvi:1 supervised:1 follow:1 rand:1 evaluated:3 though:1 done:1 correlation:3 christopher:1 lack:1 propagation:1 defines:1 mode:4 logistic:1 true:4 analytically:5 leibler:5 neal:3 avera:1 deal:1 offree:1 during:2 criterion:1 isotopic:1 novel:2 sigmoid:3 functional:2 b4:1 exponentially:1 nh:1 ncrg:1 extend:2 cambridge:1 enter:1 ai:1 rd:1 unconstrained:1 uv:1 mathematics:1 similarly:1 alwl:1 longer:1 curvature:1 posterior:25 recent:1 showed:1 lnp:4 minimum:6 george:1 somewhat:1 determine:1 ud:1 multiple:1 full:1 reduces:1 technical:2 calculation:2 controlled:1 prediction:4 involving:1 regression:4 j3:3 expectation:1 represent:1 normalization:1 addition:1 appropriately:1 unlike:2 archive:1 subject:1 vvv:1 bid:3 fit:6 reduce:1 det:1 qj:1 ul:1 suffer:1 involve:1 netherlands:2 locally:1 reduced:1 http:4 generate:1 shifted:1 write:1 hyperparameter:1 group:2 four:1 clarity:1 changing:1 wand:1 recognise:1 capturing:1 layer:5 tackled:1 quadratic:2 annual:2 integrand:1 performing:1 department:1 conjugate:1 unity:2 lid:1 modification:1 computationally:2 tractable:4 hyperpriors:2 appending:1 alternative:1 remaining:2 include:2 graphical:1 const:8 approximating:12 already:1 occurs:1 parametric:1 dependence:2 diagonal:8 september:1 gradient:1 unable:1 thank:1 berlin:1 chris:1 barber:7 reason:1 length:1 minimizing:4 kun:2 difficult:1 nijmegen:3 negative:1 rise:1 implementation:1 unknown:1 markov:2 benchmark:2 hinton:4 extended:3 david:1 introduced:2 pair:1 required:2 kl:13 specified:1 z1:1 quadratically:1 narrow:1 established:1 alternately:1 address:2 suggested:1 pattern:1 including:1 suitable:1 treated:1 oij:2 advanced:1 representing:2 aston:4 prior:4 review:1 acknowledgement:1 ina:1 lecture:1 limitation:1 proportional:1 integrate:2 degree:1 elsewhere:1 accounted:1 loa:1 supported:1 keeping:1 bias:3 allow:1 vv:2 institute:1 differentiating:1 van:4 distributed:1 calculated:1 evaluating:1 forward:1 made:1 adaptive:2 coincide:1 preprocessing:1 far:1 approximate:3 nato:1 kullback:5 iterative:1 latent:1 table:3 ca:1 icu:1 diag:1 da:1 noise:1 hyperparameters:7 quadrature:2 tl:1 xl:1 lie:1 house:1 ib:1 third:2 theorem:1 down:1 bishop:8 utoronto:1 offset:1 normalizing:1 intractable:1 boston:1 entropy:1 simply:1 prevents:1 desire:1 expressed:1 bo:2 scalar:2 springer:2 aa:1 conditional:1 determined:1 except:2 reducing:1 wt:2 called:2 ew:5 evaluate:1 ex:3

The Error Coding and Substitution PaCTs GARETH JAMES and TREVOR HASTIE Department of Statistics, Stanford University Abstract A new class of plug in classification techniques have recently been developed in the statistics and machine learning literature. A plug in classification technique (PaCT) is a method that takes a standard classifier (such as LDA or TREES) and plugs it into an algorithm to produce a new classifier. The standard classifier is known as the Plug in Classifier (PiC). These methods often produce large improvements over using a single classifier. In this paper we investigate one of these methods and give some motivation for its success. 1 Introduction Dietterich and Bakiri (1995) suggested the following method, motivated by Error Correcting Coding Theory, for solving k class classification problems using binary classifiers. ? Produce a k by B (B large) binary coding matrix, ie a matrix of zeros and ones. We will denote this matrix by Z, its i, jth component by Zij, its ith row by Zi and its j th column by zj. ? Use the first column of the coding matrix (Zl) to create two super groups by assigning all groups with a one in the corresponding element of Zl to super group one and all other groups to super group zero. ? Train your plug in classifier (PiC) on the new two class problem. ? Repeat the process for each of the B columns (Zl, Z2, ... ,ZB) to produce B trained classifiers. ? For a new test point apply each of the B classifiers to it. Each classifier will produce a 'Pi which is the estimated probability the test point comes from the jth super group one. This will produce a vector of probability estimates, f> = (PI , ih., . .. ,PB) T . The Error Coding and Substitution PaCTs 543 ? To classify the point calculate Li = 2::=1 IPi - Zij I for each of the k groups (ie for i from 1 to k). This is the LI distance between p and Zi (the ith row of Z). Classify to the group with lowest L 1 distance or equivalently argi min Li We call this the ECOC PaCT. Each row in the coding matrix corresponds to a unique (nonminimal) coding for the appropriate class. Dietterich's motivation was that this allowed errors in individual classifiers to be corrected so if a small number of classifiers gave a bad fit they did not unduly influence the final classification. Several PiC's have been tested. The best results were obtained by using tree's, so all the experiments in this paper are stated using a standard CART PiC. Note however, that the theorems are general to any Pic. In the past it has been assumed that the improvements shown by this method were attributable to the error coding structure and much effort has been devoted to choosing an optimal coding matrix. In this paper we develop results which suggest that a randomized coding matrix should match (or exceed) the performance of a designed matrix. 2 The Coding Matrix Empirical results (see Dietterich and Bakiri (1995? suggest that the ECOC PaCT can produce large improvements over a standard k class tree classifier. However, they do not shed any light on why this should be the case. To answer this question we need to explore its probability structure. The coding matrix, Z, is central to the PaCT. In the past the usual approach has been to choose one with as large a separation between rows (Zi) as possible (in terms of hamming distance) on the basis that this allows the largest number of errors to be corrected. In the next two sections we will examine the tradeoffs between a designed (deterministic) and a completely randomized matrix. Some of the results that follow will make use of the following assumption. k E[fji I Z,X] = LZiiqi = ZjT q = 1, ... ,B j (1) i=l where qi = P (Gil X) is the posterior probability that the test observation is from group i given that our predictor variable is X. This is an unbiasedness assumption. It states that on average our classifier will estimate the probability of being in super group one correctly. The assumption is probably not too bad given that trees are considered to have low bias. 2.1 Deterministic Coding Matrix Let Di = 1 - 2Ld B for i = 1 ... k. Notice that ar& min Li = arg i max Di so using Di to classify is identical to the ECOC PaCT. Theorem 3 in section 2.2 explains why this is an intuitive transformation to use. Obviously no PaCT can outperform the Bayes Classifier. However we would hope that it would achieve the Bayes Error Rate when we use the Bayes Classifier as our PiC for each 2 class problem. We have defined this property as Bayes Optimality. Bayes Optimality is essentially a consistency result It states, if our PiC converges to the Bayes Classifier, as the training sample size increases, then so will the PaCT. Definition 1 A PaCT is said to be Bayes Optimal if, for any test set, it always classifies to the bayes group when the Bayes Classifier is our PiC. For the ECOC PaCT this means that argi max qi = argi max D i , for all points in the predictor space, when we use the Bayes Classifier as our Pic. However it can be shown that in this case i = 1, ... ,k 544 G. James and T. Hastie It is not clear from this expression why there should be any guarantee that argi max Vi = argi max qi . In fact the following theorem tells us that only in very restricted circumstances will the ECOC PaCT be Bayes Optimal. Theorem 1 The Error Coding method is Bayes Optimal iff the Hamming distance between every pair of rows of the coding matrix is equal. The hamming distance between two binary vectors is the number of points where they differ. For general B and k there is no known way to generate a matrix with this property so the ECOC PaCT will not be Bayes Optimal. 2.2 Random Coding Matrix We have seen in the previous section that there are potential problems with using a deterministic matrix. Now suppose we randomly generate a coding matrix by choosing a zero or one with equal probability for every coordinate. Let Pi = E(I- 21ih - Zilll T) where T is the training set. Then Pi is the conditional expectation of Di and we can prove the following theorem. Theorem 2 For a random coding matrix, conditional on T, argi max Vi --+ argi max Pi a.s. as B --+ 00. Or in other words the classification from the ECOC PaCT approaches the classification from just using argi max Pi a.s. This leads to corollary 1 which indicates we have eliminated the main concern of a deterministic matrix. Corollary 1 When the coding matrix is randomly chosen the ECOC PaCT is asymptotically Bayes Optimal ie argi max Di --+ argi max qi a.s. as B --+ 00 This theorem is a consequence of the strong law. Theorems 2 and 3 provide motivation for the ECOC procedure. Theorem 3 Under assumption 1 for a randomly generated coding matrix E j) i = E Pi = qi i = 1 ... k This tells us that Vi is an unbiased estimate of the conditional probability so classifying to the maximum is in a sense an unbiased estimate of the Bayes classification. Now theorem 2 tells us that for large B the ECOC PaCT will be similar to classifying using argi max ILi only. However what we mean by large depends on the rate of convergence. Theorem 4 tells us that this rate is in fact exponential. Theorem 4 lfwe randomly choose Z then, conditional on T , for any fixed X Pr(argi max Vi i argi max ILi) ::; (k - 1) . e- mB for some constant m. Note that theorem 4 does not depend on assumption 1. This tells us that the error rate for the ECOC PaCT is equal to the error rate using argi max Pi plus a tenn which decreases exponentially in the limit. This result can be proved using Hoeffding's inequality (Hoeffding (1963?. Of course this only gives an upper bound on the error rate and does not necessarily indicate the behavior for smaller values of B. Under certain conditions a Taylor expansion indicates that Pr(argi maxDi i argi maxPi) :::::: 0.5 - mVE for small values of mVE. So we The Error Coding and Substitution PaCTs 545 o '"o 1/sqrt(B) convergence lIB convergence o CO> o 50 100 150 200 B Figure 1: Best fit curves for rates 1/VB and 1/ B might expect that for smaller values of B the error rate decreases as some power of B but that as B increases the change looks more and more exponential. To test this hypothesis we calculated the error rates for 6 different values of B (15,26,40,70,100,200) on the LEITER data set (available from the Irvine Repository of machine learning). Each value of B contains 5 points corresponding to 5 random matrices. Each point is the average over 20 random training sets. Figure 1 illustrates the results. Here we have two curves. The lower curve is the best fit of 1/VB to the first four groups. It fits those groups well but under predicts errors for the last two groups. The upper curve is the best fit of 1/ B to the last four groups. It fits those groups well but over predicts errors for the first two groups. This supports our hypothesis that the error rate is moving through the powers of B towards an exponential fit. We can see from the figure that even for relatively low values of B the reduction in error rate has slowed substantially. This indicates that almost all the remaining errors are a result of the error rate of argi max J-li which we can not reduce by changing the coding matrix. The coding matrix can be viewed as a method for sampling from the distribution of 1- 21pj - Zij I. If we sample randomly we will estimate J-li (its mean). It is well known that the optimal way to estimate such a parameter is by random sampling so it is not possible to improve on this by designing the coding matrix. Of course it may be possible to improve on argi max J-li by using the training data to influence the sampling procedure and hence estimating a different quantity. However a designed coding matrix does not use the training data. It should not be possible to improve on random sampling by using such a procedure (as has been attempted in the past). 3 Why does the ECOC PaCT work? The easiest way to motivate why the ECOC PaCT works, in the case of tree classifiers, is to consider a very similar method which we call the Substitution PaCT. We will show that under certain conditions the ECOC PaC!' is very similar to the Substitution PaCT and then motivate the success of the later. 546 3.1 G. James and T. Hastie Substitution PaCT The Substitution PaCT uses a coding matrix to fonn many different trees just as the ECOC PaCT does. However, instead of using the transformed training data to fonn a probability estimate for each two class problem, we now plug the original (ie k-class) training data back into the new tree. We use this training data to fonn probability estimates and classifications just as we would with a regular tree. The only difference is in how the tree is fonned. Therefore, unlike the ECOC PaCT, each tree will produce a probability estimate for each of the k classes. For each class we simply average the probability estimate for that class is the probability estimate for the Substitution PaCT, then over our B trees. So if pf 1 pf = B (2) B LPij j=l where Pij is the probability estimate for the ith group for the tree fonned from the jth column of the coding matrix. Theorem 5 shows that under certain conditions the ECOC PaCT can be thought of as an approximation to the Substitution PaCT. Theorem 5 Suppose that Pij is independentfrom the jth column of the coding matrix, for all i and j. Then as B approaches infinity the ECOC PaCT and Substitution PaCT will converge ie they will give identical classification rules. The theorem depends on an unrealistic assumption. However, empirically it is well known that trees are unstable and a small change in the data set can cause a large change in the structure of the tree so it may be reasonable to suppose that there is a low correlation. To test this empirically we ran the ECOC and Substitution PaCT's on a simulated data set. The data set was composed of 26 classes. Each class was distributed as a bivariate normal with identity covariance matrix and uniformly distributed means. The training data consisted of 10 observations from each group. Figure 2 shows a plot of the estimated probabilities for each of the 26 classes and 1040 test data points averaged over 10 training data sets. Only points where the true posterior probability is greater than 0.01 have been plotted since groups with insignificant probabilities are unlikely to affect the classification. If the two groups were producing identical estimates we would expect the data points to lie on the dotted 45 degree.line. Clearly this is not the case. The Substitution PaCT is systematically shrinking the probability estimates. However there is a very clear linear relationship (R2 :::::: 95%) and since we are only interested in the arg max for each test point we might expect similar classifications. In fact this is the. case with fewer than 4% of points correctly classified by one group but not the other. 3.2 Why does the Substitution PaCT work? pf The fact that is an average of probability estimates suggests that a reduction in variability may be an explanation for the success of the Substitution PaCT. Unfortunately it has been well shown (see for example Friedman (1996? that a reduction in variance of the probability estimates does not necessarily correspond to a reduction in the error rate. However theorem 6 provides simplifying assumptions under which a relationship between the two quantities exists. Theorem 6 Suppose that pT and pf (a[ > 0) (af > 0) (3) (4) The Error Coding and Substitution PaCTs 547 ~ II) d co ~ i ~Q. co d C .g :0 ] .... d :0 CI) '" 0 0 d 0.0 0.4 0.2 0.6 08 1.0 Eeoc probabilitJea Figure 2: Probability estimates from both the ECOC and Substitution PaCT's where eS and eT have identical joint distributions with variance 1. pT is the probability estimate of the ith group for a k class tree method, ao and al are constants and qi is the true posterior probability. Let 'V _ ,- T Var(p'!'ja ~ 1 ) S V ar(pil / a 1 ) and p = corr(pil ,Pi2) (assumed constantfor all i). Then Pr(argmaxP7 = argmaxqi) ~ Pr(argmaxp; = argmaxqi) (5) if (6) and I-p - 'Y - P B>-- (7) The theorem states that under fairly general conditions, the probability that the Substitution PaCT gives the same classification as the Bayes classifier is at least as great as that for the tree method provided that the standardized variability is low enough. It should be noted that only in the case of two groups is there a direct correspondence between the error rate and 5. The inequality in 5 is strict for most common distributions (e.g. normal, uniform, exponential and gamma) of e. Now there is reason to believe that in general p will be small. This is a result of the empirical variability of tree classifiers. A small change in the training set can cause a large change in the structure of the tree and also the final probability estimates. So by changing the super group coding we might expect a probability estimate that is fairly unrelated to previous estimates and hence a low correlation. To test the accuracy of this theory we examined the results from the simulation performed in section 3.1. We wished to estimate 'Y and p. The following table summarizes our estimates for the variance and standardizing (al) terms from the simulated data set. I Classifier Substitution PaCT Tree Method 548 G. James and T. Hastie Tree '"o ECOC s..t>stilUtion ?o ------------------------5 10 50 100 B (log scale) Figure 3: Error rates on the simulated data set for tree method, Substitution PaCf and ECOC PaCT plotted against B (on log scale) = These quantities give us an estimate for, of l' 0.227 We also derived an estimate for p of p = 0.125 We see that p is less than, so provided B ~ ~=~ ~ 8.6 we should see an improvement in the Substitution PaCT over a k class tree classifier. Figure 3 shows that the Substitution error rate drops below that of the tree classifier at almost exactly this point. 4 Conclusion The ECOC PaCT was originally envisioned as an adaption of error coding ideas to classification problems. Our results indicate that the error coding matrix is simply a method for randomly sampling from a fixed distribution. This idea is very similar to the Bootstrap where we randomly sample from the empirical distribution for a fixed data set. There you are trying to estimate the variability of some parameter. Your estimate will have two sources of error, randomness caused by sampling from the empirical distribution and the randomness from the data set itself. In our case we have the same two sources of error, error caused by sampling from 1 - 2!ftj - Zij! to estimate J-ti and error's caused by J-t itself. In both cases the first sort of error will reduce rapidly and it is the second type we are really interested in. It is possible to motivate the reduction in error rate of using argi max J-ti in terms of a decrease in variability, provided B is large enough and our correlation (p) is small enough. References Dietterich, T.G. and Bakiri G. (1995) Solving Multiclass Learning Problems via ErrorCorrecting Output Codes, Journal of Artificial Intelligence Research 2 (1995) 263-286 Diet~rich, T. G. and Kong, E. B. (1995) Error-Correcting Output Coding Corrects Bias and Variance, Proceedings of the 12th International Conference on Machine Learning pp. 313-321 Morgan Kaufmann Friedman, 1.H. (1996) On Bias, Variance, Oil-loss, and the Curse of Dimensionality, Dept of Statistics, Stanford University, Technical Report Hoeffding, W. (1963) Probability Inequalities for Sums of Bounded Random Variables. "Journal of the American Statistical Association", March, 1963 

1481 |@word kong:1 repository:1 simulation:1 covariance:1 simplifying:1 fonn:3 ld:1 reduction:5 substitution:21 contains:1 zij:4 past:3 z2:1 assigning:1 designed:3 plot:1 drop:1 tenn:1 fewer:1 intelligence:1 ith:4 provides:1 ipi:1 direct:1 prove:1 behavior:1 examine:1 ecoc:23 curse:1 pf:4 lib:1 provided:3 classifies:1 estimating:1 unrelated:1 bounded:1 lowest:1 what:1 easiest:1 substantially:1 developed:1 transformation:1 guarantee:1 every:2 ti:2 shed:1 exactly:1 classifier:25 zl:3 producing:1 limit:1 consequence:1 might:3 plus:1 examined:1 suggests:1 co:3 averaged:1 unique:1 bootstrap:1 procedure:3 empirical:4 thought:1 word:1 regular:1 suggest:2 influence:2 deterministic:4 correcting:2 rule:1 coordinate:1 pt:2 suppose:4 us:1 designing:1 hypothesis:2 element:1 predicts:2 calculate:1 decrease:3 ran:1 envisioned:1 trained:1 depend:1 solving:2 motivate:3 basis:1 completely:1 joint:1 train:1 artificial:1 tell:5 choosing:2 stanford:2 statistic:3 itself:2 final:2 obviously:1 lpij:1 eeoc:1 mb:1 rapidly:1 iff:1 achieve:1 intuitive:1 convergence:3 produce:8 argmaxp:1 converges:1 pi2:1 develop:1 wished:1 strong:1 come:1 indicate:2 differ:1 explains:1 ja:1 ao:1 really:1 considered:1 normal:2 great:1 largest:1 create:1 hope:1 clearly:1 always:1 super:6 corollary:2 derived:1 improvement:4 indicates:3 sense:1 unlikely:1 transformed:1 interested:2 arg:2 classification:13 fairly:2 equal:3 eliminated:1 sampling:7 identical:4 look:1 report:1 randomly:7 composed:1 gamma:1 individual:1 friedman:2 investigate:1 light:1 devoted:1 tree:23 taylor:1 plotted:2 column:5 classify:3 ar:2 predictor:2 uniform:1 argi:19 too:1 answer:1 unbiasedness:1 international:1 randomized:2 ie:5 corrects:1 central:1 choose:2 hoeffding:3 american:1 li:7 potential:1 standardizing:1 coding:33 caused:3 vi:4 depends:2 later:1 performed:1 bayes:16 sort:1 pil:2 accuracy:1 variance:5 kaufmann:1 correspond:1 randomness:2 classified:1 sqrt:1 trevor:1 definition:1 against:1 pp:1 james:4 di:5 hamming:3 irvine:1 proved:1 dimensionality:1 back:1 originally:1 follow:1 diet:1 just:3 correlation:3 lda:1 believe:1 oil:1 dietterich:4 consisted:1 unbiased:2 true:2 hence:2 noted:1 trying:1 recently:1 common:1 empirically:2 exponentially:1 association:1 consistency:1 moving:1 posterior:3 certain:3 inequality:3 binary:3 success:3 seen:1 morgan:1 greater:1 converge:1 ii:1 technical:1 match:1 plug:6 af:1 qi:6 essentially:1 circumstance:1 expectation:1 source:2 unlike:1 probably:1 strict:1 cart:1 call:2 exceed:1 enough:3 affect:1 fit:7 zi:3 gave:1 hastie:4 reduce:2 idea:2 tradeoff:1 multiclass:1 motivated:1 expression:1 effort:1 cause:2 clear:2 generate:2 maxpi:1 outperform:1 zj:1 notice:1 dotted:1 gil:1 estimated:2 correctly:2 group:25 four:2 pb:1 changing:2 pj:1 asymptotically:1 sum:1 you:1 almost:2 reasonable:1 separation:1 summarizes:1 vb:2 bound:1 correspondence:1 infinity:1 your:2 min:2 optimality:2 relatively:1 department:1 march:1 smaller:2 slowed:1 restricted:1 pr:4 errorcorrecting:1 fji:1 available:1 apply:1 appropriate:1 original:1 standardized:1 remaining:1 bakiri:3 question:1 quantity:3 usual:1 said:1 distance:5 simulated:3 unstable:1 reason:1 code:1 relationship:2 equivalently:1 unfortunately:1 stated:1 upper:2 observation:2 variability:5 pic:9 pair:1 unduly:1 suggested:1 below:1 ftj:1 max:18 explanation:1 power:2 unrealistic:1 pact:42 improve:3 literature:1 law:1 loss:1 expect:4 var:1 degree:1 pij:2 systematically:1 classifying:2 pi:8 row:5 course:2 repeat:1 mve:2 last:2 jth:4 bias:3 distributed:2 curve:4 calculated:1 rich:1 fonned:2 assumed:2 why:6 table:1 expansion:1 necessarily:2 did:1 main:1 motivation:3 allowed:1 attributable:1 shrinking:1 exponential:4 lie:1 theorem:19 bad:2 pac:1 r2:1 insignificant:1 concern:1 bivariate:1 exists:1 ih:2 corr:1 ci:1 illustrates:1 zjt:1 simply:2 explore:1 corresponds:1 adaption:1 gareth:1 conditional:4 viewed:1 identity:1 towards:1 change:5 corrected:2 uniformly:1 zb:1 ili:2 e:1 attempted:1 support:1 dept:1 tested:1

A Principle for Unsupervised Hierarchical Decomposition of Visual Scenes Michael C. Mozer Dept. of Computer Science University of Colorado Boulder, CO 80309-0430 ABSTRACT Structure in a visual scene can be described at many levels of granularity. At a coarse level, the scene is composed of objects; at a finer level, each object is made up of parts, and the parts of subparts. In this work, I propose a simple principle by which such hierarchical structure can be extracted from visual scenes: Regularity in the relations among different parts of an object is weaker than in the internal structure of a part. This principle can be applied recursively to define part-whole relationships among elements in a scene. The principle does not make use of object models, categories, or other sorts of higher-level knowledge; rather, part-whole relationships can be established based on the statistics of a set of sample visual scenes. I illustrate with a model that performs unsupervised decomposition of simple scenes. The model can account for the results from a human learning experiment on the ontogeny of partwhole relationships. 1 INTRODUCTION The structure in a visual scene can be described at many levels of granularity. Consider the scene in Figure I a. At a coarse level, the scene might be said to consist of stick man and stick dog. However, stick man and stick dog themselves can be decomposed further. One might describe stick man as having two components, a head and a body. The head in turn can be described in terms of its parts: the eyes, nose, and mouth. This sort of scene decomposition can continue recursively down to the level of the primitive visual features. Figure I b shows a partial decomposition of the scene in Figure I a. A scene decomposition establishes part-whole relationships among objects. For example, the mouth (a whole) consists of two parts, the teeth and the lips. If we assume that any part can belong to only one whole, the decomposition imposes a hierarchical structure over the elements in the scene. Where does this structure come from? What makes an object an object, a part a part? I propose a simple principle by which such hierarchical structure can be extracted from visual scenes and incorporate the principle in a simulation model. The principle is based on the statistics of the visual environment, not on object models or other sorts of higherlevel knowledge, or on a teacher to classify objects or their parts. 53 Hierarchical Decomposition of Visual Scenes 2 WHAT MAKES A PART A PART? Parts combine to form objects. Parts are combined in different ways to form different objects and different instances of an object. Consequently, the structural relations among different parts of an object are less regular than is the internal structure of a part. To illustrate, consider Figure 2, which depicts four instances of a box shell and lid. The components of the lid-the top and the handle-appear in a regular configuration, as do the components of the shell-the sides and base-but the relation of the lid to the shell is variable. Thus, configural regularity is an indication that components should be grouped together to form a unit. I call this the regularity principle. Other variants of the regularity principle have been suggested by Becker (1995) and Tenenbaum (1994). The regularity depicted in Figure 2 is quite rigid: one component of a part always occurs in a fixed spatial position relative to another. The regularity principle can also be cast in terms of abstract relationships such as containment and encirclement. The only difference is the featural representation that subserves the regularity discovery process. In this paper, however, I address primarily regularities that are based on physical features and fixed spatial relationships. Another generalization of the regularity principle is that it can be applied recursively to suggest not only parts of wholes, but subparts of parts . According to the regularity principle, information is implicit in the environment that can be used to establish part-whole relationships. This information comes in the form of statistical regularities among features in a visual scene. The regularity principle does not depend on explicit labeling of parts or objects. In contrast, Schyns and Murphy (1992, 1993) have suggested a theory of part ontogeny that presupposes explicit categorization of objects. They propose a homogeneity principle which states that "if a fragment of a stimulus plays a consistent role in categorization, the perceptual parts composing the fragment are instantiated as a single unit in the stimulus representation in memory." Their empirical studies with human subjects find support for the homogeneity principle. Superficially, the homogeneity and regularity principles seem quite different: while the homogeneity principle applies to supervised category learning (i.e., with a teacher to classify instances), the regularity principle applies to unsupervised discovery. But it is possible to transform one learning paradigm into the other. For example, in a category learning task, if only one category is to be learned and if the training examples are all positive instances of the category, then inducing the defining characteristics of the category is equivalent to extracting regularities in the stimulus environment. Thus, category learning in a diverse stimulus environment can be conceptualized as unsupervised regularity extraction in multiple, narrow stimulus environments (each environment being formed by taking all positive instances of a given class). (a) scene (b) ~ stick dog stick man ~ body head ~ nose mouth eyes ~ torso leg arm ~teeth lips FIGURE 1. (a) A graphical depiction of stick man and his faithful companion, stick dog; (b) a partial decomposition of the scene into its parts. FIGURE 2. Four different instances of a box with a lid M. C. Mozer 54 There are several other differences between the regularity principle proposed here and the homogeneity principle of Schyns and Murphy, but they are minor. Schyns and Murphy seem to interpret "fragment" more narrowly as spatially contiguous perceptual features. They also don't address the hierarchical nature of part-whole relationships. Nonetheless, the two principles share the notion of using the statistical structure of the visual environment to establish part-whole relations. 3 A FLAT REPRESENTATION OF STRUCTURE I have incorporated the regularity principle into a neural net that discovers partwhole relations in its environment. Neural nets, having powerful learning paradigms for unsupervised discovery, are well suited for this task. However, they have a fundamental difficulty representing complex, articulated data structures of the sort necessary to encode hierarchies (but see Pollack, 1988, and Smolensky, 1990, for promising advances). I thus begin by describing a novel representation scheme for hierarchical structures that can readily be integrated into a neural net. The tree structure in Figure I b depicts one representation of a hierarchical decomposition. The complete tree has as its leaf nodes the primitive visual features of the scene. The tree specifies the relationships among the visual features. There is another way of capturing these relationships, more connectionist in spirit than the tree structure. The idea is to assign to each primitive feature a tag-a scalar in [0, I)-such that features within a subtree have similar values. For the features of stick man, possible tags might be: eyes .1, nose .2, lips .28, teeth .32, arm .6, torso .7, leg .8. Denoting the set of all features having tags in [a, ~] by Sea, ~), one can specify any subtree of the stick man representation. For example, S(O, 1) includes all features of stick man; S(0,.5) includes all features in the subtree whose root is stick man's head, S(.5,I) his body; S(.25,.35) indicates the parts of the mouth. By a simple algorithm, tags can be assigned to the leaf nodes of any tree such that any subtree can be selected by specifying an appropriate tag range. The only requirement for this algorithm is knowledge of the maximum branching factor. There is no fixed limit to the depth of the tree that can be thus represented; however, the deeper the tree, the finer the tag resolution that wiII be needed. The tags provide a "flat" way of representing hierarchical structure. Although the tree is implicit in the representation, the tags convey all information in the tree, and thus can capture complex, articulated structures. The tags in fact convey additional information. For example in the above feature list, note that lips is closer to nose than teeth is to nose. This information can easily be ignored, but it is still worth observing that the tags carry extra baggage not present in the symbolic tree structure. It is convenient to represent the tags on a range [0, 21t) rather than [0, I]. This allows the tag to be identified with a directional-or angular-value. Viewed as part of a cyclic continuum, the directional tags are homogeneous, in contrast to the linear tags where tags near and 1 have special status by virtue of being at endpoints of the continuum. Homogeneity results in a more elegant model, as described below. The directional tags also permit a neurophysiological interpretation, albeit speculative. It has been suggested that synchronized oscillatory activities in the nervous system can be used to convey information above and beyond that contained in the average firing rate of individual neurons (e.g., Eckhorn et aI., 1988; Gray et aI., 1989; von der Malsburg, 1981). These osciIIations vary in their phase, the relative offset of the bursts. The directional tags could map directly to phases of oscillations, providing a means of implementing the tagging in neocortex. ? 4 REGULARITY DISCOVERY Many learning paradigms allow for the djscovery of regUlarity. I have used an autoencoder architecture (Plaut, Nowlan, & Hinton, 1986) that maps an input pattern-a 55 Hierarchical Decomposition of Visual Scenes representation of visual features in a scene-to an output pattern via a small layer of hidden units. The goal of this type of architecture is for the network to reproduce the input pattern over the output units. The task requires discovery of regularities because the hidden layer serves as an encoding bottleneck that limits the representational capacity of the system. Consequently, stronger regularities (the most common patterns) will be encoded over the weaker. 5 MAGIC We now need to combine the autoencoder architecture with the notion of tags such that regularity of feature configurations in the input will increase the likelihood that the features will be assigned the same tags. This goal can be achieved using a model we developed for segmenting an image into different objects using supervised learning. The model, MAGIC (Mozer, Zemel, Behrmann, & Williams, 1992), was trained on images containing several visual objects and its task was to tag features according to which object they belonged. A teacher provided the target tags. Each unit in MAGIC conveys two distinct values: a probability that a feature is present, which I will call the feature activity, and a tag associated with the feature. The tag is a directional (angular) value, of the sort suggested earlier. (The tag representation is in reality a complex number whose direction corresponds to the directional value and whose magnitude is related to the unit's confidence in the direction. As this latter aspect of the representation is not central to the present work, I discuss it no further.) The architecture is a two layer recurrent net. The input or feature layer is set of spatiotopic arrays-in most simulations having dimensions 25x25-each array containing detectors for features of a given type: oriented line segments at 0 ,45 ,90 and 135 In addition, there is a layer of hidden units. Each hidden unit is reciprocally connected to input from a local spatial patch of the input array; in the current simulations, the patch has dimensions 4x4. For each patch there is a corresponding fixed-size pool of hidden units. To achieve a translation invariant response across the image, the pools are arranged in a spatiotopic array in which neighboring pools respond to neighboring patches and the patch-to-pool weights are constrained to be the same at all locations in the array. There are interlayer connections, but no intralayer connections. The images presented to MAGIC consist of an arrangement of features over the input array. The feature activity is clamped on (i.e., the feature is present), and the initial directional tag of the feature is set at random. Feature unit activities and tags feed to the hidden units, which in turn feed back to the feature units. Through a relaxation process, the system settles on an assignment of tags to the feature units (as well as to the hidden units, although read out from the model concerns only the feature units). MAGIC is a mean-field approximation to a stochastic network of directional units with binary-gated outputs (Zemel, Williams, & Mozer, 1995). This means that a mean-field energy functional can be written that expresses the network state and controls the dynamics; consequently, MAGIC is guaranteed to converge to a stable pattern of tags. Each hidden unit detects a spatially local configuration offeatures, and it acts to reinstate a pattern of tags over the configuration. By adjusting its incoming and outgoing weights during training, the hidden unit is made to respond to configurations that are consistently tagged in the training set. For example, if the training set contains many corner junctions where horizontal and vertical lines come to a point and if the teacher tags all features composing these lines as belonging to the same object, then a hidden unit might learn to detect this configuration, and when it does so, to force the tags of the component features to be the same. In our earlier work, MAGIC was trained to map the feature activity pattern to a target pattern of feature tags, where there was a distinct tag for each object in the image. In the present work, the training objective is rather to impose uniform tags over the features. Additionally, the training objective encourages MAGIC to reinstate the feature activity 0 0 0 , 0 ? M C. Mozer 56 Iteration 1 ,, - , ---~- i I Iteration 2 Iteration 4 Iteration 6 Iteration II ----- -;,- ~ I I Directional Tag Spectrum _ AI!II.?, 1 1 1.'I!t~ FIGURE 3. The state of MAGIC as processing proceeds for an image composed of a pair of lines made out of horizontal and vertical line segments. The coloring of a segment represents the directional tag. The segments belonging to a line are randomly tagged initially; over processing iterations, these tags are brought into alignment. pattern over the feature units; that is, the hidden units must encode and propagate information back to the feature units that is sufficient to specify the feature activities (if the feature activities weren't clamped). With this training criterion, MAGIC becomes a type of autoencoder. The key property of MAGIC is that it can assign a feature configuration the same tag only if it learns to encode the configuration. If an arrangement is not encoded, there will be no force to align the feature tags. Further, fixed weak inhibitory connections between every pair of feature units serve to spread the tags apart if the force to align them is not strong enough. Note that this training paradigm does not require a teacher to tag features as belonging to one part or another. MAGIC will try to tag all features as belonging to the same part, but it is able to do so only for configurations of features that it is able to encode. Consequently, highly regular and recurring configurations will be grouped together, and irregular configurations will be pulled apart. The strength of grouping will be proportional to the degree of regularity. 6 SIMULATION EXPERIMENTS To illustrate the behavior of the model, I show a simple simulation in which MAGIC is trained on pairs of lines, one vertical and one horizontal. Each line is made up of 6 colinear line segments. The segments are primitive input features of the model. The two lines may appear in different positions relative to one another. Hence, the strongest regularity is in the segments that make up a line, not the junction between the lines. When trained with two hidden units, MAGIC has sufficient resources to encode the structure within each line, but not the relationships among the lines; because this structure is not encoded, the features of the two lines are not assigned the same tags (Figure 3). Although each "part" is made up of features having a uniform orientation and in a colinear arrangement, the composition and structure of the parts is immaterial; MAGIC's performance depends only on the regularity of the configurations. In the next set of simulations, MAGIC discovers regularities of a more arbitrary nature. 6.1 MODELING HUMAN LEARNING OF PART-WHOLE RELATIONS Schyns and Murphy (1992) studied the ontogeny of part-whole relationships by training human subjects on a novel class of objects and then examining how the subjects decomposed the objects into their parts. I briefly describe their experiment, followed by a simulation that accounts for their results. In the first phase of the experiment, subjects were shown 3-D gray level "martian rocks" on a CRT screen. The rocks were constructed by deforming a sphere, resulting in various bumps or protrusions. Subjects watched the rocks rotating on the screen, allowing them to view the rock from all sides. Subjects were shown six instances, all of which were labeled "M 1 rocks" and were then tested to determine whether they could distinguish M 1 57 Hierarchical Decomposition of Visual Scenes rocks from other rocks. Subjects continued training until they performed correctly on this task. Every Ml rock was divided into octants; the protrusions on seven of the octants were generated randomly, and the protrusions on the last octant were the same for all Ml rocks. Two groups of subjects were studied. The A group saw M I rocks all having part A, the B group saw M 1 rocks all having part B. Following training, subjects were asked to delineate the parts they thought were important on various exemplars. Subjects selected the target part from the category on which they were trained 93% of the time, and the alternative target-the target from the other category-only 8% of the time, indicating that the learning task made a part dramatically more salient. To model this phase of the experiment, I generated two dimensional contours of the same flavor as Schyns and Murphy's martian rocks (Figure 4). Each rock-can it a "venusian rock" for distinction-can be divided into four quadrants or parts. Two groups of venusian rocks were generated. Rocks of category A an contained part A (left panel, Figure 4), rocks of category B contained part B (center panel, Figure 4). One network was trained on six exemplars of category A rocks, another network was trained on six exemplars of category B rocks. Then, with learning turned off, both networks were tested on five presentations each of twelve new exemplars, six each of categories A and B. Just as the human subjects were instructed to delineate parts, we must ask MAGIC to do the same. One approach would be to run the model with a test stimulus and, once it settles, select an features having directional tags clustered tightly together as belonging to the same part. However, this requires specifying and tuning a clustering procedure. To avoid this additional step, I simply compared how tightly clustered were the tags of the target part relative to those of the alternative target. I used a directional variance measure that yields a value of 0 if all tags are identical and I if the tags are distributed uniformly over the directional spectrum. By this measure, the variance was .30 for the target part and .68 for the alternative target (F(l, 118) = 322.0, P < .001), indicating that the grouping of features of the target part was significantly stronger. This replicates, at least qualitatively, the finding of Schyns and Murphy. In a second phase of Schyns and Murphy's experiment, subjects were trained on category C rocks, which were formed by adjoining parts A and B and generating the remaining six octants at random. Following training, subjects were again asked to delineate parts. All subjects delineated A and B as distinct parts. In contrast, a naive group of subjects who were trained on category C alone always grouped A and B together as a single part. To model this phase, I generated six category C venusian rocks that had both parts A and B (right panel, Figure 4). The versions of MAGIC that had been trained on category A and B rocks alone were now trained on category C rocks. As a control condition, a third version of MAGIC was trained from scratch on category C rocks alone. I compared the tightness of clustering of the combined A-B part for the first two nets to the third. Using the same variance measure as above, the nets that first received training on parts A and B alone yielded a variance of .57, and the net that was only trained on the combined A-B part yielded a variance of .47 (F(1,88) = 7.02, P < .02). One cannot directly compare the variance of the A-B part to that of the A and B parts alone, because the measure is structured such that parts with more features always yield larger variances. However, one can compare the two conditions using the relative variance of the combined A-B part to the A 'S'~7 t.r ~ ~ -,~ (-"to. ~~" I~~ ,,~ ~~""-........ , ~~ ~ -,-./ ,~~ FIGURE 4. Three examples of the martian rock stimuli used to train MAGIC. From left to right, the rocks are of categories A, B, and C. The lighter regions are the contours that define rocks of a given category. 58 M C. Mozer and B parts alone. This yielded the same outcome as before (.21 for the first two nets, .12 for the third net, F(l,88) = 5.80, p < .02). Thus, MAGIC is also able to account for the effects of prior learning on part ontogeny. 7 CONCLUSIONS The regularity principle proposed in this work seems consistent with the homogeneity principle proposed earlier by Schyns and Murphy (1991, 1992). Indeed, MAGIC is able to model Schyns and Murphy's data using an unsupervised training paradigm, although Schyns and Murphy framed their experiment as a classification task. This work is but a start at modeling the development of part-whole hierarchies based on perceptual experience. MAGIC requires further elaboration, and I am somewhat skeptical that it is sufficiently powerful in its present form to be pushed much further. The main issue restricting it is the representation of input features. The oriented-line-segment features are certainly too primitive and inflexible a representation. For example, MAGIC could not be trained to recognize the lid and shell of Figure 2 because it encodes the orientation of the features with respect to the image plane, not with respect to one another. Minimally, the representation requires some version of scale and rotation invariance. Perhaps the most interesting computational'issue raised by MAGIC is how the pattern of feature tags is mapped into an explicit part-whole decomposition. This involves clustering together the similar tags as a unit, or possibly selecting all tags in a given range. To do so requires specification of additional parameters that are external to the model (e.g., how tight the cluster should be, how broad the range should be, around what tag direction it should be centered). These parameters are deeply related to attentional issues, and a current direction of research is to explore this relationship. 8 ACKNOWLEDGEMENTS This research was supported by NSF PYI award IRI-9058450 and grant 97-18 from the McDonnell-Pew Program in Cognitive Neuroscience. 9 REFERENCES Becker, S. (1995). JPMAX: Learning to recognize moving objects as a model-fitting problem. In G. Tesauro, D. S. Touretzky, & T. K. Leen (Eds), Advances in Neural Informatio/l ProcessinK Systems 7 (pp. 933-940). Cambridge, MA: MIT Press. Eckhorn, Roo Bauer, R., Jordan, w., Brosch, M., Kruse, w., Munk, M., & Reitboek, H. J. (1988). Coherent oscillations: A mechanism of feature linking in the visual cortex? Biological Cybernetics, 60, 121-130. Gray, C. M., Koenig, P., Engel, A. K., & Singer, W. (1989). Oscillatory responses in cat visual cortex exhibit intercolumnar synchronization which reflects global stimulus properties. Nature (London), 338, 334-337. Mozer, M. c., Zemel, R. S., Behrmann, M .. & Williams, C. K. I. (1992). Learning to segment images using dynamic feature binding. Neural Computation, 4, 650-666. Plaut, D. C. , Nowlan, S., & Hinton, G. E . (1986). Experiments 011 leaminK by back propagation (Technical report CMU-CS-86- 126). Pittsburgh, PA: Carnegie-Mellon University, Department of Computer Science. Pollack, J. B. (1988). Recursive auto-associative memory: Devising compositional distributed representations . In Proceedings of the Tenth Annual Conference of the Cognitive Science Society (pp. 33-39). Hillsdale, NJ : Erlbaum. Schyns, P. G., & Murphy, G. L. (1992). The ontogeny of units in object categories. In Proceedings of the Fourteenth Annual Conference (!fthe COKnitil'e Science Society (pp. 197-202). Hillsdale, NJ: Erlbaum. Schyns, P. G. , & Murphy, G. L. (1993). The ontogeny of transformable part representations in object concepts. In Proceedings of the Fifteenth Annual Conference of the OIKllitive Scien ce Society (pp. 917-922). Hillsdale. NJ: Erlbaum. Smolensky, P. (1990). Tensor product variable binding and the representation of symbolic structures in connectionist networks. Artificial Intelligence, 46. 159-2 I 6. Tenenbaum, J. B. (1994). Functional parts. In A. Ram & K. Eiselt (Eds .). Pmceedings (If the Sixteenth An/lual Conference of the Cognitive Science Society (pp. 864-869). Hillsdale, NJ : Erlbaum. von der Malsburg, C. (1981). The correlatioll theory of brain fun ction (Internal Report 81-2) . Goettingen: Department of Neurobiology, Max Planck Institute for Biophysical Chemistry. Zemel, R. S., Williams, C. K. I.. & Mozer, M. C. (1995). Lending direction to neural networks . Neural Networks , 8, 503-512. 

1482 |@word version:3 briefly:1 stronger:2 seems:1 simulation:7 propagate:1 decomposition:12 recursively:3 carry:1 initial:1 configuration:12 cyclic:1 fragment:3 contains:1 selecting:1 denoting:1 current:2 nowlan:2 written:1 readily:1 must:2 alone:6 intelligence:1 leaf:2 selected:2 nervous:1 devising:1 plane:1 coarse:2 plaut:2 node:2 location:1 lending:1 five:1 burst:1 constructed:1 consists:1 combine:2 fitting:1 interlayer:1 ontogeny:6 tagging:1 indeed:1 behavior:1 themselves:1 brain:1 detects:1 decomposed:2 becomes:1 begin:1 provided:1 panel:3 what:3 developed:1 finding:1 nj:4 configural:1 every:2 act:1 fun:1 stick:13 control:2 unit:26 grant:1 appear:2 planck:1 segmenting:1 positive:2 before:1 local:2 limit:2 encoding:1 firing:1 might:4 minimally:1 studied:2 specifying:2 co:1 scien:1 range:4 faithful:1 recursive:1 procedure:1 empirical:1 thought:1 significantly:1 convenient:1 confidence:1 regular:3 quadrant:1 suggest:1 symbolic:2 cannot:1 equivalent:1 map:3 center:1 conceptualized:1 primitive:5 williams:4 iri:1 resolution:1 pyi:1 reitboek:1 continued:1 array:6 his:2 handle:1 notion:2 hierarchy:2 play:1 colorado:1 target:10 lighter:1 homogeneous:1 pa:1 element:2 labeled:1 role:1 capture:1 region:1 connected:1 deeply:1 mozer:8 environment:8 asked:2 dynamic:2 trained:14 depend:1 colinear:2 segment:9 immaterial:1 tight:1 serve:1 easily:1 represented:1 various:2 cat:1 articulated:2 train:1 instantiated:1 distinct:3 describe:2 london:1 ction:1 artificial:1 zemel:4 labeling:1 outcome:1 quite:2 whose:3 encoded:3 presupposes:1 larger:1 tightness:1 roo:1 statistic:2 transform:1 associative:1 higherlevel:1 indication:1 biophysical:1 net:9 rock:27 propose:3 product:1 neighboring:2 turned:1 achieve:1 representational:1 sixteenth:1 protrusion:3 inducing:1 partwhole:2 regularity:28 requirement:1 cluster:1 sea:1 categorization:2 generating:1 object:25 illustrate:3 recurrent:1 exemplar:4 minor:1 received:1 strong:1 c:1 involves:1 come:3 synchronized:1 direction:5 stochastic:1 centered:1 human:5 crt:1 settle:2 implementing:1 munk:1 hillsdale:4 require:1 assign:2 generalization:1 weren:1 clustered:2 biological:1 goettingen:1 sufficiently:1 around:1 baggage:1 bump:1 continuum:2 vary:1 saw:2 grouped:3 engel:1 establishes:1 reflects:1 brought:1 mit:1 always:3 rather:3 avoid:1 encode:5 consistently:1 indicates:1 likelihood:1 contrast:3 detect:1 am:1 rigid:1 integrated:1 initially:1 hidden:12 relation:6 reproduce:1 issue:3 among:7 orientation:2 classification:1 development:1 spatial:3 special:1 constrained:1 raised:1 field:2 once:1 having:8 extraction:1 x4:1 represents:1 identical:1 broad:1 unsupervised:6 connectionist:2 stimulus:8 report:2 primarily:1 oriented:2 randomly:2 composed:2 tightly:2 homogeneity:7 individual:1 recognize:2 murphy:12 phase:6 highly:1 certainly:1 alignment:1 replicates:1 adjoining:1 closer:1 partial:2 necessary:1 experience:1 tree:10 rotating:1 pollack:2 instance:7 classify:2 earlier:3 modeling:2 contiguous:1 assignment:1 uniform:2 examining:1 erlbaum:4 too:1 spatiotopic:2 teacher:5 combined:4 fundamental:1 twelve:1 off:1 pool:4 michael:1 together:5 offeatures:1 von:2 central:1 venusian:3 again:1 containing:2 possibly:1 corner:1 external:1 cognitive:3 account:3 chemistry:1 includes:2 depends:1 performed:1 root:1 try:1 view:1 observing:1 start:1 sort:5 formed:2 variance:8 characteristic:1 who:1 yield:2 directional:13 weak:1 worth:1 finer:2 cybernetics:1 oscillatory:2 detector:1 strongest:1 touretzky:1 jpmax:1 ed:2 nonetheless:1 energy:1 pp:5 conveys:1 associated:1 adjusting:1 ask:1 knowledge:3 torso:2 back:3 coloring:1 feed:2 higher:1 supervised:2 specify:2 response:2 arranged:1 leen:1 box:2 delineate:3 angular:2 implicit:2 just:1 until:1 koenig:1 horizontal:3 propagation:1 gray:3 perhaps:1 effect:1 concept:1 tagged:2 assigned:3 hence:1 spatially:2 read:1 during:1 branching:1 encourages:1 intercolumnar:1 criterion:1 complete:1 performs:1 image:8 discovers:2 novel:2 common:1 speculative:1 rotation:1 functional:2 physical:1 endpoint:1 belong:1 interpretation:1 linking:1 interpret:1 mellon:1 composition:1 cambridge:1 ai:3 framed:1 tuning:1 pew:1 eckhorn:2 wiii:1 had:2 moving:1 stable:1 specification:1 cortex:2 depiction:1 base:1 align:2 apart:2 tesauro:1 binary:1 continue:1 der:2 additional:3 somewhat:1 impose:1 converge:1 paradigm:5 determine:1 kruse:1 ii:2 multiple:1 technical:1 sphere:1 elaboration:1 divided:2 award:1 watched:1 variant:1 cmu:1 fifteenth:1 iteration:6 represent:1 achieved:1 irregular:1 addition:1 extra:1 subject:15 elegant:1 spirit:1 seem:2 jordan:1 call:2 extracting:1 structural:1 near:1 granularity:2 enough:1 architecture:4 identified:1 idea:1 narrowly:1 bottleneck:1 six:6 whether:1 becker:2 compositional:1 ignored:1 dramatically:1 neocortex:1 tenenbaum:2 category:23 specifies:1 nsf:1 inhibitory:1 neuroscience:1 correctly:1 subpart:2 diverse:1 carnegie:1 transformable:1 express:1 group:5 key:1 four:3 salient:1 ce:1 tenth:1 ram:1 relaxation:1 run:1 fourteenth:1 powerful:2 respond:2 patch:5 oscillation:2 pushed:1 capturing:1 layer:5 guaranteed:1 followed:1 distinguish:1 yielded:3 activity:8 annual:3 strength:1 scene:23 flat:2 encodes:1 tag:51 aspect:1 structured:1 department:2 according:2 mcdonnell:1 belonging:5 across:1 inflexible:1 lid:5 delineated:1 leg:2 invariant:1 boulder:1 resource:1 brosch:1 turn:2 describing:1 discus:1 mechanism:1 needed:1 singer:1 nose:5 serf:1 junction:2 permit:1 hierarchical:11 appropriate:1 alternative:3 reinstate:2 top:1 clustering:3 remaining:1 graphical:1 malsburg:2 establish:2 society:4 tensor:1 objective:2 arrangement:3 occurs:1 fthe:1 said:1 exhibit:1 attentional:1 mapped:1 capacity:1 seven:1 relationship:13 providing:1 magic:25 gated:1 allowing:1 vertical:3 neuron:1 defining:1 hinton:2 incorporated:1 head:4 neurobiology:1 arbitrary:1 dog:4 cast:1 pair:3 connection:3 coherent:1 learned:1 narrow:1 distinction:1 established:1 address:2 beyond:1 suggested:4 proceeds:1 below:1 pattern:10 able:4 recurring:1 smolensky:2 belonged:1 program:1 max:1 memory:2 reciprocally:1 mouth:4 difficulty:1 force:3 arm:2 representing:2 scheme:1 eye:3 autoencoder:3 naive:1 featural:1 auto:1 prior:1 discovery:5 acknowledgement:1 relative:5 synchronization:1 interesting:1 proportional:1 degree:1 teeth:4 x25:1 sufficient:2 consistent:2 imposes:1 principle:24 share:1 translation:1 skeptical:1 supported:1 last:1 side:2 weaker:2 deeper:1 allow:1 pulled:1 institute:1 taking:1 distributed:2 bauer:1 depth:1 dimension:2 superficially:1 contour:2 instructed:1 made:6 qualitatively:1 status:1 ml:2 global:1 incoming:1 containment:1 pittsburgh:1 don:1 spectrum:2 reality:1 lip:4 promising:1 nature:3 learn:1 additionally:1 composing:2 pmceedings:1 schyns:12 complex:3 intralayer:1 spread:1 main:1 whole:13 convey:3 martian:3 body:3 depicts:2 screen:2 position:2 explicit:3 clamped:2 perceptual:3 behrmann:2 third:3 learns:1 subserves:1 down:1 companion:1 list:1 offset:1 virtue:1 concern:1 grouping:2 consist:2 albeit:1 restricting:1 magnitude:1 subtree:4 flavor:1 suited:1 depicted:1 octant:4 simply:1 explore:1 neurophysiological:1 visual:19 contained:3 scalar:1 applies:2 binding:2 corresponds:1 extracted:2 ma:1 shell:4 viewed:1 goal:2 presentation:1 consequently:4 man:9 uniformly:1 invariance:1 deforming:1 indicating:2 select:1 internal:3 support:1 latter:1 incorporate:1 dept:1 outgoing:1 tested:2 scratch:1

The Bias-Variance Tradeoff and the Randomized GACV Grace Wahba, Xiwu Lin and Fangyu Gao Dong Xiang Dept of Statistics Univ of Wisconsin 1210 W Dayton Street Madison, WI 53706 wahba,xiwu,fgao@stat.wisc.edu SAS Institute, Inc. SAS Campus Drive Cary, NC 27513 sasdxx@unx.sas.com Ronald Klein, MD and Barbara Klein, MD Dept of Ophthalmalogy 610 North Walnut Street Madison, WI 53706 kleinr,kleinb@epi.ophth.wisc.edu Abstract We propose a new in-sample cross validation based method (randomized GACV) for choosing smoothing or bandwidth parameters that govern the bias-variance or fit-complexity tradeoff in 'soft' classification. Soft classification refers to a learning procedure which estimates the probability that an example with a given attribute vector is in class 1 vs class O. The target for optimizing the the tradeoff is the Kullback-Liebler distance between the estimated probability distribution and the 'true' probability distribution, representing knowledge of an infinite population. The method uses a randomized estimate of the trace of a Hessian and mimics cross validation at the cost of a single relearning with perturbed outcome data. 1 INTRODUCTION We propose and test a new in-sample cross-validation based method for optimizing the biasvariance tradeoff in 'soft classification' (Wahba et al1994), called ranG ACV (randomized Generalized Approximate Cross Validation) . Summarizing from Wahba et al(l994) we are given a training set consisting of n examples, where for each example we have a vector t E T of attribute values, and an outcome y, which is either 0 or 1. Based on the training data it is desired to estimate the probability p of the outcome 1 for any new examples in the The Bias-Variance TradeofJand the Randomized GACV 621 future. In 'soft' classification the estimate p(t) of p(t) is of particular interest, and might be used by a physician to tell patients how they might modify their risk p by changing (some component of) t, for example, cholesterol as a risk factor for heart attack. Penalized likelihood estimates are obtained for p by assuming that the logit f(t), t E T, which satisfies p(t) = ef(t) 1(1 + ef(t?) is in some space 1{ of functions . Technically 1{ is a reproducing kernel Hilbert space, but you don't need to know what that is to read on. Let the training set be {Yi, ti, i = 1,???, n}. Letting Ii = f(td, the negative log likelihood .c{Yi, ti, fd of the observations, given f is n .c{Yi, ti, fd = 2::[-Ydi + b(li)], (1) i=1 where b(f) = log(l + ef ). The penalized likelihood estimate of the function f is the solution to: Find f E 1{ to minimize h. (I): n h.(f) = 2::[-Ydi + b(ld) + J>.(I), (2) i =1 where 1>.(1) is a quadratic penalty functional depending on parameter(s) A = (AI, ... , Aq) which govern the so called bias-variance tradeoff. Equivalently the components of A control the tradeoff between the complexity of f and the fit to the training data. In this paper we sketch the derivation of the ranG ACV method for choosing A, and present some preliminary but favorable simulation results, demonstrating its efficacy. This method is designed for use with penalized likelihood estimates, but it is clear that it can be used with a variety of other methods which contain bias-variance parameters to be chosen, and for which minimizing the Kullback-Liebler (K L) distance is the target. In the work of which this is a part, we are concerned with A having multiple components. Thus, it will be highly convenient to have an in-sample method for selecting A, if one that is accurate and computationally convenient can be found. Let P>. be the the estimate and p be the 'true' but unknown probability function and let = p(td,p>.i = p>.(ti). For in-sample tuning, our criteria for a good choice of A is Pi the KL distance KL(p,p>.) = ~ E~I[PilogP7. + (1- pdlogg~::?)]. We may replace K L(p,p>.) by the comparative K L distance (C K L), which differs from K L by a quantity which does not depend on A. Letting hi = h (ti), the C K L is given by 1 CKL(p,p>.) == CKL(A) = ;;, n 2:: [-pd>'i + b(l>.i)). (3) i=) C K L(A) depends on the unknown p, and it is desired is to have a good estimate or proxy for it, which can then be minimized with respect to A. It is known (Wong 1992) that no exact unbiased estimate of CK L(A) exists in this case, so that only approximate methods are possible. A number of authors have tackled this problem, including Utans and M90dy(1993), Liu(l993), Gu(1992). The iterative U BR method of Gu(l992) is included in GRKPACK (Wang 1997), which implements general smoothing spline ANOVA penalized likelihood estimates with multiple smoothing parameters. It has been successfully used in a number of practical problems, see, for example, Wahba et al (1994,1995). The present work represents an approach in the spirit of GRKPACK but which employs several approximations, and may be used with any data set, no matter how large, provided that an algorithm for solving the penalized likelihood equations, either exactly or approximately, can be implemented. 622 2 G. Wahba et al. THE GACV ESTIMATE In the general penalized likelihood problem the minimizer 1>,(-) of (2) has a representation M 1>.(t) =L n dv<Pv(t) v=l + L CiQ>.(ti, t) (4) i=l where the <Pv span the null space of 1>" Q>.(8, t) is a reproducing kernel (positive definite function) for the penalized part of 7-1., and C = (Cl' ... , Cn )' satisfies M linear conditions, so that there are (at most) n free parameters in 1>.. Typically the unpenalized functions <Pv are low degree polynomials. Examples of Q(ti,') include radial basis functions and various kinds of splines; minor modifications include sigmoidal basis functions, tree basis functions and so on. See, for example Wahba( 1990, 1995), Girosi, Jones and Poggio( 1995). If f>.C) is of the form (4) then 1>,(f>.) is a quadratic form in c. Substituting (4) into (2) results in h a convex functional in C and d, and C and d are obtained numerically via a Newton Raphson iteration, subject to the conditions on c. For large n, the second sum on the right of (4) may be replaced by L~=1 Cik Q>. (ti k , t), where the tik are chosen via one of several principled methods. To obtain the CACV we begin with the ordinary leaving-out-one cross validation function CV(.\) for the CKL: n [-i] CV (.\) -_ -1 "" LJ-yd>.i n + b( 1>.i) ] , (5) i=1 where fl- i ] the solution to the variational problem of (2) with the ith data point left out and fti] is the value of fl- i] at ti . Although f>.C) is computed by solving for C and d the CACV is derived in terms of the values (it"", fn)' of f at the ti. Where there is no confusion between functions f(-) and vectors (it, ... ,fn)' of values of fat tl, ... ,tn, we let f = (it, ... " fn)'. For any f(-) of the form (4), J>. (f) also has a representation as a non-negative definite quadratic form in (it, ... , fn)'. Letting L:>. be twice the matrix of this quadratic form we can rewrite (2) as 1 n h(f,Y) = L[-Ydi + b(/i)] + 2f'L:>.f. (6) i=1 Let W = W(f) be the n x n diagonal matrix with (/ii == Pi(l - Pi) in the iith position. Using the fact that (/ii is the second derivative of b(fi), we have that H = [W + L:>.] - 1 is the inverse Hessian of the variational problem (6). In Xiang and Wahba (1996), several Taylor series approximations, along with a generalization of the leaving-out-one lemma (see Wahba 1990) are applied to (5) to obtain an approximate cross validation function ACV(.\), which is a second order approximation to CV(.\) . Letting hii be the iith entry of H , the result is CV(.\) ~ ACV('\) = .!. t[-Yd>.i + b(f>.i)] + .!. t n n i= l i=1 hiiYi(Yi - P>.i) . [1 - hiwii] Then the GACV is obtained from the ACV by replacing h ii by ~ L~1 hii and replacing 1 - hiWii by ~tr[I - (Wl/2 HWl/2)], giving CACV('\) 1~ ] tr(H) == ~tr(H) L~l Yi(Yi - P>.i) (Wl/2HWl /2)] , = ;; t;;[-Yd>.i + b(1).i) + -n-tr[I _ (7) (8) where W is evaluated at 1>.. Numerical results based on an exact calculation of (8) appear in Xiang and Wahba (1996). The exact calculation is limited to small n however. 623 The Bias-Variance TradeofJand the Randomized GACV 3 THE RANDOMIZED GACV ESTIMATE Given any 'black box' which, given >., and a training set {Yi, ti} produces f>. (.) as the minimizer of (2), and thence f>. = (fA 1 , " ' , f>.n)', we can produce randomized estimates of trH and tr[! - W 1 / 2 HW 1/ 2 J without having any explicit calculations of these matrices. This is done by running the 'black box' on perturbed data {Vi + <5i , td. For the Yi Gaussian, randomized trace estimates of the Hessian of the variational problem (the 'influence matrix') have been studied extensively and shown to be essentially as good as exact calculations for large n, see for example Girard(1998) . Randomized trace estimates are based on the fact that if A is any square matrix and <5 is a zero mean random n-vector with independent components with variance then E<5' A<5 = ~ tr A. See Gong et al( 1998) and (TJ, u" references cited there for experimental results with multiple regularization parameters. Returning to the 0-1 data case, it is easy to see that the minimizer fA(') of 1;.. is continuous in Y, not withstanding the fact that in our training set the Yi take on only values 0 or 1. Letting if = UA1,"', f>.n)' be the minimizer of (6) given y = (Y1,"', Yn)', and if+O be the minimizer given data y+<5 = (Y1 +<51, ... ,Yn +<5n )' (the ti remain fixed), Xiang and Wahba (1997) show, again using Taylor series expansions, that if+O - ff ,. . ., [WUf) + ~AJ-1<5. This suggests that ~<5'Uf+O - ff) provides an estimate oftr[W(ff) + ~At1. However, u" if we take the solution ff to the nonlinear system for the original data Y as the initial value for a Newton-Raphson calculation of ff+O things become even simpler. Applying a one step Newton-Raphson iteration gives (9) -<5 + PjfUf,Y) = -<5, and [:;~f(ff,Y + <5)J - 1 y + o ,l y + o ,l 82 h(fY 8 2 h(fY [ 8?7if A' Y)J- 1 ,we have f A - fYA + [8?7if A' Y)J- 1 uJ: so that f A - fYA [WUf) + E At 1<5. The result is the following ranGACV function: Since Pjf(ff,y + <5) ranGACV(>.) = = n .!. ~[n~ 'I '+bU .)J+ Yz At At <5' (f Y+O,l A n - fY) A ",n ( ) wi=l Yi Yi - PAi . [<5'<5 - <5'WUf)Uf+O,l - ff)J (10) To reduce the variance in the term after the '+' in (10), we may draw R independent replicate vectors <51,'" ,<5R , and replace the term after the '+' in ( 1O)b 1... ",R o:(fr Hr . 1 -ff) wr=l n ranGACV(>.) function. 4 y R 2:7-1y.(y.-P>..) [O~Or-O~ W(fn(f~+Ar . l-ff)1 to obtain an R-replicated NUMERICAL RESULTS In this section we present simulation results which are representative of more extensive simulations to appear elsewhere. In each case, K < < n was chosen by a sequential clustering algorithm. In that case, the ti were grouped into K clusters and one member of each cluster selected at random. The model is fit. Then the number of clusters is doubled and the model is fit again. This procedure continues until the fit does not change. In the randomized trace estimates the random variates were Gaussian. Penalty functionals were (multivariate generalizations of) the cubic spline penalty functional>. fa1U" (X))2, and smoothing spline ANOVA models were fit. G. Wahba et at. 624 4.1 EXPERIMENT 1. SINGLE SMOOTHING PARAMETER In this experiment t E [0,1], f(t) = 2sin(10t), ti = (i - .5)/500, i = 1,???,500. A random number generator produced 'observations' Yi = 1 with probability Pi = el , /(1 + eli), to get the training set. QA is given in Wahba( 1990) for this cubic spline case, K = 50. Since the true P is known, the true CKL can be computed. Fig. l(a) gives a plot of CK L(A) and 10 replicates of ranGACV(A). In each replicate R was taken as 1, and J was generated anew as a Gaussian random vector with (115 = .001. Extensive simulations with different (115 showed that the results were insensitive to (115 from 1.0 to 10- 6 ? The minimizer of C K L is at the filled-in circle and the 10 minimizers of the 10 replicates of ranGACV are the open circles. Anyone of these 10 provides a rather good estimate of the A that goes with the filled-in circle. Fig. l(b) gives the same experiment, except that this time R = 5. It can be seen that the minimizers ranGACV become even more reliable estimates of the minimizer of C K L, and the C K L at all of the ranG ACV estimates are actually quite close to its minimum value. 4.2 EXPERIMENT 2. ADDITIVE MODEL WITH A = (Al' A2) Here t E [0,1] 0 [0,1]. n = 500 values of ti were generated randomly according to a uniform distribution on the unit square and the Yi were generated according to Pi = e li j(l + e l ,) with t = (Xl,X2) and f(t) = 5 sin 27rXl - 3sin27rX2. An additive model as a special case of the smoothing spline ANOVA model (see Wahba et al, 1995), of the form f(t) = /-l + h(xd + h(X2) with cubic spline penalties on hand h were used. K = 50, (115 = .001, R = 5. Figure l(c) gives a plot of CK L(Al' A2) and Figure l(d) gives a plot of ranGACV(Al, A2). The open circles mark the minimizer of ranGACV in both plots and the filled in circle marks the minimizer of C K L. The inefficiency, as measured by CKL()..)/minACKL(A) is 1.01. Inefficiencies near 1 are typical of our other similar simulations. 4.3 EXPERIMENT 3. COMPARISON OF ranGACV AND UBR This experiment used a model similar to the model fit by GRKPACK for the risk of progression of diabetic retinopathy given t = (Xl, X2, X3) = (duration, glycosylated hemoglobin, body mass index) in Wahba et al(l995) as 'truth'. A training set of 669 examples was generated according to that model, which had the structure f(Xl, X2, X3) = /-l + fl (xd + h (X2) + h (X3) + fl,3 (Xl, X3). This (synthetic) training set was fit by GRKPACK and also using K = 50 basis functions with ranG ACV. Here there are P = 6 smoothing parameters (there are 3 smoothing parameters in f13) and the ranGACV function was searched by a downhill simplex method to find its minimizer. Since the 'truth' is known, the CKL for)" and for the GRKPACK fit using the iterative UBR method were computed. This was repeated 100 times, and the 100 pairs of C K L values appears in Figure l(e). It can be seen that the U BR and ranGACV give similar C K L values about 90% of the time, while the ranG ACV has lower C K L for most of the remaining cases. 4.4 DATA ANALYSIS: AN APPLICATION Figure 1(f) represents part of the results of a study of association at baseline of pigmentary abnormalities with various risk factors in 2585 women between the ages of 43 and 86 in the Beaver Dam Eye Study, R. Klein et al( 1995). The attributes are: Xl = age, X2 =body mass index, X3 = systolic blood pressure, X4 = cholesterol. X5 and X6 are indicator variables for taking hormones, and history of drinking. The smoothing spline ANOVA model fitted was f(t) = /-l+dlXl +d2X2+ h(X3)+ f4(X4)+ h4(X3, x4)+d5I(x5) +d6I(x6), where I is the indicator function. Figure l(e) represents a cross section of the fit for X5 = no, X6 = no, The Bias- Variance Tradeoff and the Randomized GACV 625 X2, X3 fixed at their medians and Xl fixed at the 75th percentile. The dotted lines are the Bayesian confidence intervals, see Wahba et al( 1995). There is a suggestion of a borderline inverse association of cholesterol. The reason for this association is uncertain. More details will appear elsewhere. Principled soft classification procedures can now be implemented in much larger data sets than previously possible, and the ranG ACV should be applicable in general learning. References Girard, D. (1998), 'Asymptotic comparison of (partial) cross-validation, GCV and randomized GCV in nonparametric regression', Ann. Statist. 126, 315-334. Girosi, F., Jones, M. & Poggio, T. (1995), 'Regularization theory and neural networks architectures', Neural Computatioll 7,219-269. Gong, J., Wahba, G., Johnson, D. & Tribbia, J. (1998), 'Adaptive tuning of numerical weather prediction models: simultaneous estimation of weighting, smoothing and physical parameters', MOllthly Weather Review 125, 210-231. Gu, C. (1992), 'Penalized likelihood regression: a Bayesian analysis', Statistica Sinica 2,255-264. Klein, R., Klein, B. & Moss, S. (1995), 'Age-related eye disease and survival. the Beaver Dam Eye Study', Arch Ophthalmol113, 1995. Liu, Y. (1993), Unbiased estimate of generalization error and model selection in neural network, manuscript, Department of Physics, Institute of Brain and Neural Systems, Brown University. Utans, J. & Moody, J. (1993), Selecting neural network architectures via the prediction risk: application to corporate bond rating prediction, in 'Proc. First Int'I Conf. on Artificial Intelligence Applications on Wall Street', IEEE Computer Society Press. Wahba, G. (1990), Spline Models for Observational Data, SIAM. CBMS-NSF Regional Conference Series in Applied Mathematics, v. 59. Wahba, G. (1995), Generalization and regularization in nonlinear learning systems, in M. Arbib, ed., 'Handbook of Brain Theory and Neural Networks', MIT Press, pp. 426430. Wahba, G., Wang, Y., Gu, c., Klein, R. & Klein, B. (1994), Structured machine learning for 'soft' classification with smoothing spline ANOVA and stacked tuning, testing and evaluation, in J. Cowan, G. Tesauro & J. Alspector, eds, 'Advances in Neural Information Processing Systems 6', Morgan Kauffman, pp. 415-422. Wahba, G., Wang, Y., Gu, C., Klein, R. & Klein, B. (1995), 'Smoothing spline AN OVA for exponential families, with application to the Wisconsin Epidemiological Study of Diabetic Retinopathy' , Ann. Statist. 23, 1865-1895. Wang, Y. (1997), 'GRKPACK: Fitting smoothing spline analysis of variance models to data from exponential families', Commun. Statist. Sim. Compo 26,765-782. Wong, W. (1992), Estimation of the loss of an estimate, Technical Report 356, Dept. of Statistics, University of Chicago, Chicago, II. Xiang, D. & Wahba, G. (1996), 'A generalized approximate cross validation for smoothing splines with non-Gaussian data', Statistica Sinica 6, 675-692, preprint TR 930 available via www. stat. wise. edu/-wahba - > TRLIST. Xiang, D. & Wahba, G. (1997), Approximate smoothing spline methods for large data sets in the binary case, Technical Report 982, Department of Statistics, University of Wisconsin, Madison WI. To appear in the Proceedings of the 1997 ASA Joint Statistical Meetings, Biometrics Section, pp 94-98 (1998). Also in TRLIST as above. G, Wahba et aI, 626 CKL ranGACV 10 (0 c:i (0 c:i 0 o c:i c:i (0 10 10 c:i CKL 10 (0 10 10 .' . c:i 0 o 0 c:i 10 ~ -8 -7 -6 -5 log lambda (a) -3 -4 9.29 -8 "f \~7 0\ -7 -7 :. ? ..O-:!4!7 : -5 O. 4 .25 ...... " :0' -6 log lambda1 (c) ranGACV .' O. 7 O. 9 -3 ...0,28 .. ????????r ..... r,,6 ~, -4 -6 -5 log lambda (b) 0 .. .. . .: 0'F5 0'F8 0.[32 0'.2,4 -7 -4 0:\13 ': -6 log lambda1 (d) -4 -5 o C\I (0 c:i (0 c:i .~ =...,. .0 ca O 12! . .0 o e a.. co 10 C\I c:i c:i (0 o 10 c:i ~--------.-------.--------r--~ 0,56 0,58 0,60 ranGACV (e) 0,62 c:i ~ 100 __~____, -____. -__- .____. -__~ 150 200 250 300 Cholesterol (mg/dL) (f) 350 400 Figure 1: (a) and (b): Single smoothing parameter comparison of ranGACV and CK L . (c) and (d): Two smoothing parameter comparison of ranGACV and CK L. (e): Comparison of ranG ACV and U B R. (f): Probability estimate from Beaver Dam Study 

1483 |@word polynomial:1 replicate:2 logit:1 open:2 simulation:5 pressure:1 tr:7 ld:1 initial:1 liu:2 series:3 unx:1 efficacy:1 selecting:2 inefficiency:2 com:1 fn:5 additive:2 ronald:1 numerical:3 chicago:2 girosi:2 designed:1 plot:4 v:1 d5i:1 intelligence:1 selected:1 beaver:3 ith:1 compo:1 provides:2 attack:1 sigmoidal:1 simpler:1 along:1 h4:1 become:2 fitting:1 wuf:3 alspector:1 brain:2 td:3 provided:1 begin:1 campus:1 fti:1 mass:2 null:1 what:1 kind:1 ti:15 xd:2 exactly:1 fat:1 returning:1 control:1 unit:1 appear:4 yn:2 positive:1 walnut:1 modify:1 approximately:1 yd:3 might:2 black:2 twice:1 studied:1 suggests:1 co:1 limited:1 practical:1 testing:1 borderline:1 implement:1 differs:1 definite:2 x3:8 epidemiological:1 procedure:3 dayton:1 weather:2 convenient:2 confidence:1 radial:1 refers:1 doubled:1 get:1 close:1 selection:1 risk:5 influence:1 applying:1 dam:3 wong:2 www:1 go:1 duration:1 convex:1 cholesterol:4 population:1 target:2 exact:4 us:1 continues:1 preprint:1 wang:4 principled:2 disease:1 pd:1 govern:2 complexity:2 depend:1 solving:2 rewrite:1 asa:1 technically:1 basis:4 gacv:8 gu:5 joint:1 various:2 rxl:1 derivation:1 stacked:1 univ:1 epi:1 acv:10 artificial:1 tell:1 choosing:2 outcome:3 quite:1 larger:1 statistic:3 mg:1 propose:2 fr:1 ckl:8 cacv:3 cluster:3 produce:2 comparative:1 depending:1 gong:2 stat:2 measured:1 minor:1 sim:1 sa:3 implemented:2 lambda1:2 attribute:3 f4:1 observational:1 generalization:4 wall:1 preliminary:1 drinking:1 substituting:1 a2:3 favorable:1 estimation:2 proc:1 applicable:1 tik:1 bond:1 grouped:1 wl:2 cary:1 successfully:1 mit:1 fya:2 gaussian:4 ck:5 rather:1 derived:1 likelihood:8 baseline:1 summarizing:1 el:1 minimizers:2 typically:1 lj:1 classification:6 smoothing:17 special:1 having:2 x4:3 represents:3 pai:1 jones:2 mimic:1 future:1 report:2 minimized:1 spline:14 simplex:1 employ:1 randomly:1 tradeofjand:2 replaced:1 hemoglobin:1 consisting:1 interest:1 fd:2 highly:1 hiwii:2 evaluation:1 replicates:2 tj:1 accurate:1 partial:1 poggio:2 biometrics:1 tree:1 filled:3 taylor:2 desired:2 circle:5 fitted:1 uncertain:1 soft:6 ar:1 ordinary:1 cost:1 entry:1 uniform:1 gcv:2 johnson:1 perturbed:2 synthetic:1 grkpack:6 cited:1 randomized:13 siam:1 bu:1 dong:1 physician:1 physic:1 moody:1 ophthalmalogy:1 again:2 woman:1 f5:1 lambda:2 conf:1 derivative:1 li:2 north:1 int:1 inc:1 matter:1 depends:1 vi:1 trh:1 minimize:1 square:2 variance:10 bayesian:2 produced:1 rangacv:16 drive:1 liebler:2 history:1 simultaneous:1 ed:2 pp:3 knowledge:1 hilbert:1 actually:1 cik:1 cbms:1 appears:1 manuscript:1 x6:3 evaluated:1 box:2 done:1 arch:1 until:1 sketch:1 hand:1 ubr:2 replacing:2 nonlinear:2 hwl:2 aj:1 contain:1 true:4 unbiased:2 brown:1 regularization:3 read:1 trlist:2 sin:2 x5:3 percentile:1 criterion:1 generalized:2 confusion:1 tn:1 variational:3 wise:1 ef:3 fi:1 functional:3 physical:1 insensitive:1 association:3 numerically:1 ai:2 cv:4 tuning:3 mathematics:1 aq:1 retinopathy:2 had:1 multivariate:1 showed:1 optimizing:2 commun:1 barbara:1 tesauro:1 binary:1 meeting:1 yi:13 seen:2 minimum:1 morgan:1 ii:5 multiple:3 corporate:1 hormone:1 technical:2 calculation:5 cross:9 raphson:3 lin:1 prediction:3 regression:2 patient:1 essentially:1 iteration:2 kernel:2 interval:1 median:1 biasvariance:1 leaving:2 regional:1 subject:1 thing:1 member:1 cowan:1 spirit:1 near:1 abnormality:1 easy:1 concerned:1 variety:1 fit:10 variate:1 architecture:2 wahba:25 bandwidth:1 rang:7 reduce:1 arbib:1 cn:1 tradeoff:7 br:2 penalty:4 hessian:3 clear:1 nonparametric:1 extensively:1 statist:3 iith:2 nsf:1 dotted:1 utans:2 estimated:1 wr:1 klein:9 demonstrating:1 blood:1 wisc:2 changing:1 anova:5 f8:1 sum:1 glycosylated:1 inverse:2 eli:1 you:1 family:2 draw:1 f13:1 fl:4 hi:1 tackled:1 quadratic:4 x2:7 anyone:1 span:1 uf:2 department:2 structured:1 according:3 remain:1 wi:4 modification:1 dv:1 heart:1 taken:1 computationally:1 equation:1 previously:1 know:1 letting:5 available:1 progression:1 hii:2 original:1 running:1 include:2 clustering:1 remaining:1 madison:3 newton:3 giving:1 uj:1 yz:1 society:1 quantity:1 fa:2 md:2 grace:1 diagonal:1 distance:4 street:3 fy:3 reason:1 assuming:1 systolic:1 index:2 minimizing:1 nc:1 equivalently:1 sinica:2 trace:4 negative:2 unknown:2 observation:2 thence:1 y1:2 reproducing:2 rating:1 pair:1 kl:2 extensive:2 qa:1 kauffman:1 including:1 reliable:1 unpenalized:1 indicator:2 hr:1 representing:1 eye:3 moss:1 l992:1 review:1 xiang:6 wisconsin:3 asymptotic:1 loss:1 suggestion:1 generator:1 validation:8 at1:1 age:3 degree:1 proxy:1 pi:5 elsewhere:2 penalized:8 free:1 ovum:1 bias:7 institute:2 taking:1 author:1 adaptive:1 replicated:1 functionals:1 approximate:5 kullback:2 anew:1 handbook:1 don:1 continuous:1 iterative:2 diabetic:2 ca:1 expansion:1 cl:1 statistica:2 ciq:1 withstanding:1 repeated:1 girard:2 body:2 fig:2 representative:1 tl:1 ff:10 cubic:3 position:1 pv:3 explicit:1 downhill:1 exponential:2 xl:6 weighting:1 hw:1 survival:1 dl:1 exists:1 sequential:1 relearning:1 l993:1 gao:1 ua1:1 minimizer:10 satisfies:2 truth:2 ydi:3 ann:2 replace:2 change:1 included:1 infinite:1 except:1 typical:1 lemma:1 called:2 experimental:1 mark:2 searched:1 xiwu:2 dept:3

The Bias-Variance Tradeoff and the Randomized GACV Grace Wahba, Xiwu Lin and Fangyu Gao Dong Xiang Dept of Statistics Univ of Wisconsin 1210 W Dayton Street Madison, WI 53706 wahba,xiwu,fgao@stat.wisc.edu SAS Institute, Inc. SAS Campus Drive Cary, NC 27513 sasdxx@unx.sas.com Ronald Klein, MD and Barbara Klein, MD Dept of Ophthalmalogy 610 North Walnut Street Madison, WI 53706 kleinr,kleinb@epi.ophth.wisc.edu Abstract We propose a new in-sample cross validation based method (randomized GACV) for choosing smoothing or bandwidth parameters that govern the bias-variance or fit-complexity tradeoff in 'soft' classification. Soft classification refers to a learning procedure which estimates the probability that an example with a given attribute vector is in class 1 vs class O. The target for optimizing the the tradeoff is the Kullback-Liebler distance between the estimated probability distribution and the 'true' probability distribution, representing knowledge of an infinite population. The method uses a randomized estimate of the trace of a Hessian and mimics cross validation at the cost of a single relearning with perturbed outcome data. 1 INTRODUCTION We propose and test a new in-sample cross-validation based method for optimizing the biasvariance tradeoff in 'soft classification' (Wahba et al1994), called ranG ACV (randomized Generalized Approximate Cross Validation) . Summarizing from Wahba et al(l994) we are given a training set consisting of n examples, where for each example we have a vector t E T of attribute values, and an outcome y, which is either 0 or 1. Based on the training data it is desired to estimate the probability p of the outcome 1 for any new examples in the The Bias-Variance TradeofJand the Randomized GACV 621 future. In 'soft' classification the estimate p(t) of p(t) is of particular interest, and might be used by a physician to tell patients how they might modify their risk p by changing (some component of) t, for example, cholesterol as a risk factor for heart attack. Penalized likelihood estimates are obtained for p by assuming that the logit f(t), t E T, which satisfies p(t) = ef(t) 1(1 + ef(t?) is in some space 1{ of functions . Technically 1{ is a reproducing kernel Hilbert space, but you don't need to know what that is to read on. Let the training set be {Yi, ti, i = 1,???, n}. Letting Ii = f(td, the negative log likelihood .c{Yi, ti, fd of the observations, given f is n .c{Yi, ti, fd = 2::[-Ydi + b(li)], (1) i=1 where b(f) = log(l + ef ). The penalized likelihood estimate of the function f is the solution to: Find f E 1{ to minimize h. (I): n h.(f) = 2::[-Ydi + b(ld) + J>.(I), (2) i =1 where 1>.(1) is a quadratic penalty functional depending on parameter(s) A = (AI, ... , Aq) which govern the so called bias-variance tradeoff. Equivalently the components of A control the tradeoff between the complexity of f and the fit to the training data. In this paper we sketch the derivation of the ranG ACV method for choosing A, and present some preliminary but favorable simulation results, demonstrating its efficacy. This method is designed for use with penalized likelihood estimates, but it is clear that it can be used with a variety of other methods which contain bias-variance parameters to be chosen, and for which minimizing the Kullback-Liebler (K L) distance is the target. In the work of which this is a part, we are concerned with A having multiple components. Thus, it will be highly convenient to have an in-sample method for selecting A, if one that is accurate and computationally convenient can be found. Let P>. be the the estimate and p be the 'true' but unknown probability function and let = p(td,p>.i = p>.(ti). For in-sample tuning, our criteria for a good choice of A is Pi the KL distance KL(p,p>.) = ~ E~I[PilogP7. + (1- pdlogg~::?)]. We may replace K L(p,p>.) by the comparative K L distance (C K L), which differs from K L by a quantity which does not depend on A. Letting hi = h (ti), the C K L is given by 1 CKL(p,p>.) == CKL(A) = ;;, n 2:: [-pd>'i + b(l>.i)). (3) i=) C K L(A) depends on the unknown p, and it is desired is to have a good estimate or proxy for it, which can then be minimized with respect to A. It is known (Wong 1992) that no exact unbiased estimate of CK L(A) exists in this case, so that only approximate methods are possible. A number of authors have tackled this problem, including Utans and M90dy(1993), Liu(l993), Gu(1992). The iterative U BR method of Gu(l992) is included in GRKPACK (Wang 1997), which implements general smoothing spline ANOVA penalized likelihood estimates with multiple smoothing parameters. It has been successfully used in a number of practical problems, see, for example, Wahba et al (1994,1995). The present work represents an approach in the spirit of GRKPACK but which employs several approximations, and may be used with any data set, no matter how large, provided that an algorithm for solving the penalized likelihood equations, either exactly or approximately, can be implemented. 622 2 G. Wahba et al. THE GACV ESTIMATE In the general penalized likelihood problem the minimizer 1>,(-) of (2) has a representation M 1>.(t) =L n dv<Pv(t) v=l + L CiQ>.(ti, t) (4) i=l where the <Pv span the null space of 1>" Q>.(8, t) is a reproducing kernel (positive definite function) for the penalized part of 7-1., and C = (Cl' ... , Cn )' satisfies M linear conditions, so that there are (at most) n free parameters in 1>.. Typically the unpenalized functions <Pv are low degree polynomials. Examples of Q(ti,') include radial basis functions and various kinds of splines; minor modifications include sigmoidal basis functions, tree basis functions and so on. See, for example Wahba( 1990, 1995), Girosi, Jones and Poggio( 1995). If f>.C) is of the form (4) then 1>,(f>.) is a quadratic form in c. Substituting (4) into (2) results in h a convex functional in C and d, and C and d are obtained numerically via a Newton Raphson iteration, subject to the conditions on c. For large n, the second sum on the right of (4) may be replaced by L~=1 Cik Q>. (ti k , t), where the tik are chosen via one of several principled methods. To obtain the CACV we begin with the ordinary leaving-out-one cross validation function CV(.\) for the CKL: n [-i] CV (.\) -_ -1 "" LJ-yd>.i n + b( 1>.i) ] , (5) i=1 where fl- i ] the solution to the variational problem of (2) with the ith data point left out and fti] is the value of fl- i] at ti . Although f>.C) is computed by solving for C and d the CACV is derived in terms of the values (it"", fn)' of f at the ti. Where there is no confusion between functions f(-) and vectors (it, ... ,fn)' of values of fat tl, ... ,tn, we let f = (it, ... " fn)'. For any f(-) of the form (4), J>. (f) also has a representation as a non-negative definite quadratic form in (it, ... , fn)'. Letting L:>. be twice the matrix of this quadratic form we can rewrite (2) as 1 n h(f,Y) = L[-Ydi + b(/i)] + 2f'L:>.f. (6) i=1 Let W = W(f) be the n x n diagonal matrix with (/ii == Pi(l - Pi) in the iith position. Using the fact that (/ii is the second derivative of b(fi), we have that H = [W + L:>.] - 1 is the inverse Hessian of the variational problem (6). In Xiang and Wahba (1996), several Taylor series approximations, along with a generalization of the leaving-out-one lemma (see Wahba 1990) are applied to (5) to obtain an approximate cross validation function ACV(.\), which is a second order approximation to CV(.\) . Letting hii be the iith entry of H , the result is CV(.\) ~ ACV('\) = .!. t[-Yd>.i + b(f>.i)] + .!. t n n i= l i=1 hiiYi(Yi - P>.i) . [1 - hiwii] Then the GACV is obtained from the ACV by replacing h ii by ~ L~1 hii and replacing 1 - hiWii by ~tr[I - (Wl/2 HWl/2)], giving CACV('\) 1~ ] tr(H) == ~tr(H) L~l Yi(Yi - P>.i) (Wl/2HWl /2)] , = ;; t;;[-Yd>.i + b(1).i) + -n-tr[I _ (7) (8) where W is evaluated at 1>.. Numerical results based on an exact calculation of (8) appear in Xiang and Wahba (1996). The exact calculation is limited to small n however. 623 The Bias-Variance TradeofJand the Randomized GACV 3 THE RANDOMIZED GACV ESTIMATE Given any 'black box' which, given >., and a training set {Yi, ti} produces f>. (.) as the minimizer of (2), and thence f>. = (fA 1 , " ' , f>.n)', we can produce randomized estimates of trH and tr[! - W 1 / 2 HW 1/ 2 J without having any explicit calculations of these matrices. This is done by running the 'black box' on perturbed data {Vi + <5i , td. For the Yi Gaussian, randomized trace estimates of the Hessian of the variational problem (the 'influence matrix') have been studied extensively and shown to be essentially as good as exact calculations for large n, see for example Girard(1998) . Randomized trace estimates are based on the fact that if A is any square matrix and <5 is a zero mean random n-vector with independent components with variance then E<5' A<5 = ~ tr A. See Gong et al( 1998) and (TJ, u" references cited there for experimental results with multiple regularization parameters. Returning to the 0-1 data case, it is easy to see that the minimizer fA(') of 1;.. is continuous in Y, not withstanding the fact that in our training set the Yi take on only values 0 or 1. Letting if = UA1,"', f>.n)' be the minimizer of (6) given y = (Y1,"', Yn)', and if+O be the minimizer given data y+<5 = (Y1 +<51, ... ,Yn +<5n )' (the ti remain fixed), Xiang and Wahba (1997) show, again using Taylor series expansions, that if+O - ff ,. . ., [WUf) + ~AJ-1<5. This suggests that ~<5'Uf+O - ff) provides an estimate oftr[W(ff) + ~At1. However, u" if we take the solution ff to the nonlinear system for the original data Y as the initial value for a Newton-Raphson calculation of ff+O things become even simpler. Applying a one step Newton-Raphson iteration gives (9) -<5 + PjfUf,Y) = -<5, and [:;~f(ff,Y + <5)J - 1 y + o ,l y + o ,l 82 h(fY 8 2 h(fY [ 8?7if A' Y)J- 1 ,we have f A - fYA + [8?7if A' Y)J- 1 uJ: so that f A - fYA [WUf) + E At 1<5. The result is the following ranGACV function: Since Pjf(ff,y + <5) ranGACV(>.) = = n .!. ~[n~ 'I '+bU .)J+ Yz At At <5' (f Y+O,l A n - fY) A ",n ( ) wi=l Yi Yi - PAi . [<5'<5 - <5'WUf)Uf+O,l - ff)J (10) To reduce the variance in the term after the '+' in (10), we may draw R independent replicate vectors <51,'" ,<5R , and replace the term after the '+' in ( 1O)b 1... ",R o:(fr Hr . 1 -ff) wr=l n ranGACV(>.) function. 4 y R 2:7-1y.(y.-P>..) [O~Or-O~ W(fn(f~+Ar . l-ff)1 to obtain an R-replicated NUMERICAL RESULTS In this section we present simulation results which are representative of more extensive simulations to appear elsewhere. In each case, K < < n was chosen by a sequential clustering algorithm. In that case, the ti were grouped into K clusters and one member of each cluster selected at random. The model is fit. Then the number of clusters is doubled and the model is fit again. This procedure continues until the fit does not change. In the randomized trace estimates the random variates were Gaussian. Penalty functionals were (multivariate generalizations of) the cubic spline penalty functional>. fa1U" (X))2, and smoothing spline ANOVA models were fit. G. Wahba et at. 624 4.1 EXPERIMENT 1. SINGLE SMOOTHING PARAMETER In this experiment t E [0,1], f(t) = 2sin(10t), ti = (i - .5)/500, i = 1,???,500. A random number generator produced 'observations' Yi = 1 with probability Pi = el , /(1 + eli), to get the training set. QA is given in Wahba( 1990) for this cubic spline case, K = 50. Since the true P is known, the true CKL can be computed. Fig. l(a) gives a plot of CK L(A) and 10 replicates of ranGACV(A). In each replicate R was taken as 1, and J was generated anew as a Gaussian random vector with (115 = .001. Extensive simulations with different (115 showed that the results were insensitive to (115 from 1.0 to 10- 6 ? The minimizer of C K L is at the filled-in circle and the 10 minimizers of the 10 replicates of ranGACV are the open circles. Anyone of these 10 provides a rather good estimate of the A that goes with the filled-in circle. Fig. l(b) gives the same experiment, except that this time R = 5. It can be seen that the minimizers ranGACV become even more reliable estimates of the minimizer of C K L, and the C K L at all of the ranG ACV estimates are actually quite close to its minimum value. 4.2 EXPERIMENT 2. ADDITIVE MODEL WITH A = (Al' A2) Here t E [0,1] 0 [0,1]. n = 500 values of ti were generated randomly according to a uniform distribution on the unit square and the Yi were generated according to Pi = e li j(l + e l ,) with t = (Xl,X2) and f(t) = 5 sin 27rXl - 3sin27rX2. An additive model as a special case of the smoothing spline ANOVA model (see Wahba et al, 1995), of the form f(t) = /-l + h(xd + h(X2) with cubic spline penalties on hand h were used. K = 50, (115 = .001, R = 5. Figure l(c) gives a plot of CK L(Al' A2) and Figure l(d) gives a plot of ranGACV(Al, A2). The open circles mark the minimizer of ranGACV in both plots and the filled in circle marks the minimizer of C K L. The inefficiency, as measured by CKL()..)/minACKL(A) is 1.01. Inefficiencies near 1 are typical of our other similar simulations. 4.3 EXPERIMENT 3. COMPARISON OF ranGACV AND UBR This experiment used a model similar to the model fit by GRKPACK for the risk of progression of diabetic retinopathy given t = (Xl, X2, X3) = (duration, glycosylated hemoglobin, body mass index) in Wahba et al(l995) as 'truth'. A training set of 669 examples was generated according to that model, which had the structure f(Xl, X2, X3) = /-l + fl (xd + h (X2) + h (X3) + fl,3 (Xl, X3). This (synthetic) training set was fit by GRKPACK and also using K = 50 basis functions with ranG ACV. Here there are P = 6 smoothing parameters (there are 3 smoothing parameters in f13) and the ranGACV function was searched by a downhill simplex method to find its minimizer. Since the 'truth' is known, the CKL for)" and for the GRKPACK fit using the iterative UBR method were computed. This was repeated 100 times, and the 100 pairs of C K L values appears in Figure l(e). It can be seen that the U BR and ranGACV give similar C K L values about 90% of the time, while the ranG ACV has lower C K L for most of the remaining cases. 4.4 DATA ANALYSIS: AN APPLICATION Figure 1(f) represents part of the results of a study of association at baseline of pigmentary abnormalities with various risk factors in 2585 women between the ages of 43 and 86 in the Beaver Dam Eye Study, R. Klein et al( 1995). The attributes are: Xl = age, X2 =body mass index, X3 = systolic blood pressure, X4 = cholesterol. X5 and X6 are indicator variables for taking hormones, and history of drinking. The smoothing spline ANOVA model fitted was f(t) = /-l+dlXl +d2X2+ h(X3)+ f4(X4)+ h4(X3, x4)+d5I(x5) +d6I(x6), where I is the indicator function. Figure l(e) represents a cross section of the fit for X5 = no, X6 = no, The Bias- Variance Tradeoff and the Randomized GACV 625 X2, X3 fixed at their medians and Xl fixed at the 75th percentile. The dotted lines are the Bayesian confidence intervals, see Wahba et al( 1995). There is a suggestion of a borderline inverse association of cholesterol. The reason for this association is uncertain. More details will appear elsewhere. Principled soft classification procedures can now be implemented in much larger data sets than previously possible, and the ranG ACV should be applicable in general learning. References Girard, D. (1998), 'Asymptotic comparison of (partial) cross-validation, GCV and randomized GCV in nonparametric regression', Ann. Statist. 126, 315-334. Girosi, F., Jones, M. & Poggio, T. (1995), 'Regularization theory and neural networks architectures', Neural Computatioll 7,219-269. Gong, J., Wahba, G., Johnson, D. & Tribbia, J. (1998), 'Adaptive tuning of numerical weather prediction models: simultaneous estimation of weighting, smoothing and physical parameters', MOllthly Weather Review 125, 210-231. Gu, C. (1992), 'Penalized likelihood regression: a Bayesian analysis', Statistica Sinica 2,255-264. Klein, R., Klein, B. & Moss, S. (1995), 'Age-related eye disease and survival. the Beaver Dam Eye Study', Arch Ophthalmol113, 1995. Liu, Y. (1993), Unbiased estimate of generalization error and model selection in neural network, manuscript, Department of Physics, Institute of Brain and Neural Systems, Brown University. Utans, J. & Moody, J. (1993), Selecting neural network architectures via the prediction risk: application to corporate bond rating prediction, in 'Proc. First Int'I Conf. on Artificial Intelligence Applications on Wall Street', IEEE Computer Society Press. Wahba, G. (1990), Spline Models for Observational Data, SIAM. CBMS-NSF Regional Conference Series in Applied Mathematics, v. 59. Wahba, G. (1995), Generalization and regularization in nonlinear learning systems, in M. Arbib, ed., 'Handbook of Brain Theory and Neural Networks', MIT Press, pp. 426430. Wahba, G., Wang, Y., Gu, c., Klein, R. & Klein, B. (1994), Structured machine learning for 'soft' classification with smoothing spline ANOVA and stacked tuning, testing and evaluation, in J. Cowan, G. Tesauro & J. Alspector, eds, 'Advances in Neural Information Processing Systems 6', Morgan Kauffman, pp. 415-422. Wahba, G., Wang, Y., Gu, C., Klein, R. & Klein, B. (1995), 'Smoothing spline AN OVA for exponential families, with application to the Wisconsin Epidemiological Study of Diabetic Retinopathy' , Ann. Statist. 23, 1865-1895. Wang, Y. (1997), 'GRKPACK: Fitting smoothing spline analysis of variance models to data from exponential families', Commun. Statist. Sim. Compo 26,765-782. Wong, W. (1992), Estimation of the loss of an estimate, Technical Report 356, Dept. of Statistics, University of Chicago, Chicago, II. Xiang, D. & Wahba, G. (1996), 'A generalized approximate cross validation for smoothing splines with non-Gaussian data', Statistica Sinica 6, 675-692, preprint TR 930 available via www. stat. wise. edu/-wahba - > TRLIST. Xiang, D. & Wahba, G. (1997), Approximate smoothing spline methods for large data sets in the binary case, Technical Report 982, Department of Statistics, University of Wisconsin, Madison WI. To appear in the Proceedings of the 1997 ASA Joint Statistical Meetings, Biometrics Section, pp 94-98 (1998). Also in TRLIST as above. G, Wahba et aI, 626 CKL ranGACV 10 (0 c:i (0 c:i 0 o c:i c:i (0 10 10 c:i CKL 10 (0 10 10 .' . c:i 0 o 0 c:i 10 ~ -8 -7 -6 -5 log lambda (a) -3 -4 9.29 -8 "f \~7 0\ -7 -7 :. ? ..O-:!4!7 : -5 O. 4 .25 ...... " :0' -6 log lambda1 (c) ranGACV .' O. 7 O. 9 -3 ...0,28 .. ????????r ..... r,,6 ~, -4 -6 -5 log lambda (b) 0 .. .. . .: 0'F5 0'F8 0.[32 0'.2,4 -7 -4 0:\13 ': -6 log lambda1 (d) -4 -5 o C\I (0 c:i (0 c:i .~ =...,. .0 ca O 12! . .0 o e a.. co 10 C\I c:i c:i (0 o 10 c:i ~--------.-------.--------r--~ 0,56 0,58 0,60 ranGACV (e) 0,62 c:i ~ 100 __~____, -____. -__- .____. -__~ 150 200 250 300 Cholesterol (mg/dL) (f) 350 400 Figure 1: (a) and (b): Single smoothing parameter comparison of ranGACV and CK L . (c) and (d): Two smoothing parameter comparison of ranGACV and CK L. (e): Comparison of ranG ACV and U B R. (f): Probability estimate from Beaver Dam Study Graph Matching for Shape Retrieval Benoit Huet, Andrew D.J. Cross and Edwin R. Hancock' Department of Computer Science, University of York York, YOI 5DD, UK Abstract This paper describes a Bayesian graph matching algorithm for data-mining from large structural data-bases. The matching algorithm uses edge-consistency and node attribute similarity to determine the a posteriori probability of a query graph for each of the candidate matches in the data-base. The node feature-vectors are constructed by computing normalised histograms of pairwise geometric attributes. Attribute similarity is assessed by computing the Bhattacharyya distance between the histograms. Recognition is realised by selecting the candidate from the data-base which has the largest a posteriori probability. We illustrate the recognition technique on a data-base containing 2500 line patterns extracted from real-world imagery. Here the recognition technique is shown to significantly outperform a number of algorithm alternatives. 1 Introduction Since Barrow and Popplestone [1] first suggested that relational structures could be used to represent and interpret 2D scenes, there has been considerable interest in the machine vision literature in developing practical graph-matching algorithms [8, 3, 10]. The main computational issues are how to compare relational descriptions when there is significant structural corruption [8, 10] and how to search for the best match [3]. Despite resulting in significant improvements in the available methodology for graph-matching, there has been little progress in applying the resulting algorithms to large-scale object recognition problems. Most of the algorithms developed in the literature are evaluated for the relatively simple problem of matching a model-graph against a scene known to contain the relevant structure. A more realistic problem is that of taking a large number (maybe thousands) of scenes and retrieving the ones that best match the model. Although this problem is key to data-mining from large libraries of visual information, it has invariably been approached using low-level feature comparison techniques. Very little effort [7,4] has been devoted to matching ? corresponding author erh@cs.york.ac.uk 897 Graph Matching for Shape Retrieval higher-level structural primitives such as lines, curves or regions. Moreover, because of the perceived fragility of the graph matching process, there has been even less effort directed at attempting to retrieve shapes using relational information. Here we aim to fill this gap in the literature by using graph-matching as a means of retrieving the shape from a large data-based that most closely resembles a query shape. Although the indexation images in large data-bases is a problem of current topicality in the computer vision literature [5, 6, 9], the work presented in this paper is more ambitious. Firstly, we adopt a structural abstraction of the shape recognition problem and match using attributed relational graphs. Each shape in our data-base is a pattern of line-segments. The structural abstraction is a nearest neighbour graph for the centre-points of the line-segments. In addition, we exploit attribute information for the line patterns. Here the geometric arrangement of the line-segments is encapsulated using a histogram of Euclidean invariant pairwise (binary) attributes. For each line-segment in turn we construct a normalised histogram of relative angle and length with the remaining line-segments in the pattern. These histograms capture the global geometric context of each line-segment. Moreover, we interpret the pairwise geometric histograms as measurement densities for the line-segments which we compare using the Bhattacharyya distance. Once we have established the pattern representation, we realise object recognition using a Bayesian graph-matching algorithm. This is a two-step process. Firstly, we establish correspondence matches between the individual tokens in the query pattern and each of the patterns in the data-base. The correspondences matches are sought so as to maximise the a posteriori measurement probability. Once the MAP correspondence matches have been established, then the second step in our recognition architecture involves selecting the line-pattern from the data-base which has maximum matching probability. 2 MAP Framework Formally our recognition problem is posed as follows. Each ARG in the database is a triple, G = (Vc, Ec, Ac), where Vc is the set of vertices (nodes), Ec is the edge set (Ec C Vc x Vc), and Ac is the set of node attributes. In our experimental example, the nodes represent line-structures segmented from 2D images. The edges are established by computing the N-nearest neighbour graph for the line-centres. Each node j E Vc is characterised by a vector of attributes, ~j and hence Ac = {~j jj E Vc}. In the work reported here the attribute-vector is represents the contents of a normalised pairwise attribute histogram . The data-base of line-patterns is represented by the set of ARG's D = {G}. The goal is to retrieve from the data-base D, the individual ARG that most closely resembles a query pattern Q = (VQ' EQ, AQ). We pose the retrieval process as one of associating with the query the graph from the data-base that has the largest a posteriori probability. In other words, the class identity of the graph which most closely corresponds to the query is wQ = arg max P(G' IQ) C'EV However, since we wish to make a detailed structural comparison of the graphs, rather than comparing their overall statistical properties, we must first establish a set of best-match correspondences between each ARG in the data-base and the query Q. The set of correspondences between the query Q and the ARG G is a relation fc : Vc f-7 VQ over the vertex sets of the two graphs. The mapping function consists of a set of Cartesian pairings between the nodes of the two graphs, B. Huet, A. D. 1. Cross and E. R. Hancock 898 i.e. Ie = {(a,a);a E Ve,a E VQ} ~ Ve x VQ . Although this may appear to be a brute force method, it must be stressed that we view this process of correspondence matching as the final step in the filtering of the line-patterns. We provide more details of practical implementation in the experimental section of this paper. With the correspondences to hand we can re-state our maximum a posteriori probability recognition objective as a two step process. For each graph G in turn, we locate the maximum a posteriori probability mapping function Ie onto the query Q. The second step is to perform recognition by selecting the graph whose mapping function results in the largest matching probability. These two steps are succinctly captured by the following statement of the recognition condition wQ = arg max max P(fe,IG', Q) e'ED la' This global MAP condition is developed into a useful local update formula by applying the Bayes formula to the a posteriori matching probability. The simplification is as follows PU IG Q) e , = p(Ae, AQl/e)P(felVe, Ee, VQ, EQ)P(Ve , Ee)P(VQ, EQ) P(G)P(Q) The terms on the right-hand side of the Bayes formula convey the following meaning. The conditional measurement density p(Ae,AQl/e) models the measurement similarity of the node-sets of the two graphs. The conditional probability P(feIEe, EQ) models the structural similarity of the two graphs under the current set of correspondence matches. The assumptions used in developing our simplification of the a posteriori matching probability are as follows. Firstly, we assume that the joint measurements are conditionally independent of the structure of the two graphs provided that the set of correspondences is known, i.e. P(Ae, AQl/e, Ee, Ve, E Q, VQ) = P(Ae, AQl/e). Secondly, we assume that there is conditional independence of the two graphs in the absence of correspondences. In other words, P(Ve, Ee, VQ, EQ) = P(VQ, EQ)P(Ve, Ee) and P(G, Q) = P(G)P(Q). Finally, the graph priors P(Ve, Ee) , P(VQ, EQ) P(G) and P( Q) are taken as uniform and are eliminated from the decision making process. To continue our development, we first focus on the conditional measurement density, p(Ae, AQl/e) which models the process of comparing attribute similarity on the nodes of the two graphs. Assuming statistical independence of node attributes, the conditional measurement density p( Ae, AQ lie), can be factorised over the Cartesian pairs (a, a) E Ve x VQ which constitute the the correspondence match Ie in the following manner p(Ae, AQl/e) = II P(~a' ~ol/e(a) = a) (a,o)E/a As a result the correspondence matches may be optimised using a simple node-bynode discrete relaxation procedure. The rule for updating the match assigned to the node a of the graph G is le(a) = arg max oEVQU{4>} P(~a'~o)l/(a) = a)P(feIEe,EQ) In order to model the structural consistency of the set of assigned matches,we turn to the framework recently reported by Finch, Wilson and Hancock [2}. This work provides a framework for computing graph-matching energies using the weighted Hamming distance between matched cliques. Since we are dealing with a large-scale object recognition system, we would like to minimise the computational overheads associated with establishing correspondence matches. For this reason, rather than 899 Graph Matchingfor Shape Retrieval working with graph neighbourhoods or cliques, we chose to work with the relational units of the smallest practical size. In other words we satisfy ourself with measuring consistency at the edge level. For edge-units, the structural matching probability P(fa!Va, Ea, VQ, EQ) is computed from the formula (a,b)EEG (Ct ,(J )EEQ where Pe is the probability of an error appearing on one of the edges of the matched structure. The Sa,Ct are assignment variables which are used to represent the current state of match and convey the following meaning Sa 3 Ct , = {I 0 if fa (a) = a otherwise Histogram-based consistency We now furnish some details of the shape retrieval task used in our experimental evaluation of the recognition method. In particular, we focus on the problem of recognising 2D line patterns in a manner which is invariant to rotation, translation and scale. The raw information available for each line segment are its orientation (angle with respect to the horizontal axis) and its length (see figure 1). To illustrate how the Euclidean invariant pairwise feature attributes are computed, suppose that we denote the line segments associated with the nodes indexed a and b by the vectors Ya and Yb respectively. The vectors are directed away from their point of intersection. The pairwise relative angle attribute is given by (Ja ,b = arccos [I:: 1?1::1] From the relative angle we compute the directed relative angle. This involves giving d ~:.~~: ---------c:----;-~:--- b-! ~------. o..b ---------------. D;b Figure 1: Geometry for shape representation the relative angle a positive sign if the direction of the angle from the baseline Ya to its partner Yb is clockwise and a negative sign if it is counter-clockwise. This allows us to extend the range of angles describing pairs of segments from [0,7I"J to [-7I",7I"J. The directed relative position {}a,b is represented by the normalised length ratio between the oriented baseline vector Ya and the vector yl joining the end (b) of the baseline segment (ab) to the intersection of the segment pair (cd). {}a,b = 1 D l+~ 2 Dab B. Huet, A. D. 1. Cross and E. R. Hancock 900 The physical range of this attribute is (0, IJ. A relative position of 0 indicates that the two segments are parallel, while a relative position of 1 indicates that the two segments intersect at the middle point of the baseline. The Euclidean invariant angle and position attributes 8a,b and {)a ,b are binned in a histogram. Suppose that Sa(J-l, v) = {(a , b)18a,b E All 1\ {)a,b E Rv 1\ bE VD} is the set of nodes whose pairwise geometric attributes with the node a are spanned by the range of directed relative angles All and the relative position attribute range Rv. The contents of the histogram bin spanning the two attribute ranges is given by Ha(J-l, v) = ISa(J-l, v)l. Each histogram contains nA relative angle bins and nR length ratio bins. The normalised geometric histogram bin-entries are computed as follows Ha(J-l, v) ha(J-l, v) = "nA "nR H ( ) ~Il'=l ~v'=l a J-l, v The probability of match between the pattern-vectors is computed using the Bhattacharyya distance between the normalised histograms. P(f(a) = al~a' ~a) = I:~~l I:~~l ha(J-l, v)ha(J-l, v) L j'EQ I:nA I:nR h ( )h ( ) Il'=l v'=l a J-l, V a J-l, V = exp[-Ba ,aJ With this modelling ingredient , the condition for recognition is WQ 4 = arg~~% L L (a , b}EE~ (a,iJ}EEQ {-Ba,a-Bb,iJ+ln(I-Pe)Sa,aSb,iJ+lnPe(I-Sa,aSb,/3)} Experiments The aim in this section is to evaluate the graph-based recognition scheme on a database of real-world line-patterns. We have conducted our recognition experiments with a data-base of 2500 line-patterns each containing over a hundred lines. The line-patterns have been obtained by applying line/edge detection algorithms to the raw grey-scale images followed by polygonisation. For each line-pattern in the database, we construct the six-nearest neighbour graph . The feature extraction process together with other details of the data used in our study are described in recent papers where we have focussed on the issues of histogram representation [4J and the optimal choice of the relational structure for the purposes of recognition. In order to prune the set of line-patterns for detailed graph-matching we select about 10% of the data-base using a two-step process. This consists of first refining the data-base using a global histogram of pairwise attributes [4J . The top quartile of matches selected in this way are then further refined using a variant of the Haussdorff distance to select the set of pairwise attributes that best match against the query. The recognition task is posed as one of recovering the line-pattern which most closely resembles a digital map . The original images from which our line-patterns have been obtained are from a number of diverse sources. However , a subset of the images are aerial infra-red line-scan views of southern England. Two of these infra-red images correspond to different views of the area covered by the digital map. These views are obtained when the line-scan device is flying at different altitudes. The line-scan device used to obtain the aerial images introduces severe barrel distortions and hence the map and aerial images are not simply related via a Euclidean or affine transformation. The remaining line-patterns in the data-base have been extracted from trademarks and logos. It is important to stress that although the raw images are obtained from different sources, there is nothing salient about their associated line-pattern representations that allows us to distinguish them from one-another. Graph Matchingfor Shape Retrieval (a) Digital Map 901 (b) Target 1 (c) Target 2 Figure 2: Images from the data-base Moreover, since it is derived from a digital map rather than one of the images in the data-base, the query is not identical to any of the line-patterns in the model library. We aim to assess the importance of different attributes representation on the retrieval process. To this end, we compare node-based and the histogram-based attribute representation. \Ve also consider the effect of taking the relative angle and relative position attributes both singly and in tandem. The final aspect of the comparison is to consider the effects of using the attributes purely for initialisation purposes and also in a persistent way during the iteration of the matching process. To this end we consider the following variants of our algorithm . ? Non-Persistent Attributes: Here we ignore the attribute information provided by the node-histograms after the first iteration and attempt to maximise the structural congruence of the graphs . ? Local attributes: Here we use only the single node attributes rather than an attribute histogram to model the a posteriori matching probabilities. Graph Matching Strategy ReI. Position Attribute iInitialisation only) ReI. Angle Attribute (Initialisation only) ReI. Angle + Position Attributes (Initialisation only) 1D ReI. Position Histogram (Initialisation only) 1D ReI. Angle Histogram (Initialisation only) 2D Histogram (Initialisation only) ReI. Position Attribute (Persistent) ReI. Angle Attribute (Persistent) ReI. Angle + Position Attributes (Persistent) 1D ReI. Position Histogram (Persistent) 1D ReI. Angle Histogram (Persistent) 2D Histogram (Persistent) Retrieval Accuracy 39% 45% 58% 42% 59% 68% 63% 89% 98% 66% 92% 100% Iterations per recall 5.2 4.75 4.27 4.7 4.2 3.9 3.96 3.59 3.31 3.46 3.23 3.12 Table 1: Recognition performance of various recognition strategies averaged over 26 queries in a database of 260 line-patterns In Table 1 we present the recognition performance for each of the recognition strategies in turn. The table lists the recall performance together with the average number B. Huet, A. D. 1. Cross and E. R. Hancock 902 of iterations per recall for each of the recognition strategies in turn. The main features to note from this table are as follows . Firstly, the iterative recall using the full histogram representation outperforms each of the remaining recognition methods in terms of both accuracy and computational overheads. Secondly, it is interesting to compare the effect of using the histogram in the initialisation-only and iteration persistent modes. In the latter case the recall performance is some 32% better than in the former case. In the non-persistent mode the best recognition accuracy that can be obtained is 68%. Moreover, the recall is typically achieved in only 3.12 iterations as opposed to 3.9 (average over 26 queries on a database of 260 images) . Finally, the histogram representation provides better performance, and more significantly, much faster recall than the single-attribute similarity measure. When the attributes are used singly, rather than in tandem , then it is the relative angle that appears to be the most powerful. 5 Conclusions We have presented a practical graph-matching algorithm for data-mining in large structural libraries. The main conclusion to be drawn from this study is that the combined use of structural and histogram information improves both recognition performance and recall speed. There are a number of ways in which the ideas presented in this paper can be extended. Firstly, we intend to explore more a perceptually meaningful representation of the line patterns, using grouping principals derived from Gestalt psychology. Secondly, we are exploring the possibility of formulating the filtering of line-patterns prior to graph matching using Bayes decision trees. References [1] H. Barrow and R. Popplestone. Relational descriptions in picture processing. Machine Intelligence, 5:377- 396, 1971. [2] A. Finch, R. Wilson, and E. Hancock. Softening discrete relaxation . Advances in NIPS 9, Edited by M. Mozer, M. Jordan and T. Petsche, MIT Press, pages 438- 444, 1997. [3] S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. IEEE PAMI, 18:377- 388, 1996. [4] B. Huet and E. Hancock. Relational histograms for shape indexing. IEEE ICC V, pages 563- 569, 1998. [5] W. Niblack et al.. The QBIC project: Querying images by content using color , texture and shape. Image and Vision Storage and Retrieval, 173- 187, 1993. [6] A. P. Pentland, R. W. Picard, and S. Scarloff. Photobook: tools for contentbased manipulation of image databases. Storage and Retrieval for Image and Video Database II, pages 34- 47, February 1994. [7] K. Sengupta and K. Boyer. Organising large structural databases. IEEE PAMI, 17(4):321- 332,1995. [8] 1. Shapiro and R. Haralick. A metric for comparing relational descriptions. IEEE PAMI, 7(1):90- 94, 1985. [9] M. Swain and D. Ballard. Color indexing. International Journal of Computer Vision, 7(1) :11- 32, 1991. [10] R. Wilson and E. R. Hancock. Structural matching by discrete relaxation. IEEE PAMI, 19(6):634- 648, June 1997. 

1484 |@word middle:1 polynomial:1 replicate:2 logit:1 open:2 grey:1 simulation:5 pressure:1 tr:7 ld:1 initial:1 liu:2 series:3 unx:1 efficacy:1 selecting:5 inefficiency:2 contains:1 initialisation:7 bhattacharyya:3 outperforms:1 current:3 com:1 comparing:3 must:2 ronald:1 fn:5 numerical:3 additive:2 chicago:2 girosi:2 shape:13 realistic:1 designed:1 plot:4 update:1 v:1 d5i:1 intelligence:2 selected:2 device:2 beaver:3 ith:1 compo:1 provides:4 node:18 organising:1 attack:1 sigmoidal:1 simpler:1 firstly:5 along:1 h4:1 constructed:1 become:2 retrieving:2 pairing:1 consists:2 persistent:10 fitting:1 overhead:2 wuf:3 manner:2 pairwise:9 alspector:1 brain:2 ol:1 td:3 little:2 tandem:2 provided:3 begin:1 campus:1 fti:1 moreover:4 mass:2 matched:2 null:1 what:1 barrel:1 kind:1 developed:2 transformation:1 ti:15 xd:2 exactly:1 fat:1 returning:1 uk:2 control:1 unit:3 brute:1 appear:5 yn:2 positive:2 maximise:2 walnut:1 local:2 modify:1 despite:1 joining:1 establishing:1 optimised:1 approximately:1 yd:3 might:2 black:2 twice:1 topicality:1 studied:1 resembles:3 chose:1 suggests:1 logo:1 co:1 limited:1 range:5 averaged:1 popplestone:2 practical:5 directed:5 testing:1 borderline:1 implement:1 differs:1 definite:2 x3:8 epidemiological:1 procedure:4 dayton:1 intersect:1 area:1 significantly:2 weather:2 convenient:2 matching:26 confidence:1 radial:1 refers:1 word:3 doubled:1 get:1 close:1 selection:1 onto:1 storage:2 risk:5 influence:1 applying:4 dam:3 wong:2 www:1 context:1 map:8 go:1 primitive:1 duration:1 convex:1 rule:1 cholesterol:4 fill:1 spanned:1 retrieve:2 population:1 target:4 suppose:2 exact:4 us:2 recognition:26 updating:1 continues:1 database:8 preprint:1 wang:4 capture:1 thousand:1 region:1 counter:1 edited:1 principled:2 disease:1 pd:1 govern:2 complexity:2 mozer:1 depend:1 solving:2 rewrite:1 segment:14 asa:1 technically:1 flying:1 purely:1 basis:4 gacv:8 gu:5 edwin:1 joint:2 various:3 rxl:1 represented:2 derivation:1 stacked:1 univ:1 epi:1 acv:10 hancock:8 artificial:1 query:13 tell:1 approached:1 choosing:2 outcome:3 refined:1 quite:1 whose:2 larger:1 posed:2 distortion:1 otherwise:1 dab:1 statistic:3 final:2 mg:1 aql:6 propose:2 fr:1 ckl:8 relevant:1 gold:1 description:3 cacv:3 cluster:3 rangarajan:1 produce:2 comparative:1 object:3 depending:1 andrew:1 illustrate:2 stat:2 gong:2 pose:1 nearest:3 ac:4 measured:1 ij:4 sim:1 eq:10 minor:1 implemented:2 c:1 lambda1:2 involves:2 recovering:1 progress:1 sa:8 direction:1 rei:10 closely:4 attribute:40 f4:1 quartile:1 vc:7 observational:1 bin:4 ja:1 generalization:4 wall:1 preliminary:1 secondly:3 exploring:1 drinking:1 asb:2 exp:1 congruence:1 mapping:3 substituting:1 sought:1 adopt:1 a2:3 smallest:1 purpose:2 perceived:1 encapsulated:1 favorable:1 proc:1 applicable:1 tik:1 bond:1 estimation:2 grouped:1 wl:2 largest:3 cary:1 successfully:1 tool:1 weighted:1 mit:2 fya:2 gaussian:4 indexation:1 aim:3 ck:5 rather:6 wilson:3 derived:3 focus:2 refining:1 haralick:1 improvement:1 june:1 modelling:1 likelihood:8 indicates:2 baseline:5 summarizing:1 posteriori:9 abstraction:2 el:1 minimizers:2 typically:2 lj:1 relation:1 boyer:1 issue:2 classification:6 orientation:1 overall:1 arg:9 development:1 arccos:1 smoothing:17 special:1 sengupta:1 construct:2 once:2 having:2 extraction:1 eliminated:1 x4:3 represents:4 pai:1 jones:2 identical:1 mimic:1 future:1 minimized:1 spline:14 simplex:1 report:2 employ:1 infra:2 randomly:1 neighbour:3 oriented:1 ve:9 individual:2 tradeofjand:2 replaced:1 hemoglobin:1 consisting:1 geometry:1 ab:1 attempt:1 detection:1 invariably:1 interest:2 fd:2 highly:1 mining:3 possibility:1 picard:1 hiwii:2 evaluation:2 replicates:2 benoit:1 introduces:1 severe:1 tj:1 devoted:1 accurate:1 edge:7 partial:1 poggio:2 biometrics:1 tree:2 filled:3 taylor:2 euclidean:4 indexed:1 desired:2 circle:5 re:1 fitted:1 uncertain:1 soft:6 ar:1 measuring:1 assignment:2 ordinary:1 cost:1 vertex:2 entry:2 subset:1 uniform:2 hundred:1 swain:1 gcv:2 johnson:1 conducted:1 reported:2 perturbed:2 synthetic:1 finch:2 grkpack:6 combined:1 cited:1 density:4 randomized:13 siam:1 ie:3 international:1 bu:1 dong:1 physician:1 physic:1 yl:1 together:2 moody:1 na:3 ophthalmalogy:1 again:2 imagery:1 containing:2 opposed:1 woman:1 f5:1 lambda:2 conf:1 derivative:1 li:2 factorised:1 north:1 int:1 inc:1 matter:1 satisfy:1 depends:1 vi:1 view:4 realised:1 red:2 bayes:3 parallel:1 trh:1 minimize:1 square:2 il:2 ass:1 accuracy:3 variance:10 correspond:1 bayesian:4 raw:3 produced:1 rangacv:16 drive:1 corruption:1 liebler:2 history:1 huet:5 simultaneous:1 ed:3 against:2 realise:1 energy:1 pp:3 associated:3 attributed:1 hamming:1 recall:8 knowledge:1 color:2 improves:1 hilbert:1 actually:1 cik:1 cbms:1 appears:2 manuscript:1 ea:1 higher:1 x6:3 methodology:1 yb:2 evaluated:2 box:2 done:1 arch:1 until:1 sketch:1 hand:3 working:1 ubr:2 replacing:2 horizontal:1 nonlinear:2 hwl:2 mode:2 aj:2 effect:3 contain:2 true:4 unbiased:2 brown:1 former:1 regularization:3 hence:2 assigned:2 read:1 furnish:1 trlist:2 contentbased:1 conditionally:1 sin:2 x5:3 during:1 percentile:1 criterion:1 generalized:2 stress:1 confusion:1 tn:1 pami:4 image:16 variational:3 wise:1 ef:3 fi:1 meaning:2 recently:1 rotation:1 functional:3 physical:2 insensitive:1 association:3 extend:1 numerically:1 interpret:2 significant:2 measurement:7 ai:2 cv:4 iq:1 tuning:3 consistency:4 mathematics:1 centre:2 softening:1 aq:3 retinopathy:2 had:1 similarity:6 base:18 pu:1 multivariate:1 showed:1 recent:1 optimizing:2 commun:1 barbara:1 tesauro:1 manipulation:1 binary:2 continue:1 meeting:1 yi:13 seen:2 minimum:1 morgan:1 captured:1 prune:1 determine:1 clockwise:2 ii:7 rv:2 full:1 multiple:3 corporate:1 isa:1 hormone:1 technical:2 match:18 segmented:1 calculation:5 cross:13 raphson:3 lin:1 retrieval:10 england:1 faster:1 dept:3 va:1 yoi:1 prediction:3 variant:2 regression:2 ae:7 patient:1 metric:1 essentially:1 vision:4 iteration:8 kernel:2 histogram:29 represent:3 achieved:1 addition:1 interval:1 median:1 biasvariance:1 leaving:2 source:2 regional:1 subject:1 thing:1 member:1 cowan:1 spirit:1 jordan:1 structural:14 near:1 ee:7 abnormality:1 easy:1 concerned:1 variety:1 independence:2 fit:10 variate:1 psychology:1 architecture:3 wahba:25 bandwidth:1 rang:7 reduce:1 arbib:1 cn:1 idea:1 tradeoff:7 br:2 associating:1 minimise:1 six:1 eeq:2 effort:2 penalty:4 hessian:3 york:3 jj:1 constitute:1 useful:1 clear:1 detailed:2 covered:1 maybe:1 singly:2 nonparametric:1 extensively:1 statist:3 iith:2 shapiro:1 outperform:1 nsf:1 dotted:1 utans:2 sign:2 estimated:1 wr:1 per:2 klein:9 diverse:1 discrete:3 key:1 salient:1 demonstrating:1 blood:1 drawn:1 wisc:2 changing:1 anova:5 f8:1 graph:39 relaxation:3 sum:1 glycosylated:1 inverse:2 eli:1 you:1 angle:19 powerful:1 family:2 draw:1 decision:2 f13:1 fl:4 hi:1 ct:3 followed:1 simplification:2 tackled:1 correspondence:13 distinguish:1 quadratic:4 binned:1 x2:7 scene:3 aspect:1 speed:1 anyone:1 span:1 formulating:1 attempting:1 relatively:1 uf:2 department:3 structured:1 according:3 developing:2 project:1 aerial:3 remain:1 describes:1 wi:4 modification:1 making:1 dv:1 invariant:4 indexing:2 altitude:1 heart:1 taken:2 computationally:1 equation:1 vq:12 previously:1 ln:1 turn:5 describing:1 know:1 letting:5 end:3 available:3 progression:1 away:1 petsche:1 hii:2 appearing:1 neighbourhood:1 alternative:1 original:2 top:1 running:1 include:2 clustering:1 remaining:4 madison:3 newton:3 exploit:1 giving:2 uj:1 yz:1 establish:2 society:1 february:1 objective:1 intend:1 arrangement:1 quantity:1 fa:4 strategy:4 md:2 grace:1 diagonal:1 nr:3 southern:1 distance:9 street:3 vd:1 ourself:1 partner:1 fy:3 reason:2 spanning:1 assuming:2 systolic:1 length:4 index:2 ratio:2 minimizing:1 nc:1 equivalently:1 sinica:2 fe:1 statement:1 trace:4 negative:3 ba:2 implementation:1 ambitious:1 unknown:2 perform:1 observation:2 thence:1 barrow:2 pentland:1 relational:9 extended:1 y1:2 locate:1 reproducing:2 rating:1 pair:4 kl:2 extensive:2 established:3 nip:1 qa:1 suggested:1 pattern:26 kauffman:1 ev:1 including:1 reliable:1 max:4 unpenalized:1 niblack:1 video:1 force:1 indicator:2 hr:1 representing:1 scheme:1 eye:3 library:3 picture:1 axis:1 moss:1 l992:1 review:1 geometric:6 literature:4 prior:2 icc:1 xiang:6 wisconsin:3 asymptotic:1 loss:1 relative:14 suggestion:1 interesting:1 filtering:2 querying:1 erh:1 ingredient:1 generator:1 validation:8 at1:1 age:3 triple:1 degree:1 digital:4 affine:1 proxy:1 dd:1 pi:5 cd:1 translation:1 qbic:1 elsewhere:2 penalized:8 token:1 succinctly:1 free:1 ovum:1 bias:7 normalised:6 side:1 institute:2 taking:3 focussed:1 curve:1 world:2 author:2 adaptive:1 replicated:1 ig:2 ec:3 gestalt:1 functionals:1 bb:1 approximate:5 ignore:1 kullback:2 clique:2 dealing:1 anew:1 global:3 handbook:1 don:1 continuous:1 iterative:3 search:1 diabetic:2 table:4 ballard:1 fragility:1 ca:1 eeg:1 expansion:1 cl:1 statistica:2 ciq:1 main:3 withstanding:1 nothing:1 repeated:1 girard:2 convey:2 body:2 fig:2 representative:1 tl:1 ff:10 cubic:3 position:13 pv:3 explicit:1 downhill:1 exponential:2 xl:6 candidate:2 wish:1 lie:1 pe:2 weighting:1 hw:1 formula:4 trademark:1 list:1 survival:1 dl:1 exists:1 grouping:1 recognising:1 sequential:1 importance:1 texture:1 perceptually:1 cartesian:2 relearning:1 gap:1 intersection:2 l993:1 fc:1 simply:1 explore:1 gao:1 visual:1 ua1:1 corresponds:1 minimizer:10 satisfies:2 truth:2 extracted:2 conditional:5 ydi:3 goal:1 identity:1 graduated:1 ann:2 replace:2 absence:1 considerable:1 change:1 content:3 included:1 infinite:1 except:1 typical:1 characterised:1 lemma:1 principal:1 called:2 experimental:4 la:1 ya:3 meaningful:1 formally:1 select:2 wq:3 mark:2 searched:1 latter:1 stressed:1 assessed:1 scan:3 photobook:1 xiwu:2 evaluate:1

Dynamically Adapting Kernels in Support Vector Machines N ello Cristianini Dept. of Engineering Mathematics University of Bristol, UK nello.cristianini@bristol.ac.uk Colin Campbell Dept. of Engineering Mathematics University of Bristol, UK c.campbell@bristol.ac.uk John Shawe-Taylor Dept. of Computer Science Royal Holloway College john@dcs.rhbnc.ac.uk Abstract The kernel-parameter is one of the few tunable parameters in Support Vector machines, controlling the complexity of the resulting hypothesis. Its choice amounts to model selection and its value is usually found by means of a validation set. We present an algorithm which can automatically perform model selection with little additional computational cost and with no need of a validation set . In this procedure model selection and learning are not separate, but kernels are dynamically adjusted during the learning process to find the kernel parameter which provides the best possible upper bound on the generalisation error. Theoretical results motivating the approach and experimental results confirming its validity are presented. 1 Introduction Support Vector Machines (SVMs) are learning systems designed to automatically trade-off accuracy and complexity by minimizing an upper bound on the generalisation error provided by VC theory. In practice, however, SVMs still have a few tunable parameters which need to be determined in order to achieve the right balance and the values of these are usually found by means of a validation set. One of the most important of these is the kernel-parameter which implicitly defines the structure of the high dimensional feature space where the maximal margin hyperplane is found. Too rich a feature space would cause the system to overfit the data, Dynamically Adapting Kernels in Support Vector Machines 205 and conversely the system can be unable to separate the data if the kernels are too poor. Capacity control can therefore be performed by tuning the kernel parameter subject to the margin being maximized. For noisy datasets, yet another quantity needs to be set, namely the soft-margin parameter C. SVMs therefore display a remarkable dimensionality reduction for model selection. Systems such as neural networks need many different architectures to be tested and decision trees are faced with a similar problem during the pruning phase. On the other hand SVMs can shift from one model complexity to another by simply tuning a continuous parameter. Generally, model selection by SVMs is still performed in the standard way: by learning different SVMs and testing them on a validation set in order to determine the optimal value of the kernel-parameter. This is expensive in terms of computing time and training data. In this paper we propose a different scheme which dynamically adjusts the kernel-parameter to explore the space of possible models at little additional computational cost compared to fixed-kernel learning. Futhermore this approach only makes use of training-set information so it is more efficient in a sample complexity sense. Before proposing the model selection procedure we first prove a theoretical result, namely that the margin and structural risk minimization (SRM) bound on the generalization error depend smoothly on the kernel parameter. This can be exploited by an algorithm which keeps the system close to maximal margin while the kernel parameter is changed smoothly. During this phase, the theoretical bound given by SRM theory can be computed. The best kernel-parameter is the one which gives the lowest possible bound. In section 4 we present experimental results showing that model selection can be efficiently performed using the proposed method (though we only consider Gaussian kernels in the simulations outlined). 2 Support Vector Learning The decision function implemented by SV machines can be written as: f(x) = sign (L Yiai K(x, Xi) - B) tESV where the ai are obtained by maximising the following Lagrangian (where m is the number of patterns): m L =L i= l m ai - 1/2 L aiajYiyjK(Xi, Xj) i,j= l with respect to the ai, subject to the constraints m LaiYi = a i=l and where the functions K( x, x') are called kernels. The kernels provide an expression for dot- products in a high-dimensional feature space [1]: K( x, x') = (<I> (x) , <I>(x' )) 206 N. Cristianini, C. Campbell and 1. Shawe-Taylor and also implicitly define the nonlinear mapping <1>( x) of the training data into feature space where they may be separated using the maximal margin hyperplane. A number of choices of kernel-function can be made e.g. Gaussians kernels: K(x, x') = e-ll x-x'1 12/2(T2 The following upper bound can be proven from VC theory for the generalisation error using hyperplanes in feature space [7, 9J: where R is the radius of the smallest ball containing the training set, m the number of training points and 'Y the margin (d. [2J for a complete survey of the generalization properties of SV machines) . The Lagrange multipliers Qi are usually found by means of a Quadratic Programming optimization routine, while the kernel-parameters are found using a validation set. As illustrated in Figure 1 there is a minimum of the generalisation error for that value of the kernel-parameter which has the best trade-off between overfitting and ability to find an efficient solution. 0 13 012 0 11 01 009 008 0 07 0 06 005 0 04 2 10 Figure 1: Generalization error (y-axis) as a function of (J (x-axis) for the mirror symmetry problem (for Gaussian kernels with zero training error and maximal margin, m = 200, n = 30 and averaged over 105 examples). 3 A utomatic Model Order Selection We now prove a theorem which shows that the margin of the optimal hyperplane is a smooth function of the kernel parameter, as is the upper bound on the generalisation error. First we state the Implicit Function Theorem. Implicit Function Theorem [10]: Let F(x, y) be a continuously differentiable function, F : U ~ ~ x V ~ ~p --t ~ and let (a, b) E U x V be a solution to the equation F(x, y) = O. Let the partial derivatives matrix mi ,j = (~:;) w.r.t. y be full rank at (a, b) . Then, near (a, b), 207 Dynamically Adapting Kernels in Support Vector Machines there exists one and only one function function is continuous. y = g(x) such that F(x,g(x)) = 0, and such Theorem: The margin, of SV machines depends smoothly on the kernel parameter a. Proof: Consider the function 9 : ~ <;;; ~ --t A <;;; ~P, 9 : a ~ (aO, A) which given the data maps the choice of a to the optimal parameters aO and lagrange parameter A of the SV machine with Kernel matrix Gij = YiYjK(a; Xi, Xj )). Let p Wu(a) = l:ai - 1/2 l:aiajYiyjK(a; Xi,Xj) i,j i=l + A(l: Yiai ) be the functional that the SV machine maximizes. Fix a value of a and let aO(a) be the corresponding solution of Wu(a). Let I be the set of indices for which aj((J) =1= O. We may assume that the submatrix of G indexed by I is non-singular since otherwise the maximal margin hyperplane could be expressed in terms of a subset of indices. Now choose a maximal set of indices J containing I such that the corresponding su bmatrix of G is non-singular and all of the points indexed by J have margin 1. Now consider the function F((J ,a , A)i = (a~)j; ,i 2: 1, F((J,a,A)o = LjYjaj in the neighbourhood of (J, where ji is an enumeration of the elements of J, oWu "'_ oa. = 1 - Yj L.. aiYiK((J; Xi, Xj) J + AYj . t and satisfies the equation F((J, aO((J), A(a)) = 0 at the extremal points of Wu(a) . Then the SV function is the implicit function, (aO, A) = g((J), and is continuous (and unique) iff F is continuously differentiable and the partial derivatives matrix w.r.t. a, A is full rank. But the partial derivatives matrix H is given by Hij OP = oat = Yj;Yj}K((J;xj; ,Xj}) = Hji,i , j 2: 1, JJ for ji,iJ E J, which was non-degenerate by definition of J, while Hoo = oFo OA =0 and HOj Consider any non-zero a satisfying (a, Af H(a, A) oFo = oajJ = n = Lj ajYJ oFj OA . = Hjo,J 2: 1. = 0, and any A. We have = aTGa + 2AaT Y = aTGa > O. Hence, the matrix H is non-singular for a satisfying the given linear constraint. Hence , by the implicit function theorem 9 is a continuous function of (J. The following is proven in [2J: ,2 = (t Zif) -1 t=l which shows that, is a continuous function of (J. As the radius of the ball containing the points is also a continuous function of (J , and the generalization error bound has the form f. ~ CR(a)2llaO((J)lll for some constant C, we have the following corollary. Corollary: The bound on the generalization error is smooth in (J. This means that, when the margin is optimal, small variations in the kernel parameter will produce small variations in the margin (and in the bound on the generalisation error). Thus ,u ~ ,uHu and after updating the (J, the system will N Cristianini, C. Campbell and J. Shawe- Taylor 208 still be in a sub-optimal position. This suggests the following strategy for Gaussian kernels, for instance: Kernel Selection Procedure l. Initialize u to a very small value 2. Maximize the margin, then ? Compute the SRM bound (or observe the validation error) ? Increase the kernel parameter: u +- u + 8u 3. Stop when a predetermined value of u is reached else repeat step 2. This procedure takes advantage of the fact that for very small (J convergence is generally very rapid (overfi tting the data, of course), and that once the system is near the equilibrium, few iterations will always be sufficient to move it back to the maximal margin situation. In other words, this system is brought to a maximal margin state in the beginning, when this is computationally very cheap, and then it is actively kept in that situation by continuously adjusting the a while the kernelparameter is gradually increased. In the next section we will experimentally investigate this procedure for real-life datasets. In the numerical simulations we have used the Kernel-Adatron (KA) algorithm recently developed by two of the authors [4] which can be used to train SV machines. We have chosen this algorithm because it can be regarded as a gradient ascent procedure for maximising the Kuhn-Tucker Lagrangian L . Thus the ai for a sub-optimal state are close to those for the optimum and so little computational effort will be needed to bring the system back to a maximal margin position: The Kernel-Adatron Algorithm. l. exi = l. 2. FOR i = 1 TO m ? ,i = ? 8ex i ? IF YiZi = 17(1 _ , i ) (ex i + 8ex i ) ::; 0 THEN ex i = 0 ELSE ex i +- ex i + 8ex t . ? margin = ~ (min(z;) -max(z;)) (4 (z;) = positively (negatively) labelled patterns) 3. IF(margin 4 = 1) THEN stop, ELSE go to step 2. Experimental Results In this section we implement the above algorithm for real-life datasets and plot the upper bound given by VC theory and the generalization error as functions of (J. In order to compute the bound, E ::; R 2/m,2 we need to estimate the radius of the ball in feature space. In general his can be done explicitly by maximising the following Lagrangian w.r.t. Ai using convex quadratic programming routines: L =L subject to the constraints AiK(Xi, Xi) - L AiAjK(Xi, Xj) i,j 2:i Ai = 1 and Ai 2: O. The radius is then found from [3]: Dynamically Adapting Kernels in Support Vector Machines i,j 209 i,j However, we can also get an upper bound for this quantity by noting that Gaussian kernels always map training points to the surface of a sphere of radius 1 centered on the origin of the feature space. This can be easily seen by noting that the distance of a point from the origin is its norm: 11<I>(x)11 = J(<I>(X),<I>(X)) = JK(x,x) = Jellx-xll/2o-2 = 1 In Figure 2 we give both these bounds (the upper bound is Li adm) and generalisation error (on a test set) for two standard datasets: the aspect-angle dependent sonar classification dataset of Gorman and Sejnowski [5] and the Wisconsin breast cancer dataset [8]. As we see from these plots there is little need for the additional computational cost of determining R from the above quadratic progamming problem, at least for Gaussian kernels. In Fig. 3 we plot the bound Li adm and generalisation error for 2 figures from a United States Postal Service dataset of handwritten digits [6]. In these, and other instances we have investigated, the minimum of the bound approximately coincides with the minimum of the generalisation error. This gives a good criterion for the most suitable choice for a. Furthermore, this estimate for the best a is derived solely from training data without the need for an additional validation set . 02 ., .. Figure 2: Generalisation error (solid curves) for the sonar classification (left Fig.) and Wisconsin breast cancer datasets (right Fig.). The upper curves (dotted) show the upper bounds from VC theory (for the top curves R=l). Starting with a small a-value we have observed that the margin can be maximised rapidly. Furthermore, the margin remains close to 1 if a is incremented by a small amount. Consequently, we can study the performance of the system by traversing a range of a-values, alternately incrementing a then maximising the margin using the previous optimal set of a-values as a starting point. We have found that this procedure does not add a significant computational cost in general. For example, for the sonar classification dataset mentioned above and starting at a = 0.1 with increments ~a = 0.1 it took 186 iterations to reach a = 1.0 and 4895 to reach a = 2.0 as against 110 and 2624 iterations for learning at both these a-values. For a rough doubling of the learning time it is possible to determine a reasonable value for a for good generalisation without use of a validation set. 210 N Cristianini, C Campbell and J Shawe-Taylor .. ...... " ". O. -", '.'. '. o. \ \. 07 07 \'" O. '. O. -' \. \ \\., 0' , 02 0 0 \ "- '. 0' 10 0 ~ 0 12 Figure 3: Generalisation error (solid curve) and upper bound from VC theory (dashed curve with R=l) for digits 0 and 3 from the USPS dataset of handwritten digits. 5 Conclusion We have presented an algorithm which automatically learns the kernel parameter with little additional cost, both in a computational and sample-complexity sense. Model selection takes place during the learning process itself, and experimental results are provided showing that this strategy provides a good estimate of the correct model complexity. References [1] Aizerman, M., Braverman, E ., and Rozonoer , L. (1964) . Theoretical Foundations of the Potential Function Method in Pattern Recognition Learning, A utomations and Remote Control, 25:821-837. [2] Bartlett P., Shawe-Taylor J ., (1998). Generalization Performance of Support Vector Machines and Other Pattern Classifiers. 'Advances in Kernel Methods - Support Vector Learning', Bernhard Sch61kopf, Christopher J . C. Burges, and Alexander J . Smola (eds.), MIT Press, Cambridge, USA. [3] Burges c., (1998). A tutorial on support vector machines for pattern recognition . Data Mining and Knowledge Discovery, 2:l. [4] Friess T., Cristianini N., Campbell C. , (1998) The Kernel-Adatron Algorithm: a Fast and Simple Learning Procedure for Support Vector Machines, in Shavlik, J. , ed., Machine Learning: Proceedings of the Fifteenth International Conference, Morgan Kaufmann Publishers, San Francisco, CA. [5] Gorman R.P. & Sejnowski, T.J. (1988) Neural Networks 1:75-89. [6] LeCun, Y., Jackel, L. D. , Bottou, L., Brunot, A., Cortes, C., Denker, J . S., Drucker, H., Guyon, I., Muller, U. A., Sackinger, E ., Simard, P. and Vapnik, V., (1995) . Comparison of learning algorithms for handwritten digit recognition, International Conference on Artificial Neural Networks, Fogelman, F. and Gallinari, P. (Ed.), pp. 53-60. [7] Shawe-Taylor, J ., Bartlett, P., Williamson, R. & Anthony, M. (1996) . Structural Risk Minimization over Data-Dependent Hierarchies NeuroCOLT Technical Report NCTR-96-053 (ftp://ftp.des .rhbne .ae. uk /pub/neuroeolt/teeh_reports). [8] Ster, B ., & Dobnikar, A. (1996) Neural networks in medical diagnosis: comparison with other methods. In A. Bulsari et al. (ed.) Proceedings of the International Conference EA NN '96, p. 427-430. [9] Vapnik , V. (1995) The Nature of Statistical Learning Theory, Springer Verlag. [10] James, Robert C. (1966) Advanced calculus Belmont, Calif. : Wadsworth 

1485 |@word norm:1 calculus:1 simulation:2 solid:2 reduction:1 united:1 pub:1 ka:1 yet:1 written:1 john:2 belmont:1 numerical:1 predetermined:1 confirming:1 cheap:1 designed:1 plot:3 beginning:1 maximised:1 provides:2 postal:1 hyperplanes:1 ofo:2 prove:2 rapid:1 automatically:3 little:5 enumeration:1 lll:1 provided:2 maximizes:1 lowest:1 developed:1 proposing:1 adatron:3 classifier:1 uk:6 control:2 gallinari:1 medical:1 before:1 service:1 engineering:2 aat:1 solely:1 approximately:1 dynamically:6 conversely:1 suggests:1 range:1 averaged:1 unique:1 lecun:1 testing:1 yj:3 practice:1 implement:1 digit:4 procedure:8 sch61kopf:1 adapting:4 word:1 get:1 close:3 selection:10 risk:2 map:2 lagrangian:3 go:1 starting:3 convex:1 survey:1 adjusts:1 regarded:1 his:1 variation:2 increment:1 rozonoer:1 tting:1 hierarchy:1 controlling:1 aik:1 programming:2 hypothesis:1 origin:2 element:1 expensive:1 satisfying:2 updating:1 jk:1 recognition:3 fries:1 observed:1 remote:1 trade:2 incremented:1 mentioned:1 complexity:6 cristianini:6 depend:1 negatively:1 usps:1 easily:1 exi:1 train:1 separated:1 fast:1 sejnowski:2 artificial:1 otherwise:1 ability:1 noisy:1 itself:1 advantage:1 differentiable:2 took:1 propose:1 maximal:9 product:1 rapidly:1 iff:1 degenerate:1 achieve:1 convergence:1 optimum:1 produce:1 ftp:2 ac:3 ij:1 op:1 implemented:1 kuhn:1 radius:5 correct:1 vc:5 centered:1 ao:5 generalization:7 fix:1 adjusted:1 equilibrium:1 mapping:1 smallest:1 jackel:1 extremal:1 minimization:2 brought:1 rough:1 mit:1 gaussian:5 always:2 cr:1 corollary:2 derived:1 rank:2 sense:2 dependent:2 nn:1 lj:1 fogelman:1 adm:2 classification:3 initialize:1 wadsworth:1 once:1 t2:1 report:1 few:3 phase:2 investigate:1 mining:1 braverman:1 futhermore:1 partial:3 traversing:1 tree:1 indexed:2 taylor:6 calif:1 theoretical:4 instance:2 increased:1 soft:1 cost:5 subset:1 srm:3 too:2 motivating:1 sv:7 international:3 off:2 continuously:3 containing:3 choose:1 derivative:3 simard:1 actively:1 li:2 potential:1 de:1 explicitly:1 depends:1 performed:3 reached:1 accuracy:1 kaufmann:1 efficiently:1 maximized:1 handwritten:3 utomatic:1 bristol:4 reach:2 ed:4 definition:1 against:1 pp:1 tucker:1 james:1 proof:1 mi:1 stop:2 tunable:2 adjusting:1 dataset:5 knowledge:1 dimensionality:1 ster:1 routine:2 ea:1 campbell:6 back:2 done:1 though:1 furthermore:2 implicit:4 smola:1 overfit:1 hand:1 zif:1 christopher:1 su:1 nonlinear:1 sackinger:1 defines:1 aj:1 usa:1 validity:1 multiplier:1 hence:2 illustrated:1 ll:1 during:4 coincides:1 criterion:1 complete:1 bring:1 recently:1 functional:1 ji:2 significant:1 cambridge:1 ai:8 tuning:2 outlined:1 mathematics:2 shawe:6 dot:1 surface:1 add:1 verlag:1 yiai:2 life:2 exploited:1 muller:1 seen:1 minimum:3 additional:5 morgan:1 determine:2 maximize:1 colin:1 dashed:1 full:2 smooth:2 technical:1 af:1 sphere:1 qi:1 breast:2 ae:1 fifteenth:1 iteration:3 kernel:38 else:3 singular:3 publisher:1 ascent:1 subject:3 structural:2 near:2 noting:2 xj:7 architecture:1 drucker:1 shift:1 expression:1 rhbnc:1 bartlett:2 effort:1 cause:1 jj:1 generally:2 amount:2 svms:6 tutorial:1 dotted:1 sign:1 hji:1 diagnosis:1 nctr:1 kept:1 angle:1 place:1 reasonable:1 guyon:1 wu:3 decision:2 submatrix:1 bound:20 display:1 quadratic:3 constraint:3 aspect:1 min:1 ball:3 poor:1 hoo:1 brunot:1 gradually:1 computationally:1 equation:2 remains:1 needed:1 gaussians:1 denker:1 observe:1 neighbourhood:1 top:1 move:1 quantity:2 strategy:2 aiajyiyjk:2 gradient:1 distance:1 separate:2 unable:1 neurocolt:1 capacity:1 oa:3 nello:1 maximising:4 index:3 minimizing:1 balance:1 robert:1 hij:1 xll:1 perform:1 upper:10 datasets:5 situation:2 dc:1 yiyjk:1 namely:2 alternately:1 usually:3 pattern:5 aizerman:1 royal:1 max:1 suitable:1 advanced:1 scheme:1 axis:2 faced:1 discovery:1 determining:1 wisconsin:2 proven:2 remarkable:1 validation:8 foundation:1 ayj:1 sufficient:1 cancer:2 course:1 changed:1 repeat:1 burges:2 shavlik:1 curve:5 rich:1 author:1 made:1 san:1 pruning:1 implicitly:2 bernhard:1 keep:1 overfitting:1 francisco:1 xi:8 continuous:6 sonar:3 nature:1 ca:1 symmetry:1 williamson:1 investigated:1 bottou:1 anthony:1 ofj:1 oat:1 incrementing:1 positively:1 fig:3 sub:2 position:2 learns:1 yizi:1 theorem:5 showing:2 cortes:1 exists:1 vapnik:2 mirror:1 margin:23 gorman:2 smoothly:3 simply:1 explore:1 lagrange:2 expressed:1 doubling:1 springer:1 satisfies:1 ello:1 consequently:1 labelled:1 experimentally:1 generalisation:12 determined:1 hyperplane:4 called:1 gij:1 experimental:4 holloway:1 college:1 support:11 alexander:1 dept:3 tested:1 ex:7

Evidence for a Forward Dynamics Model in Human Adaptive Motor Control Nikhil Bhushan and Reza Shadmehr Dept. of Biomedical Engineering Johns Hopkins University, Baltimore, MD 21205 Email: nbhushan@bme.jhu.edu, reza@bme.jhu.edu Abstract Based on computational principles, the concept of an internal model for adaptive control has been divided into a forward and an inverse model. However, there is as yet little evidence that learning control by the eNS is through adaptation of one or the other. Here we examine two adaptive control architectures, one based only on the inverse model and other based on a combination of forward and inverse models. We then show that for reaching movements of the hand in novel force fields, only the learning of the forward model results in key characteristics of performance that match the kinematics of human subjects. In contrast, the adaptive control system that relies only on the inverse model fails to produce the kinematic patterns observed in the subjects, despite the fact that it is more stable. Our results provide evidence that learning control of novel dynamics is via formation of a forward model. 1 Introduction The concept of an internal model, a system for predicting behavior of a controlled process, is central to the current theories of motor control (Wolpert et al. 1995) and learning (Shadmehr and Mussa-Ivaldi 1994). Theoretical studies have proposed that internal models may be divided into two varieties: forward models, which simulate the causal flow of a process by predicting its state transition given a motor command, and inverse models, which estimate motor commands appropriate for a desired state transition (Miall and Wolpert, 1996). This classification is relevant for adaptive control because based on computational principles, it has been proposed that learning control of a nonlinear system might be facilitated if a forward model of the plant is learned initially, and then during an off-line period is used to train an inverse model (Jordan and Rumelhart, 1992). While there is no experimental evidence for this idea in the central nervous system, there is substantial evidence 4 N. Bhushan and R. Shadmehr that learning control of arm movements involves formation of an internal model. For example, practicing arm movements while holding a novel dynamical system initiates an adaptation process which results in the formation of an internal model: upon sudden removal of the force field, after-effects are observed which match the expected behavior of a system that has learned to predict and compensate for the dynamics of the imposed field (Shadmehr and Brashers-Krug, 1997). However, the computational nature of this internal model, whether it be a forward or an inverse model, or a combination of both, is not known. Here we use a computational approach to examine two adaptive control architectures: adaptive inverse model feedforward control and adaptive forward-inverse model feedback control. We show that the two systems predict different behaviors when applied to control of arm movements. While adaptation to a force field is possible with either approach, the second system with feedback control through an adaptive forward model, is far less stable and is accompanied with distinct kinematic signatures, termed "near path-discontinuities". We observe remarkably similar instability and near path-discontinuities in the kinematics of 16 subjects that learned force fields. This is behavioral evidence that learning control of novel dynamics is accomplished with an adaptive forward model of the system. 2 Adaptive Control using Internal Models Adaptive control of a nonlinear system which has large sensory feedback delays, such as the human arm, can be accomplished by using two different internal model architectures. The first method uses only an adaptive inverse dynamics model to control the system (Shadmehr and Mussa-Ivaldi, 1994). The adaptive controller is feedforward in nature and ignores delayed feedback during the movement. The control system is stable because it relies on the equilibrium properties of the muscle and the spinal reflexes to correct for any deviations from the desired trajectory. The second method uses a rapidly adapting forward dynamics model and delayed sensory feedback in addition to an inverse dynamics model to control arm movements (Miall and Wolpert, 1996). In this case, the corrections to deviations from the desired trajectory are a result of a combination of supraspinal feedback as well as spinal/muscular feedback. Since the two methods rely on different internal model and feedback structures, they are expected to behave differently when the dynamics of the system are altered. The Mechanical Model of the Human Arm For the purpose of simulating arm movements with the two different control architectures, a reasonably accurate model of the human arm is required. We model the arm as a two joint revolute arm attached to six muscles that act in pairs around the two joints. The three muscle pairs correspond to elbow joint, shoulder joint and two joint muscles and are assumed to have constant moment arms. Each muscle is modeled using a Hill parametric model with nonlinear stiffness and viscosity (Soechting and Flanders, 1997). The dynamics of the muscle can be represented by a nonlinear state function f M, such that, (1) where, Ft is the force developed by the muscle, N is the neural activation to the muscle, and X m, xm are the muscle length and velocity. The passive dynamics related to the mechanics of the two-joint revolute arm can be represented by fD, such that, x = fD(T, x, x) = D- 1 (x)[T - C(x, x)x + JT Fxl (2) Evidence for a Forward Dynamics Model in Human Adaptive Motor Control 5 where, x is the hand acceleration, T is the joint torque generated by the muscles, x, x are the hand position and velocity, D and C are the inertia and the coriolis matrices of the arm, J is the Jacobian for hand position and joint angle, and Fx is the external dynamic interaction force on the hand. Under the force field environment, the external force Fx acting on the hand is equal to Bx, where B is a 2x2 rotational viscosity matrix. The effect of the force field is to push the hand perpendicular to the direction of movement with a force proportional to the speed of the hand. The overall forward plant dynamics of the arm is a combination of JM and JD and can be repff~sented by the function Jp , (3) Adaptive Inverse Model Feedforward Control The first control architecture uses a feedforward controller with only an adaptive inverse model. The inverse model computes the neural activation to the muscles for achieving a desired acceleration, velocity and position of the hand. It can be represented as the estimated inverse, 1;1, of the forward plant dynamics, and maps the desired position Xd, velocity Xd, and acceleration Xd of the hand, into descending neural commands N c. Nc = 1;1 (Xd, Xd, Xd) (4) Adaptation to novel external dynamics occurs by learning a new inverse model of the altered external environment. The error between desired and actual hand trajectory can be used for training the inverse model. When the inverse model is an exact inverse of the forward plant dynamics, the gain of the feedforward path is unity and the arm exactly tracks the desired trajectory. Deviations from the desired trajectory occur when the inverse model does not exactly model the external dynamics. Under that situation, the spinal reflex corrects for errors in desired (Xmd, Xmd) and actual (xm,x m) muscle state, by producing a corrective neural signal NR based on a linear feedback controller with constants K1 and K 2 ? (5) Adaptive Forward-Inverse Model Feedback Control The second architecture provides feedback control of arm movements in addition to the feedforward control described above. Delays in feedback cause instability, therefore, the system relies on a forward model to generate updated state estimates of the arm. An estimated error in hand trajectory is given by the difference in desired and estimated state, and can be used by the brain to issue corrective neural signals to the muscles while a movement is being made. The forward model, written Inverse Arm Dynamics Model Td Desired Trajectory 6d(t+60) A?1 to Muscle Inverse Muscle Model f;:.,' fM T Arm Dynamics 6 to / r+----------------~ ! '".-...----v-----~ . "., l. . ...._. _. . . _. . __._..._!e.. . . . .__ . . . _. ........ . A=gO ms 6 d (I.30) Fx (external force) + . 6(1.30) A=30ms Figure 1: The adaptive inverse model feedforward control system. 6 N. Bhushan and R. Shadmehr 1\ A=120ms 1\ x, X (t+60) Desired Trajectory Td Inverse Muscle Nc Model f--L-..~ f?' M A=60 ms + NR L........_---' A~30ms A-9Oms Figure 2: A control system that provides feedback control with the use of a forward and an inverse model. as jp, mimics the forward dynamics of the plant and predicts hand acceleration from neural signal Nc, and an estimate of hand state x, ?. i, (6) U sing this equation, one can solve for x, ?at time t, when given the estimated state at some earlier time t - T, and the descending neural commands N c from time t - T to t. If t is the current time and T is the time delay in the feedback loop, then sensory feedback gives the hand state x, x at t-T. The current estimate of the hand position and velocity can be computed by assuming initial conditions x(t - T)=X(t - T) and ?(t - T)=X(t - T), and then solving Eq. 6. For the simulations, T has value of 200 msec, and is composed of 120 msec feedback delay, 60 msec descending neural path delay, and 20 msec muscle activation delay. Based on the current state estimate and the estimated error in trajectory, the desired acceleration is corrected using a linear feedback controller with constants Kp and Kv. The inverse model maps the hand acceleration to appropriate neural signal for the muscles Nc. The spinal reflex provides additional corrective feedback N R , when there is an error in the estimated and actual muscle state. + Xc = Xd + Kp(Xd - x) + Kv(Xd 1;1 (x new , x, ?) K 1 (x m - xm) + K 2 (?m d - xm) Xd ?) (7) (8) (9) When the forward model is an exact copy of the forward plant dynamics jp= jp, and the inverse model is correct j;l =1;1, the hand exactly tracks the desired trajectory. Errors due to an incorrect inverse model are corrected through the feedback loop. However, errors in the forward model cause deviations from the desired behavior and instability in the system due to inappropriate feedback action. 3 Simulations results and comparison to human behavior To test the two control architectures, we compared simulations of arm movements for the two methods to experimental human results under a novel force field environment. Sixteen human subjects were trained to make rapid point-to-point reaching Evidence for a Forward Dynamics Model in Human Adaptive Motor Control (1) Inverse Model Feedforward Control ('.::::)(:::?). ...- .."'- .,~ .""".. r<:. .~ t>4J(:: ::~ .,../ .).. ?./ !A c. .... .....v ) i.. .. :;. ..: . . . ., :.~ _.i .')-'" (2) Forward?lnverse Model Feedback Control Typical Subject ~. ",A ...... 7 ?o.. ""'.... ~ 021N\l ~:~ o:lJIffl:J Q2~ O'IT[] 04[m:J 0.5 1 1.5 O.S 1 1.5 0.5 1 1.5 0.3 0.3 0.2 0.' 0, Ql ~ffi1TI O.5 sec 1 15 o ~w o O.5 sec 1 1.5 Figure 3: Performance in field B2 after a typical subject (middle column) and each of the controllers (left and right columns) had adapted to field B 1 . (1) hand paths for 8 movement directions , (2-5) hand velocity, speed, derivative of velocity direction, and segmented hand path for the -90 0 downward movement . The segmentation in hand trajectory that is observed in our subjects is almost precisely reproduced by the controller that uses a forward model. movements with their hand while an external force field , Fx = Bx, pushed on the hand. The task was to move the hand to a target position 10 cm away in 0.5 sec. The movement could be directed in any of eight equally spaced directions. The subjects made straight-path minimum-jerk movements to the targets in the absence of any force fields. The subjects were initially trained in force field Bl with B=[O 13;-130]' until they had completely adapted to this field and converged to the straight-path minimum-jerk movement observed before the force field was applied. Subsequently, the force field was switched to B2 with B=[O -13;13 0] (the new field pushed anticlockwise, instead of clockwise), and the first three movements in each direction were used for data analysis. The movements of the subjects in field B2 showed huge deviations from the desired straight path behavior because the subjects expected clockwise force field B 1 ? The hand trajectories for the first movement in each of the eight directions are shown for a typical subject in Fig. 3 (middle column). Simulations were performed for the two methods under the same conditions as the human experiment. The movements were made in force field B 2 , while the internal models were assumed to be adapted to field B 1 . Complete adaptation to the force field Bl was found to occur for the two methods only when both N. Bhushan and R. Shadmehr 8 ? Expenmental data from ? 16 subjects Forward Model Control (a) :[[[1 I~ ~ III &' = Q A1(") d l(m) An t,(s) A,(") cJ(m/s' ) Ns Figure 4: The mean and standard deviation for segmentation parameters for each type of controller as compared to the data from our subjects. Parameters are defined in Fig. 3: Ai is angle about a seg. point, d i is the distance to the i-th seg. point, ti is time to reach the i-th seg. point, Cj is cumulative squared jerk for the entire movement, Ns is number of seg. point in the movement. Up until the first segmentation point (AI and dd, behavior of the controllers are similar and both agree with the performance of our subjects. However, as the movement progresses, only the controller that utilizes a forward model continues to agree with the movement characteristics of the subjects. the inverse and forward models expected field B I . Fig. 3 (left column) shows the simulation of the adaptive inverse model feedforward control for movements in field B2 with the inverse model incorrectly expecting B I . Fig. 3 (right column) shows the simulation of the adaptive forward-inverse model feedback control for movements in field B2 with both the forward and the inverse model incorrectly expecting B I . Simulations with the two methods show clear differences in stability and corrective behavior for all eight directions of movement. The simulations with the inverse model feedforward control seem to be stable, and converge to the target along a straight line after the initial deviation. The simulations with the forward-inverse model feedback control are more unstable and have a curious kinematic pattern with discontinuities in the hand path. This is especially marked for the downward movement. The subject's hand paths show the same kinematic pattern of near discontinuities and segmentation of movement as found with the forward-inverse model feedback control. To quantify the segmentation pattern in the hand path, we identified the "near path-discontinuities" as points on the trajectory where there was a sudden change in both the derivative of hand speed and the direction of hand velocity. The hand path was segmented on the basis of these near discontinuities. Based on the first three segments in the hand trajectory we defined the following parameters: AI, angle between the first segment and the straight path to the target; d l , the distance covered during the first segment; A2, angle between the second segment and straight path to the target from the first segmentation point; t2, time duration of the second Evidence for a Forward Dynamics Model in Human Adaptive Motor Control 9 segment; A3, angle between the second and third segments; Ns, the number of segmentation points in the movement . We also calculated the cumulative jerk CJ in the movements to get a measure of the instability in the system. The results of the movement segmentation are presented in Fig. 4 for 16 human subjects, 25 simulations of the inverse model and 20 simulations of the forward model control for three movement directions (a) -900 downward, (b) 90 0 upward and (c) 135 0 upward. We performed the different simulations for the two methods by systematically varying various model parameters over a reasonable physiological range. This was done because the parameters are only approximately known and also vary from subject to subject. The parameters of the second and third segment, as represented by A2, t2 and A3, clearly show that the forward model feedback control performs very differently from inverse model feedforward control and the behavior of human subjects is very well predicted by the former. Furthermore, this characteristic behavior could be produced by the forward-inverse model feedback control only when the forward model expected field B 1 . This could be accomplished only by adaptation of the forward model during initial practice in field B 1 ? This provides evidence for an adaptive forward model in the control of human arm movements in novel dynamic environments. We further tried to fit adaptation curves of simulated movement parameters (using forward-inverse model feedback control) to real data as subjects trained in field B 1 . We found that the best fit was obtained for a rapidly adapting forward and inverse model (Bhushan and Shadmehr, 1999). This eliminated the possibility that the inverse model was trained offline after practice. The data, however, suggested that during learning of a force field, the rate of learning of the forward model was faster than the inverse model. This finding could be paricularly relevant if it is proven that a forward model is easier to learn than an inverse model (Narendra, 1990), and could provide a computational rationale for the existence of forward model in adaptive motor control. References Bhushan N, Shadmehr R (1999) Computational architecture of the adaptive controller during learning of reaching movements in force fields. Biol Cybern, in press. Jordan MI, Flash T, Arnon Y (1994) A model of learning arm trajectories from spatial deviations Journal of Cog Neur 6:359-376 . Jordan MI, Rumelhart DE (1992) Forward model: supervised learning with a distal teacher. Cog Sc 16:307-354. Miall RC, Wolpert DM (1996) Forward models for phySiological motor control. Neural Networks 9:1265-1279. Narendra KS (1990) Identification and control of dynamical systems using neural networks. Neural Networks 1:4-27. Shadmehr R, Brashers-Krug T (1997) Functional stages in the formation of human longterm memory. J Neurosci 17:409-19. Shadmehr R, Mussa-Ivaldi FA (1994) Adaptive representation of dynamics during learning of a motor task. The Journal of Neuroscience 14:3208-3224. Soechting JF, Flanders M (1997) Evaluating an integrated musculoskeletal model of the human arm J Biomech Eng 9:93-102 . Wolpert DM, Ghahramani Z, Jordan MI (1995) An internal model for sensorimotor integration. Science 269:1880-82. 

1486 |@word longterm:1 middle:2 simulation:12 tried:1 eng:1 moment:1 initial:3 ivaldi:3 current:4 activation:3 yet:1 written:1 john:1 motor:10 nervous:1 sudden:2 provides:4 rc:1 along:1 incorrect:1 behavioral:1 expected:5 rapid:1 behavior:10 examine:2 mechanic:1 brain:1 torque:1 td:2 little:1 jm:1 actual:3 inappropriate:1 elbow:1 arnon:1 cm:1 q2:1 developed:1 finding:1 act:1 ti:1 xd:10 exactly:3 control:51 producing:1 before:1 engineering:1 despite:1 krug:2 path:16 approximately:1 might:1 k:1 perpendicular:1 range:1 directed:1 practice:2 coriolis:1 jhu:2 adapting:2 get:1 cybern:1 instability:4 descending:3 imposed:1 map:2 go:1 duration:1 stability:1 fx:4 updated:1 target:5 exact:2 us:4 velocity:8 rumelhart:2 continues:1 predicts:1 observed:4 ft:1 seg:4 movement:37 substantial:1 expecting:2 environment:4 dynamic:25 signature:1 trained:4 solving:1 segment:7 upon:1 completely:1 basis:1 joint:8 differently:2 represented:4 various:1 corrective:4 train:1 distinct:1 kp:2 sc:1 formation:4 solve:1 nikhil:1 reproduced:1 interaction:1 adaptation:7 relevant:2 loop:2 rapidly:2 kv:2 produce:1 bme:2 progress:1 eq:1 predicted:1 involves:1 quantify:1 direction:9 musculoskeletal:1 correct:2 subsequently:1 human:17 correction:1 around:1 equilibrium:1 predict:2 narendra:2 vary:1 a2:2 purpose:1 soechting:2 clearly:1 reaching:3 varying:1 command:4 contrast:1 bhushan:6 entire:1 integrated:1 initially:2 upward:2 overall:1 classification:1 issue:1 spatial:1 integration:1 field:30 equal:1 eliminated:1 mimic:1 t2:2 composed:1 delayed:2 mussa:3 huge:1 fd:2 possibility:1 kinematic:4 accurate:1 desired:16 causal:1 theoretical:1 column:5 earlier:1 deviation:8 delay:6 teacher:1 off:1 corrects:1 hopkins:1 squared:1 central:2 external:7 derivative:2 bx:2 de:1 accompanied:1 sec:3 b2:5 performed:2 characteristic:3 correspond:1 spaced:1 identification:1 produced:1 trajectory:15 straight:6 converged:1 reach:1 email:1 sensorimotor:1 dm:2 mi:3 gain:1 segmentation:8 cj:3 supervised:1 done:1 furthermore:1 biomedical:1 stage:1 until:2 hand:33 nonlinear:4 effect:2 concept:2 former:1 distal:1 during:7 m:5 hill:1 complete:1 performs:1 passive:1 novel:7 functional:1 spinal:4 reza:2 attached:1 jp:4 ai:3 had:2 stable:4 showed:1 termed:1 accomplished:3 muscle:19 minimum:2 additional:1 converge:1 period:1 signal:4 clockwise:2 segmented:2 match:2 faster:1 compensate:1 divided:2 equally:1 a1:1 controlled:1 xmd:2 controller:10 addition:2 remarkably:1 baltimore:1 subject:22 flow:1 seem:1 jordan:4 curious:1 near:5 feedforward:11 iii:1 variety:1 jerk:4 fit:2 architecture:8 fm:1 identified:1 idea:1 whether:1 six:1 cause:2 action:1 clear:1 covered:1 viscosity:2 generate:1 estimated:6 neuroscience:1 track:2 key:1 achieving:1 practicing:1 inverse:46 facilitated:1 angle:5 almost:1 reasonable:1 utilizes:1 sented:1 pushed:2 adapted:3 occur:2 precisely:1 x2:1 simulate:1 speed:3 neur:1 combination:4 unity:1 equation:1 agree:2 kinematics:2 initiate:1 brashers:2 stiffness:1 eight:3 observe:1 away:1 appropriate:2 simulating:1 jd:1 existence:1 xc:1 ghahramani:1 especially:1 k1:1 bl:2 move:1 occurs:1 parametric:1 fa:1 md:1 nr:2 distance:2 simulated:1 unstable:1 assuming:1 length:1 modeled:1 rotational:1 nc:4 ql:1 holding:1 sing:1 behave:1 incorrectly:2 situation:1 shoulder:1 pair:2 mechanical:1 required:1 learned:3 discontinuity:6 suggested:1 dynamical:2 pattern:4 xm:4 memory:1 force:22 rely:1 predicting:2 arm:23 altered:2 removal:1 revolute:2 plant:6 rationale:1 oms:1 proportional:1 proven:1 sixteen:1 switched:1 principle:2 dd:1 systematically:1 copy:1 offline:1 feedback:27 calculated:1 curve:1 transition:2 cumulative:2 evaluating:1 computes:1 sensory:3 forward:48 ignores:1 adaptive:27 inertia:1 made:3 miall:3 far:1 assumed:2 nature:2 reasonably:1 learn:1 neurosci:1 fig:5 en:1 n:3 fails:1 position:6 msec:4 flanders:2 jacobian:1 third:2 cog:2 jt:1 physiological:2 evidence:10 a3:2 downward:3 push:1 easier:1 wolpert:5 reflex:3 relies:3 marked:1 acceleration:6 flash:1 jf:1 absence:1 change:1 muscular:1 typical:3 corrected:2 shadmehr:11 acting:1 experimental:2 internal:11 dept:1 biol:1

A Reinforcement Learning Algorithm in Partially Observable Environments Using Short-Term Memory Nobuo Suematsu and Akira Hayashi Faculty of Computer Sciences Hiroshima City University 3-4-1 Ozuka-higashi, Asaminami-ku, Hiroshima 731-3194 Japan {suematsu,akira} @im.hiroshima-cu.ac.jp Abstract We describe a Reinforcement Learning algorithm for partially observable environments using short-term memory, which we call BLHT. Since BLHT learns a stochastic model based on Bayesian Learning, the overfitting problem is reasonably solved. Moreover, BLHT has an efficient implementation. This paper shows that the model learned by BLHT converges to one which provides the most accurate predictions of percepts and rewards, given short-term memory. 1 INTRODUCTION Research on Reinforcement Learning (RL) problem for partially observable environments is gaining more attention recently. This is mainly because the assumption that perfect and complete perception of the state of the environment is available for the learning agent, which many previous RL algorithms require, is not valid for many realistic environments. model-free Figure I: Three approaches One of the approaches to the problem is the model-free approach (Singh et al. 1995; Jaakkola et al. 1995) (arrow a in the Fig.l) which gives up state estimation and uses memory-less policies. We can not expect the approach to find a really effective policy when it is necessary to accumulate information to estimate the state. Model based approaches are superior in these environments. A popular model based approach is via a Partially Observable Markov Decision Process (POMDP) model which represents the decision process of the agent. In Fig.1 the approach is described by the route from "World" to "Policy" through "POMDP". The approach has two serious difficulties. One is in the learning of POMDPs (arrow b in Fig. I). Abe and N. Suematsu and A. Hayashi 1060 Warmuth (1992) shows that learning of probabilistic automata is NP-hard, which means that learning of POMDPs is also NP-hard. The other difficulty is in finding the optimal policy of a given POMDP model (arrow c in Fig. I ). Its PSAPCE-hardness is shown in Papadimitriou and Tsitsiklis (1987). Accordingly, the methods based on this approach (Chrisman 1992; McCallum 1993), will not scale well to large problems. The approach using short-term memory is computationally more tractable. Of course we can construct environments in which long-term memory is essential. However, in many environments, because of their stochasticity, the significance of the past information decreases exponentially fast as the time goes. In such environments, memories of moderate length will work fine. McCallum (1995) proposes "utile suffix memory" (USM) algorithm. USM uses a tree structure to represent short-term memories with variable length. USM's model learning is based on a statistical test, which requires time and space proportional to the learning steps. This makes it difficult to adapt USM to the environments which require long learning steps. USM suffers from the overfitting problem which is a difficult problem faced by most of model based learning methods. USM may overfit or underfit up to the significance level used for the statistical test and we can not know its proper level in advance. In this paper, we introduce an algorithm called BLHT (Suematsu et al. 1997), in which the environment is modeled as a history tree model (HTM), a stochastic model with variable memory length. Although BLHT shares the tree structured representation of short-term memory with USM, the computational time required by BLHT is constant in each step and BLHT copes with environments which require large learning steps. In addition, because BLHT is based on Bayesian Learning, the overfitting problem is solved reasonably in it. A similar version of HTMs was introduced and has been used for learning of Hidden Markov Models in Ron et at. (1994). In their learning method, a tree is grown in a similar way with USM. If we try to adapt it to our RL problem, it will face the same problems with USM. This paper shows that the HTM learned by BLHT converges to the optimal one in the sense that it provides the most accurate predictions of percepts and rewards, given shortterm memory. BLHT can learn a HTM in an efficient way (arrow d in Fig.l). And since HTMs compose a subset of Markov Decision Processes (MDPs), it can be efficiently solved by Dynamic Programming (DP) techniques (arrow e in Fig. I). So, we can see BLHT as an approach to follow an easy way from "World" to "Policy" which goes around "POMDP". 2 THE POMDP MODEL The decision process of an agent in a partially observable environment can be formulated as a POMDP. Let the finite set of states of the environment be S, the finite set of agent's actions be A, and the finite set of all possible percepts be I. Let us denote the probability of and the reward for making transition from state 8 to 8' using action a by Ps' lsa and W sas' respectively. We also denote the probability of obtaining percept i after a transition from 8 to 8' using action a by 0ilsas" Then, a POMDP model is specified by (S, A,I, P, 0, W, xo), where P = {Ps/l sa 18,8' E S,a E A}, 0 = {oilsas,18,8' E S,a E A,i E I}, W = {W sas,18, 8' E S, a E A}, and Xo = (X~l" .. , x~I SI_l) is the probability distribution of the initial state. We denote the history of actions and percepts of the agent till time t, ( ... , at-2, it-I, at-I, it) by D t . If the POMDP model, M = (S, A,I, P, 0, W, Xi) is given, one can compute the belief state, Xt = (X~l"'" x~ISI_l) from Df, which is the state estimation at time t. We denote the mapping from histories to belief states defined by POMDP model M by X M( .), that is, Xt = X M(Dt). The belief state Xt is the most precise state estimation and it is known to be the sufficient statistics for the optimal policy in POMDPs (Bertsekas 1987). It is also known that the stochastic process {Xt, t 2:: O} is an MDP in the continuous 1061 An RL Algorithm in Partially Observable Environments Using Memory 3 BAYESIAN LEARNING OF HISTORY TREE MODELS (BLHT) In this section. we summarize our RL algorithm for partially observable environments. which we call BLHT (Suematsu et at. 1997). 3.1 HISTORY TREE MODELS BLHT is Bayesian Learning on a hypothesis space which is composed of predictive models. which we call History Tree Models (HTMs). Given short-term memory. a HTM provides the probability disctribution of the next percept and the expected immediate reward for each action. A HTM is represented by a tree structure called a history tree and parameters given for each leaf of the tree. A history tree h associates history D t with a leaf as follows. Starting from the root of h. we check the most recent percept. it and follow the appropriate branch and then we check the action at-l and follow the appropriate branch. This procedure is repeated till we reach a leaf. We denote the reached leaf by Ah(D t ) and the set of leaves of h by Lh. Each leaf l E Lh has parameters Billa and Wla. Billa denotes the probability of observing i at time t + 1 when Ah(Dt} = l and the last action at was a. Wla denotes the expected immediate reward for performing a when Ah(D t ) = l. Let 8 h = {Billa liE T, l E Lh,a E A}. b (a) ---"--../--..-- ~ (b) 2 f-~- - - - it /'-.... a b f-~- - - at-l 1 a / " - . . ........ 1 2 1 2 ~ it-l Figure 2: (a) A three-state environment. in which the agent receives percept 1 in state 1 and percept 2 in states 2a and 2b. (b) A history tree which can represent the environment. Fig. 2 shows a three-state environment (a) and a history tree which can represent the environment (b). We can construct a HTM which is equivalent with the environment by setting appropriate parameters in each leaf of the history tree. 3.2 BAYESIAN LEARNING BLHT is designed as Bayesian Learning on the hypothesis space. 11.. which is a set of history trees. First we show the posterior probability of a history tree h E 11. given history D t . To derive the posterior probability we set the prior density of 8h as p(8 h lh) = II II Kia IELh aEA II B~:~a-l, iEI where Kia is the normalization constant and ailla is a hyper parameter to specify the prior density. Then we can have the posterior probabili,ty of h. P(hID 11.) = t, Ct P(hI1l.) II II IELh aEA K n? la ,E r(N~1 + a'll ) , la ~ a r(Nt + a) , I la (I) la where Ct is the normalization constant. r(?) is the gamma function. Nflla is the number of times i is observed after executing a when Ah(Dt ,) = l in the history D t ? N/ = a L.JiEI N illa ? and ala = L.JiEI ailla' "t " Next. we show the estimates of the parameters. We use the average of Billa with its posterior 1062 N. Suematsu and A. Hayashi density as the estimate, 8~lla' which is expressed as ~t + ailla t a N / a + a'a . ()"II Nflla = -'-;---- W'a is estimated just by accumulating rewards received after executing a when Ah(Dt ) = and dividing it by the number of times a was performed when Ah (D t ) = l, N/a ? That is, wIa = 1 Nt l, N/'a L Ttk+1, la k=l where tk is the k-th occurrence of execution of a when Ah(D t ) = l. 3.3 LEARNING ALGORITHM In principle, by evaluating Eq.( J) for all h E 11., we can extract the MAP model. However, it is often impractical, because a proper hypothesis space 11. is very large when the agent has little prior knowledge concerning the environment. Fortunately, we can design an efficient learning algorithm by assuming that the hypothesis space, 11., is the set of pruned trees of a large history tree h1i and the ratio of prior probabilities of a history tree h and hi obtained by pruning off subtree Llh from h is given by a known function q( Llh) I . We define function g(hIDt,1I.) by taking logarithm of the R.H.S. of Eq.(J) without the normalization constant, which can be rewritten as g(hIDt,1I.) = log P(hI1l.) + L At, (2) IEC h where [Kla ItEIreNt r(Nfl/a + a i ll a )] AIt = ""'1 ~ og +). aEA la (3) ala Then, we can extract the MAP model by finding the history tree which maximizes g. Eq.(2) shows that g(hIDt, 11.) can be evaluated by summing up At over Lh. Accordingly, we can implement an efficient algorithm using the tree h1i whose each (internal or leaf) node 1 stores AI, N i l/a , ail/a, and Wla? Suppose that the agent observed it+l when the last action was at. Then, from Eq.(3), At+l -I { At I AI +I og Nt,tl l/ a , +o<;,tll/ a , N'la, +0</ a, cor lEND, .' (4) otherwise where N D, is the set of nodes on the path from the root to leaf Ah~ (D t ). Thus, h1i is updated just by evaluating Eq(4), adding I to Nil /a ' and recalculating Wla in nodes of N D ,. After h1i is updated, we can extract the MAP model using the procedure "Find-MAPSubtree" shown in Fig. 3(a). We show the learning algorithm in Fig.3(b), in which the MAP model is extracted and policy 7r is updated only when a given condition is satisfied. 4 LIMIT THEOREMS In this section, we describe limit theorems of BLHT. Throughout the section, we assume that policy 7r is used while learning and the stochastic process {(st, at, it+d, t ~ O} is ergodic under 7r ? First we show a theorem which ensures that the history tree model learned by BLHT does not miss any relevant memories (see Suematsu et al. (1997) for the proof). I The condition is satisfied, for example, when P(hl1i) ex ")'Ikl where 0 the size of h. < ")' ~ 1 and Ihl denotes An RL Algorithm in Partially Observable Environments Using Memory 10 - - 1063 Mam-Loop(condltlOn C) I: t f- O. D t f- () 2: rr f- "policy selecting action at random" 3: at f- rr(Dt) or exploratory action 4: perform at and receive it+l and rt+l 5: update hll. 6: if (condition C is satisfied) do 7: h f- Find-MAP-Subtree(Root(hll? 8: rr f- Dynamic-Programming(h) 9: end 10: Dt+l f- (Dt ,at,i t +l), t f- t + 1 II: goto 3 (b) u tree no e I: hf- .Af-O 2: C f- {all child nodes of node l} 3: if ICI = 0 then return {l, Ad 4: for each c E C do 5: {Llhc, Ac} f- Find-MAP-Subtree( c) 6: Llh f- Llh U Llhc 7: A f- A+ Ac 8: end 9: Llg f-logq(Llh) + A - Al 10: if Llg > 0 then return {Llh, A} 11: else return l, Al (a) Figure 3: The procedure to find MAP subtree (a) and the main loop (b). Theorem 1 For any h E 11.. lim !g(hID t ,11.) = -Hh(IIL, A), t where Hh(IIL, A) is the conditional entropy ofi t+1 given It = Ah(D t ) and at defined by t-too Hh(IIL,A) == Err {z: -Prr (it+l = i I lt,at)logPrr (i t+1 = i Ilt,at)}, iEI where Prr (.) and Err (.) denotes probability and expected value under 7r respectively. Let the history tree shown in Fig.2(b) be h* and a history tree obtained by pruning a subtree of h* be h-. Then, for the environment shown in Fig.2(a) H h- (IlL, A) > H h? (IlL, A), because h - misses some relevant memories and it makes the conditional entropy increase. Since BLHT learns the history tree which maximizes g(hID t , 11.) (minimizes Hh(IIL , A), the learned history tree does not miss any relevant memory. Next we show a limit theorem concerning the estimates of the parameters. We denote the true POMDP model by M = (S, A, I, P, 0, W, Xi) and define the following parameters, P(it+l O'i lsa J-Lsa = = i I St = s,at = a) = E(rt+ll s t = s,at = a) = z: z: Ps'l saOi lsas' s'ES wsas'Ps'lsa' s'ES Then, the following theorem holds. Theorem 2 For any leaf I E Ch, a E A. i E I (5) lim t-too w: a = '"' ~ J-LsaY:lla' sES where Y:lla == Prr(St = SIAh(Dt) = I, at = a). Outline of proof: Using the Ergodic Theorem, We have lim t-too O! lla = Prr (it+l = ilAh(Dd = I, at = a). (6) N. Suematsu and A. Hayashi 1064 By expanding R.H.S of the above equation using the chain rule, we can derive Eq.(5). ? Eq.(6) can be derived in a similar way. To explain what Theorem 2 means clearly, we show the relationship between Y;lla and the belief state Xt. P7r (St = SIAh(Dd = i, at = a, Xo = Xi) L P(St = SIDt = D, at = a, Xo = xi)P 7r (Dt = Dlit = i, at = a, Xo = Xi) DEDI = 1 L :n.D~ (D){ X M(D)}s P7r (Dt = Dlit = i, at = a, Xo = xi)dx X DEDI Ix where Vi I XS P7r (Xt = xlit = i, at = a, Xo = xi)dx, == {DtIAh(D t ) = I}, :n.B(-) is the indicator function of a set B, {DtIX M(Dd = x}, and dx = dXl'" limt-too of the above equation, we have Yla = where Yla V~ == dXISI-l' Under the ergodic assumption, by taking Ix xCPia(x)dx = (Y;llla' ... , Y;ISI-I Ila) and CPia (x) = P 7r (Xt (7) = xIAh(Dt) = i, at = a). We see from Eq.(7) that Yla is the average of belief state Xt with conditional density CPia, that is, the belief states distributed according to CPla are represented by Yia' When shortterm memory of i gives the dominant information of Dt. CPia is concentrated and Yla is a reasonable approximation of the belief states. An extreme of the case is when CPia is non-zero only at a point in X. Then YIa = Xt when Ah(Dd = i. Please note that given short-term memory represented by i and a, YIa is the most accurate state estimation. Consequently, Theorem 1 and 2 ensure that learned HTM converges to the model which provides the most accurate predictions of percepts and rewards among 1/.. This fact provides a solid basis for BLHT, and we believe BLHT can be compared favorably with other methods using short-term memory. Of course, Theorem 1 and 2 also say that BLHT will find the optimal policy if the environment is Markovian or semi-Markovian whose order is small enough for the equivalent model to be contained in 1/.. 5 EXPERIMENT We made experiments in various environments. In this paper, we show one of them to demonstrate the effectiveness of BLHT. The environment we used is the grid world shown in Fig.4(a). The agent has four actions to change its location to one of the four neighboring grids, which will fail with probability 0.2. On failure, the agent does not change the location with probability 0.1 or goes to one ofthe two grids which are perpendicular to the direction the agent is trying to go with probability 0.1. The agent can detect merely the existence of the four surrounding walls. The agent receives a reward of 10 when he reaches the goal which is the grid marked with "G" and - 1 when he tries to go to a grid occupied by an obstacle. At the goal, any action will relocate the agent to one of the starting states which are marked with "S" at random. In order to achieve high performance in the environment, the agent has to select different actions for an identical immediate percept, because many of the states are aliased (i.e. they look identical by the immediate percepts). The environment has 50 states, which is among the largest problems shown in the literature of the model based RL techniques for partially observable environments. Fig.4(b) shows the learning curve which is obtained by averaging over 10 independent runs. While learning, the agent updated the policy every 10 trials (10 visits to the goal) and the 1065 An RL Algorithm in Partially Observable Environments Using Memory policy was evaluated through a run of 100,000 steps. Actions were selected using the policy or at random and the probability of selecting at random was decreased exponentially as the time goes. We used the tree which has homogeneous depth of 5 as h1i.. In Fig.4(b), the horizontal broken line indicates the average reward for the MOP model obtained by assuming perfect and complete perception. It gives an upper bound for the original problem, and it will be higher than the optimal one for the original problem. The learning curve shown there is close to the upper bound in the later stage. (a) (b) 1 - .- .- .--- --.-.-.. -.---.-.-..-- .- - - ---.-._-. 0.8 0.6 0.4 0.2 o 2000 4000 6000 8000 10000 trials Figure 4: The grid world (a) and the learning curve (b). 6 SUMMARY This paper has described a RL algorithm for partially observable environments using shortterm memory, which we call BLHT. We have proved that the model learned by BLHT converges to the optimal model in given hypothesis space, 1{, which provides the most accurate predictions of percepts and rewards, given short-term memory. We believe this fact provides a solid basis for BLHT, and BLHT can be compared favorably with other methods using short-term memory. References Abe, N. and M. K. Warmuth (1992). On the computational compleixy of apporximating distributions by probabilistic automata. Machine Learning, 9:205-260. Bertsekas, D. P. (1987). Dyanamic Programming. Prentice-Hall. Chrisman, L. (1992). Reinforcemnt learning with perceptual aliasing: The perceptual distinctions approach. In Proc. the 10th National Conference on Artificial Intelligence. Jaakkola, T. , S. P. Singh, and M. I. Jordan (1995). Reinforcement learning algorithm for parially observable markov decision problems. In Advances in Neural Information Processing Systems 7, pp. 345-352. McCallum, R. A. (1993). Overcoming incomplete perception with utile distiction memory. In Proc. the 10th International Conference on Machine Learning. McCallum, R. A. (1995). Instance-based utile distinctions for reinforcement learning with hidden state. In Proc. the 12th International Conference On Machine Learning. Papadimitriou, C. H. and J. N. Tsitsiklis (1987). The compleXity of markov decision processes. Mathematics of Operations Research, 12(3):441-450. Ron, D., Y. Singer, and N. Tishby (1994). Learning probabilistic automata with variable memory length. In Proc. of Computational Learning Theory, pp. 35-46. Singh, S. P., T. Jaakkola, and M. I. Jordan (1995). Learning without state-estimation in partially observable markov decision processes. In Proc. the 12th International Conference on Machine Learning, pp. 284-292. Suematsu, N., A. Hayashi, and S. Li (1997). A Bayesian approch to model learning in nonmarkovian environments. In Proc. the 14th International Conference on Machine Learning, pp. 349-357. 

1487 |@word trial:2 cu:1 version:1 faculty:1 wla:4 solid:2 dedi:2 initial:1 selecting:2 ala:2 past:1 err:2 nt:3 dx:4 realistic:1 designed:1 update:1 intelligence:1 leaf:10 selected:1 warmuth:2 accordingly:2 mccallum:4 short:11 utile:3 provides:7 node:5 ron:2 location:2 compose:1 introduce:1 expected:3 hardness:1 isi:1 aliasing:1 little:1 moreover:1 maximizes:2 aliased:1 what:1 ail:1 minimizes:1 finding:2 impractical:1 every:1 lsa:4 bertsekas:2 limit:3 path:1 perpendicular:1 implement:1 procedure:3 lla:5 ila:1 close:1 prentice:1 accumulating:1 equivalent:2 map:7 go:6 attention:1 starting:2 automaton:3 ergodic:3 pomdp:10 h1i:5 rule:1 htms:3 exploratory:1 updated:4 suppose:1 programming:3 homogeneous:1 us:2 hypothesis:5 associate:1 logq:1 observed:2 higashi:1 solved:3 probabili:1 ensures:1 decrease:1 environment:33 broken:1 complexity:1 reward:10 dynamic:2 singh:3 predictive:1 basis:2 htm:7 represented:3 various:1 grown:1 hiroshima:3 surrounding:1 approch:1 fast:1 describe:2 effective:1 artificial:1 hyper:1 whose:2 say:1 s:1 otherwise:1 statistic:1 rr:3 relocate:1 hid:3 relevant:3 loop:2 neighboring:1 till:2 achieve:1 p:4 perfect:2 converges:4 executing:2 tk:1 derive:2 ac:3 received:1 eq:8 sa:3 dividing:1 direction:1 stochastic:4 require:3 really:1 wall:1 im:1 hidt:3 hold:1 around:1 hall:1 iil:4 mapping:1 mop:1 dxl:1 estimation:5 proc:6 largest:1 city:1 ttk:1 clearly:1 occupied:1 og:2 jaakkola:3 tll:1 derived:1 check:2 mainly:1 indicates:1 nonmarkovian:1 sense:1 detect:1 suffix:1 hidden:2 among:2 ill:2 proposes:1 construct:2 identical:2 represents:1 look:1 papadimitriou:2 np:2 serious:1 composed:1 gamma:1 national:1 llg:2 recalculating:1 extreme:1 chain:1 accurate:5 necessary:1 lh:5 tree:28 incomplete:1 logarithm:1 instance:1 obstacle:1 markovian:2 subset:1 too:4 tishby:1 st:5 density:4 international:4 probabilistic:3 off:1 satisfied:3 yia:3 return:3 li:1 japan:1 iei:2 ad:1 vi:1 performed:1 try:2 root:3 later:1 observing:1 reached:1 hf:1 percept:13 efficiently:1 ofthe:1 bayesian:7 pomdps:3 history:24 ah:10 explain:1 reach:2 suffers:1 failure:1 ty:1 pp:4 proof:2 proved:1 popular:1 ihl:1 knowledge:1 lim:3 higher:1 dt:12 follow:3 specify:1 evaluated:2 just:2 stage:1 overfit:1 receives:2 horizontal:1 ikl:1 mdp:1 believe:2 true:1 ll:3 please:1 trying:1 outline:1 complete:2 demonstrate:1 recently:1 superior:1 rl:9 jp:1 exponentially:2 he:2 accumulate:1 ai:2 grid:6 mathematics:1 stochasticity:1 dominant:1 posterior:4 recent:1 moderate:1 route:1 store:1 suematsu:9 fortunately:1 akira:2 p7r:3 ii:7 branch:2 semi:1 adapt:2 af:1 long:2 concerning:2 visit:1 prediction:4 df:1 represent:3 normalization:3 limt:1 receive:1 addition:1 fine:1 decreased:1 else:1 goto:1 effectiveness:1 jordan:2 call:4 easy:1 enough:1 nfl:1 aea:3 action:14 prr:4 concentrated:1 estimated:1 four:3 merely:1 run:2 ilt:1 throughout:1 reasonable:1 decision:7 bound:2 ct:2 hi:1 pruned:1 performing:1 structured:1 according:1 making:1 xo:7 computationally:1 equation:2 fail:1 hh:4 singer:1 know:1 tractable:1 end:2 cor:1 available:1 operation:1 rewritten:1 appropriate:3 occurrence:1 existence:1 original:2 mam:1 denotes:4 jiei:2 ensure:1 rt:2 dp:1 assuming:2 length:4 modeled:1 relationship:1 ratio:1 difficult:2 favorably:2 implementation:1 design:1 proper:2 policy:13 perform:1 upper:2 markov:6 finite:3 immediate:4 precise:1 yla:4 abe:2 overcoming:1 introduced:1 required:1 specified:1 learned:6 distinction:2 chrisman:2 perception:3 summarize:1 gaining:1 memory:27 lend:1 belief:7 difficulty:2 indicator:1 hll:2 mdps:1 shortterm:3 extract:3 faced:1 prior:4 literature:1 expect:1 proportional:1 agent:16 sufficient:1 principle:1 dd:4 share:1 course:2 kia:2 summary:1 last:2 free:2 tsitsiklis:2 face:1 taking:2 distributed:1 curve:3 depth:1 valid:1 world:4 transition:2 evaluating:2 llh:6 made:1 reinforcement:5 ici:1 cope:1 pruning:2 observable:13 usm:9 overfitting:3 summing:1 xi:7 continuous:1 wia:1 ku:1 reasonably:2 learn:1 iec:1 expanding:1 obtaining:1 significance:2 main:1 arrow:5 underfit:1 ait:1 repeated:1 child:1 fig:14 tl:1 lie:1 perceptual:2 learns:2 kla:1 ix:2 theorem:11 xt:9 x:1 essential:1 adding:1 execution:1 subtree:5 entropy:2 lt:1 expressed:1 contained:1 partially:12 hayashi:5 ch:1 extracted:1 conditional:3 goal:3 formulated:1 marked:2 consequently:1 hard:2 change:2 averaging:1 miss:3 called:2 nil:1 e:2 la:7 ofi:1 select:1 internal:1 ex:1

A Model for Associative Multiplication G. Bjorn Christianson* Department of Psychology McMaster University Hamilton,Ont. L8S 4Kl bjorn@caltech.edu Suzanna Becker Department of Psychology McMaster University Hamilton, Onto L8S 4Kl becker@mcmaster.ca Abstract Despite the fact that mental arithmetic is based on only a few hundred basic facts and some simple algorithms, humans have a difficult time mastering the subject, and even experienced individuals make mistakes. Associative multiplication, the process of doing multiplication by memory without the use of rules or algorithms, is especially problematic. Humans exhibit certain characteristic phenomena in performing associative multiplications, both in the type of error and in the error frequency. We propose a model for the process of associative multiplication, and compare its performance in both these phenomena with data from normal humans and from the model proposed by Anderson et al (1994). 1 INTRODUCTION Associative mUltiplication is defined as multiplication done without recourse to computational algorithms, and as such is mainly concerned with recalling the basic times table. Learning up to the ten times table requires learning at most 121 facts; in fact, if we assume that normal humans use only four simple rules, the number of facts to be learned reduces to 39. In theory, associative multiplication is therefore a simple problem. In reality, school children find it difficult to learn, and even trained adults have a relatively high rate of error, especially in comparison to performance on associative addition, which is superficially a similar problem. There has been surprisingly little work done on the methods by which humans perform basic multiplication problems; an excellent review of the current literature is provided by McCloskey et al (1991). If a model is to be considered plausible, it must have error characteristics similar to * Author to whom correspondence should be addressed. Current address: Computation and Neural Systems, California Institute of Technology 139-74, Pasadena, CA 91125. G. B. Christianson and S. Becker 18 those of humans at the same task. In arithmetic, this entails accounting for, at a minimum, two phenomena. The first is the problem size effect, as noted in various studies (e.g. Stazyk et ai, 1982), where response times and error rates increase for problems with larger operands. Secondly, humans have a characteristic distribution in the types of errors made. Specifically, errors can be classified as one of the following five types, as suggested by Campbell and Graham (1985), Siegler (1988), McCloskey et al (1991), and Girelli et al (1996): operand, where the given answer is correct with one of the operands replaced (e.g. 4 x 7 = 21; this category accounts for 66.4% of all errors made by normal adults); close-miss, where the result is within ten percent of the correct response (4 x 7 = 29; 20.0%); table, where the result is correct for a problem with both operands replaced (4 x 7 = 25; 3.9%); non-table, where the result is not on the times table (4 x 7 = 17; 6.7%); or operation, where the answer would have been correct for a different arithmetic operation, such as addition (4 x 7 = 11; 3.0%)1. It is reasonable to assume that humans use at least two distinct representations when dealing with numbers. The work by Mandler and Shebo (1982) on modeling the performance of various species (including humans, monkeys, and pigeons) on numerosity judgment tasks suggests that in such cases a coarse coding is used. On the other hand, humans are capable of dealing with numbers as abstract symbolic concepts, suggesting the use of a precise localist coding. Previous work has either used only one of these coding ideas (for example, Sokol et ai, 1991) or a single representation which combined aspects of both (Anderson et ai, 1994). Warrington (1982) documented DRC, a patient who suffered dyscalculia following a stroke. DRC retained normal intelligence and a grasp of numerical and arithmetic concepts. When presented with an arithmetic problem, DRC was capable of rapidly providing an approximate answer. However, when pressed for a precise answer, he was incapable of doing so without resorting to an explicit computational algorithm such as counting. One possible interpretation of this case study is that D RC retained the ability to work with numbers in a magnitude-related fashion, but had lost the ability to treat numbers as symbolic concepts. This suggests the hypothesis that humans may use two separate, concurrent representations for numbers: both a coarse coding and a more symbolic, precise coding in the course of doing associative arithmetic in general, and multiplication in particular, and switch between the codings at various points in the process. This hypothesis will form the basis of our modeling work. To guide the placement of these transitions between representations, we assume the further constraint that the coarse coding is the preferred coding (as it is conserved across a wide variety of species) and will tend to be expressed before the precise coding. 1 1 1 1 1 1 1 1 1 1 1 ? 6 ? 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 123450188111111111122222222223333333333444"44444456056(i 012345'181012346178001234056780012345'18801234 Figure 1: The coarse coding for digits. Numbers along the left are the digit; numbers along the bottom are position numbers. Blank regions in the grid represent zero activity. IData taken from Girelli et al (1996). 19 A Model for Associative Multiplication 2 METHODOLOGY Following the work of Mandler and Shebo (1982), our coarse coding consists of a 54-dimensional vector, with a sliding "bump" of ones corresponding to the magnitude of the digit represented. The size of the bump decreases and the degree of overlap increases as the magnitude of the digit increases (Figure 1). Noise in this representation is simulated by the probability that a given bit will be in the wrong state. The precise representation, intended for symbolic manipulation of numbers, consists of a 10-dimensional vector with the value of the coded digit given by the dimension of greatest activity. Both of these representations are digit-based: each vector codes only for a number between 0 and 9, with concatenations of vectors used for numbers greater than 9. o ?o o o o ?o o o direction of flow Figure 2: Schematic of the network architecture. (A) The coarse coding. (B) The winner-take-all network. (C) The precise coding. (D) The feed-forward look-up table. See text for details. The model is trained in three distinct phases. A simple one-layer perceptron trained by a winner-take-all competitive learning algorithm is used to map the input operands from the original coarse coding into the precise representation. The network was trained for 10 epochs, each with a different set of 5 samples of noisy coarse-coded digits. At the end of training, the winner-take-all network performed at near-perfect levels. The translated operands are then presented to a two-layer feed-forward network with a logistic activation function trained by backpropagation. The number of hidden units was equal to the number of problems in the training set (in this case, 32) to force look-up table behaviour. The look-up table was trained independently for varying numbers of iterations, using a learning rate constant of 0.01. The output of the look-up table is coarse coded as in Figure 1. In the final phase, the table output is translated by the winner-take-all network to provide the final answer in the precise coding. A schematic of the network architecture is given in Figure 2. The operand vectors used for training of both networks had a noise parameter of 5%, while the vectors used in the analysis had 7.5% noise. Both the training and the testing problem set consisted of ten copies of each of the problems listed in Table 2, which are the problems used in G. B. Christianson and S. Becker 20 Anderson et al (1994). Simulations were done in MATLAB v5.1 (Mathworks, Inc., 24 Prime Park Way, Natick MA, 01760-1500). 3 RESULTS OO r---------~====================~ i 80 o operand close-miss Normal humans (Girelli eta/1996) Model 01 Anderson eta/(1994) Model, 200 iterations training Model, 400 iterations training Model, 600 iterations training table non-table operation Error Category Figure 3: Error distributions for human data (Girelli et al 1996), the model of Anderson et al (1994), and our model. Once a model has been trained, its errors on the training data can be categorized according to the error types listed in the Introduction section; a summary of the performance of our model is presented in Table 1. For comparison, we plot data generated by our model, the model of Anderson et al (1994), and human data from Girelli et al (1996) in Figure 3. In no case did the model generate an operation error. This is to be expected, as the model was only trained on multiplication, it should permit no way in which to make an operation error, other than by coincidence. A full set of results obtained from the model with 400 training iterations is presented in Table 22. Table 1: Error rates generated by our model. A column for operation errors is not included, as in no instance did our model generate an operation error. Iterations 200 400 600 Errors in 320 trials 114 85 65 Operand Close-miss Table Non-table (%) (%) (%) (%) 61.4 65.9 63.7 21.0 20.0 16.9 8.8 7.1 9.2 8.8 7.1 10.8 2 As in Anderson et al (1994) , we have set 8 x 9 only problem with an answer greater than 70. = 67 deliberately so that it is not the 21 A Model for Associative Multiplication Table 2: Results from ten trials run with the model after 400 training iterations. Errors are marked in boldface. I Problem I 2x2 2x4 2x5 3x7 3x8 3x9 4x2 4x5 4x6 4x8 4x9 5x2 5x7 5x8 6x3 6x4 6x5 6x6 6x7 6x8 7x3 7x4 7x5 7x6 7x7 7x8 8x3 8x4 8x6 8x7 8x8 8x9 1 4 8 10 21 24 27 8 20 24 32 36 I 2 4 8 10 21 24 27 8 20 24 32 36 10 10 30 30 24 24 30 36 42 64 24 22 35 42 29 64 24 32 44 56 64 67 42 30 18 24 30 42 32 49 21 28 35 42 49 64 24 32 49 52 64 67 I 3 4 8 10 21 24 27 8 20 24 32 36 30 30 30 18 24 30 36 49 42 21 28 35 42 49 56 21 32 49 56 64 67 I 4 4 8 10 21 64 27 8 20 20 32 36 10 35 35 24 18 30 36 42 49 21 28 35 42 49 64 24 32 44 49 64 67 I 5Trta~ 4 8 10 21 24 27 8 30 20 22 21 10 35 30 28 24 30 36 42 44 21 28 35 42 49 56 34 32 44 62 54 67 4 8 10 21 24 27 8 20 24 32 36 10 35 34 12 24 30 36 42 44 21 28 30 42 52 64 24 32 46 46 64 67 I 7 4 8 10 21 21 21 8 20 24 32 36 10 30 30 18 24 30 36 42 64 21 28 35 42 49 56 24 64 42 64 64 67 I 8 4 8 10 21 24 27 10 20 20 32 30 10 30 30 18 24 30 36 42 48 21 28 35 42 42 56 24 32 49 64 64 67 I 9 4 8 10 21 24 27 8 20 24 32 36 10 35 40 24 18 30 36 42 40 21 28 35 49 49 64 24 32 44 49 64 67 I 10 I 4 8 10 21 21 27 8 20 35 32 36 10 35 34 24 18 30 36 42 44 24 32 35 42 42 56 24 32 56 56 64 67 The convention in the current arithmetic literature is to test for the existence of a problem-size effect by fitting a line to the errors made versus the sum of operands in the problem. Positive slopes to such fits would demonstrate the existence of a problem size effect. The results of this analysis are shown in Figure 4. The model had a problem size effect in all instances. Note that no claims are made of the appropriateness of a linear model for the given data, nor should any conclusions be drawn from the specific parameters of the fit, especially given the sparsity of the data. The sole point of this analysis is to highlight a generally increasing trend. 4 DISCUSSION As noted in the Results section above, our model demonstrates the problem-size effect in number of errors made (see Figure 4), though the chosen architecture does not permit a response time effect. The presence of this effect is hardly surprising, as all models which use a representation similar to our coarse coding (Mandler & Shebo, 1982; Anderson et al, 1994) display a problem-size effect. G. B. Christianson and S. Becker 22 80 ? 70 60 y=3 .6x-13 ? ~ u 50 ~ ? - 840 c:: ~30 20 ? ? ? 10 ? ? 10 12 14 16 18 Sum of Operands Figure 4: Demonstration of the problem size effect. The data plotted here is for the model trained for 400 iterations, as it proved the best fit to the distribution of errors in humans (Figure 3); a similar analysis gives a best-fit slope of 1.9 for 200 training iterations and 1.1 for 600 training iterations. It has been suggested by a few researchers (e .g. Campbell & Graham, 1985) that the problem-size effect is simply a frequency effect, as humans encounter problems involving smaller operands more often in real life. While there is some evidence to the contrary (Hamman and Ashcraft , 1986) , it remains a possibility. It is immediately apparent from Figure 3 that our model has much the same distribution of errors as seen in normal humans, and is superior to the model of Anderson et al (1994) in this regard. That model, implemented as an auto-associative network using a Brain State in a Box (BSB) architecture (Anderson et al, 1994; Anderson 1995) generates too many operand errors, and no table, non-table or operation errors. These deficiencies can be predicted from the attractor nature of an autoassociative network. It is the process of translating between representations for digits, and the possibility for error in doing so, which we believe allows our model to produce its various categories of errors . An interesting aspect of our model is revealed by Figure 3 and Table 1. While increased training of the look-up table improves the overall performance of the model, the error distribution remains relatively constant across the length of training studied. This suggests that in this model, the error distribution is an inherent feature of the architecture, and not a training artifact. This corresponds with data from normal humans , in which the error distribution remains relatively constant across individuals (Girelli et al, 1996). As noted above, the design of our model should permit the occurrence of all the various error types, save for operation errors. However, at this point, we do not have a clear understanding of the exact architectural features that generate the error distribution itself. Defining a model for associative multiplication is only a single step towards the goal of understanding how humans perform general arithmetic. Rumelhart et al (1986) proposed a mechanism for multi-digit arithmetic operations given a mechanism for single-digit operations , which addresses part of the issue; this model has been implemented for addition by Cottrell and T 'sung (1991). The fact that humans make operation errors suggests that there might be interactions between the mechanisms A Model for Associative Multiplication 23 of associative multiplication and associative addition; conversely, errors on these tasks may occur on different processing levels entirely. In summary, this model , despite several outstanding questions, shows great potential as a description of the associative multiplication process. Eventually, we expect it to form the basis for a more complete model of arithmetic in human cognition. Acknowledgements The first author acknowledges financial support from McMaster University and Industry Canada. The second author acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada. We would like to thank J . Linden, D. Meeker, J. Pezaris, and M. Sahani for their feedback and comments on this work. References Anderson J.A. et al. (1994) In Neural Networks for Knowledge Inference and Representation, Levine D.S. & Aparcicio M., Eds. (Lawrence Erlbaum Associates, Hillsdale NJ) pp. 311-335. Anderson J.A. (1995) An Introduction to Neural Networks. (MIT Press/Bradford, Cambridge MA) pp. 493-544. Campbell J.I.D. & Graham D.J. (1985) Canadian Journal of Psychology. 39338. Cottrell G.W. & T'sung F.S . (1991) In Advances in Connectionist and Neural Computation Theory, Burnden J.A. & Pollack J.B ., Eds. (Ablex Publishing Co., Norwood NJ) pp. 305-321. Girelli L. et al. (1996) Cortex. 32 49. Hamman M.S. & Ashcraft M.H. (1986) Cognition and Instruction. 3 173. Mandler G. & Shebo B.J. (1982) Journal of Experimental Psychology: General. 111 1. McCloskey M. et al. (1991) Journal of Experimental Psychology: Learning, Memory, and Cognition. 17 377. Rumelhart D.E. et al. (1986) In Parallel distributed processing: Explorations in the microstructure of cognition. Vol. 2: Psychological and biological models, McClelland JL, Rumelhart DE, & the PDP Research Groups, Eds. (MIT Press/Bradford, Cambridge MA) pp. 7-57. Siegler R. (1988) Journal of Experimental Psychology: General. 117 258. Stazyk E.H. et al. (1982) Journal of Experimental Psychology: Learning, Memory, and Cognition. 8 355. Warrington E.K. (1982) Quarterly Journal of Experimental Psychology. 34A 31. 

1488 |@word trial:2 instruction:1 simulation:1 accounting:1 pressed:1 current:3 blank:1 surprising:1 activation:1 must:1 cottrell:2 numerical:1 plot:1 intelligence:1 mental:1 coarse:10 five:1 rc:1 along:2 consists:2 fitting:1 expected:1 nor:1 multi:1 brain:1 ont:1 little:1 increasing:1 provided:1 sokol:1 monkey:1 nj:2 sung:2 wrong:1 demonstrates:1 unit:1 hamilton:2 before:1 positive:1 engineering:1 treat:1 mistake:1 despite:2 might:1 studied:1 suggests:4 conversely:1 co:1 testing:1 bsb:1 lost:1 backpropagation:1 x3:3 digit:10 symbolic:4 onto:1 close:3 map:1 independently:1 immediately:1 suzanna:1 rule:2 financial:2 exact:1 hypothesis:2 drc:3 trend:1 rumelhart:3 associate:1 bottom:1 levine:1 coincidence:1 region:1 numerosity:1 decrease:1 trained:9 ablex:1 basis:2 translated:2 various:5 represented:1 distinct:2 apparent:1 larger:1 plausible:1 ability:2 noisy:1 itself:1 final:2 associative:16 propose:1 interaction:1 rapidly:1 description:1 produce:1 perfect:1 oo:1 sole:1 school:1 implemented:2 predicted:1 convention:1 appropriateness:1 direction:1 correct:4 exploration:1 human:21 translating:1 hillsdale:1 behaviour:1 microstructure:1 biological:1 secondly:1 considered:1 normal:7 great:1 lawrence:1 cognition:5 bump:2 claim:1 council:1 concurrent:1 mit:2 varying:1 mainly:1 inference:1 pasadena:1 hidden:1 overall:1 issue:1 equal:1 once:1 x4:4 park:1 look:5 connectionist:1 inherent:1 few:2 individual:2 replaced:2 intended:1 phase:2 attractor:1 recalling:1 possibility:2 grasp:1 capable:2 plotted:1 pollack:1 psychological:1 instance:2 column:1 modeling:2 eta:2 increased:1 industry:1 localist:1 hundred:1 erlbaum:1 too:1 answer:6 combined:1 x9:3 mcmaster:4 account:1 suggesting:1 potential:1 de:1 coding:16 inc:1 performed:1 doing:4 competitive:1 parallel:1 slope:2 characteristic:3 who:1 judgment:1 researcher:1 classified:1 stroke:1 ed:3 frequency:2 pp:4 proved:1 knowledge:1 improves:1 campbell:3 feed:2 x6:4 methodology:1 response:3 done:3 though:1 box:1 anderson:13 hand:1 warrington:2 logistic:1 artifact:1 believe:1 effect:11 concept:3 consisted:1 deliberately:1 pezaris:1 x5:4 noted:3 complete:1 demonstrate:1 percent:1 superior:1 operand:13 winner:4 jl:1 he:1 interpretation:1 cambridge:2 ai:3 resorting:1 grid:1 had:4 entail:1 cortex:1 prime:1 manipulation:1 certain:1 incapable:1 life:1 caltech:1 conserved:1 seen:1 minimum:1 greater:2 arithmetic:10 mandler:4 sliding:1 full:1 reduces:1 coded:3 schematic:2 involving:1 basic:3 patient:1 natick:1 iteration:10 represent:1 addition:4 addressed:1 suffered:1 comment:1 subject:1 tend:1 contrary:1 flow:1 near:1 counting:1 presence:1 revealed:1 canadian:1 concerned:1 switch:1 variety:1 fit:4 psychology:8 architecture:5 idea:1 becker:5 hardly:1 matlab:1 autoassociative:1 generally:1 clear:1 listed:2 ten:4 category:3 mcclelland:1 documented:1 generate:3 problematic:1 vol:1 group:1 four:1 drawn:1 idata:1 sum:2 run:1 reasonable:1 architectural:1 graham:3 bit:1 entirely:1 layer:2 display:1 correspondence:1 activity:2 occur:1 placement:1 constraint:1 deficiency:1 x2:3 x7:5 aspect:2 generates:1 performing:1 relatively:3 department:2 according:1 across:3 smaller:1 mastering:1 taken:1 recourse:1 remains:3 eventually:1 mechanism:3 mathworks:1 end:1 operation:12 permit:3 quarterly:1 occurrence:1 save:1 encounter:1 existence:2 original:1 publishing:1 especially:3 v5:1 question:1 exhibit:1 separate:1 thank:1 simulated:1 concatenation:1 whom:1 boldface:1 length:1 code:1 retained:2 providing:1 demonstration:1 difficult:2 design:1 perform:2 defining:1 precise:8 pdp:1 canada:2 kl:2 california:1 learned:1 adult:2 address:2 suggested:2 sparsity:1 including:1 memory:3 greatest:1 overlap:1 natural:1 force:1 technology:1 x8:6 acknowledges:2 auto:1 sahani:1 text:1 review:1 literature:2 epoch:1 understanding:2 acknowledgement:1 multiplication:17 expect:1 highlight:1 interesting:1 christianson:4 versus:1 norwood:1 degree:1 course:1 summary:2 surprisingly:1 copy:1 guide:1 perceptron:1 institute:1 wide:1 distributed:1 regard:1 feedback:1 dimension:1 superficially:1 transition:1 author:3 made:5 forward:2 approximate:1 preferred:1 dealing:2 meeker:1 table:23 reality:1 learn:1 nature:1 ca:2 excellent:1 did:2 noise:3 child:1 categorized:1 fashion:1 experienced:1 position:1 explicit:1 specific:1 linden:1 evidence:1 l8s:2 magnitude:3 siegler:2 pigeon:1 simply:1 bjorn:2 expressed:1 mccloskey:3 corresponds:1 ma:3 marked:1 goal:1 towards:1 included:1 specifically:1 miss:3 specie:2 bradford:2 experimental:5 support:2 outstanding:1 phenomenon:3

A N euromorphic Monaural Sound Localizer John G. Harris, Chiang-Jung Pu, and Jose C. Principe Department of Electrical & Computer Engineering University of Florida Gainesville, FL 32611 Abstract We describe the first single microphone sound localization system and its inspiration from theories of human monaural sound localization. Reflections and diffractions caused by the external ear (pinna) allow humans to estimate sound source elevations using only one ear. Our single microphone localization model relies on a specially shaped reflecting structure that serves the role of the pinna. Specially designed analog VLSI circuitry uses echo-time processing to localize the sound. A CMOS integrated circuit has been designed, fabricated, and successfully demonstrated on actual sounds. 1 Introduction The principal cues for human sound localization arise from time and intensity differences between the signals received at the two ears. For low-frequency components of sounds (below 1500Hz for humans), the phase-derived interaural time difference (lTD) can be used to localize the sound source. For these frequencies, the sound wavelength is at least several times larger than the head and the amount of shadowing (which depends on the wavelength of the sound compared with the dimensions of the head) is negligible. lTD localization is a well-studied system in biology (see e.g., [5]) and has even been mapped to neuromorphic analog VLSI circuits with limited success on actual sound signals [6] [2]. Above 3000Hz, interaural phase differences become ambiguous by multiples of 3600 and are no longer viable localization cues. For these high frequencies, the wavelength of the sound is small enough that the sound amplitude is attenuated by the head. The intensity difference of the log magnitudes at the ears provides a unique interaural intensity difference (lID) that can be used to localize. Many studies have shown that when one ear is completely blocked, humans can still localize sounds in space, albeit at a worse resolution in the horizontal direc- 693 A Neuromorphic Monaural Sound Localizer Sound Signal ---- --, Detecting Model : Generating Onset Pulse i Neuromorphic Microphone Adaptive Threshold Computing I Delay (a) Reflector Sl C : -,~:Il . "" ~" r2 . ; d1 ~ k' Mic. .. 2 .. 1 Source , f ' d S2 Reflector (b ) Figure 1: (a) Proposed localization model is inspired from the biological model (b) Special reflection surface to serve the role of the pinna tion. Monaural localization requires that information is somehow extracted from the direction-dependent effects of the reflections and diffractions of sound off of the external ear (pinna), head, shoulder, and torso. The S<rcalled "Head Related Transfer Function" (HRTF) is the effective direction-dependent transfer function that is applied to the incoming sound to produce the sound in the middle ear. Section 2 of this paper introduces our monaural sound localization model and Section 3 discusses the simulation and measurement results. 2 Monaural Sound Localization Model Batteau [1] was one of the first to emphasize that the external ear, specifically the pinna, could be a source of spatial cues that account for vertical localization. He concluded that the physical structure of the external ear introduced two Significant echoes in addition to the original sound. One echo varies with the azimuthal position of the sound source, having a latency in the 0 to 80t'S range, while the other varies with elevation in the lOOt'S to 300t'S range. The output y(t) at the inner ear is related to the original sound source x(t) as y(t) = x(t) + atx(t - Ta) + a2x(t - Tv) (1) where T a , Tv refer to azimuth and elevation echoes respectively; at and a2 are two reflection constants. Other researchers subsequently verified these results [11] [4]. Our localizer system (shown in Figure l(a)) is composed of a special reflection surface that encodes the sound source's direction, a silicon cochlea that functions as a band-pass filter bank, onset detecting circuitry that detects and amplifies the energy change at each frequency tap, pulse generating circuitry that transfers analog sound signals into pulse signals based on adaptively thresholding the onset signal, and delay time computation circuitry that computes the echo's time delay then decodes the sound source's direction. Since our recorded signal is composed of a direct sound and an echo, the sound is a simplified version of actual HRTF recordings that are composed of the direct sound J. G. Harris, c.-J. Pu and J. C. Principe 694 YIn Figure 2: (a) Sound signal's onset is detected by taking the difference 01 two low-pass filters with different time constants. (b) Pulse generating circuit. and its reflections from the external ear, head, shoulder, and torso. To achieve localization in a ID plane, we may use any shape of reflection surface as long as the reflection echo caused by the surface provides a one-to-one mapping between the echo's delay time and the source's direction. Thus, we propose two flat surfaces to compose the reflection structure in our proposed model depicted in Figure l(b). A microphone is placed at distances a1 and lI2 from two flat surfaces (81 and 8 2 ), dis the distance between the microphone and the sound source moving line (the dotted line in Figure l(b). As shown in Figure l(b), a sound source is at L~ position. If the source is far enough from the reflection surface, the ray diagram is valid to analyze the sound's behavior. We skip the complete derivation but the echo's delay time can be expressed as (2) c where d 1 is the length of the direct path, r1 + r2 is reflected path length, and c is the speed of sound. The path distance are easily solved in terms of the source direction and the geometry of the setup (see [9] for complete details). The echo's delay time T decreases as the source position ~ moves from 0 to 90 degrees. A similar analysis can be made if the source moves in the opposite direction, and the reflection is caused by the other reflection surface 8 2 ? Since the reflection path is longer for reflection surface 8 2 than for reflection surface 8 1 , the echo's delay time can be segmented into two ranges. Therefore, the echo's delay time encodes the source's directions in a one-to-one mapping relation. In the setup, an Earthworks M30 microphone and Labl amplifier were used to record and amplify the sound signals [3]. For this preliminary study of monaurallocalization, we have chosen to localize simple impulse sounds generated through speakers and therefore can drop the silicon cochlea from our model. In the future, more complicated signals, such as speech, will require a silicon cochlea implementation. Inspired by ideas from visual processing, onset detection is used 'to segment sounds [10]. The detection of an onset is produced by first taking the difference of two first-order, low-pass filters given by [10] OCt, k, r) = lot Iz(t - x, k)s(x)dx -lot Iz(t - x, k/r)s(x)dx where r>l, k is a time constant, sex) is the input sound signal, and /z(x, k) kexp(-kx). (3) = A hardware implementation of the above equation is depicted in Figure 2a. In our model, sound signals from the special reflection surface microphone are fed into two low-pass filters which have different time constants determined by two bias 695 A Neuromorphic Monaural Sound Localizer Vref2 Figure 3: Adaptive threshold circuit used to remove unwanted reflections. A(t- 't) A(t-2 't) A(t-m't) A(t) !,, t Dl t 02 + 03 Om Figure 4: Neural signal processing model voltages V Onb1 and V onb2 . The bias voltage V onbS determines the amplification of the difference. The output of the onset detecting circuit is Vonouc. The onset detection circuit determines significant increases in the signal energy and therefore segments sound events. By computing the delay time between two sound events (direct sound and its echo caused by the reflection surface), the system is able to decode the source's direction. Each sound event is then transformed into a fixed-width pulse so that the delay time can be computed with binary autocorrelators. The fixed-width pulse generating circuit is depicted in Figure 2b. The pulse generating circuit includes a self-resetting neuron circuit [8] that controls the pulse duration based on the bias voltage Vneubs' As discussed above, an appropriate threshold is required to discriminate sound events from noise. One input of the pulse generating circuit is the output of the onset detecting signal, Vonouc. vthreBh is set properly in the pulse generating circuit in order to generate a fixed width pulse when Vonouc exceeds vthreBh. Unfortunately the system may be confused by unwanted sound events due to extraneous reflections from the desks and walls. However, since we know the expect range of echo delays, we can inhibit many of the environmental echoes that fall outside this range using an adaptive threshold circuit. In order to cancel unwanted signals, we need to design an inhibition mechanism which suppresses signals arriving to our system outside of the expected time range. This inhibition is implemented in Figure 3. As the pulse generating circuit detects the first sound event (which is the direct sound signal), the threshold becomes high in a certain period of time to suppress the detection of the unwanted reflections (not from our reflection surfaces). The input of the adaptive threshold circuit is Vneuouc which is the output of the pulse generating circuit. The output of the threshold circuit is vthreBh which is the input of the pulse generating circuit. When the pulse generating circuit detects a sound event, Vneuouc becomes high, which increases vthreBh from V re / 2 to V re / 1 as shown in Figure 3. The higher vthreBh suppresses the detection. The suppression time is determined by the other self-resetting neuron circuit. 1. G. Harris. c.-1. Pu and 1. C. Principe 696 2 . G5':2 ,,=- 2 . 551 :- 2 . 't 51 ;- 241:2 . )51 ~ 2 It;-, , I I i I I , , i , 5 ? ~ .,. ~"r qs.? I I ~ ... ~ . I , , j , , , Ii, , -; :: 'i'f 'i5'U N2 no ~ ~ t j rJ! :.: t-~.-- -~ - -:-~ It ----:------.--4~,=-=-=-~ "~~--~~~~~~~--~ ,,:,::. l . .. II I! . ..? . ? 1 ~~~~T~~L~~~~~~~~~~r.7~~ o. TH~[ ( LBO 2? i511" Figure 5: (a) The input sound signal: impulse signal recorded in typical office environment (b) HSPICE simulation of the output of the detecting onset circuit (label 61), the output of the pulse generating circuit {label 12), and the adaptive threshold circuit response (label 11) The nervous system likely uses a running autocorrelation analysis to measure the time delay between signals. The basic neural connections are shown in Figure 4 [7]. A(t) is the input neuron, A(t - r), A(t - 2r), ... A(t - mr) is a delay chain. The original signal and the delayed signal are multiplied when A(t) and A(t - kr) feed Ck. Assuming the state of neuron A is NA(t). H each synaptic delay in the chain is r, the chain gives us NA(t) under various delays. Ck fires simultaneously when both A(t) and A(t - kr) fire. Neuron Ck connects neuron Dk. Excitation is built up at Dk by the charge and discharge of Ck' The excitation at Dk is therefore (4) Viewing the arrangement of Figure 4 as a neuron autocorrelator, the time-varying excitation at Db D2, .. Dk provides a spatial representation of the autocorrelation function. The localization resolution of this system depends on the delay time r, and the number of the correlators. As r decreases, the localization resolution is improved provided there are enough correlators. In this paper, 30 unit delay taps, and 10 correlators have been implemented on chip. The outputs of the 10 correlators display the time difference between two sound events. The delay time decodes the source's direction. Therefore, the 10 correlators provide a unit encoding of the source location in the ID plane. 3 Simulation and Measurement Results The complete system has been successfully simulated in HSPICE using database we have recorded. Figure 5(a) shows the input sound signal which is an impulse signal recording in our lab (a typical student office environment). Figure 5(b) shows the output of the onset detector (labeled 61), the pulse generating output (labeled 12), and the adaptive threshold (labeled 11). When the onset output exceeds the threshold, the output of the pulse generating circuit becomes high. Simultaneously, the high value of the generated pulse turns on the adaptive threshold circuit to increase the threshold voltage. The adaptive threshold voltage suppresses the unwanted re- A Neuromorphic Monaural Sound Localizer 697 Reflection Surface , " LED 1 dl ? a1 M30 Speaker ~ o LED 2 Labl Amp. Localizer Chip d2 - - -- > LED 3 LED 4 Speaker moving direction Figure 6: Block diagram of the test setup flection which can be seen right after the direct signal (we believe the unwanted reflection is caused by the table). Further simulation results are discussed in [9]. The single microphone sound localizer circuit has been fabricated through the MOSIS 2J.'m N-well CMOS process. Impulse signals are played through speakers to test the fabricated localizer chip. Figure 6 depicts the block diagram of the test setup. The M30 microphone picks up the direct impulse signal and echoes from the reflection surface. Since the reflection surface in our test is just a single flat surface, localization is only tested in one-half of the ID plane. The composite signals are fed into the input of the sound localizer after amplification. Our sound localizer chip receives the composite signal, computes the echo time delay, and sends out the localization result to a display circuit. The display circuit is composed of 4 LEDs with each LED representing a specific sound source location. The sound localizer sends the computational result to turn on a specific LED signifying the echo time delay. In the test, the M30 microphone and the reflection surface are placed at fixed locations. The speaker is moved along the dotted line shown in Figure 6. The M30 microphone is d1 (33cm) from the reflection surface and al (24cm) from the speaker moving line. The speaker's location is defined as ch as depicted in Figure 6. Figure 7(a) shows the theoretical echo's delay at various speaker locations. Figure 7(b) is the measurement of the setup depicted in Figure 6. The y-axis indicates LED 1 through LED 4. The x-axis represents the distance between the speaker's location (ch in Figure 6). The solid horizontal line in Figure 7(b) represents the theoretical results for which LED should respond for each displacement. The results show that localization is accurate within each region with possibilities of two LEDs responding in the overlap regions. 4 Conclusion We have developed the first monaural sound localization system. This system provides a real-time model for human sound localization and has potential use in such applications as low-cost teleconferencing. More work is needed to further develop the system. We need to characterize the accuracy of our system and to test more interesting sound signals, such as speech. Our flat reflection surface is straightforward and simple, but it lacks sufficient flexibility to encode the source's direction in more than a I-D plane. We plan to replace the flat surfaces with a more complicated surface to provide more reflections to encode a richer set of source directions. 698 J G. Harris, c.-J Pu and J C. Principe 1 0 10 20 30 40 80 50 sound so ..... d i _ f r o m _ (em) 70 80 localizer clip ............... 4 00 fil3 ....0 c: 9 9 9 9 r 9 EJ 0 0 C,l .... 0 9 99 9 e 00 e 1 0 9 9 10 20 30 40 50 80 sound so..... di_lrom _ _ _ (em) e e 70 0 80 Figure 7: Sound localizer chip test result Acknowledgments This work was supported by an ONR contract #NOOOI4-94-1-0858 and an NSF CAREER award #MIP-9502307. We gratefully acknowledge MOSIS chip fabrication and Earthworks Inc. for loaning the M30 microphone and amplifier. References [1] D. W. Batteau. The role of the pinna in human localization. Proc. R. Soc. London, Ser. B, 168:158-180,1967. [2] Neal A. Bhadkamkar. Binaural source localizer chip using subthreshold analog cmos. In Proceeding of JCNN, pages 1866-1870, 1994. [3] Earthworks, Inc., P.O. Box 517, Wilton, NH 03086. M90 Microphone. [4] y. Hiranaka and H. Yamasaki. Envelop representations of pinna impulse responses relating to three-dimensional localization of sound sources. J. Acoust. Soc. Am., 73:29, 1983. [5] E. Knudsen, G. Blasdel, and M. Konishi. Mechanisms of sound localization in the barn owl (tyto alba). J. Compo Physiol, 133:13-21, 1979. [6] J. Lazzaro and C. A. Mead. A silicon model of auditory localization. Neural Computation, 1:47-57, 1989. [7] J.C. Licklider. A duplex theory of pitch perception. Experientia, 7:128-133, 1951. [8] C. Mead. Analog VLSJ and Neural Systems. Addison-Wesley, 1989. [9] Chiang-Jung Pu. A neuromorphic microphone for sound localization. PhD thesis, University of Florida, Gainesville, FL, May 1998. [10] L.S. Smith. Sound segmentation using onsets and offsets. J. of New Music Research, 23, 1994. [11] A.J. Watkins. Psychoacoustical aspects of synthesized vertical locale cues. J. Acoust. Soc. Am., 63:1152-1165, 1978. 

1489 |@word version:1 middle:1 sex:1 d2:2 pulse:19 simulation:4 gainesville:2 azimuthal:1 pick:1 solid:1 amp:1 dx:2 john:1 physiol:1 shape:1 remove:1 designed:2 drop:1 cue:4 half:1 nervous:1 plane:4 smith:1 record:1 chiang:2 compo:1 provides:4 detecting:5 location:6 lbo:1 along:1 direct:7 become:1 viable:1 interaural:3 compose:1 ray:1 autocorrelation:2 expected:1 behavior:1 inspired:2 detects:3 actual:3 becomes:3 confused:1 provided:1 circuit:27 duplex:1 tyto:1 cm:2 suppresses:3 developed:1 acoust:2 fabricated:3 locale:1 charge:1 unwanted:6 ser:1 control:1 unit:2 negligible:1 engineering:1 encoding:1 id:3 mead:2 path:4 studied:1 hspice:2 limited:1 range:6 unique:1 acknowledgment:1 atx:1 block:2 displacement:1 composite:2 amplify:1 demonstrated:1 straightforward:1 duration:1 resolution:3 q:1 d1:2 konishi:1 discharge:1 decode:1 us:2 pinna:7 mic:1 database:1 labeled:3 role:3 electrical:1 solved:1 region:2 decrease:2 inhibit:1 environment:2 segment:2 serve:1 localization:24 completely:1 teleconferencing:1 binaural:1 easily:1 chip:7 various:2 derivation:1 describe:1 effective:1 london:1 detected:1 outside:2 richer:1 larger:1 echo:19 propose:1 flexibility:1 achieve:1 amplification:2 moved:1 amplifies:1 r1:1 produce:1 generating:14 cmos:3 develop:1 received:1 soc:3 implemented:2 skip:1 direction:13 filter:4 subsequently:1 human:7 viewing:1 owl:1 require:1 wall:1 preliminary:1 elevation:3 biological:1 experientia:1 barn:1 mapping:2 blasdel:1 circuitry:4 a2:1 proc:1 shadowing:1 label:3 successfully:2 ck:4 ej:1 varying:1 voltage:5 office:2 encode:2 derived:1 properly:1 indicates:1 suppression:1 am:2 dependent:2 integrated:1 vlsi:2 relation:1 transformed:1 extraneous:1 plan:1 spatial:2 special:3 shaped:1 having:1 biology:1 represents:2 cancel:1 future:1 composed:4 simultaneously:2 delayed:1 phase:2 geometry:1 connects:1 fire:2 envelop:1 amplifier:2 detection:5 possibility:1 introduces:1 chain:3 accurate:1 m30:6 re:3 mip:1 theoretical:2 neuromorphic:6 cost:1 delay:21 fabrication:1 azimuth:1 characterize:1 varies:2 adaptively:1 contract:1 off:1 na:2 thesis:1 ear:11 recorded:3 worse:1 external:5 account:1 potential:1 student:1 includes:1 alba:1 inc:2 caused:5 depends:2 onset:13 tion:1 lot:2 lab:1 analyze:1 complicated:2 om:1 il:1 accuracy:1 resetting:2 subthreshold:1 decodes:2 produced:1 researcher:1 detector:1 yamasaki:1 synaptic:1 energy:2 frequency:4 auditory:1 noooi4:1 torso:2 segmentation:1 amplitude:1 reflecting:1 feed:1 wesley:1 ta:1 higher:1 reflected:1 response:2 improved:1 box:1 just:1 receives:1 horizontal:2 lack:1 somehow:1 impulse:6 believe:1 effect:1 bhadkamkar:1 inspiration:1 neal:1 width:3 self:2 ambiguous:1 speaker:9 excitation:3 complete:3 reflection:29 physical:1 nh:1 analog:5 he:1 discussed:2 relating:1 synthesized:1 measurement:3 blocked:1 significant:2 silicon:4 refer:1 gratefully:1 moving:3 longer:2 surface:22 inhibition:2 pu:5 certain:1 psychoacoustical:1 binary:1 success:1 onr:1 seen:1 mr:1 period:1 signal:31 ii:2 multiple:1 sound:70 rj:1 segmented:1 exceeds:2 long:1 award:1 a1:2 pitch:1 basic:1 cochlea:3 addition:1 diagram:3 source:24 concluded:1 sends:2 specially:2 hz:2 recording:2 db:1 correlators:5 enough:3 opposite:1 inner:1 idea:1 attenuated:1 jcnn:1 ltd:2 speech:2 lazzaro:1 latency:1 amount:1 desk:1 band:1 hardware:1 clip:1 generate:1 sl:1 nsf:1 dotted:2 li2:1 iz:2 threshold:13 localize:5 verified:1 mosis:2 jose:1 i5:1 respond:1 diffraction:2 fl:2 played:1 display:3 kexp:1 flat:5 encodes:2 aspect:1 speed:1 department:1 tv:2 em:2 lid:1 equation:1 discus:1 turn:2 mechanism:2 needed:1 know:1 addison:1 fed:2 serf:1 multiplied:1 appropriate:1 florida:2 original:3 responding:1 running:1 music:1 move:2 arrangement:1 g5:1 distance:4 mapped:1 simulated:1 earthwork:3 assuming:1 length:2 setup:5 unfortunately:1 a2x:1 suppress:1 implementation:2 design:1 vertical:2 neuron:7 acknowledge:1 knudsen:1 shoulder:2 head:6 monaural:9 intensity:3 introduced:1 required:1 connection:1 tap:2 autocorrelator:1 able:1 below:1 perception:1 built:1 event:8 overlap:1 representing:1 axis:2 hrtf:2 expect:1 interesting:1 degree:1 sufficient:1 thresholding:1 bank:1 licklider:1 jung:2 placed:2 supported:1 arriving:1 dis:1 bias:3 allow:1 fall:1 taking:2 dimension:1 valid:1 computes:2 made:1 adaptive:8 simplified:1 far:1 emphasize:1 incoming:1 table:1 transfer:3 career:1 s2:1 noise:1 arise:1 n2:1 depicts:1 localizer:14 position:3 watkins:1 specific:2 r2:2 dk:4 offset:1 dl:2 albeit:1 kr:2 phd:1 magnitude:1 direc:1 kx:1 depicted:5 yin:1 led:11 wavelength:3 likely:1 visual:1 expressed:1 ch:2 determines:2 relies:1 harris:4 extracted:1 environmental:1 oct:1 replace:1 change:1 specifically:1 determined:2 typical:2 principal:1 microphone:14 pas:4 discriminate:1 principe:4 signifying:1 tested:1

57 Self Organizing Neural Networks for the Identification Problem Manoel Fernando Tenorio VVei-Tsih Lee School of Electrical Engineering Purdue University School of Electrical Engineering Purdue University VV. Lafayette, UN. 47907 VV. Lafayette, UN. 47907 lwt@ed.ecn.purdue.edu tenoriQ@ee.ecn.purdue.edu ABSTRACT This work introduces a new method called Self Organizing Neural Network (SONN) algorithm and demonstrates its use in a system identification task. The algorithm constructs the network, chooses the neuron functions, and adjusts the weights. It is compared to the Back-Propagation algorithm in the identification of the chaotic time series. The results shows that SONN constructs a simpler, more accurate model. requiring less training data and epochs. The algorithm can be applied and generalized to appilications as a classifier. I. INTRODUCTION 1.1 THE SYSTEM IDENTIFICATION PROBLEM In various engineering applications, it is important to be able to estimate, interpolate, and extrapolate the behavior of an unknown system when only its input-output pairs are available. Algorithms which produce an estimation of the system behavior based on these pairs fall under the category of system identification techniques. 1.2 SYSTEM IDENTIFICATION USING NEURAL NETWORKS A general form to represent systems, both linear and nonlinear, is the KolmogorovGarbor polynomial tGarbor. 19611 shown below: y = ao + L aixi + L L aijxiXj + ... 1 i J (1) 58 Tenorio and Lee where the y is the output. and x the input to the system. [Garbor .1961] proposed a learning method that adjusted the coefficient of (1) by minimizing the mean square error between each desired output sample and the actual output This paper describes a supervised learning algorithm for structure construction and adjustment Here. systems which can be described by (1) are presented. The computation of the function for each neuron performs a choice from a set of possible functions previously assigned to the algorithm. and it is general enough to accept a wide range of both continuous and discrete functions. In this work. the set is taken from variants of the 2-input quadratic polynomial for simplicity. although there is no requirement making it so. This approach abandons the simplistic mean-square error for perfonnance measure in favor of a modified Minimum Description Length (MOL) criterion [Rissanen,1975]. with provisions to measure the complexity of the model generated. The algorithm searches for the simplest model which generates the best estimate. The modified MDL. from hereon named the Structure Estimation Criterion (SEC). is applied hierarchically in the selection of the optimal neuron transfer function from the function set. and then used as an optimal criterion to guide the construction of the structure. The connectivity of the resulting structure is arbitrary. and under the correct conditions [Geman&Geman. 84] the estimation of the struCture is optimal in tenns of the output error and low function complexity. This approach shares the same spirit of GMDH-type algorithms. However, the concept of parameter estimation from Information Theory. combined with a stochastic search algorithm - Simulated Annealing. was used to create a new tool for system identification. This work is organized as follows: section II presents the problem formulation and the Self Organizing Neural Network (SONN) algorithm description; section III describes the results of the application of SONN to a well known problem tested before using other neural network algorithms [Lapede8&Farber. 1987; Moody. 1988]; and fmally, section IV presents a discussion of the results and future directions for this work. II. THE SELF ORGANIZING NEURAL NETWORK ALGORITHM 11.1 SELF ORGANIZING STRUCTURES The Self Organizing Neural Network (SONN) algorithm performs a search on the model space by the construction of hypersurfaces. A network of nodes. each node representing a hypersurface. is organized to be an approximate model of the real system. SONN can be fully characterized by three major components. which can be modified to incorporate knowledge about the process: (1) a generating rule of the primitive neuron transfer functions. (2) an evaluation method which accesses the quality of the model. and. (3) a structure search strategy. Below. the components of SONN are discussed. ll.2 THE ALGORITHM STRUCTURE Self Organizing Neural Networks 11.2.1 The Generating Rule Given a set of observations S: S = {(Xl, Yl),(Xl, Yl)",,(XI, YI)} Yi =f(XV + 11 generated by (2) where f(.) is represented by a Kolmogorov-Garbor polynomial. and the random variable 11 is nonnally distributed. N(O.l). The dimensions of Y is m. and the dimensions of X is n. Every component Yk of Y fonns a hypersurface Yk = fk(X) in the space of dim (X) + 1. The problem is to fmd f(.). given the observations S. which is a corrupted version of the desired function. In this work. the model which estimates f(.) is desired to be as accurate and simple (small number of parameters. and low degree of non linearity) as possible. The approach taken here is to estimate the simplest model which best describes f(.) by generating optimal functions for each neuron. which can be viewed as the construction of a hypersurface based on the observed data. It can be described as follows: given a set of observations S; use p components of the n dimensional space of X to create a hypersurface which best describes Yk =f(X). through a three step process. First, given X = [xl' x2' x3' .... xn) and Yk' and the mapping '? n: [Xl' x2' x3' .... Xn) -> [x'?(1)' x'?(2)' x,?(3)' .... x'?(n?)' construct the hypersurface hi (x'?(1)' x'?(2)' x,?(3)' .... x'?(n? (hi after the fIrst iteration) of p+ 1 dimensions. where '?n is a projection from n dimensions to p dimensions. The elements of the domain of '?n are called tenninals. Second. If the global optimality criterion is reached by the construction of hi(x'?(l)' x'?(2)' x,?(3)' .... x'?(n?' then stop. otherwise continue to the third step. Thud. generate from [Xl' x2' x3' .... xn.hl(x'?(l)' x'?(2)' x,?(3)' .... x'?(n?) a new p+l dimensional hypersurface hi+ I through the extended mapping '?n+ 1(.). and reapply the second step.The resulting model is a multilayered neural network whose topology is arbitrarily complex and created by a stochastic search guided by a structure estimation criterion. For simplicity in this work. the set of prototype functions (F) is restricted to be 2-input quadratic surfaces or smaller. with only four possible types: y y = 8o+alxl +a2x 2 = 8o+alxl +a2x2+a3 x l x2 (3) (4) Y = 3o+a l x l+ a2x1 (5) Y = 8o+alxl+a2x2+a3xlx2+~x1+a5x~ (6) 11.2.2 Evaluation or the Model Based on the MDL Criterion The selection rule (T) of the neuron transfer function was based on a modifIcation of the Minimal Description Length (MOL) information criterion. In [Rissanen. 1975] the principle of minimal description for statistical estimation was developed. The MDL provides a trade-off between the accuracy and the complexity of the model by including the structure estimation tenn of the fInal model. The final model (with the minimal 59 60 Tenorio and Lee MOL) is optimum in the sense of being a consistent estimate of the number of parameters while achieving the minimum error [Rissanen.1980]. Given a sequence of observation xl,x2,x3 ?...?xN from the random variable X. the dominant tenn of the MDL in [Rissanen. 1975] is: MDL =- log f(xI8) + 0.5 k log N where f(xI8) is the estimated probability density function of the model. k is the number of parameters. and N is the number of observations. The first tenn is actually the negative of the maximum likelihood (ML) with respect to the estimated parameter. The second term describes the structure of the models and it is used as a penalty for the complexity of the model. In the case of linear polynomial regression. the MOL is: MDL = - 0.5 N log S~ + 0.5 k log N (8) where k is the number of coefficients in the model selected. In the SONN algorithm. the MDL criterion is modified to operate both recursively and hierarchically. First. the concept of the MDL is applied to each candidate prototype surface for a given neuron. Second. the acceptance of the node. based on Simulated Annealing. uses the MDL measure as the system energy. However. since the new neuron is generated from terminals which can be the output of other neurons. the original defmition of the MDL is unable to compute the true number of system parameters of the final function. Recall that due to the arbitrary connectivity. feedback loops and other configurations it is non trivial to compute the number of parameters in the entire structure. In order to reflect the hierarchical nature of the model. a modified MDL called Structure Estimation Criterion (SEC) is used in conjunction with an heuristic estimator of the number of parameters in the system at each stage of the algorithm. A computationally efficient heuristic for the estimation of the number of parameters in the model is based on the fact that SONN creates a tree-like structure with multiple roots at the input terminals. Then k. in expression (8). can be estimated recursively by: k = kL + kR + (no. of parameters of the current node) (9) where kL and kR are the estimated number of parameters of the left and right parents of the current node. respectively. This heuristic estimator is neither a lower bound nor an upper bound of the true number of parameter in the model. 11.2.3 The SONN Algorithm To explain the algorithm. the following definitions are necessary: Node - neuron and the associated function. connections. and SEC; BASIC NODE - A node for the system input variable; FRONT NODE - A node without children; IN1ERMIDIATE NODE - The nodes that are neither front or basic nodes; STATE - The collection of nodes. and the configuration of their interconnection; INITIAL STATE (SO - The state with only basic nodes; PARENT AND CHILD STATE - The child state is equal to the parent state except for f a new node and its interconnection generated on the parent state structure; NEIGHBOR STATE - A state that is either a child or a parent state of another; ENERGY Self Organizing Neural Networks OF THE STATE (SEC-Si) - The energy of the state is defined as the minimum SEC of all the front nodes in that state. In the SONN algorithm. the search for the correct model structure is done via Simulated Annealing. Therefore the algorithm at times can accept partial structures that look less than ideal. In the same way. it is able to discard partially constructed substructures in search for better results. The use of this algorithm implies that the node accepting rule (R) varies at run-time according to a cooling temperature schedule. The SONN algorithm is as follows: m Initialize T, and S[ Repeat Repeat Sj = generate (Si), If accept ( SEC_Sj. SEC_Si, T) then Si = Sj. - application of P. - application ofR. WUiI the number of new neurons is greater than N. Decrease the temperature T. until The temperature T is smaller than tend (Terminal temperature for Simulated Annealing). Each neuron output and the system input variables are called terminals. Tenninals are viewed as potential dimensions from which a new hypersurface can be constructed. Every tenninal represents the best tentative to approximate the system function with the available infmnatioo. and are therefore treated equally. lll. EXAMPLE - THE CHAOTIC TIME SERIES In the following results. the chaotic time series generated by the Mackay-Glass differential equations was used. The SONN with the SEC. and its heuristic variant were used to obtain the approximate model of the system. The result is compared with those obtained by using the nonlinear signal processing method [LapedeS&Farber. 1987] . The advantages and disadvantages of both approaches are analyzed in the next section. 111.1 Structure of the Problem The MacKay-Glass differential equation used here can be described as: dX(t) at = a x(t - t) _ b x(t) 1 + x10(t - t) (10) By setting a =0.2. b =0.1. and t = 17. a chaotic time series with a' strange attractor of fractal dimension about 3.5 will be produced [Lapedes&Farber. 1987] . To compare the accuracy of prediction the nonnalized root mean square error is used as a perfonnance index: 61 62 Tenorio and Lee no RMSE nnalized RMSE - Standard Deviation (ll) 111.2. SONN WITH THE HEURISTIC SEC (SONN.H) In the following examples, a modified hewistic version of the SEC is used. The estimator of the number of parameters is given by (9), and the fmal configuraion is shown in figure 1. 111.2.1 Node 19 In this subsection, SONN is allowed to generate up to the 19th accepted node. In this first version of the algorithm, all neurons have the same number of interconnections. and therefore draw their transfer function from the same pool of functions .. Generalizations of the algorithm can be easily made to accommodate multiple input functions, and neuron transfer function assignment being drawn from separate pools. In this example, the first one hundred points of the time series was used for training, and samples 101 through 400 used for prediction testing. The total number of weights in the network is 27. The performance index average 0.07. The output of the network is overlapped in the figure 2 with the original time series. For comparison purposes, a GDR network with the structure used in [LapedeS&Farber, 1987] is trained for 6500 epochs. The training data consisted of the first 500 points of the time series, and the testing data ran from the 501st sample to the 832nd. The total number of weights is 165. and the fmal performance index equal to 0.12. This was done to give both algorithms similar computational resources. Figure 3 shows the original time series overlapped with the GDR network output. Ill.2.2 NODE 37 In this subsection, the model chosen was formed by the 37th accepted node. The network was trained in a similar manner to the flfSt example, sioce it is part of the same run. The final number of weights is 40, and the performance index 0.018. Figure 4 shows the output of the network overlapped with the original time series. Figure 5 shows the GDR with 11,500 epochs. Notice that in both cases, the GDR network demands 150 connections and 150 weights. as compared to 12 connections and 27 weights for the first example and 10 connections and 40 weights for the second example. The comparison of the performance of different models is listed in figure 6. IV. Conclusion and Future Work In this study, we proposed a new approach for the identification problem based on a flexible, self-organizing neural network (SONN) structure. The variable structure provides the opportunity to search and construct the optimal model based on input-output observations. The hierarchical version of the MDL, called the structure estimation criteria, Self Organizing Neural Networks was used to guide the trade-off between the model complexity and the accuracy of the estimation. The SONN approach demonstrates potential usefulness as a tool for system identification through the example of modeling a chaotic time series. REFERENCE Garber, D .? eL al. ,"A universal nonlinear filter, predicator and simulator which optimizes itsekf by a learning process," IEE Proc.,18B, pp. 422-438, 1961 RissanenJ. "Modeling by shortest data description," Automatica. vo1.14, pp. 465471.1975 Gemen, S, and Gernen D., "Stochastic relaxation, gibbs deisribution, and the bayesian restoration of images." IEEE PAMI.? PAMI-6,pp.721-741. 1984 Lapedes.A. and Farber, R. ,"Nonlinear signal processing using neural networks: Predication and system modeling," TR.. LA-UR-87-2662. 1987 Moody. J. This volume Rissanen:,,J. "Consistent order estimation of autoregressive processing by shortest description of data." Analysis and optimization of stochastic system. Jacobs et. al. Eds. N.Y. Academic. 1980 Figure 1. The 37th State Generated --------_ _ - .._---_. .. .. S"s._IO.?SOHNI_ 191 0> o. o. , ??.f..---.....---....---__.-----.------I ,.,. ,IIA '.0 0'" '. Figure 2. SONN 19th Model. P.I. = 0.06 63 64 Tenorio and Lee ..------------ -- ------. .- +----~-- - _- .- ~ -. --l nn. ~nn 'lft Figure 3. GDR after 6.500 Epochs. P.I. no =0.12 ".0 '. 00 =0.038 Figure 4. SONN 37th Model. P.I. ~------------------<; ??1_ 10.. "ICk P'''Dlq1I!O" -III II 500 EDOCM. ------ . .. ,\ ~ r r \ Ii \ I , , , .0 \ i . no l\' ~ ~o -. . no ,?...-()t,... ?,-...C)o . . . . OOD Figure 5. GDR after 11.500 Epochs. P.I. = 0.018 Comparison ollhe Perform alice Index 014~-------------------------. W' I>:'UU t: I'''~''~ 0.12 ~Vt~tllfl""Uaj l~, 0. 10 SOliN (o .. ~j ~7) BP \ I :;W Ep~I" 002 .,.-._._.;,_.", . . . . ... .o. _._._ . _~ .. _._._._._.- . -.~.,..~ 000 .............. ........ . ......... ..... . ...................... .... -_.... ..... .. ... .-. .. .. ..... . ~ ~ 002 _ _ _ _ _ _ _- - - - - - - - ' - - - 12u Figure 6. Perfonnance Index Versus the Number of Predicted Points 

149 |@word version:4 polynomial:4 nd:1 jacob:1 tr:1 accommodate:1 recursively:2 initial:1 configuration:2 series:10 lapedes:4 current:2 si:3 dx:1 tenn:3 selected:1 accepting:1 provides:2 node:22 simpler:1 constructed:2 differential:2 ood:1 manner:1 behavior:2 nor:1 simulator:1 terminal:4 actual:1 lll:1 linearity:1 developed:1 lft:1 every:2 demonstrates:2 classifier:1 before:1 engineering:3 xv:1 io:1 pami:2 alice:1 range:1 lafayette:2 testing:2 x3:4 chaotic:5 universal:1 projection:1 selection:2 primitive:1 simplicity:2 adjusts:1 rule:4 fmally:1 estimator:3 construction:5 aixi:1 us:1 overlapped:3 element:1 cooling:1 geman:2 observed:1 ep:1 electrical:2 trade:2 decrease:1 yk:4 ran:1 complexity:5 trained:2 creates:1 easily:1 various:1 represented:1 kolmogorov:1 whose:1 heuristic:5 garber:1 otherwise:1 interconnection:3 favor:1 abandon:1 final:4 sequence:1 advantage:1 xi8:2 loop:1 organizing:10 description:6 gmdh:1 parent:5 requirement:1 optimum:1 produce:1 generating:3 school:2 sonn:20 predicted:1 implies:1 uu:1 direction:1 guided:1 farber:5 correct:2 filter:1 stochastic:4 ao:1 generalization:1 adjusted:1 mapping:2 major:1 purpose:1 estimation:12 proc:1 create:2 tool:2 ick:1 modified:6 conjunction:1 likelihood:1 sense:1 glass:2 dim:1 el:1 nn:2 entire:1 accept:3 ill:1 flexible:1 initialize:1 mackay:2 equal:2 construct:4 represents:1 look:1 future:2 interpolate:1 attractor:1 acceptance:1 evaluation:2 mdl:12 introduces:1 nonnally:1 analyzed:1 accurate:2 partial:1 necessary:1 perfonnance:3 tree:1 iv:2 desired:3 minimal:3 a2x2:2 modeling:3 disadvantage:1 restoration:1 assignment:1 deviation:1 hundred:1 usefulness:1 front:3 iee:1 varies:1 corrupted:1 chooses:1 combined:1 st:1 density:1 lee:5 yl:2 off:2 pool:2 moody:2 connectivity:2 reflect:1 tenninals:2 potential:2 sec:8 coefficient:2 root:2 reached:1 substructure:1 rmse:2 square:3 formed:1 accuracy:3 identification:9 bayesian:1 tenninal:1 produced:1 explain:1 ed:2 definition:1 energy:3 pp:3 associated:1 stop:1 recall:1 knowledge:1 subsection:2 provision:1 organized:2 schedule:1 actually:1 back:1 supervised:1 formulation:1 done:2 fmal:2 stage:1 until:1 nonlinear:4 propagation:1 quality:1 requiring:1 true:2 concept:2 consisted:1 assigned:1 ll:2 self:10 criterion:10 generalized:1 performs:2 hereon:1 temperature:4 image:1 ecn:2 volume:1 discussed:1 defmition:1 gibbs:1 fk:1 iia:1 access:1 surface:2 dominant:1 optimizes:1 discard:1 arbitrarily:1 continue:1 tenns:1 vt:1 yi:2 ofr:1 minimum:3 greater:1 fernando:1 shortest:2 signal:2 ii:4 multiple:2 x10:1 characterized:1 academic:1 equally:1 prediction:2 variant:2 simplistic:1 regression:1 fonns:1 basic:3 iteration:1 represent:1 annealing:4 operate:1 tend:1 spirit:1 ee:1 ideal:1 iii:2 enough:1 topology:1 prototype:2 expression:1 flfst:1 penalty:1 fractal:1 listed:1 category:1 simplest:2 generate:3 notice:1 estimated:4 discrete:1 four:1 rissanen:5 achieving:1 drawn:1 neither:2 relaxation:1 run:2 named:1 strange:1 draw:1 bound:2 hi:4 quadratic:2 bp:1 x2:5 generates:1 optimality:1 according:1 describes:5 smaller:2 ur:1 making:1 modification:1 hl:1 restricted:1 taken:2 computationally:1 resource:1 equation:2 previously:1 available:2 hierarchical:2 reapply:1 original:4 opportunity:1 predicator:1 strategy:1 unable:1 separate:1 simulated:4 trivial:1 length:2 index:6 minimizing:1 negative:1 a2x:1 unknown:1 perform:1 upper:1 neuron:14 observation:6 purdue:4 predication:1 solin:1 extended:1 nonnalized:1 arbitrary:2 pair:2 kl:2 connection:4 tentative:1 manoel:1 able:2 below:2 including:1 treated:1 representing:1 created:1 thud:1 epoch:5 uaj:1 aijxixj:1 fully:1 versus:1 degree:1 consistent:2 principle:1 share:1 repeat:2 guide:2 vv:2 fall:1 wide:1 neighbor:1 distributed:1 feedback:1 dimension:7 xn:4 autoregressive:1 collection:1 made:1 hypersurface:7 sj:2 approximate:3 ml:1 global:1 automatica:1 xi:1 un:2 continuous:1 search:8 nature:1 transfer:5 mol:4 complex:1 domain:1 hierarchically:2 multilayered:1 child:4 allowed:1 fmd:1 x1:1 xl:6 candidate:1 third:1 a3:1 kr:2 gdr:6 demand:1 lwt:1 tenorio:5 adjustment:1 partially:1 hypersurfaces:1 viewed:2 except:1 vo1:1 called:5 total:2 accepted:2 la:1 incorporate:1 tested:1 extrapolate:1

Very Fast EM-based Mixture Model Clustering using Multiresolution kd-trees Andrew W. Moore Robotics Inst.i t. ut.e, Carnegie tl1plloll University Pittsburgh , PA 15:21:3. a\\'l11'9.'cs.cll1u.eciu Abstract Clust ering is impor ta nt in m any fi elds including m a nufac tlll'ing , bio l og~' , fin a nce , a nd astronomy. l\Iixturp models a rp a popula r approach due to their st.atist.ical found a t.ions, and EM is a very popular l1wthocl for fillding mixture models. EM, however, requires lllany accesses of the dat a , a nd thus h as been dismissed as impract ical (e .g. [9]) for d ata mining of enormous dataset.s. We present a nt' \\? algorit.hm, baspd on thp l1lultiresolution ~.'Cl- trees of [5] , whi ch dramatically reeluc ps th e cost of EtlI-baspd clusteriug , wit.h savings rising linearl:; wit.h the numb er of datapoints. Although prespnt.pd lwre for maximum likplihoocl estimation of Gaussian mixt.ure modf' ls , it. is also applicable to non-(~aussian models (provided class d ensit.i es are monotonic in Mahala nobis dist.ance ), mixed categorical/ nUllwric clusters. anel Bayesian nwthocls such as Antoclass [1] . 1 Learning Mixture Models In a Gaussian mixtur e lllod f' l (e.g. [3]) , we aSSUI1W t.hat d ata points {Xl .. . XR} ha\'p bef'n ge lw r<lt ecl incl e p e ncl e lltl~ by the following process. For each X I in turn, natlll'f' begius by randomly picking a class, c}' from a discrf' t e set of classf's {('I . . ' Cs }. T lwn nat m e dr aws X I from an .II-dimension a l Gallss ia n whosf' m ea n fI i and cO\'a riallce ~i depe nd 0 11 th e class, Thus we have . wh ere 8 d en ot ps all the p ar ameters of the mixture: the class probabilities Vi (w lwre P(Cj 18)) , the class centers fl j and th e class covariances ~j' Vi = Tlw job of a mixture m od el learn er is to find a good estimat e of t.he mod eL and Expectation MaximizRtion (EM) , also known a::l "Fuzzy ~' -m e a n::l", i::l a popular 544 A. W Moore algorit.hm for doing so. TIlt' Ith iteration of El\I begins \vith an estimatp (/ of tllP model , and ends with all il11prO\'ed pstimate ll+1. 'Write (2) E;\I iteratps over parh point.-class combination, comput.ing for pach dass Cj and Pnch datapoint Xi, thp pxtent to which Xi is "owned" by Cj. The ownership is simply tl'i) = P(Cj I Xi, (/). Throughout this paper \\iP will use thp following notation : (lij P(Xi Wlj P(Cj I Cj Jl) I Xi , (i) = (/ ijJ!.d Lt~ l (/iI.. )JJ.. (by Bayes ' Rule) Then tl1P new value of thp cpntroid, J.ij' of the jt.b rlass in thp npw modpl (l+l is sim ply tilt' \\"Pightpel t11pan of all the da t<'l point:,; , using the values {LV 1), W~j, . .. lIB.i } as t.he weights. A similar weight.eel procedure gives the new est.imat.e's of the class probabilities and the dass cov<'Iriances: sw? p' f- _ _ .J .I R 1 R tt} f- - - ~ U:i}Xi . s\\" .I L i= l . w)wre S\\'j = L~= l tl;ij . Thus each iteration of EM visit:-; ewry datapoint-rla:,;:,; pair. meaning "YR evaluations of a l\l-dilllensional Ganssian, and so needing O(J/.!.SR) arithnwtic operations ]wr iteration. This paper aims to reduce that cost. An IIIrkd-tree (Multiresolution J~D-tree), introduced in [2] and developed further in [5], is a binary tree in which each node is associateel wit.h a subset of the elatapoints. The root node owns all the datapoints. Each non-leaf-noelp has t \VO rhilelren. d efineel by a splitting dimension NO.SPLITOIM and a splitting valup NO.sPLITVAL. The two children divide their parent's datapoints between them , with the left child owing those dat.apoillts that are strictly less than the splitting \"alue in the splitting dimension, and t he right child owning tllP remaindpr of the parent's cia t apoinb: Xi E NO.LEFT <=} x i [No.sPLITDIM] Xi E NO.RIGHT <=} xdNo .SPLlTOIM] < 2: l\O.SPLITVAL anel Xi E No (4) NO .SPLITVAL and Xi E ND (.5) The distinguishing featur e of mrkcl-trees is that their nodes contain the following: ? NO.NUMPOINTS: The number of points owned by No (equivalently, the average density in No). ? NO.CENTROID: The centroid of the points owned by No (equivalently, the fir:st 1lI0ment of the density below ND) . ? No.(?ov: The cov<'lriance of the points owned by No (equivalently. the second lI10lllent. of the clensi t.y below No). ? NO. HYPERRECT: The bounding hyper-rectangle of the points below No \\"1' construct. mrkcl-trpes top-down, id ent.ifying tilt' bounding box of th e current. node , and splitting in t.hf' cent er of t.he widest dimension. A node is d eclared to be a lea L and is left unsplit , if the widest dimension of its bounding box is SOIllP threshold, JIB IV. If MB W is zero, t.hen all leaf nodes denote singleton or coincident points , the tree bas O(R) nodes and so requires O(M~ R) memory, and (with some care) the cons t ruction co ~t is O(",J'2 R+ M R log R). In practice , we :-;et MB IT' t.o 1% of t he range of the datapoint components . The tree size and construction thus cost :s Very Fast EM-Based Mixture Model Clustering Using Multiresolution Kd-Trees 545 co ns icle ra bl," If'sS t.l Ja n these b Ollnc\s b f'ca use in c\ f' nsf' reg io ns , tin~' lea f n o d es we rf' a ble t o summari ze cl ozf'lls of cl at apo ints , No t f' t oo th a t. t h e cost of trf'e-builcling is a m ol' t izecl - th f' tree must I)f' built. o nef' , .ve t E ~I p f'l' fo rms m a llY it er a tio ns , To p e rform an it.eratio n ofE1\1 with tllf' IIIrkd- tl'f'e , we ca ll t.h e functio n l\ L\K ESTAT S (d f'scril)f'c\ b e lo w) o n t. h f' root of t h e trf'f', \L-\K ESn,Ts ( No , tl) o u t puts :i N va lu es: (S\\'l , S\\':! , ' , , SWx , SWX 1, ' , ,S"'X ,V, SWXX 1, , , ,S\\,XX N ) wh f're I::: I::: Wij tl 'ij X , SWXX j = x, E NO x , E NO ((j ) X, E NO e + 1: TI1f' res ults of 1\ 1A KES TATS (RooT) pro \'idf' sU ffi c if' nt. stat isti c8 to COIl St rue t ]I) f--- S \\' t ,iI R (I) If l\ [AK ESTATS is ca llf'cl o n a If'a f n o c\ f' , \ \'P s impl)' co mpu tf', fo r f'aeh j, S te) = P (c) I x,e = P(x I t'j ,(l)p( c) I et)/ I::: P (x I Ck , et )p (Ck l et) t ) (8) 1.' =1 = whe rf' x NO, ('ENTROID , a nd when ' a ll the it f'lI1s in tllf' rig ht h a nd a r f' eas il~ ' co mput.ed , \Vf' thell r f't urn S\\'j = Il 'j X =" O,NTJ 1\IPOI NTS, Il 'j x NO,N TI1\!POINTS x X a nd s \\'xx) Il'j x );O,f\ Tli\!POINTS x No, co \' , SOil Wf' ca ll d o this is th a t, if the If'a f noclf' is W J') ' sm a ll, t.llf' rf' willlw lit tle in te l ) fo r th f' po int s o \\'l lf'd by t. h f' nod f' a nd so , fo r f'xa lll p lf' Il',j X i ~ III t h e eXpf'r ill lf'llts be lo w \\'e li se \'e r)' t iny lea f nod f' s , f' lIsurin g acc u rac,\', = L: f' qu at io n s \\,x) = The rea va ri atlo n 11'.1 LX" If \IAKEST.-\TS is ca ll e d o n a no n-If'a f-n odf', it (' a n easily co mputf' its a ns \Vf'r by r f'c llrsivf' l~' ca llin g l\ IAK ESTAT s o n its tw o chilclrf' n and t.hf' n l'f' turning the :-:UIll of tllf' t\Vo Sf' ts was th f' end co n venti o nal O(R) n od es, of a n s \\'f'l'S, In genf'r a l. th a t is f'xact ly how we will pro cef'd, If t h a t of th e sto ry , W P wo uld h aw little computa tion a l improvel1lf' nt o \'er E1\l, b ecRuse o n f' pass wo uld fully t.r al'E'rsf' t h e trf'f', which con ta ins doing O( S .11:!) \\'o rk p er node , \Ve will win if we evf'l' sp ot th at. a t so m f' int f' rm edi ate no d f' , \\'f' ca n jll '1111 t , i,f', e\'a lu a t f' th f' n o d e as if it were a If'a f. witho u t. sf' al't'hin g it.s d f' sce nd en ts , but witho u t. introduc in g si gnifica nt e rro r int o tllf' compu tatio n, T o do t hi:-: , \\,f' will CO tllput f' , fo r f'ac h j , the minimulIl a nd m aximum U'ij th a t any p o int ins idp th f' no d f' could h a \'f' , This pl'o cf'cl lll'f' is m or e complf' x than in tllf' casf' o f l o" all~ ' we ig ht pd r f'g l'eSSiOll [.5] , I{'T ax u',~nln u'T ll1 is a lo\\'f' l' b ound a nd fo r each j , wh e re o n minxi E NO /i 'i) a nd u'T <lX is a n uppf' r b o und on m a xx , E :\O (I'i,/ , T hi s is h ard b ec ause wjl11 11 is d e t ermin ed Bo t o nly by t Ilf' m ea n and co\'aria nce o f tl lf' jt.h d a s;-; but also t h e o th f'r cla:,ses , For f'x ample , in Fi g ur f' l. ti'3:! is approximatel~' 0, 5 , but it wo uld bf' much la rge r if Cl werf' f mtll er to t lr f' If'ft , o r h a d a thinll e r ('o \'a ri a nce, \\'e wi;-;h t o co mput f' Blit l'f' nw mber th a t tl lf' (/ 'j j ),1/ L: ~:~1 (/,h li h, \Vf' rpquire:, th at 1'0 1' ti'ij' S a r f' d f' fin f'cl in t e rms of (l ij'S, t hus: lI 'i) pu t bo unds o n t.l lP (li j ' :, rela tive ly <-,a sil~' , [t sim ply f'a ('h j \\,f' co mpllt f' l tllf' closf' ;-; t a nd fUl'tl lf'st po int fro m I',; within ((/11 I C o mpu t ing Ih bt' p oint:-. r equire,., non-t ri\'ia l co mpu t.a t ion a l geo Illetr,\ ' lwca u"e the co\'a ria lJ ce III a t rice:, are n ot llece""arily axis- a li gned , There i" n o space h ere fo r d el a iJ,." A. W. Moore 546 Maximizer of a 2 .~ ? ? Figure 1: The rectangle denote" a h.\"lwrrectangle in the mrkd-tn'e. The !"mall ~2 "quares denote datapoint.s "owlled" h.\? t lIe node. Suppo:se t.here are ju:;( l \\'0 ~Minimizer of a 1 claf-se!" , with the given means, and covaliances depicteel by the ellipse:;. Small circles indicate the locations wit.hin the rMinimizer of a 2 node for which (/) (i.e. P(.r I c))) would -----"----------e> NO .HYPERHECT, be extremized. using the Mahalanobis cli::otancp MHD(x, x') = (x-x/)T~.j I (x-x'). Call tllf':';P short.pst and furtllf'st squarpcl distancps illHDI11I11 and JIHD l11 ax . Then (D) is a lowpr bound for minx , nlln x, E NO END (lij , with a similar dpfinition of aTflX . Thpn write min (aij}Jj/L((ikPh'l x, E NO Wi' ) k > ajlllnpj /(ClT ll1 pj + L ar = x,min (aij}Jj/(Clij}Jj + LouiN)) E NO . kt.l 1ax Pk) = W.Tll1 h?t j wlwrp tl'T II1 is ulli' lo\\,pr bound. There is a similar definition for tl'.TflX. The iLlc'qualit.\' i;-, proved b)' elenH'ntary algebra, and requires that all qllantitips are positiw (which thpy are). vVe can often tight.en thp bounds further using a procedure that pxploits the fact. t.hat. j Wij = 1, but space does not permit further discussion. 2:: tl'T ax are close for all j. 'Vha t should be the criterion for \ \,p will prune if wjllll1 anel clospnpss? The first. idea that springs to mind is: Prune if Vj . (wj11aX - wj11lI1 < t). But such a simplp critprioll is not suitable: some classps may be accumulating very largp sums of weights, whilst others may bp accumulating vpry small Sllms. The largp-sllll1-weight clasl>ps can t.olerate far looser bounds than the small-sum-weight da.sses. Hprp, then, is a more satisfactory pruning critf'l'ion : Pnll1P ifVj . (wr ax Il',Tll1 < nC,;otal) where wjotal is the tot al weight. awarded to class j o\,pr tlw entire dataset , and T is SOI1lP small constant. Sadl~' , w.ioTal is not. known ill advan('e, but far + NO.NTTMPOINTS x lI1 , where happily we can find a lower bound on u,.~otal of Lt'jofar is the total weight awarded to cla.ss j so fa.r during the sear('h over the kcl-trpp. wr wr The algorithm as c1(>scribed so far performs c1ivide-and-conquer-\vith-cut.offs on the spt of clatapoints. In addition, it is possiblp to achieve an extra ac(,pleration by nwallS of diviclp and conquer on the class ('enters. Suppose there wpre N = 100 classps . Illstpad of considering all 100 classps at all Bodes, it is frequelltly possible t.o clPlPrmine at SOI1W node that t.he maximum possi ble \\,pight. w,Tux for som e class j is less thau a minisculp fraction of tllf' minimull1 pos:-;ible weight u't ln for sonlf' other ax < Aut lll where /\ da:-,:-, "'. Thlb if we 0\'<"1' find that in some nocle 10 -..( . tlLell class ('j is rel1lowc\ from ('onsicleration from all clescendpnt:-; ofthp Clll'l'pnt node. FrpC[uPlltly this m ea llS that nea r tllf' tree's Ipa\'ps, only a tiny fraction of thp dassps compete for o\\'nership of the datapoints, and thil> lea.ds to large time savings. wr = Very Fast EM-Based Mixture Model Clusten'ng Using Multiresolution Kd-Trees 2 547 Results \~'e h a vp subj ed pd this a pp ro ach to llum prous i\ Iont.e-Ca rlo empirical tests . Her p \VP report 0 11 on e ::::pt of Ruch tpsts . created with th e fo llowin g m eth od ology. ? We ra nd omly gP llerate a mi xt ure of Ga u::::sia ns in 1\J -dimensio ll a l : : pace (by ciefa nlt .11 = 2 ). The numb er of G a ussians , N is , by d efa ult, :20. E ach (~ a u ~ ~i a n h a ~ a m ean ly ing within the unit hypercub e, and a cova ria nce m a tri x r and omly gen erat ed with dia gonal elem ents between 0 up to 40' :! (by d efa ul t, 0' = 0.05) and rand om non-dia go nal elem ent.s t h a t ensure symm etric positive defini tene:-;s. T hus th e dist a nce from a G a ussia n cen ter t.o it.s l -::::t.andard-elevi a tion contour is of th p order of magni t ude of 0'. ? \\lp r andomly generate a d ataset fro111 t he mixt ure m odel. The number of point:::: , R , i~ (by defa ult) 160 ,000 . Figure :2 sh ow:::: a typical ge nerated set of G a u::::~i a n s a nel clat apoinb. ? We then build an I/Irkd- t ree for th e d ataset. , a nd record th e m em ory requirPlllents a nd real time to build (on a Pent.ium :200Mhz, in seconds). ? We t hpn run Ei\I on the d at a. Ei\I begin:::: wit.h a n entirely different set of (~ au ss i a n:-;, randomly ge ne ra ted using the sam e procedure. ? \Vp run 5 it era tions of the co nvent ional EM algori thm and the n ew mrkdt rpp-ba:: : pd algorithm. TllP n ew algorit.hm uses a defa ult value of 0 .1 for T . \Vp record thp rpa l t ime (ill seconds ) for each itera tion of each a lgorithm, a nd wp a lso reco rd t he m ean log-likelihood score (1/ R) L~= l log P(Xi I rl) for t. he tth m od pl fo r both algo rithm:::: . F igurf' :) :-;ho\\':-; t.h e n od es t.ha t arp visit.pd durin g It eration :2 of the Fast. EM wi t h ~y = (j cla::::ses . T a blp 1 shows t.h e d ptailecl resul ts a:::: the experim ental pa r am eters a re varied. Speedups vary fro m 8-fold to 1000-fold . There a re 100-fold speedup:" even wit.h very wiel e (no n-loca l) G a ussia ns. In oth pI' exp eriments, simil a r resul t s were also obt ain f>c\ on l'ea l d ata ~ ets t ha t disobe.y t llP Gaussia n ass umption . Ther e too, we find one- a.nd two-order- of-m agnitude computa tion al advantages with indis t. in guish able ::::tat.i :-; tical b ph a yi or (no bett.pr ancln o worse ) compared with conventi on al E i\ I. R e al Data: Prelimin a ry experiments in a pplying t his to la rge d a tasets h ave bee n encour ag ing. Fo r thrpe-dinlPnsi on al gala xy clust ering with 800 ,000 gala xies and lUOO elust ns , tr adition al El\1 need ed :3?5 minutes per iteration, while t he mrkd-trees rpquired only H SPcOl1(ls . With l. () millio n gala xies, t.radition al EM need ed 70 minut es a nd IIIrkd-trpPs required 14 seconds . 3 Conclusion Th p use of vari a ble resolution struc t ures for clustering has been suggested in m a ny pl aces (P.g . [7 ,8 , 4, !:l]). The BIRCH sys t em , in pa rt.icul a r , is popula r in the d a t.a b ase co mmunity'. BIRCH is, howpver. Iln9blp to identify seconci-m Ol11Pllt feat ures of clust,pr:; (s uch as Il on-n xis-ali gned spread). Our co ntributions h ave been the use of a ll1ulti-l'f'solut.io n a pproac h, with associa tf>d comput a tion a l benefi t s , a nd th e introducti on of a n pffi cient algo ri t hm t ha t leaves tllP sta tistica l aspects of mixture m od el estil1l a tion uncil angpd. The growth of rpcpnt d a.t. a minin g algorihm s th a t are /l ot based on st.a t istica l foundati ons has frec!pnt.ly been j ust.ified by the following statelllent: U:; illg st ate-of- t hp- Cl rt sta tisti cal techniques is too expensive because such t pchniqu ps were not d psig npel to h andle la rgp da t asets and becom e intraet a bJe with mi Ili on:'; of d a t a p oints . In earlier work we prO\ iclpd ev idence t.h a t t.hi s sta te m ent may 548 Effect of Number of Datapoints, R: As R increases so Joes the computational aJvantage, essentiall~' linearly. The tree-build time (11 seconds at worst) is a tiny cost compared with even just one iteration of Regular EM (2385 seconds, on the big dataset.) FinalSlowSecs: 238.5. FinalFastSecs: 3. A. W Moore I"" ' ' ? ~"" ,,,,, ','" ' ... ~ Num~ , ?., If' " ,_ , '" or pOinte (in thou ??nds) 300 Effect of Number of Dimensions, A/f: As with many J.:d-tree algorithms. the benefits decline as dimensionality increases, yet even in 6 dimensions, there is an 8-fold advantage. FinalSlowSecs: 2742. FinalFastSecs: 310.2.5. EHect of N umber of Classes, N: Conventional EM slows down linearly with the number of classes. Fast EM is clearly sublinear, with a 70-fold speedup even with 320 classes. Note how the tree size grows. This is because more classes mean a more uniform data distribution and fewer datapoints "sharing" tree leaves. FinalSlowSecs: 9278 . FinalFastSecs: 143.:3. Effect of Tau, T: The larger T, the more willing we are to prune during the tree search, anJ thus the faster we search, but the less accurately we mirror EM's statistical behavior. InJeeJ when T is large, the discrep<\llcy In the log likelihood is relatively large. FinalSlowSecs: .584 . .5 . FinalFastSecs: .) Effect of Standard Deviation, 17: Even with very wide Gaussians, with wide support. , we still get large savings . The nodes that are pruned in these cases are rarely nodes with one class owning all the probability, but instead are nodes where all classes have non-zero, but little varying, probability. FinalSlowSecs: 58.1. FinalFastSecs: 4.75. 1 2 3 4 5 6 Number of Inpuba 500 5 001 ' '0 20 003301 0025005 "u 01 40 80 03 09 02 04 160 320 Number of cent.,.. .Igma Table 1: In all the above results all parameters were held at their default values except for one, which varied as shown in the graphs. Each graph shows the factor by which the new E1.'1 is faster than the conventional EM. Below each graph is the time to build the mrkd-tree in seconds and the number of nodes in the t.ree. Note that although the tree builJing cost is not included in the speedup calculation, it is negligibl~ in all cases, especially considering that only one tree build is needed for all EM iterations. Does the approximate nature of this process result in inferior clusters'? The answer is no : the quality of clusters is indistinguishable between the slow and fast methods when measureJ by log-likelihood and when viewed visually. Very Fast EM-Based Mixture Model Clustering Using Multiresolution Kd-Trees 549 ,[J Figure 2: A typical set of Gaussians generated by our random procedure. They in t.urn generate the datasets upon which we compare the performance of the old and new implementations of EM. Figure 3: The ellipses show the model 8 t at the start of an EM iteration. The rectangles depict the mrkdtree nodes that were pruned. Observe larger rectangles (and larger savings) in areas with less variation in class probabilities. Note this is not merely able to only prune where the data density is low. not apply for locally weighted regression [.5] or Bayesian network learning [6], and we hope this paper provides some evidence that it also needn't apply to clustering . References (1) P Cheeseman and R. Oldford. Se lectmg Models f,'om Data: Artljioal Int elligence and S tat/shes IV. L ec tun No t es m S tattstt cs, vol. 89. Spl'inger Verlag, 1994 [:?) K Denl:) and A W Moore l\1ul t lresolutlOn Inst a nce-based Learning In Proceedl71gs of IJCAI-95. Morgan Kaufmann , 1995. [3) R O. Dud a and P E Hart Pattern C la s,<ljicatlon alld Scen r AnalYSIS John Wil ey So: Sons, [4) ]'v{ Ester , H P . Kriegel , and X . Xu A Database In Proceeding' uf th e First AAAI Press , 1~19 5. 1~173. Int e rfa,~e Int~rnatluna( Cunf~"eTlet for Clustertllg In L a r ge Spati a l D atabas",s. un I\'nowledge Discovery and D,lta .\Ilnlll~/? [s] A . W M oo re , J Schneider. all d K DEcng EfficI",nt Lo cally \\'Ec lghted P olyno mi a l R Ecg ress lon Predi Ctions In D Fisher, editor, P"UCOedlTlYS ,'f the J 9Y7 ITlt~nllltlonal !\laehllLe Le(trnmg Cunlf 7'enCf. 1\10 rga n Kaufmann, 1~19 7 [6) Andrew \V M oo re and 1\t. S. Lee Cached SuffiCient Statistics for EffiCi e nt 1\1achm'" Learnmg With Large Datasets Journal oj A,'tljicll1l Intelllyen ce Research , 8 , Marc h 1888. [7) S M . Omo h u ndro . Efficien t Algortthms With Neural Network Be haVIOu r . Systems, 1(2 ):2 73-34 7, 1987. JOU7'T1l11 of Complex [8) S. M Omohundro Burnptrees for Efficient FlmctlOn , Co nst ralllt , a nd C lassificaflon Learning. In R . P . Lippmann, J E. Moody. and D S. Touretzky, editors, Advances tTl Neural Inform a tIOn ProCfssmg S!jstElns '3 Morgan Kaufmann, 18~ll [~7) T. Zhang , R . Ramakrtshn a n, and M Llvny, BIRC H ' An Effici e nt Data Clustering Method for \'",ry Large D a taba~,"s In Proceedwgs of th e FIfteenth AC'.\J ..;'JGACT-SIGMOD-8IGART ::"!jmpollum 0'1 Pnn np ies of Database Sys t ems: PODS 1991>. Ass n for Computing l\l achme ry, EI~II). 

1490 |@word rising:1 nd:23 bf:1 willing:1 tat:3 covariance:1 cla:3 eld:1 tr:1 etric:1 score:1 imat:1 current:1 nt:9 od:6 si:1 yet:1 ust:1 must:1 tot:1 john:1 ints:1 depict:1 fewer:1 leaf:4 yr:1 ria:2 sys:2 ith:1 short:1 erat:1 record:2 lr:1 num:1 provides:1 node:17 location:1 ional:1 lx:2 zhang:1 ra:3 behavior:1 dist:2 ry:4 ol:1 sce:1 little:2 lll:3 considering:2 lib:1 itera:1 provided:1 xx:3 notation:1 begin:2 nea:1 taba:1 anj:1 ttl:1 fuzzy:1 developed:1 aximum:1 whilst:1 astronomy:1 ag:1 impl:1 computa:2 ti:2 ful:1 growth:1 estimat:1 ro:1 rm:1 bio:1 unit:1 ly:4 positive:1 io:3 rfa:1 era:1 ets:1 ak:1 id:1 ure:3 ree:2 andle:1 au:1 ecg:1 co:18 range:1 scribed:1 practice:1 tene:1 itlt:1 lf:6 ance:1 xr:1 procedure:4 area:1 empirical:1 eth:1 regular:1 atist:1 get:1 close:1 ga:1 cal:1 put:1 ilf:1 accumulating:2 mahala:1 conventional:2 center:1 go:1 l:2 pod:1 resolution:1 wit:6 splitting:5 rule:1 datapoints:6 his:1 lso:1 variation:1 pt:1 construction:1 suppose:1 gall:1 us:1 distinguishing:1 pa:3 expensive:1 ze:1 apo:1 cut:1 featur:1 database:2 tllp:4 ft:1 enters:1 worst:1 algorit:3 rig:1 pd:5 und:1 wil:1 icle:1 defini:1 inger:1 ov:1 tight:1 algo:2 ali:1 algebra:1 upon:1 po:3 easily:1 fast:7 kcl:1 minxi:1 hyper:1 ace:1 larger:3 whi:1 s:5 tlw:2 agnitude:1 statistic:1 cov:2 gp:1 ip:1 advantage:2 mb:2 gen:1 oint:1 multiresolution:5 achieve:1 ent:3 cli:1 parent:2 ijcai:1 p:5 ach:2 ecl:1 cluster:3 cached:1 tions:1 oo:3 andrew:2 ac:3 stat:1 ij:7 ard:1 sim:2 job:1 reco:1 c:3 nod:2 indicate:1 idp:1 owing:1 happily:1 nlt:1 alue:1 ja:1 strictly:1 pl:3 iny:1 visually:1 exp:1 nw:1 vith:2 cen:1 vary:1 numb:2 wre:1 estimation:1 applicable:1 ain:1 ound:1 tf:2 ere:2 weighted:1 hope:1 offs:1 clearly:1 gaussian:2 xact:1 pnt:2 spati:1 ck:2 aim:1 og:1 varying:1 ax:6 lon:1 likelihood:3 ave:2 centroid:2 wf:1 rpp:1 inst:2 bef:1 rgp:1 am:1 el:6 lj:1 entire:1 bt:1 her:1 ical:2 wij:2 rpa:1 umber:1 ill:3 loca:1 construct:1 saving:4 ng:1 ted:1 lit:1 np:1 others:1 report:1 lls:2 sta:3 randomly:2 ime:1 ve:2 baspd:2 ents:1 efa:2 mining:1 evaluation:1 mixture:9 sh:1 oth:1 llf:2 uill:1 tical:1 held:1 kt:1 xy:3 tree:22 impor:1 old:1 illg:1 divide:1 circle:1 iv:2 re:7 earlier:1 ar:2 shes:1 mhz:1 jib:1 cost:6 geo:1 deviation:1 subset:1 ory:1 uniform:1 uld:3 too:2 struc:1 answer:1 aw:1 st:7 ju:1 density:3 gaussia:1 lee:1 eel:1 picking:1 moody:1 aaai:1 algori:1 fir:1 ester:1 dr:1 worse:1 compu:1 depe:1 li:5 singleton:1 minin:1 needn:1 rsf:1 ology:1 int:8 vi:2 tli:1 root:3 tion:7 doing:2 defa:2 start:1 bayes:1 hf:2 odel:1 hpn:1 thp:8 ii1:1 alld:1 om:2 il:7 possi:1 kaufmann:3 identify:1 vp:4 bayesian:2 magni:1 eters:1 accurately:1 clust:3 lu:2 acc:1 datapoint:4 fo:10 inform:1 touretzky:1 ussia:2 ed:7 sharing:1 definition:1 esn:1 pp:1 mi:3 qualit:1 con:2 pst:1 dataset:3 popular:2 birch:2 wh:3 proved:1 rla:1 ut:1 dimensionality:1 cj:6 spt:1 ea:5 ta:2 y7:1 rand:1 box:2 xa:1 just:1 wlj:1 d:1 ntj:1 ei:3 su:1 maximizer:1 jll:1 quality:1 grows:1 effect:4 contain:1 lgorithm:1 mrkd:3 dud:1 moore:5 satisfactory:1 wp:1 mahalanobis:1 indistinguishable:1 ll:8 during:2 inferior:1 criterion:1 tt:1 omohundro:1 vo:2 tn:1 performs:1 pro:3 hin:2 meaning:1 fi:3 mixt:2 rl:1 tilt:3 jl:1 he:10 elem:2 nst:1 rd:1 hp:1 hus:2 access:1 clll:1 pu:1 awarded:2 verlag:1 nln:1 binary:1 yi:1 morgan:2 aut:1 care:1 schneider:1 ey:1 prune:4 clt:1 ii:4 thell:1 needing:1 mber:1 ing:5 faster:2 calculation:1 hart:1 e1:2 ase:1 functio:1 ellipsis:1 visit:2 y:1 das:2 va:2 regression:1 expectation:1 symm:1 lly:1 iteration:7 fifteenth:1 pnn:1 ion:3 c1:1 rea:1 addition:1 robotics:1 ures:2 lea:4 aws:1 ot:4 resul:2 extra:1 sr:1 tri:1 blp:1 ample:1 mod:1 pach:1 call:1 ter:1 iii:2 li1:1 ering:2 reduce:1 idea:1 frec:1 decline:1 ti1:1 rms:2 ul:2 wo:3 jj:4 dramatically:1 se:4 obt:1 locally:1 ph:1 nel:1 clusten:1 tth:1 bett:1 generate:2 nsf:1 ipa:1 wr:5 suppo:1 per:1 pace:1 carnegie:1 write:2 vol:1 threshold:1 enormous:1 kes:1 pj:1 ce:2 nal:2 ht:2 rectangle:4 graph:3 merely:1 fraction:2 nce:6 sum:2 compete:1 run:2 throughout:1 spl:1 looser:1 ble:3 vf:3 entirely:1 fl:1 hi:3 bound:5 fold:5 subj:1 idf:1 bp:1 ri:4 aspect:1 llin:1 min:2 ible:1 ults:1 urn:2 c8:1 spring:1 pruned:2 relatively:1 speedup:4 uf:1 combination:1 kd:4 associa:1 ate:2 llum:1 em:22 idence:1 ur:1 son:1 sam:1 tw:1 lp:2 qu:1 wi:3 pr:4 ln:1 turn:1 ffi:1 lta:1 needed:1 mind:1 ge:4 end:3 dia:2 gaussians:2 operation:1 permit:1 experim:1 apply:2 observe:1 cia:1 rac:1 ho:1 hat:2 rp:1 rlo:1 cent:2 top:1 res:1 cf:1 ensure:1 clustering:6 eriments:1 sw:1 cally:1 sigmod:1 especially:1 build:5 widest:2 ellipse:1 dat:2 conquer:2 bl:1 mpu:3 advan:1 ude:1 unsplit:1 fa:1 rt:2 ulli:1 aussian:1 minx:1 win:1 ow:1 predi:1 ause:1 gned:2 sia:1 elligence:1 ified:1 nc:1 equivalently:3 ncl:1 slows:1 uch:1 ba:2 implementation:1 l11:2 datasets:2 sm:1 fin:2 coincident:1 t:5 varied:2 thm:1 introduced:1 edi:1 pair:1 required:1 tive:1 ium:1 ther:1 able:2 suggested:1 pent:1 kriegel:1 llp:1 ev:1 below:4 pattern:1 solut:1 dismissed:1 nly:1 built:1 oj:1 memory:1 tau:1 tun:1 mall:1 rf:3 suitable:1 eration:1 including:1 ia:2 turning:1 ll1:2 ipoi:1 cheeseman:1 ne:1 axis:1 created:1 fro:2 hm:4 rela:1 categorical:1 ss:1 lij:2 discovery:1 hen:1 bee:1 fully:1 mixed:1 sublinear:1 lv:1 sil:1 scen:1 anel:3 sufficient:1 editor:2 tiny:2 pi:1 ata:3 lo:5 soil:1 aij:2 wide:2 benefit:1 dimension:7 default:1 vari:1 contour:1 ental:1 ig:1 ec:3 far:3 approximate:1 lippmann:1 pruning:1 feat:1 ons:1 pittsburgh:1 owns:1 xi:12 ctions:1 un:1 gala:3 nef:1 search:2 table:1 mtll:1 learn:1 nature:1 ca:8 ean:2 as:2 whe:1 complex:1 cl:8 som:1 rue:1 vj:1 marc:1 sp:1 pk:1 da:4 spread:1 linearly:2 bounding:3 big:1 child:3 xu:1 vve:1 cient:1 tl:11 en:3 tlll:1 rithm:1 owning:2 slow:1 sto:1 n:7 ny:1 sf:2 xl:1 lie:1 comput:2 ply:2 lw:1 tin:1 rk:1 minute:1 down:2 xt:1 jt:2 er:8 extremized:1 evidence:1 ih:1 mirror:1 aria:1 nat:1 te:3 tio:1 werf:1 lt:3 simply:1 ijj:1 bo:2 trf:3 monotonic:1 ch:1 minimizer:1 owned:4 tle:1 rice:1 coil:1 viewed:1 ownership:1 fisher:1 included:1 typical:2 except:1 rge:2 total:1 pas:1 arp:1 e:7 la:4 ili:1 est:1 ew:2 rarely:1 otal:2 illc:1 incl:1 support:1 mput:2 ult:3 tllf:9 reg:1

Kernel peA and De-Noising in Feature Spaces Sebastian Mika, Bernhard Scholkopf, Alex Smola Klaus-Robert Muller, Matthias Scholz, Gunnar Riitsch GMD FIRST, Rudower Chaussee 5, 12489 Berlin, Germany {mika, bs, smola, klaus, scholz, raetsch} @first.gmd.de Abstract Kernel PCA as a nonlinear feature extractor has proven powerful as a preprocessing step for classification algorithms. But it can also be considered as a natural generalization of linear principal component analysis. This gives rise to the question how to use nonlinear features for data compression, reconstruction, and de-noising, applications common in linear PCA. This is a nontrivial task, as the results provided by kernel PCA live in some high dimensional feature space and need not have pre-images in input space. This work presents ideas for finding approximate pre-images, focusing on Gaussian kernels, and shows experimental results using these pre-images in data reconstruction and de-noising on toy examples as well as on real world data. 1 peA and Feature Spaces Principal Component Analysis (PC A) (e.g. [3]) is an orthogonal basis transformation. The new basis is found by diagonalizing the centered covariance matrix of a data set {Xk E RNlk = 1, ... ,f}, defined by C = ((Xi - (Xk))(Xi - (Xk))T). The coordinates in the Eigenvector basis are called principal components. The size of an Eigenvalue >. corresponding to an Eigenvector v of C equals the amount of variance in the direction of v. Furthermore, the directions of the first n Eigenvectors corresponding to the biggest n Eigenvalues cover as much variance as possible by n orthogonal directions. In many applications they contain the most interesting information: for instance, in data compression, where we project onto the directions with biggest variance to retain as much information as possible, or in de-noising, where we deliberately drop directions with small variance. Clearly, one cannot assert that linear PCA will always detect all structure in a given data set. By the use of suitable nonlinear features, one can extract more information. Kernel PCA is very well suited to extract interesting nonlinear structures in the data [9]. The purpose of this work is therefore (i) to consider nonlinear de-noising based on Kernel PCA and (ii) to clarify the connection between feature space expansions and meaningful patterns in input space. Kernel PCA first maps the data into some feature space F via a (usually nonlinear) function <II and then performs linear PCA on the mapped data. As the feature space F might be very high dimensional (e.g. when mapping into the space of all possible d-th order monomials of input space), kernel PCA employs Mercer kernels instead of carrying 537 Kernel peA and De-Noising in Feature Spaces out the mapping <I> explicitly. A Mercer kernel is a function k(x, y) which for all data sets {Xi} gives rise to a positive matrix Kij = k(Xi' Xj) [6]. One can show that using k instead of a dot product in input space corresponds to mapping the data with some <I> to a feature space F [1], i.e. k(x,y) = (<I>(x) . <I>(y)). Kernels that have proven useful include Gaussian kernels k(x, y) = exp( -llx - Yll2 Ie) and polynomial kernels k(x, y) = (x?y)d. Clearly, all algorithms that can be formulated in terms of dot products, e.g. Support Vector Machines [1], can be carried out in some feature space F without mapping the data explicitly. All these algorithms construct their solutions as expansions in the potentially infinite-dimensional feature space. The paper is organized as follows: in the next section, we briefly describe the kernel PCA algorithm. In section 3, we present an algorithm for finding approximate pre-images of expansions in feature space. Experimental results on toy and real world data are given in section 4, followed by a discussion of our findings (section 5). 2 Kernel peA and Reconstruction To perform PCA in feature space, we need to find Eigenvalues A > 0 and Eigenvectors V E F\{O} satisfying AV = GV with G = (<I>(Xk)<I>(Xk)T).1 Substituting G into the Eigenvector equation, we note that all solutions V must lie in the span of <I>-images of the training data. This implies that we can consider the equivalent system A( <I>(Xk) . V) = (<I>(Xk) . GV) for all k = 1, ... ,f and that there exist coefficients Q1 , ... V = ,Ql L (1) such that l i =l Qi<l>(Xi) (2) Substituting C and (2) into (1), and defining an f x f matrix K by Kij := (<I>(Xi)? <I>(Xj)) = k( Xi, Xj), we arrive at a problem which is cast in terms of dot products: solve fAa = Ko. (3) where 0. = (Q1, ... ,Ql)T (for details see [7]). Normalizing the solutions V k , i.e. (V k . Vk) = 1, translates into Ak(o.k .o.k) = 1. To extract nonlinear principal components for the <I>-image of a test point X we compute the projection onto the k-th component by 13k := (V k . <I> (X)) = 2:f=l Q~ k(x, Xi). Forfeature extraction, we thus have to evaluate f kernel functions instead of a dot product in F, which is expensive if F is high-dimensional (or, as for Gaussian kernels, infinite-dimensional). To reconstruct the <I>-image of a vector X from its projections 13k onto the first n principal components in F (assuming that the Eigenvectors are ordered by decreasing Eigenvalue size), we define a projection operator P n by (4) k=l If n is large enough to take into account all directions belonging to Eigenvectors with nonzero Eigenvalue, we have Pn<l>(Xi) = <I>(Xi). Otherwise (kernel) PCA still satisfies (i) that the overall squared reconstruction error 2: i II Pn<l>(Xi) - <I>(xdll 2 is minimal and (ii) the retained variance is maximal among all projections onto orthogonal directions in F. In common applications, however, we are interested in a reconstruction in input space rather than in F. The present work attempts to achieve this by computing a vector z satisfying <I>(z) = Pn<l>(x). The hope is that for the kernel used, such a z will be a good approximation of X in input space. However. (i) such a z will not always exist and (ii) if it exists, I For simplicity, we assume that the mapped data are centered in F. Otherwise, we have to go through the same algebra using ~(x) := <I>(x) - (<I>(x;?). S. Mika et al. 538 it need be not unique. 2 As an example for (i), we consider a possible representation of F. One can show [7] that <I> can be thought of as a map <I> (x) = k( x, .) into a Hilbert space 1{k of functions Li ai k( Xi, .) with a dot product satisfying (k( x, .) . k(y, .)) = k( x, y). Then 1{k is caHed reproducing kernel Hilbert space (e.g. [6]). Now, for a Gaussian kernel, 1{k contains aHlinear superpositions of Gaussian bumps on RN (plus limit points), whereas by definition of <I> only single bumps k(x,.) have pre-images under <1>. When the vector P n<l>( x) has no pre-image z we try to approximate it by minimizing p(z) = 1I<I>(z) - Pn<l>(x) 112 (5) This is a special case of the reduced set method [2]. Replacing terms independent of z by 0, we obtain p(z) = 1I<I>(z)112 - 2(<I>(z) . Pn<l>(x)) +0 (6) Substituting (4) and (2) into (6), we arrive at an expression which is written in terms of dot products. Consequently, we can introduce a kernel to obtain a formula for p (and thus V' % p) which does not rely on carrying out <I> explicitly p(z) = k(z, z) - 2 3 n l k=l i=l L f3k L a~ k(z, Xi) + 0 (7) Pre-Images for Gaussian Kernels To optimize (7) we employed standard gradient descent methods. If we restrict our attention to kernels of the form k(x, y) = k(llx - Y1l2) (and thus satisfying k(x, x) == const. for all x), an optimal z can be determined as foHows (cf. [8]): we deduce from (6) that we have to maximize l p(z) = (<I>(z) . Pn<l>(x)) + 0' = L Ii k(z, Xi) + 0' (8) i=l where we set Ii = L~=l f3ka: (for some 0' independent of z). For an extremum, the gradient with respect to z has to vanish: V' %p(z) = L~=l lik'(llz - xi112)(Z - Xi) = O. This leads to a necessary condition for the extremum: z = Li tJixd Lj tJj, with tJi ,ik'(llz - xiIl 2). For a Gaussian kernel k(x, y) = exp( -lix - Yll2 jc) we get z= L~=l Ii exp( -liz - xil1 2 jC)Xi l Li=l liexp(-llz - xil1 2jc) ~ We note that the denominator equals (<I>(z) . P n<l>(X)) (cf. (8?. Making the assumption that Pn<l>(x) i= 0, we have (<I>(x) . Pn<l>(x)) = (Pn<l>(x) . Pn<l>(x)) > O. As k is smooth, we conclude that there exists a neighborhood of the extremum of (8) in which the denominator of (9) is i= O. Thus we can devise an iteration scheme for z by Zt+l = L~=l Ii exp( -llzt - xill 2 jC)Xi l Li=l li exp(-lI z t - xil1 2jc) (10) Numerical instabilities related to (<I>(z) . Pn<l>(x)) being smaH can be dealt with by restarting the iteration with a different starting value. Furthermore we note that any fixed-point of (10) will be a linear combination of the kernel PCA training data Xi. If we regard (10) in the context of clustering we see that it resembles an iteration step for the estimation of 2If the kernel allows reconstruction of the dot-product in input space, and under the assumption that a pre-image exists, it is possible to construct it explicitly (cf. [7]). But clearly, these conditions do not hold true in general. Kernel peA and De-Noising in Feature Spaces 539 the center of a single Gaussian cluster. The weights or 'probabilities' T'i reflect the (anti-) correlation between the amount of cP (x) in Eigenvector direction Vk and the contribution of CP(Xi) to this Eigenvector. So the 'cluster center' z is drawn towards training patterns with positive T'i and pushed away from those with negative T'i, i.e. for a fixed-point Zoo the influence of training patterns with smaner distance to x wi11 tend to be bigger. 4 Experiments To test the feasibility of the proposed algorithm, we run several toy and real world experiments. They were performed using (10) and Gaussian kernels of the form k(x, y) = exp( -(llx - YI12)/(nc)) where n equals the dimension of input space. We mainly focused on the application of de-noising, which differs from reconstruction by the fact that we are allowed to make use of the original test data as starting points in the iteration. Toy examples: In the first experiment (table 1), we generated a data set from eleven Gaussians in RIO with zero mean and variance u 2 in each component, by selecting from each source 100 points as a training set and 33 points for a test set (centers of the Gaussians randomly chosen in [-1, 1]10). Then we applied kernel peA to the training set and computed the projections 13k of the points in the test set. With these, we carried out de-noising, yielding an approximate pre-image in RIO for each test point. This procedure was repeated for different numbers of components in reconstruction, and for different values of u. For the kernel, we used c = 2u 2 ? We compared the results provided by our algorithm to those of linear peA via the mean squared distance of an de-noised test points to their corresponding center. Table 1 shows the ratio of these values; here and below, ratios larger than one indicate that kernel peA performed better than linear peA. For almost every choice of nand u, kernel PeA did better. Note that using alllO components, linear peA is just a basis transformation and hence cannot de-noise. The extreme superiority of kernel peA for small u is due to the fact that all test points are in this case located close to the eleven spots in input space, and linear PeA has to cover them with less than ten directions. Kernel PeA moves each point to the correct source even when using only a sman number of components. n=1 2 3 4 7 8 5 6 9 0.05 2058.42 1238.36 846.14 565.41 309.64 170.36 125.97 104.40 92.23 0.1 10.22 31.32 21.51 29.24 27.66 2:i.5:i 29.64 40.07 63.41 1.12 1.18 0.2 0.99 1.50 2.11 2.73 3.72 5.09 6.32 1.07 1.44 0.4 1.26 1.64 1.91 2.08 2.34 2.47 2.22 0.8 1.39 1.54 2.25 2.39 1.23 1.7U 1.8U 1.96 2.10 Table 1: De-noising Gaussians in RIO (see text). Performance ratios larger than one indicate how much better kernel PeA did, compared to linear PeA, for different choices of the Gaussians' std. dev. u, and different numbers of components used in reconstruction. To get some intuitive understanding in a low-dimensional case, figure 1 depicts the results of de-noising a half circle and a square in the plane, using kernel peA, a nonlinear autoencoder, principal curves, and linear PeA. The principal curves algorithm [4] iteratively estimates a curve capturing the structure of the data. The data are projected to the closest point on a curve which the algorithm tries to construct such that each point is the average of all data points projecting onto it. It can be shown that the only straight lines satisfying the latter are principal components, so principal curves are a generalization of the latter. The algorithm uses a smoothing parameter whic:h is annealed during the iteration. In the nonlinear autoencoder algorithm, a 'bottleneck' 5-layer network is trained to reproduce the input values as outputs (i.e. it is used in autoassociative mode). The hidden unit activations in the third layer form a lower-dimensional representation of the data, closely related to 540 S. Mika et al. PCA (see for instance [3]). Training is done by conjugate gradient descent. In all algorithms, parameter values were selected such that the best possible de-noising result was obtained. The figure shows that on the closed square problem, kernel PeA does (subjectively) best, followed by principal curves and the nonlinear autoencoder; linear PeA fails completely. However, note that all algorithms except for kernel PCA actually provide an explicit one-dimensional parameterization of the data, whereas kernel PCA only provides us with a means of mapping points to their de-noised versions (in this case, we used four kernel PCA features, and hence obtain a four-dimensional parameterization). kernel PCA nonlinear autoencoder Principal Curves linear PCA ~It%i 1J;:qf~~ I::f;~.;~,: '~ll Figure 1: De-noising in 2-d (see text). Depicted are the data set (small points) and its de-noised version (big points, joining up to solid lines). For linear PCA, we used one component for reconstruction, as using two components, reconstruction is perfect and thus does not de-noise. Note that all algorithms except for our approach have problems in capturing the circular structure in the bottom example. USPS example: To test our approach on real-world data, we also applied the algorithm to the USPS database of 256-dimensional handwritten digits. For each of the ten digits, we randomly chose 300 examples from the training set and 50 examples from the test set. We used (10) and Gaussian kernels with c = 0.50, equaling twice the average of the data's variance in each dimensions. In figure 4, we give two possible depictions of 1111 II iI 111111 fill 0"(1' . ___ ? ? 11m 111111111111 Figure 2: Visualization of Eigenvectors (see text). Depicted are the 2?, ... , 28 -th Eigenvector (from left to right). First row: linear PeA, second and third row: different visualizations for kernel PCA. the Eigenvectors found by kernel PCA, compared to those found by linear PCA for the USPS set. The second row shows the approximate pre-images of the Eigenvectors V k , k = 2?, ... ,2 8 , found by our algorithm. In the third row each image is computed as follows: Pixel i is the projection of the <II-image of the i-th canonical basis vector in input space onto the corresponding Eigenvector in features space (upper left <II(el) . V k , lower right <II (e256) . Vk). In the linear case, both methods would simply yield the Eigenvectors oflinear PCA depicted in the first row; in this sense, they may be considered as generalized Eigenvectors in input space. We see that the first Eigenvectors are almost identical (except for signs). But we also see, that Eigenvectors in linear PeA start to concentrate on highfrequency structures already at smaller Eigenvalue size. To understand this, note that in linear PCA we only have a maximum number of 256 Eigenvectors, contrary to kernel PCA which gives us the number of training examples (here 3000) possible Eigenvectors. This 541 Kernel peA and De-Noising in Feature Spaces ? ? ? & ? ? ? ? ? ? ? ? 3.55 ? ? ? &3 o.n 1.02 1.02 1.01 0.113 I.CI' 0.111 0.118 0.118 1.01 0.60 0.78 0.76 0.52 0.73 0.7( 0.80 0.7( 0.7( 0.72 ? ? ? ? ? ? ? ? ? ? ? ? ? 3S ? ? ? ? S! Figure 3: Reconstruction of USPS data. Depicted are the reconstructions of the first digit in the test set (original in last column) from the first n = 1, ... ,20 components for linear peA (first row) and kernel peA (second row) case. The numbers in between denote the fraction of squared distance measured towards the original example. For a small number of components both algorithms do nearly the same. For more components, we see that linear peA yields a result resembling the original digit, whereas kernel peA gives a result resembling a more prototypical 'three' also explains some of the results we found when working with the USPS set. In these experiments, linear and kernel peA were trained with the original data. Then we added (i) additive Gaussian noise with zero mean and standard deviation u = 0.5 or (ii) 'speckle' noise with probability p = 0.4 (i.e. each pixel flips to black or white with probability p/2) to the test set. For the noisy test sets we computed the projections onto the first n linear and nonlinear components, and carried out reconstruction for each case. The results were compared by taking the mean squared distance of each reconstructed digit from the noisy test set to its original counterpart. As a third experiment we did the same for the original test set (hence doing reconstruction, not de-noising). In the latter case, where the task is to reconstruct a given example as exactly as possible, linear peA did better, at least when using more than about 10 components (figure 3). This is due to the fact that linear peA starts earlier to account for fine structures, but at the same time it starts to reconstruct noise, as we will see in figure 4. Kernel PCA, on the other hand, yields recognizable results even for a small number of components, representing a prototype of the desired example. This is one reason why our approach did better than linear peA for the de-noising example (figure 4). Taking the mean squared distance measured over the whole test set for the optimal number of components in linear and kernel PCA, our approach did better by a factor of 1.6 for the Gaussian noise, and 1.2 times better for the 'speckle' noise (the optimal number of components were 32 in linear peA, and 512 and 256 in kernel PCA, respectively). Taking identical numbers of components in both algorithms, kernel peA becomes up to 8 (!) times better than linear peA. However, note that kernel PCA comes with a higher computational complexity. 5 Discussion We have studied the problem of finding approximate pre-images of vectors in feature space, and proposed an algorithm to solve it. The algorithm can be applied to both reconstruction and de-noising. In the former case, results were comparable to linear peA, while in the latter case, we obtained significantly better results. Our interpretation of this finding is as follows. Linear peA can extract at most N components, where N is the dimensionality of the data. Being a basis transform, all N components together fully describe the data. If the data are noisy, this implies that a certain fraction of the components will be devoted to the extraction of noise. Kernel peA, on the other hand, allows the extraction of up to f features, where f is the number of training examples. Accordingly, kernel peA can provide a larger number of features carrying information about the structure in the data (in our experiments, we had f > N). In addition, if the structure to be extracted is nonlinear, then linear peA must necessarily fail , as we have illustrated with toy examples. These methods, along with depictions of pre-images of vectors in feature space, provide some understanding of kernel methods which have recently attracted increasing attention. Open questions include (i) what kind of results kernels other than Gaussians will provide, 542 S. Mika et al. Figure 4: De-Noising of USPS data (see text). The left half shows: top: the first occurrence of each digit in the test set, second row: the upper digit with additive Gaussian noise (0' = 0.5), following five rows: the reconstruction for linear PCA using n = 1,4,16,64,256 components, and, last five rows: the results of our approach using the same number of components. In the right half we show the same but for 'speckle' noise with probability p = 0.4. (ii) whether there is a more efficient way to solve either (6) or (8), and (iii) the comparison (and connection) to alternative nonlinear de-noising methods (cf. [5]). References [1] B. Boser, I. Guyon, and V.N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, Proc. COLT, pages 144-152, Pittsburgh, 1992. ACM Press. [2] C.J.c. Burges. Simplified support vector decision rules. In L. Saitta, editor, Prooceedings, 13th ICML, pages 71-77, San Mateo, CA, 1996. [3] K.I. Diamantaras and S.Y. Kung. Principal Component Neural Networks. Wiley, New York,1996. [4] T. Hastie and W. Stuetzle. Principal curves. JASA, 84:502-516,1989. [5] S. Mallat and Z. Zhang. Matching Pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41(12):3397-3415, December 1993. [6] S. Saitoh. Theory of Reproducing Kernels and its Applications. Longman Scientific & Technical, Harlow, England, 1988. [7] B. Scholkopf. Support vector learning. Oldenbourg Verlag, Munich, 1997. [8] B. Scholkopf, P. Knirsch, A. Smola, and C. Burges. Fast approximation of support vector kernel expansions, and an interpretation of clustering as approximation in feature spaces. In P. Levi et. a1., editor, DAGM'98, pages 124 - 132, Berlin, 1998. Springer. [9] B. Scholkopf, A.J. Smola, and K.-R. Muller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299-1319,1998. 

1491 |@word version:2 briefly:1 polynomial:1 compression:2 open:1 covariance:1 q1:2 solid:1 contains:1 selecting:1 riitsch:1 activation:1 must:2 written:1 attracted:1 oldenbourg:1 additive:2 numerical:1 eleven:2 gv:2 drop:1 half:3 selected:1 parameterization:2 accordingly:1 plane:1 xk:7 provides:1 zhang:1 five:2 along:1 ik:1 scholkopf:4 recognizable:1 introduce:1 decreasing:1 increasing:1 becomes:1 provided:2 project:1 what:1 kind:1 eigenvector:7 finding:5 transformation:2 extremum:3 assert:1 every:1 exactly:1 classifier:1 unit:1 diamantaras:1 superiority:1 positive:2 limit:1 ak:1 joining:1 mika:5 might:1 plus:1 chose:1 resembles:1 twice:1 black:1 studied:1 mateo:1 scholz:2 unique:1 differs:1 digit:7 procedure:1 stuetzle:1 spot:1 thought:1 significantly:1 projection:7 matching:1 pre:12 get:2 onto:7 cannot:2 close:1 operator:1 noising:19 context:1 live:1 instability:1 influence:1 optimize:1 equivalent:1 map:2 center:4 annealed:1 go:1 attention:2 starting:2 resembling:2 focused:1 simplicity:1 rule:1 haussler:1 fill:1 coordinate:1 mallat:1 us:1 satisfying:5 expensive:1 located:1 std:1 database:1 bottom:1 lix:1 equaling:1 noised:3 complexity:1 rudower:1 carrying:3 trained:2 algebra:1 basis:6 completely:1 usps:6 fast:1 describe:2 klaus:2 neighborhood:1 larger:3 solve:3 reconstruct:3 otherwise:2 transform:1 noisy:3 eigenvalue:7 matthias:1 reconstruction:17 product:7 maximal:1 achieve:1 intuitive:1 cluster:2 perfect:1 measured:2 implies:2 indicate:2 come:1 direction:9 concentrate:1 closely:1 correct:1 tji:1 pea:39 centered:2 explains:1 generalization:2 clarify:1 hold:1 considered:2 liz:1 exp:6 mapping:5 bump:2 substituting:3 dictionary:1 purpose:1 estimation:1 proc:1 superposition:1 hope:1 clearly:3 gaussian:13 always:2 rather:1 pn:11 vk:3 mainly:1 detect:1 sense:1 rio:3 saitta:1 el:1 dagm:1 lj:1 nand:1 hidden:1 reproduce:1 interested:1 germany:1 pixel:2 overall:1 classification:1 among:1 colt:1 smoothing:1 special:1 equal:3 construct:3 extraction:3 identical:2 icml:1 nearly:1 employ:1 randomly:2 attempt:1 circular:1 extreme:1 yielding:1 pc:1 devoted:1 necessary:1 orthogonal:3 circle:1 desired:1 minimal:1 instance:2 kij:2 column:1 earlier:1 dev:1 cover:2 deviation:1 monomials:1 ie:1 retain:1 together:1 wi11:1 squared:5 reflect:1 knirsch:1 li:6 toy:5 account:2 de:25 coefficient:1 jc:5 explicitly:4 performed:2 try:2 closed:1 doing:1 start:3 contribution:1 square:2 variance:7 yield:3 dealt:1 handwritten:1 zoo:1 straight:1 sebastian:1 definition:1 frequency:1 f3k:1 dimensionality:1 organized:1 hilbert:2 actually:1 focusing:1 higher:1 done:1 furthermore:2 just:1 smola:4 correlation:1 working:1 hand:2 replacing:1 nonlinear:15 mode:1 scientific:1 contain:1 true:1 deliberately:1 former:1 hence:3 counterpart:1 nonzero:1 iteratively:1 illustrated:1 white:1 ll:1 during:1 faa:1 generalized:1 performs:1 cp:2 image:17 recently:1 common:2 interpretation:2 raetsch:1 llx:3 ai:1 had:1 dot:7 depiction:2 deduce:1 subjectively:1 saitoh:1 closest:1 certain:1 verlag:1 muller:2 devise:1 employed:1 maximize:1 signal:1 ii:16 lik:1 smooth:1 technical:1 england:1 bigger:1 a1:1 feasibility:1 qi:1 ko:1 denominator:2 iteration:5 kernel:63 whereas:3 addition:1 chaussee:1 fine:1 source:2 tend:1 december:1 contrary:1 iii:1 enough:1 xj:3 hastie:1 restrict:1 idea:1 prototype:1 translates:1 bottleneck:1 whether:1 expression:1 pca:31 york:1 autoassociative:1 useful:1 harlow:1 eigenvectors:13 amount:2 ten:2 gmd:2 reduced:1 exist:2 canonical:1 sign:1 gunnar:1 four:2 levi:1 drawn:1 longman:1 fraction:2 yll2:2 run:1 powerful:1 arrive:2 almost:2 guyon:1 decision:1 comparable:1 pushed:1 capturing:2 layer:2 followed:2 nontrivial:1 alex:1 whic:1 span:1 munich:1 combination:1 belonging:1 conjugate:1 smaller:1 b:1 making:1 projecting:1 equation:1 visualization:2 fail:1 flip:1 pursuit:1 gaussians:5 away:1 occurrence:1 alternative:1 original:7 top:1 clustering:2 include:2 cf:4 const:1 move:1 question:2 already:1 added:1 highfrequency:1 gradient:3 distance:5 mapped:2 berlin:2 reason:1 assuming:1 retained:1 ratio:3 minimizing:1 nc:1 ql:2 robert:1 potentially:1 negative:1 rise:2 zt:1 perform:1 upper:2 av:1 descent:2 anti:1 speckle:3 defining:1 rn:1 reproducing:2 cast:1 connection:2 boser:1 usually:1 pattern:3 xill:1 below:1 yi12:1 suitable:1 natural:1 rely:1 diagonalizing:1 representing:1 scheme:1 carried:3 extract:4 autoencoder:4 text:4 understanding:2 fully:1 interesting:2 tjj:1 prototypical:1 proven:2 jasa:1 mercer:2 llz:3 editor:3 row:10 qf:1 last:2 understand:1 burges:2 taking:3 regard:1 curve:8 dimension:2 world:4 xil1:3 preprocessing:1 projected:1 simplified:1 san:1 transaction:1 reconstructed:1 approximate:6 restarting:1 bernhard:1 pittsburgh:1 conclude:1 xi:19 why:1 table:3 ca:1 expansion:4 necessarily:1 did:6 big:1 noise:10 whole:1 allowed:1 repeated:1 biggest:2 depicts:1 oflinear:1 wiley:1 fails:1 explicit:1 lie:1 vanish:1 third:4 extractor:1 formula:1 normalizing:1 exists:3 vapnik:1 ci:1 margin:1 suited:1 depicted:4 simply:1 ordered:1 springer:1 corresponds:1 satisfies:1 extracted:1 acm:1 formulated:1 consequently:1 towards:2 infinite:2 determined:1 except:3 principal:13 called:1 experimental:2 meaningful:1 support:4 latter:4 kung:1 evaluate:1

Viewing Classifier Systems as Model Free Learning in POMDPs Akira Hayashi and Nobuo Suematsu Faculty of Information Sciences Hiroshima City University 3-4-1 Ozuka-higashi, Asaminami-ku, Hiroshima, 731-3194 Japan {akira,suematsu }@im.hiroshima-cu.ac.jp Abstract Classifier systems are now viewed disappointing because of their problems such as the rule strength vs rule set performance problem and the credit assignment problem. In order to solve the problems, we have developed a hybrid classifier system: GLS (Generalization Learning System). In designing GLS, we view CSs as model free learning in POMDPs and take a hybrid approach to finding the best generalization, given the total number of rules. GLS uses the policy improvement procedure by Jaakkola et al. for an locally optimal stochastic policy when a set of rule conditions is given. GLS uses GA to search for the best set of rule conditions. 1 INTRODUCTION Classifier systems (CSs) (Holland 1986) have been among the most used in reinforcement learning. Some of the advantages of CSs are (1) they have a built-in feature (the use of don't care symbols "#") for input generalization, and (2) the complexity of pOlicies can be controlled by restricting the number of rules. In spite of these attractive features, CSs are now viewed somewhat disappointing because of their problems (Wilson and Goldberg 1989; Westerdale 1997). Among them are the rule strength vs rule set performance problem, the definition of the rule strength parameter, and the credit assignment (BBA vs PSP) problem. In order to solve the problems, we have developed a hybrid classifier system: GLS (Generalization Learning System). GLS is based on the recent progress ofRL research in partially observable Markov decision processes (POMDPs). In POMDPs, the environments are really Markovian, but the agent cannot identify the state from the current observation. It may be due to noisy sensing or perceptual aliasing. Perceptual aliasing occurs when the sensor returns the same observation in multiple states. Note that even for a completely observable A. Hayashi and N. Suematsu 990 MDP, the use of don't care symbols for input generalization will make the process as if it were partially observable. In designing GLS, we view CSs as RL in POMDPs and take a hybrid approach to finding the best generalization, given the total number of rules. GLS uses the policy improvement procedure in Jaakkola et a!. (1994) for an locally optimal stochastic policy when a set of rule conditions is given. GLS uses GA to search for the best set of rule conditions. The paper is organized as follows. Since CS problems are easier to understand from GLS perspective, we introduce Jaakkola et a!. (1994), propose GLS, and then discuss CS problems. 2 LEARNING IN POMDPS Jaakkola et a1. (1994) consider POMDPs with perceptual aliasing and memoryless stochastic policies. Following the authors, let us call the observations messages. Therefore, a policy is a mapping from messages to probability distributions (PDs) over the actions. Given a policy 7r, the value of a state s, V7!' (s), is defined for POMDPs just as for MDPs. Then, the value of a message m under policy 7r, V7!' (m ), can be defined as follows: V7!'(m) = LP7!'(slm)V7!'(s) (1) sES where P7!' (slm) is the probability that the state is s when the message is m under the policy 7r. Then, the following holds. N lim'"' E{R(st, at) -R N-+(X)~ t=l E{V(s) Is --t lSI = s} m} (2) (3) where St and at refer to the state and the action taken at the tth step respectively, R( St, at) is the immediate reward at the tth step, R is the (unknown) gain (Le. the average reward per step). s --t m refers to all the instances where m is observed in sand E{? I s --t m} is a Monte-Carlo expectation. In order to compute E{V(s) Is --t m}, Jaakkola et a1. showed a Monte-Carlo procedure: 1 vt(m) = 'k{ Rtl +rl,IRtl+l + rl,2Rtl+2 + ... + rl ,t-tIRt + Rt2 +r2,IRt2 +l + r2,2Rt2+2 + ... + r2,t-t2Rt (4) + Rtk +rk ,IRtdl + ... + rk ,t-tkRtl where tk denotes. the time step corresponding to the kth occurrence of the message m, R t = R(st, at) - R for every t, rk,T indicates the discounting at the Tth step in the kth sequence. By estimating R and by suitably setting rk ,T, Vt(m) converges to V7!'(m). Q7!' (m, a), Q-value of the message m for the action a under the policy 7r, is also defined and computed in the same way. Jaakkola et a1. have developed a policy improvement method: Step 1 Evaluate the current policy 7r by computing V7!' (m) and Q7!' (m, a) for each m and a. Viewing Classifier Systems as Model Free Learning in POMDPs 991 Step 2 Test for any m whether max a Q1r (m, a) > V 1r (m) holds. If 110t, then return 7r. Step 3 For each m and a, define 7r 1 (alm) as follows: 7r 1 (aim) = 1.0 when a = argmaxaQ1r(m, a), 7r 1 (aim) = 0.0 otherwise. f f Then, define 7r as 7r (aim) = (1 - ? )7r( aim) + ?7r 1 (aim) Step 4 Set the new policy as 7r = 7r f , and goto Stepl. 3 GLS Each rule in GLS consists of a condition part, an action part, and an evaluation part: Rule = (Condit'i on, Action, Evaluation). The condition part is a string c over the alphabet {O, 1, #}, and is compared with a binary sensor message. # is a don't care symbol, and matches 0 and 1. When the condition c matches the message, the action is randomly selected using the PD in the action part: Action = (p(allc),p(a21c), ... ,p(a IA!lc)), I:j'!\ p(ajlc) = 1.0 where IAI is the total number ofactions. The evaluation part records the value of the condition V (c) and the Q-values of the condition action pairs Q(c, a): Evaluation = (V(c), Q(c, ad , Q(c, a2), ... ,Q(c, a lAI))' Each rule set consists of N rules, {Rulel, Rule2,"" RuleN}. N, the total number of rules in a rule set, is a design parameter to control the complexity of policies. All the rules except the last one are called standard rules. The last rule Rule N is a special rule which is called the default rule. The condition part of the default rule is a string of # 's and matches any message. Learning in GLS proceeds as follows: (1 )Initialization: randomly generate an initial population of M rule sets, (2)Policy Evaluation and Improvement: for each rule set, repeat a policy evaluation and improvement cycle for a suboptimal policy, then, record the gain of the policy for each rule set, (3)Genetic Algorithm: use the gain of each rule set as its fitness measure and produce a new generation of rule sets, (4) Repeat: repeat from the policy evaluation and improvement step with the new generation of rule sets. In (2)Policy Evaluation and Improvement, GLS repeats the following cycle for each rule set. Step 1 Set ? sufficiently small. Set t max sufficiently large. Step 2 Repeat for 1 :::; t :::; t max ? 1. Make an observation of the environment and receive a message mt from the sensor. 2. From all the rules whose condition matches the message mt, find the rule whose condition is the most specific l . Let us call the rule the active rule. 3. Select the next action at randomly according to the PD in the action part of the active rule, execute the action, and receive the reward R( St, at) from the environment. (The state St is not observable.) 4. Update the current estimate of the gain R from its previous estimate and R( St, ad . Let R t = R( St , ad - R. For each rule, consider its condition Ci as (a generalization of) a message, and update its evaluation part V (Ci ) and Q(c;, aHa E A) using Eq.(4). Step 3 Check whether the following holds. If not, exit. 3i (1 :::; i :::; N), max a Q (Ci , a) > V (cd Step 4 Improve the current policy according to the method in the previous section, and update the action part of the corresponding rules and goto Step 2. IThe most specific rule has the least number of #'s. This is intended only for saving the number of rules. A. Hayashi and N. Suematsu 992 GLS extracts the condition parts of all the rules in a rule set and concatenates them to form a string. The string will be an individual to be manipulated by the genetic algorithm (GA). The genetic algorithm used in GLS is a fairly standard one. GLS combines the SGA (the simple genetic algorithm) (Goldberg 1989) with the elitist keeping strategy. The SGA is composed of three genetic operators: selection, crossover, and mutation. The fitness proportional selection and the single-point crossover are used. The three operators are applied to an entire population at each generation. Since the original SGA does not consider #'s in the rule conditions, we modified SGA as follows. When GLS randomly generates an initial population of rule sets, it generates # at each allele position in rule conditions according to the probability P#. 4 CS PROBLEMS AND GLS In the history of classifier systems, there were two quite different approaches: the Michigan approach (Holland and Reitman 1978), and the Pittsburgh (Pitt) approach (Dejong 1988). In the Michigan approach, each rule is considered as an individual and the rule set as the population in GA. Each rule has its strength parameter, which is based on its future payoff and is used as the fitness measure in GA. These aspects of the approach cause many problems. One is the rule strength vs rule set performance problem. Can we collect only strong rules and get the best rule set performance? Not necessarily. A strong rule may cooperate with weak rules to increase its payoff. Then, how can we define and compute the strength parameter for the best rule set performance? In spite of its problems, this approach is now so much more popular than the other, that when people simply say classifier systems, they refer to Michigan type classifier systems. In the Pitt approach, the problems of the Michigan approach are avoided by requiring GA to evaluate a whole rule set. In the approach, a rule set is considered as an individual and multiple rule sets are kept as the population. The problem of the Pitt approach is its computational difficulties. GLS can be considered as a combination of the Michigan and Pitt approaches. GA in GLS works as that in the Pitt approach. It evaluates a total rule set, and completely avoids the rule strength vs rule set performance problem in the Michigan approach. As the Michigan type CSs, GLS evaluates each rule to improve the policy. This alleviates the computational burden in the Pitt approach. Moreover, GLS evaluates each rule in a more formal and sound way than the Michigan approach. The values, V(c), and Q(c, a), are defined on the basis of POMDPs, and the policy improvement procedure using the values is guaranteed to find a local maximum. Westerdale (1997) has recently made an excellent analysis of problematic behaviors of Michigan type CSs. Two popular methods for credit assignment in CSs are the bucket brigade algorithm (BBA) (Holland 1986) and the profit sharing plan (PSP) (Grefenstette 1988). Westerdale shows that BBA does not work in POMDPs. He insists that PSP with infinite time span is necessary for the right credit assignment, although he does not show how to carry out the computation. GLS does not use BBA or PSP. GLS uses the Monte Carlo procedure, Eq.(4), to compute the value of each condition action pair. The series in Eq.(4) is slow to converge. But, this is the cost we have to pay for the right credit assignment in POMDPs. Westerdale points out another CS problem. He claims that a distinction must be made between the availability and the payoff of rules. We agree with him. As he says, if the expected payoff of Rule 1 is twice as much as Rule 2, then we want to a/ways choose Rule 1. GLS makes the distinction. The probability of a stochastic policy 71'(alc) in GLS corresponds to the availability, and the value of a condition action pair Q ( c, a) corresponds to the payoff. Samuel System (Grefenstette et a1. 1990) can also be considered as a combination of the Michigan and Pitt approaches. Samuel is a highly sophisticated system which has lots of features. We conjecture, however, that Samuel is not free from the CS problems which Viewing Classifier Systems as Model Free Learning in POMDPs 993 Westerdale has analyzed. This is because Samuel uses PSP for credit assignment, and Samuel uses the payoff of each rule for action selection, and does not make a distinction between the availability and the payoff of rules. xes (Wilson 1995) seems to be an exceptionally reliable Michigan?type es. In xes, each rule's fitness is based not on its future payoff but on the prediction accuracy of its future payoff (XeS uses BBA for credit assignment). Wilson reports that xes's population tends to form a complete and accurate mapping from sensor messages and actions to payoff predictions. We conjecture that xes tries to build the most general Markovian model of the environment. Therefore, it will be difficult to apply xes when the environment is not Markovian, or when we cannot afford the number of rules enough to build a Markovian model of the environment, even if the environment itself is Markovian. As we will see in the next section, GLS is intended exactly for these situations. Kaelbling et a1. (19%) surveys methods for input generalization when reward is delayed. The methods use a function approximator to represent the value function by mapping a state description to a value . Since they use value iteration or Q?leaming anyway, it is difficult to apply the methods when the generalization violates the Markov assumption and induces a POMDP. 5 EXPERIMENTS We have tested GLS with some of the representative problems in es literature. Fig. 1 shows Grefl world (Grefenstette 1987). In Grefl world, we used GLS to find the smallest rule set which is necessary for the optimal performance. Since this is not a POMDP but an MDP, the optimal policy can easily be learned when we have a corresponding rule for each of the 16 states. However, when the total number of rules is less than that of states, the environment looks like a POMDP to the learning agent, even if the environment itself is an MDP. The graph shows how the gain of the best rule set in the population changes with the generation. We can see from the figure that four rules are enough for the optimal performance. Also note that the saving of the rules is achieved by selecting the most specific matching rule as an active rule. The rule set with this rule selection is called the defallit hierarchy in es literature. payoff 150 200 'i ~~~--------~ ISO ....................................... .. ..........~ ................. . 100 N: . l N=3 N=2 50 M ? ?? ? ? _ _ O L-~~__~~~~ ~L~~ ?I ~ .. -.~ . ~.-~ - o 10 15 10 15 30 35 40 g.!ner.a tioruJ Figure 1: LEFT: GREF1 World. States {O, 1,2, 3} are the start states and states {12.13, 14, 15 } are the end states. In each state, the agent gets the state number (4 bits) as a message, and chooses an action a,b,c, or d. When the agent reaches the end states, he receives reward 1000 in state 13, but reward 0 in other states. Then the agent is put in one of the start states with equal probability. We added 10% action errors to make the process ergodic. When an action error occurs, the agent moves to one of the 16 states with equal probability. RIGHT: Gain of the best rule set. Parameters: tma ;r =: 10000. ? =: 0.10. M =: 10. N =: 2,3 , 4, P# =: 0.33. For N =: 4, the best rule set at the 40 th generation was { if 0101 (State 5) then a 1.0, if 1010 (State 10) then c 1.0, if ##11 (States 3,7,11,15) then d 1.0, if #### (Default Rule) then b 1.0}. A. Hayashi and N. Suematsu 994 oo~~~~~~~~~--~ 80 70 60 BlillaD a II II II a a 30 N06NoS- 20 10 L -......~~~oo4---'---'!'Pz.:tim:=o?.:...1-_"".... -.-J. o W 20 30 ~ ~ 60 m 80 00 ~ aenerations Figure 2: LEFf: McCallum's Maze. We show the state numbers in the left, and the messages in the right. States 8 and 9 are the start states, and state G is the goal state. In each state, the agent receives a sensor message which is 4 bit long, Each bit in the message tells whether a wall exists in each of the four directions. From each state, the agent moves to one of the adjacent states. When the agent reaches the goal state, he receives reward 1000. The agent is then put in one of the start states with equal probability. RIGHT: Gain of the best rule set. Parameters: t mBX = 50000, ~ = 0.10, M = 10, N = 5,6, P# = 0.33. Fig. 2 is a POMDP known as as McCallum's Maze (McCallum 1993). Thanks to the use of stochastic policies, GLS achieves near optimal gain for memoryless poliCies. Note that no memoryless deterministic policy can take the agent to the goal for this problem. We have seen GLS's generalization capability for an MDP in Grefl World, the advantage of stochastic policies for a POMDP in McCallum's maze. In Woods7 (Wilson 1994), we attempt to test GLS's generalization capability for a POMDP. See Fig. 3. Since each sensor message is 16 bit long, and the conditions of GLS rules can have either O,l,or # for each of the 16 bits, there are 3 16 possible conditions in total. When we notice that there are only 92 different actual sensor messages in the environment, it seems quite difficult to discover them only by using GA. In fact, when we ran GLS for the first time, the standard rules very rarely matched the messages and the default rule took over most of the time. In order to avoid the no matching rule problem, we made the number of rules in a rule set large (N = 100), increased P# from 0.33 in the previous problems to 0.70. The problem was independently attacked by other methods. Wilson applied his ZCS, zeroth level classifier system, to Woods7 (Wilson 1994). The gain was 0.20. ZCS has a special covering procedure to tum around the no matching rule problem. The covering procedure generates a rule which matches a message when none of the current rules matches the message. We expect further improvement on the gain, if we equip GLS with some covering procedure. 6 SUMMARY In order to solve the CS problems such as the rule strength vs rule set performance problem and the credit assignment problem, we have developed a hybrid classifier system: GLS. We notice that generalization often leads to state aliasing. Therefore, in designing GLS, we view CSs as model free learning in POMDPs and take a hybrid approach to finding the best generalization, given the total number of rules. GLS uses the policy improvement procedure by Jaakkola et a1. for an locally optimal stochastic policy when a set of rule conditions is given. GLS uses GA to search for the best set of rule conditions. 995 Viewing Classifier Systems as Model Free Learning in POMDPs 0.24 ????? , ????? 0 . ? ??????...? .. ???????.??? 00 ? ? , ????? ,0 ?? . , . " ?? . or o . . . . . F . ???? .. . F .? ? .???. ,D . ????? . F ?? , ? . ? . ? . r o., ..... . .. . . ?? 0 . ? . ?? . . . 00 . . . ???. F . ? ?.?.? ?. ????? .. ? . . . .? . ? ?? . ?? ?? ? . .???? .? ??? 0 . . . . . . 0 . . . . . . . .. F ?????? 0 ?? ??? . . . r ' " .. oro ..... .. . oro .. _.. . .... F . ?.???.?? 00 ???? . ? r ... . ? . ? 00 .. . . ?? . .???. . . ? ? ? . . ? ? ? ? . .. .??. 0 . . . ?..?? . ???? ? ??. . 0 ?? . :~r~:.:: : :~~::::: : :~r: ::: : ::~::: :::: :: : ~t~ : :: :::: :~:: ::::: 'i ? ???????? ? ???.??.????????????.? ? ?.? . ?? ? ? . .??.????. 0 . ?.?? . . . . ? 00 . . .. ?. , 0 . . . . . . . . . F . . ..... r ...... . .... . . . . ~ . . . . . _ ? . 0 .?? . . ..? 00 ? . .. ? ?. 0 . " . ? ?? . . o.. . . .. . . Fo . .. . . ... F... ... .. or . . . . oro. .. . ~ .. .. . . . ... . . . . ... ... .. . .. .. . . . . ? . 0 , , , ?? . ? . . . . . ? ? . .??.?? . 0 ? . ..? . ??.??? 0 ???? . ? 0 ?????? . 0 ???? . . r . .. . ... F . ?? . ? .. ?? . . ?.. ? F . . ? ????. . Fo . ? . ... . r .? ?. ?. oF . . . . . 0 , .. _ . . ? 00 ?. . ? .. .? . . . . ? ?? 0 .?? ??.? , . ... ... .. . . . . . . . . . . .. . 0 ? .? ?? ? ? . ?.? " .. . F .. .. . ? ? ? oro ... .. . r . . . .. . . " . . . 00 . ? . ? . . ??.?? ?. ?? . 0 . . . . . . .??? ? ? ? ? 0 ? ?.? .??? .. ....... 0 . . ... . ....... . F . . ? .?.?? r . ... . 0 F .. ????.? . . . 00 . . . . ??? 0 ? .? ?? . 0 ..? ?.?? . .----~~~~~~~.....,_.__,_" 0.23 02 2 021 0.2 0.19 0.18 0.17 0.16 0.15 0.14 L.-~~~~~~~_---.J o W W ~ ~ ~ ~ ~ ~ ~ ~ geoentionJ Figure 3: LEFT: Woods7.Each cell is either empty".", contains a stone "0", or contains food "F'. The cells which contain a stone are not passable, and the cells which contain food are goals. In each cell, the agent receives a 2 * 8 = 16 bit long sensor message, which tells the contents of the eight adjacent cells. From each cell, the agent can move to one of the eight adjacent cells. When the agent reaches a cell which contains food, he receives reward 1. The agent is then put in one of the empty cells with equal probability. RlGHT:Gain of the best rule set. Parameters: t ma x = 10000, to = 0.10, M = 10, N = 100, P# 0.70. = References Dejong, K. A. (1988). Learning with genetic algorithms: An overview. Machine Learning, 3:121-138. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley. Grefenstette, J. J. (1987). Multilevel credit assignment in a genetic learning system. In Proc. Second Int. Con! on Genetic Algorithms, pp. 202-209. Grefenstette, J. J. (1988). Credit assignment in rule discovery systems based on genetic algorithms. Machine Learning, 3:225-245. Grefenstette, J. J., C. L. Ramsey, and A. C. Schultz (1990). Learning sequential decision rules using simulation and competition. Machine Learning, 5:355-381. Holland, J. H. (1986). Escaping brittleness: the possibilities of general purpose learning algorithms applied to parallel rule-based systems. In Machine Learning II, pp. 593623. Morgan Kaufmann. Holland, J. H. and J. S. Reitman (1978). Cognitive systems based on adaptive algorithms. In D. A. Waterman and F. Hayes-Roth (Eds.), Pattern-directed inference systems. Academic Press. Jaakkola, T., S. P. Singh, and M. I. Jordan (1994). Reinforcement learning algorithm for partially observable markov decision problems. In Advances of Neural Information Processing Systems 7, pp. 345-352. Kaelbling, L. P., M. L. Littman, and A. W. Moore (1996). Reinforcement learning: A survey. Journal of Artificial Intelligence Research, 4:237-285. McCallum, R. A. (1993). Overcoming incomplete perception with utile distinction memory. In Proc . the Tenth Int. Con! on Machine Learning, pp. 190-196. Westerdale, T. H. (1997). Classifier systems - no wonder they don't work. In Proc. Second Annual Genetic Programming Conference, pp. 529-537. Wilson, S. W. (1994). Zcs: A zeroth order classifier system. Evolutionary Computation , 2(1): 1-18. Wilson, S. W. (1995). Classifier fitness based on accuracy. Evolutionary Computation , 3(2): 149-175. Wilson, S. W. and D. E. Goldberg (1989). A critical review of classifier systems. In Proc . Third Int . Con! on Genetic Algorithms, pp. 244-255. 

1492 |@word cu:1 faculty:1 seems:2 suitably:1 simulation:1 profit:1 carry:1 initial:2 series:1 contains:3 selecting:1 genetic:12 ramsey:1 current:5 must:1 update:3 v:6 intelligence:1 selected:1 p7:1 mccallum:5 iso:1 record:2 utile:1 consists:2 combine:1 introduce:1 alm:1 expected:1 behavior:1 aliasing:4 food:3 actual:1 estimating:1 moreover:1 discover:1 matched:1 string:4 developed:4 dejong:2 finding:3 every:1 exactly:1 classifier:17 slm:2 control:1 ner:1 local:1 tends:1 rule2:1 zeroth:2 twice:1 initialization:1 collect:1 directed:1 procedure:9 crossover:2 sga:4 matching:3 refers:1 spite:2 get:2 cannot:2 ga:9 selection:4 operator:2 put:3 gls:41 deterministic:1 roth:1 independently:1 survey:2 pomdp:6 ergodic:1 rule:104 brittleness:1 his:1 population:7 anyway:1 cs:9 hierarchy:1 programming:1 us:10 designing:3 goldberg:4 q7:2 observed:1 higashi:1 cycle:2 ran:1 environment:10 pd:3 complexity:2 reward:8 littman:1 singh:1 ithe:1 exit:1 completely:2 basis:1 easily:1 hiroshima:3 alphabet:1 monte:3 artificial:1 tell:2 whose:2 quite:2 solve:3 say:2 s:1 otherwise:1 noisy:1 itself:2 advantage:2 sequence:1 took:1 propose:1 alleviates:1 description:1 competition:1 empty:2 produce:1 bba:5 converges:1 tk:1 tim:1 oo:1 ac:1 rt2:2 progress:1 eq:3 strong:2 c:6 direction:1 stochastic:7 allele:1 viewing:4 violates:1 sand:1 multilevel:1 generalization:13 really:1 wall:1 im:1 hold:3 sufficiently:2 credit:10 considered:4 around:1 mapping:3 pitt:7 claim:1 oro:4 achieves:1 a2:1 smallest:1 purpose:1 proc:4 him:1 city:1 sensor:8 aim:5 modified:1 avoid:1 jaakkola:8 wilson:9 improvement:10 indicates:1 check:1 inference:1 entire:1 among:2 plan:1 special:2 fairly:1 equal:4 saving:2 look:1 future:3 report:1 randomly:4 manipulated:1 composed:1 individual:3 delayed:1 fitness:5 intended:2 attempt:1 message:23 highly:1 possibility:1 evaluation:9 analyzed:1 accurate:1 necessary:2 incomplete:1 aha:1 instance:1 increased:1 markovian:5 assignment:10 cost:1 kaelbling:2 wonder:1 chooses:1 st:8 thanks:1 choose:1 v7:6 cognitive:1 return:2 japan:1 availability:3 int:3 ad:3 view:3 lot:1 try:1 start:4 capability:2 parallel:1 mutation:1 irtl:1 accuracy:2 kaufmann:1 identify:1 weak:1 none:1 carlo:3 pomdps:15 history:1 reach:3 fo:2 sharing:1 ed:1 definition:1 evaluates:3 pp:6 con:3 gain:11 popular:2 lim:1 organized:1 sophisticated:1 wesley:1 tum:1 insists:1 iai:1 execute:1 just:1 receives:5 mdp:4 requiring:1 contain:2 discounting:1 memoryless:3 moore:1 attractive:1 adjacent:3 covering:3 samuel:5 stone:2 complete:1 cooperate:1 recently:1 mt:2 rl:4 overview:1 brigade:1 jp:1 he:7 refer:2 leff:1 zcs:3 recent:1 showed:1 perspective:1 disappointing:2 binary:1 vt:2 suematsu:5 seen:1 morgan:1 akira:2 care:3 somewhat:1 q1r:1 converge:1 ii:4 multiple:2 sound:1 match:6 academic:1 long:3 lai:1 a1:6 controlled:1 prediction:2 expectation:1 iteration:1 represent:1 achieved:1 cell:9 receive:2 want:1 mbx:1 goto:2 jordan:1 call:2 near:1 enough:2 suboptimal:1 escaping:1 whether:3 cause:1 afford:1 action:20 tma:1 locally:3 induces:1 tth:3 generate:1 lsi:1 problematic:1 notice:2 rtk:1 per:1 four:2 tenth:1 kept:1 graph:1 decision:3 bit:6 pay:1 guaranteed:1 annual:1 strength:8 generates:3 aspect:1 span:1 conjecture:2 according:3 combination:2 psp:5 bucket:1 taken:1 agree:1 discus:1 addison:1 end:2 apply:2 eight:2 occurrence:1 original:1 denotes:1 build:2 move:3 added:1 occurs:2 strategy:1 evolutionary:2 kth:2 equip:1 condit:1 difficult:3 design:1 policy:32 unknown:1 observation:4 markov:3 waterman:1 attacked:1 immediate:1 payoff:11 situation:1 overcoming:1 pair:3 distinction:4 learned:1 proceeds:1 pattern:1 perception:1 built:1 max:4 reliable:1 memory:1 ia:1 critical:1 difficulty:1 hybrid:6 improve:2 mdps:1 elitist:1 extract:1 review:1 literature:2 discovery:1 expect:1 generation:5 proportional:1 approximator:1 agent:15 cd:1 summary:1 repeat:5 last:2 free:7 keeping:1 formal:1 understand:1 default:4 world:4 avoids:1 maze:3 author:1 made:3 reinforcement:3 adaptive:1 avoided:1 schultz:1 observable:5 rtl:2 active:3 hayes:1 pittsburgh:1 don:4 search:4 ku:1 concatenates:1 excellent:1 necessarily:1 whole:1 fig:3 representative:1 slow:1 lc:1 position:1 perceptual:3 third:1 rk:4 specific:3 symbol:3 sensing:1 r2:3 x:6 pz:1 burden:1 exists:1 restricting:1 sequential:1 alc:1 ci:3 easier:1 michigan:11 simply:1 partially:3 hayashi:4 holland:5 corresponds:2 ma:1 grefenstette:6 viewed:2 goal:4 leaming:1 passable:1 exceptionally:1 change:1 content:1 infinite:1 except:1 total:8 called:3 e:3 rarely:1 select:1 people:1 evaluate:2 tested:1

Multi-electrode spike sorting by clustering transfer functions Dmitry Rinberg Hanan Davidowitz N aftali Tishby* NEe Research Institute 4 Independence Way Princeton, N J 08540 E-mail: {dima,hanan, tishby }<Dreseareh. nj . nee . com Categories: spike sorting, population coding, signal processing. Abstract A new paradigm is proposed for sorting spikes in multi-electrode data using ratios of transfer functions between cells and electrodes. It is assumed that for every cell and electrode there is a stable linear relation. These are dictated by the properties of the tissue, the electrodes and their relative geometries. The main advantage of the method is that it is insensitive to variations in the shape and amplitude of a spike. Spike sorting is carried out in two separate steps. First, templates describing the statistics of each spike type are generated by clustering transfer function ratios then spikes are detected in the data using the spike statistics. These techniques were applied to data generated in the escape response system of the cockroach. 1 Introduction Simultaneous recording of activity from many neurons can greatly expand our understanding of how information is coded in neural systems[l]. 11ultiple electrodes are often used to measure the activity in neural tissue and have become a standard tool in neurophysiology [2 , 3,4]. Since every electrode is in a different position it will measure a different contribution from each of the different neurons . Simply stated, the problem is this: how can these complex signals be untangled to determine when each individual cell fired? This problem is difficult because, a) the objects being classified are very similar and often noisy, b) spikes coming from the same cell can ?Permanent address: Institute of Computer Science and Center for Neural Computation, The Hebrew University, Jerusalem, Israel. Email: tishby<Des.huji.ae.il 147 Transfer Function Spike Sorting vary in both shape and amplitude, depending on the previous activity of the cell and c) spikes can overlap in time, resulting in even more complex temporal patterns. Current approaches to spike sorting are based primarily on the presumed consistency of the spike shape and amplitude for a given cell [5, 6]. This is clearly the only possible basis for sorting using a single electrode. Multiple electrodes, however, provide additional independent information through the differences in the way the same neuron is detected by the different electrodes. The same spike measured on different electrodes can differ in amplitude, shape and its relative timing. These differences can depend on the specific cell, the electrode and the media between them. They can be characterized by linear transfer functions that are invariant to changes in the overall spike waveform. In this paper the importance of this information is highlighted by using only the differences in how signals are measured on different electrodes. It is then shown that clusters of similar differences correspond to the same neuron. It should be emphasized that in a full treatment this transfer function information will be combined with other cues to sort spikes. 2 Spikes, spectra and noise The basic assumption behind the spike sorting approach described here is that the medium between each neuron-electrode pair can be characterized by a linear system that remains fixed during the course of an experiment. This assumption is justified by the approximately linear dielectric properties of the electrode and its surrounding nerve tissues. Linear systems are described by their phase and amplitude response to pure frequencies , namely, by their complex transfer function H(w) = O(w)j I(w), where I(w) and O(w) are the complex spectra (Le. Fourier transform, henceforth called spectrum) of the input and output of the system, respectively. In the experiments described here the input signal is the spectrum of the action potential generated by cell j, denoted by Sj(w) and the output signal is the spectrum of the voltage measured at electrode Il, denoted by VJ.L(w). The transfer function of the system that links Sj(w) and VJ.L(w) is then defined as Hf(w) = VJ.L(w)jSj(w). If the transfer functions are fixed in time, the ratio between the complex spectrum of any spike from cell j as detected by electrodes Il and v , VJ.L (w) and V II (w ), is given by, Hf(w) Hj(w) , (1) which is independent of the cell action potential spectrum Sj(w), provided that the spike was detected by both electrodes. Thus, even if a spike varies in shape and amplitude, Tjll (w) will remain a fixed complex function of frequency. This ratio is also invariant with respect to time translations of the spikes. In addition, the frequency components are asymptotically un correlated for stationary processes, which justifies treating the frequency components as statistically independent[7] . The idea behind the approach described here is shown in Figure 1. In real experiments, however, noise can corrupt the invariance of Tjll. There are several possible sources of noise in experiments of this kind: a) fluctuations in the transfer function, b) changes in the spike shape, <;j and c) electrical and electrochemical noise, nJ.L. 148 D. Rinberg, H. Davidowitz and N Tishby cell-1 cell-1 cell-2 time time time time time time I~ ~ frequency frequency Q) u ::l a. E ctl Q) u ::l ::: a. E ctl c:: 0 u -- c:: ::::I 0 ~ ctl ~ Q) ( /) c:: ctl ~ frequency Figure 1: The idea behind spike sorting by clustering of transfer function ratios. Two spikes from the same cell (cell-I) may vary in shape/ amplitude during bursting activity, for example. Although the spike shapes may differ, the transfer functions relating them to the electrodes do not change so the transfer function ratios are similar (two left columns). A different cell (cell-2) has a different transfer function ratio even though the spikes shapes themselves are similar to those of cell-l (right column). Hf If varies slowly with time, the transfer function noise is small relative to and n Y ? can then be expanded to first order in <;j, nJ.l. and n Y as Try <;j, nY (2) which is independent of <;j. Since the noise, nJ.l., is un correlated with the spike signal, 5 j , the variance at each frequency component can be considered to be Gaussian with equal variances on the real and imaginary axes. Thus the mean of will be independent of 5 j , <;j and nJ.l. while its variance will be inversely proportional to 5 j . Try 3 A model system: the escape response of the cockroach These techniques were tested on a relatively simple neural system - the escape response system of the American cockroach. The escape behaviour, which has been studied extensively [9, 10, 11], is activated when the insect detects air currents 149 Transfer Function Spike Sorting it " !~ f?- ,"-, ( - I : 10 ms . t \\ .. ~ 1. lt \'\ 10.5mv 50 ms Figure 2: A schematic representation of the experiment. Typical raw data measured on two electrodes is shown at right. Relative time delays are evident in the inset, but are not a necessary condition for the sorting techniques described here. Abbreviations are: p-puffers, cg-circal ganglion, c-cerci. produced by the movements of a predator. The insect detects the approach of a predator, determines the direction of approach and starts running in an appropriate direction. The cockroach does this by detecting the movement of several hundred fine hairs located on two appendages, called cerci, protruding from the posterior end of the animal. Each of these hairs is connected to a single neuron. Axons from these cells converge on a dense neuropil called the cercal ganglion (cg), where directional information is extracted and conveyed to the rest of the body by axons in the abdominal nerve. This is shown schematically in Figure 2. This system proved to be well suited as a first test of the sorting technique. The system is simple enough so that it is not overwhelming (since only 7 neurons are known to contribute to the code) but complex enough to really test the approach . In addition, the nerve cords are linear in geometry, easily accessible and very stable. Male cockroaches (Periplaneta americana) were dissected from the dorsal side to expose the nerve cord. The left and right cords were gently separated and two tungsten wire electrodes were hooked onto the connective about 2 mm apart, separated by abdominal ganglia. The stimulus was presented by two loudspeakers driving two miniature wind tunnels pointed at the cerci, at 90 degrees from one another as shown in Figure 2. Recordings typically lasted for several hours. Data were collected with a sampling frequency of 2 . 104 Sis which was sufficient to preserve the high frequency components of the spikes. D. Rinberg, H. Davidowitz and N. Tishby 150 0.5 1 -1.5 -1 Re(T) 1 Figure 3: Real and imaginary parts of Tr v a single w. The circles have centers (radii) equal to the average (variance) of Tr v at w = 248.7 rad S-l. Note that while some clusters seem to overlap at this frequency they may be well seperated at others. Cluster-l is dispersed throughout the complex plane and its variance is well beyond the range of this plot. 4 Clustering and the detection of spikes The spike sorting algorithm described here is done is two separate stages. First, a statistical model of the individual spike types is built from "clean" examples found in the data. Only then are occurrences of these spikes detected in the multielectrode data. This two-step arrangement allows a great deal of flexibility by disconnecting the clustering from the detection. For example, here the clustering was done on transfer function ratios while the detection was done on full complex spectra. These stages are described below in more detail. 4.1 The clustering phase First, the multi-electrode recording is chopped into 3 ms long frames using a sliding window. Frames that have either too low total energy or too high energy at the window edges are discarded. This leaves frames that are energetic in their central 2 ms and are assumed to carry one spike. No attempt is made to find all spikes in the data. Instead, the idea is to generate a set of candidate spike types from clean frames. Once a large collection of candidate spikes is found, TrV(w) is calculated for every 151 Transfer Function Spike Sorting Hook #1 cluster Hook #2 frames yCluster 6 324 ~ 5 d 4 rvv V- 518 '-" 748 & 3 cs 2 ~ 1 ca & 720 -'lr -'\I z 7E 2 R & 729 ? ..- ~6 I!!!I! R g .... -- 4V E2?@?C9 500~V ~.I", I ru@ 1 w? Figure 4: Results of clustering spikes using transfer function ratios. Note that although cluster-5 and cluster-6 are similarly shaped on hook-l they are time shifted on hook-3. Cluster-l is made up of overlaps which are dealt with in the detection phase. Ttl! (w). spike. These are then grouped together into clusters containing similar Results of the clustering are shown in Figure 3 while the corresponding waveforms are shown in Figure 4. Full complex spectra are then used to build a statistical model of the different spike types, {Vj(w), af(w)}, which represent each cell's action potential as it appears on each of the electrodes. 4.2 The detection phase Once the cluster statistics are determined , an independent detection algorithm is used. The data is again broken into short frames but now the idea is to find which of the spike types (represented by the different clusters found in the previous steps) best represents the data in that frame. Each frame can contain either noise, a spike or an overlap of 2 spikes (overlaps of more than 2 spikes are not dealt with). This part is not done on transfer function ratios because dealing with overlaps is more difficult. 5 Concl usion A new method of spike sorting using transfer function ratios has been presented. In effect the sorting is done on the properties of the tissue between the neuron and 152 D. Rinberg, H Davidowitz and N. Tishby the electrode and individual spike shapes become less important. This method may be useful when dealing with bursting cells where the transfer function ratios should remain constant even though the spike amplitude can change significantly. This technique may prove to be a useful tool for analysing multi-electrode data. Acknowledgments We are grateful to Bill Bialek for numerous enlightening discussions and many useful suggestions. References [1] M. Barinaga. Listening in on the brain. Science 280, 376-378 (1998). [2] M. Abeles. Coriiconics, (Cambridge University Press, Cambridge, 1991) [3] B.L. McNaughton, J. O 'Keefe and C.A. Barnes. The stereotrode: a new technique for simultaneous isolation of several single units in the central nervous system from multiple unit records. Journal of Neuroscience Methods, 8, 391-7 (1983). [4] M.L. Reece and J. O'Keefe. The tetrode: a new technique for multi-unit extracellular recording. Society of Neuroscience Abstracts 15, 1250 (1989). [5] M.S. Fee, P. P. Mitra and D. Kleinfeld. Automatic sorting of multiple unit neuronal signals in the presence of anisotropic and non-Gaussian variability. Journal of Neuroscience Methods 69, 175-188 (1996). [6] M.S. Lewicki. A review of methods for spike sorting: the detection and classification of neural potentials. Network: Compututational Neural Systems 9 , R53-R78 (1998). [7] A. Papoulis. Probability, random variables and stochastic processes, (McGrawHill, New-York, 1965). [8] M. Abeles and G.L. Gerstein. Detecting spatio-temporal firing patterns among simultaneously recorded single neurons. Journal of Neurophysiology 60(3) , 909-924 (1988). [9] J .M. Camhi and A. Levy. The code for stimulus direction in a cell assembly in the cockroach. Journal of Comparative Physiology A 165 , 83-97 (1989). [10] L. Kolton and J.M . Camhi. Cartesian representation of stimulus direction: parallel processing by two sets of giant interneurons in the cockroach. Journal of Comparative Physiology A 176, 691-702 (1995). [11] J. Westin, J.J. Langberg and J.M. Camhi. Responses of Giant Interneurons of the cockroach Periplaneta americana to wind puffs of different directions and velocities . Journal of Comparative Physiology A 121,307-324 (1977) . 

1493 |@word neurophysiology:2 tr:2 papoulis:1 carry:1 imaginary:2 current:2 com:1 si:1 shape:10 treating:1 plot:1 stationary:1 cue:1 leaf:1 nervous:1 plane:1 short:1 record:1 lr:1 detecting:2 contribute:1 become:2 prove:1 presumed:1 themselves:1 multi:5 brain:1 detects:2 overwhelming:1 window:2 trv:1 provided:1 medium:2 israel:1 ttl:1 kind:1 connective:1 giant:2 nj:5 temporal:2 every:3 dima:1 unit:4 timing:1 mitra:1 fluctuation:1 firing:1 approximately:1 studied:1 bursting:2 tungsten:1 range:1 statistically:1 acknowledgment:1 significantly:1 physiology:3 onto:1 disconnecting:1 r78:1 davidowitz:4 bill:1 center:2 jerusalem:1 pure:1 periplaneta:2 population:1 variation:1 mcnaughton:1 velocity:1 located:1 electrical:1 cord:3 connected:1 movement:2 rinberg:4 broken:1 depend:1 grateful:1 basis:1 easily:1 represented:1 hooked:1 surrounding:1 separated:2 reece:1 detected:5 dielectric:1 statistic:3 highlighted:1 noisy:1 transform:1 advantage:1 coming:1 fired:1 flexibility:1 untangled:1 electrode:25 cluster:10 comparative:3 object:1 depending:1 measured:4 c:1 differ:2 direction:5 waveform:2 radius:1 stochastic:1 dissected:1 behaviour:1 really:1 mm:1 considered:1 great:1 miniature:1 driving:1 vary:2 expose:1 grouped:1 ctl:4 tool:2 clearly:1 gaussian:2 hj:1 voltage:1 ax:1 lasted:1 greatly:1 cerci:3 cg:2 camhi:3 typically:1 relation:1 expand:1 overall:1 hanan:2 classification:1 among:1 denoted:2 insect:2 animal:1 equal:2 once:2 shaped:1 sampling:1 represents:1 others:1 stimulus:3 escape:4 primarily:1 preserve:1 simultaneously:1 individual:3 geometry:2 phase:4 attempt:1 detection:7 abdominal:2 interneurons:2 male:1 activated:1 behind:3 usion:1 edge:1 necessary:1 re:1 circle:1 column:2 hundred:1 delay:1 tishby:6 too:2 varies:2 abele:2 combined:1 huji:1 accessible:1 together:1 again:1 central:2 recorded:1 containing:1 slowly:1 henceforth:1 american:1 potential:4 de:1 coding:1 permanent:1 mv:1 try:2 wind:2 mcgrawhill:1 start:1 sort:1 hf:3 parallel:1 predator:2 contribution:1 jsj:1 air:1 il:3 variance:5 correspond:1 directional:1 dealt:2 raw:1 produced:1 tissue:4 classified:1 simultaneous:2 email:1 energy:2 frequency:11 e2:1 proved:1 treatment:1 amplitude:8 nerve:4 appears:1 response:5 done:5 though:2 stage:2 kleinfeld:1 r53:1 effect:1 contain:1 deal:1 during:2 m:4 evident:1 insensitive:1 gently:1 anisotropic:1 relating:1 cambridge:2 automatic:1 consistency:1 similarly:1 pointed:1 concl:1 stable:2 posterior:1 dictated:1 apart:1 additional:1 determine:1 paradigm:1 converge:1 signal:7 ii:1 sliding:1 multiple:3 full:3 characterized:2 af:1 long:1 coded:1 schematic:1 basic:1 hair:2 ae:1 represent:1 cell:22 justified:1 addition:2 schematically:1 fine:1 chopped:1 source:1 rest:1 recording:4 seem:1 presence:1 enough:2 independence:1 isolation:1 idea:4 listening:1 energetic:1 york:1 action:3 tunnel:1 useful:3 extensively:1 category:1 generate:1 shifted:1 neuroscience:3 cercal:1 clean:2 appendage:1 asymptotically:1 throughout:1 gerstein:1 fee:1 multielectrode:1 barnes:1 activity:4 fourier:1 expanded:1 relatively:1 extracellular:1 loudspeaker:1 remain:2 invariant:2 remains:1 describing:1 end:1 appropriate:1 occurrence:1 clustering:9 running:1 assembly:1 build:1 society:1 arrangement:1 spike:50 bialek:1 separate:2 link:1 mail:1 collected:1 ru:1 code:2 ratio:12 hebrew:1 difficult:2 stated:1 neuron:9 wire:1 discarded:1 variability:1 frame:8 pair:1 namely:1 rad:1 nee:2 hour:1 address:1 beyond:1 below:1 pattern:2 built:1 enlightening:1 overlap:6 cockroach:8 inversely:1 numerous:1 hook:4 carried:1 review:1 understanding:1 relative:4 suggestion:1 proportional:1 degree:1 conveyed:1 sufficient:1 corrupt:1 translation:1 course:1 side:1 institute:2 template:1 protruding:1 calculated:1 made:2 collection:1 sj:3 dmitry:1 dealing:2 langberg:1 assumed:2 spatio:1 spectrum:9 un:2 seperated:1 transfer:22 ca:1 neuropil:1 complex:10 vj:5 main:1 dense:1 noise:7 body:1 neuronal:1 ny:1 axon:2 position:1 candidate:2 levy:1 specific:1 emphasized:1 inset:1 americana:2 tetrode:1 keefe:2 importance:1 justifies:1 cartesian:1 sorting:18 westin:1 suited:1 lt:1 simply:1 ganglion:3 lewicki:1 determines:1 dispersed:1 extracted:1 abbreviation:1 change:4 analysing:1 typical:1 determined:1 called:3 total:1 invariance:1 puff:1 dorsal:1 c9:1 princeton:1 tested:1 correlated:2

Where does the population vector of motor cortical cells point during reaching movements? Pierre Baraduc* pbaraduc@snv.jussieu.fr Emmanuel Guigon guigon@ccr.jussieu.fr Yves Burnod ybteam@ccr.jussieu.fr INSERM U483, Universite Pierre et Marie Curie 9 quai St Bernard, 75252 Paris cedex 05, France Abstract Visually-guided arm reaching movements are produced by distributed neural networks within parietal and frontal regions of the cerebral cortex. Experimental data indicate that (I) single neurons in these regions are broadly tuned to parameters of movement; (2) appropriate commands are elaborated by populations of neurons; (3) the coordinated action of neurons can be visualized using a neuronal population vector (NPV). However, the NPV provides only a rough estimate of movement parameters (direction, velocity) and may even fail to reflect the parameters of movement when arm posture is changed. We designed a model of the cortical motor command to investigate the relation between the desired direction of the movement, the actual direction of movement and the direction of the NPV in motor cortex. The model is a two-layer self-organizing neural network which combines broadly-tuned (muscular) proprioceptive and (cartesian) visual information to calculate (angular) motor commands for the initial part of the movement of a two-link arm. The network was trained by motor babbling in 5 positions. Simulations showed that (1) the network produced appropriate movement direction over a large part of the workspace; (2) small deviations of the actual trajectory from the desired trajectory existed at the extremities of the workspace; (3) these deviations were accompanied by large deviations of the NPV from both trajectories. These results suggest the NPV does not give a faithful image of cortical processing during arm reaching movements. ? to whom correspondence should be addressed 84 P. Baraduc, E. Guigon and Y. Burnod 1 INTRODUCTION When reaching to an object, our brain transforms a visual stimulus on the retina into a finely coordinated motor act. This complex process is subserved in part by distributed neuronal populations within parietal and frontal regions of the cerebral cortex (Kalaska and Crammond 1992). Neurons in these areas contribute to coordinate transformations by encoding target position and kinematic parameters of reaching movements in multiple frames of reference and to the elaboration of motor commands by sending directional and positional signals to the spinal cord (Georgopoulos 1996). An ubiquitous feature of cortical populations is that most neurons are broadly tuned to a preferred attribute (e.g. direction) and that tuning curves are uniformly (or regularly) distributed in the attribute space (Georgopoulos 1996). Accordingly, a powerful tool to analyse cortical populations is the NPV which describes the behavior of a whole population by a single vector (Georgopoulos 1996). Georgopoulos et al. (1986) have shown that the NPV calculated on a set of directionally tuned neurons in motor cortex points approximately (error 15?) in the direction of movement. However, the NPV may fail to indicate the correct direction of movement when the arm is in a particular posture (Scott and Kalaska 1995). These data raise two important questions: (1) how populations of broadly tuned neurons learn to compute a correct sensorimotor transformation? Previous models (Burnod et al. 1992; Bullock et al. 1993; Salinas and Abbott 1995) provided partial solutions to this problem but we still lack a model which closely matches physiological and psychophysical data on reaching movements; (2) Are cortical processes involved in the visual guidance of arm movements readable with the NPV tool? This article provides answers to these questions through a physiologically inspired model of sensorimotor transformations. f"o.J 2 MODEL OF THE VISUAL-TO-MOTOR TRANSFORMATION 2.1 ARM GEOMETRY The arm model has voluntarily been chosen simple. It is a planar, two-link arm, with limited (160 degrees) joint excursion at shoulder and elbow. An agonist/antagonist pair is attached at each joint. 2.2 INPUT AND OUTPUT eODINGS No cell is finely tuned to a specific input or output value to mimic the broad tunings or monotonic firing characteristics found in cortical visuomotor areas. 2.2.1 Arm position By analogy with the role of muscle spindles, proprioceptive sensors are assumed to code muscle length. Arm position is thus represented by the population activity of NT = 20 neurons coding for the length of each agonist or antagonist. The activity of a sensor neuron k is defined by: Tk where LIl ( k) = adLn(k)) is the length of muscle number n(k) , and ak a piecewise linear sigmoid: L ~ Ak Ak < L < Ak L? Ak Sensibility thresholds Ak are uniformly distributed in [L min , L max], and the dynamic range is Ak - Ak is taken constant, equal to Lmax - L min . 85 Population Coding ofReaching Movements 2.2.2 Desired direction The direction V of the desired movement in visual space is coded by a population of N x = 50 neurons with cosine tuning in cartesian space. Each visual neuron j thus fires as: Xj = V? Vj being the preferred direction of the cell. These 50 preferred directions are chosen uniformly distributed in 2-D space. Vj 2.2.3 Motor Command In attempt to model the existence of muscular synergies (Lemon 1988), we identified motor command with joint movement rather than with muscle contraction . A motor neuron i among Nt = 50 contributes to the effective movement M by its action on a synergy (direction in joint space) Mi. This collective effect is formally expressed by: M .= LtiMi where ti is the activity of motor neuron i. The 50 directions of action Mi are supposed uniformly distributed in joint space. 3 3.1 NETWORK STRUCTURE AND LEARNING STRUCTURE OF THE NETWORK Information concerning the position of the arm and the desired direction in cartesian space desired =. . . .~(visual) 0- ~, cY direction ~~~~----~ ..... Cf t; ~~~~t??? motor synergy Figure 1: Network Architecture is combined asymmetrically (Fig. I). First, an intermediate (somatic) layer of neurons P. Baraduc. E. Guigon and Y. Bumod 86 forms an internal representation of the arm position by a combination of the input from the NT muscle sensors and the lateral interactions inside the population. Activity in this layer is expressed by: (1) Sij = Wijk Tk + ljp Sip L L k p where the lateral connections are: ljp = cos (27r(j - p)/NT ) Equation 1 is self-referent; so calculation is done in two steps. The feed-forward input first arrives at time zero when there is no activity in the layer; iterated action of the lateral connections comes into play when this feed-forward input vanishes. The activity in the somatic layer is then combined with the visual directional information by the output sigma-pi neurons as follows: ti = L Xj Sij j 3.2 WEIGHTS AND LEARNING The only adjustable weights are the Wijk linking the proprioceptive layer to the somatic layer. Connectivity is random and not complete: only 15% of the somatic neurons receive information on arm position. The visuomotor mapping is learnt by modifying the internal representation of the arm. Motor commands issued by the network are correlated with the visual effect of the movement ("motor babbling"). More precisely, the learning algorithm is a repetition of the following cycle: 1. choice of an arm position among 5 positions (stars on Fig. 2) 2. random emission of a motor command (ti) 3. corresponding visual reafference (Xj) 4. weight modification according to a variant of the delta rule: c:'Wijk oc (tiXj - Sij) Tk The random commands are gaussian distributions of activity over the output layer. 5000 learning epochs are sufficient to obtain a stabilized performance. It must be noted that the error between the ideal response of the network and the actual performance never decreases completely to zero, as the constraints of the visuomotor transformation vary over the workspace. 4 RESULTS 4.1 NETWORK PERFORMANCE Correct learning of the mapping was tested in 21 positions in the workspace in a pointing task toward 16 uniformly distributed directions in cartesian space. Movement directions generated by the network are shown in Fig. 2 (desired direction 0 degree is shown bold). Norm of movement vectors depends on the global activity in the network which varies with arm position and movement direction. Performance of the network is maximal near the learning positions. However, a good generalization is obtained (directional error 0.3?, SD 12.1?); a bias toward the shoulder can be observed in extreme right or left positions. A similar effect was observed in psychophysical experiments (Ghilardi et a1. 1995). 87 Population Coding of Reaching Movements 90 180.0 270 Figure 2: Performance in a pointing task 4.2 4.2.1 PREFERRED DIRECTIONS AND POPULATION VECTOR Behavior of the population vector Preferred directions (PO) of output units were computed using a multilinear regression; a perfect cosine tuning was found, which is a consequence of the exact multiplication in sigma-pi neurons. Then, the population vector, the effective movement vector, and the desired movement were compared (Fig. 3) for two different arm configurations A and B marked on Fig. 2. The movement generated by the network (dashed arrow) is close to the ~, ~1' , , deSired direction contnbution of one neuron population vector ~ actual movement ......... ;:., Figure 3: Actual movement and population vector in two arm positions desired one (dotted rays) for both arm configurations. However, the population vector (solid arrow) is not always aligned with the movement. The discrepancy between movement and population vector depends both on the direction and the position of the arm: it is maximal 88 P Baraduc, E. Guigon and Y Burnod for positions near the borders of the workspace as position B. Fig. 3 (position B) shows that the deviations of the population vector are due to the anisotropic distribution of the PDs in cartesian space for given positions. 4.2.2 Difference between direction of action and preferred direction Marked anisotropy in the distribution of PDs is compatible with accurate performance. To see why, let us call "direction of action" (DA) the motor cell's contribution to the movement. The distribution of DAs presents an anisotropy due to the geometry of the arm. This anisotropy is canceled by the distribution of PDs. Mathematically, if U is a N x 2 matrix of uniformly distributed 2D vectors, the PD matrix is UJ- 1 whereas the DA matrix is UJ T , J being the jacobian of the angular-to-cartesian mapping. Difference between DA and PD has been plotted with concentric arcs for four representative neurons at 21 arm positions in Fig. 4. Sign and magnitude of the difference vary continuously over the workspace and neuron number / 4 Vi: DA, _ clockwise = counterclockwise " .. Figure 4: Difference between direction of action and preferred direction for four units. often exceed 45 degrees. It can also be noted that preferred directions rotate with the arm as was experimentally noted by (Caminiti et a1. 1991). 5 DISCUSSION We first asked how a network of broadly tuned neurons could produce visually guided arm movements. The model proposed here produces a correct behavior over the entire workspace. Biases were observed at the extreme right and left which closely resemble experimental data in humans (Ghilardi et a1. 1995). Single cells in the output layer behave as motor cortical cells do and the NPV of these cells correctly indicated the direction of movement for hand positions in the central region of the workspace (see Caminiti et al. 1991). Models of sensorimotor transformations have already been proposed. However they either considered motor synergies in cartesian coordinates (Burnod et a1. 1992), or used sharply Population Coding ofReaching Movements 89 tuned units (Bullock et al. 1993), or motor effects independent of arm position (Salinas and Abbott 1995). Next, the use of the NPV to describe cortical activity was questioned. A fundamental assumption in the calculation of the NPV is that the PD of a neuron is the direction in which the arm would move if the neuron were stimulated. The model shows that the two directions DA and PD do not necessarily coincide, which is probably the case in motor cortex (Scott and Kalaska 1995). It follows that the NPV often points neither in the actual movement direction nor in the desired movement direction (target direction), especially for unusual arm configurations. A maximum-likelihood estimator does not have these flaws; it would however accurately predict the desired movement out of the output unit activities, even for a wrong actual movement. In conclusion: (l) the NPV does not provide a faithful image of cortical visuomotor processes; (2) a correct NPV should be based on the DAs, which cannot easily be determined experimentally; (3) planning of trajectories in space cannot be realized by the successive recruitment of motor neurons whose PDs sequentially describe the movement. References Bullock, D., S. Grossberg, and F. Guenther (1993). A self-organizing neural model of motor equivalent reaching and tool use by a multijoint arm. J Cogn Neurosci 5(4), 408435. Burnod, Y., P. Grandguillaume, I. Otto, S. Ferraina, P. Johnson, and R Caminiti (1992). Visuomotor transformations underlying arm movements toward visual targets: a neural network model of cerebral cortical operations. J Neurosci 12(4), 1435-53. Caminiti, R, P. Johnson, C. Galli, S. Ferraina, and Y. Burnod (1991). Making arm movements within different parts of space: the premotor and motor cortical representation of a coordinate system for reaching to visual targets. J Neurosci 11(5), 1182-97. Georgopoulos, A (1996). On the translation of directional motor cortical commands to activation of muscles via spinal interneuronal systems. Brain Res Cogn Brain Res 3(2), 151-5. Georgopoulos, A, A Schwartz, and R Kettner (1986). Neuronal population coding of movement direction . Science 233(4771), 1416-9. Ghilardi, M. , J. Gordon, and C. Ghez (1995). Learning a visuomotor transformation in a local area of work space pr oduces directional biases in other areas. J NeurophysioI73(6), 2535-9. Kalaska, J. and D. Crammond (1992). Cerebral cortical mechanisms of reaching movements. Science 255(5051),1517-23. Lemon, R (1988). The output map of the primate motor cortex. Trends Neurosci 11 (II), 501-6. Salinas, E. and L. Abbott (1995). Transfer of coded information from sensory to motor networks. J Neurosci 15(10),6461-74. Scott, S. and J. Kalaska (1995). Changes in motor cortex activity during reaching movements with similar hand paths but different arm postures. J Neurophysioi 73(6), 2563-7. 

1494 |@word norm:1 simulation:1 contraction:1 solid:1 initial:1 configuration:3 tuned:8 nt:4 activation:1 must:1 motor:29 designed:1 accordingly:1 provides:2 contribute:1 successive:1 combine:1 ray:1 inside:1 guenther:1 behavior:3 nor:1 planning:1 brain:3 inspired:1 anisotropy:3 actual:7 elbow:1 provided:1 underlying:1 transformation:8 sensibility:1 act:1 ti:3 sip:1 wrong:1 schwartz:1 interneuronal:1 unit:4 local:1 sd:1 consequence:1 encoding:1 ak:8 extremity:1 firing:1 path:1 approximately:1 co:1 limited:1 range:1 grossberg:1 faithful:2 cogn:2 area:4 suggest:1 caminiti:4 close:1 cannot:2 equivalent:1 map:1 rule:1 estimator:1 population:23 coordinate:3 target:4 play:1 exact:1 velocity:1 trend:1 observed:3 role:1 calculate:1 cy:1 region:4 cord:1 cycle:1 movement:45 decrease:1 voluntarily:1 vanishes:1 pd:8 asked:1 dynamic:1 trained:1 raise:1 completely:1 po:1 joint:5 easily:1 represented:1 effective:2 describe:2 visuomotor:6 salina:3 whose:1 premotor:1 otto:1 analyse:1 directionally:1 interaction:1 maximal:2 fr:3 aligned:1 organizing:2 supposed:1 produce:2 perfect:1 object:1 tk:3 resemble:1 indicate:2 come:1 direction:37 guided:2 closely:2 correct:5 attribute:2 modifying:1 human:1 generalization:1 multilinear:1 mathematically:1 considered:1 visually:2 mapping:3 predict:1 pointing:2 vary:2 multijoint:1 repetition:1 tool:3 rough:1 sensor:3 gaussian:1 always:1 reaching:11 rather:1 command:10 emission:1 referent:1 likelihood:1 flaw:1 entire:1 relation:1 france:1 canceled:1 among:2 equal:1 never:1 broad:1 mimic:1 discrepancy:1 stimulus:1 piecewise:1 gordon:1 retina:1 geometry:2 fire:1 attempt:1 investigate:1 kinematic:1 wijk:3 arrives:1 extreme:2 accurate:1 partial:1 desired:12 plotted:1 re:2 guidance:1 deviation:4 johnson:2 answer:1 varies:1 learnt:1 combined:2 st:1 spindle:1 fundamental:1 workspace:8 continuously:1 ghez:1 connectivity:1 reflect:1 central:1 accompanied:1 star:1 coding:5 bold:1 coordinated:2 depends:2 vi:1 curie:1 elaborated:1 contribution:1 yves:1 characteristic:1 directional:5 iterated:1 accurately:1 produced:2 agonist:2 trajectory:4 sensorimotor:3 involved:1 universite:1 mi:2 ubiquitous:1 feed:2 planar:1 burnod:7 response:1 done:1 angular:2 hand:2 lack:1 indicated:1 effect:4 proprioceptive:3 during:3 self:3 noted:3 cosine:2 oc:1 antagonist:2 complete:1 image:2 snv:1 sigmoid:1 spinal:2 attached:1 cerebral:4 anisotropic:1 linking:1 tuning:4 cortex:7 showed:1 issued:1 muscle:6 babbling:2 signal:1 dashed:1 multiple:1 clockwise:1 ii:1 match:1 calculation:2 kalaska:5 elaboration:1 concerning:1 coded:2 a1:4 variant:1 regression:1 cell:7 receive:1 whereas:1 addressed:1 finely:2 probably:1 cedex:1 counterclockwise:1 regularly:1 call:1 near:2 ideal:1 intermediate:1 exceed:1 xj:3 architecture:1 identified:1 questioned:1 action:7 ghilardi:3 transforms:1 visualized:1 stabilized:1 dotted:1 sign:1 delta:1 correctly:1 ccr:2 broadly:5 four:2 threshold:1 marie:1 neither:1 abbott:3 recruitment:1 powerful:1 excursion:1 layer:9 correspondence:1 existed:1 activity:11 lemon:2 precisely:1 constraint:1 sharply:1 georgopoulos:6 min:2 according:1 combination:1 describes:1 bullock:3 modification:1 making:1 primate:1 sij:3 pr:1 taken:1 equation:1 fail:2 mechanism:1 sending:1 unusual:1 operation:1 appropriate:2 pierre:2 existence:1 cf:1 readable:1 emmanuel:1 uj:2 especially:1 psychophysical:2 move:1 question:2 realized:1 posture:3 already:1 link:2 lateral:3 whom:1 toward:3 code:1 length:3 sigma:2 lil:1 collective:1 adjustable:1 neuron:24 arc:1 behave:1 parietal:2 shoulder:2 frame:1 somatic:4 concentric:1 pair:1 paris:1 connection:2 scott:3 max:1 arm:32 epoch:1 multiplication:1 analogy:1 degree:3 sufficient:1 article:1 pi:2 translation:1 compatible:1 changed:1 lmax:1 bias:3 distributed:8 curve:1 calculated:1 cortical:14 sensory:1 forward:2 coincide:1 preferred:8 synergy:4 inserm:1 global:1 sequentially:1 assumed:1 physiologically:1 why:1 subserved:1 stimulated:1 learn:1 kettner:1 transfer:1 contributes:1 complex:1 necessarily:1 vj:2 da:7 neurosci:5 arrow:2 whole:1 border:1 reafference:1 neuronal:3 fig:7 representative:1 position:22 jussieu:3 jacobian:1 specific:1 physiological:1 magnitude:1 cartesian:7 visual:12 positional:1 expressed:2 monotonic:1 neurophysioi:1 marked:2 ljp:2 experimentally:2 change:1 muscular:2 determined:1 uniformly:6 crammond:2 bernard:1 asymmetrically:1 experimental:2 formally:1 internal:2 npv:15 rotate:1 frontal:2 tested:1 correlated:1

A High Performance k-NN Classifier Using a Binary Correlation Matrix Memory Ping Zhou zhoup@cs.york.ac.uk Jim Austin austin@cs.york.ac.uk John Kennedy johnk@cs.york.ac.uk Advanced Computer Architecture Group Department of Computer Science University of York, York YOW 500, UK Abstract This paper presents a novel and fast k-NN classifier that is based on a binary CMM (Correlation Matrix Memory) neural network. A robust encoding method is developed to meet CMM input requirements . A hardware implementation of the CMM is described, which gives over 200 times the speed of a current mid-range workstation, and is scaleable to very large problems. When tested on several benchmarks and compared with a simple k-NN method, the CMM classifier gave less than I % lower accuracy and over 4 and 12 times speed-up in software and hardware respectively. 1 INTRODUCTION Pattern classification is one of most fundamental and important tasks, and a k-NN rule is applicable to a wide range of classification problems. As this method is too slow for many applications with large amounts of data, a great deal of effort has been put into speeding it up via complex pre-processing of training data, such as reducing training data (Dasarathy 1994) and improving computational efficiency (Grother & Candela 1997). This work investigates a novel k-NN classification method that uses a binary correlation matrix memory (CMM) neural network as a pattern store and match engine. Whereas most neural networks need a long iterative training time, a CMM is simple and quick to train. It requires only one-shot storage mechanism and simple binary operations (Willshaw & Buneman 1969), and it has highly flexible and fast pattern search ability. Therefore, the combination of CMM and k-NN techniques is likely to result in a generic and fast classifier. For most classification problems, patterns are in the form of multi-dimensional real numbers, and appropriate quantisation and encoding are needed to convert them into binary inputs to a CMM. A robust quantisation and encoding method is developed to meet requirements for CMM input codes , and to overcome the common problem of identical data points in many applications, e .g. background of images or normal features in a diagnostic problem. Many research projects have applied the CMM successfully to commercial problems, e.g. symbolic reasoning in the AURA (Advanced Uncertain Reasoning Architecture) approach P. Zhou. J. Austin and J. Kennedy 714 (Austin 1996), chemical structure matching and post code matching. The execution of the CMM has been identified as the bottleneck. Motivated by the needs of these applications for a further high speed processing, the CMM has been implemented in dedicated hardware, i.e. the PRESENCE architecture. The primary aim is to improve the execution speed over conventional workstations in a cost-effective way. The following sections discuss the CMM for pattern classification, describe the PRESENCE architecture (the hardware implementation of CMM), and present experimental results on several benchmarks. 2 BINARY CMM k-NN CLASSIFIER The key idea (Figure I) is to use a CMM to pre-select a smaIl sub-set of training patterns from a large number of training data, and then to apply the k-NN rule to the sub-set. The CMM is fast but produces spurious errors as a side effect (Turner & Austin 1997); these are removed through the application of the k-NN rule. The architecture of the CMM classifier (Figure I) includes an encoder (detailed in 2.2) for quantising numerical inputs and generating binary codes, a CMM pattern store and match engine and a conventional kNN module as detailed below . Training patterns stored in CMM Patterns preselected by CMM ? ? ? k-NN patterns ? ? ? ? ? I I ~B~r:~~k classification Figure 1: Architecture of the binary CMM k-NN classifier 2.1 PATTERN MATCH AND CLASSIFICATION WITH CMM A correlation matrix memory is basically a single layer network with binary weights M. In the training process a unique binary vector or separator s, is generated to label an unseen input binary vector P,; the CMM learns their association by performing the following logical ORing operation: M=VS,TPi (1) i In a recall process, for a given test input vector Vk=MPJ=(ysTpl,J Pk' the CMM performs : (2) followed by thresholding v k and recovering individual separators. For speed, it is appropriate to use a fixed thresholding method and the threshold is set to a level proportional to the number of 'I' bits in the input pattern to allow an exact or partial match. To understand the recall properties of the CMM , consider the case where a known pattern Pk is represented, then Equation 2 can be written as the following when two different patterns are orthogonal to each other: (3) where np is a scalar, i.e. the number of 'I' bits in P k ' and P,P: =0 for i:;; k . Hence a perfect recall of Sk can be obtained by thresholding v, at the level n" . In practice 'partially orthogonal' codes may be used to increase the storage capacity of the CMM and the recall noise can be removed via appropriately thresholding vk (as p,p[ ~ n p for i :;; k ) RNNs Can Learn Symbol-Sensitive Counting 715 and post-processing (e.g. applying k-NN rule). Sparse codes are usually used, i.e. only a few bits in SA and P, being set to 'I' , as this maximises the number of codes and minimises the computation time (Turner & Austin 1997). These requirements for input codes are often met by an encoder as detailed below. The CMM exhibits an interesting 'partial match' property when the data dimensionality d is larger than one and input vector p; consists of d concatenated components. If two different patterns have some common components, v k also contains separators for partially matched patterns, which can be obtained at lower threshold levels. This partial or near match property is useful for pattern classification as it allows the retrieval of stored patterns that are close to the test pattern in Hamming distance. From those training patterns matched by the CMM engine, a test pattern is classified using the k-NN rule. Distances are computed in the original input space to minimise the information loss due to quantisation and noise in the above match process. As the number of matches returned by the CMM is much smaller than the number of training data, the distance computation and comparison are dramatically reduced compared with the simple k-NN method. Therefore, the speed of the classifier benefits from fast training and matching of the CMM, and the accuracy gains from the application of the k-NN rule for reducing information loss and noise in the encoding and match processes. 2.2 ROBUST UNIFORM ENCODING Figure 2 shows three stages of the encoding process. d-dimensional real numbers, xi' are quantised as y; ; sparse and orthogonal binary vectors, Ci ' are generated and concatenated to form a CMM input vector. Yd (~, Figure 2: Quantisation, code generation and concatenation CMM input codes should be distributed as uniformly as possible in order to avoid some parts of the CMM being used heavily while others are rarely used. The code uniformity is met at the quantisation stage. For a given set of N training samples in some dimension (or axis), it is required to divide the axis into Nb small intervals, called bins, such that they contain uniform numbers of data points. As the data often have a non-uniform distribution, the sizes of these bins should be different. It is also quite common for real world problems that many data points are identical. For instance, there are 11 %-99.9% identical data in benchmarks used in this work. Our robust quantisation method described below is designed to cope with the above problems and to achieve a maximal uniformity. In our method data points are first sOfted in ascending order, N , identical points are then identified, and the number of non-identical data points in each bin is estimated as N p = (N - N, )/ Nb . B in boundaries or partitions are determined as follows. The right boundary of a bin is initially set to the next N I' -th data point in the ordered data sequence; the number of identical points on both sides of the boundary is identified; these are either included in the current or next bin. If the number of non-identical data points in the last bin is N , and N,~(Np +Nb)' Np may be increased by (N, -Np)/Nb and the above partition process may be repeated to increase the uniformity. Boundaries of bins obtained become parameters of the encoder in Figure 2. In general it is appropriate to choose Nh such that each bin contains a number of samples, which is larger than k nearest neighbours for the optimal classification. P. Zhou, J. Austin and J. Kennedy 716 3 THE PRESENCE ARCHITECTURE The pattern match and store engine of the CMM k-NN classifier has been implemented using a novel hardware based CMM architecture. i.e. the PRESENCE. 3.1 ARCHITECTURE DESIGN Important design decisions include the use of cheap memory, and not embedding both the weight storage and the training and testing in hardware (VLSI). This arises because the applications commonly use CMMs with over 100Mb of weight memory. which would be difficult and expensive to implement in custom silicon. VME and PCI are chosen to host on industry standard buses and to allow widespread application. The PRESENCE architecture implements the control logic and accumulators, i.e. the core of the CMM. As shown in Figure 3a binary input selects rows from the CMM that are added, thresholded using L-max (Austin & Stonham 1987) or fixed global thresholding, and then returned to the host for further processing. The PRESENCE architecture shown in Figure 3b consists of a bus interface, a buffer memory which allows interleaving of memory transfer and operation of the PRESENCE system, a SATCON and SA TSUM combination that accumulates and thresholds the weights. The data bus connects to a pair of memory spaces, each of which contains a control block, an input block and an output block. Thus the PRESENCE card is a memory mapping device, that uses interrupts to confirm the completion of each operation. For efficiency, two memory input/output areas are provided to be acted on from the external bus and used by the card. The control memory input block feeds to the control unit, which is a FPGA device. The input data are fed to the weights and the memory area read is then passed to a block of accumulators. In our current implementation the data width of each FPGA device is 32 bits, which allows us to add a 32 bit row from the weights memory in one cycle per device Input (sparse codes) wei hts (-) p Data bus ?? Sumv Separator output s Figure 3: (a) correlation matrix memory. and (b) overall architecture of PRESENCE Currently we have 16Mb of 25ns static memory implemented on the VME card, and 128 Mb of dynamic (60ns) memory on the PCI card. The accumulators are implemented along with the thresholding logic on another FPGA device (SATSUM). To enable the SA TSUM processors to operate faster, a 5 stage pipeline architecture was used, and the data accumulation time is reduced from 175ns to 50ns. All PRESENCE operations are supported by a C++ library that is used in all AURA applications. The design of the SATCON allows many SATSUM devices to be used in parallel in a SIMD configuration. The VME implementation uses 4 devices per board giving a 128 bit wide data path. In addition the PCI version allows daisy chaining of cards allowing a 4 card set for a 512 bit wide data path. The complete VME card assembly is shown in Figure 4. The SATCON and SATSUM devices are mounted on a daughter board for simple upgrading and alteration. The weights memory, buffer memory and VME interface are held on the mother board. 717 RNNs Can Learn Symbol-Sensitive Counting Figure 4: The VME based PRESENCE card (a) motherboard, and (b) daughterboard 3.2 PERFORMANCE By an analysis of the state machines used in the SATCON device the time complexity of the approach can be calculated. Equation 4 is used to calculate the processing time, T, in seconds to recall the data with N index values, a separator size of S, R 32 bit SATSUM devices, and the clock period of C. T = C[23+(s-l)/32R+I)(N +38+2R)] (4) A comparison with a Silicon Graphics 133MHz R4600SC Indy in Table shows the speed up of the matrix operation (Equation 2) for our VME implementation (128 bits wide) using a fixed threshold. The values for processing rate are given in millions of binary weight additions per-second (MW/s). The system cycle time needed to sum a row of weights into the counters (i.e. time to accumulate one line) is SOns for the VME version and lOOns for the PCI version. In the PCI form, we will use 4 closely coupled cards, which result in a speed-up of 432. The build cost of the VME card was half the cost of the baseline SGI Indy machine, when using 4Mb of 20ns static RAM. In the PCI version the cost is greatly reduced through the use of dynamic RAM devices allowing a 128Mb memory to be used for the same cost. allowing only a 2x slower system with 32x as much memory per card (note that 4 cards used in Table I hold 512Mb of memory). Table I : Relative speed-up of the PRESENCE architecture Platform Workstation I Card VME implementation Four card PCI system (estimate) Processing_ Rate 11.8 MW/s 2557MW/s 17,114MW/s I Relative Speed I 216 432 - The training and recogmtlon speed of the system are approximately equal. This is particularly useful in on-line applications, where the system must learn to solve the problem incrementally as it is presented. In particular, the use of the system for high speed reasoning allows the rules in the system to be altered without the long training times of other systems. Furthermore our use of the system for a k-NN classifier also allows high speed operation compared with a conventional implementation of the classifier, while still allowing very fast training times. 4 RESULTS ON BENCHMARKS Performance of the robust quantisation method and the CMM classifier have been evaluated on four benchmarks consisting of large sets of real world problems from the Statlog project (Michie & Spiegelhalter 1994), including a satellite image database, letter image recognition database. shuttle data set and image segmentation data set. To visualise the result of quantisation, Figure Sa shows the distribution of numbers of data points of the 8th feature of the image segment data for equal-size bins. The distribution represents P. Zhou, J. Austin and J. Kennedy 718 the inherent characteristics of the data. Figure 5b shows our robust quantisation (RQ) has resulted in the uniform distribution desired. 400~~~-- ________ --~ 40~ 350 ~ 300 " 30 ;""2SO '" 25 20 ~ ~ 200 ~ .i1 .i1 15 g 10 E g ~ ____ ~~ ; 150 E 100 ~ _ _ _ _- -_ _ 35 1 111111 o 5 10 15 20 values o f)ll; 25 30 o 35 o 10 15 20 values of x 25 30 3S Figure 5: Distributions of the image segment data for (a) equal bins, (b) RQ bins We compared the CMM classifier with the simple k-NN method, multi-layer perceptron (MLP) and radial basis function (RBF) networks (Zhou and Austin 1997). In the evaluation we used the CMM software libraries developed in the project AURA at the University of York. Between 1 and 3 '1' bits are set in input vectors and separators. Experiments were conducted to study influences of a CMM's size on classification rate (crate) on test data sets and speed-up measured against the k-NN method (as shown in Figure 6). The speed-up of the CMM classifier includes the encoding, training and test time. The effects of the number of bins N b on the performance were also studied. ~ 0.89 i'! g 0 .88 e 0.87 '"~ 0.86 0 0.85 0 .84 0.5 I 1.5 2 2.5 3 CMM Si7.e (MBytes) 15 4 I 1.5 2 2.5 3 CMM size (MBytes) 3. 5 4 Figure 6: Effects of the CMM size on (a) c-rate and (b) speed-up on the satellite image data Choices of the CMM size and the number of bins may be application dependent, for instance, in favour of the speed or accuracy. In the experiment it was required that the speed-up is not 4 times less and c-rate is not 1% lower than that of the k-NN method. Table 2 contains the speed-up of MLP and RBF networks and the CMM on the four benchmarks. It is interesting to note that the k-NN method needed no training. The recall of MLP and RBF networks was very faster but their training was much slower than that of the CMM classifier. The recall speed-up of the CMM was 6-23 times, and the overall speed-up (including training and recall time) was 4-15x. When using the PRESENCE, i.e. the dedicated CMM hardware, the speed of the CMM was further increased over 3 times. This is much less than the speed-up of 216 given in Table 1 because of recovering separators and k-NN classification are performed in software. Table 2: Speed-up of MLP, RBF and CMM relative to the simple k-NN method Image segment method training 0.04 MLPN RBFN 0.09 simplek-NN CMM 18 test 18 9 I 9 Satellite image training 0.2 0.07 - 15.8 Test 28.4 20.3 1 5.7 Letter training 0.2 0.3 - 24.6 test 96.5 66.4 1 6.8 Shuttle training 4.2 1.8 43 test 587.2 469.7 I 23 The classification rates by the four methods are given in Table 3, which shows the CMM classifier performed only 0-1% less accurate than the k-NN method. 719 RNNs Can Learn Symbol-Sensitive Counting Table 3: Classification rates of four methods on four benchmarks MLPN RBFN simple k-NN CMM Image segment 0.950 0.939 0.956 0.948 Satellite image 0.914 0.914 0.906 0.901 Letter 0.923 0.941 0.954 0.945 Shuttle 0.998 0.997 0.999 0.999 5 CONCLUSIONS A novel classifier is presented, which uses a binary CMM for storing and matching a large amount of patterns efficiently, and the k-NN rule for classification . The RU encoder converts numerical inputs into binary ones with the maximally achievable uniformity to meet requirements of the CMM. Experimental results on the four benchmarks show that the CMM classifier, compared with the simple k-NN method , gave slightly lower classification accuracy (less than 1% lower) and over 4 times speed in software and 12 times speed in hardware. Therefore our method has resulted in a generic and fast classifier. This paper has also described a hardware implementation of a FPGA based chip set and a processor card that will support the execution of binary CMM . It has shown the viability of using a simple binary neural network to achieve high processing rates. The approach allows both recognition and training to be achieved at speeds well above two orders of magnitude faster than conventional workstations at a much lower cost than the workstation. The system is scaleable to very large problems with very large weight arrays. Current research is aimed at showing that the system is scaleable, evaluating methods for the acceleration of the pre- and post processing tasks and considering greater integration of the elements of the processor through VLSI. For more details of the AURA project and the hardware described in this paper see http://www.cs.york.ac.uk/arch/nnJaura.html. Acknowledgements We acknowledge British Aerospace and the Engineering and Physical Sciences Research Council (grant no. GRiK 41090 and GR/L 74651) for sponsoring the research. Our thanks are given to R Pack, A Moulds, Z Ulanowski. R Jennison and K Lees for their support. References Willshaw, 0.1., Buneman, O.P. & Longuet-Higgins, H .C. (1969) Non-holographic associative memory. Nature, Vol. 222, p960-962. Austin, J. (1996) AURA, A distributed associative memory for high speed symbolic reasoning. In: Ron Sun (ed), Connectionist Symbolic Integration. Kluwer. Turner, M. & Austin, J. (1997) Matching performance of binary correlation matrix memories. Neural Networks; 10:1637-1648. Dasarathy, B.V. (1994) Minimal consistent set (MCS) identification for optimal nearest neighbor decision system design. IEEE Trans. Systems Man Cybernet; 24:511-517. Grother, P.l., Candela, G.T. & Blue, J.L. (1997) Fast implementations of nearest neighbor classifiers. Pattern Recognition ; 30:459-465. Austin, J., Stonham, T.J. (1987) An associative memory for use in image recognition and occlusion analysis. Image and Vision Computing; 5:251-261. Michie, D., Spiegelhalter, 0.1. & Taylor, c.c. (1994) Machine learning, neural and statistical classification (Chapter 9). New York, Ellis Horwood. Zhou, P. & Austin J. (1998) Learning criteria for training neural network classifiers . Neural Computing and Applications Forum; 7:334-342. PART VI SPEECH, HANDWRITING AND SIGNAL PROCESSING 

1495 |@word version:4 achievable:1 shot:1 configuration:1 contains:4 current:4 written:1 must:1 john:1 numerical:2 partition:2 cheap:1 designed:1 hts:1 v:1 half:1 device:11 core:1 ron:1 along:1 become:1 consists:2 multi:2 considering:1 mpj:1 project:4 provided:1 matched:2 developed:3 willshaw:2 classifier:21 uk:5 control:4 unit:1 grant:1 engineering:1 encoding:7 accumulates:1 dasarathy:2 meet:3 path:2 yd:1 approximately:1 rnns:3 studied:1 range:2 unique:1 accumulator:3 testing:1 practice:1 block:5 implement:2 area:2 matching:5 pre:3 radial:1 symbolic:3 close:1 put:1 storage:3 applying:1 nb:4 influence:1 accumulation:1 conventional:4 www:1 quick:1 smail:1 rule:8 higgins:1 array:1 embedding:1 commercial:1 heavily:1 exact:1 us:4 element:1 expensive:1 particularly:1 recognition:4 michie:2 database:2 module:1 calculate:1 cycle:2 sun:1 counter:1 removed:2 rq:2 complexity:1 dynamic:2 uniformity:4 segment:4 mbytes:2 efficiency:2 basis:1 chip:1 represented:1 chapter:1 train:1 fast:8 effective:1 describe:1 pci:7 quite:1 larger:2 solve:1 encoder:4 ability:1 knn:1 unseen:1 associative:3 sequence:1 tpi:1 maximal:1 mb:6 achieve:2 sgi:1 requirement:4 satellite:4 produce:1 generating:1 perfect:1 ac:4 completion:1 minimises:1 measured:1 nearest:3 sa:4 indy:2 implemented:4 c:4 recovering:2 met:2 closely:1 enable:1 bin:13 statlog:1 hold:1 normal:1 great:1 mapping:1 applicable:1 label:1 currently:1 sensitive:3 council:1 successfully:1 aim:1 zhou:6 avoid:1 shuttle:3 interrupt:1 vk:2 greatly:1 baseline:1 dependent:1 nn:28 cmm:60 initially:1 spurious:1 vlsi:2 selects:1 i1:2 overall:2 classification:16 flexible:1 aura:5 html:1 platform:1 integration:2 equal:3 simd:1 identical:7 represents:1 np:4 others:1 loon:1 inherent:1 few:1 connectionist:1 neighbour:1 resulted:2 individual:1 connects:1 consisting:1 occlusion:1 scaleable:3 mlp:4 highly:1 custom:1 evaluation:1 held:1 accurate:1 partial:3 orthogonal:3 divide:1 taylor:1 desired:1 minimal:1 uncertain:1 instance:2 increased:2 industry:1 elli:1 mhz:1 cost:6 uniform:4 fpga:4 holographic:1 conducted:1 gr:1 too:1 graphic:1 stored:2 thanks:1 fundamental:1 lee:1 choose:1 external:1 alteration:1 includes:2 vi:1 performed:2 candela:2 parallel:1 daisy:1 accuracy:4 characteristic:1 efficiently:1 identification:1 basically:1 vme:10 mc:1 kennedy:4 processor:3 classified:1 ping:1 ed:1 against:1 workstation:5 hamming:1 gain:1 static:2 handwriting:1 logical:1 recall:8 dimensionality:1 segmentation:1 feed:1 wei:1 maximally:1 evaluated:1 furthermore:1 stage:3 arch:1 correlation:6 clock:1 incrementally:1 widespread:1 effect:3 contain:1 hence:1 chemical:1 read:1 deal:1 ll:1 width:1 sponsoring:1 chaining:1 criterion:1 complete:1 performs:1 dedicated:2 interface:2 reasoning:4 image:13 novel:4 common:3 physical:1 nh:1 million:1 association:1 visualise:1 kluwer:1 accumulate:1 silicon:2 mother:1 quantisation:9 add:1 store:3 buffer:2 binary:19 greater:1 period:1 signal:1 match:10 faster:3 long:2 retrieval:1 host:2 post:3 buneman:2 vision:1 achieved:1 whereas:1 background:1 addition:2 crate:1 interval:1 appropriately:1 operate:1 near:1 presence:13 counting:3 mw:4 viability:1 gave:2 architecture:14 identified:3 idea:1 minimise:1 bottleneck:1 favour:1 motivated:1 passed:1 effort:1 returned:2 speech:1 york:8 dramatically:1 useful:2 detailed:3 aimed:1 amount:2 mid:1 hardware:10 reduced:3 http:1 diagnostic:1 estimated:1 per:4 blue:1 vol:1 group:1 key:1 four:7 threshold:4 thresholded:1 ram:2 convert:2 sum:1 grother:2 quantised:1 letter:3 decision:2 investigates:1 oring:1 bit:10 layer:2 followed:1 software:4 speed:28 performing:1 acted:1 department:1 upgrading:1 combination:2 smaller:1 slightly:1 son:1 pipeline:1 equation:3 bus:5 discus:1 mechanism:1 needed:3 horwood:1 ascending:1 fed:1 operation:7 apply:1 generic:2 appropriate:3 mlpn:2 slower:2 original:1 include:1 assembly:1 giving:1 concatenated:2 build:1 forum:1 added:1 primary:1 exhibit:1 distance:3 card:15 capacity:1 concatenation:1 ru:1 code:11 index:1 difficult:1 daughter:1 implementation:9 design:4 maximises:1 allowing:4 benchmark:8 acknowledge:1 jim:1 pair:1 required:2 aerospace:1 engine:4 trans:1 below:3 pattern:24 usually:1 preselected:1 max:1 memory:26 including:2 advanced:2 turner:3 improve:1 altered:1 spiegelhalter:2 library:2 axis:2 coupled:1 speeding:1 acknowledgement:1 relative:3 loss:2 interesting:2 generation:1 proportional:1 mounted:1 consistent:1 thresholding:6 storing:1 austin:14 row:3 supported:1 last:1 side:2 allow:2 understand:1 perceptron:1 wide:4 neighbor:2 sparse:3 benefit:1 distributed:2 overcome:1 dimension:1 boundary:4 world:2 calculated:1 evaluating:1 commonly:1 cope:1 logic:2 confirm:1 global:1 xi:1 search:1 iterative:1 sk:1 table:8 nature:1 learn:4 transfer:1 robust:6 pack:1 longuet:1 improving:1 complex:1 separator:7 pk:2 noise:3 repeated:1 board:3 slow:1 n:5 sub:2 learns:1 interleaving:1 british:1 showing:1 symbol:3 rbfn:2 ci:1 magnitude:1 execution:3 likely:1 ordered:1 partially:2 scalar:1 acceleration:1 rbf:4 man:1 included:1 determined:1 reducing:2 uniformly:1 called:1 experimental:2 rarely:1 select:1 support:2 arises:1 tested:1

Coordinate Transformation Learning of Hand Position Feedback Controller by U sing Change of Position Error Norm Eimei Oyama* Mechanical Eng. Lab. Namiki 1-2, Tsukuba Science City Ibaraki 305-8564 Japan Susumu Tachi The University of Tokyo Hongo 7-3-1, Bunkyo-ku Tokyo 113-0033 Japan Abstract In order to grasp an object, we need to solve the inverse kinematics problem, i.e., the coordinate transformation from the visual coordinates to the joint angle vector coordinates of the arm. Although several models of coordinate transformation learning have been proposed, they suffer from a number of drawbacks. In human motion control, the learning of the hand position error feedback controller in the inverse kinematics solver is important. This paper proposes a novel model of the coordinate transformation learning of the human visual feedback controller that uses the change of the joint angle vector and the corresponding change of the square of the hand position error norm. The feasibility of the proposed model is illustrated using numerical simulations. 1 INTRODUCTION The task of calculating every joint angle that would result in a specific hand position is called the inverse kinematics problem. An important topic in neuroscience is the study of the learning mechanisms involved in the human inverse kinematics solver. We questioned five pediatricians about the motor function of infants suffering from serious upper limb disabilities. The doctors stated that the infants still were able to touch and stroke an object without hindrance. In one case, an infant without a thumb had a major kinematically influential surgical operation, transplanting an index finger as a thumb. After the operation, the child was able to learn how to use the index finger like a thumb [1]. In order to explain the human motor learning ? Phone:+81-298-58-7298, Fax:+81-298-58-7201, e-mail:eimei@mel.go.jp Coordinate Transformation Learning ofFeedback Controller 1039 capability, we believe that the coordinate transformation learning of the feedback controller is a necessary component. Although a number of learning models of the inverse kinematics solver have been proposed, a definitive learning model has not yet been obtained. This is from the point of view of the structural complexity of the learning model and the biological plausibility of employed hypothesis. The Direct Inverse Modeling employed by many researchers [2] requires the complex switching of the input signal of the inverse model. When the hand position control is performed, the input of the inverse model is the desired hand position, velocity, or acceleration. When the inverse model learning is performed, the input is the observed hand position, velocity, or acceleration. Although the desired signal and the observed signal could coincide, the characteristics of the two signals are very different. Currently, no research has succeesfully modeled the switching system. Furthermore, that learning model is not "goal-directed"; i.e., there is no direct way to find an action that corresponds to a particular desired result. The Forward and Inverse Modeling proposed by Jordan [3] requires the back-propagation signal, a technique does not have a biological basis. That model also requires the complex switching of the desired output signal for the forward model. When the forward model learning is performed, the desired output is the observed hand position. When the inverse kinematics solver learning is performed, the desired output is the desired hand position. The Feedback Error Learning proposed by Kawato [4] requires a pre-existing accurate feedback controller. It is necessary to obtain a learning model that possesses a number of characteristics: (1) it can explain the human learning function; (2) it has a simple structure; and (3) it is biologically plausible. This paper presents a learning model of coordinate transformation function of the hand position feedback controller. This model uses the joint angle vector change and the corresponding change of square of the hand position error norm. 2 BACKGROUND 2.1 Discrete Time First Order Model of Hand Position Controller Let 8 E Rm be the joint angle vector and x ERn be the hand position/orientation vector given by the vision system. The relationship between x and 8 is expressed as x = /(8) where / is a C 1 class function. The Jacobian of the hand position vector is expressed as J(8) = 8/(8)/88. Let Xd be the desired hand position and e = Xd - X = Xd - /(8) be the hand position error vector. In this paper, an inverse kinematics problem is assumed to be a least squares minimization problem that calculates 8 in order to minimize the square of the hand position error norm S(xd,8) = le1 2 /2 = IXd - /(8)1 2 /2. First, the feed-forward controller in the human inverse kinematics solver is disregarded and the following first order control system, consisting of a learning feedback controller, is considered: 8(k Position + Xti Desired Hand Position Error e(k) + 1) Feedback ? = 8(k) + A8(k) Di~rbance d(k) NOIse + tPp/.8, e) Joint Angle Vector ~k~ (+-~ (1) (f?-'f(8) H uman Arm Hand Position X(!i... 8(k) Figure 1: Configuration of 1-st Order Model of Hand Position Controller E. Oyama and S. Tachi 1040 a8(k) = ~fb(8(k), e(k)) + d(k) (2) e(k) = Xd - f(8(k)) (3) where d(k) is assumed to be a disturbance noise from all components except the hand position control system. Figure 1 shows the configuration of the control system. In this figure, Z-l is the operator that indicates a delay in the discrete time signal by a sampling interval of tl.t. Although the human hand position control system includes higher order complex dynamics terms which are ignored in Equation (2), McRuer's experimental model of human compensation control suggests that the term that converts the hand position error to the hand velocity is a major term in the human control system [5]. We consider Equation (2) to be a good approximate model for the analysis of human coordinates transformation learning. The learner ~ fb (8, e) E R m, which provides the hand position error feedback, is modeled using the artificial neural network. In this paper, the hand position error feedback controller learning by observing output x(k) is considered without any prior knowledge of the function f (8). 2.2 Learning Model of the Neural Network Let ~fb(8, e) be the desired output of the learner ~fb(8, e). ~fb(8, e) functions as a teacher for ~fb(8,e). Let ~jb(8 , e) be the updated output of ~fb(8,e) by the learning. Let E[t(8, e)18, e] be the expected value of a scalar, a vector, or a matrix function t(8,e) when the input vector (8 , e) is given. We assume that ~fb(8 , e) is an ideal learner which is capable of realizing the mean of the desired output signal, completely. ~+ fb(8, e) can be expressed as follows: ~jb(8, e) ~ E[~fb(8 , e)18, e] a~fb(8 , e) When the expected value of = ~fb(8 , e) + E[a~fb(8, e)18, e] = ~fb(8 , e) - a~fb(8, ~fb(8, e) (4) (5) e) is expressed as: E[a~fb(8,e)18,e] ~ Gfbe - Rfb~fb(8 , e) , (6) Rfb E Rm xm is a positive definite matrix, and the inequality I8~jb(8, e) I = I8(G fb e - (Rfb - I)~fb(8, e? e) 8~ fb(8 , e) is satisfied, the final learning result can be expressed as: ~fb(8 , e) ~ Rjb1Gfbe I< 1 8~fb(8, by the iteration of the update of 3 3.1 ~fb(8 , (7) (8) e) expressed in Equation (4). USE OF CHANGE OF POSITION ERROR NORM A Novel Learning Model of Feedback Controller The change of the square of the hand position error norm tl.S = S(Xd , 8 + a8) S(Xd , 8) reflects whether or not the change of the joint angle vector A8 is in proper direction. The propose novel learning model can be expressed as follows: ~fb(8, e) = -atl.Sa8 (9) where a is a small positive real number. We now consider a large number of trials of Equation (2) with a large variety of initial status 8(0) with learnings conducted at the point of the input space of the feedback controller (8, e) = (8(k -1), e(k -1? at time k. tl.S and a8 can be calculated as follows. tl.S a8 S(k) - S(k - 1) = = a8(k - 1) ~(le(kW -Ie(k - 1W) (10) (11) 1041 Coordinate Transformation Learning ofFeedback Controller .---------, Change of Square of Hand Position Error Norm e(k) Hand Position Change of Joint Angle Vector Input for Learning Error e(k-l) --- e(k) d8(k) Error Signal for k ' / '---,..--T----' Feedback Input for Controller Learning Input for Control (J(k-l) d(k) Dist""'-z NoiJe 8(k) Figure 2: Configuration of Learning Model of Feedback Controller Figure 2 shows the conceptual diagram of the proposed learning model. Let p(qI8, e) be the probability density function of a vector q at at the point (8, e) in the input space of ~fb(8, e). In order to simplify the analysis of the proposed learning model, d(k) is assumed to satisfy the following equation: p(dI8, e) = p( -dI8, e) (12) When d8 is small enough, the result of the learning using Equation (9) can be expressed as: (13) ~fb(8, e) ~ a(~R9JT (8)J(8) + 1)-1 R9JT (8)e R9 = E[d8d8T I8, e] (14) where JT (8)e is a vector in the steepest descent direction of S(Xd, 8). When d(k) is a non-zero vector, R9 is a positive definite symmetric matrix and (~R9JT J + 1)-1 is a positive definite matrix. When a is appropriate, ~ fb(8, e) as expressed in Equation (13) can provide appropriate output error feedback control. The derivation of the above result will be illustrated in Section 3.2. A partially modified steepest descent direction can be obtained without using the forward model or the back-propagation signal, as Jordan's forward modeling [3]. Let Rd be the covariance matrix of the disturbance noise d(k). When a is infinitesimal, R9 ~ Rd is established and an approximate solution ~fb(8,e) ~ aRdJ T (8)e is obtained. 3.2 Derivation of Learning Result The change of the square of the hand position error norm llS(Xd, 8) by d8 can be determined as: llS(xd, 8) = 8S~;, 8) d8 + ~d8T H(Xd, 8)d8 + O(d83 ) = -eT (J(8) + ~ 8~~8) i&l d8)d8 (15) + ~d8T J T (8)J(8)d8 + O(d83 ) where i&l is a 2-operand operator that indicates the Croneker's product. H(Xd,8) E is the Hessian of S(Xd, 8). O(d8 3 ) is the sum of third and higher order terms of d8 in each equation. When d8 is small enough, the following approximate equations are obtained: Rmxm 18J(8) 1 dx ~ J(8)d8 ~ J(8 + 2"d8)d8 ~ (J(8) + 2 88 i&l d8)d8 Therefore, llS can be approximated as follows: 1 llS ~ _eT J(8)d8 + 21dXI2 (16) (17) E. Oyama and S. Tachi 1042 Since e T J AOAO = AOAO T JT e and IAxI2 AO = AOAOT JT J AO are determined, tl.S AO can be approximated as: (18) Considering AO njb defined as AO njb = AO - .jb(O,e), the expected value of the product of AO and tl.S at the point (O,e) in the input space of .jb(O,e) can be approximated as follows: E[tl.SAOIO, e] TIT + 2ReJ ~ -ReJ e + 1 T T 2E[AOAO J J AOnjblO, e] J.jb(O,e) (19) When the arm is controlled according to Equation (2), AO njb is the disturbance noise d(k). Since d(k) satisfies Equation (12), the following equation is established. E[AOAO T JT JAOnjbIO,e] = 0 (20) Therefore, the expected value of A.jb(O, e) can be expressed as; TaT E[A.jb(O, e)IO, e] ~ aReJ e - (2ReJ J + I).jb(O, e) (21) When a is small enough, the condition described in Equation (7) is established. The learning result expressed as Equation (13) is obtained as described in Section 2.2. It should be noted that the learning algorithm expressed in Equation (9) is applicable not only to S(Xd,O), but also to general penalty functions of hand po- sition error norm lei. The proposed learning model synthesizes a direction that decreases S(Xd,O) by summing after weighting AO based on the increase or decrease of S(Xd, 0). The feedback controller defined in Equation (13) requires a number of iterations to find a correct inverse kinematics solution, as the coordinates transformation function of the controller is incomplete. However, by using Kawato's feedback error learning [4], the second feedback controller; the feed-forward controller; or the inverse kinematics model that has a complete coordinate transformation function can be obtained as shown in Section 4. 4 TRACKING CONTROL SYSTEM LEARNING In this section, we will consider the case where Xd changes as xd(k)(k 1,2, ... ). The hybrid controller that includes the learning feed-forward controller .ff(O(k), AXd(k)) E Rm that transforms the change of the desired hand position AXd(k) = xd(k + 1) - xd(k) to the joint angle vector space is considered: AO(k) = .ff(O(k), AXd(k)) + .,b(O(k),e(k)) e(k) xd(k) - x(k) + d(k) = (22) (23) The configuration of the hybrid controller is illustrated in Figure 3. By using the modified change of the square of the error norm expressed as: 1 2 2 tl.S = 2(lxd(k - 1) - x(k)1 - le(k - 1)1 ) (24) and AO(k) as defined in Equation (22), the feedback controller learning rule defined in Equation (9) is useful for the tracking control system. A sample holder for memorizing xd(k -1) is necessary for the calculation of tl.S. When the distribution 1043 Coordinate Transformation Learning ofFeedback Controller Error Signal for ~:,..-:--::----:-"' Fccdforword Controller Fccdforwonl CoDtrolJer iixd(k) ~ ~ 4},(8,Lix" (k?)- J*(8)Lh" (k) i----r---, :I J(8)J*(8)=1 ./i(8'. aj(8) I arr- HWIIIJIArm Position I!m>r + ~(k) Desired Hand Position e(k) - ~ Feedback ? 8(k) Hand Position x(k) X=f(O) 8(k) Figure 3: Configuration of Hybrid Controller of 4Xd(k) satisfies Equation (20), Equation (13) still holds. When 4Xd(k) has no correlation with d(k) and 4Xd(k) satisfies p(4XdI8, e) p( -4XdI8, e), Equation (20) is approximately established after the feed-forward controller learning. = Using 48(k) defined in Equation (2) and e(k) defined in Equation (23), tl.S defined in Equation (10) can be useful for the calculation of ~fb(8, e). Although the learning calculation becomes simpler, the learning speed becomes much lower. Let~' ff(8(k), 4Xd(k)) be the desired output of ~,,(8(k), 4Xd(k)). According to Kawato's feedback error learning [4], we use ~',,(8(k), 4Xd(k)) expressed as: ~',,(8(k), 4Xd(k)) = (1 - >..)~,,(8(k), 4Xd(k)) + ~fb(8(k + 1), e(k + 1)) (25) where >.. is a small, positive, real number for stabilizing the learning process and ensuring that equation ~,,(8,O) ~ 0 holds . If >.. is small enough, the learning feed-forward controller will fulfill the equation: J~,,(8, 5 4Xd) ~ 4Xd (26) NUMERICAL SIMULATION Numerical simulation experiments were performed in order to evaluate the performance of the proposed model. The inverse kinematics of a 3 DOF arm moving on a 2 DOF plane were considered. The relationship between the joint angle vector 8 = (8 1 '(h, ( 3 ) T and the hand position vector x = (x, y) T was defined as: x = Xo + Ll cos(8t} + L2 cos(81 + (2 ) + L3 cos(81 + 82 + ( 3 ) y = Yo + Ll sin(81) + L2 sin(81 + (2 ) + L3 sin(81 + 82 + ( 3 ) (27) (28) The range for 81 was (-30 0 ,1200 ); the range for 82 was (0 0 ,120 0 ); and the range for 83 was (_75 0 ,75 0 ). Ll was 0.30 m, L2 was 0.25 m and L3 was 0.15 m. Random straight lines were generated as desired trajectories for the hand. The tracking control trials expressed as Equation (22) with the learning of the feedback controller and the feed-forward controller were performed. The standard deviation of each component of d was 0.01. Learnings based on Equations (9), (22) , (24), and (25) were conducted 20 times in one tracking trial. 1,000 tracking trials were conducted to estimate the RMS(Root Mean Square) of e(k) . In order to accelerate the learning, a in Equation (9) was modified as a = 0.5/(Itl.xI 2 + 0.11tl.(12). >.. in Equation (25) was set to O.OOL Two neural networks with 4 layers were used for the simulation. The first layer had 5 neurons and the forth layer had 3 neurons. The other layers had 15 neurons each. The first layer and the forth layer consisted of linear neurons. The initial values of weights of the neural networks were generated by using uniform random numbers. The back-propagation method without optimized learning coefficients was utilized for the learning. E. Oyama and S. Tachi 1044 El00~------------------~ ....... y 0.5 ... g w 10.2 +-r--___--.........__~ ___.............-r.........f 10?101102103104105106107 o 0.5 x of Trials Figure 4: Learning Process of Controller Figure 5: One Example of Tracking Control CE: Number Figure 4 shows the progress of the proposed learning model. It can be seen that the RMS error decreases and the precision of the solver becomes higher as the number of trials increases. The RMS error became 9.31 x 1O- 3 m after 2 x 107 learning trials . Figure 5 illustrates the hand position control by the inverse kinematics solver after 2 x 107 learning trials. The number near the end point of the arm indicates the value of k. The center of the small circle in Figure 5 indicates the desired hand position. The center of the large circle indicates the final desired hand position. Through learning, a precise inverse kinematics solver can be obtained. However, for RMS error to fall below 0.02, trials must be repeated more than 106 times. In such cases, more efficient learner or a learning rule is necessary. 6 CONCLUSION A learning model of coordinate transformation of the hand position feedback controller was proposed in this paper. Although the proposed learning model may take a long time to learn, it is capable of learning a correct inverse kinematics solver without using a forward model, a back-propagation signal, or a pre-existing feedback controller. We believe that the slow learning speed can be improved by using neural networks that have a structure suitable for the coordinate transformation. A major limitation of the proposed model is the structure of the learning rule, since the learning rule requires the calculation of the product of the change of the error penalty function and the change of the joint angle vector. However, the existence of such structure in the nervous system is unknown. An advanced learning model which can be directly compared with the physiological and psychological experimental results is necessary. References [1] T. Ogino and S. Ishii, "Long-term Results after Pollicization for Congenital Hand Deformities," Hand Surgery, 2, 2,pp.79-85,1997 [2] F. H. Guenther and D. M. Barreca," Neural models for flexible control of redundant systems," in P. Morasso and V. Sanguineti (Eds.), Self-organization, Computational Maps, and Motor Control. Amsterdam: Elsevier, pp.383-421 ,1997 [3J M.1. Jordan, "Supervised Learning and Systems with Excess Degrees of Freedom," COINS Technical Report,88-27,pp.1-41 ,1988 [4J M. Kawato, K. Furukawa and R. Suzuki, "A Hierarchical Neural-network Model for Control and Learning of Voluntary Movement," Biological Cybernetics, 57, pp.169-185, 1987 [5] D.T. McRuer and H. R. Jex, "A Review of Quasi-Linear Pilot Models," IEEE Trans. on Human Factors in Electronics, HFE-8, 3, pp.38-51, 1963 

1496 |@word trial:9 norm:10 tat:1 simulation:4 eng:1 covariance:1 electronics:1 configuration:5 initial:2 existing:2 yet:1 dx:1 must:1 numerical:3 motor:3 update:1 infant:3 nervous:1 plane:1 steepest:2 realizing:1 provides:1 simpler:1 five:1 direct:2 guenther:1 expected:4 dist:1 xti:1 solver:9 considering:1 becomes:3 transformation:14 every:1 xd:32 rm:3 control:18 positive:5 io:1 switching:3 sanguineti:1 approximately:1 suggests:1 co:3 range:3 directed:1 definite:3 pre:2 operator:2 map:1 center:2 go:1 stabilizing:1 ixd:1 rule:4 atl:1 coordinate:16 updated:1 us:2 hypothesis:1 velocity:3 approximated:3 utilized:1 observed:3 lix:1 decrease:3 movement:1 complexity:1 dynamic:1 tit:1 learner:4 basis:1 completely:1 po:1 joint:11 accelerate:1 finger:2 derivation:2 artificial:1 dof:2 solve:1 plausible:1 final:2 propose:1 product:3 arr:1 forth:2 object:2 progress:1 direction:4 drawback:1 tokyo:2 correct:2 human:11 ao:11 biological:3 hold:2 rfb:3 considered:4 major:3 applicable:1 currently:1 city:1 reflects:1 minimization:1 modified:3 fulfill:1 yo:1 deformity:1 indicates:5 ishii:1 elsevier:1 quasi:1 orientation:1 flexible:1 proposes:1 sampling:1 kw:1 jb:9 report:1 simplify:1 serious:1 lls:4 njb:3 consisting:1 freedom:1 organization:1 grasp:1 accurate:1 capable:2 necessary:5 lh:1 incomplete:1 desired:17 circle:2 psychological:1 modeling:3 deviation:1 uniform:1 delay:1 conducted:3 teacher:1 st:1 density:1 ie:1 satisfied:1 kinematically:1 d8:17 japan:2 includes:2 coefficient:1 satisfy:1 performed:6 view:1 root:1 lab:1 observing:1 doctor:1 capability:1 minimize:1 square:9 holder:1 became:1 characteristic:2 surgical:1 thumb:3 trajectory:1 researcher:1 cybernetics:1 straight:1 stroke:1 explain:2 ed:1 infinitesimal:1 pp:5 involved:1 di:1 pilot:1 knowledge:1 oyama:4 back:4 feed:6 higher:3 supervised:1 improved:1 furthermore:1 correlation:1 hand:42 hindrance:1 touch:1 propagation:4 aj:1 lei:1 believe:2 consisted:1 symmetric:1 illustrated:3 ll:3 sin:3 self:1 noted:1 mel:1 complete:1 motion:1 novel:3 kawato:4 operand:1 jp:1 rd:2 axd:3 had:4 l3:3 moving:1 phone:1 inequality:1 furukawa:1 seen:1 employed:2 redundant:1 signal:12 technical:1 plausibility:1 calculation:4 long:2 feasibility:1 calculates:1 controlled:1 ensuring:1 controller:35 vision:1 sition:1 iteration:2 background:1 interval:1 diagram:1 posse:1 jordan:3 structural:1 near:1 ideal:1 enough:4 variety:1 ibaraki:1 tpp:1 whether:1 rms:4 penalty:2 ool:1 suffer:1 bunkyo:1 questioned:1 hessian:1 action:1 ignored:1 useful:2 transforms:1 neuroscience:1 discrete:2 susumu:1 ce:1 convert:1 sum:1 inverse:19 angle:11 layer:6 tsukuba:1 speed:2 jex:1 ern:1 influential:1 according:2 biologically:1 memorizing:1 xo:1 equation:30 kinematics:14 mechanism:1 end:1 operation:2 limb:1 hierarchical:1 appropriate:2 coin:1 rmxm:1 rej:3 existence:1 calculating:1 surgery:1 disability:1 topic:1 mail:1 index:2 modeled:2 relationship:2 stated:1 proper:1 unknown:1 upper:1 neuron:4 sing:1 compensation:1 descent:2 voluntary:1 precise:1 lxd:1 mechanical:1 optimized:1 established:4 trans:1 able:2 below:1 xm:1 suitable:1 hybrid:3 disturbance:3 advanced:1 arm:5 morasso:1 fax:1 prior:1 review:1 l2:3 limitation:1 degree:1 i8:3 fall:1 feedback:25 calculated:1 fb:31 forward:12 suzuki:1 coincide:1 excess:1 approximate:3 status:1 hongo:1 conceptual:1 summing:1 assumed:3 xi:1 ku:1 learn:2 synthesizes:1 complex:3 definitive:1 noise:4 suffering:1 child:1 repeated:1 tl:11 ff:3 slow:1 precision:1 position:41 congenital:1 jacobian:1 third:1 weighting:1 specific:1 jt:4 uman:1 physiological:1 illustrates:1 disregarded:1 visual:2 expressed:15 amsterdam:1 tracking:6 partially:1 scalar:1 corresponds:1 a8:7 satisfies:3 itl:1 goal:1 acceleration:2 change:16 determined:2 except:1 called:1 r9:3 experimental:2 evaluate:1

Finite-dimensional approximation of Gaussian processes Giancarlo Ferrari Trecate Dipartimento di Informatica e Sistemistica, Universita di Pavia, Via Ferrata 1, 27100 Pavia, Italy ferrari@conpro.unipv.it Christopher K. I. Williams Department of Artificial Intelligence, University of Edinburgh, 5 Forrest Hill, Edinburgh EH1 2QL, ckiw@dai.ed.ac.uk. Manfred Opper Neural Computing Research Group Division of Electronic Engineering and Computer Science Aston University, Birmingham, B4 7ET, UK m.opper@aston.ac.uk Abstract Gaussian process (GP) prediction suffers from O(n3) scaling with the data set size n. By using a finite-dimensional basis to approximate the GP predictor, the computational complexity can be reduced. We derive optimal finite-dimensional predictors under a number of assumptions, and show the superiority of these predictors over the Projected Bayes Regression method (which is asymptotically optimal). We also show how to calculate the minimal model size for a given n. The calculations are backed up by numerical experiments. 1 Introduction Over the last decade there has been a growing interest in the Bayesian approach to regression problems, using both neural networks and Gaussian process (GP) prediction, that is regression performed in function spaces when using a Gaussian random process as a prior. The computational complexity of the GP predictor scales as O(n 3), where n is the size Finite-Dimensional Approximation o/Gaussian Processes 219 of the datasetl . This suggests using a finite-dimensional approximating function space, which we will assume has dimension m < n. The use of the finite-dimensional model is motivated by the need for regression algorithms computationally cheaper than the G P one. Moreover, GP regression may be used for the identification of dynamical systems (De Nicolao and Ferrari Trecate, 1998), the next step being a model-based controller design. In many cases it is easier to accomplish this second task if the model is low dimensional. Use of a finite-dimensional model leads naturally to the question as to which basis is optimal. Zhu et al. (1997) show that, in the asymptotic regime, one should use the first m eigenfunctions of the covariance function describing the Gaussian process. We call this method Projected Bayes Regression (PBR) . The main results of the paper are: 1. Although PBR is asymptotically optimal, for finite data we derive a predictor hO(x) with computational complexity O(n 2m) which outperforms PBR, and obtain an upper bound on the generalization error of hO(x) . 2. In practice we need to know how large to make m . We show that this depends on n and provide a means of calculating the minimal m. We also provide empirical results to back up the theoretical calculation. 2 Problem statement Consider the problem of estimating an unknown function f(x) : JRd -T JR, from the noisy observations ti = f(Xi) + Ei, i = 1, ... , n where Ei are i.i.d. zero-mean Gaussian random variables with variance a 2 and the samples Xi are drawn independently at random from a distribution p(x) . The prior probability measure over the function f (.) is assumed to be Gaussian with zero mean and autocovariance function C (6,6). Moreover we suppose that f (.), Xi, Ei, are mutually independent. Given the data set 'Dn = {x , f}, where x = [Xl' . .. ' Xn] and f = [tl, ... , t n]', it is well known that the posterior probability PUI'Dn) is Gaussian and the GP prediction can be computed via explicit formula (e.g. Whittle, 1963) j(x) = E[JI'Dn](x) = C(x , Xn)] H - lf, [C(x, xd {H} ij ~C(Xi ' Xj) + a20ij where H is a n x n matrix and Oij is the Kronecker delta. In this work we are interested in approximating j in a suitable m-dimensional space that we are going to define. Consider the Mercer-Hilbert expansion of C(6, 6) r C(6,6)'Pi(6)p(6)d6 = iRd Ai'Pi(6), r 'Pi(~)'Pj(Op(~)d~ = Oij iRd (1) +00 C(6,6) = L Ai'Pi(6)'Pi(6), i=l where the eigenvalues Ai are ordered in a decreasing way. Then, in (Zhu et al., 1997) is shown that, at least asymptotically, the optimal model belongs to M = Span {'Pi, i = 1, ... , m}. This motivates the choice of this space even when dealing with a finite amount of data. Now we introduce the finite-dimensional approximator which we call Projected Bayes Regression. lO(n3) arises from the inversion of a n x n matrix. G. Ferrari- Trecate, C. K. I. Williams and M. Opper 220 Definition 1 The PBR approximator is b(x) A = (A -1 + ,8iJ>' iJ? , (A)ij=Ai6ij and k(x)= [~I{X) l' = k' (x)w, where w=,8A- 1iJ>'f, ,8=1/(12, iJ>= rpm (x) The name PBR comes from the fact that b(x) is the GP predictor when using the mis-specified prior (2) i=1 whose auto covariance function is the projection of C(6,6) on M. From the computational point of view, is interesting to note that the calculation of PBR scales with the data as O(m 2 n), assuming that n ? m (this is the cost of computing the matrix product A-I iJ>'). Throughout the paper the following measures of performance will be extensively used. Definition 2 Let s{x) be a predictor that uses only information from Dn. Then its x-error and generalization error are respectively defined as Es(n,x)=Et.,x.,l [(t* - s(x*))2] , EHn)=Ex [Es{n,x)]. An estimator SO(x) belonging to a class 11. is said x-optimal or simply optimal if, respectively, Eso(n,x) ~ Es{n,x) or E;o(n) ~ E~(n), for all the s(x) E 11. and the data sets x. Note that x-optimality means optimality for each fixed vector x of data points. Obviously, if SO(x) is x-optimal it is also simply optimal. These definitions are motivated by the fact that for Gaussian process priors over functions and a predictor s that depends linearly on 1, the computation of Es(n, x) can be carried out with finite-dimensional matrix calculations (see Lemma 4 below), although obtaining Ei{n) is more difficult, as the average over x is usually analytically intractable. 3 Optimal finite-dimensional models We start considering two classes of linear approximators, namely 11.1 {g(x) = k' (x)LIJL E jRmxn} and 11.2= {h(x) = k' (x)FiJ>'IJF E jRmxm}, where the matrices Land F are possibly dependent on the Xi samples. We point out that 11.2 C 11.1 and that the PBR predictor b(x) E 11. 2. Our goal is the characterization of the optimal predictors in 11.1 and 11. 2. Before stating the main result, two preliminary lemmas are given. The first one is proved in (Pilz, 1991) while the second follows from a straightforward calculation. = Lemma 3 Let A E jRnxn, BE jRnxr, A> O. Then it holds that inf ZERrxn Tr [(ZAZ' - ZB - B' Z')] = Tr [-B' A-I B] and the minimum is achieved for the matrix Z* = B' A-I . Lemma 4 Let g(x) E 11. 1 ? Then it holds that +00 Eg(n,x) = LAi + (12 + q{L), q(L)=Tr [LHL' - 2LiJ>A]. i=1 Finite-Dimensional Approximation o/Gaussian Processes Proof. [C(x*, xd Et< ,t In view of the C(x*, x n )]' ,it holds [(t* - k' (x*)U)2] (12 x-error 221 definition, setting r( x*) + C(X*, X*) + k' (x*)LH L' k(x*) -2k' (x*)Lr(x*) (12 + C(x*, x*) (3) +Tr [LHL'k(x*)k' (x*) - 2Lr(x*)k' (x*)] . Note that Ex< [k(x*)k' (x*)] = fm, Ex' [r(x*)k' (x*)] = ?I> A , and, from the MercerHilbert expansion (1), Ex' [C(x*, x*)] = I:~~ Ai? Then, taking the mean of (3) w.r.t. x*, the result follows.D Theorem 5 The predictors gO(x) E 11.1 given by L = ?0 = A?I>' H- 1 and hO(x) E 11.2 given by F = FO = A?I>' ?1>(<<1>' H?I?-I, Vn 2: m, are x-optimal. Moreover +00 Ego(n,x) = LAi+(12-Tr[A?I>'H-1?1>A] (4) i=1 +00 L Ai + (12 - Tr [A?I>' ?1>(<<1>' H?I?-I?I>' ?I>A] i=1 Proof. We start considering the gO(x) case. In view of Lemma 4 we need only to minimize q(L) w.r.t. to the matrix L. By applying Lemma 3 with B = ?I>A, A = H > 0, Z = L, one obtains argmlnq(L)=Lo = A?I>'H- 1 mlnq(L) = -Tr [A?I>'H- 1?1>A] (5) so proving the first result . For the second case, we apply Lemma 4 with L = F?I>' and then perform the minimization of q(F?I>'), w.r.t. the matrix F. This can be done as before noting that ?1>' H-I?I> > 0 only when n 2: m. 0 Note that the only difference between gO(x) and the GP predictor derives from the approximation of the fUnctions C(x, Xk) with I::l Ai'Pi(X)'Pi(Xk) . Moreover the complexity of gO (x) is O(n 3) the same of j(x). On the other hand hO(x) scales as O(n 2m), so having a computational cost intermediate between the GP predictor and PBR. Intuitively, the PBR method is inferior to hO as it does not take into account the x locations in setting up its prior. We can also show that the PBR predictor b(x) and hO(x) are asymptotically equivalent. l,From (4) is clear that the explicit evaluations of E%o(n) and Eho(n) are in general very hard problems, because the mean w.r.t. the Xi samples that enters in the ?I> and H matrices. In the remainder of this section we will derive an upper bound on Eho(n). Consider the class of approximators 11.3= {u(x) = k' (x) D ?1>' ~D E ffi.mxm , (D)ij = di 6ij }. Because of the inclusions 11.3 C 11.2 C 11. 1 , if UO(x) is the x-optimal predictor in 11. 3, then Ego(n) ::; Eho(n) ::; E;o(n). Due the diagonal structure of the matrix D, an upper bound to E~o (n) may be explicitly computed, as stated in the next Theorem. Theorem 6 The approximator UO(x) E 11.3 given by (<<I>' ?I> A) . (D)ij = (DO)ij = (<<I>' H?I?:: 6ij , (6) G. Ferrari-Trecate, C. K. I. Williams and M. Opper 222 is x-optimal. Moreover an upper-bound on its generalization error is given by += E;o < L Ai + m i=l L qkAk, k=l (n -l)Ak Ck n (J2 - + Ak qk = Ck JC(x,x)cp~(x)p(x)dx (7) +(J2. Proof. In order to find the x-optimal approximator in 11. 3 , we start applying the Lemma 4 with L = Dq,'. Then we need to minimize (8) w.r.t. di so obtaining (6). To bound E;o(n), we first compute the generalization error of a generic approximation u(x) that is verifying that EZ = Ex [q(Dq,')] + L~~ Ai + (J2. After we obtain from (8), assuming the d i constant, E~ += =L i=l Ai + (J2 +n m m i=l i=l L d;Ci - 2n L diAi. Minimizing EZ w.r.t. d i , and recalling that UO(x) is also simply optimal the formula (7) follows.O When C(6, 6) is stationary, the expression of the Ci coefficient becomes simply Ci = (n - l)Ai + L~~ Ai + (J2 . Remark : A naive approach to estimating the coefficients in the estimator L~lWi?i(X) would be to set Wi = n- 1 (q,'t)i as an approximation to the integral Wi = f ?i(x)f(x)p(x)dx. The effect of the matrix D is to "shrink" the wi's of the higher-frequency eigenfunctions. If there was no shrinkage it would be necessary to limit m to stop the poorly-determined Wi'S from dominating, but equation 7 shows that in fact the upper bound is improved as m increases. (In fact equation 7 can be used as an upper bound on the GP prediction error; it is tightest when m ~ 00.) This is consistent with the idea that increasing m under a Bayesian scheme should lead to improved predictions. In practice one would keep m < n, otherwise the approximate algorithm would be computationally more expensive than the O(n3) GP predictor. 4 Choosing m For large n, we can show that (9) where b(x) is the PBR approximator of Definition 1. (This arises because the matrix q,/q, becomes diagonal in the limit n ~ 00 due to the orthogonality of the eigenfunctions.) In equation 9, the factor (Ail + ,Bn)-l indicates by how much the prior variance of the ith eigenfunction ?i has been reduced by the observation of the n datapoints. (Note that this expression is exactly the same as the posterior variance of the mean 223 Finite-Dimensional Approximation o/Gaussian Processes --, , I 00/ tl " 06 .'. u.I o os - , ,, ,, , \"2, : o~ ? --------,,~,------~,,~,------~,,? log, (a) (b) Figure 1: (a) E~o(n) and detaching points for various model orders. Dashed: m dash-dot: m = 5, dotted: m = 8, solid: Ejen). (b) Eg(n) - E~ o (n) plotted against n. = 3, of a Gaussian with prior N(O , Ai) given n observations corrupted by Gaussian noise of variance {3-1 .) For an eigenfunction with Ai ? 0- 2 In, the posterior is considerably tighter than the prior, but when Ai ? 0- 2 In, the prior and posterior have almost the same width , which suggests that there is little point in including these eigenfunctions in the finite-dimensional model. By omitting all but the first m eigenfunctions we add a term L~m+1 Ai to the expected generalization error. This means that for a finite-dimensional model using the first m eigenfunctions, we expect that Eg(n) ~ Ej(n) up to a training set size n determined by n = 1/({3Am). We call n the detatching point for the m-dimensional approximator. Conversely, in practical regression problems the data set size n is known. Then, from the knowledge of the auto covariance eigenvalues, is possible to determine, via the detatching points formula, the order m of the approximation that should be used in order to guarantee Eho (n) ~ Ej(n). 5 Experimental results We have conducted experiments using the prior covariance function C(6,6) = (1 + h)e- h where h = 16 - 61/p? with p = 0.1. This covariance function corresponds to a Gaussian process which is once mean-squared differentiable, It lies in the family of stationary covariance functions C(h) = hV Kv(h) (where Kv(-) is a modified Bessel function) , with v = 3/2. The eigenvalues and eigenfunctions of this covariance kernel for the density p(x) '" U(O, 1) have been calculated in Vivarelli (1998). In our first experiment (using 0- 2 = 1) the learning curves of b(x), hO(x) and i(x) were obtained; the average over the choice of training data sets was estimated by using 100 different x samples. It was noticed that Eg (n) and Eho(n) practically coincide, so only the latter curve is drawn in the pictures. In Figure l(a) we have plotted the learning curves for GP regression and the approximation hO(x) for various model orders. The corresponding detaching points are also plotted, showing their effectiveness in determining the size of data sets for which E~ o (n) ~ Ej(n). The minimum possible error attainable is (J2 = 1.0 For finitedimensional models this is increased by L~m+l Ai; these "plateaux" can be clearly seen on the right hand side of Figure l(a). 224 G. Ferrari-Trecate, C. K. J. Williams and M. Opper Our second experiment demonstrates the differences in performance for the hO(x) and b(x) estimators, using (72 = 0.1. In Figure 1(b) we have plotted the average difference Eg(n) - Eho(n). This was obtained by averaging Eb(n,x) - Eho(n,x) (computed with the same x, i.e. a paired comparison) over 100 choices of x, for each n. Notice that hO is superior to the PBR estimator for small n (as expected), but that they are asymptotically equivalent. 6 Discussion In this paper we have shown that a finite-dimensional predictor hO can be constructed which has lower generalization error than the PBR predictor. Its computational complexity is O(n 2 m), lying between the O(n 3 ) complexity of the GP predictor and O(m 2 n) complexity of PBR. We have also shown how to calculate m, the number of basis functions required, according to the data set size. We have used finite-dimensional models to approximate GP regression. An interesting alternative is found in the work of Gibbs and MacKay (1997), where approximate matrix inversion methods that have O(n 2 ) scaling have been investigated. It would be interesting to compare the relative merits of these two methods. Acknowledgements We thank Francesco Vivarelli for his help in providing the learning curves for Ej(n) and the eigenfunctions/values in section 5. References [1) De Nicolao, G., and Ferrari Trecate, G. (1998). Identification of NARX models using regularization networks: a consistency result .. IEEE Int. Joint Conf. on Neural Networks, Anchorage, US, pp. 2407-2412. [2) Gibbs, M. and MacKay, D. J. C.'(1997). Efficient Implementation of Gaussian Processes. Cavendish Laboratory, Cambridge, UK. Draft manuscript, available from http://wol.ra.phy.cam.ac.uk/mackay/homepage.html. [3) Opper, M. (1997). Regression with Gaussian processes: Average case performance. In I. K. Kwok-Yee, M. Wong and D.-Y. Yeung (eds), Theoretical Aspects of Neural Computation: A Multidisciplinary Perspective. Springer-Verlag. [4) Pilz, J. (1991). Bayesian estimation and experimental design in linear regression models. Wiley & Sons. [5) Ripley, B. D. (1996). Pattern recognition and neural networks. CUP. [6) Wahba, G. (1990). Spline models for observational data. Society for Industrial and Applied Mathematics. CBMS-NSF Regional Conf. series in applied mathematics. [7) Whittle, P. (1963). Prediction and regUlation by linear least-square methods. English Universities Press. [8) Williams C. K. I. (1998). Prediction with Gaussian processes: from linear regression to linear prediction and beyond. In Jordan, M.I. editor, Learning and inference in graphical models. Kluwer Academic Press. [9] Vivarelli, F. (1998).Studies on generalization in Gaussian processes and Bayesian Neural Networks. Forthcoming PhD thesis, Aston University, Birmingham, UK. [10] Zhu, H., and Rohwer, R. (1996). Bayesian regression filters and the issue of priors. Neural Computing and Applications, 4:130-142. [11) Zhu, H., Williams, C. K. I. Rohwer, R. and Morciniec, M. (1997). Gaussian regression and optimal finite dimensional linear models. Tech. Rep. NCRG/97/011. Aston University, Birmingham, UK. 

1497 |@word inversion:2 bn:1 covariance:7 attainable:1 tr:7 solid:1 phy:1 series:1 outperforms:1 dx:2 numerical:1 stationary:2 intelligence:1 xk:2 ith:1 lr:2 manfred:1 draft:1 characterization:1 location:1 dn:4 constructed:1 anchorage:1 introduce:1 ra:1 expected:2 growing:1 decreasing:1 pbr:14 little:1 considering:2 increasing:1 becomes:2 estimating:2 moreover:5 homepage:1 ail:1 guarantee:1 ti:1 xd:2 exactly:1 demonstrates:1 uk:7 uo:3 superiority:1 before:2 engineering:1 morciniec:1 limit:2 ak:2 eb:1 suggests:2 conversely:1 pui:1 practical:1 practice:2 lf:1 empirical:1 projection:1 applying:2 yee:1 wong:1 equivalent:2 backed:1 williams:6 straightforward:1 go:4 independently:1 estimator:4 datapoints:1 his:1 proving:1 ferrari:7 cavendish:1 suppose:1 us:1 ego:2 expensive:1 recognition:1 enters:1 verifying:1 hv:1 calculate:2 complexity:7 cam:1 division:1 basis:3 joint:1 various:2 artificial:1 choosing:1 whose:1 dominating:1 otherwise:1 gp:14 noisy:1 obviously:1 eigenvalue:3 differentiable:1 product:1 remainder:1 j2:6 poorly:1 kv:2 help:1 derive:3 ac:3 stating:1 ij:12 op:1 come:1 fij:1 filter:1 wol:1 observational:1 generalization:7 preliminary:1 tighter:1 dipartimento:1 hold:3 practically:1 lying:1 estimation:1 birmingham:3 minimization:1 clearly:1 gaussian:20 modified:1 ck:2 ej:4 shrinkage:1 indicates:1 tech:1 industrial:1 am:1 inference:1 dependent:1 going:1 interested:1 issue:1 html:1 mackay:3 once:1 having:1 spline:1 cheaper:1 recalling:1 interest:1 evaluation:1 integral:1 necessary:1 lh:1 autocovariance:1 plotted:4 theoretical:2 minimal:2 increased:1 cost:2 predictor:19 conducted:1 corrupted:1 accomplish:1 considerably:1 density:1 squared:1 thesis:1 possibly:1 conf:2 account:1 de:2 whittle:2 coefficient:2 int:1 jc:1 explicitly:1 depends:2 performed:1 view:3 start:3 bayes:3 minimize:2 square:1 variance:4 qk:1 bayesian:5 identification:2 plateau:1 fo:1 suffers:1 ed:2 definition:5 rohwer:2 against:1 frequency:1 pp:1 naturally:1 proof:3 di:4 mi:1 stop:1 proved:1 knowledge:1 hilbert:1 back:1 cbms:1 manuscript:1 higher:1 improved:2 zaz:1 done:1 shrink:1 hand:2 christopher:1 ei:4 o:1 multidisciplinary:1 name:1 effect:1 omitting:1 analytically:1 regularization:1 laboratory:1 eg:5 width:1 inferior:1 hill:1 cp:1 superior:1 ji:1 b4:1 ncrg:1 kluwer:1 cambridge:1 gibbs:2 ai:16 cup:1 consistency:1 mathematics:2 inclusion:1 dot:1 add:1 posterior:4 perspective:1 italy:1 belongs:1 inf:1 verlag:1 rep:1 approximators:2 seen:1 minimum:2 dai:1 determine:1 bessel:1 dashed:1 jrd:1 academic:1 calculation:5 lai:2 paired:1 prediction:8 regression:15 controller:1 mxm:1 yeung:1 kernel:1 achieved:1 regional:1 eigenfunctions:8 effectiveness:1 jordan:1 call:3 noting:1 intermediate:1 xj:1 forthcoming:1 fm:1 wahba:1 idea:1 trecate:6 motivated:2 expression:2 ird:2 remark:1 clear:1 amount:1 extensively:1 informatica:1 reduced:2 http:1 nsf:1 notice:1 dotted:1 delta:1 estimated:1 group:1 drawn:2 pj:1 asymptotically:5 family:1 throughout:1 almost:1 electronic:1 forrest:1 vn:1 scaling:2 rpm:1 bound:7 giancarlo:1 dash:1 detaching:2 kronecker:1 orthogonality:1 n3:3 aspect:1 span:1 optimality:2 department:1 according:1 belonging:1 jr:1 son:1 wi:4 intuitively:1 computationally:2 equation:3 mutually:1 describing:1 ffi:1 vivarelli:3 know:1 merit:1 available:1 tightest:1 apply:1 ckiw:1 kwok:1 generic:1 alternative:1 ho:11 graphical:1 eho:7 narx:1 calculating:1 approximating:2 universita:1 society:1 noticed:1 eh1:1 question:1 diagonal:2 said:1 thank:1 d6:1 assuming:2 providing:1 minimizing:1 ql:1 difficult:1 regulation:1 statement:1 stated:1 design:2 implementation:1 motivates:1 unknown:1 perform:1 upper:6 observation:3 francesco:1 finite:19 namely:1 required:1 specified:1 eigenfunction:2 beyond:1 dynamical:1 below:1 usually:1 pattern:1 regime:1 including:1 suitable:1 oij:2 zhu:4 scheme:1 aston:4 picture:1 carried:1 auto:2 naive:1 lij:1 prior:11 acknowledgement:1 determining:1 asymptotic:1 relative:1 expect:1 interesting:3 approximator:6 consistent:1 mercer:1 dq:2 editor:1 pi:8 land:1 lo:2 last:1 english:1 side:1 taking:1 edinburgh:2 curve:4 opper:6 dimension:1 xn:2 calculated:1 finitedimensional:1 projected:3 coincide:1 approximate:4 obtains:1 keep:1 dealing:1 assumed:1 xi:6 jrnxn:1 ripley:1 decade:1 obtaining:2 expansion:2 investigated:1 main:2 linearly:1 noise:1 tl:2 wiley:1 explicit:2 xl:1 lie:1 formula:3 theorem:3 showing:1 derives:1 intractable:1 ci:3 phd:1 easier:1 simply:4 nicolao:2 ez:2 ordered:1 springer:1 corresponds:1 ijf:1 lwi:1 goal:1 lhl:2 hard:1 determined:2 averaging:1 lemma:8 zb:1 e:4 experimental:2 latter:1 arises:2 ex:5

Making Templates Rotationally Invariant: An Application to Rotated Digit Recognition Shurneet Baluja baluja@cs.cmu.edu Justsystem Pittsburgh Research Center & School of Computer Science, Carnegie Mellon University Abstract This paper describes a simple and efficient method to make template-based object classification invariant to in-plane rotations. The task is divided into two parts: orientation discrimination and classification. The key idea is to perform the orientation discrimination before the classification. This can be accomplished by hypothesizing, in turn, that the input image belongs to each class of interest. The image can then be rotated to maximize its similarity to the training images in each class (these contain the prototype object in an upright orientation). This process yields a set of images, at least one of which will have the object in an upright position. The resulting images can then be classified by models which have been trained with only upright examples. This approach has been successfully applied to two real-world vision-based tasks: rotated handwritten digit recognition and rotated face detection in cluttered scenes. 1 Introduction Rotated text is commonly used in a variety of situations, ranging from advertisements, logos, official post-office stamps, and headlines in magazines, to name a few. For examples, see Figure 1. We would like to be able to recognize these digits or characters, regardless of their rotation. Figure 1: Common examples of images which contain text that is not axis aligned include logos, post-office stamps, magazine headlines and consumer advertisements. 848 S. Baluja The focus of this paper is on the recognition of rotated digits. The simplest method for creating a system which can recognize digits rotated within the image-plane is to employ existing systems which are designed only for upright digit recognition [Le Cun et aI., 1990][Le Cun et a!., 1995a][Le Cun et ai., 1995b][Lee, 1991][Guyon et a!., 1989]. By repeatedly rotating the input image by small increments and applying the recognition system at each rotation, the digit will eventually be recognized. As will be discussed in this paper, besides being extremely computationally expensive, this approach is also errorprone. Because the classification of each digit must occur in many orientations, the likelihood of an incorrect match is high. The procedure presented in this paper to make templates rotationally invariant is significantly faster and more accurate than the one described above. Detailed descriptions of the procedure are given in Section 2. Section 3 demonstrates the applicability of this approach to a real-world vision-based task, rotated handwritten digit recognition. Section 4 closes the paper with conclusions and suggestions for future research. It also briefly describes the second application to which this method has been successfully applied, face detection in cluttered scenes. 2 Making Templates Rotationally Invariant The process to make templates rotationally invariant is easiest to describe in the context of a binary classification problem; the extension to multiple classes is discussed later in this section. Imagine a simplified version of the digit recognition task: we want a detector for a single digit. Suppose we wish to tell whether the input contains the digit '3' or not. The challenge is that the '3' can be rotated within the image plane by an arbitrary amount. Recognizing rotated objects is a two step process. In the first step, a "De-Rotation" network is applied to the input image. This network analyzes the input before it is given to a "Detection" network. If the input contains a '3', the De-Rotation network returns the digit's angle of rotation. The window can then be rotated by the negative of that angle to make the '3' upright. Note that the De-Rotation network does not require a '3' as input. If a non- ' 3' image is encountered, the De-Rotation network will return an unspecified rotation. However, a rotation of a non- '3' will yield another (perhaps different) image of a non-'3'. When the resulting image is given to the Detection network it will not detect a '3'. On the other hand, a rotated '3', which may not have been detected by the Detection network alone, will be rotated to an upright position by the De-Rotation network, and will subsequently be detected as a '3' by the Detection network. The Detection network is trained to output a positive value only if the input contains an upright '3', and a negative value otherwise (even if it contains a rotated '3 '). It should be noted that the methods described here do not require neural networks. As shown in [Le Cun et al., 1995a, Le Cun et ai., 1995b] a number of other classifiers can be used. The De-Rotation and Detection networks are used sequentially. First, the input image is processed by the De-Rotation network which returns an angle of rotation, assuming the image contains a '3'. A simple geometric transformation of the image is performed to undo this rotation. If the original image contained a '3', it would now be upright. The resulting image is then passed to the Detection network. If the original image contained a '3', it can now be successfully detected. This idea can easily be extended to multiple-class classification problems: a De-Rotation network is trained for each object class to be recognized. For the digit recognition problem, 10 De-Rotation networks are trained, one for each of the digits 0.. 9. To classify the digits once they are upright, a single classification network is used with 10 outputs (instead of the detection networks trained on individual digits - alternative approaches will be described later in this paper). The classification network is used in the standard manner; the output with the maximum value is taken as the classification. To classify a new image, the following procedure is used: 849 Making Templates Rotationally Invariant For each digitD (0 $; D $; 9): 1. Pass image through De-Rotation-network-D. This returns the rotation angle. 2. Rotate the image by (-1.0 * returned rotation angle). 3. Pass the de-rotated image to the classification network. 4. If the classification network's maximum output is output D, the activation of output D is recorded. Otherwise digit D is eliminated as a candidate. In most cases, this will eliminate all but one of the candidates. However, in some cases more than one candidate will remain. In these cases, the digit with the maximum recorded activation (from Step 4) is returned. In the unlikely event that no candidates remain, either the system can reject the sample as one it cannot classify, or it can return the maximum value which would have been recorded in Step 4 if none of the examples were rejected. 2.1 Network Specifics To train the De-Rotation networks, images of rotated digits were input, with the rotation angle as the target output. Examples of rotated digits are shown in Figure 2. Each image is 28x28 pixels. The upright data sets are from the MNIST database [Le Cun et at. , 1995a]. .?- ?~~GJmmm? mm. : ::&lIi?? ?\:? ? IJ-~~---?.-: . ?. :~.......... ?. . .?IitIIUI:.... iB ? ? ?:R llil lll :? ? 11? ? ?? ? :? ? n :tM.",.m:IWMI:II. ~R.J aLIla U ~.:II.r. ~ . I!II: ~B.::mll~II:l\Wa :...~ ...... _........~u. . . . . . . ;a . . . :. . .;aa .......... ? ? ? ?l ? ?:? ?! ? ?!:? ? ill.?.11 Figure 2: 8 examples of each of the 10 digits to be recognized. The first example in each group of eight is shown with no rotation; it is as it appears in the MNIST data set. The second through eighth examples show the same digit rotated in-plane by random amounts. In the classification network, each output represents a distinct class; therefore, the standard l-of-N output representation was used with 10 outputs. To represent a continuous variable (the angle of rotation) in the outputs of the De-Rotation network, we used a Gaussian output encoding [Pomerleau, 1992] with 90 output units. With the Gaussian encoding, instead of only training the network to activate a single output (as is done in l-of-N encoding), outputs close to the desired output are also activated in proportion to their distance from the desired output. This representation avoids the imposed discontinuities of the strict l-of-N encoding for images which are similar, but have only slight differences in rotations. Further, this representation allows finer granularity with the same number of output units than would be possible if a l-of-N encoding was used [Pomerleau, 1992]. The network architecture for both the classification and the De-Rotation networks consists of a single hidden layer. However, unlike a standard fully-connected network, each hidden unit was only connected to a small patch of the 28x28 input. The De-Rotation networks used groups of hidden units in which each hidden unit was connected to only 2x2, 3x3, 4x4 & 5x5 patches of the inputs (in each of these groups, the patches were spaced 2x2 pixels apart; therefore, the last three groups had overlapping patches). This is similar to the networks used in [Baluja, 1997][Rowley et. at, 1998a, 1998b] for face detection. Unlike the convolution networks used by [Le Cun et aI., 1990], the weights into the hidden units were not shared. 1 Note that many different local receptive field configurations were tried; almost all had equivalent performance. S. Ba/uja 850 3 Rotated Handwritten Digit Recognition To create a complete rotationally invariant digit recognition system, the first step is to segment each digit from the background. The second is to recognize the digit which has been segmented. Many systems have been proposed for segmenting written digits from background clutter [Jain & Yu, 1997][Sato et ai., 1998][Satoh & Kanade, 1997]. In this paper, we concentrate on the recognition portion of the task. Given a segmented image of a potentially rotated digit, how do we recognize the digit? The first experiment conducted was to establish the base-line performance. We used only the standard, upright training set to train a classification network (this training set consists of 60,000 digits). This network was then tested on the testing set (the testing set contains 10,000 digits) . In addition to measuring the performance on the upright testing set, the entire testing set was also rotated. As expected, performance rapidly degrades with rotation. A graph of the performance with respect to the rotation angle is shown in Figure 3. Perfonaaneeo' Network trained wtth t Jpript Dlpts and Tcwhd on Rotated Dlgtu i ] 1 J Figure 3: Performance of the classification network trained only with upright images when tested on rotated images. As the angle of rotation increases, performance degrades. Note the spike around 180 degrees, this is because some digits look the same even when they are upside-down. The peak performance is approximately 97.5% (when the digits are upright). It is interesting to note that around 1800 rotation, performance slightly rises. This is because some of the digits are symmetric across the center horizontal axis - for example the digits '8', '1', '2' & '5' can be recognized upside-down. Therefore, at these orientations, the upright detector works well for these digits. As mentioned earlier, the simplest method to make an upright digit classifier handle rotations is to repeatedly rotate the input image and classify it at each rotation. Thefirst drawback to this approach is the severe computational expense. The second drawback is that because the digit is examined at many rotations, it may appear similar to numerous digits in different orientations. One approach to avoid the latter problem is to classify the digit as the one that is voted for most often when examined over all rotations. To ensure that this process is not biased by the size of the increments by which the image is rotated, various angle increments are tried. As shown in the first row of Table I, this method yields low Table I: Exhaustive Search over all possible rotations Number of Angle IncreQ1ents Tried Exhaustive Search Method 360 100 50 (1 degree/increment) (3.6 degree/increment) (7.2 degreeslincrement Most frequent vote (over all rotations) 59.5% 66.0% 65 .0% Most frequent vote - counted onl y when votes are positi ve (over all rotations) 75.2% 74.5% 74.0% 1. Note that in the empirical comparisons presented in [Le Cun et ai., 1995aJ, convolution networks performed extremely well in the upright digit recognition task. However, due to limited computation resources, we were unable to train these networks, as each takes 14-20 days to train. The network used here was trained in 3 hours, and had approximately a 2.6% misclassification rate on the upright test set. The best networks reported in [Le Cun et ai, 1995aJ have less than 1% error. It should be noted that the De-Rotation networks trained in this study can easily be used in conjunction with any classification procedure, including convolutional networks. Making Templates Rotationally Invariant 851 classification accuracies. One reason for this is that a vote is counted even when the classification network predicts all outputs to be less than 0 (the network is trained to predict +1 when a digit is recognized, and -1 when it is not). The above experiment was repeated with the following modification: a vote was only counted when the maximum output of the classification network was above O. The result is shown in the second row of Table I. The classification rate improved by more than 10%. Given these base-line performance measures2 , we now have quantitative measurements with which to compare the effectiveness of the approach described in this paper. The performance of the procedure used here, with 10 "De-Rotation" networks and a single classification network, is shown in Figure 4. Note that unlike the graph shown in Figure 3, there is very little effect on the classification performance with the rotation angle. Figure 4: Performance ofthe combined DeRotation network and classification network system proposed in this paper. Note that the performance is largely unaffected by the rotation. The average performance, over all rotations, is 85.0%. 0.?? ~~----------------~~ 0,1. -,*10.00 -100.00 0.00 100.110 Ito.OO ~011""'1oft To provide some intuition of how the De-Rotation networks perform, Figure 5 shows examples of how each De-Rotation networks transform each digit. Each De-Rotation network suggests a rotation which makes the digit look as much like the one with which the network was trained. For example, De-Rotation-Network-5 will suggest a rotation that will make the input digit look as much like the digit '5' as possible; for example, see DeRotation-Network-5's effect on the digit '4'. o 2 Original Digit 345 6 7 8 9 Digit rotated by De-Rotation-Network-O Digit rotated by De-Rotation-Network-l Digit rotated by De-Rotation-Network-2 Digit rotated by De-Rotation-Network-3 Digit rotated by De-Rotation-Network-4 Digit rotated by De-Rotation-Network-5 Digit rotated by De-Rotation-Network-6 Digit rotated by De-Rotation-Network-7 Digit rotated by De-Rotation-Network-8 Digit rotated by De-Rotation-Network-9 Figure 5: Digits which have been rotated by the angles specified by each of the De-rotation networks. As expected (if the method is working), the digits on the diagonal (upper left to bottom right) appear upright. 2. Another approach is to train a single network to handle both rotation and classification by using rotated digits as inputs, and the digit's classification as the target output. Experiments with the approach yielded results far below the techniques presented here. 852 S. Baluja As shown in Figure 4, the average classification accuracy is approximately 85.0%. The performance is not as good as with the upright case alone, which had a peak performance of approximately 97.5% (Figure 3). The high level of performance achieved in the upright case is unlikely for rotated digits: if all rotations are admissible, some characters are ambiguous. The problem is that when working correctly, De-Rotation-Network-D will suggest an angle of rotation that will make any input image look as much like the digit D as possible through rotation. In most cases when the input image is not the digit D, the rotation will not cause the image to look like D. However, in some cases, such as those shown in Figure 6(right), the digit will be transformed enough to cause a classification error. Some of these errors will most likely never be correctable (for example, '6' and '9' in some instances); however, there is hope for correcting some of the others. Figure 6 presents the complete confusion matrix. As can be seen in the examples in Figure 6(right), the digit '4' can be rotated to appear similar to a'S'. Nonetheless, there often remain distinctive features that allow real '5's to be differentiated from the rotated '4's. However, the classification network is unable to make these distinctions because it was not trained with the appropriate examples. Remember, that since the classification network was only trained with the upright digit training set, rotated '4's are never encountered during training. This reflects a fundamental discrepancy in the training/testing procedure. The distributions of images which were used to train the classification network is different than the distributions on which the network is tested. To address this problem, the classification mechanism is modified. Rather than using the single 1-oj-1O neural network classifier used previously, 10 individual Detection networks are used. Each detection network has a single binary output that signifies whether the input contains the digit (upright) with which the network was trained. Each De-Rotation network is paired with the respective Detection network. The crucial point is that rather than training the Detection-Network-D with the original upright images in the training set, each image (whether it is a positive or negative example) is first passed through DeRotation-Network-D. Although this makes training Detection-Network-D difficult since all the digits are rotated to appear as much like upright-D's as possible by De-RotationNetwork-D, the distribution of training images matches the testing distribution more closely. In use, when a new image is presented, it is passed through the lO network pairs. Candidate digits are eliminated if the binary output from the detection network does not signal a detection. Preliminary results with this new approach are extremely promising; the classification accuracy increases dramatically - to 93% when averaged over all rotations. This is a more than a 50% reduction in error over the previously described approach. Predicted Digit o a 2 94 -- -- 90 1I -- 5 88 -- 3 88 -- -- 4 is 3 4 89 3 3 ~ 5 2 87 4 2 88 3 6 74 6 3 -- 5 3 7 25 2 B. !:a.:al liBII! 'JaBEI. c?III?BEt: "11.'111 '-? .?:?. 3 ] 10 :.! ,?.? ?!.?.?111 A. ::? ? -- .~ 7 Original Image mage Rotated to Look Like Mistake Digit r1mage Rotated to Look Like Correct Digit 2345678 :.1 :11.11111'.??~?? D. [? ? 3 -- 89 -- -- -- 64 l [? ??.la.II:.! E?l? . ? ? F. ::.:11:111 r.~~.:.:?.l i?? . ? 111:11'1:: ;??11? G. [? ? ?? Figure 6: Example errors. (LEFI) Confusion Matrix (only entries account for 2% or more entries are filled in for ease of reading). (RlGH1) some of the errors made in classification. 3 examples of each of the errors are shown. Row A: '4' mistaken as '5'. Row B: '5' mistaken as '6', Row C: '7' mistaken as '2'. Row D: '7' mistaken as '6'. Row E: '8' mistaken as '4', Row F: '9' mistaken as '5', Row G: '9' mistaken as '6'. Making Templates Rotationally Invariant 853 4 Conclusions and Future Work This paper has presented results on the difficult problem of rotated digit recognition. First, we presented base-line results with naive approaches such as exhaustively checking all rotations. These approaches are both slow and have large error rates. Second, we presented results with a novel two-stage approach which is both faster and more effective than the naive approaches. Finally, we presented preliminary results with a new approach that more closely models the training and testing distributions. We have recently applied the techniques presented in this paper to the detection of faces in cluttered scenes. In previous studies, we presented methods for finding all upright frontal faces [Rowley et aT., 1998aJ. By using the techniques presented here, we were able to detect all frontal faces, including those which were rotated within the image plane [Baluja, 1997][Rowley et al., 1998bJ. The methods presented in this paper should also be directly applicable to full alphabet rotated character recognition. In this paper, we examined each digit individually. A straight-forward method to eliminate some of the ambiguities between rotationally similar digits is to use contextual information. For example, if surrounding digits are all rotated to the same amount, this provides strong hints about the rotation of nearby digits. Further, in most real-world cases, we might expect digits to be close to upright; therefore, one method of incorporating this information is to penalize matches which rely on large rotation angles. This paper presented a general way to make template-based recognition rotation invariant. In this study, both the rotation estimation procedures and the recognition templates were implemented with neural-networks. Nonetheless, for classification, any technique which implements a form of templates, such as correlation templates, support vector machines, probabilistic networks, K-Nearest Neighbor, or principal component-based methods, could have easily been employed. Acknowledgements The author would like to thank Kaari Aagstad for her reviews of many successive drafts of this paper. References Baluja, S. (1997) "Face Detection with In-Plane Rotation: Early Concepts and Preliminary Results," Justsystem Pittsburgh Research Center Technical Report. JPRC-TR-97-001. Guyon, I, Poujaud, I., Personnaz, L, Dreyfus, G., Denker, J. LeCun, Y. (1989) "Comparing Different Neural Net Architectures for Classifying Handwritten Digits", in IlCNN II 127-132. Jain, A. & Yu, B. (1997) "Automatic Text Location in Images and Video Frames", TR: MSUCPS: TR 97-33. Le Cun, Y., Jackel, D., Bottou, L, Cortes, c., Denker, J. Drucker, J. Guyon, I, Miller, U. Sackinger, E. Simard, P. Vapnik, V. (1995a) "Learning Algorithms for Classification: A Comparison on Handwritten Digit Recognition". Neural Networks: The Statistical Mechanics Perspective, Oh, J., Kwon, C. & Cho, S. (Ed.), pp. 261-276. LeCun, Y., Jackel, L. D., Bottou, L., Brunot, A., Cortes, C., Denker, J. S., Drucker, H., Guyon, I., Muller, U. A., Sackinger, E., Simard, P. and Vapnik, V. (1995b), Comparison of learning algorithms for handwritten digit recognition," ICANN, Fogelman, F. and Gallinari, P., 1995, pp. 53-60. LeCun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W. and Jackel, L. D. (1990), "Handwritten digit recognition with a back-propagation network," Advances in Neural Information Processing Systems 2 (NIPS '89), Touretzky, David (Ed.), Morgan Kaufman. Lee, Y. (1991) "Handwritten Digit Recognition using K-NN, RBF and Backpropagation Neural Networks", Neural Computation, 3, 3. Pomerleau, D.A. (1993) Neural Network Perception for Mobile Robot Guidance, Kluwer Academic Rowley, H., Baluja, S. & Kanade, T. (1998a) "Neural Network-Based Face Detection," IEEE-Transactions on Pattern Analysis and Machine Intelligence (PAMI), Vol. 20, No.1, January, 1998. Rowley, H., Baluja, S. & Kanade, T. (1998b) "Rotation Invariant Neural Network-Based Face Detection," to appear in Proceedings of Computer Vzsion and Pattern Recognition, 1998. Sato, T, Kanade, T., Hughes, E. & Smith, M. (1998) "Video OCR for Digital News Archives" to appear in IEEE International Workshop on Content-Based Access of Image and Vzdeo Databases. Satoh, S. & Kanade, T. (1997) "Name-It: Association of face and name in Video", in Proceedings of IEEE Conference on Computer Vzsion and Pattern Recognition, 1997. 

1498 |@word version:1 briefly:1 proportion:1 tried:3 tr:3 reduction:1 configuration:1 contains:7 existing:1 contextual:1 comparing:1 activation:2 must:1 written:1 designed:1 discrimination:2 alone:2 intelligence:1 plane:6 smith:1 provides:1 draft:1 location:1 successive:1 incorrect:1 consists:2 manner:1 uja:1 expected:2 mechanic:1 little:1 window:1 lll:1 easiest:1 kaufman:1 unspecified:1 finding:1 transformation:1 quantitative:1 remember:1 demonstrates:1 classifier:3 gallinari:1 unit:6 appear:6 segmenting:1 before:2 positive:2 local:1 mistake:1 encoding:5 approximately:4 pami:1 logo:2 might:1 examined:3 suggests:1 ease:1 limited:1 averaged:1 lecun:3 testing:7 hughes:1 implement:1 x3:1 backpropagation:1 digit:86 procedure:7 empirical:1 poujaud:1 significantly:1 reject:1 suggest:2 cannot:1 close:3 context:1 applying:1 equivalent:1 imposed:1 center:3 regardless:1 cluttered:3 correcting:1 oh:1 handle:2 increment:5 imagine:1 suppose:1 target:2 magazine:2 recognition:22 expensive:1 predicts:1 database:2 bottom:1 connected:3 news:1 mentioned:1 intuition:1 rowley:5 exhaustively:1 trained:14 segment:1 distinctive:1 easily:3 various:1 alphabet:1 train:6 surrounding:1 distinct:1 jain:2 describe:1 activate:1 effective:1 detected:3 tell:1 exhaustive:2 otherwise:2 transform:1 net:1 frequent:2 aligned:1 rapidly:1 description:1 rotated:47 object:5 oo:1 nearest:1 ij:1 school:1 strong:1 implemented:1 c:1 predicted:1 concentrate:1 drawback:2 closely:2 correct:1 subsequently:1 require:2 preliminary:3 extension:1 mm:1 around:2 predict:1 bj:1 early:1 estimation:1 applicable:1 jackel:3 headline:2 individually:1 hubbard:1 create:1 successfully:3 reflects:1 hope:1 gaussian:2 modified:1 rather:2 avoid:1 bet:1 mobile:1 office:2 conjunction:1 focus:1 likelihood:1 detect:2 kaari:1 nn:1 eliminate:2 unlikely:2 entire:1 hidden:5 her:1 transformed:1 pixel:2 fogelman:1 classification:35 orientation:6 ill:1 field:1 once:1 never:2 eliminated:2 x4:1 represents:1 yu:2 look:7 hypothesizing:1 future:2 others:1 discrepancy:1 report:1 hint:1 few:1 employ:1 kwon:1 recognize:4 ve:1 individual:2 mll:1 detection:22 interest:1 severe:1 henderson:1 activated:1 accurate:1 respective:1 filled:1 rotating:1 desired:2 guidance:1 instance:1 classify:5 earlier:1 measuring:1 signifies:1 applicability:1 entry:2 recognizing:1 conducted:1 reported:1 combined:1 cho:1 peak:2 fundamental:1 international:1 lee:2 probabilistic:1 ambiguity:1 recorded:3 creating:1 lii:1 simard:2 return:5 account:1 de:35 later:2 performed:2 portion:1 voted:1 accuracy:3 convolutional:1 largely:1 miller:1 yield:3 spaced:1 ofthe:1 handwritten:8 none:1 finer:1 unaffected:1 straight:1 classified:1 detector:2 touretzky:1 ed:2 nonetheless:2 pp:2 back:1 appears:1 day:1 improved:1 done:1 rejected:1 stage:1 correlation:1 hand:1 working:2 horizontal:1 sackinger:2 overlapping:1 propagation:1 aj:3 perhaps:1 name:3 effect:2 contain:2 concept:1 symmetric:1 x5:1 during:1 ambiguous:1 noted:2 complete:2 confusion:2 image:43 ranging:1 dreyfus:1 novel:1 recently:1 common:1 rotation:76 discussed:2 slight:1 association:1 kluwer:1 mellon:1 measurement:1 ai:7 mistaken:7 automatic:1 satoh:2 had:4 robot:1 access:1 similarity:1 base:3 perspective:1 belongs:1 apart:1 binary:3 accomplished:1 muller:1 rotationally:9 analyzes:1 seen:1 morgan:1 employed:1 recognized:5 maximize:1 signal:1 ii:6 multiple:2 upside:2 full:1 segmented:2 technical:1 match:3 academic:1 faster:2 x28:2 divided:1 post:2 paired:1 vision:2 cmu:1 represent:1 achieved:1 penalize:1 background:2 want:1 addition:1 crucial:1 biased:1 unlike:3 archive:1 strict:1 undo:1 effectiveness:1 granularity:1 iii:1 enough:1 variety:1 architecture:2 idea:2 tm:1 prototype:1 drucker:2 whether:3 passed:3 returned:2 cause:2 repeatedly:2 dramatically:1 detailed:1 amount:3 clutter:1 processed:1 simplest:2 correctly:1 correctable:1 carnegie:1 vol:1 group:4 key:1 graph:2 angle:15 almost:1 guyon:4 patch:4 layer:1 encountered:2 yielded:1 sato:2 occur:1 scene:3 x2:2 nearby:1 extremely:3 wtth:1 describes:2 remain:3 slightly:1 character:3 across:1 cun:10 making:5 modification:1 brunot:1 invariant:11 taken:1 computationally:1 resource:1 previously:2 turn:1 eventually:1 mechanism:1 eight:1 denker:4 ocr:1 differentiated:1 appropriate:1 alternative:1 original:5 include:1 ensure:1 establish:1 personnaz:1 spike:1 receptive:1 degrades:2 diagonal:1 distance:1 unable:2 thank:1 reason:1 consumer:1 assuming:1 besides:1 difficult:2 onl:1 potentially:1 expense:1 negative:3 rise:1 ba:1 pomerleau:3 perform:2 upper:1 convolution:2 howard:1 january:1 situation:1 extended:1 frame:1 arbitrary:1 david:1 pair:1 specified:1 distinction:1 boser:1 hour:1 discontinuity:1 nip:1 address:1 able:2 below:1 perception:1 pattern:3 eighth:1 oft:1 challenge:1 reading:1 including:2 oj:1 video:3 event:1 misclassification:1 rely:1 numerous:1 axis:2 naive:2 text:3 review:1 geometric:1 acknowledgement:1 checking:1 fully:1 expect:1 suggestion:1 interesting:1 digital:1 degree:3 classifying:1 row:9 lo:1 last:1 allow:1 neighbor:1 template:12 face:10 world:3 avoids:1 forward:1 commonly:1 made:1 author:1 simplified:1 counted:3 far:1 transaction:1 sequentially:1 pittsburgh:2 continuous:1 search:2 table:3 kanade:5 promising:1 bottou:2 official:1 icann:1 justsystem:2 repeated:1 slow:1 position:2 wish:1 candidate:5 stamp:2 ib:1 advertisement:2 ito:1 admissible:1 down:2 specific:1 cortes:2 incorporating:1 workshop:1 mnist:2 vapnik:2 positi:1 mage:1 likely:1 contained:2 aa:1 rbf:1 shared:1 content:1 baluja:9 upright:27 ilcnn:1 principal:1 pas:2 la:1 vote:5 support:1 latter:1 rotate:2 frontal:2 tested:3

VLSI Implementation of Motion Centroid Localization for Autonomous Navigation Ralph Etienne-Cummings Dept. of ECE, Johns Hopkins University, Baltimore, MD Viktor Gruev Dept. of ECE, Johns Hopkins University, Baltimore, MD Mohammed Abdel Ghani Dept. ofEE, S. Illinois University, Carbondale, IL Abstract A circuit for fast, compact and low-power focal-plane motion centroid localization is presented. This chip, which uses mixed signal CMOS components to implement photodetection, edge detection, ON-set detection and centroid localization, models the retina and superior colliculus. The centroid localization circuit uses time-windowed asynchronously triggered row and column address events and two linear resistive grids to provide the analog coordinates of the motion centroid. This VLSI chip is used to realize fast lightweight autonavigating vehicles. The obstacle avoiding line-following algorithm is discussed. 1 INTRODUCTION Many neuromorphic chips which mimic the analog and parallel characteristics of visual, auditory and cortical neural circuits have been designed [Mead, 1989; Koch, 1995] . Recently researchers have started to combine digital circuits with neuromorphic aVLSI systems [Boahen, 1996]. The persistent doctrine, however, has been that computation should be performed in analog, and only communication should use digital circuits. We have argued that hybrid computational systems are better equipped to handle the high speed processing required for real-world problem solving , while maintaining compatibility with the ubiquitous digital computer [Etienne, 1998]. As a further illustration of this point of view, this paper presents a departure form traditional approaches for focal plane centroid localization by offering a mixed signal solution that is simultaneously high-speed, low power and compact. In addition, the chip is interfaced with an 8-bit microcomputer to implement fast autonomous navigation. Implementation of centroid localization has been either completely analog or completely digital. The analog implementations, realized in the early 1990s, used focal plane current mode circuits to find a global continuos time centroid of the pixels' intensities [DeWeerth, 1992]. Due to their sub-threshold operation, these circuits are low power, but slow. On the other hand, the digital solutions do not compute the centroid at the focal R. Etienne-Cummings, V. Grnev and M A. Ghani 686 plane. They use standard CCO cameras, AID converters and OSP/CPU to compute the intensity centroid [Mansfield, 1996]. These software approaches offer multiple centroid localization with complex mathematical processing. However, they suffer from the usual high power consumption and non-scalability of traditional digital visual processing systems. Our approach is novel in many aspects. We benefit from the low power, compactness and parallel organization of focal plane analog circuits and the speed, robustness and standard architecture of asynchronous digital circuits. Furthermore, it uses event triggered analog address read-out, which is ideal for the visual centroid localization problem. Moreover, our chip responds to moving targets only by using the ON-set of each pixel in the centroid computation. Lastly, our chip models the retina and two dimensional saccade motor error maps of superior colliculus on a single chip [Sparks, 1990] . Subsequently, this chip is interfaced with a IlC for autonomous obstacle avoidance during line-following navigation. The line-following task is similar to target tracking using the saccadic system, except that the "eye" is fixed and the "head" (the vehicle) moves to maintain fixation on the target. Control signals provided to the vehicle based on decisions made by the IlC are used for steering and accelerating/braking. Here the computational flexibility and programmability of the IlC allows rapid prototyping of complex and robust algorithms. 2 CENTROID LOCALIZATION The mathematical computation of the centroid of an object on the focal plane uses intensity weighted average of the position of the pixels forming the object [OeWeerth, 1992] . Equation (1) shows this formulation. The implementation of this representation N N LI,x, x= j =1 N L,I, LI,y; and y= ;=1 N (1) L I, ,=1 1=1 can be quite involved since a product between the intensity and position is implied. To eliminate this requirement, the intensity of the pixels can be normalized to a single value within the object. This gives equation (2) since the intensity can be factored out of the summations. Normalization of the intensity using a simple threshold is not advised since Ix, x=~ N Iy, and y=~ N (2) Intensity Image XI X l+l X 1+2 X I +3 x 1+4 Edges from pixels Figure 1: Centroid computation architecture. Figure 2: Centroid computation method. the value of the threshold is dependent on the brightness of the image and number of pixels forming the object may be altered by the thresholding process. To circumvent these problems, we take the view that the centroid of the object is defined in relation to its boundaries. This implies that edge detection (second order spatial derivative of intensity) can be used to highlight the boundaries, and edge labeling (the zero-crossing of the edges) can be used to normalize the magnitude of the edges. Subsequently, the centroid VLSI Implementation ofMotion Centroid Localization for Autonomous Navigation 687 of the zero-crossings is computed. Equation (2) is then realized by projecting the zerocrossing image onto the x- and y-axis and performing two linear centroid determinations. Figure (1) shows this process. The determination of the centroid is computed using a resistance grid to associate the position of a column (row) with a voltage. In figure 2, the positions are given by the voltages Vi . By activating the column (row) switch when a pixel of the edge image appears in that column (row), the position voltage is connected to the output line through the switch impedance, Rs. As more switches are activated, the voltage on the output line approximates equation (2). Clearly, since no buffers are used to isolate the position voltages, as more switches are activated, the position voltages will also change. This does not pose a problem since the switch resistors are design to be larger than the position resistors (the switch currents are small compared to the grid current). Equation (3) gives the error between the ideal centroid and the switch loaded centroid in the worst case when Rs = on. In the equation, N is the number of nodes, M is the number of switches set and Xl and xM are the locations of the first and last set switches, respectively. This error is improved as Rs gets larger, and vanishes as N (M~N) approaches infinity . The terms Xi represent an ascending ordered list of the activated switches; x I may correspond to column five, for example. This circuit is compact since it uses only a simple linear resistive grid and MOS switches. It is low power because the total grid resistance, N x R, can be large. It can be fast when the parasitic capacitors are kept small . It provides an analog position value, but it is triggered by fast digital signals that activate the switches. error = VmOll -Vmin M(N + 1) 1 ~[ X ?...J .=1 ? xCN+l) _ _--'-1-'--_-'--_ N + 1 + XI - Xm (3) 3 MODELING THE RETINA AND SUPERIOR COLLICULUS 3.1 System Overview The centroid computation approach presented in section 2 is used to isolate the location of moving targets on a 20 focal plane array. Consequently, a chip which realizes a neuromorphic visual target acquisition system based on the saccadic generation mechanism of primates can be implemented. The biological saccade generation process is mediated by the superior colliculus, which contains a map of the visual field [Sparks, 1990}. In laboratory experiments, cellular recordings suggest that the superior colliculus provides the spatial location of targets to be foveated. Clearly, a great deal of neural circuitry exists between the superior colliculus and the eye muscle. Horiuchi has built an analog system which replicates most of the neural circuits (including the motor system) which are believed to form the saccadic system [Horiuchi, 1996]. Staying true to the anatomy forced his implementation to be a complex multi-chip system with many control parameters. On the other hand, our approach focuses on realizing a compact single chip solution by only mimicking the behavior of the saccadic system, but not its structure. 3.2 Hardware Implementation Our approach uses a combination of analog and digital circuits to implement the functions of the retina and superior colliculus at the focal plane. We use simple digital control ideas, such as pulse-width modulation and stepper motors, to position the "eye". The retina portion of this chip uses photodiodes, logarithmic compression, edge detection and zero-crossing circuits. These circuits mimic the first three layers of cells in the retina 688 R. Etienne-Cummings, V. Grnev and M. A. Ghani with mixed sub-threshold and strong inversion circuits. The edge detection circuit is realized with an approximation of the Laplacian operator implemented using the difference between a smooth (with a resistive grid) and unsmoothed version of the image [Mead, 1989]. The high gain of the difference circuit creates a binary image of approximate zero-crossings. After this point, the computation is performed using mixed analog/digital circuits. The zero-crossings are fed to ON-set detectors (positive temporal derivatives) which signal the location of moving or flashing targets. These circuits model the behavior of some of the amacrine and ganglion cells of the primate retina [Barlow, 1982]. These first layers of processing constitute all the "direct" mimicry of the biological models. Figure 3 shows the schematic of these early processing layers. The ON-set detectors provide inputs to the model of the superior colliculus circuits. The ON-set detectors allow us to segment moving targets against textured backgrounds. This is an improvement on earlier centroid and saccade chips that used pixel intensity. The essence of the superior colliculus map is to locate the target that is to be foveated. In our case, the target chosen to be foveated will be moving. Here motion is define simply as the change in contrast over time. Motion, in this sense, can be seen as being the earliest measurable attribute of the target which can trigger a saccade without requiring any high level decision making. Subsequently, the coordinates of the motion must be extracted and provided to the motor drivers. X M otion Cenuold ~! ~A ! Edge Detc:ctlon ON-set Detecu on Figure 3: Schematic of the model of the retina. Figure 4: Schematic of the model of the superior collicu Ius. The circuits for locating the target are implemented entirely with mixed signal, nonneuromorphic circuits. The theoretical foundation for our approach is presented in section 2. The ON-set detector is triggered when an edge of the target appears at a pixel. At this time, the pixel broadcasts its location to the edge of the array by activating row and column lines. This row (column) signal sets a latch at the right (top) of the array. The latches asynchronously activate switches and the centroid of the activated positions is provided. The latches remain set until they are cleared by an external control signal. This control signal provides a time-window over which the centroid output is integrated. This has the effect of reducing noise by combining the outputs of pixels which are activated at different instances even if they are triggered by the same motion (an artifact of small fill factor focal plane image processing). Furthermore, the latches can be masked from the pixels' output with a second control signal. This signal is used to de-activate the centroid 689 VLSI Implementation of Motion Centroid Localization for Autonomous Navigation circuit during a saccade (saccadic suppression). A centroid valid signal is also generated by the chip. Figure 4 shows a portion of the schematic of the superior colliculus model. 3.3 Results In contrast to previous work, this chip provides the 2-D coordinates of the centroid of a moving target. Figure 5 shows the oscilloscope trace of the coordinates as a target moves back and forth, in and out of the chip's field of view. The y-coordinate does change while the x-coordinate increases and decreases as the target moves to the left and right, respectively. The chip has been used to track targets in 2-D by making micro-saccades . In this case, the chip chases the target as it attempts to escape from the center. The eye movement is performed by converting the analog coordinates into PWM signals, which are used to drive stepper motors. The system performance is limited by the contrast sensitivity of the edge detection circuit, and the frequency response of the edge (high frequency cut-off) and ON-set (low frequency cut-off) detectors. With the appropriate optics, it can track walking or running persons under indoor or outdoor lighting conditions at close or far distances. Table I gives a summary of the chip characteristics. VarYing x? l'Oordma te Figure 5: Oscilloscope trace of 20 centroid for a moving target. Technology 1.2um ORBIT Chip Size Array Size 4mm 1 12 x 10 Pixel Size Fill Factor llOxllOutn 11% Intensity Min . Contrast 0.lu-1OOmW/cm 2 Response Time Power (chip) 2-10 6 Hz(@1 mW/cml) 10% 5 mW (@l m W/cm~ Vdd =6V) Table I: Chip characteristics. 4 APPLICATION: OBSTACLE AVOIDANCE DURING LINEFOLLOWING AUTONA VIGATION 4.1 System Overview The frenzy of activity towards developing neuromorphic systems over the pass 10 years has been mainly driven by the promise that one day engineers will develop machines that can interact with the environment in a similar way as biological organisms. The prospect of having a robot that can help humans in their daily tasks has been a dream of science fiction for many decades. As can be expected, the key to success is premised on the development of compact systems, with large computational capabilities, at low cost (in terms of hardware and power) . Neuromorphic VLSI systems have closed the gap between dreams and reality, but we are still very far from the android robot. For all these robots, autonomous behavior, in the form of auto-navigation in natural environments, must be one of their primary skills. For miniaturization, neuromorphic vision systems performing most of the pre-processing, can be coupled with small fast computers to realize these compact yet powerful sensor/processor modules. 4.2 Navigation Algorithm The simplest form of data driven auto-navigation is the line-following task. In this task, the robot must maintain a certain relationship with some visual cues that guide its motion. In the case of the line-follower, the visual system provides data regarding the state of the R. Etienne-Cummings, V. Gruev and M A. Ghani 690 line relative to the vehicle, which results in controlling steering and/or speed. If obstacle avoidance is also required, auto-navigation is considerably more difficult. Our system handles line-following and obstacle avoidance by using two neuromorphic visual sensors that provide information to a micro-controller OlC) to steer, accelerate or decelerate the vehicle. The sensors, which uses the centroid location system outlined above, provides information on the position of the line and obstacles to the p,C, which provides PWM signals to the servos for controlling the vehicle. The algorithm implemented in the p,C places the two sensors in competition with each other to force the line into a blind zone between the sensors. Simultaneously, if an object enters the visual field from outside, it is treated as an obstacle and the p,C turns the car away from the object. Obstacle avoidance is given higher priority than line-following to avoid collisions. The p,C also keeps track of the direction of avoidance such that the vehicle can be re-oriented towards the line after the obstacle is pushed out of the field of view. Lastly, for line following, the position, orientation and velocity of drift, determined from the temporal derivative of the centroid, are used to track the line. The control strategy is to keep the line in the blind zone, while slowing down at corners, speeding up on straight aways and avoiding obstacles. The angle which the line or obstacle form with the x-axis also affects the speed. The value of the x-centroid relative to the y-centroid provides rudimentary estimate of the orientation of the line or obstacle to the vehicle. For example, angles less . Follow " AV~8id~nce ... , 0I0s1ade L Zone / ! ~ne i '.)", / : AV~na;ce ! ~\?': ~../ ;i:~~~~???. .\j ~ \;... ?? ? ?~i=~s..s Figure 6: Block diagram of the autonomous line-follower system. Figure 7: A picture of the vehicle. (greater) than +/- 45 degrees tend to have small (large) x-coordinates and large (small) ycoordinates and require deceleration (acceleration). Figure 6 shows the organization of the sensors on the vehicle and control spatial zones. Figure 7 shows the vehicle and samples of the line and obstacles. 4.3 Hardware Implementation The coordinates from the centroid localization circuits are presented to the p,C for analysis. The p,C used is the Microchip PIC16C74. This chip is chosen because of its five NO inputs and three PWM outputs. The analog coordinates are presented directly to the NO inputs. Two of the PWM outputs are connected to the steering and speed control servos. The PIC16C74 runs at 20 MHz and has 35 instructions, 4K by 8-b ROM and 80 by 20-b RAM. The program which runs on the PIC determines the control action to take, based on the signal provided by the neuromorphic visual sensors. The vehicle used is a four-wheel drive radio controlled model car (the radio receiver is disconnected) with Digital Proportional Steering (DPS) . VLSI Implementation ofMotion Centroid Localization for Autonomous Navigation 691 4.4 Results The vehicle was tested on a track composed of black tape on a gray linoleum floor with black and white obstacles. The track formed a closed loop with two sharp turns and some smooth S-curves. The neuromorphic vision chip was equipped with a 12.5 mm variable iris lens, which limited its field of view to about 100. Despite the narrow field of view , the car was able to navigate the track at an average speed of 1 mls without making any errors. On less curvy parts of the track, it accelerated to about 2 mls and slowed down at the corners. When the speed of the vehicle is scaled up, the errors made are mainly due to over steering. 5 CONCLUSION A 2D model of the saccade generating components of the superior colliculus is presented . This model only mimics the functionality the saccadic system using mixed signal focal plane circuits that realize motion centroid localization. The single chip combines a silicon retina with the superior colliculus model using compact, low power and fast circuits. Finally, the centroid chip is interfaced with an 8-bit IlC and vehicle for fast linefollowing auto navigation with obstacle avoidance. Here all of the required computation is performed at the visual sensor, and a standard IlC is the high-level decision maker. References Barlow H., The Senses: Physiology of the Retina, Cambridge University Press, Cambridge, England, 1982. Boahen K., "Retinomorphic Vision Systems II: Communication Channel Design," ISCAS 96, Atlanta, GA, 1996. DeWeerth, S. P., "Analog VLSI Circuits for Stimulus Localization and Centroid Computation," Int'l 1. Computer Vision, Vol. 8, No.2, pp. 191-202, 1992. Etienne-Cummings R., J Van der Spiegel and P. Mueller, "Neuromorphic and Digital Hybrid Systems," Neuromorphic Systems: Engineering Silicon from Neurobiology, L. Smith and A. Hamilton (Eds.), World Scientific, 1998. Horiuchi T., T. Morris, C . Koch and S. P . DeWeerth, "Analog VLSI Circuits for Attention-Based Visual Tracking," Advances in Neural Information Processing Systems, Vol. 9, Denver, CO, 1996. Koch C. and H. Li (Eds.), Vision Chips: Implementing Vision Algorithms with Analog VLSI Circuits, IEEE Computer Press, 1995. Mansfield, P., "Machine Vision Tackles Star Tracking," Laser Focus World, Vol. 30, No. 26, pp. S21 -S24, 1996. Mead C. and M. Ismail (Eds.), Analog VLSI Implementation of Neural Networks, Kluwer Academic Press, Newell, MA, 1989. Sparks D., C. Lee and W. Rohrer, "Population Coding of the Direction, Amplitude and Velocity of Saccadic Eye Movements by Neurons in the Superior Colliculus," Proc. Cold Spring Harbor Symp. Quantitative Biology, Vol. LV, 1990. 

1499 |@word version:1 inversion:1 compression:1 cco:1 instruction:1 r:3 pulse:1 cml:1 brightness:1 lightweight:1 contains:1 offering:1 detc:1 cleared:1 current:3 yet:1 follower:2 must:3 john:2 realize:3 s21:1 motor:5 designed:1 cue:1 vmin:1 slowing:1 plane:10 realizing:1 smith:1 provides:8 node:1 location:6 five:2 windowed:1 mathematical:2 direct:1 driver:1 rohrer:1 persistent:1 microchip:1 resistive:3 fixation:1 combine:2 symp:1 expected:1 rapid:1 behavior:3 oscilloscope:2 multi:1 cpu:1 equipped:2 window:1 provided:4 moreover:1 circuit:30 cm:2 microcomputer:1 temporal:2 quantitative:1 tackle:1 um:1 scaled:1 control:10 hamilton:1 positive:1 engineering:1 despite:1 id:1 mead:3 modulation:1 advised:1 black:2 co:1 limited:2 camera:1 block:1 implement:3 cold:1 physiology:1 pre:1 suggest:1 get:1 onto:1 close:1 wheel:1 operator:1 ga:1 measurable:1 map:3 center:1 attention:1 spark:3 factored:1 avoidance:7 array:4 fill:2 his:1 population:1 handle:2 autonomous:8 coordinate:10 deceleration:1 target:19 trigger:1 controlling:2 us:8 associate:1 crossing:5 velocity:2 walking:1 cut:2 module:1 enters:1 worst:1 ycoordinates:1 connected:2 decrease:1 movement:2 prospect:1 servo:2 boahen:2 vanishes:1 environment:2 vdd:1 solving:1 segment:1 localization:15 creates:1 completely:2 textured:1 accelerate:1 chip:27 laser:1 horiuchi:3 fast:8 forced:1 activate:3 labeling:1 outside:1 doctrine:1 quite:1 larger:2 spiegel:1 asynchronously:2 chase:1 triggered:5 product:1 combining:1 loop:1 flexibility:1 forth:1 ismail:1 normalize:1 scalability:1 competition:1 requirement:1 generating:1 cmos:1 staying:1 object:7 help:1 avlsi:1 develop:1 pose:1 strong:1 implemented:4 implies:1 direction:2 anatomy:1 attribute:1 functionality:1 subsequently:3 human:1 implementing:1 argued:1 require:1 activating:2 biological:3 summation:1 mm:2 koch:3 great:1 mo:1 circuitry:1 early:2 proc:1 realizes:1 radio:2 maker:1 weighted:1 clearly:2 sensor:8 amacrine:1 avoid:1 varying:1 voltage:6 earliest:1 focus:2 improvement:1 mainly:2 contrast:4 centroid:42 suppression:1 sense:1 mueller:1 dependent:1 eliminate:1 integrated:1 compactness:1 vlsi:10 relation:1 zerocrossing:1 ralph:1 compatibility:1 pixel:13 mimicking:1 orientation:2 development:1 retinomorphic:1 spatial:3 field:6 having:1 biology:1 mimic:3 stimulus:1 micro:2 escape:1 retina:10 oriented:1 composed:1 simultaneously:2 iscas:1 maintain:2 attempt:1 detection:6 organization:2 atlanta:1 replicates:1 navigation:11 activated:5 sens:1 programmability:1 edge:14 daily:1 carbondale:1 orbit:1 re:1 stepper:2 theoretical:1 android:1 instance:1 column:7 modeling:1 obstacle:15 earlier:1 steer:1 mhz:1 neuromorphic:11 cost:1 masked:1 considerably:1 person:1 sensitivity:1 lee:1 off:2 iy:1 hopkins:2 na:1 s24:1 broadcast:1 priority:1 external:1 corner:2 derivative:3 li:3 premised:1 de:1 star:1 coding:1 int:1 vi:1 blind:2 vehicle:15 performed:4 view:6 closed:2 portion:2 parallel:2 capability:1 il:1 formed:1 loaded:1 characteristic:3 interfaced:3 correspond:1 lu:1 lighting:1 researcher:1 drive:2 straight:1 processor:1 detector:5 ed:3 against:1 acquisition:1 frequency:3 involved:1 pp:2 gain:1 auditory:1 car:3 ubiquitous:1 amplitude:1 back:1 appears:2 higher:1 cummings:5 day:1 follow:1 response:2 improved:1 formulation:1 furthermore:2 lastly:2 deweerth:3 until:1 hand:2 mode:1 artifact:1 gray:1 scientific:1 effect:1 normalized:1 true:1 barlow:2 requiring:1 read:1 laboratory:1 deal:1 white:1 latch:4 during:3 width:1 essence:1 iris:1 motion:10 ilc:5 rudimentary:1 image:7 novel:1 recently:1 superior:14 denver:1 overview:2 viktor:1 analog:17 discussed:1 approximates:1 braking:1 organism:1 kluwer:1 silicon:2 cambridge:2 focal:10 grid:6 outlined:1 illinois:1 moving:7 robot:4 driven:2 mohammed:1 buffer:1 certain:1 binary:1 success:1 der:1 muscle:1 seen:1 greater:1 steering:5 floor:1 converting:1 signal:16 ii:1 multiple:1 photodiodes:1 smooth:2 determination:2 england:1 offer:1 believed:1 academic:1 laplacian:1 schematic:4 controlled:1 mansfield:2 controller:1 vision:7 normalization:1 represent:1 cell:2 addition:1 background:1 baltimore:2 diagram:1 isolate:2 recording:1 hz:1 tend:1 capacitor:1 mw:2 ideal:2 switch:13 affect:1 harbor:1 architecture:2 converter:1 idea:1 regarding:1 accelerating:1 suffer:1 locating:1 resistance:2 constitute:1 action:1 tape:1 collision:1 morris:1 hardware:3 simplest:1 fiction:1 track:8 promise:1 vol:4 key:1 four:1 threshold:4 ce:1 kept:1 ram:1 year:1 colliculus:13 nce:1 run:2 angle:2 powerful:1 place:1 decision:3 bit:2 entirely:1 layer:3 pushed:1 activity:1 optic:1 infinity:1 software:1 aspect:1 speed:8 min:1 unsmoothed:1 spring:1 performing:2 developing:1 combination:1 disconnected:1 remain:1 aways:1 primate:2 making:3 continuo:1 projecting:1 slowed:1 equation:6 turn:2 mechanism:1 ascending:1 fed:1 operation:1 decelerate:1 away:1 appropriate:1 robustness:1 top:1 pwm:4 running:1 maintaining:1 etienne:6 implied:1 move:3 realized:3 strategy:1 saccadic:7 primary:1 md:2 traditional:2 usual:1 responds:1 dp:1 distance:1 consumption:1 cellular:1 dream:2 rom:1 relationship:1 illustration:1 difficult:1 trace:2 implementation:11 design:2 av:2 neuron:1 neurobiology:1 communication:2 head:1 locate:1 sharp:1 intensity:11 drift:1 pic:1 required:3 narrow:1 address:2 able:1 prototyping:1 departure:1 xm:2 indoor:1 program:1 built:1 including:1 power:9 event:2 natural:1 hybrid:2 circumvent:1 force:1 treated:1 altered:1 technology:1 eye:5 ne:1 picture:1 axis:2 started:1 mediated:1 auto:4 coupled:1 speeding:1 relative:2 highlight:1 mixed:6 generation:2 proportional:1 lv:1 abdel:1 digital:13 foundation:1 degree:1 thresholding:1 row:6 summary:1 last:1 asynchronous:1 guide:1 allow:1 benefit:1 van:1 boundary:2 curve:1 cortical:1 world:3 valid:1 made:2 far:2 approximate:1 compact:7 skill:1 keep:2 ml:2 global:1 receiver:1 xi:3 decade:1 table:2 impedance:1 reality:1 channel:1 robust:1 interact:1 complex:3 noise:1 ghani:4 slow:1 aid:1 sub:2 position:13 resistor:2 xl:1 outdoor:1 ix:1 down:2 navigate:1 list:1 exists:1 flashing:1 magnitude:1 te:1 foveated:3 gap:1 logarithmic:1 simply:1 forming:2 ganglion:1 visual:12 ordered:1 tracking:3 saccade:7 newell:1 determines:1 extracted:1 ma:1 consequently:1 acceleration:1 towards:2 change:3 determined:1 except:1 reducing:1 engineer:1 total:1 lens:1 pas:1 ece:2 zone:4 parasitic:1 accelerated:1 dept:3 tested:1 avoiding:2

270 Correlational Strength and Computational Algebra of Synaptic Connections Between Neurons Eberhard E. Fetz Department of Physiology & Biophysics, University of Washington, Seattle, WA 98195 ABSTRACT Intracellular recordings in spinal cord motoneurons and cerebral cortex neurons have provided new evidence on the correlational strength of monosynaptic connections, and the relation between the shapes of postsynaptic potentials and the associated increased firing probability. In these cells, excitatory postsynaptic potentials (EPSPs) produce crosscorrelogram peaks which resemble in large part the derivative of the EPSP. Additional synaptic noise broadens the peak, but the peak area -- i.e., the number of above-chance firings triggered per EPSP -- remains proportional to the EPSP amplitude. A typical EPSP of 100 ~v triggers about .01 firings per EPSP. The consequences of these data for information processing by polysynaptic connections is discussed. The effects of sequential polysynaptic links can be calculated by convolving the effects of the underlying monosynaptic connections. The net effect of parallel pathways is the sum of the individual contributions. INTRODUCTION Interactions between neurons are determined by the strength and distribution of their synaptic connections. The strength of synaptic interactions has been measured directly in the central nervous system by two techniques. Intracellular recording reveals the magnitude and time course of postsynaptic potentials (PSPs) produced by synaptic connections, and crosscorrelation of extracellular spike trains measures the effect of the PSP's on the firing probability of the connected cells. The relation between the shape of excitatory postsynaptic potentials (EPSPs) and the shape of the crosscorrelogram peak they produce has been empirically investigated in cat motoneurons 2,4,5 and in neocortical cells 10. RELATION BETWEEN EPSP'S AND CORRELOGRAM PEAKS Synaptic interactions have been studied most thoroughly in spinal cord motoneurons. Figure 1 illustrates the membrane potential of a rhythmically firing motoneuron, and the effect of EPSPs on its firing. An EPSP occurring sufficiently close to threshold (8) will cause the motoneuron to fire and will advance an action potential to its rising edge (top). Mathematical analysis of this threshold-crossing process predicts that an EPSP with shape e(t) will produce a firing probability f(t), which resembles ? American Institute of Phy~ics 1988 271 rI f; :: 'I 8 'I I' /..-::-- ----.... .... .",.""..,.", .,...,."" ~-""" /' I ..,, .... .. ) \ ...",/ ,; I i: .. : \ --.----r , 'I .,.,,,, .... .... EPSP e(t) t CROSS- CORRELOGRAM f(t) TIME t Fig. 1. The relation between EPSP's and motoneuron firing. Top: membrane trajectory of rhythmically firing motoneuron, showing EPSP crossing threshold (8) and shortening the normal interspike interval by advancing a spike. V(t) is difference between membrane potential and threshold. Middle: same threshold-crossing process aligned with EPSP, with v(t) plotted as falling trajectory. Intercept (at upward arrow) indicates time of the advanced action potential. Bottom: Cross-correlation histogram predicted by threshold crossings. The peak in the firing rate f(t) above baseline (fo) is produced by spikes advanced from baseline, as indicated by the changed counts for the illustrated trajectory. Consequently, the area in the peak equals the area of the subsequent trough. 272 the derivative of the EPSP 4,8. Specifically, for smooth membrane potential trajectories approaching threshold (the case of no additional synaptic noise): f(t) =fo + (fo/v) del dt (1) v where fo is the baseline firing rate of the motoneuron and is the rate of closure between motoneuron membrane potential and threshold. This relation can be derived analytically by tranforming the process to a coordinate system aligned with the EPSP (Fig. 1, middle) and calculating the relative timing of spikes advanced by intercepts of the threshold trajectories with the EPSP 4. The above relation (1) is also valid for the correlogram trough during the falling phase of the EPSP, as long as del dt > if the EPSP falls more rapidly than the trough is limited at zero firing rate (as illustrated for the correlogram at bottom). The fact that the shape of the correlogram peak above baseline matches the EPSP derivative has been empirically confirmed for large EPSPs in cat motoneurons 4. This relation implies that the height of the correlogram peak above baseline is proportional to the EPSP rate of rise. The integral of this relationship predicts that the area between the correlogram peak and baseline is proportional to the EPSP amplitude. This linear relation further implies that the effects of simultaneously arriving EPSPs will add linearly. The presence of additional background synaptic "noise", which is normally produced by randomly occurring synaptic inputs, tends to make the correlogram peak broader than the duration of the EPSP risetime. This broadening is produced by membrane potential fluctuations which cause additional threshold crossings during the decay of the EPSP by trajectories that would have missed the EPSP (e.g., the dashed trajectory in Fig. 1, middle). On the basis of indirect empirical comparisons it has been proposed 6,7 that the broader correlogram peaks can be described by the sum of two linear functions of e(t): -v, f(t) =fo + a e(t) + b deldt -v; (2) This relation provides a reasonable match when the coefficients (a and b) can be optimized for each case 5,7, but direct empirical comparisons 2,4 indicate that the difference between the correlogram peak and the derivative is typically briefer than the EPSP. The effect of synaptic noise on the transform -between EPSP and correlogram peak has not yet been analytically derived (except for the case of However the threshold-crossing process has been Gaussian noise1). simulated by a computer model which adds synaptic noise to the trajectories intercepting the EPSP 1. The correlograms generated by the simulation match the correlograms measured empirically for small EPSP's in motoneurons 2, confirming the validity of the model. Although synaptic noise distributes the triggered firings over a wider peak, the area of the correlogram peak, i.e., the number of motoneuron firings produced by an EPSP, is essentially preserved and remains proportional to EPSP amplitude for moderate noise levels. For unitary EPSP's (produced by 273 a single afferent fiber) in cat motoneurons, the number of firings triggered per EPSP (Np) was linearly related to the amplitude (h) of the EPSP 2: Np = (O.l/mv)? h (mv) + .003 (3) The fact that the number of triggered spikes increases in proportion to EPSP amplitude has also been confirmed for neocortical neurons 10; for cells recorded in sensorimotor cortex slices (probably pyramidal cells) the coefficient of h was very similar: 0.07/mv. This means that a typical unitary EPSP with amplitude of 100 Ilv, raises the probability that the postsynaptic cell fires by less than .01. Moreover, this increase occurs during a specific time interval corresponding to the rise time of the EPSP - on the order of 1 - 2 msec. The net increase in firing rate of the postsynaptic cell is calculated by the proportional decrease in interspike intervals produced by the triggered spikes 4. (While the above values are typical, unitary EPSP's range in size from several hundred IlV down to undetectable levels of severalllv., and have risetimes of.2 - 4 msec.) Inhibitory connections between cells, mediated by inhibitory postsynaptic potentials (IPSPs), produce a trough in the cross-correlogram. This reduction of firing probability below baseline is followed by a subsequent broad, shallow peak, representing the spikes that have been delayed during the IPSP. Although the effects of inhibitory connections remain to be analyzed more quantitatively, preliminary results indicate that small IPSP's in synaptic noise produce decreases in firing probability that are similar to the increases produced by EPSP's 4,5. DISYNAPTIC LINKS The effects of polysynaptic links between neurons can be understood as combinations of the underlying monosynaptic connections. A monosynaptic connection from cell A to cell B would produce a first-order cross-correlation peak P1(BIA,t), representing the conditional probability that neuron B fires above chance at time t, given a spike in cell A at time t = O. As noted above, the shape of this first-order correlogram peak is largely proportional to the EPSP derivative (for cells whose interspike interval exceeds the duration of the EPSP). The latency of the peak is the conduction time from A to B (Fig. 2 top left). In contrast, several types of disynaptic linkages betw.een A and B, mediated by a third neuron C, will produce a second-order correlation peak between A and B. A disynaptic link may be produced by two serial monosynaptic connections, from A to C and from C to B (Fig. 2, bottom left), or by a common synaptic input from C ending on both A and B (Fig. 2, bottom right). In both cases, the second-order correlation between A and B produced by the disynaptic link would be the convolution of the two firstorder correlations between the monosynaptically connected cells: (4) 274 As indicated by the diagram, the cross-correlogram peak P2(BIA,t) would be smaller and more dispersed than the peaks of the underlying first-order correlation peaks. For serial connections the peak would appear to the right of the origin, at a latency that is the sum of the two monosynaptic latencies. The peak produced by a common input typically straddles the origin, since its timing reflects the difference between the underlying latencies. => Monosynaptic connection -----..'t- I \ t \ @ First-order correlation ~(AIB,t) LJA,,-_~_(_B_I_A_'t_)_ Disynaptic connection = ~(~IA,-t) 1 ~ Serial connection Second-order correlation Common input r--A---t t I t \ I \ : \. A ~ (C I A) @ t t t I : ll(AIC) :" V H\~ t : j \ I'" t \ \ \ \ \ t \~(BIC) P(BIA) _________ ~,_2__---- @ \. P(BIC) 1 \/\ ' J t"-_------ _ _ _/\.~(BIA) -L Fig. 2. Correlational effects of monosynaptic and disynaptic links between two neurons. Top: monosynaptic excitatory link from A to B produces an increase in firing probability of B after A (left). As with all correlograms this is the time-inverted probability of increased firing in A relative to B (right). Bottom: Two common disynaptic links between A and B are a serial connection via C (left) and a common input from C. In both cases the effect of the disynaptic link is the convolution of the underlying monosynaptic links. 275 This relation means that the probability that a spike in cell A will produce a correlated spike in cell B would be the product of the two probabilities for the intervening monosynaptic connections. Given a typical Np of .Ol/EPSP, this would reduce the effectiveness of a given disynaptic linkage by two orders of magnitude relative to a monosynaptic connection. However, the net strength of all the disynaptic linkages between two given cells is proportional to the number of mediating intemeurons (C}, since the effects of parallel pathways add. Thus, the net potency of all the disynaptic linkages between two cells could approach that of a monosynaptic linkage if the number of mediating interneurons were sufficiently large. It should also be noted that some intemeurons may fire more than once per EPSP and have a higher probability of being triggered to fire than motoneurons 11. For completeness, two other possible disynaptic links between A and B involving a third cell C may be considered. One is a serial connection from B to C to A, which is the reverse of the serial connection from A to B. This would produce a P2(BIA) with peak to the left of the origin. The fourth circuit involves convergent connections from both A and B to C; this is the only combination that would not produce any causal link between A and B. The effects of still higher-order polysynaptic linkages can be computed similarly, by convolving the effects produced by the sequential connections. For example, trisynaptic linkages between four neurons are equivalent to combinations of disynaptic and monosynaptic connections. The cross-correlograms between two cells have a certain symmetry, depending on which is the reference cell. The cross-correlation histogram of cell B referenced to A is identical to the time-inverted correlogram of A referenced to B. This is illustrated for the monosynaptic connection in Fig.2, top right, but is true for all correlograms. This symmetry represents the fact that the above-chance probability of B firing after A is the same as the probability of A firing before B: P(BIA, t) = P(AIB, -t) (5) As a consequence, polysynaptic correlational links can be computed as the same convolution integral (Eq. 4), independent of the direction of impulse propagation. P ARALLEL PATHS AND FEEDBACK LOOPS In addition to the simple combinations of pair-wise connections between neurons illustrated above, additional connections between the same cells may form circuits with various kinds of loops. Recurrent connections can produce feedback loops, whose correlational effects are also calculated by convolving effects of the underlying synaptic links. Parallel feed-forward paths can form multiple pathways between the same cells. These produce correlational effects that are the sum of the effects of the individual underlying connections. The simplest feedback loop is formed by reciprocal connections between a pair of cells. The effects of excitatory feedback can be computed by 276 successive cO?1volutions of the underlying monosynaptic connections (Fig. 3 top). Note that such a positive feedback loop would be capable of sustaining activity only if the connections were sufficiently potent to ensure postsynaptic firing. Since the probabilities of triggered firings at a single synapse are considerably less than one, reverberating activity can be sustained only if the number of interacting cells is correspondingly increased. Thus, if the probability for a single link is on the order of .01, reverberating activity can be sustained if A and B are similarly interconnected with at least a hundred cells in parallel. Connections between three neurons may produce various kinds of loops. Feedforward parallel pathways are formed when cell A is monosynaptically connected to B and in addition has a serial disynaptic connection through C, as illustrated in Fig. 3 (bottom left); the correlational effects of the two linkages from A to B would sum linearly, as shown for excitatory connections. Again, the effect of a larger set of cells {C} would be additive. Feedback loops could be formed with three cells by recurrent connections between any pair; the correlational consequences of the loop again are the convolution of the underlying links. Three cells can form another type loop if both A and B are monosynaptically connected, and simultaneously influenced by a common interneuron C (Fig. 3 bottom right). In this case the expected correlogram between A and B would be the sum of the individual components -- a common input peak around the origin plus a delayed peak produced by the serial connection. Feedback loop 1 - - - - -.. 'l;---~ ... ..../ -', \.... .... .... .. I: ' .... ....... ..........." ;'" ........ ..?.:.... .... -', .... ????? '" ???. Parallel jeedfOrward path I I : t t t Common input loop I t PI (BIA) +P 2 (BIA) PI (BIA)+P 2 (BIA) :/\ ____ ___.;.J l~ Fig. 3. Correlational effects of parallel connections between two neurons. Top: feedback loop between two neurons A and B produces higher-order effects equivalent to convolution of mono~aptic effects. Bottom: Loops formed by parallel feed forward paths (left) and by a common mput concurrent with a monosynaptic link (right) produce additive effects. 277 CONCLUSIONS Thus, a simple computational algebra can be used to derive the correlational effects of a given network structure. Effects of sequential connections can be computed by convolution and effects of parallel paths by summation. The inverse problem, of deducing the circuitry from the correlational data is more difficult, since similar correlogram features may be produced by different circuits 9. The fact that monosynaptic links produce small correlational effects on the order of .01 represents a significant constraint in the mechanisms of information processing in real neural nets. For example, secure propagation of activity through serial polysynaptic linkages requires that the small probability of triggered firing via a given link is compensated by a proportional increase in the number of parallel links. Thus, reliable serial conduction would require hundreds of neurons at each level, with appropriate divergent and convergent connections. It should also be noted that the effect of intemeurons can be modulated by changing their activity. The intervening cells need to be active to mediate the correlational effects. As indicated by eq. I, the size of the correlogram peak is proportional to the firing rate (fo) of the postsynaptic cell. This allows dynamic modulation of polysynaptic linkages. The greater the number of links, the more susceptible they are to modulation. Acknowledgements: The author thanks Mr. Garrett Kenyon for stimulating discussions and the cited colleagues for collaborative efforts. This work was supported in part by Nll-I grants NS 12542 and RR00166. REFERENCES 1. Bishop, B., Reyes, A.D., and Fetz E.E., Soc. for Neurosci Abst. 11:157 (1985). 2. Cope, T.C., Fetz, E.E., and Matsumura, M., J. Physiol. 390:161-18 (1987). 3. Fetz, E.E. and Cheney, P.D., J. Neurophysiol. 44:751-772 (1980). 4. Fetz, E.E. and Gustafsson, B., J. Physiol. 341:387-410 (1983). 5. Gustafsson, B., and McCrea, D., J. Physiol. 347:431-451 (1984). 6. Kirkwood, P.A., J. Neurosci. Meth. 1:107-132 (1979). 7. Kirkwood, P.A., and Sears, T._ J. Physiol. 275:103-134 (1978). 8. Knox, C.K., Biophys. J. 14: 567-582 (1974). 9. Moore, G.P., Segundo, J.P., Perkel, D.H. and Levitan, H., Biophys. J. 10:876900 (1970). 10. Reyes, A.D., Fetz E.E. and Schwindt, P.C., Soc. for Neurosci Abst. 13:157 (1987). 11. Surmeier, D.J. and Weinberg, R.J., Brain Res. 331:180-184 (1985). 

15 |@word middle:3 rising:1 proportion:1 closure:1 simulation:1 t_:1 reduction:1 phy:1 yet:1 physiol:4 additive:2 subsequent:2 confirming:1 interspike:3 shape:6 nervous:1 reciprocal:1 completeness:1 provides:1 cheney:1 successive:1 height:1 mathematical:1 correlograms:5 direct:1 undetectable:1 gustafsson:2 sustained:2 pathway:4 expected:1 p1:1 ol:1 perkel:1 brain:1 provided:1 monosynaptic:18 underlying:9 moreover:1 circuit:3 kind:2 firstorder:1 normally:1 grant:1 appear:1 before:1 positive:1 understood:1 timing:2 referenced:2 tends:1 consequence:3 firing:26 fluctuation:1 path:5 modulation:2 plus:1 studied:1 resembles:1 sustaining:1 co:1 limited:1 range:1 area:5 empirical:2 physiology:1 close:1 intercept:2 equivalent:2 compensated:1 duration:2 monosynaptically:3 coordinate:1 trigger:1 origin:4 crossing:6 predicts:2 bottom:8 cord:2 connected:4 decrease:2 dynamic:1 raise:1 algebra:2 volution:1 basis:1 neurophysiol:1 indirect:1 cat:3 fiber:1 various:2 train:1 sears:1 broadens:1 whose:2 larger:1 transform:1 nll:1 triggered:8 abst:2 net:5 interaction:3 product:1 epsp:41 interconnected:1 aligned:2 loop:13 rapidly:1 intervening:2 seattle:1 produce:17 wider:1 depending:1 recurrent:2 derive:1 measured:2 eq:2 p2:2 epsps:5 soc:2 predicted:1 resemble:1 implies:2 indicate:2 involves:1 direction:1 require:1 preliminary:1 summation:1 sufficiently:3 considered:1 ic:1 normal:1 around:1 circuitry:1 ilv:2 concurrent:1 reflects:1 gaussian:1 broader:2 derived:2 indicates:1 contrast:1 secure:1 baseline:7 typically:2 relation:10 upward:1 equal:1 once:1 washington:1 identical:1 represents:2 broad:1 np:3 quantitatively:1 randomly:1 simultaneously:2 individual:3 delayed:2 phase:1 fire:5 interneurons:1 analyzed:1 edge:1 integral:2 capable:1 segundo:1 re:1 plotted:1 causal:1 increased:3 hundred:3 conduction:2 considerably:1 thoroughly:1 thanks:1 cited:1 eberhard:1 peak:30 straddle:1 potent:1 knox:1 intercepting:1 again:2 central:1 recorded:1 convolving:3 derivative:5 american:1 crosscorrelation:1 potential:12 coefficient:2 trough:4 afferent:1 mv:3 parallel:10 contribution:1 collaborative:1 formed:4 largely:1 produced:14 trajectory:8 confirmed:2 fo:6 influenced:1 synaptic:15 sensorimotor:1 disynaptic:14 colleague:1 associated:1 amplitude:6 garrett:1 feed:2 higher:3 dt:2 synapse:1 correlation:9 propagation:2 del:2 indicated:3 impulse:1 effect:31 validity:1 aib:2 true:1 kenyon:1 intemeurons:3 analytically:2 moore:1 illustrated:5 ll:1 during:4 noted:3 neocortical:2 reyes:2 wise:1 common:9 empirically:3 spinal:2 cerebral:1 discussed:1 significant:1 lja:1 similarly:2 cortex:2 add:3 moderate:1 reverse:1 certain:1 inverted:2 motoneuron:14 additional:5 greater:1 mr:1 dashed:1 multiple:1 smooth:1 exceeds:1 match:3 cross:7 long:1 ipsps:1 serial:10 biophysics:1 involving:1 essentially:1 histogram:2 cell:32 preserved:1 background:1 addition:2 interval:4 diagram:1 pyramidal:1 probably:1 recording:2 effectiveness:1 unitary:3 presence:1 feedforward:1 psps:1 bic:2 een:1 approaching:1 rhythmically:2 reduce:1 linkage:10 effort:1 cause:2 action:2 latency:4 shortening:1 simplest:1 inhibitory:3 per:4 four:1 threshold:11 falling:2 mono:1 changing:1 advancing:1 sum:6 bia:10 inverse:1 fourth:1 reasonable:1 missed:1 noise1:1 followed:1 aic:1 convergent:2 activity:5 strength:5 constraint:1 ri:1 extracellular:1 department:1 combination:4 membrane:6 psp:1 remain:1 smaller:1 postsynaptic:9 shallow:1 remains:2 count:1 mechanism:1 appropriate:1 top:7 ensure:1 calculating:1 levitan:1 spike:10 occurs:1 link:21 simulated:1 relationship:1 difficult:1 mediating:2 susceptible:1 weinberg:1 rise:2 neuron:14 convolution:6 interacting:1 pair:3 connection:39 optimized:1 below:1 reliable:1 tranforming:1 ia:1 advanced:3 meth:1 representing:2 mediated:2 acknowledgement:1 relative:3 proportional:9 pi:2 excitatory:5 course:1 changed:1 supported:1 arriving:1 fetz:6 institute:1 fall:1 correspondingly:1 slice:1 feedback:8 calculated:3 kirkwood:2 valid:1 ending:1 forward:2 author:1 cope:1 active:1 reveals:1 ipsp:2 symmetry:2 broadening:1 investigated:1 linearly:3 intracellular:2 arrow:1 noise:8 neurosci:3 mediate:1 fig:12 n:1 msec:2 third:2 down:1 specific:1 bishop:1 showing:1 reverberating:2 decay:1 divergent:1 evidence:1 sequential:3 magnitude:2 illustrates:1 occurring:2 biophys:2 interneuron:1 deducing:1 polysynaptic:7 correlogram:19 chance:3 dispersed:1 stimulating:1 conditional:1 consequently:1 briefer:1 typical:4 determined:1 specifically:1 except:1 distributes:1 correlational:13 mput:1 modulated:1 betw:1 surmeier:1 correlated:1

256 AN INFORMATION THEORETIC APPROACH TO RULE-BASED CONNECTIONIST EXPERT SYSTEMS Rodney M. Goodman, John W. Miller Department of Electrical Engineering C altech 116-81 Pasadena, CA 91125 Padhraic Smyth Communication Systems Research Jet Propulsion Laboratories 238-420 4800 Oak Grove Drive Pasadena, CA 91109 Abstract We discuss in this paper architectures for executing probabilistic rule-bases in a parallel manner, using as a theoretical basis recently introduced information-theoretic models. We will begin by describing our (non-neural) learning algorithm and theory of quantitative rule modelling, followed by a discussion on the exact nature of two particular models. Finally we work through an example of our approach, going from database to rules to inference network, and compare the network's performance with the theoretical limits for specific problems. Introduction With the advent of relatively cheap mass storage devices it is common in many domains to maintain large databases or logs of data, e.g., in telecommunications, medicine, finance, etc. The question naturally arises as to whether we can extract models from the data in an automated manner and use these models as the basis for an autonomous rational agent in the given domain, i.e., automatically generate "expert systems" from data. There are really two aspects to this problem: firstly learning a model and, secondly, performing inference using this model. What we propose in this paper is a rather novel and hybrid approach to learning and inference. Essentially we combine the qu'alitative knowledge representation ideas of AI with the distributeq, computational advantages of connectionist models, using an underlying theoretical basis tied to information theory. The knowledge representation formalism we adopt is the rule-based representation, a scheme which is well supported by cognitive scientists and AI researchers for modeling higher level symbolic reasoning tasks. We have recently developed an information-theoretic algorithm called ITRULE which extracts an optimal set of probabilistic rules from a given data set [1, 2, 3]. It must be emphasised that we do not use any form of neural learning such as backpropagation in our approach. To put it simply, the ITRULE learning algorithm is far more computationally direct and better understood than (say) backpropagation for this particular learning task of finding the most informative individual rules without reference to their collective properties. Performing useful inference with this model or set of rules, is quite a difficult problem. Exact theoretical schemes such as maximum entropy (ME) are intractable for real-time applications. An Infonnation Theoretic Approach to Expert Systems We have been investigating schemes where the rules represent links on a directed graph and the nodes correspond to propositions, i.e., variable-value pairs. Our approach is characterised by loosely connected, multiple path (arbitrary topology) graph structures, with nodes performing local non-linear decisions as to their true state based on both supporting evidence and their a priori bias. What we have in fact is a recurrent neural network. What is different about this approach compared to a standard connectionist model as learned by a weight-adaptation algorithm such as BP? The difference lies in the semantics of the representation [4]. Weights such as log-odds ratios based on log transformations of probabilities possess a clear meaning to the user, as indeed do the nodes themselves. This explicit representation of knowledge is a key requirement for any system which purports to perform reasoning, probabilistic or otherwise. Conversely, the lack of explicit knowledge representation in most current connectionist approaches, i.e., the "black box" syndrome, is a major limitation to their application in critical domains where user-confidence and explanation facilities are key criteria for deployment in the field. Learning the model Consider that we have M observations or samples available, e.g., the number of items in a database. Each sample datum is described in terms of N attributes or features, which can assume values in a corresponding set of N discrete alphabets. For example our data might be described in the form of lO-component binary vectors. The requirement for discrete rather than continuous-valued attributes is dictated by the very nature of the rule-based representation. In addition it is important to note that we do not assume that the sample data is somehow exhaustive and "correct." There is a tendency in both the neural network and AI learning literature to analyse learning in terms of learning a Boolean function from a truth table. The implicit assumption is often made that given enough samples, and a good enough learning algorithm we can always learn the function exactly. This is a fallacy, since it depends on the feature representation. For any problem of interest there are always hidden causes with a consequent non-zero Bayes misclassification risk, i.e., the function is dependent on non-observable features (unseen columns of the truth table). Only in artificial problems such as game playing is "perfect" classification possible - in practical problems nature hides the real features. This phenomenon is well known in the statistical pattern recognition literature and renders invalid those schemes which simply try to perfectly classify or memorise the training data. We use the following simple model of a rule, i.e., IT Y =y then X = x with probability p where X and Yare two attributes (random variables) with "x" and "y" being values in their respective discrete alphabets. Given sample data as described earlier we pose the problem as follows: can we find the "best" rules from a given data set, say the K best rules? We will refer to this problem as that of generalised rule induction, in order to distinguish it from the special case of deriving classification 257 258 Goodman, Miller and Smyth rules. Clearly we require both a preference measure to rank the rules and a learning algorithm which uses the preference measure to find the K best rules. Let us define the information which the event y yields about the variable X, say !(Xj y). Based on the requirements that !(Xj y) is both non-negative and that its expectation with respect to Y equals the average mutual information J(Xj Y), Blachman [5] showed that the only such function is the j-measure, which is defined as i(Xj y) = p(x\y) log (p(x\y)) p(x) + p(x\y) log (p(x)~y)) p(x) More recently we have shown that i(Xj y) possesses unique properties as a rule information measure [6]. In general the j-measure is the average change in bits required to specify X between the a priori distribution (p(X)) and the a posteriori distribution (p(X\y)). It can also be interpreted as a special case of the cross-entropy or binary discrimination (Kullback [7]) between these two distributions. We further define J(Xj y) as the average information content where J(X; y) = p(Y)-i(Xj y). J(Xj y) simply weights the instantaneous rule information i(X; y) by the probability that the left-hand side will occur, i.e., that the rule will be fired. This definition is motivated by considerations of learning useful rules in a resource-constrained environment. A rule with high information content must be both a good predictor and have a reasonable probability of being fired, i.e., p(y) can not be too small. Interestingly enough our definition of J(Xj y) possesses a well-defined interpretation in terms of classical induction theory, trading off hypothesis simplicity with the goodness-of-fit of the hypothesis to the data [8]. The ITRULE algorithm [1, 2, 3] uses the J-measure to derive the most informative set of rules from an input data set. The algorithm produces a set of K probabilistic rules, ranked in order of decreasing information content. The parameter K may be user-defined or determined via some statistical significance test based on the size of the sample data set available. The algorithm searches the space of possible rules, trading off generality of the rules with their predictiveness, and using informationtheoretic bounds to constrain the search space. Using the Model to Perform Inference Having learned the model we now have at our disposal a set of lower order constraints on the N-th order joint distribution in the form of probabilistic rules. This is our a priori model. In a typical inference situation we are given some initial conditions (i.e., some nodes are clamped), we are allowed to measure the state of some other nodes (possibly at a cost), and we wish to infer the state or probability of one more goal propositions or nodes from the available evidence. It is important to note that this is a much more difficult and general problem than classification of a single, fixed, goal variable, since both the initial conditions and goal propositions may vary considerably from one problem instance to the next. This is the inference problem, determining an a posteriori distribution in the face of incomplete and uncertain information. The exact maximum entropy solution to this problem is in- An Information Theoretic Approach to Expert Systems tractable and, despite the elegance of the problem formulation, stochastic relaxation techniques (Geman [9]) are at present impractical for real-time robust applications. Our motivation then is to perform an approximation to exact Bayesian inference in a robust manner. With this in mind we have developed two particular models which we describe as the hypothesis testing network and the uncertainty network. Principles of the Hypothesis Testing Network In the first model under consideration each directed link from Y to x is assigned a weight corresponding to the weight of evidence of yon x. This idea is not necessarily new, although our interpretation and approach is different to previous work [10, 4]. Hence we have W -1 :r.y - p{xIY) -1 og p(x) p(:xIY) og p(x) R and :r. = -log p(x) p(x) and the node x is assigned a threshold term corresponding to a priori bias. We use a sigmoidal activation function, i.e., a(x) = 1 --~7'""E=-t----;;R'--, l+e n where l:J.E:r. = I: W:r.y; . q(y,) - R:r. ,=1 T based on multiple binary inputs Y1 ... Yn to x. Let 8 be the set of all Yi which are hypothesised true (Le., a{yd = 1), so that AE = L.l:r. Iog p(x) + '" (1 p(xlYd _ 1 p(x IY,)) p(x) L- og p(x) og p(x) y;ES If each y, is conditionally independent given x then we can write p(xIS) = p(x) p(xIS) p(x) II y;ES p(xIY,) p(xlYd Therefore the updating rule for conditionally independent y, is: T . log a(x) = log 1 - a(x) p(xI8) 1 - p(x/S) Hence a(x) > ~ iff p{xI8) > ~ and if T == 1, a(x) is exactly p(xIS). In terms of a hypothesis test, a(x) is chosen true iff: 'L" Iog p(XIYi) > - Iog-p{x) p(XIYi) - p(x) Since this describes the Neyman-Pearson decision region for independent measurements (evidence or yd with R:r. = -log :~~~ [11], this model can be interpreted as a distributed form of hypothesis testing. 259 260 Goodman, Miller and Smyth Principles of the Uncertainty Network For this model we defined the weight on a directed link from Yi to x as W XYi = . ( p(XIYi) _ p(xIYi))) si.1(XjYi) = Si? p(XIYi}log( p(x) ) + p(xly,)log( p(x) where Si = ?1 and the threshold is the same as the hypothesis model. We can interpret W:Z lli as the change in bits to specify the a posteriori distribution of x. H P(XIYi) > p{x), w:ZYi has positive support for x, i.e., Si = +1. H P{XIYi) < p(x), W:Z lli has negative support for x, Le., Si = -1. IT we interpret the activation a(Yi) as an estimator (p(y)) for p(Yi), then for multiple inputs, i ~ .. ( ) P(XIYi) (_ P(XIYi) ) - ~ p(Yi).Si. p(XIYi log( p{x) ) + P xly,) log( p(x) ) ? This sum over input links weighted by activation functions can be interpreted as the total directional change in bits required to specify x, as calculated locally by the node x. One can normalise !:1Ex to obtain an average change in bits by dividing by a suitable temperature T. The node x can make a local decision by recovering p(x) from an inverse J-measure transformation of !:1E (the sigmoid is an approximation to this inverse function). Experimental Results and Conclusions In this section we show how rules can be generated from example data and automatically incorporated into a parallel inference network that takes the form of a multi-layer neural network. The network can then be "run" to perform parallel inference. The domain we consider is that of a financial database of mutual funds, using published statistical data [12]. The approach is, however, typical of many different real world domains. Figure 1 shows a portion of a set of typical raw data on no-load mutual funds. Each line is an instance of a fund (with name omitted), and each column represents an attribute (or feature) of the fund. Attributes can be numerical or categorical. Typical categorical attributes are the fund type which reflect the investment objectives of the fund (growth, growth and income, balanced, and agressive growth) and a typical numerical attribute is the five year return on investment expressed as a percentage. There are a total of 88 fund examples in this data set. From this raw data a second quantized set of the 88 examples is produced to serve as the input to ITRULE (Figure 2). In this example the attributes have been categorised to binary values so that they can be directly implemented as binary neurons. The ITRULE software then processes this table to produce a set of rules. The rules are ranked in order of decreasing information according to the J-measure. Figure 3 shows a An Infonnation Theoretic Approach to Expert Systems portion (the top ten rules) of the ITRULE output for the mutual fund data set. The hypothesis test log-likelihood metric h(Xj y), the instantaneous j-measure j(Xj y), and the average J-measure J(Xj y), are all shown, together with the rule transition probability p{x/y). In order to perform inference with the ITRULE rules we need to map the rules into a neural inference net. This is automatically done by ITRULE which generates a network file that can be loaded into a neural network simulator. Thus rule information metrics become connection weights. Figure 4 shows a typical network derived from the ITRULE rule output for the mutual funds data. For clarity not all the connections are shown. The architecture consists of two layers of neurons (or "units"): an input layer and an output layer, both of which have an activation within the range {O,l}. There is one unit in the input layer (and a corresponding unit in the output layer) for each attribute in the mutual funds data. The output feeds back to the input layer, and each layer is synchronously updated. The output units can be considered to be the right hand sides of the rules and thus receive inputs from many rules, where the strength of the connection is the rule's metric. The output units implement a sigmoid activation function on the sum of the inputs, and thus compute an activation which is an estimator of the right hand side posteriori attribute value. The input units simply pass this value on to the output layer and thus have a linear activation. To perform inference on the network, a probe vector of attribute values is loaded into the input and output layers. Known values are clamped and cannot change while unknown or desired attribute values are free to change. The network then relaxes and after several feedback cycles converges to a solution which can be read off the input or output units. To evaluate the models we setup fo~r standard classification tests with varying number of nodes clamped as inPlits. Undamped nodes were set to their a priori probability. After relaxing the network, the activation of the "target" node was compared with the true attribute values for that sample in order to determine classification performance. The two models were each trained on 10 randomly selected sets of 44 samples. The performance results given in Table 1 are the average classification rate of the models on the other 44 unseen samples. The Bayes risk (for a uniform loss matrix) of each classification test was calculated from the 88 samples. The actual performance of the networks occasionally exceeded this value due to small sample variations on the 44/44 cross validations. Table 1 Units Cramped Uncertainty Test 9 5 2 1 66.8% 70.1% 48.2% 51.4% HYPOthesis Test 70.4% 70.1% 63.0% 65.7% 1 - Bayes' Risk 88.6% 80.6% 63.6% 64.8% 261 262 Goodman, Miller and Smyth We conclude from the performance of the networks as classifiers that they have indeed learned a model of the data using a rule-based representation. The hypothesis network performs slightly better than the uncertainty model, with both being quite close to the estimated optimal rate (the Bayes' risk). Given that we know that the independence assumptions in both models do not hold exactly, we coin the term robust inference to describe this kind of accurate behaviour in the presence of incomplete and uncertain information. Based on these encouraging initial results, our current research is focusing on higher-order rule networks and extending our theoretical understanding of models of this nature. Acknowledgments This work is supported in part by a grant from Pacific Bell, and by Caltech's program in Advanced Technologies sponsored by Aerojet General, General Motors and TRW. Part of the research described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. John Miller is supported by NSF grant no. ENG-8711673. References 1. R. M. Goodman and P. Smyth, 'An information theoretic model for rule-based expert systems,' presented at the 1988 International Symposium on Information Theory, Kobe, Japan. 2. R. M. Goodman and P. Smyth, 'Information theoretic rule induction,' Proceedings of the 1988 European Conference on AI, Pitman Publishing: London. 3. R. M. Goodman and P. Smyth, 'Deriving rules from databases: the ITRULE algorithm,' submitted for publication. 4. H. Geffner and J. Pearl, 'On the probabilistic semantics of connectionist networks,' Proceedings of the 1987 IEEE ICNN, vol. II, pp. 187-195. 5. N. M. Blachman, 'The amount of information that y gives about X,' IEEE Transactions on Information Theory, vol. IT-14 (1), 27-31, 1968. 6. P. Smyth and R. M. Goodman, 'The information content of a probabilistic rule,' submitted for publication. 7. S. Kullback, Information Theory and Statistics, New York: Wiley, 1959. 8. D. Angluin and C. Smith, 'Inductive inference: theory and methods,' ACM Computing Surveys, 15(9), pp. 237-270, 1984. 9. S. Geman, 'Stochastic relaxation methods for image restoration and expert systems,' in Maximum Entropy and Bayesian Methods in Science and Engineering (Vol. 2), 265-311, Kluwer Academic Publishers, 1988. 10. G. Hinton and T. Sejnowski, 'Optimal perceptual inference,' Proceedings of the IEEE CVPR 1989. 11. R. E. Blahut, Principles and Practice of Information Theory, Addison-Wesley: Reading, MA, 1987. 12. American Association of Investors, The individual investor's guide to no-load mutual funds, International Publishing Corporation: Chicago, 1987. An Infonnation Theoretic Approach to Expert Systems Fund Type 5 Year Diver- Beta Bull Bear Stocks Return sity (Risk) Perf. Perf. 0/0 0/0 Balanced 136 C 0.8 B D 87 Growth 32 .5 C 1.05 E B 81 Growth& Income 88.3 A 0.96 C 82 D Agressive -24 A 1.23 E E 95 Growth&lncome 172 0.59 A E 73 B Balanced 144 C 0.71 B B 51 Flgure1. Type Type Type Type 5 Year A B G (?J Return 0/0 S&P=1380/0 above S&P below S&P no no yes no below no no yes no below no no no yes below no no no yes above no no no yes below no no yes no above Beta under1 over1 under1 under1 under1 under1 Figure 2. Invest- Net Distri- Expense Turn- Total ment Asset butions Ratio % over Assets Incm. $ Value $ (%NAV\ Rate %$M 0.67 37 .3 17 . 63 0 .79 34 415 - 0.02 12.5 0.88 1.4 200 16 0.14 11.9 4 .78 1.34 127 27 0.02 6.45 9 .30 1.4 1 61 64 0.53 13.6 9.97 1.09 31 113 0.72 13 10 .44 0 .98 239 190 Raw Mutual Funds Data Stocks Turn>90% over no no no no yes no IF IF IF IF IF IF IF IF IF IF 5yrRebS&P BullJ)erf Assets BullJ)erf typeA BullJ)erf typeGl BullJ)erf typeG Assets above low large high yes low yes high yes small Diver- Bull Bear sity Perf. Perf. <100% <$100M <150/0NAV C.D.E C.D.E C.D.E >100% >$100M >150/0NAV A.B AB A,B large high low low high low high small low low low high high small low high low low low large low low high high low small high high low high large high high high low low Quantized Mutual Funds Data ITRULE rule output: Mutual Funds 1 2 3 4 5 6 7 8 9 10 Distributions Assets lHEN lHEN lHEN lHEN lHEN lHEN lHEN lHEN lHEN lHEN BullJ)erf 5yrRet>S&P BullJ)erf 5yrRet>s&P typeG Assets typeG Assets typeA Bull perf high below high above no small no large no low p(x/y) j(X;y) 0.97 0.98 0.81 0.40 0 .04 0.18 0.05 0.72 0.97 0 .26 0.75 0.41 0.28 0.25 0 .50 0.25 0 .49 0.21 0.27 0.19 J(X;y) h(X;y) 0.235 0 .201 0.127 0.127 0 .123 0.121 0.109 0.109 0.108 0.103 4.74 4.31 2.02 -1.71 -3 . 87 -1 . 95 -3.74 1.64 3 .54 -1.57 Figure 3. Top Ten Mutual Funds Rules nfo2atl~ DD 00~ metric connection weights one unit per attribute I I I ~ ~ Input layer - linear units Feedback connections weight 1 = o DOD 0 DD0 o I I output layer - sigmoid units Figure 4. Rule Network 263 

150 |@word agressive:2 eng:1 initial:3 xiy:3 interestingly:1 current:2 activation:8 xiyi:10 si:6 must:2 john:2 chicago:1 numerical:2 informative:2 cheap:1 motor:1 sponsored:1 fund:16 discrimination:1 selected:1 device:1 item:1 smith:1 quantized:2 node:12 preference:2 firstly:1 oak:1 sigmoidal:1 five:1 direct:1 become:1 symposium:1 beta:2 consists:1 combine:1 manner:3 indeed:2 themselves:1 multi:1 simulator:1 decreasing:2 automatically:3 actual:1 encouraging:1 begin:1 distri:1 underlying:1 mass:1 advent:1 what:3 nav:3 kind:1 interpreted:3 developed:2 finding:1 transformation:2 corporation:1 impractical:1 quantitative:1 growth:6 finance:1 exactly:3 classifier:1 unit:11 grant:2 yn:1 generalised:1 positive:1 engineering:2 scientist:1 understood:1 local:2 limit:1 despite:1 path:1 yd:2 black:1 might:1 conversely:1 relaxing:1 deployment:1 range:1 directed:3 practical:1 unique:1 acknowledgment:1 testing:3 investment:2 practice:1 implement:1 backpropagation:2 bell:1 confidence:1 symbolic:1 cannot:1 close:1 storage:1 put:1 risk:5 map:1 survey:1 simplicity:1 rule:49 estimator:2 deriving:2 financial:1 autonomous:1 xjyi:1 variation:1 updated:1 target:1 user:3 smyth:8 exact:4 us:2 hypothesis:10 recognition:1 updating:1 aerojet:1 geman:2 database:5 electrical:1 region:1 connected:1 hypothesised:1 cycle:1 balanced:3 environment:1 trained:1 serve:1 basis:3 joint:1 stock:2 alphabet:2 describe:2 london:1 sejnowski:1 artificial:1 pearson:1 exhaustive:1 quite:2 valued:1 cvpr:1 say:3 otherwise:1 statistic:1 erf:6 unseen:2 analyse:1 advantage:1 net:2 xi8:2 propose:1 ment:1 adaptation:1 iff:2 fired:2 invest:1 requirement:3 extending:1 produce:2 perfect:1 executing:1 converges:1 sity:2 derive:1 recurrent:1 pose:1 dividing:1 recovering:1 implemented:1 trading:2 xly:2 correct:1 attribute:14 stochastic:2 require:1 behaviour:1 really:1 icnn:1 itrule:11 proposition:3 secondly:1 hold:1 predictiveness:1 considered:1 major:1 vary:1 adopt:1 omitted:1 infonnation:3 weighted:1 clearly:1 always:2 rather:2 og:4 varying:1 publication:2 derived:1 modelling:1 rank:1 likelihood:1 posteriori:4 inference:16 dependent:1 pasadena:2 hidden:1 going:1 semantics:2 classification:7 priori:5 constrained:1 special:2 mutual:11 field:1 equal:1 having:1 represents:1 connectionist:5 kobe:1 randomly:1 national:1 individual:2 blahut:1 maintain:1 ab:1 interest:1 accurate:1 grove:1 respective:1 incomplete:2 loosely:1 desired:1 theoretical:5 uncertain:2 instance:2 formalism:1 modeling:1 boolean:1 column:2 classify:1 earlier:1 goodness:1 restoration:1 bull:3 cost:1 predictor:1 uniform:1 dod:1 too:1 considerably:1 international:2 probabilistic:7 off:3 contract:1 together:1 iy:1 reflect:1 padhraic:1 possibly:1 geffner:1 cognitive:1 expert:8 american:1 return:3 japan:1 depends:1 try:1 portion:2 bayes:4 investor:2 parallel:3 rodney:1 loaded:2 miller:5 correspond:1 yield:1 directional:1 yes:10 bayesian:2 raw:3 produced:1 xyi:1 lli:2 drive:1 researcher:1 published:1 asset:7 submitted:2 fo:1 definition:2 pp:2 naturally:1 elegance:1 rational:1 knowledge:4 back:1 trw:1 focusing:1 feed:1 disposal:1 higher:2 exceeded:1 wesley:1 specify:3 formulation:1 done:1 box:1 generality:1 implicit:1 hand:3 lack:1 somehow:1 name:1 true:4 facility:1 hence:2 assigned:2 inductive:1 read:1 laboratory:2 conditionally:2 game:1 criterion:1 butions:1 theoretic:9 performs:1 temperature:1 reasoning:2 meaning:1 image:1 instantaneous:2 novel:1 recently:3 consideration:2 sigmoid:3 common:1 association:1 interpretation:2 kluwer:1 interpret:2 refer:1 measurement:1 ai:4 etc:1 base:1 aeronautics:1 hide:1 dictated:1 showed:1 occasionally:1 binary:5 yi:5 caltech:1 syndrome:1 determine:1 ii:2 multiple:3 infer:1 jet:2 academic:1 cross:2 iog:3 ae:1 essentially:1 expectation:1 metric:4 represent:1 receive:1 addition:1 publisher:1 goodman:8 posse:3 file:1 odds:1 presence:1 enough:3 relaxes:1 automated:1 xj:12 fit:1 independence:1 architecture:2 topology:1 perfectly:1 idea:2 administration:1 whether:1 motivated:1 render:1 york:1 cause:1 useful:2 clear:1 amount:1 locally:1 ten:2 generate:1 angluin:1 percentage:1 nsf:1 estimated:1 per:1 discrete:3 write:1 vol:3 key:2 threshold:2 clarity:1 graph:2 relaxation:2 sum:2 year:3 run:1 inverse:2 uncertainty:4 telecommunication:1 reasonable:1 decision:3 bit:4 bound:1 layer:12 followed:1 datum:1 distinguish:1 strength:1 occur:1 constraint:1 constrain:1 bp:1 software:1 generates:1 aspect:1 performing:3 relatively:1 department:1 pacific:1 according:1 describes:1 slightly:1 qu:1 computationally:1 resource:1 neyman:1 discus:1 describing:1 turn:2 mind:1 know:1 addison:1 tractable:1 available:3 yare:1 probe:1 coin:1 top:2 publishing:2 medicine:1 emphasised:1 classical:1 objective:1 question:1 link:4 normalise:1 propulsion:2 me:1 induction:3 ratio:2 difficult:2 setup:1 expense:1 memorise:1 negative:2 collective:1 unknown:1 perform:6 observation:1 neuron:2 supporting:1 situation:1 hinton:1 communication:1 incorporated:1 y1:1 synchronously:1 arbitrary:1 introduced:1 pair:1 required:2 connection:5 california:1 learned:3 pearl:1 below:6 pattern:1 reading:1 program:1 explanation:1 critical:1 misclassification:1 event:1 hybrid:1 ranked:2 suitable:1 dd0:1 advanced:1 scheme:4 technology:2 carried:1 perf:5 categorical:2 extract:2 literature:2 understanding:1 determining:1 loss:1 bear:2 limitation:1 validation:1 undamped:1 agent:1 principle:3 dd:1 playing:1 lo:1 supported:3 free:1 bias:2 side:3 guide:1 institute:1 face:1 pitman:1 diver:2 distributed:1 feedback:2 calculated:2 world:1 transition:1 made:1 far:1 income:2 transaction:1 observable:1 informationtheoretic:1 kullback:2 categorised:1 investigating:1 conclude:1 xi:3 continuous:1 fallacy:1 search:2 table:5 learn:1 nature:4 robust:3 ca:2 necessarily:1 european:1 domain:5 significance:1 motivation:1 allowed:1 wiley:1 purport:1 explicit:2 wish:1 lie:1 clamped:3 tied:1 perceptual:1 zyi:1 load:2 specific:1 consequent:1 evidence:4 intractable:1 entropy:4 simply:4 expressed:1 truth:2 acm:1 ma:1 goal:3 invalid:1 content:4 change:6 characterised:1 determined:1 typical:6 called:1 total:3 pas:1 tendency:1 e:2 experimental:1 support:2 arises:1 evaluate:1 phenomenon:1 ex:1

Outcomes of the Equivalence of Adaptive Ridge with Least Absolute Shrinkage Yves Grandvalet Stephane Canu Heudiasyc, UMR CNRS 6599, Universite de Technologie de Compiegne, BP 20.529, 60205 Compiegne cedex, France Yves.Grandvalet@hds.utc.fr Abstract Adaptive Ridge is a special form of Ridge regression, balancing the quadratic penalization on each parameter of the model. It was shown to be equivalent to Lasso (least absolute shrinkage and selection operator), in the sense that both procedures produce the same estimate. Lasso can thus be viewed as a particular quadratic penalizer. From this observation, we derive a fixed point algorithm to compute the Lasso solution. The analogy provides also a new hyper-parameter for tuning effectively the model complexity. We finally present a series ofpossible extensions oflasso performing sparse regression in kernel smoothing, additive modeling and neural net training. 1 INTRODUCTION In supervised learning, we have a set of explicative variables x from which we wish to predict a response variable y. To solve this problem, a learning algorithm is used to produce a predictor x) from a learning set Sf. = {(Xi, yd of examples. The goal of prediction may be: 1) to provide an accurate prediction of future responses, accuracy being measured by a user-defined loss function; 2) to quantify the effect of each explicative variable in the response; 3) to better understand the underlying phenomenon. J( H=l Penalization is extensively used in learning algorithms. It decreases the predictor variability to improve the prediction accuracy. It is also expected to produce models with few non-zero coefficients if interpretation is planned. Ridge regression and Subset Selection are the two main penalization procedures. The former is stable, but does not shrink parameters to zero, the latter gives simple models, but is unstable [1]. These observations motivated the search for new penalization techniques such as Garrotte, Non-Negative Garrotte [1], and Lasso (least absolute shrinkage and selection operator) [10]. Y. Grandvalet and S. Canu 446 Adaptive Ridge was proposed as a means to automatically balance penalization on different coefficients. It was shown to be equivalent to Lasso [4]. Section 2 presents Adaptive Ridge and recalls the equivalence statement. The following sections give some of the main outcomes ofthis connection. They concern algorithmic issues in section 3, complexity control in section 4, and some possible generalizations oflasso to non-linear regression in section 5. 2 ADAPTIVE RIDGE REGRESSION For clarity of exposure, the formulae are given here for linear regression with quadratic loss. The predictor is defined as j( x) = rff x, with rff = (f31, ... , f3d). Adaptive Ridge is a modification of the Ridge estimate, which is defined by the quadratic constraint ~~ = 1 f3; ~ C applied to the parameters. It is usually computed by minimizing the Lagrangian l jj = Argmin i=l {3 d L (L d f3j Xij - Yi) 2 + A j=l L f3; , (1) j=l where A is the Lagrange multiplier varying with the bound C on the norm of the parameters. When the ordinary least squares (OLS) estimate maximizes likelihood 1 , the Ridge estimate may be seen as a maximum a posteriori estimate. The Bayes prior distribution is a centered normal distribution, with variance proportional to 1/ A. This prior distribution treats all covariates similarly. It is not appropriate when we know that all covariates are not equally relevant. ----0 The garrotte estimate [1] is based on the OLS estimate,8 . The standard quadratic constraint is replaced by ~~ = 1 f3] (iJf ~ C. The coefficients with smaller OLS estimate are thus more heavily penalized. Other modifications are better explained with the prior distribution viewpoint. Mixtures of Gaussians may be used to cluster different set of covariates. Several models have been proposed, with data dependent clusters [9], or classes defined a priori [7]. The Automatic Relevance Determination model [8] ranks in the latter type. In [4], we propose to use such a mixture, in the form t d d Xij - Yi) 2 ,8= ArgmmL.,., ---. " " (L.,.,f3j "" {3 i=l j=l + "L.,.,Ajf3j " 2 (2) j=l Here, each coefficient has its own prior distribution. The priors are centered normal distributions with variances proportional to 1/ Aj. To avoid the simultaneous estimation of these d hyper-parameters by trial, the constraint 1 d 1 j=l J 1 dL ~ = ~ , Aj > 0 (3) is applied on A = (A1, .. . , Ad)T, where A is a predefined value. This constraint is a link between the d prior distributions. Their mean variance is proportional to 1/ A. The values of Aj are automatically2 induced from the sample, hence the qualifieradaptative. Adaptativity refers here to the penalization balance on {Pj }, not to the tuning of the hyper-parameter A. {(.C,} are independently and identically drawn from some distribution, and that some{3" exists, such that Y. = {3" T (.C, + e, where c is a centered normal random variable, then the empirical cost 1 If "'0 based on the quadratic loss is proportional to the log-likelihood of the sample. The OLS estimate{3 is thus the maximum likelihood estimate offJ'. 2 Adaptive Ridge, as Ridge or Lasso, is not scale invariant, so that the covariates should be normalized to produce sensible estimates. 447 Equivalence of Adaptive Ridge with Least Absolute Shrinkage It was shown [4] that Adaptive Ridge and least absolute value shrinkage are equivalent, in the sense that they yield the same estimate. We remind that the Lasso estimate is defined by e j3 = Argmin j3 d L (L (3j i=l d Xij - Yi ) 2 L subject to j =l l{3j l ~ f{ (4) . j =l The only difference in the definition of the Adaptive Ridge and the Lasso estimate is that the Lagrangian form of Adaptive Ridge uses the constraint CL1=1l{3j 1) 2/ d ~ f{ 2. 3 OPTIMIZATION ALGORITHM Tibshirani [10] proposed to use quadratic programming to find the l,asso solution, with 2d variables (positive and negative parts of (3j ) and 2d + 1 constraints (signs of positive and negative parts of (3j plus constraint (4)). Equations (2) and (3) suggest to use a fixed point (FP) algorithm. At each step s, the FP algorithm estimates the optimal parameters ). of .y) the Bayes prior based on the estimate (3) S -1 ) , and then maximizes the posterior to compute the current estimate (3) S ) ? As the parameterization (j3, A) may lead to divergent solutions, we define new variables and Cj = Vr;: I; .= 1, .. . , d for J (5) The FP algorithm updates alternatively c and -y as follows: (S)2 { cj d ,jS -1 )2 = ,,",d (s -1 )2 (6) L.., k =l /k -y (s) = (diag( c (s) )XT X diag( c (s) ) + AI) -1 diag( c (S) )XT y where Xi j = X i j , I is the identity matrix, and diag( c) is the square matrix with the vector c on its diagonal. The algorithm can be initialized by the Ridge or the OLS estimate. In the latter case,,B(1) is the garrotte estimate. Practically, 'if ,ys) lys-1 )is small compared to numerical accuracy, then c~s) is set to zero. In turn, is zero, and the system to be solved in the second step to determine -y can be reduced to the other var~ables. If cJ' is set to zero at any time during the optimization process, the final estimate {3j will be zero. The computations are simplified, but it is not clear whether global convergence can be obtained with this algorithm. It is easy to show the convergence towards a local minimum, but we did not find general conditions ensuring global convergence. If these conditions exist, they rely on initial conditions. Finally, we stress that the optimality conditions for c (or in a less rigorous sense for A) do not depend on the first part of the cost minimized in (2). In consequence, the equivalence between Adaptive Ridge and lasso holds/or any model or loss/unction . The FP algorithm can be applied to these other problems, without modifying the first step. 4 COMPLEXITY TUNING The Adaptive Ridge estimate depends on the learning set Sf. and on the hyper-parameter A. When the estimate is defined by (2) and (3), the analogy with Ridge suggests A as the 448 Y. Grandvalet and S. Canu ~ "natural" hyper-parameter for tuning the complexity of the regressor. As ..\ goes to zero, j3 r-<> approac~es the OLS estimatej3 , and the number of effective parameters is d. As ..\ goes to infinity, (3 goes to zero and the number of effective parameters is zero. When the estimate is defined by (4), there is no obvious choice for the hyper-parameter controlling complexity. Tibshirani [10] proposed to use v = 'Lf=1 ~ r-<> l.8j 1/ 'Lf=l ~ I. As v goes ~ to one,{3 approaches{3 ; as v goes to infinity, {3goes to zero. The weakness of v is that it is explicitly defined from the OLS estimate. As a result, it is variable when the design matrix is badly conditioned. The estimation of v is thus harder, and the overall procedure looses in stability. This is illustrated on an experiment following Breiman's benchmark [1] with 30 highly correlated predictors lE(XjX k ) = plj-k l , with p = 1 - 10- 3 . We generate 1000 Li.d. samples of size ? = 60. For each sampie s1, the modeling error (ME) is computed for several values of v and'\. We select v k and ,\k achieving the lowest ME. For one sample, there is a one to one mapping from v to'\. Thus ME is the same for v k and ,\k. Then, we compute v* and ..\* achieving the best average ME on the 1000 samples. As v k and ,\k achieve the lowest ME for s1, the ME for is higher or equal for v* and ,\ *. Due to the wide spread of {Vk }, the average loss encountered is twice for v* than for ,\*: 1/1000 'L!~10 (ME(s~, v*) - ME(s; , v k )) = 4.6 10- 2 , and 1/1000'L!~010 (ME(s~ , ..\ * ) - ME(s1 , ,\k)) = 2.310- 2 . The average modeling error are ME(v*) = 1.910- 1 and ME("\*) = 1.710- 1. s1 The estimates of prediction error, such as leave-one-out cross-validation tend to be variable. Hence, complexity tuning is often based on the minimization of some estimate of the mean prediction error (e.g bootstrap, K-fold cross-validation). Our experiment supports that, regarding mean prediction error, the optimal ,\ performs better than the optimal v . Thus, ,\ is the best candidate for complexity tuning. Although,\ and v are respectively the control parameter of the FP and QP algorithms, the preceding statement does not imply that we should use the FP algorithm. Once the solution 73 is known, v or ,\ are easily computed. The choice of one hyper-parameter is not linked to the choice of the optimization algorithm. 5 APPLICATIONS Adaptive Ridge may be applied to a variety of regression techniques. They include kernel smoothing, additive and neural net modeling. 5.1 KERNEL SMOOTHING Soft-thresholding was proved to be efficient in wavelet functional e~timation [2]. Kernel smoothers [5] can also benefit from the sparse representation given by soft-thresholding methods. For these regressors, l( x) = 'L1=1 f3i K(x , xd+f3o, there are as many covariates as pairs in the sample. The quadratic procedure of Lasso with 2? + 1 constraints becomes computationally expensive, but the FP algorithm of Adaptive Ridge is reasonably fast to converge. An example of least squares fitting is shown in fig. 1 for the motorcycle dataset [5]. On this example, the hyperparameter ,\ has been estimated by .632 bootstrap (with 50 bootstrap replicates) for Ridge and Adaptive Ridge regressions. For tuning..\, it is not necessary to determine the coefficients {3 with high accuracy. Hence, compared to Ridge regression, 449 Equivalence ofAdaptive Ridge with Least Absolute Shrinkage the overall amount of computation required to get the Adaptive Ridge estimate was about six times more important. For evaluation, Adaptive Ridge is ten times faster as Ridge regression as the final fitting uses only a few kernels (11 out of 133). -AR -- - - R + "'+ -1+ + ++ +t+ + + + + x Figure 1: Adaptive Ridge (AR) and Ridge (R) in kernel smoothing on the motorcycle data. The + are data points, and. are the prototypes corresponding to the kernels with non-zero coefficients in AR. The Gaussian kernel used is represented dotted in the lower right-hand corner. Girosi [3] showed an equivalence between a version of least absolute shrinkage applied to kernel smoothing, and Support Vector Machine (SVM). However, Adaptive Ridge, as applied here, is not equivalent to SVM, as the cost minimized is different. The fit and prototypes are thus different from the fit and support vectors that would be obtained from SVM. 5.2 ADDITIVE MODELS LJ= Additive models [6] are sums of univariate functions , f( x) = 1 fj (x j ). In the non- parametric setting, {fj} are smooth but unspecified functions. Additive models are easily represented and thus interpretable, but they require the ch~ice of the relevant covariates to be included in the model, and of the smoothness of each Ij. In the form presented in the two previous sections, Adaptive Ridge regression penalizes differently each individual coefficient, but it is easily extended to the pooled penalization of coefficients. Adaptive Ridge may th~ be used as an alternative to BRUTO [6] to balance the penalization parameters on each Ij . A classical choice for fj is cubic spline smoothing. Let B j denote the ? x (? + 2) matrix of the unconstrained B-spline basis, evaluated at Xij. Let 51 j be the (? + 2) x (f + 2) matrix corresponding to the penalization of the second derivative of J;. The coefficients of fj in the unconstrained B-spline basis are noted /3j. The "natural" extension of Adaptive Ridge is to minimize d II L d B j/3j - YI12 + L j=l ,\jf3]' 51 j /3j (7) , j=l subject to constraint (3). This problem is easily shown to have the same solution as the minimization of I t, Bjf3j - yll' + A (t, Jf3J fljf3j) 2 (8) Note that if the cost (8) is optimized with respect to a single covariate, the solution is a usual smoothing spline regression (with quadratic penalization). In the multidimensional case, 450 Y. Grandvalet and S. Canu J ex] =rif o'j/3j = {Ij'(t)}2dt may be used to summarize the non-linearity of Ij, thus lajl can be interpreted as a relevance index operating besides linear dependence of feature j. The penalizer in (8) is a least absolute shrinkage operator applied to ex j. Hence, formula (8) may be interpreted as "quadratic penalization within, and soft-thresholding between covariates".The FP algorithm of section 3 is easily modified to minimize (8), and backfitting may be used to solve the second step of this procedure. A simulated example in dimension five is shown in fig. 2. The fitted univariate functions are plotted for five values of'\. There is no dependency between the the explained variable and the last covariate. The other covariates affect the response, but the dependency on the first features is smoother, hence easier to capture and more relevant for the spline smoother. For a small value of '\, the univariate functions are unsmooth, and the additive model is interpolating the data. For,\ 10- 4 , the dependencies are well estimated on all covariates. As ,\ increases, the cov~riates with higher coordinate number are more heavily penalized, and the corresponding Ij tend to be linear. = ~ I 0 II "<t I 0 " . " " ....< " ....< .., ...... .. ....+-t-. ' ~----~-------+------~------~----~ t +. .. .... :.:... :. .. .. " ....<~:______J -_ _ _ _ _ _~_ _ _ _ _ _~_ _ _ _~_ _ _ _ _ _- J Figure 2: Adaptive Ridge in additive modeling on simulated data. The true model is y = Xl + cos( rrx2) + cOS(2rrx3) + cos(3rrx4) + E. The covariates are independently drawn from a uniform distribution on [-1, 1] and E is a Gaussian noise of standard deviation (j 0.3. The solid curves are the estimated univariate functions for different values of '\, and + are partial residuals. Linear trends are not penalized in cubic spline smoothing. Thus, when after convergence = ~T ~ /3j n j/3j = 0, the jth covariate is not eliminated. This can be corrected by applying Adap- bve Ridge a second time. To test if a significant linear trend can be detected, a linear (pek #- j being cubic splines. nalized) model may be used for 1;, the remaining h, 5.3 MLP FITTING The generalization to the pooled penalization of coefficients can also be applied to MultiLayered Perceptrons to control the complexity of the fit. If weights are penalized individually, Adaptive Ridge is equivalent to the Lasso. If weights are pooled by layer, Adaptive Ridge automatically tunes the amount of penalization on each layer, thus avoiding the multiple hyper-parameter tuning necessary in weight-decay [7]. Equivalence ofAdaptive Ridge with Least Absolute Shrinkage 451 Figure 3: groups of weights for two examples of Adaptive Ridge in MLP fitting. Left: hidden node soft-thresholding. Right: input penalization and selection, and individual smoothing coefficient for each output unit. Two other interesting configurations are shown in fig. 3. If weights are pooled by incoming and outcoming weights of a unit, node penalization/pruning is performed. The weight groups may also gather the outcoming weights from each input unit, orthe incoming weights from each output unit (one set per input plus one per output). The goal here is to penalize/select the input variables according to their relevance, and each output variable according to the smoothness of the corresponding mapping. This configuration proves itself especially useful in time series prediction, where the number of inputs to be fed into the network is not known in advance. There are also more complex choices of pooling, such as the one proposed to encourage additive modeling in Automatic Relevance Determination [8]. References [1] L. Breiman. Heuristics of instability and stabilization in model selection. The Annals of Statistics, 24(6):2350-2383, 1996. [2] D.L Donoho and I.M. Johnstone. Minimax estimation via wavelet shrinkage. Ann. Statist., 26(3):879--921,1998. [3] F. Girosi. An equivalence between sparse approximation and support vector machines. Technical Report 1606, M.LT AI Laboratory, Cambridge, MA., 1997. [4] Y. Grandvalet. Least absolute shrinkage is equivalent to quadratic penalization. In L. Niklasson, M. Boden, and T Ziemske, editors, ICANN'98, volume 1 of Perspectives in Neural Computing, pages 201-206. Springer, 1998. [5] W. HardIe. Applied Nonparametric Regression, volume 19 of Economic Society Monographs. Cambridge University Press, New York, 1990. [6] TJ. Hastie and R.J. Tibshirani. Generalized Additive Models, volume 43 of Monographs on Statistics and Applied Probability. Chapman & Hall, New York, 1990. [7] D.J.C. MacKay. A practical Bayesian framework for backprop networks. Neural Computation, 4(3):448-472,1992. [8] R. M. Neal. Bayesian Learning for Neural Networks. Lecture Notes in Statistics. Springer, New York, 1996. [9] S.I. Nowlan and G.E. Hinton. Simplifying neural networks by soft weight-sharing. Neural Computation, 4(4):473-493, 1992. [10] R.I. Tibshirani. Regression shrinkage and selection via the lasso. Journal ofthe Royal Statistical Society, B, 58(1):267-288, 1995. 

1500 |@word trial:1 version:1 norm:1 simplifying:1 solid:1 harder:1 initial:1 configuration:2 series:2 current:1 unction:1 nowlan:1 additive:9 numerical:1 girosi:2 interpretable:1 update:1 parameterization:1 provides:1 node:2 five:2 backfitting:1 fitting:4 expected:1 utc:1 automatically:2 becomes:1 underlying:1 linearity:1 maximizes:2 lowest:2 argmin:2 unspecified:1 interpreted:2 loos:1 multidimensional:1 xd:1 lys:1 control:3 unit:4 positive:2 ice:1 local:1 treat:1 consequence:1 yd:1 plus:2 umr:1 twice:1 equivalence:8 suggests:1 co:3 practical:1 lf:2 bootstrap:3 procedure:5 empirical:1 refers:1 suggest:1 get:1 selection:6 operator:3 applying:1 instability:1 equivalent:6 lagrangian:2 penalizer:2 exposure:1 go:6 independently:2 hds:1 stability:1 coordinate:1 pek:1 controlling:1 annals:1 heavily:2 user:1 programming:1 us:2 trend:2 expensive:1 solved:1 capture:1 decrease:1 monograph:2 complexity:8 covariates:10 technologie:1 depend:1 basis:2 easily:5 differently:1 represented:2 fast:1 effective:2 detected:1 hyper:8 outcome:2 heuristic:1 solve:2 cov:1 statistic:3 itself:1 final:2 net:2 propose:1 fr:1 relevant:3 motorcycle:2 achieve:1 cl1:1 rff:2 cluster:2 convergence:4 adap:1 produce:4 leave:1 bve:1 derive:1 measured:1 ij:5 quantify:1 stephane:1 modifying:1 centered:3 stabilization:1 backprop:1 require:1 generalization:2 extension:2 hold:1 practically:1 hall:1 normal:3 algorithmic:1 predict:1 mapping:2 estimation:3 ofpossible:1 individually:1 minimization:2 gaussian:2 modified:1 avoid:1 shrinkage:12 breiman:2 varying:1 heudiasyc:1 vk:1 rank:1 likelihood:3 rigorous:1 sense:3 posteriori:1 dependent:1 cnrs:1 lj:1 hidden:1 france:1 issue:1 overall:2 priori:1 smoothing:9 special:1 mackay:1 equal:1 once:1 f3:3 eliminated:1 chapman:1 future:1 minimized:2 report:1 spline:7 few:2 individual:2 replaced:1 mlp:2 highly:1 ziemske:1 evaluation:1 replicates:1 weakness:1 mixture:2 tj:1 predefined:1 accurate:1 encourage:1 partial:1 necessary:2 initialized:1 penalizes:1 plotted:1 fitted:1 modeling:6 soft:5 planned:1 ar:3 ordinary:1 cost:4 deviation:1 subset:1 predictor:4 uniform:1 dependency:3 regressor:1 corner:1 derivative:1 li:1 de:2 pooled:4 coefficient:11 explicitly:1 ad:1 depends:1 performed:1 linked:1 bayes:2 minimize:2 yves:2 square:3 accuracy:4 variance:3 yield:1 ofthe:1 bayesian:2 simultaneous:1 sharing:1 definition:1 obvious:1 universite:1 proved:1 dataset:1 recall:1 f3j:2 cj:3 rif:1 higher:2 dt:1 supervised:1 response:4 evaluated:1 shrink:1 hand:1 hardie:1 aj:3 effect:1 normalized:1 multiplier:1 true:1 former:1 hence:5 laboratory:1 orthe:1 neal:1 illustrated:1 during:1 noted:1 generalized:1 compiegne:2 stress:1 ridge:41 performs:1 l1:1 fj:4 ols:7 niklasson:1 functional:1 qp:1 volume:3 interpretation:1 significant:1 cambridge:2 ai:2 smoothness:2 tuning:8 automatic:2 unconstrained:2 canu:4 similarly:1 stable:1 operating:1 j:1 posterior:1 own:1 showed:1 perspective:1 unsmooth:1 yi:3 seen:1 minimum:1 preceding:1 determine:2 converge:1 ii:2 smoother:3 multiple:1 asso:1 technical:1 faster:1 smooth:1 determination:2 cross:2 f3o:1 equally:1 y:1 a1:1 ensuring:1 j3:4 prediction:7 regression:14 kernel:9 penalize:1 cedex:1 pooling:1 induced:1 subject:2 tend:2 identically:1 easy:1 variety:1 affect:1 fit:3 hastie:1 lasso:12 f3i:1 economic:1 regarding:1 prototype:2 whether:1 motivated:1 six:1 york:3 jj:1 useful:1 clear:1 tune:1 amount:2 nonparametric:1 extensively:1 ten:1 statist:1 reduced:1 generate:1 xij:4 exist:1 dotted:1 sign:1 estimated:3 tibshirani:4 per:2 hyperparameter:1 group:2 achieving:2 drawn:2 clarity:1 pj:1 sum:1 bound:1 layer:2 fold:1 quadratic:11 encountered:1 badly:1 constraint:9 infinity:2 bp:1 ables:1 optimality:1 performing:1 according:2 smaller:1 explicative:2 modification:2 s1:4 explained:2 invariant:1 computationally:1 equation:1 turn:1 know:1 fed:1 gaussians:1 appropriate:1 alternative:1 remaining:1 include:1 plj:1 prof:1 especially:1 classical:1 garrotte:4 society:2 parametric:1 dependence:1 usual:1 diagonal:1 link:1 simulated:2 sensible:1 me:12 unstable:1 besides:1 index:1 remind:1 balance:3 minimizing:1 statement:2 negative:3 design:1 observation:2 benchmark:1 extended:1 variability:1 hinton:1 pair:1 required:1 connection:1 optimized:1 yll:1 usually:1 fp:8 summarize:1 yi12:1 royal:1 natural:2 rely:1 residual:1 minimax:1 improve:1 imply:1 prior:7 f3d:1 loss:5 lecture:1 interesting:1 proportional:4 analogy:2 var:1 penalization:16 validation:2 gather:1 thresholding:4 viewpoint:1 editor:1 grandvalet:6 balancing:1 penalized:4 last:1 jth:1 understand:1 johnstone:1 wide:1 absolute:10 sparse:3 benefit:1 f31:1 curve:1 dimension:1 adaptive:27 regressors:1 simplified:1 pruning:1 global:2 incoming:2 xi:2 alternatively:1 search:1 timation:1 reasonably:1 correlated:1 boden:1 interpolating:1 complex:1 diag:4 did:1 icann:1 main:2 spread:1 multilayered:1 noise:1 fig:3 cubic:3 vr:1 wish:1 sf:2 xl:1 candidate:1 wavelet:2 formula:2 xt:2 covariate:3 divergent:1 svm:3 decay:1 concern:1 dl:1 ofthis:1 exists:1 effectively:1 conditioned:1 easier:1 lt:1 univariate:4 lagrange:1 springer:2 ch:1 outcoming:2 ijf:1 ma:1 viewed:1 goal:2 identity:1 donoho:1 ann:1 towards:1 included:1 corrected:1 e:1 perceptrons:1 select:2 support:4 latter:3 relevance:4 phenomenon:1 avoiding:1 ex:2

Learning curves for Gaussian processes Peter Sollich * Department of Physics, University of Edinburgh Edinburgh EH9 3JZ, U.K. Email: P.Sollich<Oed.ac . uk Abstract I consider the problem of calculating learning curves (i.e., average generalization performance) of Gaussian processes used for regression. A simple expression for the generalization error in terms of the eigenvalue decomposition of the covariance function is derived, and used as the starting point for several approximation schemes. I identify where these become exact, and compare with existing bounds on learning curves; the new approximations, which can be used for any input space dimension, generally get substantially closer to the truth. 1 INTRODUCTION: GAUSSIAN PROCESSES Within the neural networks community, there has in the last few years been a good deal of excitement about the use of Gaussian processes as an alternative to feedforward networks [lJ. The advantages of Gaussian processes are that prior assumptions about the problem to be learned are encoded in a very transparent way, and that inference-at least in the case of regression that I will consider-is relatively straightforward. One crucial question for applications is then how 'fast' Gaussian processes learn, i.e., how many training examples are needed to achieve a certain level of generalization performance. The typical (as opposed to worst case) behaviour is captured in the learning curve, which gives the average generalization error ? as a function of the number of training examples n. Several workers have derived bounds on ?(n) [2,3, 4J or studied its large n asymptotics. As I will illustrate below, however, the existing bounds are often far from tight; and asymptotic results will not necessarily apply for realistic sample sizes n. My main aim in this paper is therefore to derive approximations to ?( n) which get closer to the true learning curves than existing bounds, and apply both for small and large n. In its simplest form, the regression problem that I am considering is this: We are trying to learn a function 0* which maps inputs x (real-valued vectors) to (realvalued scalar) outputs O*(x) . We are given a set of training data D, consisting of n 'Present address: Department of Mathematics, King's College London, Strand, London WC2R 2LS, U.K. Email peter.sollicMlkcl.ac . uk 345 Learning Curves for Gaussian Processes input-output pairs (Xl, yt); the training outputs Yl may differ from the 'clean' target outputs 9* (xL) due to corruption by noise. Given a test input x, we are then asked to come up with a prediction 9(x) for the corresponding output, expressed either in the simple form of a mean prediction 9(x) plus error bars, or more comprehensively in terms of a 'predictive distribution' P(9(x)lx, D). In a Bayesian setting, we do this by specifying a prior P(9) over our hypothesis functions, and a likelihood P(DI9) with which each 9 could have generated the training data; from this we deduce the posterior distribution P(9ID) ex P(DI9)P(9). In the case of feedforward networks, where the hypothesis functions 9 are parameterized by a set of network weights, the predictive distribution then needs to be extracted by integration over this posterior, either by computationally intensive Monte Carlo techniques or by approximations which lead to analytically tractable integrals. For a Gaussian process, on the other hand, obtaining the predictive distribution is trivial (see below); one reason for this is that the prior P(9) is defined directly over input-output functions 9. How is this done? Any 9 is uniquely determined by its output values 9(x) for all x from the input domain, and for a Gaussian process, these are simply assumed to have a joint Gaussian distribution (hence the name). This distribution can be specified by the mean values (9(x))o (which I assume to be zero in the following, as is commonly done), and the covariances (9(x)9(x ' ))o = C(x, x'); C(x, x') is called the covariance function of the Gaussian process. It encodes in an easily interpretable way prior assumptions about the function to be learned. Smoothness, for example, is controlled by the behaviour of C(x, x') for x' -+ x: The OrnsteinUhlenbeck (OU) covariance function C(x, x') ex exp( -IX-X'l/l) produces very rough (non-differentiable) functions, while functions sampled from the squared exponential (SE) prior with C(X,X') ex exp(-Ix - x ' 12/(2l2)) are infinitely differentiable. The 'length scale' parameter l, on the other hand, corresponds directly to the distance in input space over which we expect our function to vary significantly. More complex properties can also be encoded; by replacing l with different length scales for each input component, for example, relevant (smalll) and irrelevant (large l) inputs can be distinguished. How does inference with Gaussian processes work? I only give a brief summary here and refer to existing reviews on the subject (see e.g. [5, 1]) for details. It is simplest to assume that outputs yare generated from the 'clean' values of a hypothesis function 9(x) by adding Gaussian noise of x-independent variance 0'2. The joint distribution of a set of training outputs {yd and the function values 9(x) is then also Gaussian, with covariances given by here I have defined an n x n matrix K and x-dependent n-component vectors k(x). The posterior distribution P(9ID) is then obtained by simply conditioning on the {Yl}. It is again Gaussian and has mean and variance (B(x))OID ( (9(x) - 9(X))2) 9(x) = k(X)TK-1y C(x, x) - k(X)TK-lk(x) (1) (2) OlD Eqs. (1,2) solve the inference problem for Gaussian process: They provide us directly with the predictive distribution P(9(x)lx, D). The posterior variance, eq. (2), in fact also gives us the expected generalization error at x. Why? If the teacher is 9*, the squared deviation between our mean prediction and the teacher output is 1 (9(x) - 9* (X))2; averaging this over the posterior distribution of teachers P(9* ID) just gives (2). The underlying assumption is that our assumed Gaussian process lOne can also one measure the generalization by the squared deviation between the prediction O(x) and the noisy teacher output; this simply adds a term 0'2 to eq. (3). P. Sollich 346 prior is the true one from which teachers are actually generated (and that we are using the correct noise model). Otherwise, a more complicated expression for the expected generalization error results; in line with most other work on the subject, I only consider the 'correct prior' case in the following. Averaging the generalization error at x over the distribution of inputs gives then (3) This form of the generalization error (which is well known [2, 3, 4, 5]) still depends on the training inputs (the fact that the training outputs have dropped out already is a signature of the fact that Gaussian processes are linear predictors, compare (1)). Averaging over data sets yields the quantity we are after, ? (4) = (t(D)}D? This average expected generalization error (I will drop the 'average expected' in the following) only depends on the number of training examples n; the function ?(n) is called the learning curve. Its exact calculation is difficult because of the joint average in eqs. (3,4) over the training inputs Xl and the test input x. 2 LEARNING CURVES As a starting point for an approximate calculation of ?(n), I first derive a representation of the generalization error in terms of the eigenvalue decomposition of the covariance function. Mercer's theorem (see e.g. [6]) tells us that the covariance function can be decomposed into its eigenvalues Ai and eigenfunctions (/Ji(x): 00 C(x, x') = L Ai<Pi(X)qJi(x') (5) i=1 This is simply the analogue of the eigenvalue decomposition of a finite symmetric matrix; the eigenfunctions can be taken to be normalized such that (<Pi (x) <Pj (x)} x = Oij . Now write the data-dependent generalization error (3) as ?(D) = (C(x,x)}xtr (k(x)k(x)T)x K- 1 and perform the x-average in the second term: ?(k(x)k(x)T)lm)x = L AiAj<Pi(XI) (<Pi (x)<pj (x)} <pj(xm) = L A;<Pi (Xt}<Pi (x ij m) i This suggests introducing the diagonal matrix (A)ij = AiOij and the 'design matrix' (<J?li = <Pi (xt), so that (k(x)k(x)T)x = <J>A2<J>T. One then also has (C(x,x)}x = tr A, and the matrix K is expressed as K = a 21 + <J>A<J>T, 1 being the identity matrix. Collecting these results, we have ?(D) = tr A - tr (a 21 + <J>A<J>T)-I<J>A 2<J>T This can be simplified using the Woodbury formula for matrix inverses (see e.g. [7]), which applied to our case gives (a 2I+<J>A<J>T)-1 = a- 2[I-<J>(a 21+A<J>T<J?-1 A<J>TJ; after a few lines of algebra, one then obtains the final result t= (t(D))D' ?(D) =tra 2A(a 21+A<J>T<J?-1 =tr(A- 1 +a- 2 <J>T<J?-1 (6) This exact representation of the generalization error is one of the main results of this paper. Its advantages are that the average over the test input X has already been carried out, and that the remainingf dependence on the training data is contained entirely in the matrix <J> T <J>. It also includes as a special case the well-known result for linear regression (see e.g. [8]); A-I and <J> T <J> can be interpreted as suitably generalized versions of the weight decay (matrix) and input correlation matrix. Starting from (6), one can now derive approximate expressions for the learning 347 Learning Curves for Gaussian Processes curve I:(n). The most naive approach is to entirely neglect the fluctuations in cJ>TcJ> over different data sets and replace it by its average, which is simply (( cJ> T cJ> )ij ) D = I:l (?i(Xt)?j(XI)) D = n8ij . This leads to the Naive approximation I:N(n) = tr (A -1 + O'- 2 nI)-1 (7) which is not, in general, very good. It does however become exact in the large noise limit 0'2 -t 00 at constant nlO' 2 : The fluctuations of the elements of the matrix O'- 2cJ>TcJ> then become vanishingly small (of order foO'- 2 = (nlO' 2 )/fo -t 0) and so replacing cJ> T cJ> by its average is justified. To derive better approximations, it is useful to see how the matrix 9 = (A -1 + O'- 2cJ>TcJ?-1 changes when a new example is added to the training set. One has 9(n + 1) - 9(n) = [9- 1 (n) + O'- 2 1j11j1 T r l - 9(n) =_ 9(n)1jI1jI T 9(n) + 1jIT 9(n)1jI (8) 0'2 in terms of the vector 1jI with elements (1jI)i = ?i(Xn+I); the second identity uses again the Woodbury formula. To get exact learning curves, one would have to average this update formula over both the new training input Xn+1 and all previous ones. This is difficult, but progress can be made by again neglecting some fluctuations: The average over Xn +1 is approximated by replacing 1jI1jIT by its average, which is simply the identity matrix; the average over the previous training inputs by replacing 9(n) by its average G(n) = (9(n)) D' This yields the approximation G 2 (n) G(n + 1) - G(n) = - 2 G() (9) a +tr n Iterating from G(n = 0) = A, one sees that G(n) remains diagonal for all n, and so (9) is trivial to implement numerically. I call the resulting I:D(n) = tr G(n) the Discrete approximation to the learning curve, because it still correctly treats n as a variable with discrete, integer values. One can further approximate (9) by taking n as continuously varying, replacing the difference on the left-hand side by the derivative dG( n) 1dn. The resulting differential equation for G( n) is readily solved; taking the trace, one obtains the generalization error I:uc(n) = tr (A -1 + O'- 2 n'I)-1 (10) with n' determined by the self-consistency equation n' + tr In(I + O'- 2 n' A) = n. By comparison with (7), n' can be thought of as an 'effective number of training examples'. The subscript DC in (10) stands for Upper Continuous approximation. As the name suggests, there is another, lower approximation also derived by treating n as continuous. It has the same form as (10), but a different self-consistent equation for n', and is derived as follows. Introduce an auxiliary offset parameter v (whose usefulness will become clear shortly) by 9- 1 = vI+A -1 +O'- 2cJ>TcJ>; at the end ofthe calculation, v will be set to zero again. As before, start from (8)-which also holds for nonzero v-and approximate 1jI1jIT and tr 9 by their averages, but retain possible fluctuations of 9 in the numerator. This gives G(n+ 1) - G(n) = - (9 2 (n)) 1[0'2 + tr G(n)]. Taking the trace yields an update formula for the generalization error 1:, where the extra parameter v lets us rewrite the average on the right-hand side as -tr (9 2 ) = (olov)tr (9) = ol:lov. Treating n again as continuous, we thus arrive at the partial differential equation Eh{on = (oI:l ov) 1 (0'2 + 1:). This can be solved using the method of characteristics [8 and (for v = 0) gives the Lower Continuous approximation to the learning curve, I:Lc(n) = tr (A -1 + O'- 2 n'I)-1 , n' = nO' 2 0'2 + I:LC (11) By comparing derivatives w.r.t. n, it is easy to show that this is always lower than the DC approximation (10). One can also check that all three approximations that I have derived (D, LC and DC) converge to the exact result (7) in the large noise limit as defined above. P. Sol/ich 348 3 COMPARISON WITH BOUNDS AND SIMULATIONS I now compare the D, LC and UC approximations with existing bounds, and with the 'true' learning curves as obtained by simulations. A lower bound on the generalization error was given by Michelli and Wahba [2J as ?(n) ~ ?Mw(n) = 2::n+l Ai This is derived for the noiseless jections of 0* (x) along the first to be tight for the case of 'real' information theoretic methods, a (12) case by allowing 'generalized observations' (pron eigenfunctions of C (x, x') ), and so is unlikely observations at discrete input points. Based on different Lower bound was obtained by Opper [3J: 1 ?(n) ~ ?Lo(n) = 4"tr (A -1 + 2a- 2 nl)-1 x [I + (I + 2a- 2 nA)-lJ This is always lower than the naive approximation (7); both incorrectly suggest that ? decreases to zero for a 2 -+ 0 at fixed n, which is clearly not the case (compare (12)). There is also an Upper bound due to Opper [3J, i(n) ~ ?uo(n) = (a- 2 n)-1 tr In(1 + a- 2 nA) + tr (A -1 + a- 2 nl)-1 (13) Here i is a modified version of ? which (in the rescaled version that I am using) becomes identical to ? in the limit of small generalization errors (? ? a 2 ), but never gets larger that 2a 2 ; for small n in particular, ?(n) can therefore actually be much larger than i(n) and its bound (13). An upper bound on ?(n) itself was derived by Williams and Vivarelli [4J for one-dimensional inputs and stationary covariance functions (for which C(x, x') is a function of x - x' alone). They considered the generalization error at x that would be obtained from each individual training example, and then took the minimum over all n examples; the training set average of this 'lower envelope' can be evaluated explicitly in terms of integrals over the covariance function [4J. The resulting upper bound, ?wv(n), never decays below a 2 and therefore complements the range of applicability of the UO bound (13). In the examples in Fig. 1, I consider a very simple input domain, x E [0, 1Jd, with a uniform input distribution. I also restrict myself to stationary covariance functions, and in fact I use what physicists call periodic boundary conditions. This is simply a trick that makes it easy to calculate the required eigenvalue spectra of the covariance function, but otherwise has little effect on the results as long as the length scale of the covariance function is smaller than the size of the input domain 2 , l ? 1. To cover the two extremes of 'rough' and 'smooth' Gaussian priors, I consider the OU [C(x, x') = exp( -lx-xll/l)J and SE [C(x, x') = exp( -lx-x' 12 /2l 2 )J covariance functions. The prior variance of the values of the function to be learned is simply C (x, x) = 1; one generically expects this 'prior ignorance' to be significantly larger than the noise on the training data, so I only consider values of a 2 < 1. I also fix the covariance function length scale to l = 0.1; results for l = 0.01 are qualitatively similar. Several observations can be made from Figure 1. (1) The MW lower bound is not tight, as expected. (2) The bracket between Opper's lower and upper bounds (LO /UO) is rather wide (1-2 orders of magnitude); both give good representations of the overall shape of the learning curve only in the asymptotic regime (most clearly visible for the SE covariance function), i. e., once ? has dropped below a 2 . (3) The WV upper bound (available only in d = 1) works 21n d = 1 dimension, for example, a 'periodically continued' stationary covariance function on [0,1] can be written as C(X,X') = 2:::_ooc(x - x' + r). For I ? 1, only the r = 0 term makes a significant contribution, except when x and x' are within ~ I of opposite ends of the input space. With this definition, the eigenvalues of C(x, x') are given dx c(x) exp( -2rriqx), for integer q. by the Fourier transform 1:"00 349 Learning Curves for Gaussian Processes 2 OU, d=l, 1=0.1 , cr =10 -3 2 -3 SE, d=l, 1=0.1, cr =10 10? 10? E (b) 10-1 10- 1 -_WV 10-2 10-3 10-2 \. MW--y~ - - -- ' ,L? ,\ 'i- , 10-4 , - __ 'MW --- \ 10-3 10-5 0 200 400 600 ~O 50 0 2 100 200 150 2 OU, d=l, 1=0.1, cr =0.1 SE, d=l , 1=0.1, cr =0.1 10? 10? (c) E ___ 10- 1 -~~-::::.-::::.- 10-1 D/uC , \\~ WV 10-2 --VO LC ''''',--- 0 ,- - - - - - - - - - - --- - - _ _ -VO- , 10-3 '-.\. 10-2 '-. (d) wv - _ _ -l-O IMW - ..!-O- - - -- 10-4 200 400 2 OU, d=2, 1=0.1, cr =10 600 0 200 -3 400 2 600 -3 SE, d=2, 1=0.1, cr =10 10? 10? (e) E \. '-. -- -- - -~ LC vo--- --- 10- 1 \ 0 10-2 10-3 , - ___ ~o 200 \. - n 400 - -- 600 '.MW , , \ , \~---,~- 10-4 -- - 10-2 (D 10-1 DIVC \ 10-5 0 200 n 400 600 Figure 1: Learning curves c(n): Comparison of simulation results (thick solid lines; the small fluctuations indicate the order of magnitude of error bars), approximations derived in this paper (thin solid lines; D = discrete, UC/LC = upper/lower continuous) , and existing upper (dashed; UO = upper Opper, WV = Williams-Vivarelli) and lower (dot-dashed; LO = lower Opper, MW = Michelli-Wahba) bounds. The type of covariance function (Ornstein-Uhlenbeck/Squared Exponential), its length scale l, the dimension d of the input space, and the noise level (72 are as shown. Note the logarithmic y-axes. On the scale of the plots, D and UC coincide (except in (b)); the simulation results are essentially on top of the LC curve in (c-e) . 350 P'Sollich well for the OU covariance function, but less so for the SE case. As expected, it is not useful in the asymptotic regime because it always remains above (72. (4) The discrete (D) and upper continuous (UC) approximations are very similar, and in fact indistinguishable on the scale of most plots. This makes the UC version preferable in practice, because it can be evaluated for any chosen n without having to step through all smaller values of n. (5) In all the examples, the true learning curve lies between the UC and LC curves. In fact I would conjecture that these two approximations provide upper and lower bounds on the learning curves, at least for stationary covariance functions. (6) Finally, the LC approximation comes out as the clear winner: For (72 = 0.1 (Fig. 1c,d), it is indistinguishable from the true learning curves. But even in the other cases it represents the overall shape of the learning curves very well, both for small n and in the asymptotic regime; the largest deviations occur in the crossover region between these two regimes. In summary, I have derived an exact representation of the average generalization c error of Gaussian processes used for regression, in terms of the eigenvalue decomposition of the covariance function. Starting from this, I have obtained three different approximations to the learning curve c(n) . All of them become exact in the large noise limit; in practice, one generically expects the opposite case ((72 /C(x, x) ? 1), but comparison with simulation results shows that even in this regime the new approximations perform well. The LC approximation in particular represents the overall shape of the learning curves very well, both for 'rough' (OU) and 'smooth' (SE) Gaussian priors, and for small as well as for large numbers of training examples n. It is not perfect, but does get substantially closer to the true learning curves than existing bounds. Future work will have to show how well the new approximations work for non-stationary covariance functions and/or non-uniform input distributions, and whether the treatment of fluctuations in the generalization error (due to the random selection of training sets) can be improved, by analogy with fluctuation corrections in linear perceptron learning [8]. Acknowledgements: I would like to thank Chris Williams and Manfred Opper for stimulating discussions, and for providing me with copies of their papers [3,4] prior to publication. I am grateful to the Royal Society for financial support through a Dorothy Hodgkin Research Fellowship. [1] See e.g. D J C MacKay, Gaussian Processes, Tutorial at NIPS 10, and recent papers by Goldberg/Williams/Bishop (in NIPS 10), Williams and Barber/Williams (NIPS 9), Williams/Rasmussen (NIPS 8). [2] C A Michelli and G Wahba. Design problems for optimal surface interpolation. In Z Ziegler, editor, Approximation theory and applications, pages 329-348. Academic Press, 1981. [3] M Opper. Regression with Gaussian processes: Average case performance. In I K Kwok-Yee, M Wong, and D-Y Yeung, editors, Theoretical Aspects of Neural Computation: A Multidisciplinary Perspective. Springer, 1997. [4] C K I Williams and F Vivarelli. An upper bound on the learning curve for Gaussian processes. Submitted for publication. [5] 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. Kluwer Academic. In press. [6] E Wong. Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York, 1971. [7] W H Press, S A Teukolsky, W T Vetterling, and B P Flannery. Numerical Recipes in C (2nd ed.). Cambridge University Press, Cambridge, 1992. [8] P Sollich. Finite-size effects in learning and generalization in linear perceptrons. Journal of Physics A, 27:7771- 7784, 1994. 

1501 |@word version:4 nd:1 suitably:1 simulation:5 decomposition:4 covariance:21 tr:17 solid:2 existing:7 comparing:1 dx:1 written:1 readily:1 visible:1 periodically:1 numerical:1 realistic:1 shape:3 drop:1 interpretable:1 update:2 treating:2 plot:2 stationary:5 alone:1 manfred:1 lx:4 dn:1 along:1 become:5 differential:2 introduce:1 lov:1 expected:6 ol:1 decomposed:1 little:1 considering:1 becomes:1 underlying:1 what:1 interpreted:1 substantially:2 lone:1 collecting:1 preferable:1 uk:2 uo:4 before:1 dropped:2 treat:1 limit:4 physicist:1 id:3 subscript:1 fluctuation:7 interpolation:1 yd:1 plus:1 studied:1 specifying:1 suggests:2 range:1 woodbury:2 practice:2 implement:1 ooc:1 asymptotics:1 crossover:1 significantly:2 thought:1 suggest:1 get:5 selection:1 yee:1 wong:2 map:1 yt:1 straightforward:1 williams:9 starting:4 l:1 continued:1 financial:1 aiaj:1 target:1 exact:8 us:1 goldberg:1 hypothesis:3 trick:1 element:2 approximated:1 solved:2 worst:1 calculate:1 region:1 sol:1 decrease:1 rescaled:1 asked:1 oed:1 signature:1 ov:1 tight:3 rewrite:1 algebra:1 grateful:1 predictive:4 tcj:4 easily:1 joint:3 fast:1 effective:1 london:2 monte:1 tell:1 whose:1 encoded:2 larger:3 valued:1 solve:1 otherwise:2 transform:1 noisy:1 itself:1 final:1 advantage:2 eigenvalue:7 differentiable:2 took:1 vanishingly:1 relevant:1 achieve:1 recipe:1 produce:1 perfect:1 tk:2 illustrate:1 derive:4 ac:2 ij:3 progress:1 eq:4 auxiliary:1 imw:1 come:2 indicate:1 differ:1 thick:1 correct:2 stochastic:1 myself:1 wc2r:1 behaviour:2 transparent:1 generalization:21 fix:1 correction:1 hold:1 considered:1 exp:5 lm:1 vary:1 a2:1 ziegler:1 largest:1 rough:3 xtr:1 clearly:2 gaussian:27 always:3 aim:1 modified:1 rather:1 cr:6 varying:1 publication:2 derived:9 ax:1 likelihood:1 check:1 am:3 inference:4 dependent:2 vetterling:1 lj:2 unlikely:1 overall:3 integration:1 special:1 uc:8 mackay:1 once:1 never:2 having:1 identical:1 represents:2 thin:1 future:1 few:2 dg:1 individual:1 consisting:1 generically:2 extreme:1 nl:2 bracket:1 tj:1 integral:2 closer:3 worker:1 neglecting:1 partial:1 old:1 michelli:3 theoretical:1 cover:1 applicability:1 introducing:1 deviation:3 expects:2 predictor:1 usefulness:1 uniform:2 teacher:5 periodic:1 my:1 retain:1 physic:2 yl:2 continuously:1 na:2 squared:4 again:5 opposed:1 derivative:2 li:1 includes:1 tra:1 explicitly:1 depends:2 vi:1 pron:1 ornstein:1 start:1 di9:2 complicated:1 contribution:1 oi:1 ni:1 variance:4 characteristic:1 yield:3 identify:1 ofthe:1 bayesian:1 carlo:1 corruption:1 submitted:1 fo:1 ed:1 email:2 definition:1 sampled:1 treatment:1 cj:8 ou:7 actually:2 improved:1 done:2 evaluated:2 just:1 correlation:1 hand:4 replacing:5 multidisciplinary:1 name:2 effect:2 normalized:1 true:6 analytically:1 hence:1 symmetric:1 nonzero:1 deal:1 ignorance:1 indistinguishable:2 numerator:1 self:2 uniquely:1 generalized:2 trying:1 hill:1 theoretic:1 vo:3 oid:1 ornsteinuhlenbeck:1 ji:4 conditioning:1 winner:1 kluwer:1 numerically:1 refer:1 significant:1 cambridge:2 ai:3 smoothness:1 consistency:1 mathematics:1 dot:1 surface:1 deduce:1 add:1 posterior:5 recent:1 perspective:1 irrelevant:1 certain:1 wv:6 captured:1 minimum:1 converge:1 dashed:2 smooth:2 academic:2 calculation:3 nlo:2 long:1 controlled:1 prediction:6 regression:7 noiseless:1 essentially:1 yeung:1 uhlenbeck:1 justified:1 fellowship:1 dorothy:1 crucial:1 extra:1 envelope:1 eigenfunctions:3 subject:2 jordan:1 call:2 integer:2 mw:6 feedforward:2 easy:2 wahba:3 restrict:1 opposite:2 intensive:1 whether:1 expression:3 peter:2 york:1 generally:1 useful:2 se:8 iterating:1 clear:2 simplest:2 tutorial:1 correctly:1 write:1 discrete:5 pj:3 clean:2 year:1 inverse:1 parameterized:1 ich:1 hodgkin:1 arrive:1 eh9:1 bound:20 entirely:2 occur:1 encodes:1 fourier:1 aspect:1 relatively:1 conjecture:1 department:2 smaller:2 sollich:5 taken:1 computationally:1 equation:4 remains:2 vivarelli:3 excitement:1 needed:1 tractable:1 end:2 available:1 apply:2 yare:1 kwok:1 distinguished:1 alternative:1 shortly:1 jd:1 top:1 graphical:1 qji:1 calculating:1 neglect:1 society:1 question:1 already:2 quantity:1 added:1 dependence:1 diagonal:2 distance:1 thank:1 chris:1 me:1 barber:1 trivial:2 reason:1 jit:1 length:5 providing:1 difficult:2 trace:2 design:2 xll:1 perform:2 allowing:1 upper:12 observation:3 finite:2 incorrectly:1 dc:3 community:1 complement:1 pair:1 required:1 specified:1 learned:3 nip:4 address:1 beyond:1 bar:2 below:4 dynamical:1 xm:1 regime:5 royal:1 analogue:1 eh:1 oij:1 scheme:1 brief:1 realvalued:1 lk:1 carried:1 naive:3 prior:12 review:1 l2:1 acknowledgement:1 asymptotic:4 expect:1 analogy:1 consistent:1 mercer:1 editor:3 pi:7 lo:3 summary:2 last:1 copy:1 rasmussen:1 side:2 perceptron:1 wide:1 comprehensively:1 taking:3 edinburgh:2 curve:27 dimension:3 xn:3 stand:1 opper:7 boundary:1 commonly:1 made:2 qualitatively:1 simplified:1 coincide:1 far:1 approximate:4 obtains:2 mcgraw:1 assumed:2 xi:2 spectrum:1 continuous:6 why:1 jz:1 learn:2 obtaining:1 necessarily:1 complex:1 domain:3 main:2 noise:8 fig:2 foo:1 lc:11 exponential:2 xl:3 lie:1 ix:2 theorem:1 formula:4 xt:3 bishop:1 offset:1 decay:2 adding:1 magnitude:2 flannery:1 logarithmic:1 simply:8 infinitely:1 expressed:2 strand:1 contained:1 scalar:1 springer:1 corresponds:1 truth:1 extracted:1 teukolsky:1 stimulating:1 identity:3 king:1 replace:1 change:1 typical:1 determined:2 except:2 averaging:3 called:2 perceptrons:1 college:1 support:1 ex:3

Adding Constrained Discontinuities to Gaussian Process Models of Wind Fields Ian T. Nabney Christopher K. I. Williams t Neural Computing Research Group Aston University, BIRMINGHAM, B4 7ET, UK d.comford@aston.ac.uk Dan Cornford* Abstract Gaussian Processes provide good prior models for spatial data, but can be too smooth. In many physical situations there are discontinuities along bounding surfaces, for example fronts in near-surface wind fields. We describe a modelling method for such a constrained discontinuity and demonstrate how to infer the model parameters in wind fields with MCMC sampling. 1 INTRODUCTION We introduce a model for wind fields based on Gaussian Processes (GPs) with 'constrained discontinuities'. GPs provide a flexible framework for modelling various systems. They have been adopted in the neural network community and are interpreted as placing priors over functions. Stationary vector-valued GP models (Daley, 1991) can produce realistic wind fields when run as a generative model; however, the resulting wind fields do not contain some features typical of the atmosphere. The most difficult features to include are surface fronts. Fronts are generated by complex atmospheric dynamics and are marked by large changes in the surface wind direction (see for example Figures 2a and 3b) and temperature. In order to account for such features, which appear discontinuous at our observation scale, we have developed a model for vector-valued GPs with constrained discontinuities which could also be applied to surface reconstruction in computer vision, and geostatistics. In section 2 we illustrate the generative model for wind fields with fronts. Section 3 explains what we mean by GPs with constrained discontinuities and derives the likelihood of data under the model. Results of Bayesian estimation of the model parameters are given, ?To whom correspondence should be addressed. tNowat: Division of Informatics, University of Edinburgh, 5 Forrest Hill, Edinburgh EHI 2QL, Scotland, UK D. Com/ord, I. T. Nabney and C. K. I. Williams 862 using a Markov Chain Monte Carlo (MCMC) procedure. In the final section, the strengths and weaknesses of the model are discussed and improvements suggested. 2 A GENERATIVE WIND FIELD MODEL We are primarily interested in retrieving wind fields from satellite scatterometer observations of the ocean surface!. A probabilistic prior model for wind fields will be used in a Bayesian procedure to resolve ambiguities in local predictions of wind direction. The generative model for a wind field including a front is taken to be a combination of two vector-valued GPs with a constrained discontinuity. A common method for representing wind fields is to put GP priors over the velocity potential ~ and stream function 'It, assuming the processes are uncorrelated (Daley, 1991). The horizontal wind vector u = (u, v) can then be derived from: 8'lt 8y 8~ u=--+-, (1) 8x This produces good prior models for wind fields when a suitable choice of covariance function for ~ and 'It is made. We have investigated using a modified Bessel function based covariance2 (Handcock and Wallis, 1994) but found, using three years of wind data for the North Atlantic, that the maximum a posteriori value for the smoothness paramete~ in this covariance function was'" 2.5. Thus we used the correlation function: p(r) = (1 + .!:. + ~) L 3L2 exp (-.!:..) L (2) where L is the correlation length scale, which is equivalent to the modified Bessel function and less computationally demanding (Cornford, 1998). N Simulate Frontal Position. Orientation and Direction Simulate Along Both Sides of Front using GPl Simulate 'Mnd Raids Either Side of Front Conditionally on that Sides Frontal 'Mnds using GP2 (a) Origin (b) Figure 1: (a) Flowchart describing the generative frontal model. See text for full description. (b) A description of the frontal model. The generative model has the form outlined in Figure 1a. Initially the frontal position and orientation are simulated. They are defined by the angle clockwise from north (?/) that the front makes and a point on the line (x/, Y/). Having defined the position of the front, lS~ http://www.ncrg.aston.ac.uk/Projects/NEUROSAT/NEUROSAT.htm1 for details of the scatterometer work. Technical reports describing, in more detail, methods for generating prior wind field models can also be accessed from the same page. 2The modified Bessel function allows us to control the differentiability of the sample realisations through the 'smoothness parameter', as well as the length scales and variances. 3This varies with season, but is the most temporally stable parameter in the covariance function. 863 Adding Constrained Discontinuities to GP Models o/Wind Fields the angle of the wind across the front (a J) is simulated from a distribution covering the range [0,71"). This angle is related to the vertical component of vorticity ?() across the front through ( = k? V x u ex: cos and the constraint a J E [0,71") ensures cyclonic vorticity at the front. It is assumed that the front bisects a J. The wind speed (8 J) is then simulated at the front. Since there is generally little change in wind speed across the front, one value is simulated for both sides of the front. These components 8 f = (? J , x J , YJ, a J, 8 J) define the line of the front and the mean wind vectors just ahead of and just behind the front (Figure Ib): (? ) A realistic model requires some variability in wind vectors along the front. Thus we use a GP with a non-zero mean (mla or mlb) along the line of the front. In the real atmosphere we observe a smaller variability in the wind vectors along the line of the front compared with regions away from fronts . Thus we use different GP parameters along the front (G Pl ), from those used in the wind field away from the front (GP2 ), although the same GPl parameters are used both sides of the front, just with different means. The winds just ahead of and behind the front are assumed conditionally independent given ml a and mlb, and are simulated at a regular 50 km spacing. The final step in the generative model is to simulate wind vectors using G P2 in both regions either side of the front, conditionally on the values along that side of the front. This model is flexible enough to represent fronts, yet has the required constraints derived from meteorological principles, for example that fronts should always be associated with cyclonic vorticity and that discontinuities at the model scale should be in wind direction but not in wind speed4 . To make this generative model useful for inference, we need to be able to compute the data likelihood, which is the subject of the next section. 3 GPs WITH CONSTRAINED DISCONTINUITIES " . ;.... -]. 1 . ! I D2 > Dl (a) (b) Figure 2: (a) The discontinuity in one ofthe vector components in a simulation. (b) Framework for GPs with boundary conditions. The curve Dl has nl sample points with values Zt. The domain D2 has n2 points with values Z2. 4The model allows small discontinuities in wind speed, which are consistent with frontal dynamics. D. Cornford, 1. T Nabney and C. K. 1. Williams 864 We consider data from two domains D1 and D2 (Figure 2b), where in this case D1 is a curve in the plane which is intended to be the front and D2 is a region of the plane. We obtain n1 variables Zl at points Xl along the curve, and we assume these are generated under G P1 (a GP which depends on parameters 81 and has mean m1 = m1l which will be determined by (3) or (4?. We are interested in determining the likelihood of the variables Z2 observed at n2 points X2 under GP2 which depends on parameters 82, conditioned on the 'constrained discontinuities' at the front. We evaluate this by calculating the likelihood of Z2 conditioned on the n1 values of Zl from G P1 along the front and marginalising out Zl: i: p(Z2182,81) = p(Z2I Z 1,82,81,m1)p(ZlI81,m1) dZ1. (5) From the definition ofthe likelihood of a GP (Cressie, 1993) we find: p(Z2IZ1,82,81,m1) = ~ (271") ISd'2 1 exp 2 (--21Z;'S2;lZ;) (6) where: To understand the notation consider the joint distribution of Zl, Z2 and in particular its covariance matrix: (7) where K 1112 is the n1 x n1 covariance matrix between the points in D1 evaluated using 8 2, K1212 = K~112 the n1 x n2 (cross) covariance matrix between the points in D1 and D2 evaluated using 8 2 and K2212 is the usual n2 x n2 covariance for points in D 2. Thus we can see that S22 is the n2 x n2 modified covariance for the points in D2 given the points along D 1 , while the Z; is the corrected mean that accounts for the values at the points in D 1 ? which have non-zero mean. We remove the dependency on the values Zl by evaluating the integral in (5). p(ZlI8 1, m1) is given by: p(ZlI81, m1) = (271") ~1 1. IK111112 exp (--21(Zl - m1)' Kill1 (Zl - m 1 ?) (8) where K 1111 is the n1 x n1 covariance matrix between the points in D1 evaluated under the covariance given by 8 1 . Completing the square in Zl in the exponent, the integral (5) can be evaluated to give: p (z 188m ) 2 2, 1, 1 - 1 (271")~ _1_ 1 _1_ x IS221 t IK11111t IBlt exp (~ (C' B- 1C - (9) Z2' S2;l Z2 - m1' Kill1 m1) ) where: B C' ' K-1 )'S-lK' K- 1 K- 1 (K 1212 1112 22 1212 1112 + 1111 1 1 Z 2'S-lK' 22 1212 K1112 + m1 'K1111 The algorithm has been coded in MATLAB and can deal with reasonably large numbers of points quickly. For a two dimensional vector-valued GP with n1 = 12 and n2 = 200 5 and 5This is equivalent to nl = 24 and n2 = 400 for a scalar GP. 865 Adding Constrained Discontinuities to GP Models of Wind Fields a covariance function given by (2), computation of the log likelihood takes 4.13 seconds on an SGI Indy R5000. The mean value just ahead and behind the front define the mean values for the constrained discontinuity (i.e. m1 in (9?. Conditional on the frontal parameters the wind fields either side (Figure 3a) are assumed independent: p(Z2a, Z2b\02, 01, Of) = p(Z2a\02, 01, m1a)p(m1a\Of) x p(Z2b\02, 01, m1b)p(m1b\Of) where we have performed the integration (5) to remove the dependency on Z1a and Z1b. Thus the likelihood of the data Z2 = (Z2a, Z2b) given the model parameters O2,01 , Of is simply the product of the likelihoods of two GPs with a constrained discontinuity which can be computed using (9). SOIl _ _- . . . " " " " , , - - "" ---' ......,"--.............. ," -von " , .... , ' - - - - _II :::: .- ,!. 100 --- ' , .... , , -- - - - , ''I. \ , - - -.... , - ' ,,,'\--_ "" _-......... ", , ' \ \, \, - - - ..... , , , , \, ' " _--....'''''' " I , "DC Front (a) (b) Figure 3: (a) The division of the wind field using the generative frontal model. Z1a, Z1b are the wind fields just ahead and behind the front, along its length, respectively. Z2a, Z2b are the wind fields in the regions ahead of and behind the front respectively. (b) An example from the generative frontal model: the wind field looks like a typical 'cold front'. The model outlined above was tested on simulated data generated from the model to assess parameter sensitivity. We generated a wind field ZO = (Z2a' Z2b) using known model parameters (e.g. Figure 3b). We then sampled the model parameters from the posterior distribution: (10) where p( ( 2), p( ( 1 ), p( Of) are prior distributions over the parameters in the GPs and front models. This brings out one advantage of the proposed model. All the model parameters have a physical interpretation and thus expert knowledge was used to set priors which produce realistic wind fields. We will also use (10) to help set (hyper)priors using real data in Zoo MCMC using the Metropolis algorithm (Neal, 1993) is used to sample from (to) using the NETLAB 6 library. Convergence of the Markov chain is currently assessed using visual inspection of the univariate sample paths since the generating parameters are known, although other diagnostics could be used (Cowles and Carlin, 1996). We find that the procedure is insensitive to the initial value of the GP parameters, but that the parameters describing the location ofthe front (1/>" d,) need to be initialised 'close' to the correct values if the chain is to converge on a reasonable time-scale. In the application some preliminary analysis of the wind field would be necessary to identify possible fronts and thus set the initial parameters to 'sensible' values. We intend to fit a vector-valued GP without any discontinuities 6Available from http://www.ncrg.aston.ac . uk/netlab/index. html. 866 D. Comjord, I. T. Nabney and C. K. 1. Williams 2 3 Sample nurrber 4 ' 5 2 3 4 Sample number ? In' w 104 (b) (a) Figure 4: Examples from the Markov chain of the posterior distribution (10). (a) The energy = negative log posterior probability. Note that the energy when the chain was initialised was 2789 and the first 27 values are outside the range of the y-axis. (b) The angle of the front relative to north (?> I) ' and then measure the 'strain' or misfit of the locally predicted winds with the winds fitted by the GP. Lines of large 'strain' will be used to initialise the front parameters. 3000 1000 500 2 3 sample number (a) ~ ~- ~~-an1.5~uw~2ww~2.~5~~3L-~3. 5 Angle of wind (radians) (b) Figure 5: Examples from the Markov chain of the posterior distribution (10). (a) The angle of the wind across the front (01 ). (b) Histogram of the posterior distribution of 01 allowing a 10000 iteration bum-in period. Examples of samples from the Markov chain from the simulated wind field shown in Figure 3a can be seen in Figures 4 and 5. Figure 4a shows that the energy level (= negative log posterior probability) falls very rapidly to near its minimum value from its large starting value of 2789. In these plots the true parameters for the front were ?> I = 0.555,01 = 2.125 while the initial values were set at ?>I = 0.89,01 = 1.49. Other parameters were also incorrectly set. The Metropolis algorithm seems to be able to find the minimum and then stays in it. Figure 4b and 5a show the Markov chains for ?>I and 0/ ' Both converge quickly to an apparently stationary distributions, which have mean values very close to the 'true' generating parameters. The histogram of the distribution of 01 is shown in Figure 5b. Adding Constrained Discontinuities to GP Models of Wind Fields 867 4 DISCUSSION AND CONCLUSIONS Simulations from our model are meteorologically plausible wind fields which contain fronts. It is possible similar models could usefully be applied to other modelling problems where there are discontinuities with known properties. A method for the computation of the likelihood of data given two GP models, one with non-zero mean on the boundary and another in the domain in which the data is observed, has been given. This allows us to perform inference on the parameters in the frontal model using a Bayesian approach of sampling from the posterior distribution using a MCMC algorithm. There are several weaknesses in the model specifically for fronts, which could be improved with further work. Real atmospheric fronts are not straight, thus the model would be improved by allowing 'curved' fronts. We could represent the position of the front, oriented along the angle defined by ?, using either another smooth GP, B-splines or possibly polynomials. Currently the points along the line of the front are simulated at the mean observation spacing in the rest of the wind field ('" 50 km). Interesting questions remain about the (in-fill) asymptotics (Cressie, 1993) as the distance between the points along the front tends to zero. Empirical evidence suggests that as long as the spacing along the front is 'much less' than the length scale of the GP along the front (which is typically'" 1000 km) then the spacing does not significantly affect the results. Although we currently use a Metropolis algorithm for sampling from the Markov chain, the derivative of (9) with respect to the GP parameters 8 1 and 8 2 could be computed analytically and used in a hybrid Monte Carlo procedure (Neal, 1993). These improvements should lead to a relatively robust procedure for putting priors over wind fields which will be used with real data when retrieving wind vectors from scatterometer observations over the ocean. Acknowledgements This work was partially supported by the European Union funded NEUROSAT programme (grant number ENV 4 CT96-0314) and also EPSRC grant GRlL03088 Combining Spatially Distributed Predictions from Neural Networks. References Cornford, D. 1998. Flexible Gaussian Process Wind Field Models. Technical Report NCRG/98/017, Neural Computing Research Group, Aston University, Aston Triangle, Birmingham, UK. Cowles, M. K. and B. P. Carlin 1996. Markov-Chain Monte-Carlo Convergence Diagnostics-A Comparative Review. Journal of the American Statistical Association 91, 883-904. Cressie, N. A. C. 1993. Statistics for Spatial Data. New York: John Wiley and Sons. Daley, R. 1991. Atmospheric Data Analysis. Cambridge: Cambridge University Press. Handcock, M. S. and J. R. Wallis 1994. An Approach to Statistical Spatio-Temporal Modelling of Meteorological Fields. Journal of the American Statistical Association 89, 368-378. Neal, R. M. 1993. Probabilistic Inference Using Markov Chain Monte Carlo Methods. Technical Report CRG-TR-93-1, Department of Computer Science, University of Toronto. URL: http://www.cs.utoronto.ca/ ... radford. 

1502 |@word polynomial:1 seems:1 km:3 d2:6 simulation:2 dz1:1 covariance:11 tr:1 initial:3 o2:1 atlantic:1 com:1 z2:7 yet:1 john:1 realistic:3 remove:2 plot:1 stationary:2 generative:10 plane:2 inspection:1 scotland:1 location:1 toronto:1 accessed:1 along:16 mla:1 retrieving:2 dan:1 gpl:2 introduce:1 p1:2 scatterometer:3 resolve:1 little:1 project:1 notation:1 what:1 interpreted:1 developed:1 temporal:1 usefully:1 uk:6 control:1 zl:8 grant:2 appear:1 local:1 tends:1 path:1 suggests:1 co:1 range:2 yj:1 union:1 procedure:5 cold:1 asymptotics:1 empirical:1 significantly:1 regular:1 close:2 put:1 www:3 equivalent:2 williams:4 starting:1 l:1 d1:5 fill:1 initialise:1 gps:9 cressie:3 origin:1 velocity:1 observed:2 epsrc:1 cornford:4 region:4 ensures:1 dynamic:2 division:2 triangle:1 joint:1 various:1 zo:1 describe:1 monte:4 hyper:1 outside:1 valued:5 z2i:1 plausible:1 statistic:1 gp:18 final:2 advantage:1 reconstruction:1 product:1 combining:1 rapidly:1 description:2 sgi:1 convergence:2 satellite:1 produce:3 generating:3 comparative:1 help:1 illustrate:1 ac:3 ex:1 p2:1 indy:1 predicted:1 c:1 direction:4 discontinuous:1 correct:1 explains:1 atmosphere:2 preliminary:1 crg:1 pl:1 exp:4 nurrber:1 estimation:1 birmingham:2 currently:3 bisects:1 gaussian:4 always:1 modified:4 season:1 derived:2 improvement:2 modelling:4 likelihood:9 posteriori:1 inference:3 typically:1 initially:1 interested:2 flexible:3 orientation:2 html:1 exponent:1 constrained:13 spatial:2 integration:1 field:32 having:1 sampling:3 env:1 placing:1 s22:1 look:1 report:3 spline:1 realisation:1 primarily:1 oriented:1 intended:1 n1:8 weakness:2 nl:2 diagnostics:2 behind:5 z2b:5 bum:1 chain:11 integral:2 necessary:1 fitted:1 front:54 too:1 dependency:2 varies:1 ehi:1 sensitivity:1 stay:1 probabilistic:2 informatics:1 quickly:2 von:1 ambiguity:1 possibly:1 expert:1 derivative:1 american:2 account:2 potential:1 north:3 depends:2 stream:1 performed:1 wind:51 apparently:1 ass:1 square:1 variance:1 ofthe:3 identify:1 misfit:1 bayesian:3 carlo:4 zoo:1 straight:1 definition:1 energy:3 initialised:2 associated:1 radian:1 sampled:1 knowledge:1 improved:2 evaluated:4 marginalising:1 just:6 correlation:2 horizontal:1 christopher:1 meteorological:2 brings:1 contain:2 true:2 analytically:1 spatially:1 neal:3 deal:1 conditionally:3 covering:1 hill:1 m1l:1 demonstrate:1 temperature:1 common:1 physical:2 b4:1 insensitive:1 ncrg:3 discussed:1 interpretation:1 m1:11 association:2 cambridge:2 smoothness:2 outlined:2 handcock:2 mlb:2 funded:1 stable:1 surface:6 vorticity:3 posterior:7 seen:1 minimum:2 gp2:3 converge:2 period:1 bessel:3 clockwise:1 ii:1 full:1 infer:1 smooth:2 technical:3 cross:1 long:1 coded:1 prediction:2 vision:1 histogram:2 represent:2 iteration:1 spacing:4 addressed:1 rest:1 subject:1 near:2 enough:1 affect:1 carlin:2 fit:1 url:1 york:1 matlab:1 generally:1 useful:1 locally:1 differentiability:1 http:3 group:2 putting:1 uw:1 isd:1 year:1 run:1 angle:7 reasonable:1 forrest:1 z2a:5 netlab:2 completing:1 nabney:4 correspondence:1 strength:1 ahead:5 constraint:2 x2:1 speed:3 simulate:4 relatively:1 department:1 combination:1 across:4 smaller:1 meteorologically:1 remain:1 son:1 metropolis:3 taken:1 computationally:1 describing:3 adopted:1 available:1 observe:1 away:2 ocean:2 include:1 calculating:1 intend:1 question:1 mnd:1 usual:1 distance:1 simulated:8 sensible:1 whom:1 assuming:1 length:4 index:1 difficult:1 ql:1 negative:2 zt:1 perform:1 allowing:2 ord:1 observation:4 vertical:1 markov:9 curved:1 incorrectly:1 situation:1 variability:2 strain:2 dc:1 ww:1 community:1 atmospheric:3 required:1 flowchart:1 discontinuity:19 geostatistics:1 able:2 suggested:1 including:1 suitable:1 demanding:1 hybrid:1 representing:1 aston:6 library:1 temporally:1 lk:2 axis:1 text:1 prior:10 review:1 l2:1 acknowledgement:1 determining:1 relative:1 interesting:1 consistent:1 principle:1 uncorrelated:1 soil:1 supported:1 side:8 understand:1 fall:1 edinburgh:2 distributed:1 boundary:2 curve:3 evaluating:1 made:1 lz:1 programme:1 ml:1 assumed:3 spatio:1 an1:1 reasonably:1 robust:1 ca:1 investigated:1 complex:1 european:1 domain:3 bounding:1 s2:2 n2:9 raid:1 wiley:1 position:4 daley:3 xl:1 ib:1 ian:1 utoronto:1 evidence:1 derives:1 dl:2 adding:4 conditioned:2 lt:1 simply:1 univariate:1 visual:1 partially:1 scalar:1 radford:1 conditional:1 marked:1 change:2 typical:2 determined:1 corrected:1 specifically:1 wallis:2 htm1:1 assessed:1 frontal:10 evaluate:1 mcmc:4 tested:1 cowles:2

Learning from Dyadic Data Thomas Hofmann?, Jan Puzicha+, Michael I. Jordan? ? Center for Biological and Computational Learning, M .I.T Cambridge , MA , {hofmann , jordan}@ai.mit.edu + Institut fi.ir Informatik III , Universitat Bonn, Germany, jan@cs.uni-bonn.de Abstract Dyadzc data refers to a domain with two finite sets of objects in which observations are made for dyads , i.e., pairs with one element from either set. This type of data arises naturally in many application ranging from computational linguistics and information retrieval to preference analysis and computer vision . In this paper, we present a systematic, domain-independent framework of learning from dyadic data by statistical mixture models. Our approach covers different models with fiat and hierarchical latent class structures. We propose an annealed version of the standard EM algorithm for model fitting which is empirically evaluated on a variety of data sets from different domains. 1 Introduction Over the past decade learning from data has become a highly active field of research distributed over many disciplines like pattern recognition, neural computation , statistics, machine learning, and data mining. Most domain-independent learning architectures as well as the underlying th eories of learning have been focusing on a feature-based data representation by vectors in an Euclidean space. For this restricted case substantial progress has been achieved. However, a variety of important problems does not fit into this setting and far less advances have been made for data types based on different representations. In this paper, we will present a general framework for unsupervised learning from dyadic data . The notion dyadic refers to a domain with two (abstract) sets of objects, ;r = {Xl , ... , XN} and Y = {YI, ... , YM} in which observations S are made for dyads (Xi, Yk). In the simplest case - on which we focus - an elementary observation consists just of (Xi, Yk) itself, i.e., a co-occurrence of Xi and Yk, while other cases may also provide a scalar value Wik (strength of preference or association). Some exemplary application areas are: (i) Computational linguistics with the corpus-based statistical analysis of word co-occurrences with applications in language modeling , word clustering, word sense disambiguation , and thesaurus construction. (ii) Textbased znJormatzon retrieval, where ,:{, may correspond to a document collection , Y 467 Learningfrom Dyadic Data to keywords , and (Xi, Yk) would represent the occurrence of a term Yk in a document Xi. (iii) Modeling of preference and consumption behavior by identifying X with individuals and Y with obj ects or stimuli as in collaborative jilterzng. (iv) Computer VIS tOn , in particular in the context of image segmentation, where X corresponds to imagE' loc ations , y to discretized or categorical feature values , and a dyad (Xi , Yk) represents a feature Yk observed at a particular location Xi. 2 Mixture Models for Dyadic Data Across different domains there are at least two tasks which playa fundamental role in unsupervised learning from dyadic data: (i) probabilistic modeling, i.e., learning a joint or conditional probability model over X xY , and (ii) structure discovery, e.g. , identifying clusters and data hierarchies. The key problem in probabilistic modeling is the data sparseness: How can probabilities for rarely observed or even unobserved co-occurrences be reliably estimated? As an answer we propose a model-based approach and formulate latent class or mixture models . The latter have the further advantage to offer a unifying method for probabilistic modeling and structure discovery. There are at least three (four, if both variants in (ii) are counted) different ways of defining latent class models: The most direct way is to introduce an (unobserved) mapping c : X X Y --+ {Cl , . . . , CK} that partitions X x Y into K classes. This type of model is called aspect-based and the pre-image c- l (cO') is referred to as an aspect. n. Alternatively, a class can be defined as a subset of one of the spaces X (or Y by symmetry, yielding a different model) , i.e., C : X --+ {Cl, . .. , CK} which induces a unique partitioning on X x Y by C(Xi , yk) == C(Xi) . This model is referr ed to as on e-szded clustering and c-l(c a ) ~ X is called a cluster. Ill. If latent classes are defined for both sets, c : X --+ {ci , .. . , cK} and C : Y --+ {cI , . .. , cD, respectively, this induces a mapping C which is a K . L partitioning of X x y. This model is called two-sided clustering. I. 2.1 Aspect Model for Dyadic Data In order to sp ecify an aspect model we make the assumption that all co-occurrences in the sample set S are i.i .d. and that Xi and Yk are conditionally independent given the class. With parameters P(x i lca ), P(Yklca) for the class-conditional distributions and prior probabilities P( cO' ) the complete data probability can be written as P(S , c) = IT [P(Cik)P(Xilcik)P(Yklcik)t (x"Yk) (1) , i,k where n(xi, Yk) are the empirical counts for dyads in Sand Cik == C(Xi, Yk) . By summing over the latent variables C the usual mixture formulation is obtained P(S) = IT P(Xi, Ykt(X"Yk), i,k where P(Xi , Yk) =L P(ca)P(xilca)P(Yk Ic a ) . (2) a Following the standard Expectation Maximi zation approach for maximum likelihood t's timation [DE'mpster et al .. 1977], the E-step equations for the class posterior probabilities arE' given byl (3) 1 In the case of multiple observations of dyads it has been assumed that each observation may have a different latent class. If only one latent class variable is introduced for each dyad, slightly different equations are obtained. T. Hofmann, J puzicha and M. 1. Jordan 468 ?? ~ ..... . . . . . _0 . . . . . . . . ............... -- ..... __ ......... Ill , U.UU4 114, U.UU:l ! P (Ca) : maximal !P(XiICcx) : maximal !p(YklcCX> two 0.18 seven 0.10 tbree 0.10 four 0.06 five 0.06 years 0.11 bousand 0.1 buodred 0.1 days 0.07 cubits 0.05 bave batb bad bast 0.38 0.22 0.11 0.09 be 0.02 up 0.40 dowoO.17 fortb 0.15 out 0.09 ioO.Ol I"', U.U~9 sbalt 0.18 bast 0.08 wilt 0.08 art 0.07 if 0.05 tbe 0.95 bis 0.006 my 0.005 our 0.003 tby 0.003 tbou 0.85 DOt 0.01 also 0.004 ndeed 0.00 aooiot 0.003 lord 0.09 bildreo 0.0 SOD 0.02 land 0.02 o Ie 0.02 <.> <:> <,> <;> <?> 0.52 0.16 0.14 0.07 0.04 aDd 0.33 for 0.08 but 0.07 ben 0.0 so 0.02 ee O. me 0.03 him 0.03 it 0.02 you 0.02 <?> <,> <.> <:> <.> 0.27 0.23 0.12 0.06 0.04 Figure 1: Some aspects of the Bible (bigrams) . It is straightforward to derive the M-step re-estimation formulae P(c a) ex L n(xi' Yk)P{Cik = Ca }, P(xilca) ex L n(xi, Yk)P{Cik = Ca }, i,k (4) k and an analogous equation for P(Yk Ic a ). By re-parameterization the aspect model can also be characterized by a cross-entropy criterion. Moreover, formal equivalence to the aggregate Markov model, independently proposed for language modeling in [Saul, Pereira, 1997], has been established (cf. [Hofmann, Puzicha, 1998] for details). 2.2 One-Sided Clustering Model The complete data model proposed for the one-sided clustering model is P(S, c) = P( c)P(SIc) = (If P( c(x;)) ) (IT [P( x;)P(Y' Ic( X;))]n(x",,)) , (5) where we have made the assumption that observations (Xi, Yk) for a particular Xi are conditionally independent given c( xd . This effectively defines the mixture P(S) = IT P(S;) , P(S;) = L P(c a ) IT [P(XdP(Yklea)r(X"Yk) a , (6) k where Si are all observations involving Xi. Notice that co-occurrences in Si are not independent (as they are in the aspect model) , but get coupled by the (shared) latent variable C(Xi). As before, it is straightforward to derive an EM algorithm with update equations P{ c( Xi) =Ca } ex P( Ca ) IT P(Yk Icat(x. ,Yk), P(Yk lea) ex L n(Xi, Yk )P{ c( Xi) =ca } (7) k and P(c a ) ex Li P{C(Xi ) = cal, P(Xi) ex Lj n(xi,Yj)? The one-sided clustering mod el is similar to the distributional clustering model [Pereira et al. , 1993], however, there are two important differences: (i) the number of likelihood contributions in (7) scales with the number of observations - a fact which follows from Bayes' rule - and (ii) mixing proportions are rpissing in the original distributional clustering model. The one-sided clustering model corresponds to an unsupervised version of the naive Bayes' classifier, if we interpret Y as a feature space for objects Xi EX . There are also ways to weaken the conditional independence assumption, e.g., by utilizing a mixture of tree dependency models [Meila, Jordan, 1998] . 2.3 Two-Sided Clustering Model The latent variable structure of the two-sided clustering model significantly reduces the degrees of freedom in the specification of the class conditional distribution. We Learning from Dyadic Data 469 Figure 2: Exemplary segmentation results on Aerial by one-sided clustering. propose the following complete data model P(S, c) = II P(C(Xi))P(C(Yk)) [P(xi)P(Yk)1Tc(xi),c(YIc)f(x"yIc) (8) i,k where 1Tc:r: cll are cluster association parameters. In this model the latent variables in the X and Y space are coupled by the 1T-parameters. Therefore, there exists no simple mixture model representation for P(S). Skipping some of the technical details (cf. [Hofmann, Puzicha, 1998]) we obtain P(Xi) ex Lk n(xi,Yk), P(Yk) ex Li n(xi' Yk) and the M-step equations 0" "Y L i k n(xi, Yk)P{C(Xi) = c~ /\ C(Yk) = c~} 1Tc~.c~ = [Li P{C(Xi) = ~;} Lk n(xi, Yk)] [Lk P{C(Yk) = cn Li n(xi, Yk)] = = = (9) = as well as P(c~) L i P{C(Xi) c~} and P(c~) Lk P{C(Xk) cn . To preserve tractability for the remaining problem of computing the posterior probabilities in the E-step , we apply a factorial approximation (mean field approximation), i.e., P{C(Xi ) = c~ /\ C(Yk) = ~ P{C(Xi) = c~}P{C(Yk) = This results in the following coupled approximation equations for the marginal posterior probabilities cO P{ c(x;) cn. = c~} ex P(c~) exp [~n(x;, y,) ~ PI cry,) = c'(} log "'~"~1 (10) and a similar equation for P {C(Yk) = c~}. The resulting approximate EM algorithm performs updates according to the sequence (CX- post., 1T, cLpost., 1T). Intuitively the (probabilistic) clustering in one set is optimized in alternation for a given clustering in the other space and vice versa. The two-sided clustering model can also be shown to maximize a mutual information criterion [Hofmann, Puzicha, 1998] . Discussion: Aspects and Clusters To better understand the differences of the presented models it is elucidating to systematically compare the conditional probabilities P( CO' Ixd and P( CO' IYk): 2.4 Aspect Model P(colxd P(CoIYk ) P{x.ico' W{co' 2 P(x,) P~lf.k Ic", W( c'" 2 P(Y k) Two-sided Clustering One-sided One-sided X Clustering Y Clustering P{xdcO' W{ CO' 2 P{C(Xi) P{C(Yk) = cO'} P{C(Yk) = c~} P{c(xd = cO'} P(lf.kl cO' )P(cO' 2 P(Yk) P(x.) = c~} As can be seen from the above table, probabilities P(CoIXi) and P(CaIYk) correspond to posterior probabilities of latent variables if clusters are defined in the X-and Y-space, respectively. Otherwise, they are computed from model parameters. This is a crucial difference as, for example, the posterior probabilities are approaching T. Hofmann, J. Puzicha and M. I. Jordan 470 . . ...- '''' P' '' ''' '~ I ... 1? ' ~. .~ , f ..... I~ . --" .... . . .r... ...........,..... .. < ?? I~ h .. .. ' ? ??? " 'dol _.ie _ ??? 1 Il10' ' '' t. .,... c r ..... ~:.:' ~ I?? I''''''.' d?,. .".... h'" .. , .. t._~.. ..10 ..' .,..... ? ... ~ . " ... , ~.'''< .I ?? ~:.:: ~.".. ~.::;,:'::.I ~:::: .. IT . h .~. ..... ~.... .. . .,_. .... I"". ... n, ??10' ''' f.:~I:.:,., ,.1.? ~::. ~i... " ? ..... _ . i,. .. .....,. ?u. _" .... ..? .. .i'....... f . . .. U.tl~ .~ ~" cl~ 1& . _11 "'7 ' '''.,. I ~ .. '''. " ...... , .... ~ .i ul ::.7.:,., to., " ' ..tol. Io'.,f.. , ? ?? 1-. ::':~., t~?.?t ~;:.;::.' . 1. . .... ,' " .... ,. oa." ...., ~~?:;?i:? ~, . .c; ?? "" ...!:~:I::' ,,,II " . .. .. ,... , .. h lt. ..."... , ...it, ::::,:,:;"??1::;::;. .Iln., ... ... w , . .. .. ?? ? ? ? L ::,':~~;~~ ~::~' 1. '10 ? ,n., ?? ".d . D~~~ "" I~ .1 I?? ,. U ..,.<. Ho, .. ..-~ "?:~::,~~:. ..,fa_. ,.1.... _ ;' ,117 I d ..... u u ll ? ?? ,,,??.U, ........ .......... ... ,,<., .,,"??ua, " ? , " , , h .... , ., " ??? , .. .. . ,~,-." u d d . .. y .. .. .... hi II ~,," .< ~ ,."... . . ~.c, .. .h. ..... ?.. Ii., " ,e. ..... ?? , , ..?... 1111 , ???? .~I. 1<1 ~ :,~:::. 'It b ..... , u ."'~' )O ,.i. ~ ._ .. u :~::. "'~h" ........... 'u.. .. ., <? ?? .,. <. . . .11 . . .. i . .. " .flu . . , .. , . . . .. n, ..." ,........ _.. ',.<. :~::~ 1, ? ? - .. ... , ? . . ,, '" fl ???? . .... ]0 :I~;:;?.~' ",oi, ~ ~~.:: ", 1" .... ....'.,"10 ... ~., ,... ,, 1. . . .0 31 1.<0' ~ I."""'''' .. ~Ii.,<oJ ~:~::~', .. ;:~:,. ~~:::~.".~'. :::~'.~~~~' .??;::., :::;!..'., Figure 3: Two-sided clustering of LOB: 7r matrix and most probable words. Boolean values in the infinite data limit and P(Yklxd = Lo P{C(Xi)=Co}P(Yk!co) are converging to one of the class-conditional distributions. Yet, in the aspect model P(Yklxd = Lo P(CoIXi)P(Yk!co) and P(CoIXi) ex: P(Co)P(Xi!co) are typically not peaking more sharply with an increasing number of observations. In the aspect model, conditionals P(Yk IXi) are inherently a weighted sum of the 'prototypical' distributions P(Yk Ic o ). Cluster models in turn ultimately look for the 'best' classconditional and weights are only indirectly induced by the posterior uncertainty. 3 The Cluster-Abstraction Model The models discussed in Section 2 all define a non-hierarchical, 'flat' latent class structure. However, for structure discovery it is important to find hierarchical data organizations. There are well-known architectures like the Hierarchical Mixtures of Experts [Jordan, Jacobs , 1994] which fit hierarchical models. Yet, in the case of dyadic data there is an alternative possibility to define a hierarchical model. The Cluster-Abstraction Model (CAM) is a clustering model (e.g., in X) where the conditionals P(Yk Ico) are itself xi-specific aspect mixtures, P(Yk leo, Xi) = LII P(Yk la ll )P( alllc o , Xi) with a latent aspect mapping a. To obtain a hierarchical organization, clusters Co are identified with the terminal nodes of a hierarchy (e.g., a complete binary tree) and aspects all with inner and terminal nodes. As a compatibility constraint it is imposed that P( all/co, xd = 0 whenever the node corresponding to all is not on the path to the terminal node co. Intuitively, conditioned on a 'horizontal' clustering c all observations (Xi, Yk) E Si for a particular Xi have to be generated from one of the 'vertical' abstraction levels on the path to c( Xi)' Since different clusters share aspects according to their topological relation , this favors a meaningful hierarchical organization of clusters. Moreover, aspects at inner nodes do not simply represent averages over clusters in their subtree as they are forced to explicitly represent what is common to all subsequent clusters. Skipping the technical details, the E-step is given by P{a(xi,Yk) = all/c(xi) = co} ex: P(alllco,xi)P(Yk/all) P{ C(Xi) = co} ex: P( co) II L [P( alllc o , Xi)P(Yk /a ll (11) )r(X"Yk) (12) II and the M-step formulae are P(Yk/all) ex: LiP{a(xi,Yk) = all}n(xi,Yk), P(c o ) ex: Li P{C(Xi) = co}, and P(alllco , Xi) ex: Lk P{a(xi ' Yk) = all/c(xi) = co}n(xi, Yk)' k 471 Learning from Dyadic Data network neural function ....... :f;:::::m?????? function : weiCht '. .: learn \ .' error '. .:' example \ nd ,talc neuron ~onlinu <lrbitr~ri rundl.l.ln : ~idd~:r i learn train problem d .. s~ir 'r:lin taleet frtttur method ~rror cener learn perform im.{ j ctntr ne~.~ork model ~~rs~ , !'in ;~=fit ~ :.:~ update-: '_tur .pUm bound; ute cCHWtr-.in, lin?... It.n:d .....-. 1 ~c~. i\, '1M1"I1 1 /~ i """ ~:!:' ~.r ./ \ ... :-e:rsion demnln i ~;IKIU' ! . . . : :. . . .: .-: . ~;~-.// \=" appt"e("u. m .... th~d 1MOt00itnn 'lUlu ;;yne~nu::=t ~:::n ~i;~ .... UrUyers rglat deted ,.b.1' theta tnnde!' -II ~t~n ?-? ?? ?? ?-? process author Infonn dy~.~m,ic control ; : / ;?k. i ;~~:...; ~~~!~-tc machin : : ...... ...... -, dynamjc :' Umt '" .: modd \ synaptic.. f nruron ", neuron nrtwork actiy nrural spike learn r~pons aicorithm i control neural desi", infonn C);,I;\sin 1I11~'rilh~ "",wI ptrfonn : ntlwork: i process ~ : data : j modtl : . ,.,"'/ / raw .plim m.".... ? -;;:.. .. 7?~d~_" " ~""_-::;'bll = t i ~ nrtwork \ ; map \ visual memori imae associ motion (papc ctll pattern p~~tSS lra~ ~pt INtt:w. j i ~~:nl :::;eri map paramll : naI~ certic object ' =!:. ..... ~ ,,.,....w (:...., ',.... ~ni;~ ::: ::~d'.;:;" ';~'';ari thre:m.td mKhin Gscil 1.caI ,.,-...et :;::~~ :~it.ri steM,....t ? learn. :' a1corithm i rult ,..oIen: r::kl' ::!::-' pedsJ"a.pt nop) ".ndud. ~tin ~:;:: repoa m: krn (...J "tat t,..nsfonll Figure 4: Parts of the top levels of a hierarchical clustering solution for the Neural document collection, aspects are represented by their 5 most probable word stems. 4 Annealed Expectation Maximization Annealed EM is a generalization of EM based on the idea of deterministic annealing [Rose et al., 1990] that has been successfully applied as a heuristic optimization technique to many clustering and mixture problems. Annealing reduces the sensitivity to local maxima, but, even more importantly in this context , it may also improve the generalization performance compared to maximum likelihood estimation. 2 The key idea in annealed EM is to introduce an (inverse temperature) parameter (3 , and to replace the negative (averaged) complete data log-likelihood by a substitute known as the fre e energy (both are in fact equivalent at f3 = 1) . This effectively results in a simple modifi cation of the E-step by taking the likelihood contribution in Bayes ' rul e to the power of ;3. In order to determine the optimal value for f3 we used an additional validation set in a cross validation procedure. 5 Results and Conclusions In our experiments we have utilized the following real-world data sets: (i) Cranfield: a standard test collection from information retrieval (N = 1400, M = 4898) , (ii) Penn : adjective-noun co-occurrences from the Penn Treebank corpus (N = 6931 , M = 4995) and the LOB corpus (N = 5448, M = 6052) , (iii) Neural: a document collection with abstracts of journal papers on neural networks (N = 1278 , M = 6065) , (iv) Bzble: word bigrams from the bible edition of the Gutenberg project (N = M = 12858) , (v) Aerial: Textured aerial images for segmentation (N = 128x128, M = 192). In Fig. 1 we have visualized an aspect model fitted to the Bible bigram data. Notice that although X = Y the role of the preceding and the subsequent words in bigrams is quite different . Segmentation results obtained on Aerial applying the one-sided clustering model are depicted in Fig. 2. A multi-scale Gabor filter bank (3 octaves, 4 orientations) was utilized as an image representation (cf. [Hofmann et al. , 1998]) . In Fig. 3 a two- sided clustering solution of LOB is shown. Fig. 4 shows the top levels of the hierarchy found by the Cluster-Abstraction Model in Neural. The inner node distributions provide resolution-specific descriptors for the documents in the conesponding subtree which can be utilized , e.g., in interactive browsing for information retrieval, Fig. 5 shows typical test set perplexity curves of the 2 Moreover, the tree topology for the CAM is heuristically grown via phase transitions. T. Hofmann, 1. Puzicha and M 1. Jordan 472 (b) (a) (c) ".;----=-~---;:--~-,:;---:~ .~EJoII,"""', .... Figure 5: Perplexity curves for annealed EM (aspect (a), (b) and one-sided clustering model (c)) on the Bible and Gran data. Aspect K 1 8 16 32 64 128 f3 'P Cran - 685 0.88 482 0.72 255 0.83 386 0.79 360 0.78 353 X-duster f3 'P CAM f3 'P X /Y-c1uster 'P f3 Aspect f3 X-cluster CAM 'P f3 'P f3 'P 639 312 255 205 182 166 0.08 0.07 0.07 0.07 0.06 352 302 254 223 231 0.13 0.10 0.08 0.07 0.06 322 268 226 204 179 X /Y-cluster 'P f3 Penn 0.09 0.07 0.07 0.06 0.04 527 302 452 527 663 0.18 0.10 0.12 0.11 0.10 511 268 438 422 410 0.67 0.51 0.53 OA8 OA5 0.73 0.72 0.71 0.69 0.68 615 335 506 477 462 0.55 0.51 0.46 0.44 DAD 394 335 286 272 241 Table 1: Perplexity results for different models on the Gran (predicting words conditioned on documents) and Penn data (predicting nouns conditioned on adjectives). annealed EM algorithm for the aspect and clustering model (P = e- 1 where I is the per-observation log-likelihood). At {J 1 (standard EM) overfitting is clearly visible, an effect that vanishes with decreasing (J. Annealed learning performs also better than standard EM with early stopping. Tab. 1 systematically summarizes perplexity results for different models and data sets. = In conclusion mixture models for dyadic data have shown a broad application potential. Annealing yields a substantial improvement in generalization performance compared to standard EM, in particular for clustering models, and also outperforms a complexity control via J{. In terms of perplexity, the aspect model has the best performance. Detailed performance studies and comparisons with other state-of-the-art techniques will appear in forthcoming papers. References [Dempster et al., 1977] Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm . J. Royal Statist. Soc. B , 39 , 1-38. [Hofmann , Puzicha, 1998] Hofmann, T., Puzicha, J. 1998. Statistical models for cooccurrence data. Tech. rept . Artifical Intelligence Laboratory Memo 1625, M.LT. [Hofmann et al., 1998] Hofmann, T., Puzicha, J ., Buhmann, J.M. (1998). Unsupervised texture segmentation in a deterministic annealing framework. IEEE Transactions on Pattern Analysis and Machine Intelligence , 20(8) , 803-818. [Jordan, Jacobs, 1994] Jordan, M.L, Jacobs, R.A. (1994). Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6(2), 181-214. [Meila, Jordan , 1998] Meila, M., Jordan, M. L 1998. Estimating Dependency Structure as a Hidden Variable. In: Advances in Neural Information Processing Systems 10. [Pereira et al., 1993] Pereira, F.e.N., Tishby, N.Z., Lee, L. 1993. Distributional clustering of English words. Pages 189-190 of: Proceedings of the A CL. [Rose et al., 1990] Rose, K., Gurewitz , E., Fox, G. (1990). Statistical mechanics and phase transitions in clustering. Physical Review Letters, 65(8), 945-948. [Saul, Pereira, 1997] Saul, 1., Pereira, F. 1997. Aggregate and mixed-order Markov models for statistical language processing. In: Proceedings of the 2nd International Conference on Empirical Methods in Natural Language Processing. 

1503 |@word version:2 bigram:4 proportion:1 nd:2 heuristically:1 r:1 tat:1 jacob:3 loc:1 document:6 past:1 outperforms:1 thre:1 skipping:2 si:3 yet:2 written:1 visible:1 subsequent:2 partition:1 hofmann:13 update:3 intelligence:2 parameterization:1 xk:1 node:6 location:1 preference:3 x128:1 five:1 direct:1 become:1 ect:1 consists:1 fitting:1 introduce:2 behavior:1 pum:1 mechanic:1 multi:1 ol:1 discretized:1 terminal:3 decreasing:1 td:1 ua:1 increasing:1 project:1 estimating:1 underlying:1 moreover:3 what:1 unobserved:2 xd:3 interactive:1 classifier:1 partitioning:2 control:3 penn:4 appear:1 before:1 local:1 rept:1 limit:1 io:1 flu:1 path:2 equivalence:1 co:30 bi:1 averaged:1 unique:1 yj:1 lf:2 procedure:1 jan:2 area:1 empirical:2 significantly:1 gabor:1 word:9 pre:1 refers:2 get:1 cal:1 context:2 applying:1 equivalent:1 imposed:1 map:2 center:1 deterministic:2 annealed:7 straightforward:2 independently:1 formulate:1 resolution:1 identifying:2 ixd:1 rule:1 utilizing:1 importantly:1 notion:1 analogous:1 construction:1 hierarchy:3 machin:1 pt:2 ixi:1 element:1 recognition:1 utilized:3 distributional:3 observed:2 role:2 tur:1 yk:57 substantial:2 rose:3 vanishes:1 dempster:2 complexity:1 cooccurrence:1 cam:4 ultimately:1 lord:1 textured:1 joint:1 represented:1 grown:1 leo:1 train:1 infonn:2 forced:1 aggregate:2 quite:1 heuristic:1 otherwise:1 favor:1 statistic:1 itself:2 laird:1 advantage:1 sequence:1 exemplary:2 cai:1 propose:3 maximal:2 mixing:1 cluster:16 ben:1 object:4 derive:2 keywords:1 progress:1 soc:1 c:1 uu:1 filter:1 sand:1 generalization:3 biological:1 elementary:1 probable:2 im:1 ic:6 exp:1 mapping:3 early:1 estimation:2 him:1 vice:1 successfully:1 weighted:1 mit:1 clearly:1 ck:3 focus:1 improvement:1 likelihood:7 tech:1 sense:1 textbased:1 el:1 abstraction:4 stopping:1 lj:1 typically:1 hidden:1 relation:1 i1:1 germany:1 compatibility:1 ill:2 orientation:1 art:2 noun:2 mutual:1 marginal:1 field:2 f3:10 represents:1 broad:1 look:1 unsupervised:4 stimulus:1 preserve:1 individual:1 phase:2 freedom:1 organization:3 tss:1 plim:1 highly:1 mining:1 possibility:1 elucidating:1 mixture:12 nl:1 yielding:1 bible:4 xy:1 institut:1 fox:1 tree:3 iv:2 euclidean:1 incomplete:1 re:2 zation:1 weaken:1 fitted:1 modeling:6 boolean:1 cover:1 ations:1 pons:1 maximization:1 tractability:1 subset:1 sod:1 gutenberg:1 tishby:1 universitat:1 dependency:2 answer:1 my:1 fundamental:1 cll:1 sensitivity:1 ie:2 international:1 systematic:1 probabilistic:4 lee:1 discipline:1 michael:1 ym:1 lii:1 expert:2 li:5 potential:1 de:2 wilt:1 explicitly:1 vi:1 dad:1 tab:1 bayes:3 collaborative:1 contribution:2 oi:1 ir:2 ni:1 descriptor:1 bave:1 correspond:2 yield:1 raw:1 informatik:1 cation:1 umt:1 whenever:1 ed:1 synaptic:1 energy:1 naturally:1 fre:1 fiat:1 segmentation:5 cik:4 focusing:1 day:1 formulation:1 evaluated:1 just:1 cran:1 horizontal:1 defines:1 sic:1 effect:1 laboratory:1 conditionally:2 ll:3 sin:1 lob:3 iln:1 criterion:2 octave:1 complete:5 performs:2 motion:1 temperature:1 ranging:1 image:5 fi:1 ari:1 common:1 empirically:1 physical:1 association:2 discussed:1 m1:1 interpret:1 cambridge:1 versa:1 ai:1 meila:3 associ:1 language:4 dot:1 ute:1 specification:1 add:1 playa:1 nop:1 posterior:6 perplexity:5 lra:1 binary:1 alternation:1 yi:1 seen:1 additional:1 preceding:1 determine:1 maximize:1 ii:13 bast:2 multiple:1 ico:2 reduces:2 stem:2 intt:1 technical:2 characterized:1 offer:1 cross:2 retrieval:4 lin:2 post:1 converging:1 variant:1 involving:1 vision:1 expectation:2 represent:3 lulu:1 achieved:1 lea:1 conditionals:2 annealing:4 crucial:1 induced:1 mod:1 idd:1 jordan:11 obj:1 ee:1 iii:3 variety:2 independence:1 fit:3 forthcoming:1 architecture:2 approaching:1 identified:1 topology:1 inner:3 idea:2 cn:3 dyad:6 ul:1 modd:1 tol:1 byl:1 detailed:1 factorial:1 statist:1 induces:2 visualized:1 simplest:1 notice:2 estimated:1 per:1 key:2 four:2 year:1 tbe:1 sum:1 ork:1 inverse:1 letter:1 you:1 uncertainty:1 disambiguation:1 thesaurus:1 dy:1 summarizes:1 fl:1 hi:1 bound:1 topological:1 strength:1 constraint:1 sharply:1 ri:2 flat:1 bonn:2 aspect:23 lca:1 according:2 gran:2 aerial:4 across:1 slightly:1 em:13 wi:1 intuitively:2 restricted:1 peaking:1 classconditional:1 sided:16 ln:1 equation:7 turn:1 count:1 apply:1 hierarchical:10 indirectly:1 occurrence:7 alternative:1 ho:1 thomas:1 original:1 top:2 clustering:30 linguistics:2 cf:3 remaining:1 eri:1 substitute:1 unifying:1 learningfrom:1 nai:1 spike:1 usual:1 oa:1 consumption:1 me:1 seven:1 negative:1 memo:1 reliably:1 perform:1 vertical:1 observation:11 neuron:2 markov:2 finite:1 defining:1 bll:1 introduced:1 pair:1 kl:2 optimized:1 established:1 nu:1 pattern:3 adjective:2 oj:1 royal:1 power:1 natural:1 predicting:2 buhmann:1 wik:1 improve:1 iyk:1 theta:1 ne:1 lk:5 categorical:1 coupled:3 naive:1 gurewitz:1 prior:1 review:1 discovery:3 mixed:1 prototypical:1 validation:2 degree:1 rubin:1 treebank:1 bank:1 systematically:2 pi:1 cd:1 land:1 cry:1 lo:2 share:1 cranfield:1 english:1 formal:1 understand:1 saul:3 taking:1 distributed:1 curve:2 xn:1 world:1 transition:2 author:1 made:4 collection:4 counted:1 far:1 transaction:1 approximate:1 uni:1 active:1 rul:1 overfitting:1 corpus:3 summing:1 assumed:1 xi:62 alternatively:1 latent:13 decade:1 timation:1 yic:2 table:2 lip:1 learn:5 ca:7 inherently:1 symmetry:1 ioo:1 cl:4 domain:6 sp:1 edition:1 dyadic:12 fig:5 referred:1 tl:1 pereira:6 xl:1 tin:1 krn:1 formula:2 bad:1 specific:2 dol:1 ton:1 exists:1 i11:1 effectively:2 ci:2 texture:1 subtree:2 conditioned:3 sparseness:1 browsing:1 entropy:1 tc:4 cx:1 lt:2 simply:1 depicted:1 visual:1 ykt:1 scalar:1 rror:1 corresponds:2 ma:1 conditional:6 shared:1 replace:1 infinite:1 typical:1 called:3 la:1 meaningful:1 rarely:1 puzicha:10 tby:1 latter:1 arises:1 artifical:1 ex:16

Example Based Image Synthesis of Articulated Figures Trevor Darrell Interval Research. 1801C Page Mill Road. Palo Alto CA 94304 trevor@interval.com, http://www.interval.com/-trevor/ Abstract We present a method for learning complex appearance mappings. such as occur with images of articulated objects. Traditional interpolation networks fail on this case since appearance is not necessarily a smooth function nor a linear manifold for articulated objects. We define an appearance mapping from examples by constructing a set of independently smooth interpolation networks; these networks can cover overlapping regions of parameter space. A set growing procedure is used to find example clusters which are well-approximated within their convex hull; interpolation then proceeds only within these sets of examples. With this method physically valid images are produced even in regions of parameter space where nearby examples have different appearances. We show results generating both simulated and real arm images. 1 Introduction Image-based view synthesis is.an important application of learning networks. offering the ability to render realistic images without requiring detailed models of object shape and illumination effects. To date. much attention has been given to the problem of view synthesis under varying camera pose or rigid object transformation. Several successful solutions have been proposed in the computer graphics and vision literature. including view morphing [12], plenoptic modeling/depth recovery [8], "lightfields" [7], and recent approaches using the trifocal tensor for view extrapolation [13]. For non-rigid view synthesis. networks for model-based interpolation and manifold learning have been used successfully in some cases [14. 2. 4. 11]. Techniques based on Radial Basis Function (RBF) interpolation or on Principle Components Analysis (peA), have been able to interpolate face images under varying pose. expression and identity [1.5, 6]. How- Example-Based Image Synthesis ofArticulated Figures 769 extends the notion of example clustering to the case of coupled shape and texture appearance models. Our basic method is to find sets of examples which can be well-approximated from their convex hull in parameter space. We define a set growing criterion which enforces compactness and the good-interpolation property. To add a new point to an example set, we require both that the new point must be well approximated by the previous set alone and that all interior points in the resulting set be well interpolated from the exterior examples. We define exterior examples to be those on the convex hull of the set in parameter space. Given a training subset s C 0 and new point p E 0, E(s,p) = max(E/(s U {p}),EE(S,p)) , with the interior and extrapolation error defined as 1ix (s) is the subset of s whose x vectors lie on the convex hull of all such vectors in s. To add a new point, we require E < E, where E is a free parameter of the clustering method. Given a seed example set, we look to nearest neighbors in appearance space to find the next candidate to add. Unless we are willing to test the extrapolation error of the current model to all points, we have to rely on precomputed non-vectorized appearance distance (e.g., MSE between example images). If the examples are sparse in the appearance domain, this may not lead to effective groupings. If examples are provided in sequence and are based on observations from an object with realistic dynamics, then we can find effective groupings even if observations are sparse in appearance space. We make the assumption that along the trajectory of example observations over time, the underlying object is likely to remain smooth and locally span regions of appearance which are possible to interpolate. We thus perform set growing along examples on their input trajectory. Specifically, in the results reported below, we select K seed points on the trajectory to form initial clusters. At each point p we find the set s which is the smallest interval on the example trajectory which contains p, has a non-zero interior region (s -1i x (s)), and for which E / (s) < ?. If such set exists, we continue to expand it, growing the set along the example trajectory until the above set growing criterion is violated. Once we can no longer grow any set, we test whether any set is a proper subset of another, and delete it if so. We keep the remaining sets, and use them for interpolation as described below. 4 Synthesis using example sets We generate new views using sets of examples: interpolation is restricted to only occur inside the convex hull of an example set found as above for which E/(s) ::; E. Given a new parameter vector x, we test whether it is in the convex hull of parameters in any example set. If the point does not lie in the convex hull of any example set, we find the nearest point on the convex hull of one of the example sets, and use that instead. This prevents erroneous extrapolation. If a new parameter is in the convex hull of more than one example set, we first select the set whose median example parameter is closest to the desired example parameter. Once a set has been selected, we interpolate a new function value from examples using the RBF method summarized above. To enforce temporal consistency of rendered images over time, 770 T. Darrell (b) (c) Figure 2: (a) Images of a real arm (from a sequence of 33 images) with changing appearance and elbow configuration. (b,c) Interpolated shape of arms tracked in previous figure. (b) shows results using all examples in a single interpolation network; (c) shows results using example sets algorithm. Open contours show arms example locations; filled contour shows interpolation result. Near regions of appearance singularity in parameter space the full network method generates physically-invalid arm shapes; the example sets method produces realistic images. The method presented below for grouping examples into locally valid spaces is generally applicable to both the PCA and RBF-based view synthesis techniques. However our initial implementation, and the results reported in this paper, have been with RBF-based models. 3 Finding consistent example sets Given examples from a complicated (non-linear, non-smooth) appearance mapping, we find local regions of appearance which are well-behaved as smooth, possibly linear, functions. We wish to cluster our examples into sets which can be used for successful interpolation using our local appearance mode\. Conceptually, this problem is similar to that faced by Bregler and Omohundro [2], who built image manifolds using a mixture of local PCA models. Their work was limited to modeling shape (lip outlines); they used K-means clustering of image appearance to form the initial groupings for PCA analysis. However this approach had no model of texture and performed clustering using a mean-squared-error distance metric in simple appearance. Simple appearance clustering drastically over-partitions the appearance space compared to a model that jointly represent shape and texture. Examples which are distant in simple appearance can often be close when considered in 'vectorized' representation. Our work 771 Example-Based Image Synthesis ofArticulated Figures ", '" " .. : ... . .... . .. .. . ... ", " .............. ...... '.. . '" (b) ,-------.c'-,-' '_' . _' ' _--, (C)'----________ " " " . .. .. .. .:.... .. , ' :': ',::':: _ _ _ _ _ _ _ ___" , '" .. .. . . : .... - ' , ' 0 ? ?? :::~:.:.: :-:.: ' 0 '~1.)'. ..... . .. . ' '-----"--'---'--_ _ _ _---' '------"--'---'--_ _--' ~L " " '------"~_ _- ' '_____ _~_ _ _ _~ '-----'_ _ _ _ _ _- - ' ..' . '-----"--'---'--_ _ _ _---' '----~ _ _ _ _ _ _--' Figure 1: Arm appearance interpolated from examples using approximation network. (a) A 2DOF planar arm. Discontinuities in appearance due to workspace constraints make this a difficult function to learn from examples; the first and last example are very close in parameter space, but far in appearance space. (b) shows results using all examples in a single network; (c) using the example sets algorithm described in text. Note poor approximation on last two examples in (a); appearance discontinuities and extrapolation cause problems for full network, but are handled well in examples sets method. In peA-based approaches, G projects a portion of u onto a optimal linear subspace found from D, and F projects a portion of u onto a subspace found from T [6, 5]. For example G D (u) = PI) 59 U , where 59 is a diagonal boolean matrix which selects the texture parameters from u and PI) is a matrix containing the m-th largest principle components of D . F warps the reconstructed texture according to the given shape: FT(u, s) = [PT 5 t u] 0 s. While interpolation is simple using a peA approach, the parameters used in peA models often do not have any direct physical interpretation. For the task of view synthesis, an additional mapping u = H(x) is needed to map from task parameters to peA input values; a backpropogation neural net was used to perform this function for the task of eye gaze analysis [10]. Using the RBF-based approach [1], the application to view synthesis is straightforward. Both G and F are networks which compute locally-weighted regression, and parameters are used directly (u = x) . G computes an interpolated shape, and F warps and blends the example texture images according to that shape: G D(X) = Ei cd(x - xd, FT(X, s) = [Ei cU(x - Xi)] os , where f is a radial basis function. The coefficients c and c' are derived from D and T, respectively: C = D R+ , where rij = f (x i-X j) and C is the matrix of row vectors Ci; similarly C' = T R+ [9] . We have found both vector norm and Gaussian basis functions give good results when appearance data is from a smooth function; the results below use f(r) = Ilrll. 772 T. Darrell ever, these methods are limited in the types of object appearance they can accurately model. PCA-based face analysis typically assumes images of face shape and texture fall in a linear subspace; RBF approaches fare poorly when appearance is not a smooth function. We want to extend non-rigid interpolation networks to handle cases where appearance is not a linear manifold and is not a smooth function, such as with articulated bodies. The mapping from parameter to appearance for articulated bodies is often one-to-many due to the multiple solutions possible for a given endpoint. It will also be discontinuous when constraints call for different solutions across a boundary in parameter space, such as the example shown in Figure 1. Our approach represents an appearance mapping as a set of piecewise smooth functions. We search for sets of examples which are well approximated by the examples on the convex hull of the set's parameter values. Once we have these 'safe' sets of examples we perform interpolation using only the examples in a single set. The clear advantage of this approach is that it will prevent inconsistent examples from being combined during interpolation. It also can reduce the number of examples needed to fully interpolate the function, as only those examples which are on the convex hull of one or more example sets are needed. If a new example is provided and it falls within and is well-approximated by the convex hull of an existing set, it can be safely ignored. The remainder of this paper proceeds as follows. First, we will review methods for modeling appearance when it can be well approximated with a smooth and/or linear function. Next, we will present a technique for clustering examples to find maximal subsets which are well approximated in their interior. We will then detail how we select among the subsets during interpolation, and finally show results with both synthetic and real imagery. 2 Modeling smooth and/or linear appearance functions Traditional interpolation networks work well when object appearance can be modeled either as a linear manifold or as a smooth function over the parameters of interest (describing pose, expression, identity, configuration, etc.). As mentioned above, both peA and RBF approaches have been successfully applied to model facial expression. In both approaches, a key step in modeling non-rigid shape appearance from examples is to couple shape and texture into a single representation. Interpolation of shape has been well studied in the computer graphics literature (e.g., splines for key-frame animation) but does not alone render realistic images. PCA or RBF models of images without a shape model can only represent and interpolate within a very limited range of pose or object configuration. In a coupled representation, texture is modeled in shape-normalized coordinates, and shape is modeled as disparity between examples or displacement from a canonical example to all examples. Image warping is used to generate images for a particular texture and shape. Given a training set n = {(Yi, Xi, d i ), 0 ~ i ~ n}, where Yi is the image of example i, Xi is the associated pose or configuration parameter, and di is a dense correspondence map relative to a canonical pose, a set of shape-aligned texture images can be computed such that texture ti warped with displacement di renders example image Yi: Yi = ti 0 d i [5, 1,6]. A new image is constructed using a coupled shape model G and texture model F, based on input u: y(n,U) = FT(GD(U),u) , where D, T are the matrices [dodl ... d n ], [totl ... t n ], respectively. Example-Based Image Synthesis ofArticulated Figures 773 (c (b) Figure 3: Interpolated shape and texture result. (a) shows exemplar contours (open) and interpolated shape (filled). (b) shows example texture images. (c) shows final interpolated image. we can use a simple additional constraint on subsequent frames. Once we have selected an example set, we keep using it until the desired parameter value leaves the valid region (convex hull) of that set. When this occurs, we allow transitions only to "adjacent" example sets; adjacency is defined as those pairs of sets for which at least one example on each convex hull are sufficiently close (11Yi - Yj II < E) in appearance space. S Results First we show examples using a synthetic arm with several workspace constraints. Figure l(a) shows examples of a simple planar 2DOF ann and the inverse kinematic solution for a variety of endpoints. Due to an artificial obstacle in the world, the ann is forced to switch between ann-up and ann-down configurations to avoid collision. We trained an interpolation network using a single RBF to model the appearance of the ann as a function of endpoint location. Appearance was modeled as the vector of contour point locations, obtained from the synthetic ann rendering function. We first trained a single RBF network on a dense set of examples of this appearance function. Figure l(b) shows results interpolating new ann images from these examples; results are accurate except where there are regions of appearance discontinuity due to workspace constraints, or when the network extrapolates erroneously. We applied our clustering method described above to this data, yielding the results shown in Figure 1(c). None of the problems with discontinuities or erroneous extrapolation can be seen in these results, since our method enforces the constraint that an interpolated result must be returned from on or within the convex hull of a valid example set. Next we applied our method to the images of real anns shown in Figure 2(a). Ann contours were obtained in a sequence of 33 such images using a semi-automated defonnable contour tracker augmented with a local image distance metric [3]. Dense correspondences were interpolated from the values on the contour. Figure 2(b) shows interpolated ann shapes using a single RBF on all examples; dramatic errors can be seen near where multiple different 774 T. Darrell appearances exist within a small region of parameter space. Figure 2( c) shows the results on the same points using sets of examples found using our clustering method; physically realistic arms are generated in each case. Figure 3 shows the final interpolated result rendered with both shape and texture. 6 Conclusion View-based image interpolation is a powerful paradigm for generating realistic imagery without full models of the underlying scene geometry. Current techniques for non-rigid interpolation assume appearance is a smooth function. We apply an example clustering approach using on-line cross validation to decompose a complex appearance mapping into sets of examples which can be smoothly interpolated. We show results on real imagery of human arms, with correspondences recovered from deformable contour tracking. Given images of an arm moving on a plane with various configuration conditions (elbow up and elbow down), and with associated parameter vectors marking the hand location, our method is able to discover a small set of manifolds with a small number of exemplars each can render new examples which are always physically correct. A single interpolating manifold for this same data has errors near the boundary between different arm configurations, and where multiple images have the same parameter value. References [1] D. Beymer, A. Shashua and T. Poggio, Example Based Image Analysis and Synthesis, MIT AI Lab Memo No. 1431, MIT, 1993. also see D. Beymer and T. Poggio, Science 272:1905-1909, 1996. [2] C. Breg1er and S. Omohundro, Nonlinear Image Interpolation using Manifold Learning, NIPS7, MIT Press, 1995. [3] T. DarrelI, A Radial Cumulative Similarity Transform for Robust Image Correspondence, Proc. CVPR-98. Santa Barbara, CA, IEEE CS Press, 1998. [4] M. Jagersand, Image Based View Synthesis of Articulated Agents, Proc. CVPR-97, San Jaun, Pureto Rico, pp. 1047-1053, IEEE CS Press, 1997. [5] M. Jones and T Poggio, Multidimensional Morphable Models, Proc. ICCV-98, Bombay, India, pp. 683-688, 1998. [6] A. Lanitis, C.J. Taylor, TF. Cootes, A Unified Approach to Coding and Interpreting Fa:::e Images, Proc. ICCV-95, pp. 368-373, Cambridge, MA, 1995. [7] M. Levoy and P. Hanrahan, Light Field Rendering, In SIGGRAPH-96, pp. 31-42,1996. [8] L. McMillan and G. Bishop, Plenoptic Modeling: An image-based rendering system. In Proc. SIGGRAPH-95, pp. 39-46, 1995. [9] T Poggio and F. Girosi , A Theory of Networks for Approximation and Learning, MIT AI Lab Memo No. 1140. 1989. [10] T Rikert and M. Jones, Gaze Estimation using Morphable Models, Proc. IEEE Conf. Face and Gesture Recognition '98, pp. 436-441, Nara, Japan, IEEE CS Press, 1998. [II] L. Saul and M. Jordan, A Variational Principle for Model-based Morphing, NIPS-9, MIT Press, 1997. [12] S. Seitz and C. Dyer, View Morphing, in Proc. SIGGRAPH-96, pp. 21-30,1996. [13] A. Shashua and M. Werman, Trilinearity of Three Perspective Views and its Associated Tensor, in Proc. ICCV-95, pp. 920-935, Cambridge, MA, IEEE CS Press, 1995. [14] 1. Tenenbaum, Mapping a manifold of perceptual observations, NIPS-IO, MIT Press, 1998. 

1504 |@word cu:1 norm:1 open:2 willing:1 seitz:1 dramatic:1 initial:3 configuration:7 contains:1 disparity:1 offering:1 existing:1 current:2 com:2 recovered:1 must:2 distant:1 subsequent:1 realistic:6 partition:1 shape:23 girosi:1 alone:2 selected:2 leaf:1 plane:1 location:4 along:3 constructed:1 direct:1 inside:1 nor:1 growing:5 elbow:3 provided:2 project:2 underlying:2 discover:1 alto:1 unified:1 finding:1 transformation:1 temporal:1 safely:1 multidimensional:1 ti:2 xd:1 local:4 io:1 interpolation:22 studied:1 limited:3 range:1 camera:1 enforces:2 yj:1 procedure:1 displacement:2 road:1 radial:3 onto:2 interior:4 close:3 www:1 map:2 straightforward:1 attention:1 independently:1 convex:15 recovery:1 handle:1 notion:1 coordinate:1 pt:1 approximated:7 recognition:1 ft:3 rij:1 region:9 mentioned:1 dynamic:1 trained:2 basis:3 siggraph:3 various:1 articulated:6 forced:1 effective:2 artificial:1 dof:2 whose:2 cvpr:2 ability:1 jointly:1 transform:1 final:2 sequence:3 advantage:1 net:1 maximal:1 remainder:1 aligned:1 date:1 poorly:1 deformable:1 cluster:3 darrell:4 produce:1 generating:2 object:9 pose:6 exemplar:2 nearest:2 c:4 safe:1 discontinuous:1 correct:1 hull:15 pea:6 human:1 adjacency:1 require:2 decompose:1 singularity:1 bregler:1 sufficiently:1 considered:1 tracker:1 seed:2 mapping:8 werman:1 smallest:1 estimation:1 proc:8 applicable:1 palo:1 largest:1 tf:1 successfully:2 weighted:1 mit:6 gaussian:1 always:1 avoid:1 varying:2 derived:1 rigid:5 typically:1 compactness:1 expand:1 selects:1 among:1 field:1 once:4 represents:1 look:1 jones:2 spline:1 piecewise:1 interpolate:5 geometry:1 interest:1 kinematic:1 mixture:1 yielding:1 light:1 accurate:1 poggio:4 facial:1 unless:1 filled:2 taylor:1 desired:2 delete:1 modeling:6 boolean:1 obstacle:1 bombay:1 cover:1 subset:5 successful:2 graphic:2 reported:2 lanitis:1 synthetic:3 combined:1 gd:1 workspace:3 gaze:2 synthesis:13 squared:1 imagery:3 containing:1 possibly:1 conf:1 warped:1 japan:1 summarized:1 coding:1 coefficient:1 performed:1 view:13 extrapolation:6 lab:2 portion:2 shashua:2 complicated:1 who:1 conceptually:1 accurately:1 produced:1 none:1 trajectory:5 anns:1 trevor:3 pp:8 associated:3 di:2 couple:1 rico:1 planar:2 until:2 hand:1 ei:2 o:1 overlapping:1 nonlinear:1 mode:1 behaved:1 effect:1 requiring:1 normalized:1 adjacent:1 during:2 criterion:2 outline:1 omohundro:2 interpreting:1 image:42 variational:1 physical:1 tracked:1 endpoint:3 extend:1 interpretation:1 fare:1 cambridge:2 backpropogation:1 ai:2 consistency:1 similarly:1 had:1 moving:1 longer:1 similarity:1 morphable:2 etc:1 add:3 closest:1 recent:1 perspective:1 barbara:1 continue:1 yi:5 hanrahan:1 seen:2 additional:2 paradigm:1 ii:2 semi:1 full:3 multiple:3 smooth:13 gesture:1 cross:1 nara:1 basic:1 regression:1 vision:1 metric:2 physically:4 represent:2 want:1 interval:4 grow:1 median:1 inconsistent:1 jordan:1 call:1 ee:1 near:3 rendering:3 variety:1 switch:1 automated:1 reduce:1 whether:2 expression:3 pca:5 handled:1 render:4 returned:1 cause:1 ignored:1 generally:1 collision:1 detailed:1 clear:1 santa:1 locally:3 tenenbaum:1 http:1 generate:2 exist:1 canonical:2 key:2 changing:1 prevent:1 inverse:1 powerful:1 cootes:1 extends:1 correspondence:4 extrapolates:1 occur:2 constraint:6 scene:1 nearby:1 interpolated:12 generates:1 erroneously:1 span:1 rendered:2 marking:1 according:2 poor:1 remain:1 across:1 restricted:1 iccv:3 describing:1 precomputed:1 fail:1 needed:3 dyer:1 apply:1 enforce:1 assumes:1 remaining:1 clustering:9 tensor:2 warping:1 occurs:1 blend:1 fa:1 traditional:2 diagonal:1 subspace:3 distance:3 simulated:1 manifold:9 modeled:4 difficult:1 memo:2 implementation:1 proper:1 perform:3 observation:4 ever:1 frame:2 mcmillan:1 pair:1 discontinuity:4 nip:2 able:2 proceeds:2 below:4 built:1 including:1 max:1 rely:1 arm:12 eye:1 coupled:3 faced:1 morphing:3 literature:2 text:1 review:1 relative:1 fully:1 validation:1 agent:1 vectorized:2 consistent:1 principle:3 pi:2 cd:1 row:1 last:2 free:1 drastically:1 allow:1 warp:2 india:1 neighbor:1 fall:2 face:4 saul:1 sparse:2 boundary:2 depth:1 valid:4 transition:1 contour:8 computes:1 world:1 cumulative:1 levoy:1 san:1 far:1 reconstructed:1 keep:2 xi:3 search:1 lip:1 learn:1 robust:1 ca:2 exterior:2 mse:1 complex:2 necessarily:1 constructing:1 domain:1 interpolating:2 dense:3 animation:1 body:2 augmented:1 wish:1 lie:2 candidate:1 perceptual:1 ix:1 down:2 erroneous:2 bishop:1 grouping:4 exists:1 ci:1 texture:16 illumination:1 jagersand:1 smoothly:1 mill:1 appearance:42 likely:1 beymer:2 prevents:1 tracking:1 ma:2 identity:2 ann:9 rbf:11 invalid:1 specifically:1 except:1 select:3 plenoptic:2 violated:1

Call-based Fraud Detection in Mobile Communication Networks using a Hierarchical Regime-Switching Model Jaakko Hollmen Helsinki University of Technology Lab. of Computer and Information Science P.O. Box 5400, 02015 HUT, Finland laakko.Hollmen@hut.fi Volker Tresp Siemens AG, Corporate Technology Dept. Information and Communications 81730 Munich, Germany Volker.Tresp@mchp.siemens.de Abstract Fraud causes substantial losses to telecommunication carriers. Detection systems which automatically detect illegal use of the network can be used to alleviate the problem. Previous approaches worked on features derived from the call patterns of individual users. In this paper we present a call-based detection system based on a hierarchical regime-switching model. The detection problem is formulated as an inference problem on the regime probabilities. Inference is implemented by applying the junction tree algorithm to the underlying graphical model. The dynamics are learned from data using the EM algorithm and subsequent discriminative training. The methods are assessed using fraud data from a real mobile communication network. 1 INTRODUCTION Fraud is costly to a network carrier both in terms of lost income and wasted capacity. It has been estimated that the telecommunication industry looses approximately 2-5% of its total revenue to fraud. The true losses are expected to be even higher since telecommunication companies are reluctant to admit fraud in their systems. A fraudulent attack causes lots of inconveniences to the victimized subscriber which might motivate the subscriber to switch to a competing carrier. Furthermore, potential new customers would be very reluctant to switch to a carrier which is troubled with fraud. Mobile communication networks -which are the focus of this work- are particularly appealing to fraudsters as the calling from the mobile terminal is not bound to a physical place and a subscription is easy to get. This provides means for an illegal high-profit business requiring minimal investment and relatively low risk of getting caught. Fraud is 890 J. Hollmen and V. Tresp usually initiated by a mobile phone theft, by cloning the mobile phone card or by acquiring a subscription with false identification. After intrusion the subscription can be used for gaining free services either for the intruder himself or for his illegal customers in form of call-selling. In the latter case, the fraudster sells calls to customers for reduced rates. The earliest means of detecting fraud were to register overlapping calls originating from one subscription, evidencing card cloning. While this procedure efficiently detects cloning, it misses a large share of other fraud cases. A more advanced system is a velocity trap which detects card cloning by using an upper speed limit at which a mobile phone user can travel. Subsequent calls from distant places provide evidence for card cloning. Although a velocity trap is a powerful method of detecting card cloning, it is ineffective against other types of fraud. Therefore there is great interest in detection systems which detect fraud based on an analysis of behavioral patterns (Barson et aI., 1996, Burge et aI., 1997, Fawcett and Provost, 1997, Taniguchi et aI., 1998). In an absolute analysis, a user is classified as a fraudster based on features derived from daily statistics summarizing the call pattern such as the average number of calls. In a differential analysis, the detection is based on measures describing the changes in those features capturing the transition from a normal use to fraud. Both approaches have the problem of finding efficient feature representations describing normal and fraudulent behavior. As they usually derive features as summary statistics over one day, they are plagued with a latency time of up to a day to detect fraudulent behavior. The resulting delay in detection can already lead to unacceptable losses and can be exploited by the fraudster. For these reasons real-time fraud detection is seen to be the most important development in fraud detection (Pequeno, 1997). In this paper we present a real-time fraud detection system which is based on a stochastic generative model. In the generative model we assume a variable victimized which indicates if the account has been victimized by a fraudster and a second variable fraud which indicates if the fraudster is currently performing fraud. Both variables are hidden. Furthermore, we have an observed variable call which indicates if a call being is performed or not. The transition probabilities from no-call to call and from call to no-call are dependent on the state of the variable fraud. Overall, we obtain a regime-switching time-series model as described by Hamilton (1994), with the modifications that first, the variables in the time series are not continuous but binary and second, the switching variable has a hierarchical structure. The benefit of the hierarchical structure is that it allows us to model the time-series at different time scales. At the lowest hierarchical level, we model the dynamical behavior of the individual calls, at the next level the transition from normal behavior to fraudulent behavior and at the highest level the transition to being victimized. To be able to model a time-series at different temporal resolutions was also the reason for introducing a hierarchy into a hidden Markov model for Jordan, Ghahramani and Saul (1997). Fortunately, our hidden variables have only a small number of states such that we do not have to work with the approximation techniques those authors have introduced. Section 2 introduces our hierarchical regime-switching fraud model. The detection problem is formulated as an inference problem on the regime probabilities based on subscriber data. We derive iterative algorithms for estimating the hidden variables fraud and victimized based on past and present data (filtering) or based on the complete set of observed data (smoothing). We present EM learning rules for learning the parameters in the model using observed data. We develop a gradient based approach for fine tuning the emission probabilities in the non-fraud state to enhance the discrimination capability of the model. In Section 3 we present experimental results. We show that a system which is fine-tuned on real data can be used for detecting fraudulent behavior on-line based on the call patterns. In Section 4 we present conclusions and discuss further applications and extensions of our fraud model. Fraud Detection Using a Hierarchical Regime-Switching Model 2 891 THE HIERARCHICAL REGIME-SWITCHING FRAUD MODEL 2.1 THE GENERATIVE MODEL The hierarchical regime-switching model consists of three variables which evolve in time stochastically according to first-order Markov chains. The first binary variable Vt (victimized) is equal to one if the account is currently being victimized by a fraudster and zero otherwise. The states of this variable evolve according to the state transition probabiliP(Vt = ilVt_l j); i,j 0,1. The second binary variable St (fraud) is ties pij equal to one if the fraudster currently performs fraud and is equal to zero if the fraudster is inactive. The change between actively performing fraud and intermittent silence is typical for a victimized account as is apparent from Figure 3. Note that this transient bursty behavior of a victimized account would be difficult to capture with a pure feature based approach. The states of this variable evolve following the state transition probabilities pfjk = P(St ilvt j,St-l = k,);i,j,k 0,1. Finally, the binary variable Yt (call) is equal to one if the mobile phone is being used and zero otherwise with state transition matrix pfjk P(Yt ilst j, Yt-l k); i, j, k = 0,1. Note that this corresponds to the assumption of exponentially distributed call duration. Although not quite realistic, this is the general assumption in telecommunications. Typically, both the frequency of calls and the lengths of the calls are increased when fraud is executed. The joint probability of the time series up to time T is then = = = = = = = = P(VT' ST, YT) = P(vo, So, Yo) = = T T T t=l t=l t=l II P(Vt!Vt-l) II P(stlvt, St-l) II P(Ytlst, Yt-l) (1) where in the experiments we used a sampling time of one minute. Furthermore, VT {vo, ... , VT }, ST = {so, ... , ST }, YT = {Yo, ... , YT} and P(vo, So, Yo) is the prior distribution of the initial states. Figure 1: Dependency graph of the hierarchical regime-switching fraud model. The square boxes denote hidden variables and the circles observed variables. The hidden variable Vt on the top describes whether the subscriber account is victimized by fraud. The hidden variable St indicates if fraud is currently being executed. The state of St determines the statistics of the variable call Yt. 2.2 INFERENCE: FILTERING AND SMOOTIDNG When using the fraud detection system, we are interested to estimate the probability that an account is victimized or that fraud is currently occurring based on the call patterns up to the current point in time (filtering). We can calculate the probabilities of the states of the hidden variables by applying the following equations recursively with t = 1, ... , T. J. Hol/men and V Tresp 892 P(Vt = i, St-1 = k!Yt-1) = '2::prlP(Vt-1 = l, St-1 = kIYt- 1) I = j!Yt-1) = '2:: PjikP(Vt = i, St-1 = kIYt-1) P(Vt = i, St k where c is a scaling factor. These equations can be derived from the junction tree algorithm for the Bayesian networks (Jensen, 1996). We obtain the probability of victimization and fraud by simple marginalization P(Vt = ilYt ) = L = i, = jlYr) ; P(St = jlYd = L P(Vt St P(Vt = i, St = jlYd? i j In some cases -in particular for the EM learning rules in the next section- we might be interested in estimating the probabilities of the hidden states at some time in the past (smoothing). In this case we can use a variation of the smoothing equations described in Hamilton (1994) and Kim (1994). After performing the forward recursion, we can calculate the probability of the hidden states at time tf given data up to time T > tf iterating the following equations with t = T, T - 1, ... ,1. P(Vt+1 """' P(Vt+1 = k,St+1 = lIYT) P( _ k -ll?') P(Vt+l Vt+1 - ,St+1 t . = k,St = JIYT) = ~ I , ?IV) P( Vt=z,St=JI.T 2.3 . s = k,St = JIYt}Plkj """' P(Vt+1 = k,St =jIYT)p( . ?Iv") v P( -k _ 'I?,) Vt=z,St=)I.tPki k Vt+1 - ,St - J t =~ EM LEARNING RULES Parameter estimation in the regime-switching model is conveniently formulated as an incomplete data problem, which can be solved using the EM algorithm (Hamilton, 1994). Each iteration of the EM algorithm is guaranteed to increase the value of the marginal loglikelihood function until a fixed point is reached. This fixed point is a local optimum of the marginal log-likelihood function. In the M-step the model parameters are optimized using the estimates of the hidden states using the current parameter estimates. Let 0 {prj, Pijk' P;kj} denote the current parameter estimates. The new estimates are obtained using = 2:;=1 P( Vt v Pij = s Pijk = Y Pikj = = i, Vt-1 = jIYT; 0) ",T ( 'I ," L....t=l P Vt-1 = J }T; 0) 2:;-1 P(St = i, Vt = j, St-1 = kIYT; 0) T . 2:t=l P( Vt = J, St-1 = kIYT; 0) 2:;=l,if Yt=i andYt_l=j P(St-1 = kIYT;O) T 2:t=l, if Yt-l=j P(St-1 kIYT; 0) = Fraud Detection Using a Hierarchical Regime-Switching Model 893 The E-step determines the probabilities on the right sides of the equations using the current parameter estimates. These can be determined using the smoothing equations from the previous section directly by marginalizing P(Vt = k, St = l, Vt+l = i, St+1 = jIYT) where the terms on the right side are obtained from the equations in the last Section. 2.4 DISCRIMINATIVE TRAINING In our data setting, it is not known when the fraudulent accounts were victimized by fraud. This is why we use the EM algorithm to learn the two regimes from data in an incomplete data setting. We know, however, which accounts were victimized by fraud. After EM learning the discrimination ability of the model was not satisfactory. We therefore used the labeled sequences to improve the model. The reason for the poor performance was credited to unsuitable call emission probabilities in the normal state. We therefore mini- L:i (maXt P(v!i)I}~(i)) - t(i?)2 with regard to the parameter mize the error function E P;=O,j=O ,k=O' where the t(i) = {O, I} is the label for the sequence i. The error function was minimized with Quasi-Newton procedure with numerical differentiation. = 3 EXPERIMENTS To test our approach we used a data set consisting of 600 accounts which were not affected by fraud and 304 accounts which were affected by fraud. The time period for non-fraud and fraud accounts were 49 and 92 days, respectively. We divided the data equally into training data and test data. From the non-fraud data we estimated the parameters describing the normal calling behavior, i.e. pr,j =O,k' Next, we fixed the probability that an account is victimized from one time step to the next to PY=l,j=O = 10- 5 and the probability that a victimized account becomes de-victimized as pi=O,j=l = 5 X 10- 4 ? Leaving those parameters fixed the remaining parameters were trained using the fraudulent accounts and the EM algorithm described in Section 2. We had to do unsupervised training since it was known by velocity check that the accounts were affected but it was not clear when the intrusion occurred. After unsupervised training, we further enhanced the discrimination capability of the system which helped us reduce the amount of false alarms. The final model parameters can be found in the Appendix. After training, the system was tested using the test data. Unfortunately, it is not known when the accounts were attacked by fraud, but only on per-account basis if an account was at some point a victim of fraud. Therefore, we declare an account to be victimized if the victimized variable at some point exceeds the threshold. Also, it is interesting to study the results shown in Figure 3. We show data and posterior time-evolving probabilities for an account which is known to be victimized. From the call pattern it is obvious that there are periods of suspiciously high traffic at which the probability of victimization is recognized to be very high. We also see that the variable fraud St follows the bursty behavior of the fraudulent behavior correctly. Note, that for smoothing which is important both for a retrospective analysis of call data and for learning, we achieve smoother curves for the victimized variable. J. Hollmen and V. Tresp 894 ?? .. ,- - ' , " 00 aDI 01? OeD O.Dt OIlS 0,08 om o.oa 001 0.1 O.D1 0,(2 OeD 001 0.05 001 oar Q.O& 001 01 Figure 2: The Receiver Operating Characteristic (ROC) curves are shown for on-line detection (left figure) and for retrospective classification (right figure). In the figures, detection probability is plotted against the false alarm probability. The dash-dotted lines are results before, the solid lines after discriminative training. We can see that the discriminative training improves the model considerably. After EM training and discriminative training, we tested the model both in on-line detection mode (filtering) and in retrospective classification (smoothing) with smoothed probabilities. The detection results are shown in Figure 2. With a fixed false alarm probability of 0.003, the detection probabilities for the training set were found to be 0.974 and 0.934 using on-line detection mode and with smoothed probabilities, respectively. With a testing set and a fixed false alarm probability of 0.020, we obtain the detection probabilities of 0.928 and 0.921, for the on-line detection and for retrospective classification, respectively. 4 CONCLUSIONS We presented a call-based on-line fraud detection system which is based on a hierarchical regime-switching generative model. The inference rules are obtained from the junction tree algorithm for the underlying graphical model. The model is trained using the EM algorithm in an incomplete data setting and is further refined with gradient-based discriminative training, which considerably improves the results. A few extensions are in the process of being implemented. First of all , it makes sense to use more than one fraud model for the different fraud scenarios and several user models to account for different user profiles. For these more complex models we might have to rely on approximations techniques such as the ones introduced by Jordan, Ghahramani and Saul (1997). Appendix The model parameters after EM training and discriminative training. Note that entering the fraud state without first entering the victimized state is impossible. 0.9559 pY. . ,)= . Ok , = ( 0.3533 1.0000 pi,j=O,k = ( 0.0000 0.0441 ) 0.6467 0.0000 ) 1.0000 0.9292 p?t ,J= . 1 , k = ( 0.0570 0.9979 pi,j=l ,k 0.0086 =( 0.0708 ) 0.9430 0.0021 ) 0.9914 895 Fraud Detection Using a Hierarchical Regime-Switching Model _-o:t 111111111111U 11111] 111I111111[IIIIIIIIJIillIIIIIIII 0.5 1.5 ~o:~: 1.5 :\J 2 2.5 1 .5 2 2.5 0.5 0.5 1 : 2 j 1 2 .5 Figure 3: The first line shows the calling data Yt from a victimized account. The second and third lines show the states of the victimized and fraud variables, respectively. Both are calculated with the filtering equations. The fourth and fifth lines show the same variables using the smoothing equations. The displayed time window period is seventeen days. References Barson P., Field, S., Davey, N., McAskie, G., and Frank, R. (1996). The Detection of Fraud in Mobile Phone Networks. Neural Network World, Vol. 6, No.4. Bengio, Y. (1996). Markovian Models for Sequential Data. Technical Report # 1049, Universite de Montreal. Burge, P., Shawe-Taylor J., Moreau Y., Verrelst, H., Stormann C. and Gosset, P. (1997). BRUTUS - A Hybrid Detection Tool. Proc. of ACTS Mobile Telecommunications Summit, Aalborg, Denmark. Fawcett, T. and Provost, F. (1997). Adaptive Fraud Detection. Journal of Data Mining and Knowledge Discovery, , Vol. I, No.3, pp. 1-28. Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press. Jensen, Finn V. (1996). Introduction to Bayesian Networks. UCL Press. Jordan, M. I, Ghahramani, Z. and Saul, L. K. (1997). Hidden Markov Decision Trees, in Advances in Neural Information Processing Systems: Proceedings of the 1996 Conference (NIPS'9), MIT-Press, pp. 501-507. Kim, c.-J. (1994). Dynamical linear models with Markov-switching. Journal of Econometrics, Vol. 60, pp. 1-22. Pequeno, K. A.(1997). Real-Time fraud detection: Telecom's next big step. Telecommunications (America Edition), Vol. 31, No.5, pp. 59-60. Taniguchi, M., Haft, M., Hollmen, J. and Tresp, V. (1998). Fraud detection in communications networks using neural and probabilistic methods. Proceedings of the 1998 IEEE Int. Con! in Acoustics, Speech and Signal Processing (ICASSP'98). Vol. 2. pp. 1241-1244. 

1505 |@word verrelst:1 subscriber:4 profit:1 solid:1 recursively:1 initial:1 series:6 tuned:1 past:2 current:4 distant:1 realistic:1 numerical:1 subsequent:2 discrimination:3 generative:4 inconvenience:1 provides:1 detecting:3 attack:1 unacceptable:1 differential:1 consists:1 behavioral:1 expected:1 behavior:10 terminal:1 detects:2 automatically:1 company:1 window:1 becomes:1 estimating:2 underlying:2 lowest:1 loos:1 ag:1 finding:1 differentiation:1 temporal:1 act:1 tie:1 hamilton:4 carrier:4 service:1 declare:1 local:1 before:1 haft:1 limit:1 switching:14 initiated:1 approximately:1 credited:1 might:3 testing:1 lost:1 investment:1 procedure:2 evolving:1 illegal:3 fraud:56 get:1 risk:1 applying:2 impossible:1 py:2 customer:3 yt:13 caught:1 duration:1 resolution:1 pure:1 rule:4 his:1 variation:1 enhanced:1 hierarchy:1 user:5 velocity:3 particularly:1 econometrics:1 summit:1 labeled:1 observed:4 solved:1 capture:1 calculate:2 highest:1 substantial:1 jaakko:1 dynamic:1 oed:2 hol:1 motivate:1 trained:2 basis:1 selling:1 icassp:1 joint:1 america:1 evidencing:1 refined:1 apparent:1 quite:1 victim:1 loglikelihood:1 tested:2 otherwise:2 ability:1 statistic:3 final:1 sequence:2 ucl:1 achieve:1 getting:1 optimum:1 prj:1 ilvt:1 derive:2 develop:1 montreal:1 implemented:2 stochastic:1 transient:1 alleviate:1 extension:2 hut:2 normal:5 plagued:1 great:1 bursty:2 finland:1 estimation:1 proc:1 travel:1 label:1 currently:5 tf:2 tool:1 mit:1 mobile:10 volker:2 earliest:1 derived:3 focus:1 emission:2 yo:3 indicates:4 cloning:6 likelihood:1 intrusion:2 check:1 kim:2 detect:3 summarizing:1 sense:1 inference:5 dependent:1 typically:1 hidden:12 originating:1 quasi:1 interested:2 germany:1 overall:1 classification:3 development:1 smoothing:7 marginal:2 equal:4 field:1 sampling:1 sell:1 unsupervised:2 minimized:1 report:1 few:1 aalborg:1 individual:2 consisting:1 detection:29 interest:1 mining:1 introduces:1 chain:1 daily:1 tree:4 iv:2 incomplete:3 taylor:1 circle:1 plotted:1 minimal:1 increased:1 industry:1 markovian:1 introducing:1 delay:1 dependency:1 considerably:2 st:32 probabilistic:1 enhance:1 admit:1 stochastically:1 actively:1 account:22 potential:1 de:3 int:1 register:1 performed:1 helped:1 lot:1 lab:1 traffic:1 reached:1 capability:2 om:1 square:1 characteristic:1 efficiently:1 identification:1 bayesian:2 classified:1 against:2 frequency:1 pp:5 obvious:1 universite:1 con:1 seventeen:1 reluctant:2 knowledge:1 improves:2 ok:1 higher:1 dt:1 day:4 box:2 furthermore:3 until:1 overlapping:1 mode:2 oil:1 requiring:1 true:1 entering:2 satisfactory:1 ll:1 subscription:4 complete:1 vo:3 performs:1 fi:1 physical:1 ji:1 exponentially:1 occurred:1 ai:3 tuning:1 shawe:1 had:1 operating:1 taniguchi:2 posterior:1 phone:5 scenario:1 binary:4 vt:30 exploited:1 seen:1 fortunately:1 recognized:1 period:3 signal:1 mize:1 ii:3 smoother:1 corporate:1 exceeds:1 technical:1 divided:1 equally:1 himself:1 iteration:1 fawcett:2 fine:2 leaving:1 ineffective:1 jordan:3 call:27 bengio:1 easy:1 switch:2 marginalization:1 competing:1 reduce:1 inactive:1 whether:1 retrospective:4 speech:1 cause:2 latency:1 iterating:1 clear:1 amount:1 reduced:1 dotted:1 estimated:2 per:1 correctly:1 vol:5 affected:3 threshold:1 wasted:1 graph:1 powerful:1 telecommunication:6 fraudulent:8 fourth:1 place:2 intruder:1 decision:1 appendix:2 scaling:1 capturing:1 bound:1 guaranteed:1 dash:1 pijk:2 worked:1 helsinki:1 calling:3 speed:1 performing:3 relatively:1 munich:1 according:2 poor:1 describes:1 em:12 appealing:1 modification:1 pr:1 equation:9 describing:3 discus:1 know:1 finn:1 junction:3 hierarchical:13 top:1 remaining:1 graphical:2 newton:1 unsuitable:1 ghahramani:3 already:1 costly:1 gradient:2 card:5 capacity:1 oa:1 reason:3 denmark:1 length:1 mini:1 difficult:1 executed:2 unfortunately:1 frank:1 upper:1 markov:4 attacked:1 displayed:1 communication:5 intermittent:1 provost:2 smoothed:2 introduced:2 optimized:1 acoustic:1 learned:1 nip:1 able:1 usually:2 pattern:6 dynamical:2 regime:15 gaining:1 business:1 rely:1 hybrid:1 recursion:1 advanced:1 improve:1 technology:2 tresp:6 kj:1 prior:1 discovery:1 evolve:3 marginalizing:1 loss:3 men:1 interesting:1 filtering:5 revenue:1 i111111:1 pij:2 share:1 pi:3 maxt:1 summary:1 last:1 free:1 silence:1 side:2 saul:3 absolute:1 mchp:1 fifth:1 benefit:1 distributed:1 regard:1 curve:2 calculated:1 transition:7 davey:1 world:1 moreau:1 author:1 forward:1 adaptive:1 income:1 receiver:1 discriminative:7 continuous:1 iterative:1 why:1 learn:1 adi:1 complex:1 big:1 alarm:4 profile:1 edition:1 telecom:1 roc:1 burge:2 third:1 minute:1 jensen:2 evidence:1 trap:2 false:5 sequential:1 occurring:1 conveniently:1 acquiring:1 corresponds:1 determines:2 formulated:3 change:2 typical:1 determined:1 miss:1 total:1 experimental:1 siemens:2 latter:1 assessed:1 dept:1 princeton:1 d1:1

Applications of multi-resolution neural networks to mammography Clay D. Spence and Paul Sajda Sarnoff Corporation CN5300 Princeton, NJ 08543-5300 {cspence, psajda }@sarnoff.com Abstract We have previously presented a coarse-to-fine hierarchical pyramid/neural network (HPNN) architecture which combines multiscale image processing techniques with neural networks. In this paper we present applications of this general architecture to two problems in mammographic Computer-Aided Diagnosis (CAD). The first application is the detection of microcalcifications. The <:oarse-to-fine HPNN was designed to learn large-scale context information for detecting small objects like microcalcifications. Receiver operating characteristic (ROC) analysis suggests that the hierarchical architecture improves detection performance of a well established CAD system by roughly 50 %. The second application is to detect mammographic masses directly. Since masses are large, extended objects, the coarse-to-fine HPNN architecture is not suitable for this problem. Instead we construct a fine-to-coarse HPNN architecture which is designed to learn small-scale detail structure associated with the extended objects. Our initial results applying the fine-to-coarse HPNN to mass detection are encouraging, with detection performance improvements of about 36 %. We conclude that the ability of the HPNN architecture to integrate information across scales, both coarse-to-fine and fine-to-coarse, makes it well suited for detecting objects which may have contextual clues or detail structure occurring at scales other than the natural scale of the object. 1 Introduction In a previous paper [8] we presented a coarse-to-fine hierarchical pyramid/neural network (HPNN) architecture that combines multi-scale image processing tech- Applications of Multi-Resolution Neural Networks to Mammography 939 niques with neural networks to search for small targets in images (see figure IA). To search an image we apply the network at a position and use its output as an estimate of the probability that a target (an object of the class we wish to find) is present there. We then repeat this at each position in the image. In the coarseto-fine HPNN , the hidden units of networks operating at low resolution or coarse scale learn associated context information, since the targets themselves are difficult to detect at low resolution. The context is then passed to networks searching at higher resolution. The use of context can significantly improve detection performance since small objects have few distinguishing features. In the HPNN each of the networks receives information directly from only a small part of several feature images , and so the networks can be relatively simple. The network at the highest resolution integrates the contextual information learned at coarser resolutions to detect the object of interest. The HPNN architecture can be extended by considering the implications of inverting the information flow in the coarse-to-fine architecture. This fine-to-coarse HPNN would have networks extracting detail structure at fine resolutions of the image and then passing this detail information to networks operating at coarser scales (see figure IB). For many types of objects, information about the fine structure is important for discriminating between different classes. The fine-to-coarse HPNN is therefore a natural architecture for exploiting fine detail information for detecting extended objects. In this paper , we present our experiences in applying the HPNN framework to two problems in mammographic Computer-Aided Diagnosis (CAD); that of detecting microcalcifications in mammograms and that of detecting malignant masses in mammograms. The coarse-to-fine HPNN architecture is well-suited for the microcalcification problem , while the fine-to-coarse HPNN is suited for mass detection. We evaluate the performance and utility of the HPNN framework by considering its effects on reducing false positive rates in a well characterized CAD system. The University of Chicago (UofC) has been actively developing ' mammographic CAD systems for micro calcification and mass detection [6] and has been evaluating their performance clinically. A general block diagram showing the basic processing elements of these CAD systems is shown in figure 2. First, a pre-processing step is used to segment the breast area and increase the overall signal-to-noise levels in the image. Regions of interest (ROIs) are defined at this stage , representing local areas of the breast which potentially contain a cluster of micro calcifications or a mass. The next stage typically involves feature extraction and rule-based/heuristic analysis , in order to prune false positives. The remaining ROIs are classified as positive or negative by a statistical classifier or neural network. The CAD system is used as a "second reader", aiding the radiologist by pointing out spots to double check. One of the key requirements of CAD is that false positive rates be low enough that radiologists will not ignore the CAD system output. Therefore it is critical to reduce false positive rates of CAD systems without significant reductions in sensitivity. In this paper we evaluate the HPNN framework within the context of reducing the false positive rates of the UofC CAD systems for microcalcification and mass detection. In both cases the HPNN acts as a post-processor of the UofC CAD system. 2 Microcalcification detection Microcalcifications are calcium deposits in breast tissue that appear as very small bright dots in mammograms. Clusters of microcalcifications frequently occur around tumors. Unfortunately microcalcification clusters are sometimes missed, since they 940 C. D. Spence and P. Sqjda P(t) P(t) Figure 1: Hierarchical pyramid/neural network architectures for (A) detecting microcalcifications and (B) detecting masses. In (A) context is propagated from low to high resolution via the hidden units of low resolution networks. In (B) small scale detail information is propagated from high to low resolution. In both cases the output of the last integration network is an estimate of the probability that a target is present. Mammogram -1 Feature extraction Pre-processing and rule-based/ heuristic analysis Statistical/NN classifier Mass or Cluster locations Figure 2: Block diagram for a typical CAD detection system. can be quite subtle and the radiologists can only spend about a minute evaluating a patient's mammograms. Data used for the micro calcification experiments was provided by The University of Chicago. The first set of data consists of 50 true positive and 86 false positive ROls_ These ROIs are 99x99 pixels and digiti7,ed at 100 micron resolution. A second set of data from the UofC clinical testing database included 47 true positives and 103 false positives, also 99x99 and sampled at 100 micron resolution. We trained the coarse-to-fine HPNN architecture in figure 1A as a detector for individual calcifications. For each level in the pyramid a network is trained, beginning with the network at low resolution. The network at a particular pyramid level is applied to one pixel at a time in the image at that resolution , and so produces an output at each pixel. All of the networks are trained to detect micro calcifications, however, at low resolutions the micro calcifications are not directly detectable. To achieve better than chance performance, the networks at those levels must learn something about the context in which micro calcifications appear. To integrate context information with the other features the outputs of hidden units from low resolution networks are propagated hierarchically as inputs to networks operating at higher resolutions. Input to the neural networks come from an integrated feature pyramid (IFF) [lJ. To construct the IFP, we used steerable filters [3J to compute local orientation energy. The steering properties of these filters enable the direct computation of the orientation having maximum energy. We constructed features which represent, at each pixel location, the maximum energy (energy at 8rnax) , the energy at the Applications of Multi-Resolution Neural Networks to Mammography cc Az (7 Az HPNN FPF (7FPF Az Chicago NN FPF (7 Az TPF=l.O 1 2 3 4 5 .93 .94 .94 .93 .93 .03 .02 .03 .03 .03 .24 .21 .39 .48 .51 941 (7FPF TPF=l.O .11 .11 .19 .15 .06 .88 .91 .91 .90 .88 .04 .02 .03 .05 .05 .50 .43 .48 .56 .68 .11 .10 .19 .21 .21 Table 1: Comparison of HPNN and Chicago networks . orientation perpendicular to emu;]; (ernux - 90?), and the energy at the diagonal (energy at ernux - 45 0 ).l The resulting features are input into the coarse-to-fine network hierarchy. In examining the truth data for the ROI data set , we found that the experts who specified the microcalcification positions often made errors in these positions of up to ?2 pixels of the correct position. To take this uncertainty in position into account , we used the following error function Euop = - L pEPos log( 1 - IT (1 xEp y(X))) - L 10g(1 - y(x)) (1) x ENe y which we have called the Uncertain Object Position (UOP) error function [7].2 (y(x) is the network's output when applied to position x.) It is essentially the crossentropy error, but for positive examples the probability of generating a positive output (y( x), in this case) has been replaced by the probability of generating at least one positive output in a region or set of pixels p in the image. In our case each p is a five-by-five pixel square centered on the location specified by the expert. To this we added the standard weight decay regularization term. The regularization constant was adjusted to minimize the ten-fold cross-validation error. The coarse-to-fine HPNN was applied to each input ROI , and an image was constructed from the output of the Level 0 network at each pixel. Each of these pixel values is the network 's estimate of the probability that a microcalcification is present there. Training and testing were done using as jackknife protocol [5], whereby one half of the data (25 TPs and 43 FPs) was used for training and the other half for testing. We used five different random splits of the data into training and test sets. For a given ROI, the probability map produced by the network was thresholded at a given value to produce a binary detection map. Region growing was used to count the number of distinct detected regions. The ROI was classified as a positive if the number of regions was greater than or equal to a certain cluster criterion. Table 1 compares ROC results for the HPNN and another network that had been used in the University of Chicago CAD system [9] using five different cluster criterion (cc). Reported are the area under the ROC curve (Az), the standard deviation of A z across the subsets of the jackknife ((7 AJ, the false posit ive fraction at a true positive fraction of 1.0 (FPF@TPF= 1.0) and the standard deviation of the FPF across the subsets of the jackknife ((7FPF). A z and FPF@TPF = 1.0 represent 1 We found that the energies in the two diagonal directions were nearly identical. 2Keeler et al. [4] developed a network for object recognition that had some similarities to the UOP error. In fact the way in which the outputs of units are combined for their error function can be shown to be an approximation to the UOP error. 942 C. D. Spence and P Sajda the averages of the subsets of the jackknife. Note that both networks operate best when the cluster criterion is set to two. For this case the HPNN has a higher Az than the Chicago network while also halving the false positive rate. This difference, between the two networks ' A z and F P F values , is statistically significant (z-test; PAz = .0018, PFPF = .00001). A second set of data was also tested. 150 ROls taken from a clinical prospective study and classified as positive by the full Chicago CAD system (including the Chicago neural network) were used to test the HPNN. Though the Chicago CAD system classified all 150 ROls as positive, only 47 were in fact positive while 103 were negatives. We applied the HPNN trained on the entire previous data set to this new set of ROls. The HPNN was able to reclassify 47/103 negatives as negative, without loss in sensitivity (no false negatives were introduced). On examining the negative examples rejected by the coarse-to-fine HPNN, we found that many of these ROls contained linear, high-contrast structure which would otherwise be false positives for the Chicago network. The Chicago neural network presumably interprets the "peaks" on the linear structure as calcifications. However because the coarse-to-fine HPNN also integrates information from low resolution it can associate these "peaks" with the low-resolution linear structure and reject them. 3 Mass detection Although microcalcifications are an important cue for malignant masses in mammograms, they are not visible or even present in all cases. Thus mammographic CAD systems include algorithms to directly detect the presence of masses. We have started to apply a fine-to-coarse HPNN architecture to detect malignant masses in digitized mammograms. Radiologists often distinguish malignant from benign masses based on the detailed shape of the mass border and the presence of spicules alone the border. Thus to integrate this high resolution information to detect malignant masses, which are extended objects, we apply the fine-to-coarse HPNN of figure lB. As for microcalcifications, we apply the HPNN as a post-processor, but hei'e it processes the output of the mass-detection component of UofC CAD system. The data in our study consists of 72 positive and 100 negative ROls. These are 256-by256 pixels and are sampled at 200 micron resolution. At each level of the fine-to-coarse HPNN several hidden units process the feature images. The outputs of each unit at all of the positions in an image make up a new feature image. This is reduced in resolution by the usual pyramid blur-andsubsample operation to make an input feature image for the network units at the next lower resolution. We trained the entire fine-to-coarse HPNN as one network instead of training a network for each level, one level at a time. This training is quite straightforward. Back-propagating error through the network units is the same as in conventional networks. We must also back-propagate through the pyramid reduction operation, but this is linear and therefore quite simple. In addition we use the same UOP error function (Equation 1) used to train the coarse-to-fine architecture. The rationale for this application of the UOP error function is that the truth data specifies the location of the center of the mass at the highest resolution. However, because of the sub-sampling the center cannot be unambiguously assigned to a particular pixel at low resolution . The features input to the fine-to-coarse HPNN are filtered versions of the image, with filter kernels given by . 0/' (r e) = ( 'l/q,]1' q! 71"(q+ lp l)! )1 / 2r IPle-r2 / 2LIP I(r2)etp1> in polar q Applications of Multi-Resolution Neural Networks to Mammography Sensitivity 100% 95% 90% 80% Coarse-to-Fine HPNN Microcalcification 45% 47% 63% 69% 943 Fine-to-Coarse HPNN Mass 32% 36% 40% 78% Table 2: Detector Specificity (% reduction in false positive rate of UofC CAD system) . coordinates, with (q, p) E {(O, 1) , (1,0), (0, 2)}. These are combinations of derivatives of Gaussians, and can be written as combinations of separable filter kernels (products of purely horizontal and vertical filters) , so they can be computed at relatively low cost. They are also easy to steer, since this is just multiplication by a complex phase factor. We steered these in the radial and tangential directions relative to the tentative mass centers, and used the real and imaginary parts and their squares and products as features. The center coordinates of the are generated by the earlier stages of the CAD system. These features were extracted at each level of the Gaussian pyramid representation of the mass ROI, and used as inputs only to the network units at the same level. The fine-to-coarse HPNN is quite similar to the convolution network proposed by Le Cun, et al [2], however with a few notable differences. The fine-to-coarse HPNN receives as inputs preset features extracted from the image (in this case radial and tangential gradients) at each resolution, compared to the convolution network, whose inputs are the original pixel values at the highest resolution. Secondly, in the fine-to- coarse HPNN , the inputs to a hidden unit at a particular position are the pixel values at that position in each of the feature images , one pixel value per feature image. Thus the HPN N's hidden units do not learn linear filters, except as linear combinations of the filters used to form the features. Finally the fine-tocoarse HPNN is trained using the UOP error function , which is not used in the Le Cun network. Currently our best performing fine-to-coarse HPNN system for mass detection has two hidden units per pyramid level. This gives an ROC area of A z = 0.85 and eliminates 36 % of the false-positives at a cost of missing 5 % of the actual positives. To improve performance further , we are investigating different regularizers, richer feature sets, and more complex architectures, i.e., more hidden units. 4 Conclusion We have presented the application of multi-resolution neural network architectures to two problems in computer-aided diagnosis , the detection of micro calcifications in mammograms and the direct detection of malignant masses in mammograms. A summary of the performance of these architectures is given in Table 2. In the case of microcalcifications , the coarse-to-fine HPNN architecture successfully discovered large-scale context information that improves the system's performance in detecting small objects. A coarse-to-fine HPNN has been directly integrated with the UofC CAD system for micro calcification detection and the complete system is undergoing clinical evaluation. In the case of malignant masses, a fine-to-coarse HPNN architecture was used to exploit information from fine resolution detail which could be used to differentiate C. D. Spence and P Sajda 944 malignant from benign masses. The results of this network are encouraging, but additional improvement is needed. In general, we have found that the multi-resolution HPNNs are a useful class of network architecture for exploiting and integrating information at multiple scales. 5 Acknowledgments This work was funded by the National Information Display Laboratory, DARPA through ONR contract No. N00014-93-C-0202, and the Murray Foundation. We would like to thank Drs. Robert Nishikawa and Maryellen Giger of The University of Chicago for useful discussions and providing the data. References [1] Peter Burt. Smart sensing within a pyramid vision machine. Proceedings IEEE, 76(8):1006-1015, 1988. Also in Neuro- Vision Systems, Gupta and Knopf, eds., 1994. [2] Y. Le Cun, B. Boser, J . S. Denker, and D. Henderson. Handwritten digit recognition with a back-propagation network. In David S. Touretzky, editor, Advances in Neural Information Processing Systems 2, pages 396- 404, 2929 Campus Drive, San Mateo: CA 94403 , 1991. Morgan-Kaufmann Publishers. [3] William T. Freeman and Edward H. Adelson. The design and use of steerable filters. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI13(9):891- 906, 1991. [4] James D. Keeler, David E. Rumelhart, and Wee-Keng Leow. Integrated segmentation and recognition of hand-printed numerals. In Richard P. Lippmann, John E. Moody, and David S. Touretzky, editors, Advances in Neural Information Processing Systems 3, pages 557- 563, 2929 Campus Drive, San Mateo, CA 94403, 1991. Morgan-Kaufmann Publishers. [5] Charles Metz. Current problems in ROC analysis. In Proceedings of the Chest Imaging Conference, pages 315- 33, Madison, WI, November 1988. [6] R. M. Nishikawa, R. C. Haldemann, J. Papaioannou, M. L. Giger, P. Lu, R. A. Schmidt, D. E. Wolverton, U. Bick, and K. DoL Initial experience with a prototype clinical intelligent mammography workstation for computer-aided diagnosis. In Murray H. Loew and Kenneth M. Hanson, editors, Medical Imaging 1995, volume 2434, pages 65- 71, P .O. Box 10, Bellingham WA 98227-0010, 1995. SPIE. [7] Clay D. Spence. Supervised learning of detection and classification tasks with uncertain training data. In Image Understanding Workshop. ARPA, 1996. This Volume. [8] Clay D. Spence, John C. Pearson, and Jim Bergen. Coarse-to-fine image search using neural networks. In Gerald Tesauro, David S. Touretzky, and Todd K. Leen, editors, Advances in Neural Information Processing Systems 7, pages 981988, Massachusetts Institute of Technology, Cambridge, MA 02142, 1994. MIT Press. [9] W. Zhang, K. Doi, M. L. Giger, Y. Wu , R. M. Nishikawa, and R. Schmidt. Computerized detection of clustered micro calcifications in digital mammograms using a shift-invariant artificial neural network. Medical Physics, 21(4):517-524 , April 1994. 

1506 |@word version:1 propagate:1 leow:1 reduction:3 initial:2 iple:1 imaginary:1 current:1 com:1 contextual:2 cad:21 must:2 written:1 john:2 visible:1 chicago:12 blur:1 benign:2 shape:1 designed:2 alone:1 half:2 cue:1 intelligence:1 beginning:1 filtered:1 coarse:34 detecting:8 location:4 zhang:1 five:4 constructed:2 direct:2 fps:1 consists:2 uop:6 combine:2 roughly:1 themselves:1 frequently:1 growing:1 multi:7 freeman:1 encouraging:2 actual:1 considering:2 provided:1 campus:2 mass:26 developed:1 corporation:1 nj:1 act:1 classifier:2 rols:5 unit:13 medical:2 appear:2 positive:24 local:2 todd:1 mateo:2 suggests:1 sarnoff:2 perpendicular:1 statistically:1 acknowledgment:1 testing:3 spence:6 block:2 bellingham:1 digit:1 spot:1 steerable:2 area:4 significantly:1 reject:1 printed:1 pre:2 radial:2 integrating:1 specificity:1 cannot:1 context:9 applying:2 conventional:1 map:2 center:4 missing:1 straightforward:1 resolution:33 mammography:5 rule:2 searching:1 coordinate:2 target:4 hierarchy:1 distinguishing:1 associate:1 element:1 rumelhart:1 recognition:3 cspence:1 coarser:2 database:1 region:5 highest:3 gerald:1 trained:6 segment:1 smart:1 purely:1 darpa:1 sajda:3 train:1 distinct:1 doi:1 detected:1 artificial:1 pearson:1 quite:4 heuristic:2 spend:1 ive:1 whose:1 richer:1 otherwise:1 ability:1 differentiate:1 product:2 iff:1 achieve:1 az:6 exploiting:2 cluster:7 double:1 requirement:1 produce:2 generating:2 nishikawa:3 object:14 propagating:1 edward:1 involves:1 come:1 direction:2 posit:1 correct:1 filter:8 centered:1 enable:1 numeral:1 clustered:1 secondly:1 adjusted:1 keeler:2 around:1 fpf:8 roi:8 presumably:1 pointing:1 wolverton:1 polar:1 integrates:2 coarseto:1 currently:1 successfully:1 mit:1 gaussian:1 crossentropy:1 improvement:2 check:1 tech:1 contrast:1 detect:7 bergen:1 nn:2 typically:1 integrated:3 lj:1 entire:2 hidden:8 pixel:14 overall:1 classification:1 orientation:3 integration:1 mammographic:5 equal:1 construct:2 extraction:2 having:1 sampling:1 identical:1 adelson:1 nearly:1 intelligent:1 micro:9 few:2 tangential:2 richard:1 wee:1 national:1 individual:1 tpf:4 replaced:1 phase:1 william:1 detection:19 interest:2 evaluation:1 henderson:1 regularizers:1 radiologist:4 implication:1 experience:2 uncertain:2 arpa:1 earlier:1 steer:1 cost:2 deviation:2 subset:3 examining:2 paz:1 reported:1 combined:1 peak:2 sensitivity:3 discriminating:1 contract:1 physic:1 moody:1 expert:2 derivative:1 steered:1 actively:1 account:1 notable:1 metz:1 hpn:1 psajda:1 minimize:1 bright:1 square:2 kaufmann:2 characteristic:1 who:1 handwritten:1 produced:1 lu:1 computerized:1 cc:2 drive:2 processor:2 classified:4 tissue:1 detector:2 touretzky:3 ed:2 energy:8 james:1 associated:2 spie:1 workstation:1 propagated:3 sampled:2 massachusetts:1 improves:2 segmentation:1 clay:3 subtle:1 back:3 higher:3 supervised:1 unambiguously:1 april:1 leen:1 done:1 though:1 box:1 rejected:1 stage:3 just:1 hand:1 receives:2 horizontal:1 multiscale:1 propagation:1 aj:1 effect:1 contain:1 true:3 regularization:2 assigned:1 laboratory:1 whereby:1 criterion:3 tps:1 complete:1 image:21 charles:1 ifp:1 volume:2 significant:2 cambridge:1 cn5300:1 dot:1 had:2 funded:1 similarity:1 operating:4 something:1 tesauro:1 certain:1 n00014:1 binary:1 onr:1 morgan:2 greater:1 additional:1 steering:1 prune:1 signal:1 full:1 multiple:1 hpnn:45 characterized:1 clinical:4 cross:1 post:2 halving:1 neuro:1 basic:1 breast:3 vision:2 essentially:1 patient:1 represent:2 kernel:2 sometimes:1 pyramid:11 addition:1 fine:40 diagram:2 publisher:2 operate:1 eliminates:1 flow:1 chest:1 extracting:1 presence:2 split:1 enough:1 easy:1 architecture:21 interprets:1 reduce:1 prototype:1 shift:1 utility:1 passed:1 peter:1 passing:1 useful:2 detailed:1 aiding:1 ten:1 reduced:1 specifies:1 per:2 diagnosis:4 key:1 thresholded:1 kenneth:1 imaging:2 fraction:2 micron:3 uncertainty:1 reader:1 wu:1 missed:1 distinguish:1 display:1 fold:1 occur:1 performing:1 separable:1 relatively:2 jackknife:4 developing:1 clinically:1 combination:3 across:3 wi:1 lp:1 cun:3 invariant:1 ene:1 taken:1 equation:1 previously:1 hei:1 detectable:1 malignant:8 count:1 needed:1 drs:1 operation:2 gaussians:1 apply:4 denker:1 hierarchical:4 schmidt:2 original:1 remaining:1 include:1 madison:1 exploit:1 murray:2 added:1 usual:1 diagonal:2 gradient:1 thank:1 prospective:1 providing:1 difficult:1 unfortunately:1 robert:1 potentially:1 negative:7 design:1 calcium:1 vertical:1 convolution:2 november:1 extended:5 digitized:1 jim:1 discovered:1 lb:1 burt:1 introduced:1 inverting:1 david:4 specified:2 tentative:1 hanson:1 learned:1 boser:1 established:1 emu:1 able:1 pattern:1 rnax:1 including:1 ia:1 suitable:1 critical:1 natural:2 representing:1 improve:2 technology:1 started:1 understanding:1 multiplication:1 relative:1 deposit:1 loss:1 bick:1 rationale:1 validation:1 foundation:1 integrate:3 digital:1 editor:4 summary:1 repeat:1 last:1 institute:1 curve:1 evaluating:2 made:1 clue:1 san:2 transaction:1 lippmann:1 ignore:1 investigating:1 receiver:1 conclude:1 search:3 table:4 lip:1 learn:5 ca:2 complex:2 protocol:1 hierarchically:1 keng:1 border:2 noise:1 paul:1 roc:5 sub:1 position:11 wish:1 ib:1 mammogram:10 minute:1 showing:1 undergoing:1 r2:2 decay:1 sensing:1 gupta:1 dol:1 workshop:1 false:13 niques:1 occurring:1 suited:3 contained:1 truth:2 chance:1 extracted:2 ma:1 aided:4 included:1 typical:1 except:1 reducing:2 preset:1 tumor:1 called:1 evaluate:2 princeton:1 tested:1

Lazy Learning Meets the Recursive Least Squares Algorithm Mauro Birattari, Gianluca Bontempi, and Hugues Bersini Iridia - Universite Libre de Bruxelles Bruxelles, Belgium {mbiro, gbonte, bersini} @ulb.ac.be Abstract Lazy learning is a memory-based technique that, once a query is received, extracts a prediction interpolating locally the neighboring examples of the query which are considered relevant according to a distance measure. In this paper we propose a data-driven method to select on a query-by-query basis the optimal number of neighbors to be considered for each prediction. As an efficient way to identify and validate local models, the recursive least squares algorithm is introduced in the context of local approximation and lazy learning. Furthermore, beside the winner-takes-all strategy for model selection, a local combination of the most promising models is explored. The method proposed is tested on six different datasets and compared with a state-of-the-art approach. 1 Introduction Lazy learning (Aha, 1997) postpones all the computation until an explicit request for a prediction is received. The request is fulfilled interpolating locally the examples considered relevant according to a distance measure. Each prediction requires therefore a local modeling procedure that can be seen as composed of a structural and of a parametric identification . The parametric identification consists in the optimization of the parameters of the local approximator. On the other hand, structural identification involves, among other things, the selection of a family of local approximators, the selection of a metric to evaluate which examples are more relevant, and the selection of the bandwidth which indicates the size of the region in which the data are correctly modeled by members of the chosen family of approximators. For a comprehensive tutorial on local learning and for further references see Atkeson et al. (1997). As far as the problem of bandwidth selection is concerned, different approaches exist. The choice of the bandwidth may be performed either based on some a priori assumption or on the data themselves. A further sub-classification of data-driven approaches is of interest M. Birattari, G. Bontempi and H. Bersini 376 here. On the one hand, a constant bandwidth may be used; in this case it is set by a global optimization that minimizes an error criterion over the available dataset. On the other hand, the bandwidth may be selected locally and tailored for each query point. In the present work, we propose a method that belongs to the latter class of local data-driven approaches. Assuming a given fixed metric and local linear approximators, the method we introduce selects the bandwidth on a query-by-query basis by means of a localleave-oneout cross- validation. The problem of bandwidth selection is reduced to the selection of the number k of neighboring examples which are given a non-zero weight in the local modeling procedure. Each time a prediction is required for a specific query point, a set of local models is identified, each including a different number of neighbors. The generalization ability of each model is then assessed through a local cross-validation procedure. Finally, a prediction is obtained either combining or selecting the different models on the basis of some statistic of their cross-validation errors. The main reason to favor a query-by-query bandwidth selection is that it allows better adaptation to the local characteristics of the problem at hand. Moreover, this approach is able to handle directly the case in which the database is updated on-line (Bontempi et at., 1997). On the other hand, a globally optimized bandwidth approach would, in principle, require the global optimization to be repeated each time the distribution of the examples changes. The major contribution of the paper consists in the adoption of the recursive least squares algorithm in the context of lazy learning. This is an appealing and efficient solution to the intrinsically incremental problem of identifying and validating a sequence of local linear models centered in the query point, each including a growing number of neighbors. It is worth noticing here that a leave-one-out cross-validation of each model considered does not involve any significant computational overload, since it is obtained though the PRESS statistic (Myers, 1990) which simply uses partial results returned by the recursive least squares algorithm. Schaal and Atkeson (1998) used already the recursive least squares algorithm for the incremental update of a set of local models. In the present paper, we use for the first time this algorithm in a query-by-query perspective as an effective way to explore the neighborhood of each query point. As a second contribution, we propose a comparison, on a local scale, between a competitive and a cooperative approach to model selection. On the problem of extracting a final prediction from a set of alternatives, we compared a winner-takes-all strategy with a strategy based on the combination of estimators (Wolpert, 1992). In Section 5 an experimental analysis of the recursive algorithm for local identification and validation is presented. The algorithm proposed, used in conjunction with different strategies for model selection or combination, is compared experimentally with Cubist, the rule-based tool developed by Ross Quinlan for generating piecewise-linear models. 2 Local Weighted Regression Given two variables x E lR m and y E lR, let us consider the mapping only through a set of n examples {(Xi, yd} ~=l obtained as follows: f: lR m --t lR, known (1) where Vi, Ci is a random variable such that E[ciJ = 0 and E[ciCjJ = 0, Vj =1= i, and such that E[ciJ = I-lm(Xi), Vm ~ 2, where I-lmO is the unknown mth moment of the distribution of Ci and is defined as a function of Xi. In particular for m = 2, the last of the above mentioned properties implies that no assumption of global homoscedasticity is made. 377 Lazy Learning Meets the Recursive Least Squares Algorithm The problem of local regression can be stated as the problem of estimating the value that the regression function f(x) = E[Ylx] assumes for a specific query point x, using information pertaining only to a neighborhood of x. Given a query point x q , and under the hypothesis of a local homoscedasticity of Ci, the parameter (3 of a local linear approximation of f (.) in a neighborhood of Xq can be obtained solving the local polynomial regression: (2) where, given a metric on the space Rm, d( Xi, Xq) is the distance from the query point to the example, K (.) is a weight function, h is the bandwidth, and where a constant value 1 has been appended to each vector Xi in order to consider a constant term in the regression. ith In matrix notation, the solution of the above stated weighted least squares problem is given by: /3 = (X'W'WX)-lX'W'Wy = (Z'Z)-lZ'V = PZ'v, (3) where X is a matrix whose ith row is x~, y is a vector whose ith element is Yi, W is a diagonal matrix whose ith diagonal element is Wii = JK (d(Xi,Xq)jh), Z = WX, v = Wy, and the matrix X'W'WX = Z'Z is assumed to be non-singular so that its inverse P = (Z'Z)-l is defined. Once obtained the local linear polynomial approximation, a prediction of Yq = f(x q), is finally given by: Yq=X~/3 . (4) Moreover, exploiting the linearity of the local approximator, a leave-one-out crossvalidation estimation of the error variance E[ (Yq - Yq)2] can be obtained without any significant overload. In fact, using the PRESS statistic (Myers, 1990), it is possible to = Yj - xj /3 _ j' without explicitly identifying the parameters /3- j calculate the error from the examples available with the ph removed. The formulation of the PRESS statistic for the case at hand is the following: er cv _ ej - ,A _ Yj - xjPZ'v _ Yj - xj/3 Yj - x j {3 _ j - 1 - 'P Zj Zj - 1 - where zj is the ph row of Z and therefore Zj = WjjXj, and where h jj is the e1ementoftheHatmatrixH = ZPZ' = Z(Z'Z) - lZ' . 3 (5) h jj ' ph diagonal Recursive Local Regression In what follows, for the sake of simplicity, we will focus on linear approximator. An extension to generic polynomial approximators of any degree is straightforward. We will assume also that a metric on the space R m is given. All the attention will be thus centered on the problem of bandwidth selection. If as a weight function K(-) the indicator function K (d(Xi'X q)) = h {I 0 ifd(xi,xq)::; h, otherwise; (6) is adopted, the optimization of the parameter h can be conveniently reduced to the optimization of the number k of neighbors to which a unitary weight is assigned in the local M. Birattari, G. Bontempi and H. Bersini 378 regression evaluation. In other words, we reduce the problem of bandwidth selection to a search in the space of h( k) = d( x( k), Xq), where x( k) is the kth nearest neighbor of the query point. The main advantage deriving from the adoption of the weight function defined in Eq. 6, is that, simply by updating the parameter /3(k) of the model identified using the k nearest neighbors, it is straightforward and inexpensive to obtain /3 (k + 1). In fact, performing a step of the standard recursive least squares algorithm (Bierman, 1977), we have: P(k + 1) = P(k) _ P(k)x(k + l)x'(k + l)P(k) 1 + x'(k + l)P(k)x(k + 1) ,(k + 1) = P(k + l)x(k e(k + 1) = /3(k + 1) = + 1) y(k + 1) - x' (k + l)/3(k) /3(k) + ,(k + l)e(k + 1) where P(k) = (Z'Z)-l when h = h(k), and where x(k neighbor of the query point. (7) + 1) is the (k + l)th nearest Moreover, once the matrix P(k + 1) is available, the leave-one-out cross-validation errors can be directly calculated without the need of any further model identification: cv _ Yj - xj/3(k + 1) ej (k + 1) - 1 _ xjP(k + l)x/ (8) It will be useful in the following to define for each value of k the [k x 1] vector e CV (k) that contains all the leave-one-out errors associated to the model {3(k). Once an initialization /3(0) = jj and P(O) = P is given, Eq. 7 and Eq. 8 recursively evaluate for different values of k a local approximation of the regression function f(?), a prediction of the value of the regression function in the query point, and the vector of leave-one-out errors from which it is possible to extract an estimate of the variance of the prediction error. Notice that jj is an a priqri estimate of the parameter and P is the covariance matrix that reflects the reliabi!ity of f3 (Bierman, 1977). For non-reliable initialization, the following is usually adopted: P = >'1, with>. large and where I is the identity matrix. 4 Local Model Selection and Combination The recursive algorithm described by Eq. 7 and Eq. 8 returns for a given query point x q , a set of predictions Yq (k) = x~/3(k), together with a set of associated leave-one-out error vectors e Cv (k) . From the information available, a final prediction f)q of the value of the regression function can be obtained in different ways. Two main paradigms deserve to be considered: the first is based on the selection of the best approximator according to a given criterion, while the second returns a prediction as a combination of more local models. If the selection paradigm, frequently called winner-takes-all, is adopted, the most natural way to extract a final prediction Yq, consists in comparing the prediction obtained for each value of k on the basis of the classical mean square error criterion: with k = argmin MSE(k) = argmin A k k "k L.J' Wi t=l "k (e?CV(k))2 t . L.Ji=l W t . ' (9) 379 Lazy Learning Meets the Recursive Least Squares Algorithm Table 1: A summary of the characteristics of the data sets considered. Dataset Number of examples Number of regressors I Housing I Cpu I Prices I Mpg I Servo I Ozone I 506 209 159 392 167 330 l3 6 16 7 8 8 where Wi are weights than can be conveniently used to discount each error according to the distance from the query point to the point to which the error corresponds (Atkeson et at., 1997). As an alternative to the winner-takes-all paradigm, we explored also the effectiveness of local combinations of estimates (Wolpert, 1992). Adopting also in this case the mean square error criterion, the final prediction of the value Yq is obtained as a weighted average of the best b models, where b is a parameter of the algorithm. Suppose the predictions il q (k) and the error vectors e Cv (k) have been ordered creating a sequence of integers {k i } so that MSE( ki ) ::; MSE( kj ), Vi < j. The prediction of Yq is given by ~ Yq = L~-l (iYq(kd ",b r. L..-i=l ,>z ' (10) where the weights are the inverse of the mean square errors: (i = l/MSE(ki ). This is an example of the generalized ensemble method (Perrone & Cooper, 1993). 5 Experiments and Results The experimental evaluation ofthe incremental local identification and validation algorithm was performed on six datasets. The first five, described by Quinlan (1993), were obtained from the VCI Repository of machine learning databases (Merz & Murphy, 1998), while the last one was provided by Leo Breiman. A summary ofthe characteristics of each dataset is presented in Table 1. The methods compared adopt the recursive identification and validation algorithm, combined with different strategies for model selection or combination. We considered also two approaches in which k is selected globally: Ibl: Local bandwidth selection for linear local models. The number of neighbors is selected on a query-by-query basis and the prediction returned is the one of the best model according to the mean square error criterion. IbO: Local bandwidth selection for constant local models. The algorithm for constant models is derived directly from the recursive method described in Eq. 7 and Eq. 8. The best model is selected according to the mean square error criterion. IbC: Local combination of estimators. This is an example, of the method described in Eq. 10. On the datasets proposed, for each query the best 2 linear local models and the best 2 constant models are combined. gbl: Global bandwidth selection for linear local models. The value of k is obtained minimizing the prediction error in 20-fold cross-validation on the dataset available. This value is then used for all the query points. gbO: Global bandwidth selection for constant local models. As in gbl, the value of k is optimized globally and kept constant for all the queries. M. Birattarl. G. Bontempi and H. Bersini 380 Table 2: Mean absolute error on unseen cases. Method I Housing I Cpu I Prices I Mpg I Servo I Ozone Ibl 2.21 28.38 1509 1.94 0.48 3.52 2.60 0.32 3.33 1.97 IbO 31.54 1627 IbC 2.12 26.79 1488 1.83 0.29 3.31 2.30 28.69 1492 1.92 0.52 3.46 gbl 2.59 32.19 1639 1.99 3.19 gbO 0.34 Cubist 2.17 28.37 1331 1.90 0.36 3.15 Table 3: Relative error (%) on unseen cases. I Method I Housing I Ibl IbO IbC gb1 gbO 12.63 18.06 12.35 13.47 17.99 Cpu 9.20 20.37 9.29 9.93 21.43 Cubist 16.02 12.71 I Prices I 15.87 22.19 17.62 15.95 22.29 Mpg 12.65 12.64 11.82 12.83 13.48 11.67 12.57 I Servo I Ozone 28.66 22.04 19.72 30.46 24.30 35.25 31.11 30.28 32.58 28.21 18.53 26.59 As far as the metric is concerned, we adopted a global Euclidean metric based on the relative influence (relevance) ofthe regressors (Friedman, 1994). We are confident that the adoption of a local metric could improve the performance of our lazy learning method. The results of the methods introduced are compared with those we obtained, in the same experimental settings, with Cubist, the rule-based tool developed by Quinlan for generating piecewise-linear models. Each approach was tested on each dataset using the same 10-fold cross-validation strategy. Each dataset was divided randomly into 10 groups of nearly equal size. In turn, each of these groups was used as a testing set while the remaining ones together were providing the examples. Thus all the methods performed a prediction on the same unseen cases, using for each of them the same set of examples. In Table 2 we present the results obtained by all the methods, and averaged on the 10 cross-validation groups. Since the methods were compared on the same examples in exactly the same conditions, the sensitive one-tailed paired test of significance can be used. In what follows, by "significantly better" we mean better at least at a 5% significance level. The first consideration about the results concerns the local combination of estimators. According to Table 2, the method IbC performs in average always better than the winnertakes-all linear and constant. On two dataset IbC is significantly better than both Ibl and IbO; and on three dataset it is significantly better than one of the two, and better in average than the other. The second consideration is about the comparison between our query-by-query bandwidth selection and a global optimization of the number of neighbors: in average Ibl and IbO performs better than their counterparts gbl and gbO. On two datasets Ibl is significantly better than gbl, while is about the same on the other four. On one dataset IbO is significantly better than gbO. As far as the comparison with Cubist is concerned, the recursive lazy identification and validation proposed obtains results comparable with those obtained by the state-of-the-art method implemented in Cubist. On the six datasets, IbC performs one time significantly better than Cubist, and one time significantly worse. Lazy Learning Meets the Recursive Least Squares Algorithm 381 The second index of performance we investigated is the relative error, defined as the mean square error on unseen cases, normalized by the variance of the test set. The relative errors are presented in Table 3 and show a similar picture to Table 2, although the mean square errors considered here penalize larger absolute errors. 6 Conclusion and Future Work The experimental results confirm that the recursive least squares algorithm can be effectively used in a local context. Despite the trivial metric adopted, the local combination of estimators, identified and validated recursively, showed to be able to compete with a state-of-the-art approach. Future work will focus on the problem of local metric selection. Moreover, we will explore more sophisticated ways to combine local estimators and we will extend this work to polynomial approximators of higher degree. Acknowledgments The work of Mauro Birattari was supported by the FIRST program of the Region Wallonne, Belgium. The work of Gianluca Bontempi was supported by the European Union TMR Grant FMBICT960692. The authors thank Ross Quinlan and gratefully acknowledge using his software Cubist. For more details on Cubist see http://www.rulequest.com. We also thank Leo Breiman for the dataset ozone and the UCI Repository for the other datasets used in this paper. References Aha D. W. 1997. Editorial. Artificial Intelligence Review, 11(1-5), 1-6. Special Issue on Lazy Learning. Atkeson C. G. , Moore A. W. & Schaal S. 1997. Locally weighted learning. Artificial Intelligence Review, 11(1-5), 11-73. Bierman G. 1. 1977. Factorization Methodsfor Discrete Sequential Estimation. New York, NY: Academic Press. Bontempi G., Birattari M. & Bersini H. 1997. Lazy learning for local modeling and control design. International Journal of Control. Accepted for publication. Friedman 1. H. 1994. Flexible metric nearest neighbor classification. Tech. rept. Department of Statistics, Stanford University. Merz C. J. & Murphy P. M . 1998. UCI Repository of machine learning databases. Myers R. H. 1990. Classical and Modern Regression with Applications. Boston, MA: PWS-KENT. Perrone M. P. & Cooper L. N. 1993. When networks disagree: Ensemble methods for hybrid neural networks. Pages 126-142 of Mammone R. J. (ed), Artificial Neural Networks for Speech and Vision. Chapman and Hall. Quinlan 1. R. 1993. Combining instance-based and model-based learning. Pages 236-243 of Machine Learning. Proceedings of the Tenth International Conference. Morgan Kaufmann. Schaal S. & Atkeson C. G. 1998. Constructive incremental learning from only local information. Neural Computation, 10(8), 2047-2084. Wolpert D. 1992. Stacked Generalization. Neural Networks, 5, 241-259. 

1507 |@word repository:3 polynomial:4 covariance:1 kent:1 recursively:2 moment:1 contains:1 selecting:1 comparing:1 com:1 wx:3 update:1 intelligence:2 selected:4 cubist:9 ith:4 lr:4 lx:1 five:1 consists:3 combine:1 introduce:1 homoscedasticity:2 themselves:1 frequently:1 growing:1 mpg:3 globally:3 cpu:3 hugues:1 provided:1 estimating:1 moreover:4 notation:1 linearity:1 what:2 argmin:2 minimizes:1 developed:2 exactly:1 rm:1 control:2 grant:1 local:47 rept:1 despite:1 meet:4 yd:1 initialization:2 factorization:1 adoption:3 averaged:1 acknowledgment:1 yj:5 testing:1 recursive:16 union:1 procedure:3 significantly:7 word:1 selection:22 context:3 influence:1 www:1 straightforward:2 attention:1 simplicity:1 identifying:2 ozone:4 estimator:5 rule:2 deriving:1 his:1 ity:1 handle:1 updated:1 suppose:1 us:1 hypothesis:1 element:2 jk:1 updating:1 database:3 cooperative:1 calculate:1 region:2 removed:1 servo:3 mentioned:1 solving:1 basis:5 leo:2 stacked:1 effective:1 pertaining:1 query:29 artificial:3 neighborhood:3 mammone:1 whose:3 larger:1 stanford:1 otherwise:1 ability:1 statistic:5 favor:1 unseen:4 final:4 housing:3 sequence:2 myers:3 advantage:1 propose:3 adaptation:1 neighboring:2 relevant:3 combining:2 uci:2 validate:1 crossvalidation:1 exploiting:1 generating:2 incremental:4 leave:6 ac:1 nearest:4 received:2 eq:8 implemented:1 involves:1 implies:1 centered:2 require:1 generalization:2 extension:1 considered:8 hall:1 mapping:1 lm:1 major:1 adopt:1 belgium:2 estimation:2 ross:2 sensitive:1 tool:2 weighted:4 reflects:1 always:1 ej:2 breiman:2 publication:1 conjunction:1 derived:1 focus:2 validated:1 schaal:3 indicates:1 tech:1 ibl:6 mth:1 bierman:3 selects:1 issue:1 among:1 classification:2 flexible:1 priori:1 art:3 special:1 equal:1 once:4 f3:1 chapman:1 nearly:1 ibc:6 future:2 piecewise:2 modern:1 randomly:1 composed:1 comprehensive:1 murphy:2 friedman:2 interest:1 evaluation:2 bontempi:7 partial:1 aha:2 euclidean:1 rulequest:1 instance:1 modeling:3 combined:2 confident:1 international:2 vm:1 together:2 tmr:1 worse:1 creating:1 return:2 de:1 explicitly:1 vi:2 performed:3 competitive:1 contribution:2 appended:1 square:18 il:1 variance:3 characteristic:3 kaufmann:1 ensemble:2 identify:1 ofthe:3 identification:8 worth:1 ed:1 inexpensive:1 universite:1 associated:2 dataset:10 intrinsically:1 sophisticated:1 higher:1 formulation:1 though:1 furthermore:1 until:1 hand:6 vci:1 normalized:1 counterpart:1 assigned:1 moore:1 criterion:6 generalized:1 performs:3 consideration:2 ji:1 winner:4 extend:1 significant:2 cv:6 gratefully:1 l3:1 showed:1 perspective:1 belongs:1 driven:3 approximators:5 yi:1 seen:1 morgan:1 paradigm:3 gb1:1 academic:1 cross:8 divided:1 paired:1 prediction:21 regression:11 vision:1 metric:10 editorial:1 tailored:1 adopting:1 penalize:1 singular:1 gbl:5 validating:1 thing:1 member:1 effectiveness:1 birattari:5 integer:1 extracting:1 structural:2 unitary:1 concerned:3 xj:3 winnertakes:1 bandwidth:17 identified:3 reduce:1 six:3 returned:2 speech:1 york:1 jj:4 useful:1 involve:1 ylx:1 discount:1 locally:4 ph:3 reduced:2 http:1 exist:1 zj:4 tutorial:1 notice:1 fulfilled:1 correctly:1 discrete:1 group:3 four:1 tenth:1 kept:1 compete:1 inverse:2 noticing:1 family:2 comparable:1 ki:2 fold:2 software:1 sake:1 performing:1 department:1 according:7 combination:10 request:2 perrone:2 kd:1 wi:2 appealing:1 turn:1 adopted:5 available:5 wii:1 generic:1 alternative:2 assumes:1 remaining:1 quinlan:5 bersini:6 classical:2 already:1 strategy:6 ulb:1 parametric:2 diagonal:3 bruxelles:2 kth:1 distance:4 thank:2 mauro:2 trivial:1 reason:1 assuming:1 modeled:1 index:1 providing:1 minimizing:1 cij:2 oneout:1 stated:2 design:1 unknown:1 disagree:1 datasets:6 acknowledge:1 introduced:2 required:1 optimized:2 deserve:1 able:2 wy:2 usually:1 xjp:1 program:1 including:2 memory:1 reliable:1 natural:1 hybrid:1 indicator:1 improve:1 yq:9 picture:1 extract:3 kj:1 xq:5 review:2 relative:4 beside:1 approximator:4 validation:12 degree:2 principle:1 row:2 gianluca:2 summary:2 supported:2 last:2 jh:1 neighbor:10 absolute:2 calculated:1 author:1 made:1 regressors:2 atkeson:5 far:3 lz:2 obtains:1 confirm:1 global:7 assumed:1 xi:8 search:1 tailed:1 table:8 promising:1 mse:4 investigated:1 interpolating:2 european:1 vj:1 significance:2 main:3 libre:1 repeated:1 cooper:2 ny:1 sub:1 lmo:1 explicit:1 pws:1 specific:2 methodsfor:1 er:1 explored:2 pz:1 concern:1 sequential:1 effectively:1 ci:3 boston:1 wolpert:3 simply:2 explore:2 conveniently:2 lazy:12 ordered:1 corresponds:1 ma:1 identity:1 price:3 change:1 experimentally:1 called:1 accepted:1 experimental:4 merz:2 select:1 latter:1 assessed:1 relevance:1 overload:2 constructive:1 evaluate:2 tested:2

Markov processes on curves for automatic speech recognition Lawrence Saul and Mazin Rahim AT&T Labs - Research Shannon Laboratory 180 Park Ave E-171 Florham Park, NJ 07932 {lsaul,rnazin}Gresearch.att.com Abstract We investigate a probabilistic framework for automatic speech recognition based on the intrinsic geometric properties of curves. In particular, we analyze the setting in which two variables-one continuous (~), one discrete (s )-evolve jointly in time. We suppose that the vector ~ traces out a smooth multidimensional curve and that the variable s evolves stochastically as a function of the arc length traversed along this curve. Since arc length does not depend on the rate at which a curve is traversed, this gives rise to a family of Markov processes whose predictions, Pr[sl~]' are invariant to nonlinear warpings of time. We describe the use of such models, known as Markov processes on curves (MPCs), for automatic speech recognition, where ~ are acoustic feature trajectories and s are phonetic transcriptions. On two tasks-recognizing New Jersey town names and connected alpha-digits- we find that MPCs yield lower word error rates than comparably trained hidden Markov models. 1 Introduction Variations in speaking rate currently present a serious challenge for automatic speech recognition (ASR) (Siegler & Stern, 1995). It is widely observed , for example, that fast speech is more prone to recognition errors than slow speech. A related effect, occurring at the phoneme level, is that consonants (l,re more frequently botched than vowels. Generally speaking, consonants have short-lived, non-stationary acoustic signatures; vowels, just the opposite. Thus, at the phoneme level, we can view the increased confusability of consonants as a consequence of locally fast speech. L. Saul and M. Rahim 752 set) =s 1 START t=O x(t) END t='t Figure 1: Two variables-one continuous (x), one discrete (s )- evol ve jointly in time. The trace of s partitions the curve of x into different segments whose boundaries occur where s changes value. In this paper , we investigate a probabilistic framework for ASR that models variations in speaking rate as arising from nonlinear warpings of time (Tishby, 1990) . Our framework is based on the observation that acoustic feature vectors trace out continuous trajectories (Ostendorf et aI , 1996). We view these trajectories as multidimensional curves whose intrinsic geometric properties (such as arc length or radius) do not depend on the rate at which they are traversed (do Carmo, 1976). We describe a probabilistic model whose predictions are based on these intrinsic geometric properties and-as such-are invariant to nonlinear warpings of time. The handling of this invariance distinguishes our methods from traditional hidden Markov models (HMMs) (Rabiner & Juang, 1993). The probabilistic models studied in this paper are known as Markov processes on curves (MPCs). The theoretical framework for MPCs was introduced in an earlier paper (Saul , 1997), which also discussed the problems of decoding and parameter estimation. In the present work, we report the first experimental results for MPCs on two difficult benchmark problems in ASR. On these problems- recognizing New Jersey town names and connected alpha-digits- our results show that MPCs generally match or exceed the performance of comparably trained HMMs. The organization of this paper is as follows . In section 2, we review the basic elements of MPCs and discuss important differences between MPCs and HMMs. In section 3, we present our experimental results and evaluate their significance. 2 Markov processes on curves Speech recognizers take a continuous acoustic signal as input and return a sequence of discrete labels representing phonemes, syllables, or words as output. Typically the short-time properties of the speech signal are summarized by acoustic feature vectors. Thus the abstract mathematical problem is to describe a multidimensional trajectory {x(t) It E [0, T]} by a sequence of discrete labels S1 S2 . . . Sn. As shown in figure 1, this is done by specifying consecutive time intervals such that s(t) Sk for t E [tk-1, tk] and attaching the labels Sk to contiguous arcs along the trajectory. To formulate a probabilistic model of this process, we consider two variables-one continuous (x), one discrete (s )-that evolve jointly in time. Thus the vector x traces out a smooth multidimensional curve, to each point of which the variable s attaches a discrete label. = Markov processes on curves are based on the concept of arc length. After reviewing how to compute arc lengths along curves, we introduce a family of Markov processes whose predictions are invariant to nonlinear warpings of time. We then consider the ways in which these processes (and various generalizations) differ from HMMs. Markov Processes on Curves for Automatic Speech Recognition 2.1 753 Arc length Let g(~) define a D x D matrix-valued function over x E RP. If g(~) is everywhere non-negative definite, then we can use it as a metric to compute distances along curves. In particular, consider two nearby points separated by the infinitesimal vector d~. We define the squared distance between these two points as: (1) Arc length along a curve is the non-decreasing function computed by integrating these local distances. Thus, for the trajectory x(t), the arc length between the points x(t!) and X(t2) is given by: f= l t2 [~Tg(x)i:]~, dt (2) tl it where i: = [~(t)] denotes the time derivative of~ . Note that the arc length defined by eq. (2) is invariant under reparameterizations of the trajectory, ~(t) -t ~(J(t)) , where f(t) is any smooth monotonic function of time that maps the interval [tl, t2] into itself. In the special case where g(~) is the identity matrix, eq. (2) reduces to the standard definition of arc length in Euclidean space. More generally, however, eq. (1) defines a non-Euclidean metric for computing arc lengths. Thus, for example, if the metric g(x) varies as a function of~, then eq. (2) can assign different arc lengths to the trajectories x(t) and x(t) + ~o, where ~o is a constant displacement. 2.2 States and lifelengths We now return to the problem of segmentation, as illustrated in figure 1. We refer to the possible values of s as states. MPCs are conditional random processes that evolve the state variable s stochastically as a function of the arc length traversed along the curve of~. In MPCs, the probability of remaining in a particular state decays exponentially with the cumulative arc length traversed in that state. The signature of a state is the particular way in which it computes arc length. To formalize this idea, we associate with each state i the following quantities: (i) a feature-dependent matrix gi (x) that can be used to compute arc lengths, as in eq. (2); (ii) a decay parameter Ai that measures the probability per unit arc length that s makes a transition from state i to some other state; and (iii) a set of transition probabilities aij, where aij represents the probability that-having decayed out of state i-the variable s makes a transition to state j . Thus, aij defines a stochastic transition matrix with zero elements along the diagonal and rows that sum to one: aii 0 and 2: j aij 1. A Markov process is defined by the set of differential equations: = = d Pi dt = \ -/liPi ).]:1 + ~ \ [. L.J /ljpjaji [X. T gi ( X X 1 1 T ( ) ? ] :I ~ 9j x ~ , (3) #i where Pi(t) denotes the (forward) probability that s is in state i at time t, based on its history up to that point in time. The right hand side of eq. (3) consists of two competing terms. The first term computes the probability that s decays out of state i; the second computes the probability that s decays into state i. Both terms are proportional to measures of arc length, making the evolution of Pi along the curve of x invariant to nonlinear warpings of time. The decay parameter, Ai, controls the typical amount of arc length traversed in state i ; it may be viewed as L. Saul and M. Rahim 754 an inverse lifetime or-to be more precise-an inverse lifelength. The entire process is Markovian because the evolution of Pi depends only on quantities available at time t. 2.3 Decoding Given a trajectory x(t), the Markov process in eq. (3) gives rise to a conditional probability distribution over possible segmentations, s(t). Consider the segmentation in which s(t) takes the value Sk between times tk-l and tk, and let fSk = jtk dt [XTgsk(X) X ]% (4) tk-l denote the arc length traversed in state Sk. By integrating eq. (3), one can show that the probability of remaining in state Sk decays exponentially with the arc length f Sk ' Thus, the conditional probability of the overall segmentation is given by: Pr[s,flx] n n k=l k=O = II ASke->'Sklsk II aSkSk+ll I / (5) where we have used So and Sn+1 to denote the START and END states of the Markov process. The first product in eq. (5) multiplies the probabilities that each segment traverses exactly its observed arc length. The second product multiplies the probabilities for transitions between states Sk and Sk+l' The leading factors of ASk are included to normalize each state's distribution over observed arc lengths. There are many important quantities that can be computed from the distribution, Pr[ S Ix]. Of particular interest for ASR is the most probable segmentation: s* (x) = argmaxs,l {In Pr[s, fix]}. As described elsewhere (Saul, 1997), this maximization can be performed by discretizing the time axis and applying a dynamic programming procedure. The resulting algorithm is similar to the Viterbi procedure for maximum likelihood decoding (Rabiner & Juang, 1993). 2.4 Parameter estimation The parameters {Ai, aij, gi (x)} in MPCs are estimated from training data to maximize the log-likelihood of target segmentations. In our preliminary experiments with MPCs, we estimated only the metric parameters, gi(X); the others were assigned the default values Ai = 1 and aij = 1/ Ii, where Ii is the fanout of state i. The metrics gi (x) were assumed to have the parameterized form: (6) where (ji is a positive definite matrix with unit determinant, and cI>i (x) is a nonnegative scalar-valued function of x. For the experiments in this paper, the form of cI>i(X) was fixed so that the MPCs reduced to HMMs as a special case, as described in the next section. Thus the only learning problem was to estimate the matrix parameters (ji. This was done using the reestimation formula: (ji ~ C J ~xT dt. T ? 1 cI>i(x(t)), [x (ji-1X]"2 (7) where the integral is over all speech segments belonging to state i, and the constant C is chosen to enforce the determinant constraint l(ji I = 1. For fixed cI>i (x), we have shown previously (Saul, 1997) that this iterative update leads to monotonic increases in the log-likelihood. Markov Processes on Curves for Automatic Speech Recognition 2.5 755 Relation to HMMs and previous work There are several important differences between HMMs and MPCs. HMMs parameterize joint distributions of the form: Pr[s, z] = Dt Pr[st+1lsd Pr[zt Isd. Thus, in HMMs, parameter estimation is directed at learning a synthesis model, Pr[zls]' while in MPCs, it is directed at learning a segmentation model, Pr[s,flz]. The direction of conditioning on z is a crucial difference. MPCs do not attempt to learn anything as ambitious as a joint distribution over acoustic feature trajectories. \ HMMs and MPCs also differ in how they weight the speech signal. In HMMs, each state contributes an amount to the overall log-likelihood that grows in proportion to its duration in time. In MPCs, on the other hand, each state contributes an amount that grows in proportion to its arc length. Naturally, the weighting by arc length attaches a more important role to short-lived but non-stationary phonemes, such as consonants. It also guarantees the invariance to nonlinear warpings of time (to which the predictions of HMMs are quite sensitive). In terms of previous work,\mr motivation for MPCs resembles that of Tishby (1990), who several years ago proposed a dynamical systems approach to speech processing. Because MPCs exploit the continuity of acoustic feature trajectories, they also bear some resemblance to so-called segmental HMMs (Ostendorf et aI, 1996). MPCs nevertheless differ from segmental HMMs in two important respects: the invariance to nonlinear warpings of time , and the emphasis on learning a segmentation model Pr[s , flz], as opposed to a synthesis model, Pr[xls]. Finally, we note that admitting a slight generalization in the concept of arc length, we can essentially realize HMMs as a special case of MPCs. This is done by computing arc lengths along the spacetime trajectories z(t) = {x(t),t}-that is to say, replacing eq. (1) by dL 2 = [zTg(z) z]dt 2 , where z = {:il, 1} and g(z) is a spacetime metric. This relaxes the invariance to nonlinear warpings of time and incorporates both movement in acoustic feature space and duration in time as measures of phonemic evolution. Moreover, in this setting, one can mimic the predictions of HMMs by setting the (J'i matrices to have only one non-zero element (namely, the diagonal element for delta-time contributions to the arc length) and by defining the functions <l>i(X) in terms of HMM emission probabilities P(xli) as: P(zli) ] <l>i(X) = -In [ 2::k P(xlk) . (8) This relation is important because it allows us to initialize the parameters of an MPC by those of a continuous-density HMM,. This initialization was used in all the experiments reported below. 3 Automatic speech recognition Both HMMs and MPCs were used to, build connected speech recognizers. Training and test data came from speaker-independent databases of telephone speech. All data was digitized at the caller's local switch and transmitted in this form to the receiver. For feature extraction, input telephone signals (sampled at 8 kHz and band-limited between 100-3800 Hz) were pre-emphasized and blocked into 30ms frames with a frame shift of 10ms. Each frame was Hamming windowed , autocorrelated, and processed by LPC cepstral analysis to produce a vector of 12 liftered cepstral coefficients (Rabiner & Juang, 1993). The feature vector was then augmented by its normalized log energy value, as well as temporal derivatives of first and second order. Overall, each frame of speech was described by 39 features . These features were used diffe:.;ently by HMMs and MPCs, as described below. L. Saul and M. Rahim 756 NJ town names 22 ~ , Mixtures 2 4 8 16 32 64 HMM (%) 22.3 18.9 16.5 14.6 13.5 11.7 MPC ('fo) 20.9 17.5 15.1 13.3 12.3 11.4 14 - 0- 12 o 1000 2000 3000 4000 5000 parameters pe r state Table 1: Word error rates for HMMs (dashed) and MPCs (solid) on the task of recognizing NJ town names. The table shows the error rates versus the number of mixture components; the graph , versus the number of parameters per hidden state. Recognizers were evaluated on two tasks. The first task was recognizing New Jersey town names (e.g., Newark) . The training data for this task (Sachs et aI , 1994) consisted of 12100 short phrases, spoken in the seven major dialects of American English . These phrases, ranging from two to four words in length, were selected to provide maximum phonetic coverage. The test data consisted of 2426 isolated utterances of 1219 New Jersey town names and was collected from nearly 100 speakers. Note that the training and test data for this task have non-overlapping vocabularies . Baseline recognizers were built using 43Ieft-to-right continuous-density HMMs, each corresponding to a context-independent English phone. Phones were modeled by three-state HMMs , with the exception of background noise , which was modeled by a single state. State emission probabilities were computed by mixtures of Gaussians with diagonal covariance matrices. Different sized models were trained using M 2, 4, 8, 16, 32, and 64 mixture components per hidden state; for a particular model , the number of mixture components was the same across all states. Parameter estimation was handled by a Viterbi implementation of the Baum-Welch algorithm. = MPC recognizers were built using the same overall grammar. Each hidden state in the MPCs was assigned a metric gi(~) = O';l<I>l(~). The functions <I>i(~) were initialized (and fixed) by the state emission probabilities of the HMMs , as given by eq. (8). The matrices O'i were estimated by iterating eq. (7). We computed arc lengths along the 14 dimensional spacetime trajectories through cepstra, log-energy, and time . Thus each O'i was a 14 x 14 symmetric matrix applied to tangent vectors consisting of delta-cepstra, delta-log-energy, and delta-time. The table in figure 1 shows the results of these experiments comparing MPCs to HMMs. For various model sizes (as measured by the number of mixture components), we found the MPCs to yield consistently lower error rates than the HMMs. The graph in figure 1 plots these word error rates versus the number of modeling parameters per hidden state. This graph shows that the MPCs are not outperforming the HMMs merely because they have extra modeling parameters (i .e. , the O'i matrices). The beam widths for the decoding procedures in these experiments were chosen so that corresponding recognizers activated roughly equal numbers of arcs. The second task in our experiments involved the recognition of connected alphadigits (e.g ., N Z 3 V J 4 E 3 U 2). The training and test data consisted of 757 Markov Processes on Curves for Automatic Speech Recognition 13 12 Mixtures 2 4 8 HMM (%) 12.5 10.7 10.0 MPC (%) 10.0 8.8 8.2 ..... , ~11 ~ g10 '0 CD 9 ~oo 400 600 800 1000 parameters per state 1200 1400 Figure 2: Word error rates for HMMs and MPCs on the task of recognizing connected alpha-digits. The table shows the error rates versus the number of mixture components; the graph , versus the number of parameters per hidden state. 14622 and 7255 utterances, respectively. Recognizers were built from 285 sub-word HMMs/MPCs, each corresponding to a context-dependent English phone. The recognizers were trained and evaluated in the same way as the previous task. Results are shown in figure 2. While these results demonstrate the viability of MPCs for automatic speech recognition, several issues require further attention . The most important issues are feature selection-how to define meaningful acoustic trajectories from the raw speech signal-and learning- how to parameterize and estimate the hidden state metrics gi (~) from sampled trajectories {z (t)}. These issues and others will be studied in future work. References M. P. do Carmo (1976) . Differential Geometry of Curves and Surfaces. Prentice Hall. M. Ostendorf, V. Digalakis, and O. Kimball (1996). From HMMs to segment models: a unified view of stochastic modeling for speech recognition. IEEE Transactions on Acoustics, Speech and Signal Processing, 4:360-378. L. Rabiner and B . Juang (1993) . Fundamentals of Speech Recognition. Prentice Hall, Englewood Cliffs, NJ. R. Sachs, M. Tikijian, and E. Roskos (1994). United States English subword speech data. AT&T unpublished report. L. Saul (1998) . Automatic segmentation of continuous trajectories with invariance to nonlinear warpings of time . In Proceedings of the Fifteenth International Conference on Machine Learning, 506- 514. M. A. Siegler and R . M. Stern (1995). On the effects of speech rate in large vocabulary speech recognition systems. In Proceedings of th e 1995 IEEE International Conference on Acoustics, Speech, and Signal Processing, 612-615. N. Tishby (1990). A dynamical system approach to speech processing. In Proceedings of the 1990 IEEE International Conference on Acoustics, Speech, and Signal Processing, 365-368 . PART VII VISUAL PROCESSING 

1508 |@word determinant:2 proportion:2 aske:1 covariance:1 solid:1 att:1 united:1 subword:1 com:1 comparing:1 realize:1 partition:1 plot:1 update:1 stationary:2 selected:1 short:4 traverse:1 mathematical:1 along:10 windowed:1 differential:2 consists:1 introduce:1 roughly:1 frequently:1 decreasing:1 moreover:1 spoken:1 unified:1 nj:4 guarantee:1 temporal:1 multidimensional:4 rahim:4 exactly:1 control:1 unit:2 positive:1 local:2 consequence:1 lsd:1 cliff:1 emphasis:1 initialization:1 studied:2 resembles:1 specifying:1 hmms:28 limited:1 directed:2 definite:2 digit:3 procedure:3 displacement:1 word:7 integrating:2 pre:1 selection:1 prentice:2 context:2 applying:1 map:1 baum:1 attention:1 duration:2 formulate:1 welch:1 evol:1 variation:2 caller:1 target:1 suppose:1 programming:1 associate:1 element:4 recognition:14 database:1 observed:3 role:1 parameterize:2 connected:5 movement:1 jtk:1 reparameterizations:1 flz:2 dynamic:1 signature:2 trained:4 depend:2 reviewing:1 segment:4 aii:1 joint:2 jersey:4 various:2 autocorrelated:1 dialect:1 separated:1 fast:2 describe:3 whose:5 quite:1 widely:1 valued:2 say:1 grammar:1 florham:1 gi:7 jointly:3 itself:1 sequence:2 product:2 argmaxs:1 normalize:1 juang:4 produce:1 tk:5 oo:1 measured:1 phonemic:1 eq:12 coverage:1 differ:3 direction:1 radius:1 stochastic:2 require:1 assign:1 fix:1 generalization:2 preliminary:1 probable:1 traversed:7 hall:2 lawrence:1 viterbi:2 major:1 consecutive:1 estimation:4 label:4 currently:1 sensitive:1 kimball:1 emission:3 consistently:1 likelihood:4 ave:1 baseline:1 dependent:2 typically:1 entire:1 lsaul:1 hidden:8 relation:2 overall:4 issue:3 multiplies:2 special:3 initialize:1 equal:1 asr:4 having:1 extraction:1 represents:1 park:2 nearly:1 mimic:1 future:1 t2:3 report:2 others:2 serious:1 distinguishes:1 ve:1 geometry:1 consisting:1 vowel:2 attempt:1 organization:1 interest:1 englewood:1 investigate:2 mixture:8 admitting:1 activated:1 integral:1 euclidean:2 initialized:1 re:1 isolated:1 theoretical:1 increased:1 earlier:1 modeling:3 markovian:1 contiguous:1 maximization:1 tg:1 phrase:2 recognizing:5 tishby:3 reported:1 varies:1 st:1 decayed:1 density:2 fundamental:1 international:3 probabilistic:5 decoding:4 synthesis:2 squared:1 town:6 opposed:1 stochastically:2 american:1 derivative:2 leading:1 return:2 summarized:1 coefficient:1 depends:1 performed:1 view:3 lab:1 analyze:1 start:2 contribution:1 il:1 phoneme:4 who:1 yield:2 rabiner:4 xli:1 raw:1 comparably:2 trajectory:16 ago:1 history:1 fo:1 definition:1 infinitesimal:1 energy:3 involved:1 mpc:4 naturally:1 hamming:1 sampled:2 ask:1 diffe:1 segmentation:9 formalize:1 dt:6 done:3 evaluated:2 lifetime:1 just:1 hand:2 ostendorf:3 replacing:1 nonlinear:9 overlapping:1 continuity:1 defines:2 resemblance:1 grows:2 name:6 effect:2 concept:2 normalized:1 consisted:3 evolution:3 assigned:2 symmetric:1 laboratory:1 illustrated:1 ll:1 width:1 speaker:2 anything:1 m:2 demonstrate:1 ranging:1 ji:5 conditioning:1 exponentially:2 khz:1 discussed:1 slight:1 refer:1 blocked:1 ai:7 automatic:10 recognizers:8 surface:1 segmental:2 phone:3 phonetic:2 carmo:2 discretizing:1 came:1 outperforming:1 transmitted:1 mr:1 maximize:1 signal:8 ii:5 dashed:1 reduces:1 smooth:3 match:1 prediction:5 basic:1 essentially:1 metric:8 fifteenth:1 beam:1 background:1 interval:2 crucial:1 extra:1 hz:1 incorporates:1 exceed:1 iii:1 viability:1 relaxes:1 switch:1 competing:1 opposite:1 idea:1 shift:1 handled:1 fanout:1 speech:30 speaking:3 ieft:1 generally:3 iterating:1 fsk:1 amount:3 locally:1 band:1 processed:1 reduced:1 mazin:1 sl:1 estimated:3 arising:1 per:6 delta:4 discrete:6 four:1 nevertheless:1 isd:1 graph:4 merely:1 sum:1 year:1 inverse:2 everywhere:1 parameterized:1 zli:1 flx:1 family:2 syllable:1 spacetime:3 nonnegative:1 occur:1 constraint:1 cepstra:2 nearby:1 zls:1 belonging:1 across:1 evolves:1 making:1 s1:1 invariant:5 pr:11 equation:1 previously:1 discus:1 mpcs:32 end:2 available:1 gaussians:1 enforce:1 rp:1 denotes:2 remaining:2 exploit:1 build:1 warping:9 quantity:3 traditional:1 diagonal:3 distance:3 hmm:4 seven:1 collected:1 length:30 modeled:2 difficult:1 trace:4 negative:1 rise:2 implementation:1 stern:2 lived:2 zt:1 ambitious:1 observation:1 markov:15 arc:31 benchmark:1 defining:1 precise:1 digitized:1 frame:4 introduced:1 namely:1 unpublished:1 acoustic:12 dynamical:2 below:2 lpc:1 challenge:1 built:3 confusability:1 digalakis:1 representing:1 axis:1 utterance:2 sn:2 review:1 geometric:3 tangent:1 evolve:3 bear:1 attache:2 proportional:1 versus:5 pi:4 cd:1 row:1 prone:1 elsewhere:1 english:4 aij:6 side:1 saul:8 cepstral:2 attaching:1 curve:21 boundary:1 default:1 transition:5 cumulative:1 vocabulary:2 computes:3 forward:1 transaction:1 alpha:3 transcription:1 reestimation:1 receiver:1 assumed:1 consonant:4 continuous:8 iterative:1 sk:8 table:4 learn:1 contributes:2 significance:1 sachs:2 s2:1 motivation:1 noise:1 augmented:1 tl:2 slow:1 sub:1 xl:1 pe:1 weighting:1 ix:1 formula:1 xt:1 emphasized:1 decay:6 dl:1 intrinsic:3 g10:1 ci:4 occurring:1 siegler:2 vii:1 visual:1 scalar:1 monotonic:2 conditional:3 identity:1 viewed:1 sized:1 xlk:1 change:1 included:1 typical:1 telephone:2 called:1 invariance:5 experimental:2 shannon:1 meaningful:1 exception:1 newark:1 evaluate:1 handling:1

Experimental Results on Learning Stochastic Memoryless Policies for Partially Observable Markov Decision Processes John K. Williams Department of Mathematics University of Colorado Boulder, CO 80309-0395 jkwillia@euclid.colorado.edu Satinder Singh AT &T Labs-Research 180 Park Avenue Florham Park, NJ 07932 baveja@research.att.com Abstract Partially Observable Markov Decision Processes (pO "MOPs) constitute an important class of reinforcement learning problems which present unique theoretical and computational difficulties. In the absence of the Markov property, popular reinforcement learning algorithms such as Q-Iearning may no longer be effective, and memory-based methods which remove partial observability via state-estimation are notoriously expensive. An alternative approach is to seek a stochastic memoryless policy which for each observation of the environment prescribes a probability distribution over available actions that maximizes the average reward per timestep. A reinforcement learning algorithm which learns a locally optimal stochastic memoryless policy has been proposed by Jaakkola, Singh and Jordan, but not empirically verified. We present a variation of this algorithm, discuss its implementation, and demonstrate its viability using four test problems. 1 INTRODUCTION Reinforcement learning techniques have proven quite effective in solving Markov Decision Processes ("MOPs), control problems in which the exact state of the environment is available to the learner and the expected result of an action depends only on the present state [10]. Algorithms such as Q-Iearning learn optimal deterministic policies for "MOPs----rules which for every state prescribe an action that maximizes the expected future reward. In many important problems, however, the exact state of the environment is either inherently unknowable or prohibitively expensive to obtain, and only a limited, possibly stochastic observation of the environment is available. Such 1074 1. K. Williams and S. Singh Partially Observable Markov Decision Processes (POMDPs) [3 ,6] are often much more difficult than MDPs to solve [4]. Distinct sequences of observations and actions preceding a given observation in a POMDP may lead to different probabilities of occupying the underlying exact states of the MDP. If the efficacy of an action depends on the hidden exact state of the environment, an optimal choice may require knowing the past history as well as the current observation, and the problem is no longer Markov. In light of this difficulty, one approach to solving POMDPs is to explore the environment while building up a memory of past observations, actions and rewards which allows estimation of the current hidden state [1]. Such methods produce deterministic policies, but they are computationally expensive and may not scale well with problem size. Furthermore, policies that require state-estimation using memory may be complicated to implement. Memoryless policies are particularly appropriate for problems in which the state is expensive to obtain or inherently difficult to estimate, and they have the advantage of being extremely simple to act upon. For a POMDP, the optimal memoryless policy is generally a stochastic policy-one which for each observation of the environment prescribes a probability distribution over the available actions. In fact, examples of POMDPs can be constructed for which a stochastic policy is arbitrarily better than the optimal deterministic policy [9] . An algorithm proposed by Jaakkola, Singh and Jordan OSJ) [2], which we investigate here, learns memoryless stochastic policies for POMDPs. 2 POMDPs AND DIFFERENTIAL-REWARD Q-VALUES We assume that the environment has discrete states S = {s1, S2, .. IV}, and the learner chooses actions from a set f4. State transitions depend only on the current state s and the action a taken (the Markov property); they occur with probabilities r(s,sl) and result in expected rewards K'(s,s} In a POMDP, the learner cannot sense exactly the state s of the enVironment, but rather perceives only an observation--or "message"-from a set :M = {m 1, m 2 , .. m M } according to a conditional probability distribution P(mls). The learner will in general not know the size of the underlying state space, its transition probabilities, reward function, or the conditional distributions of the messages. In MDPs, there always exists a policy which simultaneously maximizes the expected future reward for all states, but this is not the case for POMDPs [9]. An appropriate alternative measure of the merit of a stochastic POMDP policy 7Z{alm) is the asymptotic average reward per timestep, R 7r, that it achieves. In seeking an optimal stochastic policy, the JSJ algorithm makes use of Q-values determined by the infinite-horizon differential reward for each observation-action pair (m,a). In particular, if rr denotes the reward obtained at time t, we may define the differential-reward Q-values by Q7r(s,a)= LE7r [Ii _R 7r I S1 =s,a 1 = a]; Q7r(m,a)= E s [Q7r(s,a)IM(s)=m](l) 1=1 where M is the observation operator. Note that E[rr] ~ R7r as t ~ so the summand converges to ~ero. The value functions V7r(s) and V7r(m) may be defined similarly. 00, 3 POLICY IMPROVEMENT The JSJ algorithm consists of a method for evaluating Q7r and V7r and a mechanism for using them to improve the current policy. Roughly speaking, if Q7r(m,a) > V7r(m), then action a realized a higher differential reward than the average for observation m, and assigning it a slightly greater probability will increase the average reward per timestep, R7r. We interpret the quantities ~m(a) = Q 7r(m,a) - V7r(m) as comprising a "gradient" of R7r in policy space. Their projections onto the probability simplexes may then be written An Algorithm which Learns Stochastic Memoryless Policies for POMDPs 1075 as 8m = Llm -<Llm,l> 11/JIl, where 1 is the one-vector (1,1, ... ,1), <, > is the inner product, and IJIl is the number of actions, or 1 R 8mea) = Llm(a) - IAI1 LLlm (a') = Q~ (m,a) - -IAI LQ (m, a'). a'EA For sufficiently small E;n, 1l'(a/m) = 1l(alm) + (2) a'EA an improved policy 1l'(alm) may be obtained by the increments E;n 8m(a) . (3) In practice, we also enforce 1l'(alm) ~ Pmin for all a and m to guarantee continued exploration. The original JSJ algorithm prescribed using Llm(a) in place of 8m(a) in equation (3), followed by renormalization [2]. Our method has the advantage that a given value of Ll yields the same incremeiu regardless of the current value of the policy, and it ensures that the step is in the correct direction. We also do not require the differential-reward value estimate, yR. 4 Q-EVALUATION As the POMDP is simulated under a fixed stochastic policy 1l, every occurrence of an observation-action pair (m, a) begins a sequence of rewards which can be used to estimate QR(m, a). Exploiting the fact that the QR(m, a) are defined as sums, the JSJ Qevaluation method recursively averages the estimates from all such sequences using a socalled "every-visit" Monte-Carlo method. In order to reduce the bias and variance caused by the dependence of the evaluation sequences, a factor fJ is used to discount their shared "tails". Specifically, at time t the learner makes observation m r , takes action ar , and obtains reward rr. The number of visits K(mr,ar) is incremented, the tail discount rate rem, a) = 1- K(m, arl/4, and the following updates are performed (the indicator function x.:Cm, a) is 1 if (m,a) = (mr,a r) and 0 otherwise). fJ %r(m,a) fJ (m,a)= [1- %r(m,a)] K(m,a) r(m,a) (m,a)+ K(m,a) Q(m,a)= C(m,a)= [1- ~~::~ [1- ~f:::~ =Q(m, a) - (4) ]Q(m, a) + fJ(m,a)[Ti - R] (5) ]c(m,a) + fJ(m, a) (6) R = (1 - lIt)R + (lit) rr Q(m, a) (tail discount factor) (cumulative discount effect) (R~-estimate) C(m, a) [R - R old ]; Rold (7) =R (QR-estimate correction) (8) Other schedules for rem, a) are possible----see [2~and the correction provided by (8) need not be performed at every step, but can be delayed until the Q~-estirnate is needed. This evaluation method can be used as given for a policy-iteration type algorithm in which independent T-step evaluations of Q~ are interspersed with policy improvements as prescribed in section 3. However, an online version of the algorithm which performs policy improvement after every step requires that old experience be gradually "forgotten" so that the QR-estimate can respond to more recent experience. To achieve this, we multiply the previous estimates of fJ, Q, and C at each timestep by a "decay" factor a, 0 < a< 1, before they are updated via equations (4)-(6), and replace equation (7) by R = a(l - lit) R + [1 - a(1 - lit)] r l . An alternative method, which also works reasonably well, is to multiply K and t by each timestep instead. (9) a at J K. Williams and S. Singh 1076 0. ' (a) r -- - r -- - r -- ----;,.-- - - - - ; -- - - - - ; (b) A .,... .. .. .. .. j...... . B 10000 20000 3 0000 40000 50000 number of iterations +1 +1 (c) 0 .8 f \ 06 ~::"==_:::_-=-__~~.~=~.-....:'.. . .'-"'-" '-"'-" . ---..;...--- \..-:' ::-0 ' [0.4 0. 2 ?o!:----,,;-;;c oo=o-=o --'2::::0~ 00:::0- -'3 ""0""00""'0- --;: 40:;';,00"0 " --= 50000 number Of Iterations Figure 1: (a) Schematic of confounded two-state POMDP, (b) evolution of the R7r_ estimate, and (c) evolution of n(A) (solid) and nCB) (dashed) for e= 0.0002, a= 0.9995 . 5 EMPIRICAL RESULTS We present only results from single runs of our online algorithm, including the modified policy improvement and Q-evaluation procedures described above. Results from the policy iteration version are qualitatively similar, and statistics performed on multiple runs verify that those shown are representative of the algorithm's behavior. To simplify the presentation, we fix a constant learning rate, e, and decay factor, a, for each problem, and we use Pmin = 0.02 throughout. Note, however, that appropriate schedules or online heuristics for decreasing e and Pmin while increasing a would improve performance and are necessary to ensure convergence. Except for the first problem, we choose the initial policy n to be uniform. In the last two problems, values of n(alm) < 0.03 are rounded down to zero, with renormalization, before the learned policy is evaluated. ]S] 5.1 CONFOUNDED TWO-STA TE PROBLEM The two-state MDP diagrammed in Figure l(a) becomes a POMDP when the two states are confounded into a single observation. The learner may take action A or B, and receives a reward of either +1 or -1; the state transition is deterministic, as indicated in the diagram. Note that either stationary deterministic policy results in R7r = -1 , whereas the optimal stochastic policy assigns each action the probability 112, resulting in R7r = O. The evolution of the R7r-estimate and policy, starting from the initial policy n(A) = 0.1 and nCB) = 0.9, is shown in Figure 1. Clearly the learned policy approaches the optimal stochastic policy n =(112,112). 5.2 MATRIX GAME: SCISSORS-PAPER-STONE-GLASS-WATER Scissors-Paper-Stone-Glass-Water (SPSGW), an extension of the well-known ScissorsPaper-Stone, is a symmetric zero-sum matrix game in which the learner selects a row i, the opponent selects a column j, and the learner' s payoff is determined by the matrix entry M(i,j). A game-theoretic solution is a stochastic (or "mixed") policy which guarantees the learner an expected payoff of at least zero. It can be shown using linear programming that the unique optimal strategy for SPSGW, yielding R7r = 0, is to play stone and water with probability 1/3, and to play scissors, paper, and glass with probability 119 [7]. Any stationary deterministic policy results in R7r = -1, since the opponent eventually learns to anticipate the learner's choice and exploit it. An Algorithm which Learns Stochastic Memory/ess Policiesfor POMDPs (a) (c) stone water or:---I---\--~ paper - 0. 4 [0 -1 1 0 -1 -1 1 1 1 M= -1 . ... -0 5 O~--='-=OO::::OO:-----:::20=:':O=OO"-----:::300'-!:'OO:::::---:-::40::':::OOO:::-----:5;-;::'OOOO number of iterations scissors (b) 1077 (d) -1] 1 1 1 -1 -1 1 0 -1 1 0 -1 -1 1 0 0. 8 -___ ~ ~.= _______ .______________ __ _ s __ _ %~-~1=OO~OO~~2~OO~OO~~3~OO~OO~~4~OO~OO~~50000 number of iteratio ns Figure 2: (a) Diagram of Scissors-Paper-Stone-Glass-Water, (b) the payoff matrix, (c) evolution of the RJr-estimate, and (d) evolution of n(stone) and n(water) (solid) and n(scissors), n(paper), and n(glass) (dashed) for ?= 0.00005, a= 0.9995. In formulating SPSGW as a POMDP, it is necessary to include in the state sufficient information to allow the opponent to exploit any sub-optimal strategy. We thus choose as states the learner's past action frequencies, multiplied at each timestep by the decay factor, a. There is only one observation, and the learner acts by selecting the "row" scissors, paper, stone, glass or water, producing a deterministic state transition. The simulated opponent plays the column which maximizes its expected payoff against the estimate of the learner's strategy obtained from the state. The learner's reward is then obtained from the appropriate entry of the payoff matrix. The policy n = (0.1124,0.1033,0.3350,0.1117,0.3376) learned after 50,000 iterations (see Figure 2) is very close to the optimal policy 7i = (119, 119,113,119,1/3). 5.3 PARR AND RUSSELL'S GRID WORLD Parr and Russell's grid world [S] consists of 11 states in a 4x3 grid with a single obstacle as shown in Figure 3(a). The learner senses only walls to its immediate east or west and whether it is in the goal state (upper right comer) or penalty state (directly below the goal), resUlting in the 6 possible observations (0-3, G and P) indicated in the diagram. The available actions are to move N, E, S, or W, but there is a probability 0.1 of slipping to either side and only O.S of moving in the deSired direction; a movement into a wall results in bouncing back to the original state. The learner receives a reward of + 1 for a transition into the goal state, -1 for a transition into the penalty state, and -0.04 for all other transitions. The goal and penalty states are connected to a cost-free absorbing state; when the learner reaches either of them it is teleported immediately to a new start state chosen with uniform probability. The results are shown in Figure 3. A separate 106 -step evaluation of the final learned policy resulted in RJr = 0.047. In contrast, the optimal deterministic policy indicated by arrows in Figure 3(a) yields R Jr = 0.024 [5], while Parr and Russell's memory-based SPOVA-RL algorithm achieved RJr = 0.12 after learning for 400,000 iterations [S]. 5.4 MULTI-SERVER QUEUE At each timestep, an arriving job having type 1, 2, or 3 with probability 112, 113 or 116, respectively, must be assigned to server A, B or C; see Figure 4(a). Each server is optimized for a particular job type which it can complete in an expected time of 2.6 J K. Williams and S. Singh 1078 0.06 (a) (b) 0 04 ~ "... 0 .0 2 ;j' ~ 0 t? 3 a: - 0 .04 - 0 .06 -1 -0.0 8 (c) t 0 ~ ~ 2 2 1 0 40000 20000 60000 80000 100000 nurrber of itera1iorlS; P 0 ~ 0 ~ - 0 .02 2 2 t ~ +1 ~ 91 rO. 7r(alm) = 8:8i 0.02 0.21 0.34 0.02 0.43 0.52J 0.36 0.60 0.18 0.02 0.11 0.02 0.19 Figure 3: (a) Parr and Russell's grid world, with observations shown in lower right corners and the optimal deterministic memoryless policy represented by arrows, (b) evolution of the R7r-estimate, and (c) the resulting learned policy (observations 0-3 across columns, actions N, E, S, W down rows) for E= 0.02, a= 0.9999. timesteps, while the other job types require 50% longer. All jobs in a server's queue are handled in parallel, up to a capacity of 10 for each server; they finish with probability Ilf at each timestep, where f is the product of the expected time for the job and the number of jobs in the server's queue. The states for this POMDP are all combinations of waiting jobs and server occupancies of the three job types, but the learner's observation is restricted to the type of the waiting job. The state transition is obtained by removing all jobs which have finished and adding the waiting job to the chosen server if it has space available. The reward is + 1 if the job is successfully placed, or 0 if it is dropped. The results are shown in Figure 4. A separate 106- step evaluation of the learned policy obtained R7r = 0.95, corresponding to 95% success in placing jobs. In contrast, the optimal deterministic policy, which assigns each job to the server optimized for it, attained only 87% success. Thus the learned policy more than halves the drop rate! 6 CONCLUSION Our online version of an algorithm proposed by Jaakkola, Singh and Jordan efficiently learns a stochastic memoryless policy which is either provably optimal or at least superior to any deterministic memoryless policy for each of four test problems. Many enhancements are possible, including appropriate learning schedules to improve performance and ensure convergence, estimation of the time between observation-action visits to obtain better discount rates r and thereby enhance Q7r-estimate bias and variance reduction (see [2]), and multiple starts or simulated annealing to avoid local minima. In addition, observations could be extended to include some past history when appropriate. Most POMDP algorithms use memory and attempt to learn an optimal deterministic policy based on belief states. The stochastic memoryless policies learned by the JSJ algorithm may not always be as good, but they are simpler to act upon and can adapt smoothly in non-stationary environments. Moreover, because it searches the space of stochastic policies, the JS] algorithm has the potential to find the optimal memoryless policy. These considerations, along with the success of our simple implementation, suggest that this algorithm may be a viable candidate for solving real-world POMDPs, including distributed control or network admission and routing problems in which the numbers of states are enormous and complete state information may be difficult to obtain or estimate in a timely manner. An AlgOrithm which Learns Stochastic Memoryless Policiesjor POMDPs (a) Server A (b) 095 TA = (2.6,3.9,3.9) Job arrival of type 1,2,or 3 1079 I . 09 I a: Server B TB = (3.9,2.6,3.9) o. 8o'----=20~00::-:0-----,-: 4oo ~00 =---60~00-=-0--::8~00'":c00::------:-=-::' 1 00000 number of iterations Server C Tc =(3.9,3.9,2.6) (c) [0.73 0.02 0.02] n(alm) = 0.02 0.96 0.09 0.25 0.02 0.89 Figure 4: (a) Schematic of the multi-server queue, (b) evolution of the R71-estimate, and (c) the resulting learned policy (observations I, 2, 3 across columns, actions A, B, C down rows) for ?= 0.005, a= 0.9999. Acknowledgements We would like to thank Mike Mozer and Tim Brown for helpful discussions. Satinder Singh was funded by NSF grant IIS-9711753. References [1] Chrisman, L. (1992). Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. In Proceedings of the Tenth National Conference on Artificial Intelligence. [2] Jaakkola, T., Singh, S. P., and Jordan, M. I. (1995). Reinforcement learning algorithm for partially observable Markov decision problems. In Advances in Neural Information Processing Systems 7. [3] Littman, M., Cassandra, A., and Kaelbling, L. (1995). Learning poliCies for partially observable environments: Scaling up. In Proceedings of the Twelfth International Conference on Machine Learning. [4] Littman, M. L. (1994). Memoryless policies: Theoretical limitations and practical results. Proceedings of the Third International Conference on Simulation of Adaptive Behavior: From Animals to Animats. [5] Loch, J., and Singh, S. P. (1998). Using eligibility traces to find the best memoryless policy in partially observable Markov decision processes. In Machine Learning: Proceedings of the Fifteenth International Conference. [6] Lovejoy, W. S. (1991). A survey of algorithmic methods for partially observable Markov decision processes. In Annals of Operations Research, 28. [7] Morris, P. (1994). Introduction to Game Theory. Springer-Verlag, New York. [8] Parr, R. and Russell, S. (1995). Approximating optimal poliCies for partially In Proceedings of the International Joint observable stochastic domains. Conference on Artificial Intelligence. [9] Singh, S. P., Jaakkola, T., and Jordan, M. I. (1994). Learning without stateestimation in partially observable Markovian decision processes. In Machine Learning: Proceedings of the Eleventh International Conference. [10] Sutton, R. S. and Barto, A. G. (1998). Reinforcement Learning: An Introduction. MIT Press. 

1509 |@word version:3 twelfth:1 simulation:1 seek:1 thereby:1 solid:2 recursively:1 reduction:1 initial:2 att:1 efficacy:1 selecting:1 past:4 current:5 com:1 assigning:1 written:1 must:1 john:1 remove:1 drop:1 update:1 stationary:3 half:1 intelligence:2 yr:1 es:1 simpler:1 admission:1 along:1 constructed:1 differential:5 viable:1 consists:2 eleventh:1 manner:1 alm:7 expected:8 behavior:2 roughly:1 multi:2 aliasing:1 rem:2 decreasing:1 increasing:1 perceives:1 begin:1 provided:1 underlying:2 becomes:1 maximizes:4 moreover:1 cm:1 nj:1 guarantee:2 forgotten:1 every:5 act:3 ti:1 iearning:2 exactly:1 unknowable:1 prohibitively:1 ro:1 control:2 grant:1 producing:1 before:2 dropped:1 local:1 sutton:1 co:1 limited:1 unique:2 practical:1 practice:1 implement:1 x3:1 procedure:1 empirical:1 r7r:10 projection:1 suggest:1 cannot:1 onto:1 close:1 operator:1 mea:1 ilf:1 deterministic:12 williams:4 regardless:1 starting:1 pomdp:10 survey:1 assigns:2 immediately:1 rule:1 continued:1 variation:1 increment:1 updated:1 annals:1 play:3 colorado:2 exact:4 programming:1 prescribe:1 expensive:4 particularly:1 mike:1 ensures:1 connected:1 russell:5 incremented:1 movement:1 mozer:1 environment:11 reward:20 littman:2 diagrammed:1 prescribes:2 singh:11 solving:3 depend:1 upon:2 learner:18 comer:1 po:1 joint:1 represented:1 distinct:1 effective:2 q7r:6 monte:1 artificial:2 quite:1 heuristic:1 solve:1 otherwise:1 florham:1 statistic:1 final:1 online:4 sequence:4 advantage:2 rr:4 ero:1 product:2 achieve:1 qr:4 exploiting:1 convergence:2 enhancement:1 produce:1 converges:1 tim:1 oo:14 job:15 arl:1 direction:2 correct:1 f4:1 stochastic:20 exploration:1 routing:1 require:4 fix:1 wall:2 anticipate:1 im:1 c00:1 extension:1 correction:2 sufficiently:1 algorithmic:1 mop:3 parr:5 nurrber:1 achieves:1 estimation:4 occupying:1 successfully:1 mit:1 clearly:1 always:2 modified:1 rather:1 avoid:1 barto:1 jaakkola:5 improvement:4 contrast:2 sense:1 glass:6 helpful:1 lovejoy:1 hidden:2 comprising:1 selects:2 provably:1 socalled:1 animal:1 having:1 placing:1 park:2 lit:4 future:2 simplify:1 summand:1 sta:1 simultaneously:1 resulted:1 national:1 delayed:1 attempt:1 message:2 investigate:1 multiply:2 evaluation:7 yielding:1 light:1 sens:1 partial:1 necessary:2 experience:2 iv:1 old:2 desired:1 ncb:2 theoretical:2 column:4 obstacle:1 jil:1 markovian:1 ar:2 cost:1 kaelbling:1 entry:2 uniform:2 chooses:1 international:5 rounded:1 enhance:1 choose:2 possibly:1 corner:1 pmin:3 potential:1 caused:1 scissors:7 depends:2 performed:3 lab:1 start:2 complicated:1 parallel:1 timely:1 jsj:5 variance:2 efficiently:1 v7r:5 yield:2 euclid:1 carlo:1 notoriously:1 pomdps:10 history:2 reach:1 against:1 simplexes:1 frequency:1 popular:1 oooo:1 schedule:3 ea:2 back:1 higher:1 attained:1 ta:1 improved:1 iai:1 ooo:1 evaluated:1 furthermore:1 until:1 receives:2 indicated:3 mdp:2 building:1 effect:1 verify:1 brown:1 evolution:7 assigned:1 memoryless:15 symmetric:1 ll:1 game:4 eligibility:1 stone:8 theoretic:1 demonstrate:1 complete:2 performs:1 fj:6 consideration:1 superior:1 absorbing:1 empirically:1 rl:1 interspersed:1 tail:3 interpret:1 grid:4 mathematics:1 similarly:1 baveja:1 funded:1 moving:1 longer:3 j:1 recent:1 verlag:1 server:13 arbitrarily:1 success:3 slipping:1 minimum:1 greater:1 preceding:1 mr:2 dashed:2 ii:2 multiple:2 adapt:1 visit:3 schematic:2 fifteenth:1 iteration:8 achieved:1 whereas:1 addition:1 annealing:1 diagram:3 jordan:5 viability:1 finish:1 timesteps:1 observability:1 inner:1 reduce:1 avenue:1 knowing:1 whether:1 handled:1 penalty:3 queue:4 speaking:1 york:1 constitute:1 action:21 generally:1 discount:5 locally:1 morris:1 sl:1 nsf:1 per:3 discrete:1 waiting:3 four:2 enormous:1 verified:1 tenth:1 timestep:8 sum:2 run:2 respond:1 bouncing:1 place:1 throughout:1 decision:8 scaling:1 followed:1 occur:1 extremely:1 prescribed:2 formulating:1 department:1 according:1 teleported:1 combination:1 jr:1 across:2 slightly:1 s1:2 gradually:1 restricted:1 boulder:1 taken:1 computationally:1 equation:3 discus:1 eventually:1 mechanism:1 needed:1 know:1 merit:1 confounded:3 available:6 operation:1 opponent:4 multiplied:1 appropriate:6 enforce:1 occurrence:1 alternative:3 original:2 denotes:1 ensure:2 include:2 exploit:2 approximating:1 seeking:1 move:1 realized:1 quantity:1 strategy:3 dependence:1 gradient:1 separate:2 thank:1 simulated:3 capacity:1 water:7 loch:1 difficult:3 trace:1 implementation:2 policy:57 animats:1 upper:1 observation:22 markov:10 immediate:1 payoff:5 extended:1 spova:1 pair:2 optimized:2 learned:9 distinction:1 chrisman:1 below:1 tb:1 including:3 memory:6 belief:1 difficulty:2 indicator:1 occupancy:1 improve:3 mdps:2 finished:1 acknowledgement:1 asymptotic:1 mixed:1 limitation:1 proven:1 sufficient:1 row:4 placed:1 last:1 free:1 arriving:1 bias:2 allow:1 side:1 distributed:1 transition:8 evaluating:1 cumulative:1 world:4 qualitatively:1 reinforcement:7 adaptive:1 observable:9 obtains:1 satinder:2 ml:1 llm:4 search:1 osj:1 learn:2 reasonably:1 inherently:2 domain:1 arrow:2 s2:1 arrival:1 representative:1 west:1 renormalization:2 n:1 sub:1 lq:1 candidate:1 perceptual:2 third:1 learns:7 down:3 removing:1 decay:3 exists:1 adding:1 te:1 horizon:1 cassandra:1 smoothly:1 tc:1 explore:1 partially:9 springer:1 conditional:2 goal:4 presentation:1 shared:1 absence:1 replace:1 determined:2 infinite:1 specifically:1 except:1 experimental:1 east:1

703 WINNER-TAKE-ALL NETWORKS OF O(N) COMPLEXITY J. Lazzaro, S. Ryckebusch, M.A. Mahowald, and C. A. Mead California Institute of Technology Pasadena, CA 91125 ABSTRACT We have designed, fabricated, and tested a series of compact CMOS integrated circuits that realize the winner-take-all function. These analog, continuous-time circuits use only O(n) of interconnect to perform this function. We have also modified the winner-take-all circuit, realizing a circuit that computes local nonlinear inhibition. Two general types of inhibition mediate activity in neural systems: subtractive inhibition, which sets a zero level for the computation, and multiplicative (nonlinear) inhibition, which regulates the gain of the computation. We report a physical realization of general nonlinear inhibition in its extreme form, known as winner-take-all. We have designed and fabricated a series of compact, completely functional CMOS integrated circuits that realize the winner-take-all function, using the full analog nature of the medium. This circuit has been used successfully as a component in several VLSI sensory systems that perform auditory localization (Lazzaro and Mead, in press) and visual stereopsis (Mahowald and Delbruck, 1988). Winnertake-all circuits with over 170 inputs function correctly in these sensory systems. We have also modified this global winner-take-all circuit, realizing a circuit that computes local nonlinear inhibition. The circuit allows multiple winners in the network, and is well suited for use in systems that represent a feature space topographically and that process several features in parallel. We have designed, fabricated, and tested a CMOS integrated circuit that computes locally the winner-take-all function of spatially ordered input. 704 Lazzaro, Ryckebusch, Mahowald and Mead THE WINNER-TAKE-ALL CmCUIT Figure 1 is a schematic diagram of the winner-take-all circuit. A single wire, associated with the potential Vc, computes the inhibition for the entire circuit; for an n neuron circuit, this wire is O(n) long. To compute the global inhibition, each neuron k contributes a current onto this common wire, using transistor T2 a.' To apply this global inhibition locally, each neuron responds to the common wire voltage Vc, using transistor Tla.' This computation is continuous in time; no clocks are used. The circuit exhibits no hysteresis, and operates with a time constant related to the size of the largest input. The output representation of the circuit is not binary; the winning output encodes the logarithm of its associated input. Figure 1. Schematic diagram of the winner-take-all circuit. Each neuron receives a unidirectional current input 11;; the output voltages VI ?.. VB represent the result of the winner-take-all computation. If II; = max(II ??? I B ), then VI; is a logarithmic function of 11;; if Ii <: 11;, then Vi ~ O. A static and dynamic ana.lysis of the two-neuron circuit illustrates these system properties. Figure 2 shows a schematic diagram of a two-neuron winner-take-all circuit. To understand the beha.vior of the circuit, we first consider the input condition II = 12 1m. Transistors TIl ~d T12 have identical potentials at gate and source, and are both sinking 1m; thus, the drain potentials VI and V2 must be equal. Transistors T21 and T22 have identical source, drain, and gate potentials, and therefore must sink the identical current ICI = IC2 = I c/2. In the subthreshold region of operation, the equation 1m = 10 exp(Vc/Vo) describes transistors Til and T 12 , where 10 is a fabrication parameter, and Vo = kT/qlt. Likewise, the equation Ic/2 = 10 exp((Vm - Vel/Volt where Vm VI = V2, describes transistors T21 and T22 . Solving for Vm(Im, Ie) yields = = Vm = Voln(~:) + Voln(:;). (1) Winner-Take-All Networks ofO(N) Complexity Thus, for equal input currents, the circuit produces equal output voltages; this behavior is desirable for a winner-take-all circuit. In addition, the output voltage Vm logarithmically encodes the magnitude of the input current 1m. Figure 2. Schematic diagram of a two-neuron winner-take-all circuit. The input condition II = 1m + Oi, 12 = 1m illustrates the inhibitory action of the circuit. Transistor Til must sink 0, more current than in the previous example; as a result, the gate voltage of Til rises. Transistors Tit and TI2 share a common gate, howeverj thus, TI2 must also sink 1m + 0,. But only 1m is present at the drain of T12 ? To compensate, the drain voltage of T12 , V2, must decrease. For small OiS, the Early effect serves to decrease the current through Th , decreasing V2 linearly with 0,. For large o's, TI2 must leave saturation, driving V2 to approximately 0 volts. As desired, the output associated with the smaller input diminishes. For large OiS, Ie2 $!:::f 0, and Iel $!:::f Ie. The equation 1m + 0, = 10 exp(Ve/Vo) describes transistor Til' and the equation Ie = 10 exp((VI - Vel/Yo) describes transistor T21 ? Solving for VI yields (2) The winning output encodes the logarithm of the associated input. The symmetrical circuit topology ensures similar behavior for increases in 12 relative to II. Equation 2 predicts the winning response of the circuit; a more complex expression, derived in (Lazzaro et.al., 1989), predicts the losing and crossover response of the circuit. Figure 3 is a plot of this analysis, fit to experimental data. Figure 4 shows the wide dynamic range and logarithmic properties of the circuitj the experiment in Figure 3 is repeated for several values of 12 , ranging over four orders of magnitude. The conductance of transistors Til and T1:a determines the losing response of the circuit. Variants of the winner-take-all circuit shown in (Lazzaro et. aI., 1988) achieve losing responses wider and narrower than Figure 3, using circuit and mask layout techniques. 705 706 Lazzaro, Ryckebusch, Mahowald and Mead WINNER-TAKE-ALL TIME RESPONSE A good winner-take-all circuit should be stable, and should not exhibit damped oscillations ("ringing") in response to input changes. This section explores these dynamic properties of our winner-take-all circuit, and predicts the temporal response of the circuit. Figure 8 shows the two-neuron winner-take-all circuit, with capacitances added to model dynamic behavior. o T 102 Vo Ie Figure 8. Schematic diagram of a two-neuron winner-take-all circuit, with capacitances added for dynamic analysis. 0 is a large MOS capacitor added to each neuron for smoothingj 0., models the parasitic capacitance contributed by the gates of Tu and T 12 , the drains of T21 and T22, and the interconnect. (Lazzaro et. al., 1988) shows a small-signal analysis of this circuit. The transfer function for the circuit has real poles, and thus the circuit is stable and does not ring, if 10 > 41(Oe/O), where 11 RlI2 Rl 1. Figure 9 compares this bound with experimental data. H Ie > 41(0 0 /0), the circuit exhibits first-order behavior. The time constant OVo/I sets the dynamics of the winning neuron, where Vo = A:T /qK. Rl 40 mV. The time constant OVE/I sets the dynamics of the losing neuron, where VE Rl 50 v. Figure 10 compares these predictions with experimental data. Winner-Take-All Networks ofO(N) Complexity 2.6 Vl,V, (V) 2.4 2.2 2.0 I.S 1.6 1.4 1.2 1.0+--+--+--+--.....~I----t~--t---f 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.0S II/I, Figure 8. Experimental data (circles) and theory (solid lines) for a two-neuron winner-take-all circuit. II, the input current of the first neuron, is swept about the value of 12, the input current of the second neuron; neuron voltage outputs VI and V2 are plotted versus normalized input current. 2.6 I.S 1.6 1.4 1.2 10- 11 10- 10 10- 0 IdA) 10- 8 Figure 4. The experiment of Figure 3 is repeated for several values of 12; experimental data of output voltage response are plotted versus absolute input current on a log scale. The output voltage VI = V2 is highlighted with a circle for each experiment. The dashed line is a theoretical expression confirming logarithmic behavior over four orders of magnitude (Equation 1). 707 708 Lazzaro, Ryckebusch, Mahowald and Mead 1 Figure 9. Experimental data (circles) and theoretical statements (solid line) for a two-neuron winner-take-all circuit, showing the smallest 10 , for a given I, necessary for a first-order response to a small-signal step input. Figure 10. Experimental data (symbols) and theoretical statements (solid line) for a two-neuron winner-take-all circuit, showing the time constant of the first-order response to a small-signal step input. The winning response (filled circles) and losing response (triangles) of a winner-take-a.ll circuit are shownj the time constants differ by several orders of magnit ude. Winner~Take~AlI Networks ofO(N) Complexity THE LOCAL NONLINEAR INHIBITION CIRCUIT The winner-take-all circuit in Figure 1, as previously explained, locates the largest input to the circuit. Certain applications require a gentler form of nonlinear inhibition. Sometimes, a circuit that can represent multiple intensity scales is necessary. Without circuit modification, the winner-take-all circuit in Figure 1 can perform this task. (Lazzaro et. al., 1988) explains this mode of operation. Other applications require a local winner-take-all computation, with each winner having inHuence over only a limited spatial area. Figure 12 shows a circuit that computes the local winner-taite-all function. The circuit is identical to the original winner-take-all circuit, except that each neuron connects to its nearest neighbors with a nonlinear resistor circuit (Mead, in press). Each resistor conducts a current Ir in response to a voltage ~V across it, where Ir = I.tanh(~V/(2Vo)). 1., the saturating current of the resistor, is a controllable parameter. The current source, 10, present in the original winner-take-all circuit, is distributed between the resistors in the local winner-take-all circuit. Figure 11. Schematic diagram of a section of the local winner-take-all circuit. Each neuron i receives a unidirectional current input Iii the output voltages Vi represent the result of the local winner-take-all computation. To understand the operation of the local winner-take-all circuit, we consider the circuit response to a spatial impulse, defined as 1" :> 1, where 1 == h~". 1,,:> 1"-1 and 1,,:> 1"+1, so Ve:,. is much larger than Ve:,._l and Ve:lI+l' and the resistor circuits connecting neuron 1: with neuron 1: - 1 and neuron 1: + 1 saturate. Each resistor sinks 1. current when saturatedj transistor T2,. thus conducts 21. + Ie: current. In the subthreshold region of operation, the equation 1" = 10 exp(Ve:,. /Vo) describes transistor TI ,., and the equation 21. + Ie = Ioexp((V" - Ve:,.)/Vo) describes transistor 709 710 Lazzaro, Ryckebusch, Mahowald and Mead T2,.. Solving for VA: yields VA: = voln((2I. + 10 )/10 ) + voln(IA:/lo). (4) As in the original winner-take-all circuit, the output of a winning neuron encodes the logarithm of that neuron's associated input. As mentioned, the resistor circuit connecting neuron Ie with neuron Ie - 1 sinks 1. CUlTent. The current sources 10 associated with neurons Ie -1, Ie - 2, ... must supply this current. If the current source 10 for neuron Ie - 1 supplies part of this current, the transistor T2,._1 carries no current, and the neuron output VA:-l approaches zero. In this way, a winning neuron inhibits its neighboring neurons. This inhibitory action does not extend throughout the network. Neuron Ie needs only 1. current from neurons Ie - 1, Ie - 2, .... Thus, neurons sufficiently distant from neuron Ie maintain the service of their current source 10, and the outputs of these distant neurons can be active. Since, for a spatial impulse, all neurons Ie - 1, Ie - 2, ... have an equal input current I, all distant neurons have the equal output (5) Similar reasoning applies for neurons Ie + 1, Ie + 2, .... The relative values of 1. and 10 determine the spatial extent of the inhibitory action. Figure 12 shows the spatial impulse response of the local winner-take-all circuit, for different settings of 1./10 , o I 2 4 I 8 10 6 Ie (Pollition) I 12 I 14 I 16 Figure 12. Experimental data showing the spatial impulse response of the local winner-take-all circuit, for values of 1./10 ranging over a factor of 12.7. Wider inhibitory responses correspond to larger ratios. For clarity, the plots are vertically displaced in 0.25 volt increments. Winner-Take-All Networks ofO(N) Complexity CONCLUSIONS The circuits described in this paper use the full analog nature of MOS devices to realize an interesting class of neural computations efficiently. The circuits exploit the physics of the medium in many ways. The winner-take-all circuit uses a single wire to compute and communicate inhibition for the entire circuit. Transistor TI,. in the winner-take-all circuit uses two physical phenomena in its computation: its exponential current function encodes the logarithm of the input, and the finite conductance of the transistor defines the losing output response. As evolution exploits all the physical properties of neural devices to optimize system performance, designers of synthetic neural systems should strive to harness the full potential of the physics of their media. Acknow ledgments John Platt, John Wyatt, David Feinstein, Mark Bell, and Dave Gillespie provided mathematical insights in the analysis of the circuit. Lyn Dupre proofread the document. We thank Hewlett-Packard for computing support, and DARPA and MOSIS for chip fabrication. This work was sponsored by the Office of Naval Research and the System Development Foundation. References Lazzaro, J. P., Ryckebusch, S., Mahowald, M.A., and Mead, C.A. (1989). WinnerTake-All Networks of O(N) Oomplexity, Caltech Computer Science Department Technical Report Caltech-CS-TR-21-88. Lazzaro, J. P., and Mead, C.A. {in press}. Silicon Models of Auditory Localization, Neural Oomputation. Mahowald, M.A., and Delbruck, T.I. (1988). An Analog VLSI Implementation of the Marr-Poggio Stereo Correspondence Algorithm, Abstracts of the First Annual INNS Meeting, Boston, 1988, Vol. I, Supplement I, p. 392. Mead, C. A. (in press). Analog VLSI and Neural Systems. Reading, MA: AddisonWesley. 711 

151 |@word tr:1 solid:3 carry:1 series:2 document:1 current:26 ida:1 must:7 john:2 realize:3 distant:3 confirming:1 designed:3 plot:2 sponsored:1 device:2 realizing:2 ofo:4 mathematical:1 supply:2 mask:1 behavior:5 decreasing:1 provided:1 circuit:69 medium:3 ringing:1 fabricated:3 temporal:1 ti:2 platt:1 t1:1 service:1 local:11 vertically:1 mead:10 approximately:1 limited:1 range:1 area:1 crossover:1 bell:1 onto:1 optimize:1 cmcuit:1 layout:1 insight:1 marr:1 increment:1 losing:6 us:2 logarithmically:1 predicts:3 t12:3 region:2 ensures:1 oe:1 decrease:2 mentioned:1 complexity:5 dynamic:7 solving:3 tit:1 topographically:1 ove:1 ali:1 localization:2 completely:1 sink:5 triangle:1 darpa:1 chip:1 larger:2 highlighted:1 transistor:17 inn:1 t21:4 tu:1 neighboring:1 realization:1 achieve:1 produce:1 cmos:3 leave:1 ring:1 wider:2 nearest:1 ois:2 c:1 differ:1 vc:3 ana:1 explains:1 require:2 im:1 sufficiently:1 ic:1 exp:5 mo:2 driving:1 early:1 smallest:1 diminishes:1 tanh:1 largest:2 successfully:1 t22:3 modified:2 voltage:11 office:1 derived:1 yo:1 naval:1 interconnect:2 vl:1 integrated:3 entire:2 pasadena:1 vlsi:3 vior:1 development:1 spatial:6 equal:5 having:1 identical:4 report:2 t2:4 ve:7 connects:1 maintain:1 conductance:2 extreme:1 hewlett:1 damped:1 kt:1 necessary:2 poggio:1 addisonwesley:1 filled:1 conduct:2 logarithm:4 desired:1 circle:4 plotted:2 theoretical:3 wyatt:1 delbruck:2 mahowald:8 pole:1 fabrication:2 synthetic:1 explores:1 ie:21 vm:5 physic:2 connecting:2 strive:1 til:6 li:1 potential:5 hysteresis:1 mv:1 vi:10 multiplicative:1 parallel:1 unidirectional:2 oi:1 ir:2 qk:1 efficiently:1 likewise:1 subthreshold:2 yield:3 correspond:1 dave:1 associated:6 static:1 gain:1 auditory:2 harness:1 response:18 vel:2 clock:1 receives:2 nonlinear:7 defines:1 mode:1 impulse:4 effect:1 normalized:1 evolution:1 spatially:1 volt:3 ll:1 gentler:1 vo:8 reasoning:1 ranging:2 common:3 functional:1 physical:3 regulates:1 rl:3 winner:45 ti2:3 analog:5 sinking:1 extend:1 silicon:1 ai:1 winnertake:2 stable:2 inhibition:12 certain:1 binary:1 meeting:1 swept:1 proofread:1 caltech:2 determine:1 signal:3 ii:8 dashed:1 full:3 multiple:2 desirable:1 technical:1 long:1 compensate:1 locates:1 va:3 schematic:6 beha:1 variant:1 prediction:1 represent:4 sometimes:1 addition:1 diagram:6 source:6 capacitor:1 iii:1 ledgments:1 fit:1 topology:1 expression:2 stereo:1 lazzaro:12 action:3 locally:2 inhibitory:4 designer:1 correctly:1 vol:1 four:2 clarity:1 mosis:1 communicate:1 throughout:1 oscillation:1 vb:1 bound:1 correspondence:1 annual:1 activity:1 encodes:5 inhibits:1 magnit:1 department:1 describes:6 smaller:1 across:1 modification:1 explained:1 tla:1 equation:8 previously:1 feinstein:1 serf:1 operation:4 apply:1 v2:7 gate:5 original:3 exploit:2 capacitance:3 added:3 ude:1 ryckebusch:6 responds:1 exhibit:3 thank:1 extent:1 ratio:1 statement:2 acknow:1 rise:1 implementation:1 perform:3 contributed:1 wire:5 displaced:1 neuron:40 finite:1 lyn:1 intensity:1 david:1 california:1 reading:1 saturation:1 max:1 packard:1 gillespie:1 ia:1 technology:1 drain:5 relative:2 interesting:1 versus:2 foundation:1 ovo:1 subtractive:1 share:1 lo:1 understand:2 institute:1 wide:1 neighbor:1 absolute:1 voln:4 distributed:1 computes:5 sensory:2 ici:1 compact:2 global:3 active:1 symmetrical:1 stereopsis:1 continuous:2 nature:2 transfer:1 ca:1 controllable:1 contributes:1 complex:1 ic2:1 linearly:1 mediate:1 taite:1 repeated:2 resistor:7 winning:7 exponential:1 saturate:1 showing:3 symbol:1 supplement:1 magnitude:3 iel:1 illustrates:2 boston:1 suited:1 logarithmic:3 visual:1 ordered:1 saturating:1 applies:1 determines:1 ma:1 narrower:1 change:1 except:1 operates:1 experimental:8 parasitic:1 mark:1 support:1 tested:2 phenomenon:1

The Effect of Correlations on the Fisher Information of Population Codes Hyoungsoo Yoon hyoung@fiz.huji.ac.il Haim Sompolinsky hairn@fiz.huji.ac.il Racah Institute of Physics and Center for Neural Computation Hebrew University, Jerusalem 91904, Israel Abstract We study the effect of correlated noise on the accuracy of population coding using a model of a population of neurons that are broadly tuned to an angle in two-dimension. The fluctuations in the neuronal activity is modeled as a Gaussian noise with pairwise correlations which decays exponentially with the difference between the preferred orientations of the pair. By calculating the Fisher information of the system, we show that in the biologically relevant regime of parameters positive correlations decrease the estimation capability of the network relative to the uncorrelated population. Moreover strong positive correlations result in information capacity which saturates to a finite value as the number of cells in the population grows. In contrast, negative correlations substantially increase the information capacity of the neuronal population. 1 Introduction In many neural systems , information regarding sensory inputs or (intended) motor outputs is found to be distributed throughout a localized pool of neurons. It is generally believed that one of the main characteristics of the population coding scheme is its redundancy in representing information (Paradiso 1988; Snippe and Koenderink 1992a; Seung and Sompolinsky 1993). Hence the intrinsic neuronal noise, which has detrimental impact on the information processing capability, is expected to be compensated by increasing the number of neurons in a pool. Although this expectation is universally true for an ensemble of neurons whose stochastic variabilities are statistically independent, a general theory of the efficiency of population coding when the neuronal noise is correlated within the population, has been lacking. The conventional wisdom has been that the correlated variability limits the information pro cessing capacity of neuronal ensembles (Zohary, Shadlen, and Newsome 1994). H. Yoon and H. Sompolinsky 168 However, detailed studies of simple models of a correlated population that code for a single real-valued parameter led to apparently contradicting claims. Snippe and Koenderink (Snippe and Koenderink 1992b) conclude that depending on the details of the correlations, such as their spatial range, they may either increase or decrease the information capacity relative to the un correlated one. Recently, Abbott and Dayan (Abbott and Dayan 1998) claimed that in many cases correlated noise improves the accuracy of population code. Furthermore, even when the information is decreased it still grows linearly with the size of the population. If true, this conclusion has an important implication on the utility of using a large population to improve the estimation accuracy. Since cross-correlations in neuronal activity are frequently observed in both primary sensory and motor areas (Fetz, Yoyama, and Smith 1991 ; Lee, Port, Kruse, and Georgopoulos 1998), understanding the effect of noise correlation in biologically relevant situations is of great importance. In this paper we present an analytical study of the effect of noise correlations on the population coding of a pool of cells that code for a single one-dimensional variable, an angle on a plane, e.g. , an orientation of a visual stimulus, or the direction of an arm movement. By assuming that the noise follows the multivariate Gaussian distribution, we investigate analytically the effect of correlation on the Fisher information. This model is similar to that considered in (Snippe and Koenderink 1992b; Abbott and Dayan 1998). By analyzing its behavior in the biologically relevant regime of tuning width and correlation range, we derive general conclusions about the effect of the correlations on the information capacity of the population. 2 Population Coding with Correlated Noise We consider a population of N neurons which respond to a stimulus characterized by an angle (), where -1r < () ~ 1r . The activity of each neuron (indexed by i) is assumed to be Gaussian with a mean h((}) which represents its tuning curve, and a uniform variance a . The noise is assumed to be pairwise-correlated throughout the population. Hence the activity profile of the whole population, R = {rl, r2, .. . , r N } , given a stimulus () , follows the following multivariate Gaussian distribution. P(RI(}) = Nexp (-~ l:(ri - h((}?)Gi-/(rj - fj((})) (1) t ,J where N is a normalization constant and Cj is the correlation matrix. G ij = ac5ij + bij (1 - c5ij ). (2) It is assumed that the tuning curves of all the neurons are identical in form but peaked at different angles, that is fi((}) = f((} - ?i) where the preferred angles ?i are distributed uniformly from -1r to 1r with a lattice spacing, w , which is equal to 21r IN. We further assume that the noise correlation between a pair of neurons is only a function of their preferred angle difference, i.e., bij ((}) = b(ll?i - ?jll) where lI(}l - (}211 is defined to be the relative angle between (}l and (}2, and hence its maximum value is 1r. A decrease in the magnitude of neuronal correlations with the dissimilarity in the preferred stimulus is often observed in cortical areas. We model this by exponentially decaying correlations bij = b exp( _1I?i - p where p specifies the angular correlation length. ?j II) (3) 169 Fisher Information of Correlated Population Codes The amount of information that can be extracted from the above population will depend on the decoding scheme. A convenient measure of the information capacitv in the population is given by the Fisher information, which in our case is (for it given stimulus 8) J(8) =L (4) giGi-/ gj i ,j where .(e) = gt - {)Iiae(8) . (5) The utility of this measure follows from the well known Cramer-Rao bound for the variance of any unbiased estimators, i.e., ((8 - iJ)2) 2: 1/ J(8). For the rest of this paper, we will concentrate on the Fisher information as a function of the noise correlation parameters, band p, as well as the population size N. 3 Results In the case of un correlated population (b by (Seung and Sompolinsky 1993) 0) , the Fisher information is given (6) n \vhere gn is the Fourier transform of gj, defined by 1 - gn = N Le . A. -'l.n'P] (7) gj. j The mode number n is an integer running from _N:;l to N:;l (for odd N) and = -7f(N + 1)/N + iw, i = 1, .. . , N. Likewise, in the case of b ::j:. 0, J is given by ?i J = NL Ignl G n 2 (8) n where Gn are the eigenvalues of the covariance matrix, t,] (a _ 2&) + 2b 1 - N+I .\ cos(nuJ) - ( _ 1)11.\ -y- C'os(nw)(1 - .\) 1 - 2,\ cos(nw) + .\2 (9) 7J, .\ where w = = e- w / p , and N is assumed to be an odd integer. Note that thE' covariance matrix Gij remains positive definite as long as (10) where the lower bound holds for general N while the upper bound is valid for large N. To evaluate the effect of correlations in a large population it is important to specify the appropriate scales of the system parameters. We consider here the biologically relevant case of broadly tuned neurons that have a smoothly varying tuning curve with a single peak. When the tuning curve is smoothly varying, Ignl 2 will be a rapidly decaying function as n increases beyond a characteristic value which is H. Yoon and H. Sompolinsky 170 proportional to the inverse of the tuning width, a. We further assume a broad tuning, namely that the tuning curve spans a substantial fraction of the angular extent. This is consistent with the observed typical values of half-width at half height in visual and motor areas, which range from 20 to 60 degrees . Likewise , it is reasonable to assume that the angular correlation length p spans a substantial fraction of the entire angular range. This broad tuning of correlations with respect to the difference in the preferred angles is commonly observed in cortex (Fetz, Yoyama, and Smith 1991 ; Lee , Port, Kruse, and Georgopoulos 1998). To capture these features we will consider the limit of large N while keeping the parameters p and a constant. Note that keeping a of order 1 implies that substantial contributions to Eq. (8) come only from n which remain of order 1 as N increases. On the other hand, given the enormous variability in the strength of the observed crosscorrelations between pairs of neurons in cortex, we do not restrict the value of b at this point. Incorporating the above scaling we find that when N is large 1 is given by N 2 p- 2 + n 2 , 1=~~19nl p-2+n2+(~)(1-(-1)ne-7I'/p) ' (11) Inspection of the denominator in the above equation clearly shows that for all positive values of b, 1 is smaller than 1 0 , On the other hand, when b is negative 1 is larger than 10 , To estimate the magnitude of these effects we consider below three different regimes. 1.0 """"'----,-----r---.....,------, D.8 .I .10 0.6 0.4 0.2 0.0 () 1000 2000 3000 400() N Figure 1: Normalized Fisher information when p '" 0(1) (p 0.257r was used). a = 1 and b = 0.1 , 0.01, and 0.001 from the bottom. We used a circular Gaussian tuning curve, Eq. (13), with fmax = 10 and a = 0.27r. = Strong positive correlations: We first discuss the regime of strong positiw correlations, by which we mean that a < b/a "" 0(1). In this case the second term in the denominator of Eq. (11) is of order Nand Eq. (11) becomes 7rp 2 p-2 + n 2 1=b 19n1 1 _ (-1)ne-7I'/P' (12) L n This result implies that in this regime the Fisher information in the entire population does not scale linearly with the population size N but saturates to a sizeindependent finite limit . Thus, for these strong correlations, although the number of neurons in the population may be large, the number of independent degrees of freedom is small. We demonstrate the above phenomenon by a numerical evaluation of 1 for the following choice of tuning curve f(O) = fmax exp ((cos(O) - 1)/a 2 ) (13) 171 Fisher Information of Correlated Population Codes with (J = 0.211". The results are shown in Fig. 1 and Fig. 2. The results of Fig. 1 clearly show the substantial decrease in J as b increases. The reduction in J I J o when b '" 0(1) indicates that J does not scale with N in this limit. Fig. 2 shows t.he saturation of J when N increases. For p = 0.1 and 1 ((c) and (d)), J saturates at. about N = 100, which means that for these parameter values the network contains at most 100 independent degrees of freedom. When the correlation range becomes either smaller or bigger, the saturation becomes less prominent (( a) and (b)) , which is further explained later in the text. 40 .30 J 20 (c) 10 (ti) 200 400 600 800 N Figure 2: Saturation of Fisher information with the correlation coefficient kept fixed: a = 1 and b = 0.5. Both p '" 0(1) ((c) p = 0.1 and (d) p = 1) and other extreme limits ((a) p = 0.01 and (b) p = 10) are shown. Tuning curve with fmax = 1 and (J = 0.211" was used for all four curves . Weak positive correlations: This regime is defined formally by positive values In this case, while J is still smaller than .10 the of b which scale as bla '" O( suppressive effects of the correlations are not as strong as in the first case. This is shown in Fig. 3 (bottom traces) for N = 1000. While J is less than J o , it is still a substantial fraction of J o , indicating J is of order N. -k). 2.3 2.0 .:L J" 1.3 1.0 0.3 0 1 2 3 4 P Figure 3: Normalized Fisher information when p '" 0(1) and bla '" O(~). N = 1000, a = 1, fmax = 10, and (J = 0.211". The top curves represent negative h (b = -0.005 and -0.002 from the top) and the bottom ones positive b (b = 0.01 and 0.005 from the bottom). Weak negative correlations: So far we have considered the case of positive b. As stated above, Eq. (11) implies that when b < 0, J > J o . The lower bound of b (Eq. (10)) means that when the correlations are negative and p is of order 1 th!' amplitude of the c:orrelations must be small. It scales as bla = biN with b which is of order 1 and is larger than bmin = -(11"lp)/(I- exp(-11"lp)). In this regime (.] - .10 )IN retains a finite positive value even for large N. This enhancement call H. Yoon and H. Sompolinsky 172 , , be made large if b comes close to bmin . This behavior is shown in Fig. 3 (upper traces). Note that, for both positive and negative weak correlations, the curves have peaks around a characteristic length scale p '" a, which is 0.211" in this figure. Extremely long and short range correlations: Calculation with strictly uniform correlations, i. e., bij = b, shows that in this case the positive correlations enhance the Fisher information of the system, leading to claims that this might be a gelleri<: result (Abbott and Dayan 1998). Here we show that this behavior is special to cases where the correlations essentially do not vary in strength. We consider the case p '" O(N). This means that the strength of the correlations is the same for all the neurons up to a correction of order liN. In this limit Eq. (11) is not valid, and the Fisher information is obtained from Eq. (8) and Eq. (9), (14) where {} = wpl4. Note that even in this extreme regime, only for {} > 1 is 1 guaranteed to be always larger than .10' Below this value the sign of 1 -.10 depends on the particular shape of the tuning curve and the value of b. In fact, a more detailed analysis (Yoon and Sompolinsky 1998) shows that as soon as p? O(VN), 1 - 10 < 0, as in the case of p rv 0(1) discussed above. The crossover between these two opposite behaviors is shown in Fig. 4. For comparison the case with p rv 0(1) is also shown. 4.0 3.0 .J .In 2.0 1.D n.o n.2 n.D 0.4 0.8 D.G 1.0 b Figure 4: Normalized Fisher information when bla rv 0(1). N = 1000 and a = 1. When p '" 0(1), increasing b always decreases the Fisher information (bottom curve p = 0.2511"). However, this trend is reversed when p ,. . ., O(VN) and when p > ~N .1 - .10 becomes always positive. From the top p = 400, 50, and 25 . Another extreme regime is where the correlation length p scales as 1IN but the tuning width remains of order 1. This means that a given neuron is correlated with a small number of its immediate neighbors, which remains finite as N ~ 00. In this limit , the Fishel' information becomes, again from Eq. (8) and Eq. (9), _ N(>..-l_l) 2 1 - a(>..-1_1)+2bI:19nl. (15 ) " In this case, the behavior of 1 is similar to the cases of weak correlations discussed above. The information remains of order N but the sign of 1 - 10 depends on the sign of b. Thus, when the amplitude of the positive correlation function is 0(1), .] increases linearly with N in the two opposite extremes of very large and very small p as shown in Fig. 2 ((a) and (b)). Fisher Information of Correlated Population Codes 4 173 Discussion In this paper we have studied the effect of correlated variability of neuronal activity OIl the maximum accuracy of the population coding. We have shown that the effect of correlation on the information capacity of the population crucially depends on the scale of correlation length. We argue that for the sensory and motor areas which are presumed to utilize population coding, the tuning of both the correlations and the mean response profile is broad and of the same order. This implies that each neuron is correlated with a finite fraction of the total number of neurons, N, and a given stimulus activates a finite fraction of N. We show that in this regime positive correlations always decrease the information. When they are strong enough in amplitude they reduce the number of independent degrees of freedom to a finite number even for large population. Only in the extreme case of almost uniform correlations the information capacity is enhanced. This is reasonable since to overcome the positive correlations one needs to subtract the responses of different neurons. But in general this will reduce their signal by a larger amount. When the correlations are uniform, the reduction of the correlated noise by subtraction is perfect and can be made in a manner that will little affect the signal component. Acknow ledgments H.S. acknowledges helpful discussions with Larry Abbott and Sebastian Seung. This research is partially supported by the Fund for Basic Research of the Israeli Academy of Science and by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. References L. F. Abbott and P. Dayan (1998). The effect of correlated variability on the accuracy of a population code. Neural Camp., in press. E. Fetz , K. Yoyama, and W. Smith (1991). Synaptic interactions between cortical neurons. In A. Peters and E. G. Jones (Eds.) , Cerebral Cortex, Volume 9. New York: Plenum Press. D. Lee, N. L. Port, W. Kruse, and A. P. Georgopoulos (1998). Variability and correlated noise in the discharge of neurons in motor and parietal areas of the primate cortex. J. Neurosci. 18, 1161- 1170. M. A. Paradiso (1988). A theory for the use of visual orientation informatioll which exploits the columnar structure of striate cortex. BioI. Cybern . 58, 35- 49. H. S. Seung and H. Sompolinsky (1993). Simple models for reading neuronal population codes. Proc . Natl. Acad. Sci. USA 90, 10749- 10753. H. P. Snippe and J. J. Koenderink (1992a). Discrimination thresholds for channelcoded Hystems. Biol. Cybern. 66, 543- 551. H. P. Snippe and J. J. Koenderink (1992b). Information in chClnnel-coded system: correlated receivers . Biol. Cybern. 67, 183- 190. H. Yoon and H. Sompolinsky (1998). Population coding in neuronal systems with correlated noise, preprint. E. Zohary, M. N. Shadlen, and W. T. Newsome (1994). Correlated neuronal discharge rate and its implications for psychophysical performance. Nature 370, 140- 143. 

1510 |@word crucially:1 covariance:2 reduction:2 contains:1 united:1 tuned:2 must:1 numerical:1 shape:1 motor:5 fund:1 discrimination:1 half:2 plane:1 inspection:1 smith:3 short:1 height:1 manner:1 pairwise:2 presumed:1 expected:1 behavior:5 frequently:1 little:1 zohary:2 increasing:2 becomes:5 moreover:1 israel:3 substantially:1 ti:1 grant:1 positive:16 limit:7 acad:1 analyzing:1 fluctuation:1 might:1 studied:1 co:3 range:6 statistically:1 bi:1 bla:4 definite:1 area:5 crossover:1 convenient:1 close:1 cybern:3 conventional:1 center:1 compensated:1 jerusalem:2 estimator:1 bsf:1 racah:1 population:37 plenum:1 discharge:2 enhanced:1 trend:1 observed:5 bottom:5 yoon:6 preprint:1 capture:1 sompolinsky:9 decrease:6 movement:1 substantial:5 seung:4 depend:1 efficiency:1 whose:1 larger:4 valued:1 gi:1 transform:1 eigenvalue:1 analytical:1 interaction:1 relevant:4 rapidly:1 fmax:4 academy:1 enhancement:1 perfect:1 depending:1 derive:1 ac:2 ij:2 odd:2 eq:11 strong:6 paradiso:2 implies:4 come:2 direction:1 concentrate:1 snippe:6 stochastic:1 larry:1 bin:1 strictly:1 correction:1 hold:1 around:1 considered:2 cramer:1 exp:3 great:1 nw:2 claim:2 vary:1 estimation:2 proc:1 iw:1 clearly:2 activates:1 gaussian:5 c5ij:1 always:4 varying:2 indicates:1 contrast:1 camp:1 helpful:1 dayan:5 entire:2 nand:1 orientation:3 spatial:1 special:1 equal:1 identical:1 represents:1 broad:3 jones:1 peaked:1 stimulus:6 intended:1 fiz:2 n1:1 freedom:3 investigate:1 circular:1 evaluation:1 extreme:5 nl:3 natl:1 implication:2 indexed:1 rao:1 gn:3 newsome:2 retains:1 lattice:1 uniform:4 peak:2 huji:2 lee:3 physic:1 decoding:1 pool:3 enhance:1 again:1 koenderink:6 leading:1 li:1 coding:8 coefficient:1 depends:3 later:1 apparently:1 decaying:2 capability:2 contribution:1 il:2 accuracy:5 variance:2 characteristic:3 likewise:2 ensemble:2 wisdom:1 weak:4 sebastian:1 synaptic:1 ed:1 improves:1 cj:1 amplitude:3 specify:1 response:2 furthermore:1 angular:4 correlation:46 hand:2 o:1 jll:1 mode:1 grows:2 oil:1 effect:12 usa:1 normalized:3 true:2 unbiased:1 hence:3 analytically:1 ll:1 width:4 prominent:1 bmin:2 demonstrate:1 pro:1 fj:1 recently:1 fi:1 rl:1 binational:1 exponentially:2 cerebral:1 volume:1 discussed:2 he:1 cessing:1 tuning:15 nexp:1 cortex:5 gj:3 gt:1 multivariate:2 claimed:1 subtraction:1 kruse:3 signal:2 ii:1 rv:3 rj:1 characterized:1 calculation:1 believed:1 cross:1 long:2 lin:1 bigger:1 coded:1 impact:1 basic:1 denominator:2 essentially:1 expectation:1 normalization:1 represent:1 cell:2 spacing:1 decreased:1 suppressive:1 rest:1 integer:2 call:1 enough:1 ledgments:1 affect:1 restrict:1 opposite:2 reduce:2 regarding:1 utility:2 peter:1 york:1 generally:1 detailed:2 amount:2 band:1 crosscorrelations:1 specifies:1 sign:3 broadly:2 redundancy:1 four:1 threshold:1 enormous:1 abbott:6 kept:1 utilize:1 fraction:5 angle:8 inverse:1 respond:1 throughout:2 reasonable:2 almost:1 vn:2 scaling:1 bound:4 haim:1 guaranteed:1 activity:5 strength:3 georgopoulos:3 ri:2 fourier:1 span:2 extremely:1 remain:1 smaller:3 lp:2 biologically:4 primate:1 explained:1 equation:1 remains:4 discus:1 appropriate:1 rp:1 top:3 running:1 calculating:1 exploit:1 psychophysical:1 primary:1 striate:1 detrimental:1 reversed:1 sci:1 capacity:7 argue:1 extent:1 assuming:1 length:5 code:9 modeled:1 hebrew:1 trace:2 stated:1 negative:6 acknow:1 upper:2 neuron:18 finite:7 parietal:1 immediate:1 situation:1 saturates:3 variability:6 pair:3 namely:1 israeli:1 beyond:1 below:2 regime:10 reading:1 saturation:3 arm:1 representing:1 scheme:2 improve:1 ne:2 acknowledges:1 text:1 understanding:1 relative:3 lacking:1 proportional:1 localized:1 foundation:1 degree:4 consistent:1 shadlen:2 port:3 uncorrelated:1 supported:1 keeping:2 soon:1 institute:1 fetz:3 neighbor:1 distributed:2 curve:13 dimension:1 cortical:2 valid:2 overcome:1 sensory:3 commonly:1 made:2 universally:1 far:1 preferred:5 receiver:1 conclude:1 assumed:4 un:2 nature:1 main:1 linearly:3 neurosci:1 whole:1 noise:15 profile:2 n2:1 contradicting:1 neuronal:11 fig:8 bij:4 r2:1 decay:1 intrinsic:1 incorporating:1 importance:1 magnitude:2 dissimilarity:1 columnar:1 subtract:1 smoothly:2 led:1 visual:3 partially:1 extracted:1 bioi:1 fisher:17 typical:1 uniformly:1 total:1 gij:1 indicating:1 formally:1 vhere:1 phenomenon:1 evaluate:1 biol:2 correlated:21

Learning multi-class dynamics A. Blake, B. North and M. Isard Department of Engineering Science, University of Oxford, Oxford OXl 3P J, UK. Web: http://www.robots.ox.ac.uk/ ... vdg/ Abstract Standard techniques (eg. Yule-Walker) are available for learning Auto-Regressive process models of simple, directly observable, dynamical processes. When sensor noise means that dynamics are observed only approximately, learning can still been achieved via Expectation-Maximisation (EM) together with Kalman Filtering. However, this does not handle more complex dynamics, involving multiple classes of motion. For that problem, we show here how EM can be combined with the CONDENSATION algorithm, which is based on propagation of random sample-sets. Experiments have been performed with visually observed juggling, and plausible dynamical models are found to emerge from the learning process. 1 Introduction The paper presents a probabilistic framework for estimation (perception) and classification of complex time-varying signals, represented as temporal streams of states. Automated learning of dynamics is of crucial importance as practical models may be too complex for parameters to be set by hand. The framework is particularly general, in several respects, as follows. 1. Mixed states: each state comprises a continuous and a discrete component. The continuous component can be thought of as representing the instantaneous position of some object in a continuum. The discrete state represents the current class of the motion, and acts as a label, selecting the current member from a set of dynamical models. 2. Multi-dimensionality: the continuous component of a state is, in general, allowed to be multi-dimensional. This could represent motion in a higher dimensional continuum, for example , two-dimensional translation as in figure 1. Other examples include multi-spectral acoustic or image signals, or multi-channel sensors such as an electro-encephalograph. 390 A. Blake. B. North and M Isard Figure 1: Learning the dynamics of juggling. Three motion classes, emerging from dynamical learning, turn out to correspond accurately to ballistic motion (mid grey), catch/throw (light grey) and carry (dark grey). 3. Arbitrary order: each dynamical system is modelled as an Auto-Regressive Process (ARP) and allowed to have arbitrary order (the number of time-steps of "memory" that it carries.) 4. Stochastic observations: the sequence of mixed states is "hidden" - not observable directly, but only via observations, which may be multi-dimensional, and are stochastically related to the continuous component of states. This aspect is essential to represent the inherent variability of response of any real signal sensing system. Estimation for processes with properties 2,3,4 has been widely discussed both in the control-theory literature as "estimation" and "Kalman filtering" (Gelb, 1974) and in statistics as ''forecasting'' (Brockwell and Davis, 1996). Learning of models with properties 2,3 is well understood (Gelb, 1974) and once learned can be used to drive pattern classification procedures, as in Linear Predictive Coding (LPC) in speech analysis (Rabiner and Bing-Hwang, 1993), or in classification of EEG signals (Pardey et al., 1995). When property 4 is added, the learning problem becomes harder (Ljung, 1987) because the training sets are no longer observed directly. Mixed states (property 1) allow for combining perception with classification. Allowing properties 2,4, but restricted to a Oth order ARP (in breach of property 3), gives 391 Learning Multi-Class Dynamics Hidden Markov Models (HMM) (Rabiner and Bing-Hwang, 1993), which have been used effectively for visual classification (Bregler , 1997). Learning HMMs is accomplished by the "Baum-Welch" algorithm, a form of Expectation-Maximisation (EM) (Dempster et al., 1977). Baum-Welch learning has been extended to "graphicalmodels" of quite general topology (Lauritzen, 1996). In this paper, graph topology is a simple chain-pair as in standard HMMs, and the complexity of the problem lies elsewhere - in the generality of the dynamical model. Generally then, restoring non-zero order to the ARPs (property 3), there is no exact algorithm for estimation. However the estimation problem can be solved by random sampling algorithms , known variously as bootstrap filters (Gordon et al., 1993), particle filters (Kitagawa, 1996), and CONDENSATION (Blake and Isard, 1997). Here we show how such algorithms can be used, with EM, in dynamical learning theory and experiments (figure 1). 2 Multi-class dynamics Continuous dynamical systems can be specified in terms of a continuous state vector Xt E nNcr. In machine vision , for example, Xt represents the parameters of a timevarying shape at time t . Multi-class dynamics are represented by appending to the continuous state vector Xt, a discrete state component Yt to make a "mixed" state Xt = ( ~: ) , where Yt E Y = {I, . .. , Ny} is the discrete component of the state, drawn from a finite set of integer labels. Each discrete state represents a class of motion , for example "stroke", "rest" and "shade" for a hand engaged in drawing. Corresponding to each state Yt = Y there is a dynamical model, taken to be a Markov model of order KY that specifies Pi (Xt IXt-l, . .. Xt-KY ) . A linear-Gaussian Markov model of order K is an Auto-Regressive Process (ARP) defined by K Xt = LAkxt-k + d + BWt k=1 in which each Wt is a vector of N x independent random N(O, 1) variables and are independent for t ? t'. The dynamical parameters of the model are Wi, W t' ? deterministic parameters AI, A 2 , ... , AK ? stochastic parameters B, which are multipliers for the stochastic process Wt, and determine the "coupling" of noise Wt into the vector valued process Xt. For convenience of notation, let Each state Y E Y has a set {AY, BY, dY} of dynamical parameters, and the goal is to learn these from example trajectories. Note that the stochastic parameter BY is a first-class part of a dynamical model, representing the degree and the shape of uncertainty in motion, allowing the representation of an entire distribution of possible motions for each state y. In addition, and independently, state transitions are governed by the transition matrix for a 1st order Markov chain: P(Yt = y'IYt-1 = y) = My,y" A. Blake. B. North and M. Isard. 392 Observations Zt are assumed to be conditioned purely on the continuous part x of the mixed state, independent of Yt, and this maintains a healthy separation between the modelling of dynamics and of observations. Observations are also assumed to be independent, both mutually and with respect to the dynamical process. The observation process is defined by specifying, at each time t, the conditional density p(ZtIXt) which is taken to be Gaussian in experiments here. 3 Maximum Likelihood learning When observations are exact, maximum likelihood estimates (MLE) for dynamical parameters can be obtained from a training sequence Xi ... X T of mixed states. The well known Yule-Walker formula approximates MLE (Gelb, 1974; Ljung, 1987), but generalisations are needed to allow for short training sets (small T), to include stochastic parameters B, to allow a non-zero offset d (this proves essential in experiments later) and to encompass multiple dynamical classes. The resulting MLE learning rule is as follows. AY RY = BY0' dY = where (omitting the 1 (R Y _ AYRY) CY TY _ KY 0 , Y = TY _1 KY (iW ""0,0 _ AY('QY)T) .L"O' superscripts for clarity) C = BBT and and the first-order moments Ri and (offset-invariant) auto correlations Ri,j, for each class y, are given by Rf = where L y;=y x;_i and RL = RL - T ~ KRfRrT, Y RL = L X;_iX;_j T; Ty = Ht : Y; = y} == L 1. t:Yt=Y Yt=Y The MLE for the transition matrix M is constructed from relative frequencies as: h T y,y' = ll{t? * M Y,Y' = "" Ty,y'T , were II . Yt-l 6y'EY 4 = y, Yt* = Y'} . Y,Y Learning with stochastic observations To allow for stochastic observations, direct MLE is no longer possible, but an EM learning algorithm can be formulated. Its M-step is simply the MLE estimate of the previous section. It might be thought that the E-step should consist simply of computing expectations, for instance [[xtIZ[J, (where Zi = (Zl,"" Zt) denotes a sequence of observations) and treating them as training values x;. This would be incorrect however because the log-likelihood function I:- for the problem is not linear in the x; but quadratic. Instead, we need expectations Learning Multi-Class Dynamics 393 conditioned on the entire training set Z'[ of observations, given that ? is linear in the R i , Ri,j etc. (Shumway and Stoffer, 1982). These expected values of autocorrelations and frequencies are to be used in place of actual auto correlations and frequencies in the learning formulae of section 3. The question is, how to compute them. In the special case y = {I} of single-class dynamics, and assuming a Gaussian observation density, exact methods are available for computing expected moments, using Kalman and smoothing filters (Gelb, 1974), in an "augmented state" filter (North and Blake, 1998). For multi-class dynamics, exact computation is infeasible, but good approximations can be achieved based on propagation of sample sets, using CONDENSATION. Forward sampling with backward chaining For the purposes of learning, an extended and generalised form of the CONDENSATION algorithm is required. The generalisations allow for mixed states, arbitrary order for the ARP, and backward-chaining of samples. In backward chaining, sample-sets for successive times are built up and stored together with a complete state history back to time t = O. The extended CONDENSATION algorithm is given in figure 2. Note that the algorithm needs to be initialised. This requires that the Yo and (X~~lo' k = 0, ... ,KYO - 1) be drawn from a suitable (joint) prior for the multi-class process. One way to do this is to ensure that the training set starts in a known state and to fix the initial sample-values accordingly. Normally, the choice of prior is not too important as it is dominated by data. At time t = T, when the entire training sequence has been processed, the final sample set is (n)} ,n -- 1 , ... , N} { ( X(n) TIT'? .. , X(n?) OIT' 7rT represents fairly (in the limit, weakly, as N -+ 00) the posterior distribution for the entire state sequence X O , .?? ,XT, conditioned on the entire training set Z'[ of observations. The expectations of the autocorrelation and frequency measures required for learning can be estimated from the sample set, for example: An alternative algorithm is a sample-set version of forward-backward propagation (Kitagawa, 1996). Experiments have suggested that probability densities generated by this form of smoothing converge far more quickly with respect to sample set size N, but at the expense of computational complexity - O(N2) as opposed to O(N log N) for the algorithm above. 5 Practical applications Experiments are reported briefly here on learning the dynami(:s of juggling using the EM-Condensation algorithm, as in figure 1. An offset d Y is learned for each class in Y = {I, 2, 3}; other dynamical parameters are fixed such that that learning d Y amounts to learning mean accelerations a Y for each class. The transition matrix is also learned. From a more or ?less neutral starting point, learned structure emerges as in figure 3. Around 60 iterations of EM suffice, with N = 2048, to learn dynamics in this case. It is clear from the figure that the learned structure is an altogether plausible model for the juggling process. 394 A. Blake, B. North and M. Isard Iterate for t = 1, ... , T. (n?) (n)} . Construct t he sampIe-set {(X (n) l i t " ' " X tit ,7r t ,n = 1, ... , N for time t. For each n: 1. Choose (with replacement) mE {I, .. . , N} with prob. 7ri~{' 2. Predict by sampling from P (x t I vt-l 1"\.1 - (X(m) llt-l"'" X(m?)) t-llt-1 to choose X~~). For multi-class ARPs this is done in two steps. Discrete: Choose y~n) = y' E Y with probability My,y" where y = y~~i. Continuous: Compute K (n) _ ~AY (m) x tit - ~ kXt-klt-l k=l +d + Bw~n), where y = y~n) and w~n) is a vector of standard normal r.v. 3. Observation weights 7r~n) are computed from the observation density, evaluated for the current observations Zt: (n?) 7rt(n) = P(I Zt Xt = x tit ' then normalised multiplicatively so that En 7ri n ) = 1. 4. Update sample history: X ti(n)lt - x(m) tilt-I' I t = 1, ... , t - 1. Figure 2: The CONDENSATION algorithm for forward propagation with backward chaining. Acknowledgements We are grateful for the support of the EPSRC (AB,BN) and Magdalen College Oxford (MI). References Blake, A. and Isard, M. (1997) . The Condensation algorithm - conditional density propagation and applications to visual tracking. In Advances in Neural Information Processing Systems 9, pages 361-368. MIT Press. 395 Learning Multi-Class Dynamics (:0 0.01 a = ( 0.0 ) -9.7 0.04 Ballistic pat:::) ~ Cony a=(-;:) ~ Catchlthrow J Figure 3: Learned dynamical model for juggling. The three motion classes allowed in this experiment organise themselves into: ballistic motion (acceleration a ~ -g),- catch/throw,- carry. As expected, life-time in the ballistic state is longest, the transition probability of 0.95 corresponding to 20 time-steps or about 0.7 seconds. Transitions tend to be directed, as expected,- for example ballistic motion is more likely to be followed by a catch/throw (p = 0.04) than by a carry (p = 0.01). (Acceleration a shown here in units of m/ S2 .) Bregler, C. (1997). Learning and recognising human dynamics in video sequences. In Proc. Conf. Computer Vision and Pattern Recognition. Brockwell, P. and Davis, R. (1996). Introduction to time-series and forecasting. SpringerVerlag. Dempster, A., Laird, M., and Rubin, D. (1977) . Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. B ., 39:1-38. Gelb, A., editor (1974). Applied Optimal Estimation. MIT Press, Cambridge, MA. Gordon, N., Salmond, D., and Smith, A. (1993). Novel approach to nonlinear/nonGaussian Bayesian state estimation. lEE Proc. F, 140(2):107- 113. Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5(1) :1- 25 . Lauritzen, S. (1996). Graphical models. Oxford. Ljung, L. (1987). System identification: theory for the user. Prentice-Hall. North, B. and Blake, A. (1998). Learning dynamical models using expectationmaximisation. In Proc. 6th Int. Conf. on Computer Vision, pages 384-389. Par dey, J., Roberts, S., and Tarassenko, L. (1995). A review of parametric modelling techniques for EEG analysis. Medical Engineering Physics, 18(1):2- 1l. Rabiner, L. and Bing-Hwang, J. (1993) . Fundamentals of speech recognition. Prentice-Hall. Shumway, R. and Stoffer, D. (1982) . An approach to time series smoothing and forecasting USing the EM algorithm. J. Time Series Analysis, 3:253-226 . 

1511 |@word briefly:1 version:1 grey:3 bn:1 harder:1 carry:4 moment:2 initial:1 series:3 selecting:1 current:3 shape:2 treating:1 update:1 isard:6 accordingly:1 smith:1 short:1 bwt:1 regressive:3 successive:1 constructed:1 direct:1 incorrect:1 autocorrelation:1 expected:4 themselves:1 multi:14 ry:1 actual:1 becomes:1 notation:1 suffice:1 emerging:1 temporal:1 act:1 ti:1 uk:2 control:1 zl:1 normally:1 unit:1 medical:1 generalised:1 engineering:2 understood:1 limit:1 ak:1 oxford:4 approximately:1 might:1 specifying:1 hmms:2 directed:1 practical:2 restoring:1 maximisation:2 bootstrap:1 procedure:1 thought:2 convenience:1 prentice:2 www:1 deterministic:1 yt:9 baum:2 starting:1 independently:1 welch:2 rule:1 handle:1 user:1 exact:4 roy:1 recognition:2 particularly:1 tarassenko:1 observed:3 epsrc:1 solved:1 cy:1 dempster:2 complexity:2 dynamic:15 weakly:1 grateful:1 tit:4 predictive:1 purely:1 joint:1 represented:2 monte:1 quite:1 widely:1 plausible:2 valued:1 drawing:1 statistic:2 laird:1 superscript:1 final:1 sequence:6 kxt:1 combining:1 brockwell:2 ky:4 object:1 coupling:1 ac:1 stat:1 lauritzen:2 throw:3 soc:1 filter:5 stochastic:7 human:1 fix:1 kitagawa:3 bregler:2 around:1 hall:2 blake:8 normal:1 visually:1 predict:1 continuum:2 purpose:1 estimation:7 proc:3 label:2 ballistic:5 iw:1 healthy:1 mit:2 sensor:2 gaussian:4 varying:1 timevarying:1 yo:1 klt:1 longest:1 modelling:2 likelihood:4 entire:5 hidden:2 classification:5 smoothing:3 special:1 fairly:1 once:1 construct:1 sampling:3 represents:4 gordon:2 inherent:1 variously:1 replacement:1 bw:1 ab:1 stoffer:2 light:1 oth:1 chain:2 incomplete:1 instance:1 neutral:1 too:2 ixt:1 stored:1 reported:1 my:2 combined:1 st:1 density:5 fundamental:1 probabilistic:1 lee:1 physic:1 together:2 pardey:1 quickly:1 nongaussian:1 opposed:1 choose:3 stochastically:1 conf:2 coding:1 north:6 int:1 stream:1 performed:1 later:1 start:1 maintains:1 correspond:1 rabiner:3 bbt:1 modelled:1 bayesian:1 identification:1 accurately:1 carlo:1 trajectory:1 drive:1 history:2 stroke:1 llt:2 ty:4 frequency:4 initialised:1 mi:1 emerges:1 dimensionality:1 back:1 higher:1 response:1 done:1 ox:1 evaluated:1 generality:1 dey:1 correlation:2 hand:2 web:1 nonlinear:2 propagation:5 autocorrelations:1 hwang:3 omitting:1 multiplier:1 eg:1 ll:1 davis:2 chaining:4 ay:4 complete:1 motion:11 image:1 instantaneous:1 novel:1 rl:3 tilt:1 discussed:1 he:1 approximates:1 cambridge:1 ai:1 particle:1 robot:1 longer:2 etc:1 posterior:1 vt:1 life:1 accomplished:1 ey:1 determine:1 converge:1 signal:4 ii:1 smoother:1 multiple:2 encompass:1 mle:6 involving:1 vision:3 expectation:5 iteration:1 represent:2 achieved:2 qy:1 condensation:8 addition:1 oxl:1 walker:2 crucial:1 rest:1 tend:1 electro:1 member:1 integer:1 automated:1 iterate:1 zi:1 topology:2 oit:1 forecasting:3 juggling:5 speech:2 generally:1 clear:1 amount:1 dark:1 mid:1 cony:1 processed:1 http:1 specifies:1 estimated:1 discrete:6 drawn:2 clarity:1 ht:1 backward:5 graph:1 prob:1 uncertainty:1 place:1 separation:1 dy:2 followed:1 quadratic:1 ri:5 dominated:1 aspect:1 department:1 em:9 wi:1 restricted:1 invariant:1 taken:2 mutually:1 bing:3 turn:1 needed:1 available:2 spectral:1 appending:1 alternative:1 altogether:1 denotes:1 include:2 ensure:1 graphical:2 prof:1 added:1 question:1 parametric:1 rt:2 hmm:1 me:1 assuming:1 kalman:3 multiplicatively:1 iyt:1 robert:1 expense:1 zt:4 allowing:2 observation:16 markov:4 finite:1 gelb:5 pat:1 extended:3 variability:1 arbitrary:3 pair:1 required:2 specified:1 acoustic:1 learned:6 suggested:1 dynamical:18 perception:2 pattern:2 lpc:1 rf:1 built:1 memory:1 video:1 suitable:1 representing:2 catch:3 auto:5 breach:1 prior:2 literature:1 acknowledgement:1 review:1 relative:1 shumway:2 par:1 ljung:3 mixed:7 filtering:2 degree:1 rubin:1 editor:1 pi:1 translation:1 lo:1 elsewhere:1 infeasible:1 allow:5 normalised:1 organise:1 salmond:1 emerge:1 transition:6 forward:3 far:1 observable:2 assumed:2 xi:1 continuous:9 expectationmaximisation:1 channel:1 learn:2 eeg:2 complex:3 s2:1 noise:2 n2:1 allowed:3 augmented:1 en:1 dynami:1 ny:1 position:1 comprises:1 lie:1 governed:1 ix:1 yule:2 formula:2 shade:1 xt:10 sensing:1 offset:3 essential:2 consist:1 recognising:1 effectively:1 importance:1 conditioned:3 lt:1 simply:2 likely:1 visual:2 tracking:1 ma:1 conditional:2 goal:1 formulated:1 acceleration:3 springerverlag:1 generalisation:2 wt:3 arp:6 engaged:1 college:1 support:1

Replicator Equations, Maximal Cliques, and Graph Isomorphism Marcello Pelillo Dipartimento di Informatica Universita Ca' Foscari di Venezia Via Torino 155, 30172 Venezia Mestre, Italy E-mail: pelillo@dsi.unive.it Abstract We present a new energy-minimization framework for the graph isomorphism problem which is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid-1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. To solve the program we use "replicator" equations, a class of simple continuous- and discrete-time dynamical systems developed in various branches of theoretical biology. We show how, despite their inability to escape from local solutions, they nevertheless provide experimental results which are competitive with those obtained using more elaborate mean-field annealing heuristics. 1 INTRODUCTION The graph isomorphism problem is one of those few combinatorial optimization problems which still resist any computational complexity characterization [6]. Despite decades of active research, no polynomial-time algorithm for it has yet been found. At the same time, while clearly belonging to N P, no proof has beel1 provided that it is NP-complete. Indeed, there is strong evidence that this cannot be the case for, otherwise, the polynomial hierarchy would collapse [5]. The current belief is that the problem lies strictly between the P and NP-complete classes. Because of its theoretical as well as practical importance, the problem has attracted much attention in the neural network community, and various powerful heuristics have been developed [11, 18, 19, 20]. Following Hopfield and Tank's seminal work [10], the typical approach has been to write down a (continuous) energy function whose minimizers correspond to the (discrete) solutions being sought, and then construct a dynamical system which converges toward them. Almost invariably, all the algorithms developed so far are based on techniques borrowed from statistical mechanics, in particular mean field theory, which allow one to escape from poor Replicator Equations, Maximal Cliques, and Graph Isomorphism 551 local solutions. In this paper, we develop a new energy-minimization framework for the graph isomorphism problem which is based on the idea of reducing it to the maximum clique problem, another well-known combinatorial optimization problem. Central to our approach is a powerful result originally proved by Motzkin and Straus [13], and recently extended in various ways [3, 7, 16], which allows us to formulate the maximum clique problem in terms of an indefinite quadratic program. We then present a class of straightforward continuous- and discrete-time dynamical systems known in mathematical biology as replicator equations, and show how, thanks to their dynamical properties, they naturally suggest themselves as a useful heuristic for solving the proposed graph isomorphism program. The extensive experimental results presented show that, despite their simplicity and their inherent inability to escape from local optima, replicator dynamics are nevertheless competitive with more sophisticated deterministic annealing algorithms. The proposed formulation seems therefore a promising framework within which powerful continuous-based graph matching heuristics can be developed, and is in fact being employed for solving practical computer vision problems [17J. More details on the work presented here can be found in [15J. 2 A QUADRATIC PROGRAM FOR GRAPH ISOMORPHISM 2.1 GRAPH ISOMORPHISM AS CLIQUE SEARCH Let G = (V, E) be an undirected graph, where V is the set of vertices and E ~ V x V is the set of edges. The order of G is the number of its vertices, and its size is the number of edges. Two vertices i,j E V are said to be adjacent if (i,j) E E. The adjacency matrix of G is the n x n symmetric matrix A = (aij) defined as follows: aij = 1 if (i,j) E E, aij = a otherwise. Given two graphs G' = (V', E') and Gil = (V", E") having the same order and size, an isomorphism between them is any bijection ? : V' -t V" such that (i,j) E E' {:} (?(i),?(j)) E E", for all i,j E V'. Two graphs are said to be isomorphic if there exists an isomorphism between them. The graph isomorphism problem is therefore to decide whether two graphs are isomorphic and, in the affirmative, to find an isomorphism. Barrow and Burstall [IJ introduced the notion of an association graph as a useful auxiliary graph structure for solving general graphjsubgraph isomorphism problems. The association graph derived from G' and Gil is the undirected graph G = (V, E), where V = V' X V" and E = {((i, h), (j, k)) E V x V : i:f= j, h:f= k, and (i,j) E E' {:} (h, k) E E"} . Given an arbitrary undirected graph G = (V, E), a subset of vertices C is called a clique if all its vertices are mutually adjacent , i.e. , for all i,j E C we have (i,j) E E. A clique is said to be maximal if it is not contained in any larger clique, and maximum if it is the largest clique in the graph. The clique number, denoted by w(G), is defined as the cardinality of the maximum clique. The following result establishes an equivalence between the graph isomorphism problem and the maximum clique problem (see [15J for proof). Theorem 2.1 Let G' and Gil be two graphs of order n , and let G be the corresponding association graph . Then, G' and Gil are isomorphic if and only if w(G) = n. In this case, any maximum clique of G induces an isomorphism between G' and Gil , and vice versa. 552 2.2 M. Pelillo CONTINUOUS FORMULATION OF MAX-CLIQUE Let G = (V, E) be an arbitrary undirected graph of order n, and let Sn denote the standard simplex of lRn : Sn={xElR n : Xi~O foralli=l. .. n, and tXi=I}. z== 1 Given a subset of vertices C of G, we will denote by XC its characteristic vector which is the point in Sn defined as xI = 1/ICI if i E C, xi = 0 otherwise, where ICI denotes the cardinality of C. Now, consider the following quadratic function: f(x) = x T Ax (1) where "T" denotes transposition. The Motzkin-Straus theorem [13] establishes a remarkable connection between global (local) maximizers of fin Sn and maximum (maximal) cliques of G. Specifically, it states that a subset of vertices C of a graph G is a maximum clique if and only if its characteristic vector XC is a global maximizer of the function f in Sn. A similiar relationship holds between (strict) local maximizers and maximal cliques [7, 16]. One drawback associated with the original Motzkin-Straus formulation relates to the existence of spurious solutions, i.e., maximizers of f which are not in the form of characteristic vectors [16]. In principle, spurious solutions represent a problem since, while providing information about the order of the maximum clique, do not allow us to extract the vertices comprising the clique. Fortunately, there is straightforward solution to this problem which has recently been introduced and studied by Bomze [3]. Consider the following regularized version of function f: j (x) = x T Ax + ~ X T X (2) . The following is the spurious-free counterpart of the original Motzkin-Straus theorem (see [3] for proof). Theorem 2.2 Let C be a subset of vertices of a graph G, and let XC be its charac- teristic vector. Then the following statements hold: (a) C is a maximum clique of G if and only if XC is a global maximizer of the simplex Sn. Its order is then given by ICI = 1/2(1 - f(x C ) ) . (b) C is a maximal clique of G if and only if XC is a local maximizer of (c) All local (and hence global) maximizers of j j j over in Sn. over Sn are strict. Unlike the Motzkin-Straus formulation, the previous result guarantees that all maximizers of j on Sn are strict, and are characteristic vectors of maximal/maximum cliques in the graph. In an exact sense, therefore, a one-to-one correspondence exists between maximal cliques and local maximizers of j in Sn on the one hand, and maximum cliques and global maximizers on the other hand. 2.3 A QUADRATIC PROGRAM FOR GRAPH ISOMORPHISM Let G' and Gil be two arbitrary graphs of order n, and let A denote the adjacency matrix of the corresponding association graph, whose order is assumed to be N. The graph isomorphism problem is equivalent to the following program: maXImIze subject to j(x) = x T (A x E SN + ~ IN)X (3) 553 Replicator Equations. Maximal Cliques. and Graph Isomorphism More precisely, the following result holds, which is a straightforward consequence of Theorems 2.1 and 2.2. Theorem 2.3 Let G' and Gil be two graphs of order n, and let x* be a global solution of program (3), where A is the adjacency matrix of the association graph of G' and Gil . Then, G' and Gil are isomorphic if and only if j(x*) = 1 - 1/2n. In this case, any global solution to (3) induces an isomorphism between G' and Gil, and vice versa. In [15] we discuss the analogies between our objective function and those proposed in the literature (e.g., [18, 19]). 3 REPLICATOR EQUATIONS AND GRAPH ISOMORPHISM Let W be a non-negative n x n matrix, and consider the following dynamical system: ~Xi(t) = Xi(t) ("i(t) - t.X;(t)";(t)) where 7ri(t) , i = 1. . . n (4) = 2:.7=1 WijXj(t), i = 1 . . . n , and its discrete-time counterpart: Xi(t)7ri (t) xi(t+1)=2:. nj = l x] () t 7r]. (t ) ' i = l .. . n. (5) It is readily seen that the simplex Sn is invariant under these dynamics, which means that every trajectory starting in Sn will remain in Sn for all future times. Both (4) and (5) are called replicator equations in theoretical biology, since they are used to model evolution over time of relative frequencies of interacting, selfreplicating entities [9]. The discrete-time dynamical equations turn also out to be a special case of a general class of dynamical systems introduced by Baum and Eagon [2] in the context of Markov chain theory. Theorem 3.1 If W is symmetric, then the quadratic polynomial F(x) = xTWx is strictly increasing along any non-constant trajectory of both continuous-time (4) and discrete-time (5) replicator equations. Furthermore, any such trajectory converges to a (unique) stationary point. Finally, a vector x E Sn is asymptotically stable under (4) and (5) if and only if x is a strict local maximizer of F on Sn. The previous result is known in mathematical biology as the Fundamental Theorem of Natural Selection [9, 21]. As far as the discrete-time model is concerned, it can be regarded as a straightforward implication of the more general Baum-Eagon theorem [2]. The fact that all trajectories of the replicator dynamics converge to a stationary point is proven in [12]. Recently, there has been much interest in evolutionary game theory around the following exponential version of replicator equations , which arises as a model of evolution guided by imitation [8, 21]: :t Xi (t) = Xi(t) (L:7~1 ~:;;;~ .. '(t) - 1), i = l... n (6) where K, is a positive constant. As K, tends to 0, the orbits of this dynamics approach those of the standard, first-order replicator model (4), slowed down by the factor 554 M Pelillo K. Hofbauer [8] has recently proven that when the matrix W is symmetric, the quadratic polynomial F defined in Theorem 3.1 is also strictly increasing, as in the first-order case. After discussing various properties of this, and more general dynamics, he concluded that the model behaves essentially in the same way as the standard replicator equations, the only difference being the size of the basins of attraction around stable equilibria. A customary way of discretizating equation (6) is given by the following difference equations: Xi(t + 1) = xi(t)e"1l';(t) L: n. )=1 ( ) X)? t e"1l'J (t)' i = l. .. n (7) which enjoys many of the properties of the first-order system (5), e.g., they have the same set of equilibria. The properties discussed above naturally suggest using replicator equations as a useful heuristic for the graph isomorphism problem. Let G' and G" be two graphs of order n, and let A denote the adjacency matrix of the corresponding N-vertex association graph G. By letting 1 W = A + "2IN we know that the replicator dynamical systems, starting from an arbitrary initial state, will iteratively maximize the function j(x) = xT(A + !IN)x in SN, and will eventually converge to a strict local maximizer which, by virtue of Theorem 2.2 will then correspond to the characteristic vector of a maximal clique in the association graph. This will in turn induce an isomorphism between two subgraphs of G' and G" which is "maximal," in the sense that there is no other isomorphism between subgraphs of G' and G" which includes the one found. Clearly, in theory there is no guarantee that the converged solution will be a global maximizer of j, and therefore that it will induce an isomorphism between the two original graphs . Previous work done on the maximum clique problem [4, 14], and also the results presented in this paper, however, suggest that the basins of attraction of global maximizers are quite large, and very frequently the algorithm converges to one of them. 4 EXPERIMENTAL RESULTS In the experiments reported here, the discrete-time replicator equation (5) and its exponential counterpart (7) with K = 10 were used. The algorithms were started from the barycenter of the simplex and they were stopped when either a maximal clique was found or the distance between two successive points was smaller than a fixed threshold, which was set to 10- 17 . In the latter case the converged vector was randomly perturbed, and the algorithm restarted from the perturbed point . Because of the one-to-one correspondence between local maximizers and maximal cliques, this situation corresponds to convergence to a saddle point. All the experiments were run on a Sparc20. Undirected 100-vertex random graphs were generated with expected connectivities ranging from 1% to 99%. For each connectivity value, 10'0 graphs were produced and each of them had its vertices randomly permuted so as to obtain a pair of isomorphic graphs. Overall, therefore, 1500 pairs of isomorphic graphs were used. Each pair was given as input to the replicator models and, after convergence, a success was recorded when the cardinality of the returned clique was equal to the order of the graphs given as input (Le., 100) .1 Because of the stopping criterion employed, this 1 Due to the high computational time required, in the 1% and 99% cases the first-order replicator algorithm (5) was tested only on 10 pairs, instead of 100. Replicator Equations, Maximal Cliques, and Graph Isomorphism f .i '00 ~, / - - --- . ~--.-- -- . . - I ---.--.. -. .-. -. . . . ? \ - - - \ \ 75 1 t .i c 50 \ I U ! 555 j/ I 25 I ' I i iii 001 003 0 05 0' 0.2 0.3 0 " as 06 aa 0.7 09 0 95 0 97 099 001 003 0 .05 01 0 .2 Expecled connectivity 03 0 " as 06 07 08 0.9 095 a 97 0 99 Expected connecllvlty Figure 1: Percentage of correct isomorphisms obtained using the first-order (left) and the exponential (right) replicator equations, as a function of the expected connectivity. 100000 -- - - (?I ? - - .~~om<l~~ ;;- 10000 (?21~IIIK) !!!. ! - ~1U7 (1) 1000 1?I""'17) (:t294KfI) (:t201 0) ! I & ~ '-' (N ~ Ill) '00 \~t!%) '0 ' ,- ... 02 03 t ..e (:t226) ----.----(1<)MI (tI07) 001 0030 05 0.1 . I '1<)58) a 5 ,00 '0 (:to 94) (i069) 0" 1000 06 07 Expected connectivity 08 09 095 0 , 9 7 a 99 001 003 0 .05 a 1 0 2 03 0 " 0.5 06 0.7 08 0.9 095 097 099 Expected connectivity Figure 2: Average computational time taken by the first-order (left) and the exponential (right) replicator equations, as a function of the expected connectivity. The vertical axes are in logarithmic scale, and the numbers in parentheses represent the standard deviation. guarantees that a maximum clique, and therefore a correct isomorphism, was found. The proportion of successes as a function of the expected connectivities for both replicator models is plotted in Fig. 1, whereas Fig. 2 shows the average CPU time taken by the two algorithms to converge (in logarithmic scale). Notice how the exponential replicator system (7) is dramatically faster and also performs bet.ter than the first-order model (5). These results are significantly superior to those reported by Simic [20] who obtained poor results at connectivities less than 40% even on smaller graphs (Le. , up to 75 vertices). They also compare favorably with the results obtained more recently by Rangarajan et ai. [18] on 100-vertex random graphs for connectivities up to 50%. Specifically, at 1% and 3% connectivities they report a percentage of correct isomorphisms of about 30% and 0%, respectively. Using our approach we obtained, on the same kind of graphs, a percentage of success of 80% and 11%, respectively. Rangarajan and Mjolsness [19] also ran experiments on 100-vertex random graphs with various connectivities, using a powerful Lagrangian relaxation network. Except for a few instances , they always obtained a correct solution. The computational time required by their model, however, turns out to largely exceed ours. As an example, the average time taken by their algorithm to match two 100-vertex 50%connectivity graphs was about 30 minutes on an SGI workstation. As shown in Fig. 2, we obtained identical results in about 3 seconds. It should be emphasized that all the algorithms mentioned above do incorporate sophisticated annealing mechanisms to escape from poor local minima. By contrast, in the presented work no attempt was made to prevent the algorithms from converging to such solutions. 556 M Pelillo Acknowledgments. This work has been done while the author was visiting the Department of Computer Science at the Yale University. Funding for this research has been provided by the Consiglio Nazionale delle Ricerche , Italy. The author would like to thank 1. M. Bomze, A. Rangarajan, K. Siddiqi , and S. W. Zucker for many stimulating discussions. References [1) H. G . Barrow and R. M. Burstall, "Subgraph isomorphism, matching relational structures and maximal cliques," Inform. Process. Lett., vol. 4, no. 4, pp. 83- 84, 1976. [2) L. E. Baum and J. A. Eagon, "An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology," Bull. Amer. Math. Soc., vol. 73, pp. 360- 363, 1967. [3) 1. M. Bomze, "Evolution towards the maximum clique," J. Global Optim., vol. 10, pp. 143- 164, 1997. [4) I. M. Bomze, M. Pelillo, and R. Giacomini, "Evolutionary approach to the maximum clique problem: Empirical evidence on a larger scale," in Developments in Global Optimization, I. M. Bomze et al., eds., Kluwer , The Netherlands, 1997, pp. 95- 108. [5) R. B. Boppana, J. Hastad, and S. Zachos, "Does co-NP have short interactive proofs?" Inform . Process. Lett., vol. 25, pp. 127-132, 1987. [6) M. R. Garey and D . S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco , CA , 1979. [7) L. E. Gibbons, D. W. Hearn, P. M. Pardalos, and M. V. Ramana, "Continuous characterizations of the maximum clique problem ," Math . Oper. Res., vol. 22 , no. 3, pp. 754- 768, 1997. [8) J. Hofbauer, "Imitation dynamics for games," Collegium Budapest, preprint, 1995. [9) J . Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, UK, 1988. [10) J . J . Hopfield and D. W. Tank, "Neural computation of decisions in optimization problems," Biol. Cybern ., vol. 52, pp. 141- 152, 1985. [11) R. Kree and A. Zippelius, "Recognition of topological features of graphs and images in neural networks ," J. Phys. A : Math . Gen., vol. 21 , pp . L813- L818 , 1988. [12) V. Losert and E . Akin, "Dynamics of games and genes: Discrete versuS continuous time," J . Math. Biol., vol. 17, pp . 241- 251 , 1983. [13) T . S. Motzkin and E . G. Straus, "Maxima for graphs and a new proof of a theorem of'I\min," Canad. J. Math., vol. 17, pp . 533- 540, 1965 . [14) M. Pelillo, "Relaxation labeling networks for the maximum clique problem," J. Artij. Neural Networks, vol. 2, no. 4, pp. 313- 328, 1995. [15) M. Pelillo, "Replicator equations, maximal cliques, and graph isomorphism," Neural Computation, to appear. [16) M. Pelillo and A. Jagota, "Feasible and infeasible maxima in a quadratic program for maximum clique," J. Artij. Neural N etworks, vol. 2, no. 4, pp . 411- 420, 1995. [17) M. Pelillo, K. Siddiqi , and S. W Zucker, "Matching hierarchical structures using association graphs," in Computer Vision - ECCV '98, Vol. II, H. Burkhardt and B. Neumann, eds. , Springer-Verlag, Berlin, 1998, pp. 3- 16. [18) A. Rangarajan , S. Gold, and E . Mjolsness, "A novel optimizing network architecture with applications," Neural Computation, vol. 8, pp. 1041- 1060, 1996. [19) A. Rangarajan and E. Mjolsness, "A Lagrangian relaxation network for graph matching," IEEE Trans. Neural Networks, vol. 7, no. 6, pp. 1365- 1381, 1996. [20) P. D . Simir, "Constrained nets for graph matching and other quadratic assignment problems," Neural Computation, vol. 3, pp. 268-281, 1991. [21) J. W . Weibull , Evolutionary Game Th eory. MIT Press, Cambridge, MA, 1995. 

1512 |@word version:2 polynomial:4 seems:1 proportion:1 initial:1 ours:1 hearn:1 current:1 optim:1 yet:1 attracted:1 readily:1 stationary:2 short:1 transposition:1 completeness:1 characterization:2 math:5 bijection:1 successive:1 mathematical:2 along:1 expected:7 indeed:1 themselves:1 frequently:1 mechanic:1 freeman:1 cpu:1 cardinality:3 increasing:2 provided:2 kind:1 affirmative:1 weibull:1 developed:4 nj:1 guarantee:3 zippelius:1 bomze:5 every:1 interactive:1 uk:1 appear:1 unive:1 positive:1 local:12 tends:1 consequence:1 despite:3 studied:1 equivalence:1 co:1 collapse:1 practical:2 unique:1 acknowledgment:1 foralli:1 empirical:1 significantly:1 matching:5 induce:2 suggest:3 cannot:1 selection:1 context:1 cybern:1 seminal:1 equivalent:2 deterministic:1 lagrangian:2 baum:3 straightforward:4 attention:1 starting:2 formulate:2 simplicity:1 subgraphs:2 attraction:2 regarded:1 notion:1 hierarchy:1 exact:1 recognition:1 preprint:1 mjolsness:3 ran:1 mentioned:1 nazionale:1 complexity:1 gibbon:1 dynamic:7 solving:3 hopfield:2 various:6 venezia:2 labeling:1 whose:2 heuristic:5 larger:2 solve:1 quite:1 otherwise:3 net:1 maximal:16 budapest:1 gen:1 subgraph:1 gold:1 sgi:1 convergence:2 optimum:1 rangarajan:5 neumann:1 converges:3 develop:1 ij:1 pelillo:10 borrowed:1 strong:1 soc:1 auxiliary:1 guided:1 drawback:1 correct:4 centered:1 pardalos:1 adjacency:4 dipartimento:1 strictly:3 hold:3 around:3 equilibrium:2 sought:1 estimation:1 combinatorial:2 largest:1 vice:2 establishes:2 minimization:2 mit:1 clearly:2 always:1 bet:1 derived:1 ax:3 contrast:1 sense:2 minimizers:1 stopping:1 spurious:3 comprising:1 tank:2 overall:1 ill:1 denoted:1 development:1 constrained:1 special:1 field:2 construct:1 equal:1 having:1 biology:4 identical:1 marcello:1 future:1 simplex:4 np:4 report:1 escape:4 few:2 inherent:1 randomly:2 attempt:1 ecology:1 invariably:1 interest:1 lrn:1 chain:1 implication:1 edge:2 orbit:1 plotted:1 re:1 theoretical:3 stopped:1 instance:1 hastad:1 delle:1 assignment:1 bull:1 vertex:16 subset:4 deviation:1 johnson:1 reported:2 perturbed:2 thanks:1 fundamental:2 probabilistic:1 mestre:1 connectivity:13 central:1 recorded:1 oper:1 includes:1 sparc20:1 competitive:2 om:1 characteristic:5 who:1 largely:1 correspond:2 produced:1 trajectory:4 converged:2 inform:2 phys:1 ed:2 energy:3 frequency:1 pp:16 straus:7 garey:1 naturally:2 proof:5 di:2 associated:1 mi:1 workstation:1 proved:2 sophisticated:2 originally:1 formulation:5 done:2 amer:1 furthermore:1 hand:2 maximizer:6 counterpart:3 evolution:4 hence:1 symmetric:3 iteratively:1 adjacent:2 game:4 simic:1 criterion:1 complete:2 txi:1 performs:1 ranging:1 image:1 sigmund:1 recently:6 funding:1 novel:1 superior:1 replicator:24 burstall:2 behaves:1 permuted:1 association:8 he:1 discussed:1 kluwer:1 versa:2 cambridge:3 ai:1 u7:1 had:1 stable:2 zucker:2 italy:2 optimizing:1 verlag:1 inequality:1 success:3 discussing:1 jagota:1 seen:1 minimum:1 fortunately:1 wijxj:1 employed:2 ricerche:1 converge:3 maximize:2 ii:1 branch:1 relates:1 match:1 faster:1 parenthesis:1 converging:1 vision:2 essentially:1 represent:2 whereas:1 annealing:3 concluded:1 unlike:1 strict:5 subject:1 undirected:5 ter:1 exceed:1 iii:1 concerned:1 architecture:1 idea:1 whether:1 isomorphism:30 akin:1 returned:1 dramatically:1 useful:3 burkhardt:1 netherlands:1 mid:1 induces:2 siddiqi:2 informatica:1 eory:1 percentage:3 notice:1 gil:10 discrete:9 write:1 vol:15 indefinite:1 nevertheless:2 threshold:1 prevent:1 graph:57 asymptotically:1 relaxation:3 run:1 powerful:4 almost:1 decide:1 decision:1 collegium:1 correspondence:2 yale:1 quadratic:9 topological:1 foscari:1 precisely:1 ri:2 min:1 expanded:1 department:1 poor:3 belonging:1 remain:1 smaller:2 slowed:1 invariant:1 taken:3 equation:19 mutually:1 etworks:1 discus:1 turn:3 eventually:1 mechanism:1 know:1 letting:1 ramana:1 hierarchical:1 eagon:3 customary:1 existence:1 original:3 denotes:2 xc:5 hofbauer:3 universita:1 objective:1 torino:1 barycenter:1 canad:1 said:3 evolutionary:3 visiting:1 distance:1 thank:1 berlin:1 entity:1 mail:1 toward:1 relationship:1 providing:1 charac:1 statement:1 favorably:1 negative:1 vertical:1 markov:2 fin:1 similiar:1 barrow:2 situation:1 extended:1 relational:1 interacting:1 arbitrary:4 community:1 introduced:3 pair:4 required:2 extensive:1 connection:1 resist:1 trans:1 dynamical:9 program:9 max:1 belief:1 natural:1 regularized:1 started:1 extract:1 sn:17 literature:1 relative:1 dsi:1 analogy:1 proven:2 remarkable:1 versus:1 basin:2 principle:1 intractability:1 eccv:1 free:1 infeasible:1 enjoys:1 aij:3 guide:1 allow:2 lett:2 author:2 made:1 ici:3 san:1 far:2 gene:1 clique:41 global:11 active:1 assumed:1 francisco:1 xi:11 imitation:2 continuous:8 search:1 decade:1 promising:1 ca:2 fig:3 elaborate:1 exponential:5 lie:1 down:2 theorem:12 minute:1 xt:1 emphasized:1 virtue:1 evidence:2 maximizers:9 exists:2 importance:1 logarithmic:2 saddle:1 motzkin:7 contained:1 restarted:1 springer:1 aa:1 corresponds:1 ma:1 stimulating:1 boppana:1 towards:1 feasible:1 typical:1 specifically:2 reducing:1 except:1 called:2 isomorphic:6 experimental:3 latter:1 inability:2 arises:1 incorporate:1 tested:1 biol:2

Convergence Rates of Algorithms for Visual Search: Detecting Visual Contours A.L. Yuille Smith-Kettlewell Inst . San Francisco, CA 94115 James M. Coughlan Smith-Kettlewell Inst. San Francisco, CA 94115 Abstract This paper formulates the problem of visual search as Bayesian inference and defines a Bayesian ensemble of problem instances . In particular, we address the problem of the detection of visual contours in noise/clutter by optimizing a global criterion which combines local intensity and geometry information. We analyze the convergence rates of A * search algorithms using results from information theory to bound the probability of rare events within the Bayesian ensemble. This analysis determines characteristics of the domain , which we call order parameters, that determine the convergence rates. In particular, we present a specific admissible A * algorithm with pruning which converges, with high probability, with expected time O(N) in the size of the problem. In addition, we briefly summarize extensions of this work which address fundamental limits of target contour detectability (Le. algorithm independent results) and the use of non-admissible heuristics. 1 Introduction Many problems in vision, such as the detection of edges and object boundaries in noise/clutter, see figure (1), require the use of search algorithms . Though many algorithms have been proposed, see Yuille and Coughlan (1997) for a review, none of them are clearly optimal and it is difficult to judge their relative effectiveness. One approach has been to compare the results of algorithms on a representative dataset of images. This is clearly highly desirable though determining a representative dataset is often rather subjective. In this paper we are specifically interested in the convergence rates of A * algorithms (Pearl 1984). It can be shown (Yuille and Coughlan 1997) that many algorithms proposed to detect visual contours are special cases of A * . We would like to understand what characteristics of the problem domain determine the convergence 642 A. L. Yuille and J. M Coughlan Figure 1: The difficulty of detecting the target path in clutter depends, by our theory (Yuille and Coughlan 1998), on the order parameter K. The larger K the less computation required. Left, an easy detection task with K = 3.1. Middle, a hard detection task K = 1.6. Right, an impossible task with K = -0.7. rates. We formulate the problem of detecting object curves in images to be one of statistical estimation. This assumes statistical knowledge of the images and the curves, see section (2). Such statistical knowledge has often been used in computer vision for determining optimization criteria to be minimized. We want to go one step further and use this statistical knowledge to determine good search strategies by defining a Bayesian ensemble of problem instances. For this ensemble, we can prove certain curve and boundary detection algorithms, with high probability, achieve expected time convergence in time linear with the size of the problem. Our analysis helps determine important characteristics of the problem, which we call order parameters, which quantify the difficulty of the problem. The next section (2) of this paper describes the basic statistical assumptions we make about the domain and describes the mathematical tools used in the remaining sections. In section (3) we specify our search algorithm and establish converEence rates. We conclude by placing this work in a larger context and summarizing recent extensions. 2 Statistical Background Our approach assumes that both the intensity properties and the geometrical shapes of the target path (i.e. the edge contour) can be determined statistically. This path can be considered .to be a set of elementary path segments joined together. We first consider the intensity properties along the edge and then the geometric properties. The set of all possible paths can be represented by a tree structure, see figure (2). The image properties at segments lying on the path are assumed to differ, in a statistical sense, from those off the path. More precisely, we can design a filter ?(.) with output {Yx = ?(I(x))} for a segment at point x so that: P(Yx) = Pon(Yx), if "XII lies on the true path P(Yx) = Poff(Yx), if "X'I lies off the true path. (1) For example, we can think of the {Yx} as being values of the edge strength at point x and Pon, Poll being the probability distributions of the response of ?(.) on and off an edge. The set of possible values of the random variable Yx is the alphabet with alphabet size M (Le. Yx can take any of M possible values). See (Geman and Jedynak 1996) for examples of distributions for Pon, Pol I used in computer vision applications. We now consider the geometry of the target contour. We require the path to be made up of connected segments Xl, X2, ... , x N. There will be a Markov probability distribution Pg(Xi+I!Xi) which specifies prior probabilistic knowledge of the target. Convergence Rates ofAlgorithmsfor Visual Search: Detecting Visual Contours 643 It is convenient, in terms of the graph search algorithms we will use, to consider that each point x has a set of Q neighbours. Following terminology from graph theory, we refer to Q as the branching factor. We will assume that the distribution P g depends only on the relative positions of XHI and Xi. In other words, Pg(XHllxi) = PLlg(XHl - Xi). An important special case is when the probability distribution is uniform for all branches (Le. PLlg(Ax) = U(Ax) = I/Q, VAx). The joint distribution P(X, Y) of the road geometry X and filter responses Y determines the Bayesian Ensemble. By standard Bayesian analysis, the optimal path X* = {xi, ... , XN} maximizes the sum of the log posterior: (2) where the sum i is taken over all points on the target. U(Xi+l - Xi) is the uniform distribution and its presence merely changes the log posterior E(X) by a constant value. It is included to make the form of the intensity and geometric terms similar, which simplifies our later analysis. We will refer to E(X) as the reward of the path X which is the sum of the intensity rewards log Pon (Y(~jl) and the geometric rewards log PL:>.g (Xi+l -Xi) Poll (Y(~i? U(Xi+l -Xi) It is important to emphasize that our results can be extended to higher-order Markov chain models (provided they are shift-invariant). We can, for example, define the x variable to represent spatial orientation and position of a small edge segment. This will allow our theory to apply to models, such as snakes, used in recent successful vision applications (Geman and Jedynak 1996). (It is straightforward to transform the standard energy function formulation of snakes into a Markov chain by discretizing and replacing the derivatives by differences. The smoothness constraints, such as membranes and thin plate terms, will transform into first and second order Markov chain connections respectively). Recent work by Zhu (1998) shows that Markov chain models of this type can be learnt using Minimax Entropy Learning theory from a representative set of examples. Indeed Zhu goes further by demonstrating that other Gestalt grouping laws can be expressed in this framework and learnt from representative data. Most Bayesian vision theories have stopped at this point. The statistics of the problem domain are used only to determine the optimization criterion to be minimized and are not exploited to analyze the complexity of algorithms for performing the optimization. In this paper, we go a stage further. We use the statistics ofthe problem domain to define a Bayesian ensemble and hence to determine the effectiveness of algorithms for optimizing criteria such as (2). To do this requires the use of Sanov's theorem for calculating the probability of rare events (Cover and Thomas 1991). For the road tracking problem this can be re-expressed as the following theorem, derived in (Yuille and Coughlan 1998): Theorem 1. The probabilities that the spatially averaged log-likelihoods on, and off, the true curve are above, or below, threshold T are bounded above as follows: Pr{.!. n t i=l < T} :s; (n + I)M2-nD(PTlfPon) (3) > T}:S; (n+ I)M2-nD(PTIIPOI/) , (4) {log Pon(y(Xi?) }on Poff (Y(Xi?) Pr{.!. t{lOg Pon(Y(Xi?) }off n i=l POff(Y(Xi?) A. L. Yuille and J. M. Coughlan 644 where PT(y) on the mined the subscripts on and off mean that the data is generated by Pon, Po", = p;;;>'(T) (y)P;;p jZ(T) where a ::; "\(T) ::; 1 is a scalar which depends threshold T and Z(T) is a normalization factor. The value of "\(T) is deterby the constraint 2: y PT (y) log ;'?In}(~) = T. In the next section, we will use Theorem 1 to determine a criterion for pruning the search based on comparing the intensity reward to a threshold T (pruning will also be done using the geometric reward). The choice of T involves a trade-off. If T is large (Le. close to D(PonllPoff)) then we will rapidly reject false paths but we might also prune out the target (true) path. Conversely, if T is small (close to -D(PoffllPon)) then it is unlikely we will prune out the target path but we may waste a lot of time exploring false paths. In this paper we choose T large and write the fall-off factors (Le. the exponents in the bounds of equations (3,4)) as D(PTllPon) = tl (T), D(PTilPoff) = D(PonilPoff) - t2(T) where tl (T), t2(T) are positive and (tl(T),t2(T)) t-+ (0,0) as T t-+ D(PonilPoff ). We perform a similar analysis for the geometric rewards by substituting P6.g , U for Pon , Pol I' We choose a threshold T satisfying -D(UIIP6.g) < T < D(P6.gllU). The results of Theorem 1 apply with the obvious substitutions. In particular, the alphabet factor becomes Q (the branching factor). Once again, in this paper, we choose T to be large and obtain fall-off factors D(Pt'IIP6.g) = El (T), D(Pt'IIU) = D(P6.gllU) - E2(T). 3 Tree Search: A *, heuristics, and block pruning We now consider a specific example, motivated by Geman and Jedynak (1996), of searching for a path through a search tree. In Geman and Jedynak the path corresponds to a road in an aerial image and they assume that they are given an initial point and direction on the target path. They have a branching factor Q = 3 and, in their first version, the prior probability of branching is considered to be the uniform distribution (later they consider more sophisticated priors). They assume that no path segments overlap which means that the search space is a tree of size QN where N is the size of the problem (Le. the longest length). The size of the problem requires an algorithm that converges in O(N) time and they demonstrate an algorithm which empirically performs at this speed. But no proof of convergence rates are given in their paper. It can be shown, see (Yuille and Coughlan 1997), that the Geman and Jedynak algorithm is a close approximation to A * which uses pruning. (Observe that Geman and Jedynak's tree representation is a simplifying assumption of the Bayesian model which assumes that once a path diverges from the true path it can never recover, although we stress that the algorithm is able to recover from false starts - for more details see Coughlan and Yuille 1998). We consider an algorithm which uses an admissible A * heuristic and a pruning mechanism. The idea is to examine the paths chosen by the A * heuristic. As the length of the candidate path reaches an integer multiple of No we prune it based on its intensity reward and its geometric reward evaluated on the previous No segments, which we call a segment block. The reasoning is that few false paths will survive this pruning for long but the target path will survive with high probability. We prune on the intensity by eliminating all paths whose intensity reward, averaged over the last No segments, is below a threshold T (recall that -D(PoffllPon) < T < D(PonllPoff) and we will usually select T to take values close to D(PonllPoff)). In addition, we prune on the geometry by eliminating all paths whose geometric rewards, averaged over the last No segments, are below T (where -D(UIIP6.g) < T < D(P6.gllU) with T typically being close to D(P6.gllU)). More precisely, we 645 Convergence Rates ofAlgOrithms for Visual Search : Detecting Visual Contours discard a path provided (for any integer z 1 (z~o I Pon(Yi) ~ og No i=zNo+l Poff(yd < T, ~ 0): 1 (z+l)No or No L log i=zNo+l PLlg(Llxi) U(Llx.) < T. (5) t There are two important issues to address: (i) With what probability will the algorithm converge?, (ii) How long will we expect it take to converge? The next two subsections put bounds on these issues. 3.1 Probability of Convergence Because of the pruning, there is a chance that there will be no paths which survive pruning. To put a bound on this we calculate the probability that the target (true) path survives the pruning. This gives a lower bound on the probability of convergence (because there could be false paths which survive even if the target path is mistakenly pruned out). The pruning rules removes path segments for which the intensity reward r I or the geometric reward r 9 fails the pruning test. The probability of failure by removing a block segment of the true path, with rewards r~, r~, is Pr(r~ < T or r~ < T) ::; Pr(r~ < T) + Pr(r~ < T) ::; (No + 1)M2- NoE1 (T) + (No + 1)Q2-NoilCT), where we have used Theorem 1 to put bounds on the probabilities. The probability of pruning out any No segments of the true path can therefore be made arbitrarily small by choosing No, T, T so as to make Notl and NOtl large. It should be emphasized that the algorithm will not necessarily converge to the exact target path. The admissible nature of the heuristic means that the algorithm will converge to the path with highest reward which has survived the pruning. It is highly probable that this path is close to the target path. Our recent results (Coughlan and Yuille 1998, Yuille and Coughlan 1998) enable us to quantify this claim. 3.2 Bounding the Number of False Paths Suppose we face a Q-nary tree. We can order the false paths by the stage at which they diverge from the target (true) path, see figure (2). For example, at the first branch point the target path lies on only one of the Q branches and there are Q - 1 false branches which generate the first set of false paths Fl' Now consider all the Q -1 false branches at the second target branch, these generate set F 2 . As we follow along the true path we keep generating these false sets F i . The set of all paths is therefore the target path plus the union of the Fi (i = 1, ... , N). To determine convergence rates we must bound the amount of time we spend searching the Fi. If the expected time to search each Fi is constant then searching for the target path will at most take constant? N steps. Consider the set Fi of false paths which leave the true path at stage i. We will apply our analysis to block segments of Fi which are completely off the true path. If (i -1) is an integer multiple of No then all block segments of Fi will satisfy this condition. Otherwise, we will start our analysis at the next block and make the worse case assumption that all path segments up till this next block will be searched. Since the distance to the next block is at most No - 1, this gives a maximum number of QNo-l starting blocks for any branch of F i . Each Fi also has Q - 1 branches and so this gives a generous upper bound of (Q - l)Q N o-l starting blocks for each Fi . A. L. Yuille and J. M. Coughlan 646 Figure 2: The target path is shown as the heavy line. The false path sets are labelled as Fl ,F2 , etc. with the numbering depending on how soon they leave the target path. The branching factor Q = 3. For each starting block, we wish to compute (or bound) the expected number of blocks that are explored thereafter. This requires computing the fertility of a block, the average number of paths in the block that survive pruning. Provided the fertility is smaller than one, we can then apply results from the theory of branching processes to determine the expected number of blocks searched in F i . The fertility q is the number of paths that survive the geometric pruning times the probability that each survives the intensity pruning. This can be bounded (using Theorem 1) by q q where: :s q= QN0(No + I)Q2-No{D(hgIIU)-?2(T)}(No + I)M2-No{D(PonIIPoff)-E2(T)} = (No + I)Q+M 2- No {D(Pon IIPof! )-H(Pa g)-E2(T)-?2(T)}, (6) where we used the fact that D(PLlgIIU) = 10gQ - H(PLlg). Observe that the condition q < 1 can be satisfied provided D(PonllPolf )-H(PLlg) > O. This condition is intuitive, it requires that the edge detector information, quantified by D(PonIIPolf )' must be greater than the uncertainty in the geometry measured by H(PLlg). In other words, the better the edge detector and the more predictable the path geometry then the smaller q will be. We now apply the theory of branching processes to determine the expected number of blocks explored from a starting block in Fi 'L~o qZ = 1/(1 - q). The number of branches of Fi is (Q - 1), the total number of segments explored per block is at most QNo , and we explore at most QNo-l segments before reaching the first block. The total number of Fi is N. Therefore the total number of segments wastefully explored is at most N(Q - 1) 1~qQ2No-1. We summarize this result in a theorem: Theorem 2. Provided q = (No + I)Q+M2- N oK < 1, where the order parameter K = D(PonllPolf) - H(PLlg) - ?2(T) - ?2(T), then the expected number of false segments explored is at most N(Q - 1) 1~qQ2No-1. Comment The requirement that q < 1 is chiefly determined by the order parameter K = D (Pon IlPolf ) - H (PLlg) - ?2 (T) - f2 (T). Our convergence proofrequires that K > 0 and will break down if K < O. Is this a limitation of our proof? Or does it correspond to a fundamental difficulty in solving this tracking problem? In more recent work (Yuille and Coughlan 1998) we extend the concept of order parameters and show that they characterize the difficulty of visual search problem independently of the algorithm. In other words, as K 1----7 0 the problem becomes impossible to solve by any algorithm. There will be too many false paths which have better rewards than the target path. As K 1----7 0 there is a phase transition in the ease of solving the problem. Convergence Rates ofAlgorithmsfor Visual Search: Detecting Visual Contours 4 647 Conclusion Our analysis shows it is possible to detect certain types of image contours in linear expected time (with given starting points). We have shown how the convergence rates depend on order parameters which characterize the problem domain. In particular, the entropy of the geometric prior and the Kullback-Leibler distance between Pon and Pof f allow us to quantify intuitions about the power of geometrical assumptions and edge detectors to solve these tasks. Our more recent work (Yuille and Coughlan 1998) has extended this work by showing that the order parameters can be used to specify the intrinsic (algorithm independent) difficulty of the search problem and that phase transitions occur when these order parameters take critical values. In addition, we have proved convergence rates for A * algorithms which use inadmissible heuristics or combinations of heuristics and pruning (Coughlan and Yuille 1998). As shown in (Yuille and Coughlan 1997) many of the search algorithms proposed to solve vision search problems, such as (Geman and Jedynak 1996), are special cases of A * (or close approximations). We therefore hope that the results of this paper will throw light on the success of the algorithms and may suggest practical improvements and speed ups. Acknow ledgements We want to acknowledge funding from NSF with award number IRI-9700446, from the Center for Imaging Sciences funded by ARO DAAH049510494, and from an ASOSRF contract 49620-98-1-0197 to ALY. We would like to thank L. Xu, D. Snow, S. Konishi, D. Geiger, J. Malik, and D. Forsyth for helpful discussions. References [1] J .M. Coughlan and A.L. Yuille. "Bayesian A * Tree Search with Expected O(N) Convergence Rates for Road Tracking." Submitted to Artificial Intelligence. 1998. [2] T.M. Cover and J.A. Thomas. Elements of Information Theory. Wiley Interscience Press. New York. 1991. [3] D. Geman. and B. Jedynak. "An active testing model for tracking roads in satellite images". IEEE Trans. Patt. Anal. and Machine Intel. Vol. 18. No.1, pp 1-14. January. 1996. [4] J. Pearl. Heuristics. Addison-Wesley. 1984. [5] A.L. Yuille and J. Coughlan. " Twenty Questions, Focus of Attention, and A *" . In Energy Minimization Methods in Computer Vision and Pattern Recognition. Ed. M. Pellilo and E. Hancock. Springer-Verlag. (Lecture Notes in Computer Science 1223). 1997. [6] A.L. Yuille and J .M~ Coughlan. "Visual Search: Fundamental Bounds, Order Parameters, Phase Transitions, and Convergence Rates." Submitted to Pattern Analysis and Machine Intelligence. 1998. [7] S.C. Zhu. "Embedding Gestalt Laws in Markov Random Fields". Submitted to IEEE Computer Society Workshop on Perceptual Organization in Computer Vision. 

1513 |@word middle:1 version:1 briefly:1 eliminating:2 nd:2 simplifying:1 pg:2 initial:1 substitution:1 subjective:1 comparing:1 must:2 shape:1 remove:1 intelligence:2 smith:2 coughlan:19 detecting:6 mathematical:1 along:2 kettlewell:2 prove:1 combine:1 interscience:1 indeed:1 expected:9 examine:1 becomes:2 provided:5 pof:1 bounded:2 maximizes:1 what:2 q2:2 sanov:1 positive:1 before:1 local:1 limit:1 subscript:1 path:61 yd:1 might:1 plus:1 quantified:1 conversely:1 ease:1 statistically:1 averaged:3 jedynak:8 practical:1 testing:1 union:1 block:19 survived:1 reject:1 convenient:1 ups:1 word:3 road:5 suggest:1 close:7 put:3 context:1 impossible:2 pon:12 center:1 go:3 straightforward:1 starting:5 independently:1 iri:1 attention:1 formulate:1 iiu:1 m2:5 rule:1 konishi:1 searching:3 embedding:1 target:22 pt:4 suppose:1 exact:1 us:2 pa:1 element:1 satisfying:1 recognition:1 geman:8 calculate:1 connected:1 trade:1 highest:1 intuition:1 predictable:1 complexity:1 pol:2 reward:15 depend:1 solving:2 segment:20 yuille:19 f2:2 completely:1 po:1 joint:1 represented:1 alphabet:3 hancock:1 artificial:1 choosing:1 whose:2 heuristic:8 larger:2 spend:1 solve:3 otherwise:1 statistic:2 think:1 transform:2 aro:1 gq:1 rapidly:1 till:1 achieve:1 intuitive:1 convergence:18 requirement:1 diverges:1 satellite:1 generating:1 converges:2 leave:2 object:2 help:1 depending:1 measured:1 throw:1 involves:1 judge:1 quantify:3 differ:1 direction:1 snow:1 filter:2 enable:1 require:2 probable:1 elementary:1 extension:2 pl:1 exploring:1 lying:1 considered:2 claim:1 substituting:1 generous:1 estimation:1 tool:1 survives:2 hope:1 minimization:1 clearly:2 rather:1 reaching:1 og:1 ax:2 derived:1 focus:1 longest:1 improvement:1 likelihood:1 detect:2 summarizing:1 inst:2 inference:1 sense:1 helpful:1 el:1 snake:2 unlikely:1 typically:1 interested:1 issue:2 orientation:1 exponent:1 spatial:1 special:3 field:1 once:2 never:1 placing:1 survive:6 thin:1 minimized:2 t2:3 xhi:1 few:1 neighbour:1 geometry:6 phase:3 detection:5 organization:1 highly:2 light:1 chain:4 edge:9 tree:7 re:1 stopped:1 instance:2 cover:2 formulates:1 rare:2 uniform:3 successful:1 too:1 characterize:2 learnt:2 fundamental:3 probabilistic:1 off:10 contract:1 diverge:1 together:1 again:1 satisfied:1 choose:3 worse:1 derivative:1 waste:1 forsyth:1 satisfy:1 depends:3 later:2 break:1 lot:1 analyze:2 start:2 recover:2 characteristic:3 ensemble:6 correspond:1 ofthe:1 bayesian:10 none:1 submitted:3 detector:3 reach:1 ed:1 failure:1 energy:2 pp:1 james:1 obvious:1 e2:3 proof:2 dataset:2 proved:1 recall:1 knowledge:4 subsection:1 sophisticated:1 ok:1 wesley:1 higher:1 follow:1 specify:2 response:2 formulation:1 done:1 though:2 evaluated:1 stage:3 p6:5 mistakenly:1 replacing:1 defines:1 concept:1 true:12 hence:1 spatially:1 leibler:1 branching:7 criterion:5 plate:1 stress:1 demonstrate:1 performs:1 geometrical:2 reasoning:1 image:7 fi:11 funding:1 empirically:1 jl:1 extend:1 refer:2 llx:1 smoothness:1 funded:1 etc:1 posterior:2 recent:6 optimizing:2 discard:1 verlag:1 certain:2 discretizing:1 arbitrarily:1 success:1 yi:1 exploited:1 greater:1 prune:5 determine:10 converge:4 ii:1 branch:9 multiple:2 desirable:1 long:2 award:1 basic:1 vision:8 represent:1 normalization:1 addition:3 want:2 background:1 nary:1 comment:1 effectiveness:2 call:3 integer:3 presence:1 easy:1 simplifies:1 idea:1 shift:1 motivated:1 york:1 amount:1 clutter:3 generate:2 specifies:1 nsf:1 per:1 patt:1 xii:1 write:1 detectability:1 ledgements:1 vol:1 thereafter:1 terminology:1 demonstrating:1 poll:2 threshold:5 imaging:1 graph:2 merely:1 sum:3 uncertainty:1 geiger:1 bound:10 fl:2 mined:1 strength:1 occur:1 precisely:2 constraint:2 x2:1 speed:2 pruned:1 performing:1 numbering:1 combination:1 aerial:1 membrane:1 describes:2 smaller:2 invariant:1 pr:5 taken:1 equation:1 mechanism:1 addison:1 apply:5 observe:2 thomas:2 assumes:3 remaining:1 yx:8 calculating:1 establish:1 society:1 malik:1 question:1 strategy:1 distance:2 thank:1 length:2 difficult:1 acknow:1 design:1 anal:1 twenty:1 perform:1 upper:1 markov:6 poff:4 acknowledge:1 january:1 defining:1 extended:2 intensity:11 aly:1 required:1 connection:1 pearl:2 trans:1 address:3 able:1 below:3 usually:1 pattern:2 summarize:2 power:1 event:2 overlap:1 difficulty:5 critical:1 zhu:3 minimax:1 vax:1 fertility:3 review:1 geometric:10 prior:4 determining:2 relative:2 law:2 expect:1 lecture:1 limitation:1 heavy:1 last:2 soon:1 allow:2 understand:1 fall:2 face:1 boundary:2 curve:4 xn:1 transition:3 contour:10 qn:1 made:2 san:2 gestalt:2 pruning:18 emphasize:1 kullback:1 keep:1 global:1 active:1 conclude:1 francisco:2 assumed:1 xi:15 search:21 qz:1 jz:1 nature:1 ca:2 necessarily:1 domain:6 bounding:1 noise:2 xu:1 representative:4 intel:1 tl:3 wiley:1 fails:1 position:2 wish:1 xl:1 lie:3 candidate:1 perceptual:1 admissible:4 theorem:9 removing:1 down:1 specific:2 emphasized:1 showing:1 explored:5 grouping:1 intrinsic:1 workshop:1 false:15 entropy:2 explore:1 visual:13 expressed:2 tracking:4 joined:1 scalar:1 springer:1 corresponds:1 determines:2 chance:1 chiefly:1 llxi:1 labelled:1 hard:1 change:1 included:1 specifically:1 determined:2 inadmissible:1 total:3 select:1 searched:2

Coding time-varying signals using sparse, shift-invariant representations Terrence J. Sejnowski terryCsalk.edu Michael S. Lewicki* lewickiCsalk.edu Howard Hughes Medical Institute Computational Neurobiology Laboratory The Salk Institute 10010 N. Torrey Pines Rd. La Jolla, CA 92037 Abstract A common way to represent a time series is to divide it into shortduration blocks, each of which is then represented by a set of basis functions. A limitation of this approach, however, is that the temporal alignment of the basis functions with the underlying structure in the time series is arbitrary. We present an algorithm for encoding a time series that does not require blocking the data. The algorithm finds an efficient representation by inferring the best temporal positions for functions in a kernel basis. These can have arbitrary temporal extent and are not constrained to be orthogonal. This allows the model to capture structure in the signal that may occur at arbitrary temporal positions and preserves the relative temporal structure of underlying events. The model is shown to be equivalent to a very sparse and highly over complete basis. Under this model, the mapping from the data to the representation is nonlinear, but can be computed efficiently. This form also allows the use of existing methods for adapting the basis itself to data. This approach is applied to speech data and results in a shift invariant, spike-like representation that resembles coding in the cochlear nerve. 1 Introduction Time series are often encoded by first dividing the signal into a sequence of blocks. The data within each block is then fit with a standard basis such as a Fourier or wavelet. This has a limitation that the components of the bases are arbitrarily aligned with respect to structure in the time series. Figure 1 shows a short segment of speech data and the boundaries of the blocks. Although the structure in the signal is largely periodic, each large oscillation appears in a different position within the blocks and is sometimes split across blocks. This problem is particularly present for acoustic events with sharp onset, such as plosives in speech. It also presents ?To whom correspondence should be addressed. Coding Time-Varying Signals Using Sparse, Shift-Invariant Representations 731 difficulties for encoding the signal efficiently, because any basis that is adapted to the underlying structure must represent all possible phases. This can be somewhat circumvented by techniques such as windowing or averaging sliding blocks, but it would be more desirable if the representation were shift invariant. time Figure 1: Blocking results in arbitrary phase alignment the underlying structure. 2 The Model Our goal is to model a signal by using a small set of kernel functions that can be placed at arbitrary time points. Ultimately, we want to find the minimal set of functions and time points that fit the signal within a given noise level. We expect this type of model to work well for signals composed of events whose onset can occur at arbitrary temporal positions. Examples of these include, musical instruments sounds with sharp attack or plosive sounds in speech. We assume time series x(t) is modeled by (1) where Ti indicates the temporal position of the ith kernel function, <Pm [i) , which is scaled by Si. The notation m[i] represents an index function that specifies which of the M kernel functions is present at time Ti. A single kernel function can occur at multiple times during the time series. Additive noise at time t is given by E(t). A more general way to express (1) is to assume that the kernel functions exist at all time points during the signal, and let the non-zero coefficients determine the positions of the kernel functions. In this case, the model can be expressed in convolutional form x(t) L / Sm(T)<Pm(t - T)dT + E(t) L sm(t) * <Pm(t) + E(t) , (2) m (3) m where Sm(T) is the coefficient at time T for kernel function <Pm. It is also helpful to express the model in matrix form using a discrete sampling of the continuous time series: x = As + E. (4) M. S. Lewicki and T. J. Sejnowski 732 The basis matrix, A, is defined by (5) where C(a) is an N-by-N circulant matrix parameterized by the vector a. This matrix is constructed by replicating the kernel functions at each sample position [~ C(a) = al an a2 a3 aN-2 aN-l aN-3 aN-2 ao al aN-I a2 al 1 (6) aN-l ao The kernels are zero padded to be of length N . The length of each kernel is typically much less than the length of the signal, making A very sparse. This can be viewed as a special case of a Toeplitz matrix. Note that the size of A is M N-by-N, and is thus an example of an overcomplete basis, i.e. a basis with more basis functions than dimensions in the data space (Simoncelli et al., 1992; Coifman and Wickerhauser, 1992; Mallat and Zhang, 1993; Lewicki and Sejnowski, 1998) . 3 A probabilistic formulation The optimal coefficient values for a signal are found by maximizing the posterior distribution s = argmaxP(slx,A) = argmaxP(xIA,s)P(s) 8 (7) 8 where s is the most probable representation of the signal. Note that omission of the normalizing constant P(xIA) does not change the location of the maximum. This formulation of the problem offers the advantage that the model can fit more general types of distributions and naturally "denoises" the signal. Note that the mapping from x to s is nonlinear with non-zero additive noise and an overcomplete basis (Chen et al., 1996; Lewicki and Sejnowski, 1998). Optimizing (7) essentially selects out the subset of basis functions that best account for the data. To define a probabilistic model, we follow previous conventions for linear generative models with additive noise (Cardoso, 1997; Lewicki and Sejnowski, 1998). We assume the noise, to, to have a Gaussian distribution which yields a data likelihood for a given representation of 1 logP(xIA,s) ex - 2u 2 (x - As)2. (8) The function P(s) describes the a priori distribution of the coefficients. Under the assumption that P(s) is sparse (highly -peaked around zero), maximizing (7) results in very few nonzero coefficients. A compact representation of s is to describe the values of the non-zero coefficients and their temporal positions M P(s) = n", II P(Um,Tm) = II II P(Um,i)P(Tm ,i), m (9) m=l i =l where the prior for the non-zero coefficient values, Um,i, is assumed to be Laplacian, and the prior for the temporal positions (or intervals), Tm,i, is assumed to be a gamma distribution. Coding Time-Varying Signals Using Sparse, Shift-Invariant Representations 4 733 Finding the best encoding A difficult challenge presented by the proposed model is finding a computationally tractable method for fitting it to the data. The brute-force approach of generating the basis matrix A generates an intractable number basis functions for signals of any reasonable length, so we need to look for ways of making the optimization of (7) more efficient. The gradient of the log posterior is given by a as 10gP(sIA,x) oc AT(x - As) + z(s) , (10) where z(s) = (logP(s)),. A basic operation required is v = AT u. We saw that x = As can be computed efficiently using convolution (2). Because AT is also block circulant AT = [ C.(~.D 1 (11) C(?'u ) where ?'(1 : N) = ?(N : -1 : 1). Thus, terms involving AT can also be computed efficiently using convolution v = AT U = [ ?1 (-~~ ~ u(t) ?M( -t) * u(t) 1 (12) Obtaining an initial representation An alternative approach to optimizing (7) is to make use of the fact that if the kernel functions are short enough in length, direct multiplication is faster than convolution, and that, for this highly overcomplete basis, most of the coefficients will be zero after being fit to the data. The central problem in encoding the signal then is to determine which coefficients are non-zero, ideally finding a description of the time series with the minimal number of non-zero coefficients. This is equivalent to determining the best set of temporal positions for each of the kernel functions (1). A crucial step in this approach is to obtain a good initial estimate of the coefficients. One way to do this is to consider the projection of the signal onto each of the basis functions, i.e. AT x. This estimate will be exact (i.e. zero residual error) in the case of zero noise and A orthogonal. For the non-orthogonal, overcomplete case the solution will be approximate, but for certain choices of the basis matrix, an exact representation can still be obtained efficiently (Daubechies, 1990; Simoncelli et aI., 1992). Figure 2 shows examples of convolving two different kernel functions with data. One disadvantage with this initial solution is that the coefficient functions s~(t) are not sparse. For example, even though the signal in figure 2a is composed of only three instances of the kernel function, the convolution is mostly non-zero. A simple procedure for obtaining a better initial estimate of the most probable coefficients is to select the time locations of the maxima (or extrema) in the convolutions. These are positions where the kernel functions capture the greatest amount of signal structure and where the optimal coefficients are likely to be non-zero. This generates a large number of positions, but their number can be reduced further by selecting only those that contribute significantly, i.e. where the average power is greater than some fraction of the noise level. From these, a basis for the entire signal is constructed by replicating the kernel functions at the appropriate time positions. 734 M. S. Lewicki and T J Sejnowski ~Z'C7'C71 V1 I fJVSNSM ~ I Figure 2: Convolution using the fast Fourier transform is an efficient way to select an initial solution for the temporal positions of the kernel functions. (a) The convolution of a sawtooth-shaped kernel function, ?J(t), with a sawtooth waveform, x(t). (b) A single period sine-wave kernel function convolved with a speech segment. Once an initial estimate and basis are formed, the most probable coefficient values are estimated using a modified conjugate gradient procedure. The size of the generated basis does not pose a problem for optimization, because it is has very few non-zero elements (the number of which is roughly constant per unit time). This arises because each column is non-zero only around the position of the kernel function, which is typically much shorter in duration than the data waveform. This structure affords the use of sparse matrix routines for all the key computations in the conjugate gradient routine. After the initial fit, there typically are a large number of basis functions that give a very small contribution. These can be pruned to yield, after refitting, a more probable representation that has significantly fewer coefficients. 5 Properties of the representation Figure 3 shows the results of fitting a segment of speech with a sine wave kernel. The 64 kernel functions were constructed using a single period of a sine function whose log frequencies were evenly distributed between 0 and Nyquist (4 kHz), which yielded kernel functions that were minimally correlated (they are not orthogonal because each has only one cycle and is zero elsewhere). The kernel function lengths varied between 2 and 64 samples. The plots show the positions of the non-zero coefficients superimposed on the waveform. The residual errors curves from the fitted waveforms are shown offset, below each waveform. The right axes indicate the kernel function number which increase with frequency. The dots show the starting position of the kernels with non-zero coefficients, with the dot size scaled according to the mean power contribution. This plot is essentially a time/frequency analysis, similar to a wavelet decomposition, but on a finer temporal scale. Figure 3a shows that the structure in the coefficients repeats for each oscillation in the waveform. Adding a delay leaves the relative temporal structure of the nonzero coefficients mostly unchanged (figure 3b). The small variations between the two sets of coefficients are due to variations in the fitting of the small-magnitude coefficients. Representing the signal in figure 3b with a standard complete basis would result in a very different representation. 735 Coding Time- Varying Signals Using Sparse, Shift-Invariant Representations a .. . . .: ... .: . 0.2 : ? e? ? ? : : 0.1 : 53 o 14 ~. 1 o 20 40 60 60 100 120 time 14 ~. 1 o 20 40 60 80 100 120 time Figure 3: Fitting a shift-invariant model to a segment of speech, x(t). Dots indicate positions of kernels (right axis) with size scaled by the mean power contribution. Fitting error is plotted below speech signal. M S. Lewicki and T. J. Sejnowski 736 6 Discussion The model presented here can be viewed as an extension of the shiftable transforms of Simoncelli et al. (1992). One difference is that here no constraints are placed on the kernel functions. Furthermore, this model accounts for additive noise, which yields automatic signal denoising and provides sensible criteria for selecting significant coefficients. An important unresolved issue is how well the algorithm works for increasingly non-orthogonal kernels. One interesting property of this representation is that it results in a spike-like representation . In the resulting set of non-zero coefficients, not only is their value important for representing the signal, but also their relative temporal position, which indicate when an underlying event has occurred. This shares many properties with cochlear models. The model described here also has capacity to have an over complete representation at any given timepoint, e.g. a kernel basis with an arbitrarily large number of frequencies. These properties make this model potentially useful for binaural signal processing applications. The effectiveness of this method for efficient coding remains to be proved. A trivial example of a shift-invariant basis is a delta-function model. For a model to encode information efficiently, the representation should be non-redundant. Each basis function should "grab" as much structure in the data as possible and achieve the same level of coding efficiency for arbitrary shifts of the data. The matrix form of the model (4) suggests that it is possible to achieve this optimum by adapting the kernel functions themselves using the methods of Lewicki and Sejnowski (1998). Initial results suggest that this approach is promising. Beyond this, it is evident that modeling the higher-order structure in the coefficients themselves will be necessary both to achieve an efficient representation and to capture structure that is relevant to such tasks as speech recognition or auditory stream segmentation. These results are a step toward these goals. Acknowledgments. We thank Tony Bell, Bruno Olshausen, and David Donoho for helpful discussions. References Cardoso, J.-F. (1997). Infomax and maximum likelihood for blind source separation. IEEE Signal Processing Letters, 4:109- 11 I. Chen, S., Donoho, D. L., and Saunders, M. A. (1996). Atomic decomposition by basis pursuit. Technical report, Dept. Stat., Stanford Univ., Stanford, CA. Coifman, R. R. and Wickerhauser, M. V. (1992). Entropy-based algorithms for best basis selection. IEEE Transactions on Information Theory, 38(2) :713- 718. Daubechies, I. (1990). The wavelet transform, time-frequency localization, and signal analysis. IEEE Transactions on Information Theory, 36(5):961- 1004. Lewicki, M. S. and Sejnowski, T. J. (1998). Learning overcomplete representations. Neural Computation. submitted. Mallat, S. G. and Zhang, Z. F. (1993). Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41(12):3397-3415. Simoncelli, E. P., Freeman, W. T., Adelson, E. H., and J ., H. D. (1992). Shiftable multiscale transforms. IEEE Trans . Info . Theory, 38:587- 607. 

1514 |@word decomposition:2 initial:8 series:9 selecting:2 existing:1 si:1 must:1 additive:4 plot:2 generative:1 fewer:1 leaf:1 ith:1 short:2 provides:1 contribute:1 location:2 attack:1 zhang:2 constructed:3 direct:1 fitting:5 coifman:2 roughly:1 themselves:2 freeman:1 underlying:5 notation:1 finding:3 extremum:1 temporal:14 ti:2 um:3 scaled:3 brute:1 unit:1 medical:1 encoding:4 minimally:1 resembles:1 suggests:1 slx:1 acknowledgment:1 atomic:1 hughes:1 block:8 procedure:2 bell:1 adapting:2 significantly:2 projection:1 matching:1 suggest:1 onto:1 selection:1 equivalent:2 maximizing:2 starting:1 duration:1 variation:2 mallat:2 exact:2 element:1 recognition:1 particularly:1 blocking:2 capture:3 cycle:1 ideally:1 ultimately:1 segment:4 localization:1 efficiency:1 basis:28 binaural:1 represented:1 univ:1 fast:1 describe:1 sejnowski:9 saunders:1 whose:2 encoded:1 stanford:2 toeplitz:1 torrey:1 gp:1 itself:1 transform:2 sequence:1 advantage:1 unresolved:1 aligned:1 relevant:1 achieve:3 description:1 optimum:1 argmaxp:2 generating:1 stat:1 pose:1 ex:1 dividing:1 indicate:3 convention:1 waveform:6 require:1 ao:2 timepoint:1 probable:4 extension:1 around:2 mapping:2 pine:1 dictionary:1 a2:2 saw:1 gaussian:1 modified:1 varying:4 encode:1 ax:1 indicates:1 likelihood:2 superimposed:1 helpful:2 typically:3 entire:1 selects:1 issue:1 priori:1 constrained:1 special:1 once:1 shaped:1 sampling:1 represents:1 look:1 adelson:1 peaked:1 report:1 few:2 composed:2 preserve:1 gamma:1 phase:2 highly:3 alignment:2 necessary:1 shorter:1 orthogonal:5 divide:1 plotted:1 overcomplete:5 minimal:2 fitted:1 instance:1 column:1 modeling:1 disadvantage:1 logp:2 subset:1 shiftable:2 delay:1 periodic:1 refitting:1 probabilistic:2 terrence:1 infomax:1 michael:1 daubechies:2 central:1 convolving:1 denoises:1 account:2 coding:7 coefficient:25 onset:2 stream:1 blind:1 sine:3 wave:2 contribution:3 formed:1 convolutional:1 musical:1 largely:1 efficiently:6 yield:3 finer:1 submitted:1 c7:1 frequency:6 naturally:1 auditory:1 proved:1 segmentation:1 routine:2 nerve:1 appears:1 higher:1 dt:1 follow:1 formulation:2 though:1 furthermore:1 nonlinear:2 multiscale:1 olshausen:1 laboratory:1 nonzero:2 during:2 oc:1 criterion:1 evident:1 complete:3 common:1 khz:1 occurred:1 significant:1 ai:1 rd:1 automatic:1 pm:4 bruno:1 replicating:2 dot:3 base:1 posterior:2 optimizing:2 jolla:1 certain:1 arbitrarily:2 greater:1 somewhat:1 determine:2 period:2 redundant:1 signal:30 ii:3 sliding:1 windowing:1 desirable:1 sound:2 multiple:1 simoncelli:4 technical:1 faster:1 offer:1 laplacian:1 involving:1 basic:1 essentially:2 represent:2 kernel:32 sometimes:1 want:1 addressed:1 interval:1 source:1 crucial:1 effectiveness:1 split:1 enough:1 fit:5 tm:3 shift:9 nyquist:1 speech:9 useful:1 cardoso:2 amount:1 transforms:2 reduced:1 specifies:1 exist:1 affords:1 estimated:1 delta:1 per:1 discrete:1 express:2 key:1 v1:1 grab:1 padded:1 wickerhauser:2 fraction:1 parameterized:1 letter:1 reasonable:1 separation:1 oscillation:2 correspondence:1 yielded:1 adapted:1 occur:3 constraint:1 generates:2 fourier:2 pruned:1 circumvented:1 according:1 conjugate:2 across:1 describes:1 increasingly:1 making:2 invariant:8 computationally:1 remains:1 tractable:1 instrument:1 pursuit:2 operation:1 appropriate:1 alternative:1 convolved:1 include:1 tony:1 unchanged:1 spike:2 gradient:3 thank:1 capacity:1 sensible:1 evenly:1 cochlear:2 whom:1 extent:1 trivial:1 toward:1 length:6 sia:1 modeled:1 index:1 difficult:1 mostly:2 potentially:1 info:1 convolution:7 sm:3 howard:1 neurobiology:1 varied:1 arbitrary:7 sharp:2 omission:1 david:1 required:1 acoustic:1 trans:1 beyond:1 below:2 challenge:1 greatest:1 event:4 power:3 difficulty:1 force:1 residual:2 representing:2 axis:1 prior:2 multiplication:1 determining:1 relative:3 sawtooth:2 expect:1 interesting:1 limitation:2 share:1 elsewhere:1 placed:2 repeat:1 institute:2 circulant:2 sparse:9 distributed:1 boundary:1 dimension:1 xia:3 curve:1 transaction:3 approximate:1 compact:1 assumed:2 continuous:1 promising:1 ca:2 obtaining:2 plosive:2 noise:8 salk:1 inferring:1 position:19 wavelet:3 offset:1 a3:1 normalizing:1 intractable:1 adding:1 magnitude:1 chen:2 entropy:1 likely:1 expressed:1 lewicki:9 goal:2 viewed:2 donoho:2 change:1 averaging:1 denoising:1 la:1 select:2 arises:1 dept:1 correlated:1

Almost Linear VC Dimension Bounds for Piecewise Polynomial Networks Peter L. Bartlett Department of System Engineering Australian National University Canberra, ACT 0200 Australia Peter.Bartlett@anu.edu.au Vitaly Maiorov Department of Mathematics Technion, Haifa 32000 Israel Ron Meir Department of Electrical Engineering Technion, Haifa 32000 Israel rmeir@dumbo.technion.ac.il Abstract We compute upper and lower bounds on the VC dimension of feedforward networks of units with piecewise polynomial activation functions. We show that if the number of layers is fixed, then the VC dimension grows as W log W, where W is the number of parameters in the network. This result stands in opposition to the case where the number of layers is unbounded, in which case the VC dimension grows as W 2 ? 1 MOTIVATION The VC dimension is an important measure of the complexity of a class of binaryvalued functions, since it characterizes the amount of data required for learning in the PAC setting (see [BEHW89, Vap82]). In this paper, we establish upper and lower bounds on the VC dimension of a specific class of multi-layered feedforward neural networks. Let F be the class of binary-valued functions computed by a feed forward neural network with W weights and k computational (non-input) units, each with a piecewise polynomial activation function. Goldberg and Jerrum [GJ95] Cl(W 2 + Wk) = O(W2), where Cl is a constant. have shown that VCdim(F) Moreover, Koiran and Sontag [KS97] have demonstrated such a network that has VCdim(F) ~ C2 W 2 = O(W2), which would lead one to conclude that the bounds :s Almost Linear VC Dimension Bounds for Piecewise Polynomial Networks 191 are in fact tight up to a constant. However, the proof used in [KS97] to establish the lower bound made use of the fact that the number of layers can grow with W. In practical applications, this number is often a small constant. Thus, the question remains as to whether it is possible to obtain a better bound in the realistic scenario where the number of layers is fixed. The contribution of this work is the proof of upper and lower bounds on the VC dimension of piecewise polynomial nets. The upper bound behaves as O(W L2 + W L log W L), where L is the number of layers. If L is fixed, this is O(W log W), which is superior to the previous best result which behaves as O(W2). Moreover, using ideas from [KS97] and [GJ95] we are able to derive a lower bound on the VC dimension which is O(WL) for L = O(W). Maass [Maa94] shows that three-layer networks with threshold activation functions and binary inputs have VC dimension O(W log W), and Sakurai [Sak93] shows that this is also true for two-layer networks with threshold activation functions and real inputs. It is easy to show that these results imply similar lower bounds if the threshold activation function is replaced by any piecewise polynomial activation function f that has bounded and distinct limits limx-t - oo f(x) and limx-too f(x). We thus conclude that if the number oflayers L is fixed, the VC dimension of piecewise polynomial networks with L ~ 2 layers and real inputs, and of piecewise polynomial networks with L ~ 3 layers and binary inputs, grows as W log W. We note that for the piecewise polynomial networks considered in this work, it is easy to show that the VC dimension and pseudo-dimension are closely related (see e.g. [Vid96]), so that similar bounds (with different constants) hold for the pseudo-dimension. Independently, Sakurai has obtained similar upper bounds and improved lower bounds on the VC dimension of piecewise polynomial networks (see [Sak99]). 2 UPPER BOUNDS We begin the technical discussion with precise definitions of the VC-dimension and the class of networks considered in this work. Definition 1 Let X be a set, and A a system of subsets of X. A set S = { Xl, . .. ,xn} is shattered by A if, for every subset B ~ S, there exists a set A E A such that SnA = B. The VC-dimension of A, denoted by VCdim(A), is the largest integer n such that there exists a set of cardinality n that is shattered by A. Intuitively, the VC dimension measures the size, n, of the largest set of points for which all possible 2n labelings may be achieved by sets A E A. It is often convenient to talk about the VC dimension of classes of indicator functions F. In this case we simply identify the sets of points X E X for which f(x) = 1 with the subsets of A, and use the notation VCdim(F). A feedforward multi-layer network is a directed acyclic graph that represents a parametrized real-valued function of d real inputs. Each node is called either an input unit or a computation unit. The computation units are arranged in L layers. Edges are allowed from input units to computation units. There can also be an edge from a computation unit to another computation unit, but only if the first unit is in a lower layer than the second. There is a single unit in the final layer, called the output unit. Each input unit has an associated real value, which is One of the components of the input vector x E Rd. Each computation unit has an associated real value, called the unit's output value. Each edge has an associated real parameter, as does each computation unit. The output of a computation unit is given by (7 CEe weze + wo), where the sum ranges over the set of edges leading to 192 P L. Bartlett, V. Maiorov and R. Meir the unit, We is the parameter (weight) associated with edge e, Ze is the output value of the unit from which edge e emerges, Wo is the parameter (bias) associated with the unit, and a : R -t R is called the activation function of the unit. The argument of a is called the net input of the unit. We suppose that in each unit except the output unit, the activation function is a fixed piecewise polynomial function of the form for i = 1, ... ,p+ 1 (and set to = -00 and tp+1 = 00), where each cPi is a polynomial of degree no more than l. We say that a has p break-points, and degree l. The activation function in the output unit is the identity function. Let k i denote the number of computational units in layer i and suppose there is a total of W parameters (weights and biases) and k computational units (k = k1 + k2 + ... + k L - 1 + 1). For input x and parameter vector a E A = R w, let f(x, a) denote the output of this network, and let F = {x f-t f(x,a) : a E RW} denote the class of functions computed by such an architecture, as we vary the W parameters. We first discuss the computation of the VC dimension, and thus consider the class of functions sgn(F) = {x f-t sgn(f(x, a)) : a E RW}. Before giving the main theorem of this section, we present the following result, which is a slight improvement of a result due to Warren (see [ABar], Chapter 8). Lemma 2.1 Suppose II (.), h (.), .. , ,fm (-) are fixed polynomials of degree at most 1 in n ~ m variables. Then the number of distinct sign vectors {sgn(Jl (a)), ... ,sgn(Jm (a))} that can be generated by varying a ERn is at most 2(2eml/n)n. We then have our main result: Theorem 2.1 For any positive integers W, k ~ W, L ~ W, l, and p, consider a network with real inputs, up to W parameters, up to k computational units arranged in L layers, a single output unit with the identity activation function, and all other computation units with piecewise polynomial activation functions of degree 1 and with p break-points. Let F be the class of real-valued functions computed by this network. Then VCdim(sgn(F)) ~ 2WLlog(2eWLpk) + 2WL2log(1 + 1) + 2L. Since Land k are O(W), for fixed 1 and p this implies that VCdim(sgn(F)) = O(WLlogW + WL2). Before presenting the proof, we outline the main idea in the construction. For any fixed input x, the output of the network f(x, a) corresponds to a piecewise polynomial function in the parameters a, of degree no larger than (l + I)L-1 (recall that the last layer is linear). Thus, the parameter domain A = R W can be split into regions, in each of which the function f(x,?) is polynomial. From Lemma 2.1, it is possible to obtain an upper bound on the number of sign assignments that can be attained by varying the parameters of a set of polynomials. The theorem will be established by combining this bound with a bound on the number of regions. PROOF OF THEOREM 2.1 For an arbitrary choice of m points Xl, X2, ..? ,xm , we wish to bound K = I{(sgn(f(Xl ,a)), . .. ,sgn(J(xm, a))) : a E A }I. Almost Linear VC Dimension Bounds for Piecewise Polynomial Networks 193 Fix these m points, and consider a partition {SI, S2, ... , S N} of the parameter domain A. Clearly N K ~ L I{(sgn(J(xl , a?, ... , sgn(J(xm, a?) : a ESdi? i=1 We choose the partition so that within each region Si, f (Xl, .), ... ,f (x m, .) are all fixed polynomials of degree no more than (1 + I)L-1. Then, by Lemma 2.1, each term in the sum above is no more than 2 (2em(1;' I)L - l) W (1) The only remaining point is to construct the partition and determine an upper bound on its size. The partition is constructed recursively, using the following procedure. Let 51 be a partition of A such that, for all S E 51, there are constants bh,i,j E {0,1} for which for all a E S, where j E {I, ... ,m}, h E {I, ... ,kd and i E {1, ... ,pl. Here ti are the breakpoints of the piecewise polynomial activation functions, and Ph,x) is the affine function describing the net input to the h-th unit in the first layer, in response to X j. That is, where ah E R d, ah,O E R are the weights of the h-th unit in the first layer. Note that the partition 51 is determined solely by the parameters corresponding to the first hidden layer, as the input to this layer is unaffected by the other parameters. Clearly, for a E S, the output of any first layer unit in response to an Xj is a fixed polynomial in a. Now, let WI, ... , W L be the number of variables used in computing the unit outputs up to layer 1, ... , L respectively (so W L = W), and let k l , . .. , kL be the number of computation units in layer 1, ... , L respectively (recall that kL = 1). Then we can choose 51 so that 151 1is no more than the number of sign assignments possible with mk l P affine functions in WI variables. Lemma 2.1 shows that 151 ~ 2 (2e~~IP) WI 1 Now, we define 5 n (for n > 1) as follows. Assume that for all S in 5 n - 1 and all Xj, the net input of every unit in layer n in response to Xj is a fixed polynomial function of a E S, of degree no more than (1 + l)n-1 . Let 5 n be a partition of A that is a refinement of 5 n- 1 (that is, for all S E 5 n , there is an S' E 5 n- 1 with S ~ S'), such that for all S E 5 n there are constants bh,i,j E {O, I} such that sgn(Ph,x) (a) - ti ) = bh,i,j for all a E S, (2) where Ph ,x) is the polynomial function describing the net input of the h-th unit in the n-th layer, in response to Xj, when a E S. Since S ~ S' for some S' E 5 n- 1 , (2) implies that the output of each n-th layer unit in response to an X j is a fixed polynomial in a of degree no more than l (l + 1) n-l, for all a E S. Finally, we can choose 5 n such that, for all S' E 5 n- 1 we have I{S E 5 n : S ~ S'}I is no more than the number of sign assignments of mknP polynomials in Wn variables of degree no more than (l + 1)n- l, and by Lemma 2.1 this is no more than 2 (2emkn~n+lr-I ) Wn . Notice also that the net input of every unit in layer n + 1 in P. L. Bartlett, V Maiorov and R. Meir 194 response to (l + l)n. Xj is a fixed polynomial function of a ESE Sn of degree no more than Proceeding in this way we get a partition SL-l of A such that for S E SL-l the network output in response to any Xj is a fixed polynomial of a E S of degree no more than l(l + 1)L-2. Furthermore, JSL-d < 2 < Ce;:,P) W, TI eemk'p~,+ 1)'-') 2 TI 2CemkiP~,+ 1)'-') W, W; Multiplying by the bound (1) gives the result ~ K IT 2 (2emkip(l .+ l)i-l) W. i=l Wt Since the points Xl, ... ,X m were chosen arbitrarily, this .gives a bound on the maximal number of dichotomies induced by a E A on m points. An upper bound on the VC-dimension is then obtained by computing the largest value of m for which this number is at least 2 m , yielding m < L+ t. w, log Cempk'~,+ 1)i-1 ) < L [1 + (L - l)W log(l + 1) + W log(2empk)] , where all logarithms are to the base 2. We conclude (see for example [Vid96] Lemma 4.4) that VCdim(F) ~ 2L [(L -l)W log(l + 1) + W log (2eWLpk) + 1]. ? We briefly mention the application of this result to the problem of learning a regression function E[YIX = x], from n input/output pairs {(Xi, Yi)}i=l' drawn independently at random from an unknown distribution P(X, Y). In the case of quadratic loss, L(f) = E(Y - f(X))2, one can show that there exist constants Cl ;::: 1 and C2 such that EL(f~n ) < - 8 2 ? f L- (f) logn + Cl JET In + C2 MPdim(F) , n where 8 2 = E [Y - E[YIX]]2 is the noise variance, i(f) = E [(E[YIX] - f(X))2] is the approximation error of f, and is a function from the class F that approximately minimizes the sample average of the quadratic loss. Making use of recently derived bounds [MM97] on the approximation error, inf JET i(f), which are equal, up to logarithmic factors, to those obtained for networks of units with the standard sigmoidal function u{u) = (1 + e-u)-l , and combining with the considerably lower pseudo-dimension bounds for piecewise polynomial networks, we obtain much better error rates than are currently available for sigmoid networks. in 3 LOWER BOUND We now compute a lower bound on the VC dimension of neural networks with continuous activation functions. This result generalizes the lower bound in [KS97], since it holds for any number of layers. Almost Linear VC Dimension Bounds for Piecewise Polynomial Networks 195 Theorem 3.1 Suppose f : R -+ R has the following properties: 1. limo-too f(a) = 1 and limo-t-oo f(a) = 0, 2. f is differentiable at some point Xo and with derivative f'(xo) =1= O. Then for any L ~ 1 and W ~ 10L - 14, there is a feedforward network with the following properties: The network has L layers and W parameters, the output unit is a linear unit, all other computation units have activation function f, and the set sgn(F) of functions computed by the network has VCdim(sgn(F? l l ~ ~ J ~ J' where l u J is the largest integer less than or equal to u. PROOF As in [KS97], the proof follows that of Theorem 2.5 in [GJ95], but we show how the functions described in [GJ95] can be computed by a network, and keep track of the number of parameters and layers required. We first prove the lower bound for a network containing linear threshold units and linear units (with the identity activation function), and then show that all except the output unit can be replaced by units with activation function f, and the resulting network still shatters the same set. For further details of the proof, see the full paper [BMM98]. Fix positive integers M, N E N. We now construct a set of M N points, which may be shattered by a network with O(N) weights and O(M) layers. Let {ad, i = 1,2, ... ,N denote a set of N parameters, where each ai E [0,1) has an M -bit binary representation ai = E~l 2-jai,j, ai,j E {O, I}, i.e. the M-bit base two representation of ai is ai = O.ai,l ai,2 ... ai,M. We will consider inputs in B N X B M, where BN = {ei : 1 ~ i ~ N}, ei E {O, I}N has i-th bit 1 and all other bits 0, and BM is defined similarly. We show how to extract the bits of the ai, so that for input x = (el' ern) the network outputs al,rn. Since there are N M inputs of the form (el,e rn ), and al,rn can take on all possible 2MN values, the result will follow. There are three stages to the computation of al,rn: (1) computing ai, (2) extracting al,k from ai, for every k, and (3) selecting al,rn among the al,ks. Suppose the network input is x = ((Ul,'" ,UN),(Vt, ... ,VM? = (el,e rn ). Using one linear unit we can compute E~l Uiai = al. This involves N + 1 parameters and one computation unit in one layer. In fact, we only need N parameters, but we need the extra parameter when we show that this linear unit can be replaced by a unit with activation function f. Consider the parameter Ck = O.al,k ... al,M, that is, Ck = E~k 2k-1-jal,j for k = 1, ... ,M. Since Ck ~ 1/2 iff al,k = 1, clearly sgn(ck - 1/2) = al,k for all k. Also, Cl = al and Ck = 2Ck-l - al ,k-l' Thus, consider the recursion Ck = 2Ck-l - al,k-l = sgn(ck - 1/2)' with initial conditions CI = al and au = sgn(al - 1/2). Clearly, we can compute al,l, ... ,al,M-l and C2,' .. ,CM-l in another 2(M - 2) + 1 layers, using 5(M - 2) + 2 parameters in 2(M - 2) + 1 computational units. al,k We could compute al,M in the same way, but the following approach gives fewer layers. Set b = sgn (2C M - 1 - al,M - l then the input vector (VI, ... ,VM) b = sgn(cM) = sgn(O.al,M) = al,M. E~~I Vi)' = eM, If m =1= M then b = O. If m and thus E~~lvi = 0, =M implying that P L. Bartlett, V. Maiorov and R. Meir 196 In order to conclude the proof, we need to show how the variables al,m may be recovered, depending on the inputs (VI, V2, ... ,VM). We then have al,m = b V V';~I(al,i/\vi). Since for boolean x and y, x/\y = sgn(x+y-3/2), and V';I Xi = sgn(2:,;1 Xi - 1/2), we see that the computation of al,m involves an additional 5M parameters in M + 1 computational units, and adds another 2 layers. In total, there are 2M layers and 10M + N -7 parameters, and the network shatters a set of size N M. Clearly, we can add parameters and layers without affecting the function of the network. So for any L, WEN, we can set M = lL/2J and N = W + 7 - 10M, which is at least lW/2J provided W :2: 10L - 14. In that case, the VC-dimension is at least l L /2 J l W /2 J . The network just constructed uses linear threshold units and linear units. However, it is easy to show (see [KS97], Theorem 5) that each unit except the output unit can be replaced by a unit with activation function f so that the network still shatters the set of size M N. For linear units, the input and output weights are scaled so that the linear function can be approximated to sufficient accuracy by f in the neighborhood of the point Xo. For linear threshold units, the input weights are scaled so that the behavior of f at infinity accurately approximates a linear threshold function. ? References M. Anthony and P. L. Bartlett. Neural Network Learning: Theoretical Foundations. Cambridge University Press, 1999 (to appear). [BEHW89] A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. J. ACM, 36(4):929965, 1989. [BMM98] P. L. Bartlett, V. Maiorov, and R. Meir. Almost linear VC-dimension bounds for piecewise polynomial networks. Neural Computation, 10:2159- 2173, 1998. P.W. Goldberg and M.R. Jerrum. Bounding the VC Dimension of [GJ95] Concept Classes Parameterized by Real Numbers. Machine Learning, 18:131 - 148, 1995. P. Koiran and E.D. Sontag. Neural Networks with Quadratic VC Di[KS97] . mension. Journal of Computer and System Science, 54:190- 198, 1997. [ABar] [Maa94] W. Maass. Neural nets with superlinear VC-dimension. Neural Computation, 6(5):877- 884, 1994. [MM97] V. Maiorov and R. Meir. On the Near Optimality of the Stochastic Approximation of Smooth Functions by Neural Networks. Submitted for publication, 1997. A. Sakurai. Tighter bounds on the VC-dimension of three-layer networks. In World Congress on Neural Networks, volume 3, pages 540543, Hillsdale, NJ, 1993. Erlbaum. [Sak93] [Sak99] A. Sakurai. Tight bounds for the VC-dimension of piecewise polynomial networks. In Advances in Neural Information Processing Systems, volume 11. MIT Press, 1999. [Vap82] V. N. Vapnik. Estimation of Dependences Based on Empirical Data. Springer-Verlag, New York, 1982. M Vidyasagar. A Theory of Learning and Generalization. Springer Verlag, New York, 1996. [Vid96] 

1515 |@word behw89:2 briefly:1 polynomial:31 bn:1 mention:1 recursively:1 initial:1 selecting:1 chervonenkis:1 recovered:1 activation:18 si:2 realistic:1 partition:8 implying:1 fewer:1 warmuth:1 lr:1 node:1 ron:1 sigmoidal:1 unbounded:1 c2:4 constructed:2 prove:1 behavior:1 multi:2 jm:1 cardinality:1 begin:1 provided:1 moreover:2 bounded:1 notation:1 israel:2 cm:2 minimizes:1 nj:1 pseudo:3 every:4 act:1 ti:4 k2:1 scaled:2 unit:60 appear:1 before:2 positive:2 engineering:2 congress:1 limit:1 solely:1 approximately:1 au:2 k:1 range:1 directed:1 practical:1 eml:1 procedure:1 empirical:1 binaryvalued:1 convenient:1 get:1 superlinear:1 layered:1 bh:3 demonstrated:1 independently:2 haussler:1 construction:1 suppose:5 goldberg:2 us:1 ze:1 approximated:1 electrical:1 region:3 complexity:1 tight:2 chapter:1 talk:1 distinct:2 dichotomy:1 neighborhood:1 limo:2 larger:1 valued:3 say:1 jerrum:2 maiorov:6 final:1 ip:1 differentiable:1 net:7 oflayers:1 maximal:1 combining:2 iff:1 sna:1 derive:1 oo:2 ac:1 depending:1 involves:2 implies:2 australian:1 closely:1 stochastic:1 vc:29 australia:1 sgn:21 hillsdale:1 vcdim:8 fix:2 generalization:1 tighter:1 pl:1 hold:2 considered:2 koiran:2 vary:1 estimation:1 currently:1 largest:4 wl:1 mit:1 clearly:5 yix:3 ck:9 varying:2 publication:1 derived:1 improvement:1 el:4 shattered:3 hidden:1 labelings:1 among:1 denoted:1 logn:1 equal:2 construct:2 represents:1 dumbo:1 piecewise:19 wen:1 national:1 replaced:4 limx:2 yielding:1 edge:6 wl2:1 logarithm:1 haifa:2 theoretical:1 mk:1 boolean:1 tp:1 sakurai:4 assignment:3 subset:3 technion:3 erlbaum:1 too:2 learnability:1 considerably:1 vm:3 containing:1 choose:3 derivative:1 leading:1 wk:1 ad:1 vi:4 break:2 characterizes:1 contribution:1 il:1 accuracy:1 variance:1 identify:1 accurately:1 multiplying:1 unaffected:1 ah:2 submitted:1 definition:2 proof:8 associated:5 di:1 cee:1 recall:2 emerges:1 feed:1 attained:1 follow:1 response:7 improved:1 arranged:2 furthermore:1 just:1 stage:1 ei:2 grows:3 concept:1 true:1 maass:2 ehrenfeucht:1 ll:1 presenting:1 outline:1 recently:1 superior:1 sigmoid:1 behaves:2 volume:2 jl:1 slight:1 approximates:1 cambridge:1 ai:11 rd:1 mathematics:1 similarly:1 base:2 add:2 inf:1 scenario:1 verlag:2 binary:4 arbitrarily:1 vt:1 yi:1 additional:1 determine:1 ii:1 full:1 smooth:1 technical:1 jet:2 regression:1 achieved:1 affecting:1 grow:1 w2:3 extra:1 induced:1 vitaly:1 integer:4 extracting:1 near:1 feedforward:4 split:1 easy:3 wn:2 xj:6 architecture:1 fm:1 idea:2 whether:1 ese:1 bartlett:7 ul:1 wo:2 peter:2 sontag:2 jal:1 york:2 amount:1 ph:3 rw:2 meir:6 sl:2 exist:1 notice:1 sign:4 rmeir:1 track:1 threshold:7 drawn:1 shatters:3 ce:1 graph:1 sum:2 parameterized:1 almost:5 bit:5 bound:34 layer:38 opposition:1 breakpoints:1 quadratic:3 infinity:1 x2:1 argument:1 optimality:1 ern:2 department:3 kd:1 em:2 wi:3 making:1 intuitively:1 xo:3 remains:1 discus:1 describing:2 available:1 generalizes:1 v2:1 remaining:1 giving:1 k1:1 establish:2 question:1 dependence:1 parametrized:1 unknown:1 upper:9 precise:1 rn:6 arbitrary:1 pair:1 required:2 kl:2 vap82:2 established:1 able:1 xm:3 vidyasagar:1 indicator:1 recursion:1 mn:1 imply:1 extract:1 sn:1 jai:1 l2:1 loss:2 acyclic:1 foundation:1 degree:11 affine:2 sufficient:1 jsl:1 land:1 last:1 bias:2 warren:1 dimension:32 xn:1 stand:1 world:1 forward:1 made:1 refinement:1 bm:1 keep:1 conclude:4 xi:3 continuous:1 un:1 cl:5 anthony:1 domain:2 lvi:1 main:3 motivation:1 s2:1 noise:1 bounding:1 allowed:1 cpi:1 canberra:1 wish:1 xl:6 lw:1 theorem:7 specific:1 pac:1 exists:2 vapnik:2 ci:1 anu:1 logarithmic:1 simply:1 springer:2 corresponds:1 acm:1 identity:3 blumer:1 determined:1 except:3 wt:1 lemma:6 called:5 total:2

Probabilistic Image Sensor Fusion Ravi K. Sharma1 , Todd K. Leen 2 and Misha Pavel 1 1 Department of Electrical and Computer Engineering 2Department of Computer Science and Engineering Oregon Graduate Institute of Science and Technology P.O. Box 91000 , Portland , OR 97291-1000 Email: {ravi,pavel} @ece.ogi.edu, tleen@cse .ogi.edu Abstract We present a probabilistic method for fusion of images produced by multiple sensors . The approach is based on an image formation model in which the sensor images are noisy, locally linear functions of an underlying, true scene. A Bayesian framework then provides for maximum likelihood or maximum a posteriori estimates of the true scene from the sensor images. Maximum likelihood estimates of the parameters of the image formation model involve (local) second order image statistics, and thus are related to local principal component analysis. We demonstrate the efficacy of the method on images from visible-band and infrared sensors . 1 Introduction Advances in sensing devices have fueled the deployment of multiple sensors in several computational vision systems [1, for example]. Using multiple sensors can increase reliability with respect to single sensor systems. This work was motivated by a need for an aircraft autonomous landing guidance (ALG) system [2, 3] that uses visible-band, infrared (IR) and radar-based imaging sensors to provide guidance to pilots for landing aircraft in low visibility. IR is suitable for night operation, whereas radar can penetrate fog. The application requires fusion algorithms [4] to combine the different sensor images . Images from different sensors have different characteristics arising from the varied physical imaging processes. Local contrast may be polarity reversed between visibleband and IR images [5 , 6] . A particular sensor image may contain local features not found in another sensor image , i.e., sensors may report complementary features . Finally, individual sensors are subj ect to noise. Fig . l(a) and l(b) are visible-band and IR images respectively, of a runway scene showing polarity reversed (rectangle) 825 Probabilistic Image Sensor Fusion and complementary (circle) features. These effects pose difficulties for fusion. An obvious approach to fusion is to average the pixel intensities from different sensors. Averaging, Fig. l(c), increases the signal to noise ratio, but reduces the contrast where there are polarity reversed or complementary features [7]. Transform-based fusion methods [8, 5, 9] selectfrom one sensor or another for fusion. They consist of three steps: (i) decompose the sensor images using a specified transform e.g. a multiresolution Laplacian pyramid, (ii) fuse at each level of the pyramid by selecting the highest energy transform coefficient, and (iii) invert the transform to synthesize the fused image. Since features are selected rather than averaged, they are rendered at full contrast, but the methods are sensitive to sensor noise, see Fig. l(d). To overcome the limitations of averaging or selection methods, and put sensor fusion on firm theoretical grounds, we explicitly model the production of sensor images from the true scene, including the effects of sensor noise. From the model, and sensor images, one can ask What is the most probable true scene? This forms the basis for fusing the sensor images. Our technique uses the Laplacian pyramid representation [5], with the step (ii) above replaced by our probabilistic fusion. A similar probabilistic framework for sensor fusion is discussed in ([10]). 2 The lInage Forlnation Model The true scene, denoted s, gives rise to a sensor image through a noisy, non-linear transformation. For ALG, s would be an image of the landing scene under conditions of uniform lighting, unlimited visibility, and perfect sensors. We model the map from the true scene to a sensor image by a noisy, locally affine transformation whose parameters are allowed to vary across the image (actually across the Laplacian pyramid) ai(~ t) = (3i(~ t) s(~ t) + O'i(~ t) + Ei(~ t) (1) r where, s is the true scene, ai is ith sensor image, == (x, y, k) is the hyperpixel location, with x, y the pixel coordinates and k the level of the pyramid, t is the time, 0' is the sensor offset, {3 is the sensor gain (which includes the effects of local polarity reversals and complementarity), and E is the (zero-mean) sensor noise. To simplify notation, we adopt the matrix form a = (3s = + 0' + l (2) = where a = [al,a2, . . . ,aqr, f3 [(31,(32, ... , (3qr, Q' [0'1,0'2, ... ,O'qr, s is a and we have dropped the reference to location and scalar and l = [El,E2, ... ,E q time. r, Since the image formation parameters f3, Q', and the sensor noise covariance E~ can vary from hyperpixel to hyperpixel, the model can express local polarity reversals, complementary features, spatial variation of sensor gain, and noise. We do assume, however, that the image formation parameters and sensor noise distribu tion vary slowly with location 1 . Hence, a particular set of parameters is considered to hold true over a spatial region of several square hyperpixels. We will use this assumption implicitly when we estimate these parameters from data. The model (2) fits the framework of the factor analysis model in statistics [11, 12] . Here the hyperpixel values of the true scene s are the latent variables or 1 Specifically the parameters vary slowly on the spatia-temporal scales over which the true scene s may exhibit large variations. R. K. Sharma, T. K. Leen and M. Pavel 826 common factors, f3 contains the factor loadings, and the sensor noise ? values are the independent factors. Estimation of the true scene is equivalent to estimating the common factors from the observations a. 3 Bayesian Fusion Given the sensor intensities a, we will estimate the true scene s by appeal to a Bayesian framework. We assume that the probability density function of the latent variables s is a Gaussian with local mean so(~ t) and local variance u;(~ t). An attractive benefit of this setup is that the prior mean So might be obtained from knowledge in the form of maps, or clear-weather images of the scene. Thus, such database information can be folded into the sensor fusion in a natural way. The density on the sensor images conditioned on the true scene, P(als), is normal with mean f3 s+a and covariance E? :::: diag[u;l' U;2" .. ,u;J The marginal density P(a) is normal with mean I'm :::: f3 So + a and covariance C :::: E? + u;f3f3 T (3) Finally, the posterior density on s, given the sensor data a, P(sla) is also normal with mean M- 1 (f3T E;l (a -a)+ so/u;), and covariance M- 1 :::: (f/ E;l f3+ l/u;fl. Given these densities, there are two obvious candidates for probabilistic fusion : maximum likelihood (ML) 5 :::: max. P(als), and maximum a posteriori (MAP) 5:::: max. P(sla) . The MAP fusion estimate is simply the posterior mean 5:::: [f3TE;If3+1/u;r 1 (f3 TE ;l(a_a) + so/un (4) (5) To obtain the ML fusion estimate we take the limit u; -+ 00 in either (4) or (5). For both ML and MAP, the fused image 5 is a locally linear combination of the sensor images that can, through the spatio-temporal variations in f3, a, and E?, properly respond to changes in the sensor characteristics that tax averaging or selection schemes. For example, if the second sensor has a polarity reversal relative to the first, then f32 is negative and the two sensor contributions are properly subtracted. If the first sensor has high noise (large u;J, its contribution to the fused image is attenuated. Finally, a feature missing from sensor 1 corresponds to f31 :::: O. The model compensates by accentuating the contribution from sensor 2. 4 Model Parameter Estimates We need to estimate the local image formation model parameters a(~ t), f3(~ t) and the local sensor noise covariance?E?(~ t). We estimate the latter from successive, motion compensated video frames from each sensor. First we estimate the average value at each hyperpixel (ai(t)), and the average square (a;(t)) by exponential moving averages . We next estimate the noise variance by the difference (t) :::: a; (t) - U;i ai 2 (t). To estimate f3 and a, we assume that f3, a, E?, So and u; are nearly constant over small spatial regions (5 x 5 blocks) surrounding the hyperpixel for which the Probabilistic Image Sensor Fusion 827 parameters are desired. Essentially we are invoking a spatial analog of ergodicity, where ensemble averages are replaced by spatial averages, carried out locally over regions in which the statistics are approximately constant. To form a maximum likelihood (ML) estimate of a, we extremize the data loglikelihood C = Z=;;=llog[P(an)] with respect to a to obtain a ML = I'a - f3 so (6) , where I'a is the data mean, computed over a 5 x 5 hyperpixellocal region (N points). = 25 To obtain a ML estimate of f3, we set the derivatives of C with respect to to zero and recover equal (C - Ea)C -1 f3 = 0 f3 (7) where Ea is the data covariance matrix, also computed over a 5 x 5 hyperpixel local region . The only non-trivial solution to (7) is f3 ML = E,!-(X-l)t U r u~ (8) where U , A are the principal eigenvector and eigenvalue of the weighted data co_ _1. _1. variance matrix, Ea == E, 2 EaE ? and r = ?l. 2, An alternative to maximum likelihood estimation is the least squares (LS) approach [11] . We obtain the LS estimate aLS by minimizing (9) with respect to a . This gives aLS The least squares estimate f3 LS f3 . The (10) is obtained by minimizing E{3 with respect to = I'a - f3 so . = II Ea - C W (11) solution to this minimization is f3LS = At -Ur u~ (12) where U, A are the principal eigenvector and eigenvalue of the noise-corrected covariance matrix (Ea - E f ), and r = ? l. 2 The estimation procedures cannot provide values of the priors u~ and So. Were we dealing with a single global model, this would pose no problem. But we must impose a constraint in order to smoothly piece together our local models. We impose that 11.811 1 everywhere, or by (12) = A. Recall that A is the leading eigenvalue of ~a - ~, and thus captures the scale of variations in a that arise from variations in s . Thus we would expect A ex u~. Our constraint insures that the proportionality constant be the same for each local model. Next, note that changing So causes a shift = u; 2The least squares and maximum likelihood solutions are identical when the model is exact Ea == C, i.e. the observed data covariance is exactly of the form dictated by the model. Under this condition, U = (U TE;lU)-1/2Ee -1/2U and (~- 1) = ~(UTE;lU). The LS and ML solutions are also identical when the noise covariance is homoscedastic Ee = I, even if the model is not exact. (1; R. K. Sharma, T. K. Leen and M. Pavel 828 in s. To maintain consistency between local regions, we take So These choices for 11'; and So constrain the parameter estimates to f3 LS r V aLS Pa . = 0 everywhere. and (13) In (5) 11'; and So are defined at each hyperpixel. However, to estimate f3 and a, we used spatial averages to compute the sample mean and covariance. This is somewhat inconsistent, since the spatial variation of So (e.g. when there are edges in the scene) is not explicitly captured in the model mean and covariance. These variations are, instead , attributed to 11';, resulting in overestimation of the latter. A more complete model would explicitly model the spatial variations of So, though we expect this will produce only minor changes in the results . Finally, the sign parameter r is not specified. In order to properly piece together our local models , we must choose r at each hyperpixel in such a way that f3 changes direction slowly as we move from hyperpixel to hyperpixel and encounter changes in the local image statistics. That is, large direction changes due to arbitrary sign reversals are not allowed . We use a simple heuristic to accomplish this. 5 Relation to peA The MAP and ML fusion rules are closely related to PCA. To see this, assume that the noise is homoscedastic EE = 11';1 and use the parameter estimates (13) in the MAP fusion rule (5), reducing the latter to 1 s= 1+I1'UI1'; T Va(a-Pa) 1 + 1+11';;11'~ So (14) where Va is the principal eigenvector of the data covariance matrix Ea. The MAP estimate s is simply a scaled and shifted local PCA projection of the sensor data. Both the scaling and shift arise because the prior distribution on s tends to bias s towards So. When the prior is flat 11'; -+ 00, (or equivalently when using the ML fusion estimate), or when the noise variance vanishes, the fused image is given by a simple local PCA projection (15 ) 6 Experilllents and Results We applied our fusion method to visible-band and IR runway images, Fig. 1, containing additive Gaussian noise. Fig. l(e) shows the result of ML fusion with f3 and a estimated using (13) . ML fusion performs better than either averaging or selection in regions that contain local polarity reversals or complementary features. ML fusion gives higher weight to IR in regions where the features in the two images are common , thus reducing the effects of noise in the visible-band image. ML fusion gives higher weight to the appropriate sensor in regions with complementary features. Fig. l(f) shows the result of MAP fusion (5) with the priors 11'; and So those dictated by th e consistency requirements discussed in section 4. Clearly, the MAP image is less noisy than the ML image. In regions of low sensor image contrast, 11'; is low (since>. is low), thus the contribution from the sensor images is attenuated compared to the ML fusion rule. Hence the noise is attenuated. In regions containing features such as edges, 11'; is high (since>. is high); hence the contribution from the sensor images is similar to that in ML fusion. Probabilistic Image Sensor Fusion 829 (a) Visible-band image (b) IR image (c) Averaging (d) Selection (e) ML (f) MAP Figure 1: Fusion of visible-band and IR images containing additive Gaussian noise In Fig. 2 we demonstrate the use of a database image for fusion. Fig. 2(a) and 2(b) are simulated noisy sensor images from visible-band and JR, that depict a runway with an aircraft on it. Fig. 2(c) is an image of the same scene as might be obtained from a terrain database. Although this image is clean, it does not show the actual situation on the runway. One can use the database image pixel intensities as the prior mean So in the MAP fusion rule (5). The prior variance u; in (5) can be regarded as a m-e asure of confidence in the database image - it's value controls the relative contribution of the sensors vs. the database image in the fused image. (The parameters f3 and a, and the sensor noise covariance EIE were estimated exactly as before.) Fig. 2(d), 2(e) and 2(f) show the MAP-fused image as a function of increasing 0";. Higher values of 0"; accentuate the contribution of the sensor images, whereas lower values of 0"; accentuate the contribution of the database. 7 Discussion We presented a model-based probabilistic framework for fusion of images from multiple sensors and exercised the approach on visible-band and IR images. The approach provides both a rigorous framework for PCA-like fusion rules, and a principled way to combine information from a terrain database with sensor images. We envision several refinements of the approach given here. Writing new image formation models at each hyperpixel produces an overabundance of models. Early experiments show that this can be relaxed by using the same model parameters over regions of several square hyperpixels, rather than recalculating for each hyperpixel. A further refinement could be provided by adopting a mixture of linear models to build up the non-linear image formation model. Finally, we have used multiple frames from a video sequence to obtain ML and MAP fused sequences, and one should be able to produce superior parameter estimates by suitable use of the video sequence. R. K. Sharma, T. K. Leen and M Pavel 830 (a) Visible-band image (b) IR image (c) Database image Figure 2: Fusion of simulated visible-band and IR images using database image Acknowledgments - This work was supported by NASA Ames Research Center grant NCC2-S11. TKL was partially supported by NSF grant ECS-9704094. References [1] L. A. Klein. Sensor and Data Fusion Concepts and Applications. SPIE, 1993. [2] J. R. Kerr, D. P. Pond, and S. Inman. Infrared-optical muItisensor for autonomous landing guidance. Proceedings of SPIE, 2463:38-45, 1995. [3] B. Roberts and P. Symosek. Image processing for flight crew situation awareness. Proceedings of SPIE, 2220:246-255, 1994. [4] M. Pavel and R. K. Sharma. Model-based sensor fusion for aviation. In J. G. Verly, editor, Enhanced and Synthetic Vision 1997, volume 3088, pages 169-176. SPIE, 1997. [5] P. J. Burt and R. J. Kolczynski. Enhanced image capture through fusion. In Fourth Int. Conf. on Computer Vision, pages 173-182. IEEE Compo Soc., 1993. [6] H. Li and Y. Zhou. Automatic visual/IR image registration. Optical Engineering, 35(2):391-400, 1996. ' [7] M . Pavel, J. Larimer, and A. Ahumada. Sensor fusion for synthetic vision. In Proceedings of the Society for Information Display, pages 475-478. SPIE, 1992. [8] P. Burt. A gradient pyramid basis for pattern-selective image fusion. In Proceedings of the Society for Information Display, pages 467-470. SPIE, 1992. [9] A. Toet. Hierarchical image fusion. Machine Vision and Applications, 3:1-11, 1990. [10] J. J. Clark and A. L. Yuille. Data Fusion for Sensory Information Processing Systems. Kluwer, Boston, 1990. [11] A. Basilevsky. Statistical Factor Analysis and Related Methods. Wiley, 1994. [12] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Technical report, NCRG/97/01O, Neural Computing Research Group, Aston University, UK,1997. 

1516 |@word aircraft:3 f32:1 loading:1 proportionality:1 covariance:13 pavel:7 invoking:1 contains:1 efficacy:1 selecting:1 envision:1 must:2 additive:2 visible:11 visibility:2 depict:1 v:1 selected:1 device:1 ith:1 compo:1 provides:2 cse:1 location:3 successive:1 ames:1 ect:1 combine:2 actual:1 increasing:1 provided:1 estimating:1 notation:1 underlying:1 what:1 eigenvector:3 transformation:2 temporal:2 exactly:2 scaled:1 uk:1 control:1 grant:2 before:1 ncc2:1 engineering:3 local:19 todd:1 dropped:1 limit:1 tends:1 approximately:1 might:2 deployment:1 graduate:1 averaged:1 acknowledgment:1 block:1 procedure:1 weather:1 projection:2 confidence:1 cannot:1 selection:4 put:1 writing:1 landing:4 equivalent:1 map:14 missing:1 compensated:1 center:1 l:5 penetrate:1 rule:5 regarded:1 autonomous:2 coordinate:1 variation:8 enhanced:2 exact:2 us:2 runway:4 complementarity:1 synthesize:1 pa:2 infrared:3 database:10 observed:1 electrical:1 capture:2 region:12 highest:1 principled:1 vanishes:1 overestimation:1 radar:2 yuille:1 basis:2 surrounding:1 formation:7 firm:1 whose:1 heuristic:1 loglikelihood:1 compensates:1 statistic:4 transform:4 noisy:5 sequence:3 eigenvalue:3 multiresolution:1 tax:1 qr:2 requirement:1 produce:3 perfect:1 pose:2 minor:1 soc:1 crew:1 direction:2 closely:1 pea:1 decompose:1 probable:1 hold:1 considered:1 ground:1 normal:3 vary:4 adopt:1 a2:1 homoscedastic:2 early:1 estimation:3 sensitive:1 exercised:1 weighted:1 minimization:1 clearly:1 sensor:66 gaussian:3 rather:2 zhou:1 properly:3 portland:1 likelihood:6 contrast:4 rigorous:1 posteriori:2 el:1 relation:1 selective:1 i1:1 pixel:3 denoted:1 spatial:8 marginal:1 equal:1 tkl:1 f3:24 identical:2 nearly:1 report:2 simplify:1 individual:1 replaced:2 maintain:1 recalculating:1 mixture:1 misha:1 fog:1 edge:2 circle:1 desired:1 guidance:3 theoretical:1 fusing:1 f3t:1 uniform:1 accomplish:1 synthetic:2 density:5 probabilistic:10 together:2 fused:7 containing:3 choose:1 slowly:3 conf:1 derivative:1 leading:1 li:1 includes:1 coefficient:1 int:1 oregon:1 explicitly:3 piece:2 tion:1 recover:1 contribution:8 square:6 ir:12 variance:5 characteristic:2 ensemble:1 bayesian:3 produced:1 lu:2 lighting:1 eie:1 email:1 energy:1 obvious:2 e2:1 attributed:1 spie:6 gain:2 pilot:1 ask:1 recall:1 knowledge:1 actually:1 ea:7 nasa:1 higher:3 tipping:1 leen:4 box:1 though:1 ergodicity:1 flight:1 night:1 ei:1 effect:4 contain:2 true:13 concept:1 hence:3 attractive:1 ogi:2 complete:1 demonstrate:2 performs:1 motion:1 image:73 common:3 superior:1 accentuating:1 physical:1 volume:1 ncrg:1 discussed:2 analog:1 kluwer:1 ai:4 automatic:1 consistency:2 reliability:1 moving:1 ute:1 spatia:1 posterior:2 dictated:2 s11:1 captured:1 somewhat:1 impose:2 relaxed:1 sharma:4 signal:1 ii:3 multiple:5 full:1 reduces:1 technical:1 laplacian:3 va:2 vision:5 essentially:1 ui1:1 accentuate:2 adopting:1 pyramid:6 invert:1 whereas:2 inconsistent:1 ee:3 iii:1 fit:1 attenuated:3 shift:2 motivated:1 pca:4 cause:1 clear:1 involve:1 locally:4 band:11 extremize:1 nsf:1 shifted:1 sign:2 estimated:2 arising:1 klein:1 express:1 group:1 sla:2 changing:1 clean:1 ravi:2 registration:1 rectangle:1 imaging:2 fuse:1 everywhere:2 fourth:1 fueled:1 respond:1 scaling:1 tleen:1 fl:1 display:2 subj:1 constraint:2 constrain:1 scene:17 flat:1 unlimited:1 optical:2 rendered:1 department:2 combination:1 jr:1 across:2 ur:1 kerr:1 reversal:5 operation:1 hierarchical:1 appropriate:1 subtracted:1 alternative:1 encounter:1 build:1 society:2 move:1 pond:1 exhibit:1 gradient:1 reversed:3 simulated:2 trivial:1 polarity:7 ratio:1 minimizing:2 equivalently:1 setup:1 robert:1 negative:1 rise:1 observation:1 situation:2 frame:2 varied:1 arbitrary:1 intensity:3 burt:2 specified:2 able:1 pattern:1 including:1 max:2 video:3 suitable:2 difficulty:1 natural:1 scheme:1 technology:1 aston:1 carried:1 prior:7 relative:2 expect:2 limitation:1 clark:1 awareness:1 affine:1 editor:1 production:1 supported:2 distribu:1 bias:1 institute:1 benefit:1 f31:1 overcome:1 toet:1 sensory:1 refinement:2 ec:1 implicitly:1 dealing:1 ml:19 global:1 spatio:1 terrain:2 un:1 latent:2 alg:2 ahumada:1 diag:1 noise:21 arise:2 allowed:2 complementary:6 fig:10 wiley:1 exponential:1 candidate:1 bishop:1 showing:1 sensing:1 offset:1 appeal:1 fusion:42 consist:1 te:2 conditioned:1 boston:1 smoothly:1 simply:2 insures:1 visual:1 partially:1 scalar:1 corresponds:1 towards:1 change:5 specifically:1 folded:1 corrected:1 reducing:2 averaging:5 llog:1 aviation:1 principal:5 ece:1 latter:3 ex:1

On the optimality of incremental neural network algorithms Ron Meir* Department of Electrical Engineering Technion, Haifa 32000, Israel rmeir@dumbo.technion.ac.il Vitaly Maiorov t Department of Mathematics Technion, Haifa 32000, Israel maiorov@tx.technion.ac.il Abstract We study the approximation of functions by two-layer feedforward neural networks, focusing on incremental algorithms which greedily add units, estimating single unit parameters at each stage . As opposed to standard algorithms for fixed architectures, the optimization at each stage is performed over a small number of parameters, mitigating many of the difficult numerical problems inherent in high-dimensional non-linear optimization. We establish upper bounds on the error incurred by the algorithm, when approximating functions from the Sobolev class, thereby extending previous results which only provided rates of convergence for functions in certain convex hulls of functional spaces. By comparing our results to recently derived lower bounds, we show that the greedy algorithms are nearly optimal. Combined with estimation error results for greedy algorithms, a strong case can be made for this type of approach. 1 Introduction and background A major problem in the application of neural networks to real world problems is the excessively long time required for training large networks of a fixed architecture. Moreover, theoretical results establish the intractability of such training in the worst case [9][4]. Additionally, the problem of determining the architecture and size of the network required to solve a certain task is left open. Due to these problems, several authors have considered incremental algorithms for constructing the network by the addition of hidden units, and estimation of each unit's parameters incrementally. These approaches possess two desirable attributes: first, the optimization is done step-wise, so that only a small number of parameters need to be optimized at each stage; and second, the structure of the network -This work was supported in part by the a grant from the Israel Science Foundation tThe author was partially supported by the center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel. 296 R. Meir and V Maiorov is established concomitantly with the learning, rather than specifying it in advance. However, until recently these algorithms have been rather heuristic in nature, as no guaranteed performance bounds had been established. Note that while there has been a recent surge of interest in these types of algorithms, they in fact date back to work done in the early seventies (see [3] for a historical survey). The first theoretical result establishing performance bounds for incremental approximations in Hilbert space, was given by Jones [8]. This work was later extended by Barron [2], and applied to neural network approximation of functions characterized by certain conditions on their Fourier coefficients. The work of Barron has been extended in two main directions. First, Lee et at. [10] have considered approximating general functions using Hilbert space techniques, while Donahue et al. [7] have provided powerful extensions of Jones' and Barron's results to general Banach spaces. One of the most impressive results of the latter work is the demonstration that iterative algorithms can, in many cases, achieve nearly optimal rates of convergence, when approximating convex hulls. While this paper is concerned mainly with issues of approximation, we comment that it is highly relevant to the statistical problem of learning from data in neural networks. First, Lee et at. [10] give estimation error bounds for algorithms performing incremental optimization with respect to the training error. Under certain regularity conditions, they are able to achieve rates of convergence comparable to those obtained by the much more computationally demanding algorithm of empirical error minimization. Moreover, it is well known that upper bounds on the approximation error are needed in order to obtain performance bounds, both for parametric and nonparametric estimation, where the latter is achieved using the method of complexity regularization. Finally, as pointed out by Donahue et al. [7], lower bounds on the approximation error are crucial in establishing worst case speed limitations for learning. The main contribution of this paper is as follows. For functions belonging to the Sobolev class (see definition below), we establish, under appropriate conditions, near-optimal rates of convergence for the incremental approach, and obtain explicit bounds on the parameter values of the network. The latter bounds are often crucial for establishing estimation error rates. In contrast to the work in [10] and [7], we characterize approximation rates for functions belonging to standard smoothness classes, such as the Sobolev class. The former work establishes rates of convergence with respect to the convex hulls of certain subsets of functions, which do not relate in a any simple way to standard functional classes (such as Lipschitz, Sobolev, Holder, etc.). As far as we are aware, the results reported here are the first to report on such bounds for incremental neural network procedures. A detailed version of this work, complete with the detailed proofs, is available in [13]. 2 Problem statement We make use of the nomenclature and definitions from [7]. Let H be a Banach space of functions with norm II . II. For concreteness we assume henceforth that the norm is given by the Lq norm, 1 < q < 00, denoted by II . Ilq. Let linn H consist of all sums of the form L~=l aigi , gi E H and arbitrary ai, and COn H is the set of such sums with ai E [0,1] and L~=l ai = 1. The distances, measured in the Lq norm, from a function f are given by dist(1in n H,f) = inf {l lh - fllq : hE linnH}, dist(conH , f) = inf {l lh - fllq : hE conH}. The linear span of H is given by linH = Un linn H, while the convex-hull of H is coH = Unco n H. We follow standard notation and denote closures of sets by a bar, e.g. coH is the closure of the convex hull of H. In this work we focus on the special case where H = H1} ~ {g : g(x) = eCJ(a T x + b), lei::; 1}, IICJ(?)llq ::; I}, (1) 297 On the Optimality of Incremental Neural Network Algorithms corresponding to the basic building blocks of multilayer neural networks. The restriction 110-011 ::; 1 is not very demanding as many sigmoidal functions can be expressed as a sum of functions of bounded norm. It should be obvious that linn 1l1) corresponds to a two-layer neural network with a linear output unit and o--activation functions in the single hidden layer, while COn 1l1) is equivalent to a restricted form of such a network, where restrictions are placed on the hidden-to-output weights. In terms of the definitions introduced above, the by now well known property of universal function approximation over compacta can be stated as lin1l = C(M), where C(M) is the class of continuous real valued functions defined over M , a compact subset of Rd . A necessary and sufficient condition for this has been established by Leshno et af. [11], and essentially requires that 0-(.) be locally integrable and non-polynomial. We comment that if 'T} = 00 in (l), and c is unrestricted in sign, then co1l= = lin1l=. The distinction becomes important only if 'T} < 00 , in which case co1l1) C lin1l1). For the purpose of incremental approximation, it turns out to be useful to consider the convex hull co1l, rather than the usual linear span, as powerful algorithms and performance bounds can be developed in this case. In this context several authors have considered bounds for the approximation of a function 1 belonging to co1l by sequences of functions belonging to COn 1l . However, it is not clear in general how well convex hulls of bounded functions approximate general functions . One contribution of this work is to show how one may control the rate of growth of the bound 'T} in (1), so that general functions, belonging to certain smoothness classes (e.g. Sobolev), may be well approximated. In fact, we show that the incremental approximation scheme described below achieves nearly optimal approximation error for functions in the Sobolev space. Following Donahue et af. [7], we consider c:-greedy algorithms. Let E = (El ' E2, ... ) be a positive sequence, and similarly for (ai, a 2, ... ), 0 < an < 1. A sequence of functions hI, h2 ' ... is E-greedy with respect to 1 if for n = 0, 1, 2, .. . , Ilhn+I - Illq< inf {llanhn + (1 - an)g - Illq : 9 E 1l1)} + En , (2) where we set ho = O. For simplicity we set an = (n - l)/n , although other schemes are also possible. It should be clear that at each stage n, the function h n belongs to COn 1l1). Observe also that at each step, the infimum is taken with respect to 9 E 1l1)' the function h n being fixed. In terms of neural networks, this implies that the optimization over each hidden unit parameters (a, b, c) is performed independently of the others. We note in passing, that while this greatly facilitates the optimization process in practice, no theoretical guarantee can be made as to the convexity of the single-node error function (see [1] for counter-examples). The variables En are slack variables, allowing the extra freedom of only approximate minimization . In this paper we do not optimize over an, but rather fix a sequence in advance, forfeiting some generality at the price of a simpler presentation. In any event, the rates we obtain are unchanged by such a restriction. In the sequel we consider E-greedy approximations of smooth functions belonging to the Sobolev class of function s, W; = {I : m?x WDk 1112 ::; ?1} , OS k S r where k = (k 1 , . ?? , k d ) , k i 2:: 0 and Ikl = ki + ... k d . Here Vk is the partial derivative operator of order k. All functions are defined over a compact domain K C Rd. 3 Upper bound for the L2 norm First, we consider the approximation of functions from WI using the L2 norm. In distinction with other Lq norms, there exists an inner product in this case, defined through R. Meir and V Maiorov 298 (".) = II?II~. This simp1ification is essential to the proof in this case. We begin by recalling a result from [12], demonstrating that any function in L2 may be exactly expressed as a convex integral representation of the form f(x) =Q J (3) h(x, O)w(O)dO, where 0 < Q < 00 depends on f, and w( 0) is a probability density function (pdf) with respect to the multi-dimensional variable O. Thus, we may write f(x) = QEw{h(x, e)}, where Ew denotes the expectation operator with respect to the pdf w . Moreover, it was shown in [12], using the Radon and wavelet transforms, that the function h(x, 0) can be taken to be a ridge function with 0 = (a, b, e) and h(x, 0) = ea(a T x + b). In the case of neural networks, this type of convex representation was first exploited by Barron in [2], assuming f belongs to a class of functions characterized by certain moment conditions on their Fourier transforms. Later, Delyon et al. [6] and Maiorov and Meir [12] extended Barron's results to the case of wavelets and neural networks, respectively, obtaining rates of convergence for functions in the Sobolev class. The basic idea at this point is to generate an approximation, hn(x), based on n draws of random variables en = {e l , e 2, ... ,en}, e i ,....., w(?), resulting in the random function hn(x; en) Q =- n 2: h(x, e n (4) i ). i=l Throughout the paper we conform to standard notation, and denote random variables by uppercase letters, as in e, and their realization by lower case letters, as in O. Let w n = TI~=l Wi represent the product pdf for {e l , ... ,en}. Our first result demonstrates that, on the average, the above procedure leads to good approximation of functions belonging to W{. Theorem 3.1 Let K C Rd be a compact set. Then/or any there exists a constant e > 0, such that Ew " Ilf - f hn(x; en)112 :S en- rld +E , E W{, n > 0 and c > 0 (5) where Q < en(1/ 2- r l d)+, and (x)+ = max(O,x). The implication of the upper bound on the expected value, is that there exists a set of values o* ,n = {Or, ... , O~}, for which the rate (5) can be achieved. Moreover, as long as the functions h(x, Od in (4) are bounded in the L2 norm, a bound on Q implies a bound on the size of the function h n itself. Proof sketch The proof proceeds by expressing f as the sum of two functions, iI and 12 . The function iI is the best approximation to f from the class of multi-variate splines of degree r. From [12] we know that there exist parameters on such that IliI (.) - h n {-, on)112 :S en-rid. Moreover, using the results of [5] it can be shown that 1112112 :S en-rid. Using these two observations, together with the triangle inequality Ilf - h n l1 2:S IliI - h nl1 2+ 1112112, yields the desired result. I Next, we show that given the approximation rates attained in Theorem 3.1, the same rates may be obtained using an c -greedy algorithm. Moreover, since in [12] we have established the optimality of the upper bound (up to a logarithmic factor in n), we conclude that greedy approximations can indeed yield near-optimal perfonnance, while at the same time being much more attractive computationally. In fact, in this section we use a weaker algorithm, which does not perform a full minimization at each stage. On the Optimality of Incremental Neural Network Algorithms Incremental algorithm: (q = 2) Let an 1. Let 0i be chosen to satisfy =1- lin, 6: n 299 =1- an = lin. Ilf(x) - Qh(x,Onll~ = EWl {llf(x) - Qh(x,edIID? 2. Assume that 0i , f(x) - ?2,... ::~; ,O~-l have been generated. Select O~ to obey n-l 2 i=l 2 L h(x,On - 6:nQh(x,O~) f(x) - Qa n n-l "h(x, On - 6: n Qh(x, en) 22} . n-lL..,; i=l Define which measures the error incurred at the n-th stage by this incremental procedure. The main result of this section then follows. Theorem 3.2 For any bounded as f E WI and c > 0, the error of the incremental algorithm above is for some finite constant c. Proof sketch The claim will be established upon showing that (6) namely, the error incurred by the incremental procedure is identical to that of the nonincremental one described preceding Theorem 3.1. The result will then follow upon using Holder's inequality and the upper bound (5) for the r.h.s. of (6). The remaining details are I straightforward, but tedious, and can be found in the full paper [13]. 4 Upper bound for general Lq norms Having established rates of convergence for incremental approximation of WI in the L2 norm, we move now to general Lq norms. First, note that the proof of Theorem 3.2 relies heavily on the existence on an inner product. This useful tool is no longer available in the case of general Banach spaces such as L q . In order to extend the results to the latter norm, we need to use more advanced ideas from the theory of the geometry of Banach spaces. In particular, we will make use of recent results from the work of Donahue et al. [7]. Second, we must keep in mind that the approximation of the Sobolev space WI using the Lq norm only makes sense if the embedding condition rid> (1/2 - l/q)+ holds, since otherwise the Lq norm may be infinite (the embedding condition guarantees its finiteness; see [14] for details). We first present the main result of this section, followed by a sketch of the proof. The full details of the rather technical proof can be found in [13]. Note that in this case we need to use the greedy algorithm (2) rather than the algorithm of Section 3. R. Meir and V Maiorov 300 Theorem 4.1 Let the embedding condition r / d 0< r < r*, r* = JEW; andf. ~ + (~- ~)+ > (1/2 - 1/ q) + hold for 1 < q < 00, andassumethatllh(-,O)llq:S IforallO. Thenforany >0 where ;I- (~-P !+% _qd q > 2, (7) ~ q :s 2, c = c(r, d , K) and h n(-, on) is obtained via the incremental greedy algorithm (2) with cn = O. , = { Proof sketch The main idea in the proof of Theorem 4.1 is a two-part approximation scheme. First, based on [13], we show that any JEW; may be well approximated by functions in the convex class COn ('111/) for an appropriate value of TJ (see Lemma 5.2 in [13]), where R1/ is defined in (1). Then, it is argued, making use of results from [7] (in particular, using Corollary 3.6) , that an incremental greedy algorithm can be used to approximate the closure of the class co( R 1/) by the class COn (111/). The proof is completed by using the triangle inequality. The proof along the above lines is done for the case q > 2. In the case q 2, a simple use of the Holder inequality in the form Ilfllq ~ IKI 1 /q- l/21IfI12, where IKI is the volume of the region K, yields the desired result, which, given the lower I bounds in [12], is nearly optimal. :s 5 Discussion We have presented a theoretical analysis of an increasingly popular approach to incremental learning in neural networks . Extending previous results , we have shown that near-optimal rates of convergence may be obtained for approximating functions in the Sobolev class These results extend and clarify previous work dealing solely with the approximation of functions belonging to the closure of convex hulls of certain sets of functions. Moreover, we have given explicit bounds on the parameters used in the algorithm , and shown that the restriction to COn 111/ is not too stringent. In the case q ~ 2 the rates obtained are as good (up to logarithmic factors) to the rates obtained for general spline functions, which are known to be optimal for approximating Sobolev spaces. The rates obtained in the case q > 2 are sub-optimal as compared to spline functions, but can be shown to be provably better than any linear approach. In any event, we have shown that the rates obtained are equal, up to logarithmic factors, to approximation from linn 111/' when the size of TJ is chosen appropriately, implying that positive input-to-output weights suffice for approximation. An open problem remaining at this point is to demonstrate whether incremental algorithms for neural network construction can be shown to be optimal for every value of q. In fact, this is not even known at this stage for neural network approximation in general. W; . References [1] P. Auer, M. Herbster, and M. Warmuth. Exponentially many local minima for single neurons. In D.S. Touretzky, M .e. Mozer, and M.E. Hasselmo, editors, Advances in Neural Information Processing Systems 8, pages 316-322. MIT Press, 1996. [2] AR. Barron. Universal approximation bound for superpositions of a sigmoidal function. IEEE Trans. In! Th., 39:930-945, 1993. [3] AR. Barron and R.L. Barron. Statistical learning networks: a unifying view. In E. Wegman , editor, Computing Science and Statistics: Proceedings 20th Symposium Interface , pages 192-203, Washington D.e. , 1988. Amer. Statis. Assoc. On the Optimality of Incremental Neural Network Algorithms 301 [4] A. Blum and R. Rivest. Training a 3-node neural net is np-complete. In D.S. Touretzky, editor, Advances in Neural Information Processing Systems I, pages 494-50l. Morgan Kaufmann, 1989. [5] c. de Boor and G. Fix. Spline approximation by quasi-interpolation. J. Approx. Theory , 7:19-45,1973 . [6] B. Delyon, A. Juditsky, and A. Benveniste. Accuracy nalysis for wavelet approximations. IEEE Transaction on Neural Networks, 6:332-348, 1995. [7] M .J. Donahue, L. Gurvits, C. Darken, and E. Sontag. Rates of convex approximation in non-hilbert spaces. Constructive Approx. , 13: 187-220, 1997. [8] L. Jones. A simple lemma on greedy approximation in Hilbert space and convergence rate for projection pursuit regression and neural network training. Ann. Statis. , 20:608-613, 1992. [9] S. Judd. Neural Network Design and the Complexity of Learning. MIT Press, Boston, USA,1990. [10] W.S . Lee, P.S. Bartlett, and R.c. Williamson. Efficient Agnostic learning of neural networks with bounded fan-in . IEEE Trans. In! Theory, 42(6):2118-2132, 1996. [11] M. Leshno, V. Lin, A. Pinkus, and S. Schocken. Multilayer Feedforward Networks with a Nonpolynomial Activation Function Can Approximate any Function. Neural Networks, 6:861-867,1993. [12] V.E. Maiorov and R. Meir. On the near optimality of the stochastic approximation of smooth functions by neural networks. Technical Report CC-223, Technion, Department of Electrical Engineering, November ]997. Submitted to Advances in Computational Mathematics. [13] R. Meir and V. Maiorov. On the optimality of neural network approximaSubmitted for publication, October 1998. tion using incremental algorithms. ftp://dumbo.technion.ac.il/pub/PAPERSlincrementa] .pdf. [14] H. Triebel. Theory of Function Spaces. Birkhauser, Basel, 1983. 

1517 |@word version:1 polynomial:1 norm:15 tedious:1 open:2 closure:4 iki:2 thereby:1 moment:1 pub:1 comparing:1 od:1 activation:2 must:1 numerical:1 statis:2 ilii:2 juditsky:1 implying:1 greedy:11 warmuth:1 node:2 ron:1 forfeiting:1 sigmoidal:2 simpler:1 along:1 symposium:1 boor:1 indeed:1 expected:1 surge:1 dist:2 multi:2 leshno:2 becomes:1 provided:2 estimating:1 moreover:7 notation:2 bounded:5 begin:1 suffice:1 rivest:1 israel:4 agnostic:1 developed:1 guarantee:2 every:1 ti:1 growth:1 exactly:1 demonstrates:1 assoc:1 control:1 unit:6 grant:1 positive:2 engineering:2 local:1 establishing:3 solely:1 interpolation:1 specifying:1 co:1 practice:1 block:1 procedure:4 universal:2 empirical:1 projection:1 operator:2 context:1 ilf:3 restriction:4 equivalent:1 optimize:1 center:1 straightforward:1 independently:1 convex:12 survey:1 simplicity:1 embedding:3 qh:3 construction:1 heavily:1 ilq:1 approximated:2 electrical:2 worst:2 region:1 counter:1 mozer:1 convexity:1 complexity:2 upon:2 triangle:2 tx:1 wdk:1 heuristic:1 solve:1 valued:1 schocken:1 otherwise:1 statistic:1 gi:1 itself:1 maiorov:8 sequence:4 net:1 product:3 relevant:1 realization:1 date:1 achieve:2 ecj:1 convergence:9 regularity:1 extending:2 r1:1 incremental:22 ftp:1 ac:3 measured:1 strong:1 implies:2 qd:1 direction:1 attribute:1 hull:8 stochastic:1 stringent:1 argued:1 tthe:1 fix:2 absorption:2 extension:1 clarify:1 hold:2 considered:3 claim:1 major:1 achieves:1 early:1 purpose:1 estimation:5 superposition:1 hasselmo:1 establishes:1 tool:1 minimization:3 mit:2 rather:6 nqh:1 publication:1 corollary:1 derived:1 focus:1 vk:1 mainly:1 greatly:1 contrast:1 greedily:1 sense:1 el:1 hidden:4 quasi:1 mitigating:1 provably:1 issue:1 denoted:1 special:1 equal:1 aware:1 gurvits:1 having:1 washington:1 identical:1 seventy:1 jones:3 nearly:4 dumbo:2 report:2 others:1 spline:4 inherent:1 np:1 geometry:1 recalling:1 freedom:1 llq:2 interest:1 highly:1 uppercase:1 llf:1 tj:2 implication:1 integral:1 partial:1 necessary:1 lh:2 perfonnance:1 concomitantly:1 haifa:2 desired:2 theoretical:4 ar:2 subset:2 technion:6 too:1 characterize:1 reported:1 combined:1 density:1 herbster:1 sequel:1 lee:3 together:1 opposed:1 hn:3 henceforth:1 derivative:1 linh:1 de:1 coefficient:1 satisfy:1 depends:1 performed:2 later:2 h1:1 view:1 tion:1 contribution:2 il:3 holder:3 accuracy:1 kaufmann:1 yield:3 cc:1 submitted:1 touretzky:2 definition:3 obvious:1 e2:1 proof:12 con:7 rld:1 popular:1 hilbert:4 ea:1 back:1 auer:1 focusing:1 attained:1 follow:2 amer:1 done:3 generality:1 stage:7 until:1 sketch:4 o:1 ikl:1 incrementally:1 infimum:1 lei:1 usa:1 building:1 excessively:1 former:1 regularization:1 attractive:1 ll:1 pdf:4 complete:2 ridge:1 demonstrate:1 l1:6 interface:1 wise:1 recently:2 functional:2 exponentially:1 pinkus:1 banach:4 extend:2 he:2 volume:1 expressing:1 nonpolynomial:1 ai:4 smoothness:2 rd:3 approx:2 mathematics:2 similarly:1 pointed:1 had:1 impressive:1 longer:1 etc:1 add:1 recent:2 inf:3 belongs:2 certain:8 inequality:4 exploited:1 integrable:1 morgan:1 ministry:1 unrestricted:1 minimum:1 preceding:1 ewl:1 ii:7 full:3 desirable:1 smooth:2 technical:2 characterized:2 af:2 long:2 lin:3 basic:2 regression:1 multilayer:2 essentially:1 expectation:1 represent:1 achieved:2 background:1 addition:1 finiteness:1 crucial:2 appropriately:1 extra:1 posse:1 comment:2 facilitates:1 vitaly:1 near:4 feedforward:2 concerned:1 variate:1 architecture:3 inner:2 idea:3 cn:1 triebel:1 whether:1 bartlett:1 sontag:1 nomenclature:1 passing:1 useful:2 detailed:2 clear:2 transforms:2 nonparametric:1 locally:1 generate:1 meir:7 exist:1 sign:1 rmeir:1 conform:1 write:1 demonstrating:1 blum:1 linn:4 nalysis:1 concreteness:1 sum:4 letter:2 powerful:2 throughout:1 sobolev:11 draw:1 radon:1 comparable:1 layer:3 bound:24 hi:1 guaranteed:1 ki:1 followed:1 fan:1 fourier:2 speed:1 optimality:7 span:2 performing:1 jew:2 coh:2 department:3 belonging:8 increasingly:1 wi:5 making:1 restricted:1 taken:2 computationally:2 turn:1 slack:1 needed:1 know:1 mind:1 available:2 pursuit:1 observe:1 obey:1 barron:8 appropriate:2 ho:1 existence:1 denotes:1 remaining:2 completed:1 unifying:1 establish:3 approximating:5 unchanged:1 move:1 parametric:1 usual:1 distance:1 assuming:1 demonstration:1 difficult:1 october:1 statement:1 relate:1 stated:1 design:1 basel:1 perform:1 allowing:1 upper:7 observation:1 neuron:1 darken:1 finite:1 november:1 wegman:1 extended:3 arbitrary:1 introduced:1 immigrant:1 namely:1 required:2 optimized:1 distinction:2 established:6 qa:1 trans:2 able:1 bar:1 proceeds:1 below:2 max:1 event:2 demanding:2 advanced:1 scheme:3 l2:5 determining:1 limitation:1 foundation:1 h2:1 incurred:3 degree:1 sufficient:1 nonincremental:1 editor:3 intractability:1 benveniste:1 supported:2 placed:1 weaker:1 judd:1 world:1 author:3 made:2 historical:1 far:1 transaction:1 compacta:1 approximate:4 compact:3 keep:1 dealing:1 rid:3 conclude:1 un:1 iterative:1 continuous:1 additionally:1 nature:1 obtaining:1 williamson:1 constructing:1 domain:1 main:5 en:12 sub:1 explicit:2 lq:7 wavelet:3 donahue:5 theorem:7 showing:1 consist:1 exists:3 essential:1 delyon:2 boston:1 illq:2 logarithmic:3 expressed:2 partially:1 corresponds:1 relies:1 presentation:1 ann:1 lipschitz:1 price:1 infinite:1 birkhauser:1 lemma:2 ew:2 select:1 latter:4 constructive:1 andf:1

Contrast adaptation in simple cells by changing the transmitter release probability Peter Adorjan Klaus Obennayer Dept. of Computer Science, FR2-1, Technical University Berlin Franklinstrasse 28/2910587 Berlin, Germany {adp, oby} @cs.tu-berlin.de http://www.ni.cs.tu-berlin.de Abstract The contrast response function (CRF) of many neurons in the primary visual cortex saturates and shifts towards higher contrast values following prolonged presentation of high contrast visual stimuli. Using a recurrent neural network of excitatory spiking neurons with adapting synapses we show that both effects could be explained by a fast and a slow component in the synaptic adaptation. (i) Fast synaptic depression leads to saturation of the CRF and phase advance in the cortical response to high contrast stimuli. (ii) Slow adaptation of the synaptic transmitter release probability is derived such that the mutual information between the input and the output of a cortical neuron is maximal. This component-given by infomax learning rule-explains contrast adaptation of the averaged membrane potential (DC component) as well as the surprising experimental result, that the stimulus modulated component (Fl component) of a cortical cell's membrane potential adapts only weakly. Based on our results, we propose a new experiment to estimate the strength of the effective excitatory feedback to a cortical neuron, and we also suggest a relatively simple experimental test to justify our hypothesized synaptic mechanism for contrast adaptation. 1 Introduction Cells in the primary visual cortex have to encode a wide range of contrast levels, and they still need to be sensitive to small changes in the input intensities. Because the signaling capacity is limited, this paradox can be resolved only by a dynamic adaptation to changes in the input intensity distribution: the contrast response function (CRF) of many neurons in the primary visual cortex shifts towards higher contrast values following prolonged presentation of high contrast visual stimuli (Ahmed et al. 1997, Carandini & Ferster 1997). On the one hand, recent experiments, suggest that synaptic plasticity has a major role 77 Contrast Adaptation and Infomax in contrast adaptation. Because local application of GABA does not mediate adaptation (Vidyasagar 1990) and the membrane conductance does not increase significantly during adaptation (Ahmed et al. 1997, Carandini & Ferster 1997), lateral inhibition is unlikely to account for contrast adaptation. In contrast, blocking glutamate (excitatory) autoreceptors decreases the degree of adaptation (McLean & Palmer 1996). Furthermore, the adaptation is stimulus specific (e.g. Carandini et al. 1998), it is strongest if the adapting and testing stimuli are the same. On the other hand, plasticity of synaptic weights (e.g. Chance et al. 1998) cannot explain the weak adaptation of the stimulus driven modulations in the membrane potential (FI component) (Carandini & Ferster 1997) and the retardation of the response phase after high contrast adaptation (Saul 1995). These experimental findings motivated us to explore how presynaptic factors, such as a long term plasticity mediated by changes in the transmitter release probability (Finlayson & Cynader 1995) affect contrast adaptation. 2 The single cell and the cortical circuit model The cortical cells are modeled as leaky integrators with a firing threshold of -55 mY. The interspike membrane potential dynamics is described by 8Vi(t) Cm~ = -91eak (Vi(t) - Eresd - ~ L9ij(t) (Vi(t) - Esyn) . (I) J The postsynaptic conductance 9ij (t) is the integral over the previous presynaptic events and is described by the alpha-function (2) t; where is the arrival time of spike number s from neuron j. Including short term synaptic depressIOn, the effective conductance is weighted by the portion of the synaptic resource Pij (t) . Rij (t) that targets the postsynaptic side. The model parameters are C m = 0.5 nF, 91e ak = 31 nS, E rest = -65 mY, Esyn = -5 mY, 9'::':~x = 7.8 nS, and Tpeak = 1 ms, and the absolute refractory period is 2 ms, and after a spike, the membrane potential is reset 1 mV below the resting potential. Following Tsodyks & Markram (1997) a synapse between neurons j and i is characterized by the relative portion of the available synaptic transmitter or resource Rij. After a presynaptic event, Rij decreases by Pij Rij, and recovers exponentially, where Pij is the transmitter release probability. The time evolution of Rij between two presynaptic spikes is then Rij(t) = 1 - (1 - (Rij{t) - pij(t)Rij{t))) exp A ? (-(t -i)) , (3) Tree where ? is the last spike time, and the recovery time constant Tree = 200 ms. Assuming Poisson distributed presynaptic firing, the steady state of the expected resource is (4) The stationary mean excitatory postsynaptic current (EPSC) Ii] (fj , Pij) is proportional to the presynaptic firing frequency fj and the activated transmitter Pij Ri] (fj , Pij ) Ii] (fj, Pij) ex f Pij Ri] (fj, Pij) . (5) The mean current saturates for high input rates /j and it also depends on the transmitter release probability Pij: with a high release probability the function is steeper at low presynaptic frequencies but saturates earlier than for a low release probability. P Adorjem and K. Obermayer 78 _ g .. ? 80 r = = = = - '- -- - - , --- p=O.55 ,'. 60 - p=O.24 ,A .? ,,/" e 40 ~. OD J!" 20 # -641.. - - - --.====:::::::;-] :' --- p = 0.55 - -64.2 , '\ .? > p= 02 .4 S -64.4 : \ ;\ .... -- 0 0 ~ -64.6' 0 t-" ......... " 0 : \ :\. ~ ,- (5 ~ 0 ... .. -64.8 ? OF'='-4-~o~~-----~ 10? (a) 10 ' Firing rate [Hz 1 -{)5 0 102 (b) 50 100 150 200 Time [ms[ Figure 1: Short term synaptic dynamics at high and low transmitter release probability, (a) The estimated transfer function O(f, p) for the cortical cells (Eq. 7) (solid and dashed lines) in comparison with data obtained by the integrate and fire model (Eq. 1, circles and asterisks). (b) EPSP trains for a series of presynaptic spikes at intervals of 31 ms (32 Hz). p=O.55 (0.24) corresponds to adaptation to 1% (50% ) contrast (see Section 4). In order to study contrast adaptation. 30 leaky-integrator neurons are connected fully via excitatory fast adapting synapses. Each "cortical" leaky integrator neuron receives its "geniculate" input through 30 synapses. The presynaptic geniculate spike-trains are independent Poisson-processes. Modeling visual stimulation with a drifting grating, their rates are modulated sinusoidally with a temporal frequency of2 Hz. The background activity for each individual "geniculate" source is drawn from a Gaussian distribution with a mean of 20 Hz and a standard deviation of 5 Hz. In the model the mean geniculate firing rate (Fig. 2b) and the amplitude of modulation (Fig. 2a) increases with stimulus log contrast according to the experimental data (Kaplan et al. 1987). In the following simulations CRFs are determined according to the protocol of Carandini & Ferster (1997). The CRFs are calculated using an initial adaptation period of 5 s and a subsequent series of interleaved test and re-adaptation stimuli (1 s each). 3 The learning rule We propose that contrast adaptation in a visual cortical cell is a result of its goal to maximize the amount of information the cell's output conveys about the geniculate input l . Following (Bell & Sejnowski 1995) we derive a learning rule for the transmitter release probability p to maximize the mutual information between a cortical cell's input and output. Let O(f, p) be the average output firing rate, f the presynaptic firing rate, and p the synaptic transmitter release probability. Maximizing the mutual information is then equivalent to maximizing the entropy of a neuron's output if we assume only additive noise: H [O(f,p)] -E[ In Prob(O(J, p))] Prob(f)] [ -E In I(}O(f ,p)/(}f l E [In 1(}O~;, p) I] - E[ In Prob(f)] (6) (In the following all equations apply locally to a synapse between neurons j and i.) In order to derive an analytic expression for the relation between 0 and f we use the fact that the EPSP amplitude converges to its steady state relatively fast compared to the modulation of the geniculate input to the visual cortex, and that the average firing rates of the I A different approach of maximizing mutual information between input and output of a single spiking neuron has been developed by Stemmler & Koch (1999). For non-spiking neurons this strategy has been demonstrated experimentally by. e.g. Laughlin (1994). Contrast Adaptation and Infomax 79 presynaptic neurons are approximately similar. Thus we approximate the activation function by Ire O(f,p) ex S(f)pRoo (f ,p), (7) where S(f) = accounts for the frequency dependent summation of EPSCs. The parameters a = 1.8 and = 15 Hz are determined by fitting O(f, p) to the firing rate of our integrate and fire single cell model (see Fig. 1a) . The objective function is then maximized by a stochastic gradient ascent learning rule for the release probability p e op _ oH [O(f, p)] _ ~ 1 Tadapt ot - Op - Op n IoO(f, p) I of . (8) Evaluating the derivatives we obtain a non-Hebbian learning rule for the transmitter release probability p, op Tadapt = }- ot - 2 Tre e fR 1 Tree(fa - 1) - 1) + p- + a + TreeP(fa (9) = where a I~e' and the adaptation time constant Tadapt 7 s (Ohzawa et al. 1985). This is similar in spirit to the anti-Hebbian learning mechanism for the synaptic strength proposed by Barlow & Foldiak (1989) to explain adaptation phenomena. Here, the first term is proportional to the presynaptic firing rate f and to the available synaptic resource R, suggesting a presynaptic mechanism for the learning. Because the amplitude of the EPSP is proportional to the available synaptic resource, we could interpret R as an output related quantity and -2Tree f R as an anti-Hebbian learning rule for the "strength of the synapse", i.e. the probability p of the transmitter release. The second term ensures that pis always larger than O. In the current model setup for the operating range of the presynaptic geniculate cells p also stays al ways less than 1. The third term modulates the adaptation slightly and increases the release probability p most if the input firing rate is close to 20 Hz, i.e. the stimulus contrast is low. Image contrast is related to the standard deviation of the luminance levels normalized by the mean . Because ganglion cells adapt to the mean luminance, contrast adaptation in the primary visual cortex requires only the estimation of the standard deviation. In a free viewing scenario with an eye saccade frequency of 2-3 Hz, the standard deviation can be estimated based on 10-20 image samples. Thus the adaptation rate can be fast (Tadapt = 7 s), and it should also be fast in order to maintain good a representation whenever visual contrast changes, e.g. by changing light conditions. Higher order moments (than the standard deviation) of the statistics of the visual world express image structure and are represented by the receptive fields' profiles. The statistics of the visual environment are relatively static, thus the receptive field profiles should be determined and constrained by another less plastic synaptic parameter. such as the maximal synaptic conductance 9max. 4 Results Figure 2 shows the average geniculate input, the membrane potential, the firing rate and the response phase of the modeled cortical cells as a function of stimulus contrast. The CRFs were calculated for two adapting contrasts 1% (dashed line) and 50% (solid line). The cortical CRF saturates for high contrast stimuli (Fig. 2e). This is due to the saturation ofthe postsynaptic current (cf. Fig. I a) and thus induced by the short term synaptic depression. In accordance with the experimental data (e.g. Carandini et al. 1997) the delay of the cortical response (Fig. 2f) decreases towards high contrast stimuli. This is a consequence of fast synaptic depression (c.f. Chance et al. 1998). High modulation in the input firing rate leads to a fast transient rise in the EPSC followed by a rapid depression. 80 P. Ador}an and K. Obermayer LGN >E <L) --a fJj ~ ~ .... ~ ~(f 5 ~ 0.0 ....0 <L) ~ 0 100 U' 40 102 .----~---, <L) 102 ~20 ~-58 10 1 Contrast [%] 102 ~..o--_ ,P.-El~-~ 101 102 -0 0- 62 L--_~ 100 101 ~ ~ ro _ _- - ' Contrast [%] , I -40 o I ~ -60 0.. (d) p--e' ~-20 '0 U ,,- - o ~ <c1o u o 0 l...-_~~_~ ,, ,, (e) 10 ....... >-54 ~30 .S oa ~ (c) .s 100 ~ 101 ,, 10 .S ...... t:l.. 10 1 fJj (b) --~ 20 ,, C 0.0 .S (a) <L) fJj ....... :-20 u:: ,......, 30 (,) ....... 10 U'40 102 -80 l...-_~~_---' 100 10 1 102 (0 Contrast [%] Figure 2: The DC (a) and the FI (b) component of the geniculate input, and the response of the cortical units in the model with strong recurrent lateral connections and slow adaptation of the release probability on both the geniculocortical and lateral synapses. The Fl (c) and the DC (d) component of the subthreshold membrane potential of a single cortical unit, the Fl component of the firing rate (e). and the response phase (0 are plotted as a function of stimulus contrast after adaptation to 1% (solid lines) and to 50% (dashed lines) contrast stimuli. The CRF for the membrane potential (c, d) is calculated by integrating Eg. I without spikes and without reset after spikes. The cortical circuitry involves strong recurrent lateral connections. The model predicts a shift of 3-5 mV in the DC component of the subthreshold membrane potential (Fig. 2d)- a smaller amount than measured by Carandini & Ferster (1997). Nevertheless, in accordance with the data the shift caused by the adaptation is larger than the change in the DC component of the membrane potential from I % contrast to 100% contrast. The largest shift in the DC membrane potential during adaptation occurs for small contrast stimuli because an alteration in the transmitter release probability has the largest effect on the postsynaptic current if the presynaptic firing rate is close to the geniculate background activity of 20 Hz. The maximal change in the Fl component (Fig. 2c) is around 5mV and it is half of the increase in the FI component of the membrane potential from 1% contrast to 100% contrast. The CRF for the cortical firing rate (Fig. 2e) shifts to the right and the slope decreases after adaptation to high contrast. The model predicts that the probability p for the transmitter release decreases by approximately a factor of two. The Fl component of the cortical firing rate decreases after adaptation because after tonic decrease in the input modulated membrane potential, the over-threshold area of its FI component decreases. The adaptation in the Fl firing rate is fed back via the recurrent excitatory connections resulting in the observable adaptation in the FI membrane potential. Without lateral feedback (Fig. 3) the Fl component of the membrane potential is basically independent of the contrast adaptation. At high release probability a steep rise of the EPSC to a high amplitude peak is followed by rapid depression if the input is increasing. At low release probability the current increases slower to a lower amplitude, but the depression is Contrast Adaptation and lnfomax 81 ,......, u ,......, u 25 30 -aE 20 tU tU -. <Jl .? 20 ......... ~ - 10 .-1Z ~ ~ btl btl t: t: ?c 0 102 (c) 10? ,......, 102 10 1 Contrast [%] 102 0 i-20 ~ U 10 1 0 >-54 E ......... o0.. 5 0 (a) 10? ~ 1Z .-.5-58 .... 15 10 '"0 ....... tU ~ ...... -40 r..-...---..-__ -.l ~ -60 -o--a-o ___ .(). __ 0-62 10? (b) Figure 3 ~ 10 1 Contrast [%] 102 -80 10? (d) 10 1 Contrast [%] 102 -60 L--~~_~...-l 10? 10 1 102 (b) Contrast [%] Figure 4 Figure 3: The membrane potential (a, b), the phase (d) of the Fl component of the firing rate, and the Fl component (c) averaged for the modeled cortical cells after adaptation to 1% (dashed lines) and 50% (solid lines) contrast. The weight of cortical connections is set to zero. The CRF for the membrane potential (a, b) is calculated by integrating Eq. 1 without spikes and without reset after spikes. Figure 4: Hysteresis curve revealed by following the ramp method protocol (Carandini & Ferster 1997). After adaption to I % contrast, test stimuli of 2 s duration were applied with a contrast successively increasing from 1% to 100% (asterisks). and then decreasing back to 1% (circles). less pronounced too. As a consequence, the power at the first harmonic (Fl component) of the subthreshold membrane potential does not change if the release probability is modulated. It is modulated to a large extent by the recurrent excitatory feedback . The adaptation of the Fl component of the firing rate could therefore be used to measure the effective strength of the recurrent excitatory input to a simple ceIl in the primary visual cortex. Additional simulations (data not shown) revealed that changing the transmitter release probability of the geniculocortical synapses is responsible for the adaptation in our model network. Fixing the value of p for the geniculocortical synapses abolishes contrast adaptation, while fixing the release probability p for the lateral synapses has no effect. Simulations show that increasing the release probability of the recurrent excitatory synapses leads to oscillatory activity (e.g. Senn et al. 1996) without altering the mean activity of simple cells. These results suggest an efficient functional segregation of feedforward and recurrent excitatory connections. Plasticity of the geniculocortical connections may playa key role in contrast adaptation, while-without affecting the CRF-plasticity of the recurrent excitatory synapses could could playa key role in dynamic feature binding and segregation in the visual cortex (e.g. Engel et al. 1997). Figure 4 shows the averaged CRF of the cortical model neurons revealed by the ramp method (see figure caption) for strong recurrent feedback and adapting feedforward and recurrent synapses. We find hysteresis curves for the Fl component of the firing rate simi- 82 P. Adorjan and K. Obermayer lar to the results reported by Carandini & Ferster (1997), and for the response phase. In summary, by assuming two different dynamics for a single synapse we explain the saturation of the CRFs, the contrast adaptation, and the increase in the delay of the cortical response to low contrast stimuli. For the visual cortex of higher mammals, adaptation of release probability p as a substrate for contrast adaptation is so far only a hypothesis. This hypothesis, however, is in agreement with the currently available data, and could additionally be justified experimentally by intracellular measurements of EPSPs evoked by stimulating the geniculocortical axons. The model predicts that after adaptation to a low contrast stimulus the amplitude of the EPSPs decreases steeply from a high value, while it shows only small changes after adaptation to a high contrast stimulus (cf. Fig. 1b). Acknowledgments The authors are grateful to Christian Piepenbrock for fruitful discussions. Funded by the German Science Foundation (Ob 102/2-1, GK 120-2). References Ahmed, B., Allison, J. D., Douglas, R. 1. & Martin, K A. C. (1997), 'Intracellular study of the contrast-dependence of neuronal activity in cat visual cortex.' , Cerebral Cortex 7,559-570. Barlow, H. B. & F6ldiak, P. (1989), Adaptation and decorrelation in the cortex, in R. Durbin, C. Miall & Mitchison, eds, 'The computing neuron', Workingham: Addison-Wesley, pp. 54-72. c. Bell, A. 1. & Sejnowski, T. J. (1995), 'An information-maximization approach to blind sepertation and blind deconvolution', Neur. Comput. 7(6),1129-1159. Carandini, M. & Ferster, D. (1997), 'A tonic hyperpolarization underlying contrast adaptation in cat visual cortex', Science 276, 949-952. Carandini, M., Heeger, D. J. & Movshon, 1. A. (1997), 'Linearity and normalization in simple cells of the macaque primary visual cortex' , J. Neurosci. 17,8621-8644. Carandini, M., Movshon, J. A. & Ferster, D. (1998), 'Pattern adaptation and cross-orientation interactions in the primary visual cortex' , Neuropharmacology 37, 501-51 I. Chance, F. S., Nelson, S. B. & Abbott, L. F. (1998), 'Synaptic depression and the temporal response characteristics of V I cells', 1. Neurosci. 18,4785-4799. Engel, A. K, Roelfsema, P. R., Fries, P.. Brecht, M. & Singer. W (1997), 'Role of the temporal domain for response selection and perceptual binding', Cerebral Cortex 7,571-582. Finlayson, P. G. & Cynader, M. S. (I 995), 'Synaptic depression in visual cortex tissue slices: am in vitro model for cortical neuron adaptation', Exp. Brain Res. 106, 145-155. Kaplan, E., Purpura, K & Shapley, R. M. (1987), 'Contrast affects the transmission of visual information through the mammalian lateral geniculate nucleus', J. PhysioL. 391, 267-288. Laughlin, S. B. (1994), 'Matching coding, circuits, cells, and molecules to signals: general principles of retinal design in the fly's eye', Prog. Ret. Eye Res. 13, 165-196. McLean, 1. & Palmer, L. A. (1996), 'Contrast adaptation and excitatory amino acid receptors in cat striate cortex', Vis. Neurosci. 13, 1069-1087. Ohzawa, I., ScJar, G . & Freeman , R. D . (1985), 'Contrast gain control in the cat's visual system', 1. Neurophysiol. 54, 651-667. Saul, A. B. (1995), 'Adaptation in single units in visual cortex: response timing is retarted by adapting', Vis. Neurosci. 12, 191-205. Senn, W, Wyler, K, Streit, J., Larkum, M., Luscher, H.-R., H. Mey, L. M. a. D. S ., Vogt, K & Wannier, T. (1996), 'Dynamics of a random neural network with synaptic depression', Neural Networks 9, 575-588. Stemmler, M. & Koch, C. (1999), Information maximization in single neurons, in 'Advances in Neural Information Processing Systems NIPS II'. same volume. Tsodyks , M. V. & Markram, H. (1997), 'The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability', Proc. NatL. Acad. Sci. 94, 719-723. Vidyasagar, T. R. (1990), 'Pattern adaptation in cat visual cortex is a co-operative phenomenon', Neurosci. 36, 175-179. 

1518 |@word vogt:1 simulation:3 mammal:1 solid:4 moment:1 initial:1 series:2 current:6 od:1 surprising:1 activation:1 physiol:1 additive:1 subsequent:1 interspike:1 plasticity:5 analytic:1 christian:1 piepenbrock:1 stationary:1 half:1 short:3 ire:1 fitting:1 shapley:1 wannier:1 expected:1 rapid:2 integrator:3 brain:1 freeman:1 decreasing:1 prolonged:2 increasing:3 underlying:1 linearity:1 circuit:2 cm:1 developed:1 ret:1 finding:1 temporal:3 nf:1 ro:1 control:1 unit:3 local:1 accordance:2 ceil:1 timing:1 consequence:2 acad:1 receptor:1 ak:1 firing:21 modulation:4 approximately:2 evoked:1 co:1 limited:1 palmer:2 range:2 averaged:3 acknowledgment:1 responsible:1 testing:1 signaling:1 area:1 bell:2 adapting:6 significantly:1 matching:1 integrating:2 suggest:3 cannot:1 close:2 selection:1 www:1 equivalent:1 fruitful:1 demonstrated:1 crfs:4 maximizing:3 duration:1 recovery:1 rule:6 oh:1 target:1 caption:1 substrate:1 hypothesis:2 agreement:1 fr2:1 mammalian:1 predicts:3 blocking:1 role:4 fly:1 rij:8 tsodyks:2 epsc:3 ensures:1 connected:1 decrease:9 environment:1 dynamic:6 weakly:1 grateful:1 cynader:2 neurophysiol:1 resolved:1 represented:1 cat:5 neurotransmitter:1 stemmler:2 train:2 fast:8 effective:3 sejnowski:2 oby:1 klaus:1 larger:2 ramp:2 statistic:2 propose:2 interaction:1 maximal:3 epsp:3 adaptation:54 reset:3 tu:5 fr:1 adapts:1 pronounced:1 transmission:1 converges:1 derive:2 recurrent:11 fixing:2 measured:1 ij:1 op:4 eq:3 strong:3 epsps:2 grating:1 c:2 involves:1 stochastic:1 viewing:1 transient:1 explains:1 summation:1 koch:2 around:1 exp:2 circuitry:1 major:1 estimation:1 proc:1 geniculate:11 currently:1 sensitive:1 largest:2 engel:2 weighted:1 gaussian:1 always:1 encode:1 release:25 derived:1 transmitter:15 steeply:1 contrast:62 am:1 dependent:1 el:1 unlikely:1 relation:1 lgn:1 germany:1 adorjan:2 orientation:1 constrained:1 mutual:4 field:2 stimulus:20 individual:1 phase:6 fire:2 maintain:1 conductance:4 allison:1 light:1 activated:1 natl:1 integral:1 tree:4 circle:2 re:3 plotted:1 earlier:1 modeling:1 sinusoidally:1 altering:1 maximization:2 deviation:5 delay:2 too:1 reported:1 my:3 peak:1 stay:1 infomax:3 successively:1 mclean:2 derivative:1 account:2 potential:19 suggesting:1 de:2 retinal:1 alteration:1 coding:1 hysteresis:2 caused:1 mv:3 vi:5 depends:2 blind:2 steeper:1 portion:2 slope:1 ni:1 acid:1 characteristic:1 maximized:1 subthreshold:3 ofthe:1 weak:1 plastic:1 basically:1 tissue:1 explain:3 synapsis:10 strongest:1 oscillatory:1 whenever:1 synaptic:21 ed:1 frequency:5 pp:1 conveys:1 recovers:1 static:1 gain:1 carandini:12 amplitude:6 back:2 wesley:1 higher:4 response:13 synapse:4 furthermore:1 hand:2 receives:1 tre:1 lar:1 effect:3 hypothesized:1 ohzawa:2 normalized:1 barlow:2 evolution:1 eg:1 during:2 steady:2 m:5 crf:9 neocortical:1 fj:5 image:3 harmonic:1 fi:5 stimulation:1 spiking:3 hyperpolarization:1 vitro:1 functional:1 refractory:1 exponentially:1 volume:1 jl:1 cerebral:2 adp:1 resting:1 interpret:1 neuropharmacology:1 measurement:1 of2:1 funded:1 cortex:19 operating:1 inhibition:1 playa:2 recent:1 foldiak:1 driven:1 scenario:1 additional:1 maximize:2 period:2 dashed:4 ii:4 signal:1 hebbian:3 technical:1 characterized:1 ahmed:3 adapt:1 long:1 cross:1 fjj:3 ae:1 poisson:2 normalization:1 cell:18 justified:1 background:2 affecting:1 operative:1 interval:1 pyramidal:1 source:1 ot:2 rest:1 ascent:1 hz:9 induced:1 spirit:1 revealed:3 feedforward:2 affect:2 simi:1 brecht:1 shift:6 motivated:1 expression:1 o0:1 movshon:2 peter:1 depression:10 amount:2 locally:1 http:1 senn:2 estimated:2 express:1 key:2 threshold:2 nevertheless:1 drawn:1 changing:3 douglas:1 abbott:1 btl:2 luminance:2 prob:3 prog:1 roelfsema:1 ob:1 interleaved:1 fl:12 followed:2 durbin:1 activity:5 strength:4 ri:2 relatively:3 martin:1 according:2 neur:1 gaba:1 membrane:19 smaller:1 slightly:1 postsynaptic:5 explained:1 resource:5 equation:1 segregation:2 german:1 mechanism:3 singer:1 addison:1 fed:1 available:4 apply:1 fry:1 slower:1 drifting:1 cf:2 retardation:1 objective:1 quantity:1 spike:10 occurs:1 strategy:1 primary:7 fa:2 receptive:2 dependence:1 striate:1 obermayer:3 gradient:1 berlin:4 capacity:1 lateral:7 oa:1 sci:1 nelson:1 presynaptic:15 extent:1 geniculocortical:5 finlayson:2 assuming:2 code:1 modeled:3 esyn:2 setup:1 steep:1 gk:1 obennayer:1 kaplan:2 rise:2 f6ldiak:1 design:1 neuron:19 anti:2 saturates:4 tonic:2 paradox:1 dc:6 intensity:2 connection:6 macaque:1 nip:1 below:1 pattern:2 saturation:3 including:1 max:1 vidyasagar:2 event:2 power:1 decorrelation:1 glutamate:1 eye:3 mediated:1 relative:1 fully:1 proportional:3 asterisk:2 foundation:1 integrate:2 nucleus:1 degree:1 pij:11 lnfomax:1 principle:1 pi:1 excitatory:12 summary:1 last:1 free:1 side:1 laughlin:2 wide:1 saul:2 markram:2 absolute:1 leaky:3 distributed:1 slice:1 feedback:4 calculated:4 cortical:23 evaluating:1 world:1 curve:2 author:1 far:1 miall:1 eak:1 alpha:1 approximate:1 observable:1 mitchison:1 purpura:1 additionally:1 transfer:1 molecule:1 ioo:1 protocol:2 domain:1 intracellular:2 neurosci:5 noise:1 mediate:1 arrival:1 profile:2 amino:1 neuronal:1 fig:11 slow:3 axon:1 n:2 heeger:1 comput:1 perceptual:1 third:1 c1o:1 specific:1 deconvolution:1 modulates:1 epscs:1 entropy:1 explore:1 ganglion:1 visual:24 saccade:1 binding:2 corresponds:1 chance:3 adaption:1 stimulating:1 goal:1 presentation:2 towards:3 ferster:9 change:8 experimentally:2 determined:3 justify:1 experimental:5 modulated:5 tpeak:1 dept:1 phenomenon:2 ex:2

Controlling the Complexity of HMM Systems by Regularization Christoph Neukirchen, Gerhard Rigoll Department of Computer Science Gerhard-Mercator-University Duisburg 47057 Duisburg, Germany email: {chn.rigoll}@fb9-ti.uni-duisburg.de Abstract This paper introduces a method for regularization ofHMM systems that avoids parameter overfitting caused by insufficient training data. Regularization is done by augmenting the EM training method by a penalty term that favors simple and smooth HMM systems. The penalty term is constructed as a mixture model of negative exponential distributions that is assumed to generate the state dependent emission probabilities of the HMMs. This new method is the successful transfer of a well known regularization approach in neural networks to the HMM domain and can be interpreted as a generalization of traditional state-tying for HMM systems. The effect of regularization is demonstrated for continuous speech recognition tasks by improving overfitted triphone models and by speaker adaptation with limited training data. 1 Introduction One general problem when constructing statistical pattern recognition systems is to ensure the capability to generalize well, i.e. the system must be able to classify data that is not contained in the training data set. Hence the classifier should learn the true underlying data distribution instead of overfitting to the few data examples seen during system training. One way to cope with the problem of overfitting is to balance the system's complexity and flexibility against the limited amount of data that is available for training. In the neural network community it is well known that the amount of information used in system training that is required for a good generalization performance should be larger than ,the number of adjustable weights (Baum, 1989). A common method to train a large size neural network sufficiently well is to reduce the number of adjustable parameters either by removing those weights that seem to be less important (in (Ie Cun, 1990) the sensitivity of individual network weights is estimated by the second order gradient) or by sharing C. Neukirchen and G. Rigoll 738 the weights among many network connections (in (Lang, 1990) the connections that share identical weight values are determined in advance by using prior knowledge about invariances in the problem to be solved). A second approach to avoid overfitting in neural networks is to make use of regularization methods. Regularization adds an extra term to the training objective function that penalizes network complexity. The simplest regularization method is weight decay (Plaut, 1986) that assigns high penalties to large weights. A more complex regularization term is used in soft weight-sharing (Nowlan, 1992) by favoring neural network weights that fall into a finite set of small weight-clusters. The traditional neural weight sharing technique can be interpreted as a special case of soft weight-sharing regularization when the cluster variances tend towards zero. In continuous speech recognition the Hidden Markov Model (HMM) method is common. When using detailed context-dependent triphone HMMs, the number ofHMM-states and parameters to estimate in the state-dependent probability density functions (pdfs) is increasingly large and overfitting becomes a serious problem. The most common approach ,to balance the complexity of triphone HMM systems against the training data set is to reduce the number of parameters by tying, i.e. parameter sharing (Young, 1992). A popular sharing method is state-tying with selecting the HMM-states to be tied in advance, either by data-driven state-clustering based on a pdf-dependent distance metric (Young, 1993), or by constructing binary decision trees that incorporate higher phonetic knowledge (Bahl, 1991). In these methods, the number of state-clusters and the decision tree sizes, respectively, must be chosen adequately to match the training data size. However, a possible drawback of both methods is that two different states may be selected to be tied (and their pdfs are forced to be identical) although there is enough training data to estimate the different pdfs of both states sufficiently well. In the following, a method to reduce the complexity of general HMM systems based on a regularization term is presented. Due to its close relationship to the soft weight-sharing method for neural networks this novel approach can be interpreted as soft state-tying. 2 Maximum likelihood training in HMM systems Traditionally, the method most commonly used to determine the set of adjustable parameters 8 in a HMM system is maximum likelihood (ML) estimation via the expectation maximization (EM) algorithm. If the training observation vector sequence is denoted as X = (x(l), ... ,x(T)) and the corresponding HMM is denoted as W the ML estimator is given by: {)ML = argmax {logpe(XIW)} (1) () In the following, the total number of different HMM states is given by K. The emission pdf ,of the k-th state is denoted as bk (x); for continuous HMMs bk (x) is a mixture of Gaussian pdfs most commonly; in the case of discrete HMMs the observation vector x is mapped by a vector quantizer (VQ) on the discrete VQ-Iabel m(x) and the emission pdfis replaced by the discrete output probability bk (m). By the forward-backward algorithm the probabilistic state counts rdt) can be determined for each training observation and the log-likelihood over the training data can be decomposed into the auxiliary function Q(8) optimized in the EM steps (state transition probabilities are neglected here): T Q(8) = K L L rk(t) ?logbk(x(t)) (2) t=l k=l Sometimes, the observation vector x is split up into several independent streams. If the total number of streams is given by Z, the features in the z-th stream comprise the subvector x(z) and in the case of application ofa VQ the corresponding VQ label is denoted as m(z) (x(z?). 739 Controlling the Complexity ofHMM Systems by Regularization The observation subvectors in different streams are assumed to be statistically independent thus the states' pdfs can be written as: Z bk(x) = II b~z)(x(z?) (3) z=l 3 A complexity measure for HMM systems When using regularization methods to train the HMM system, the traditional objective training function Q(0) is augmented by a complexity penalization term 0 and the new optimization problem becomes: (reg = argmax {Q(0) (} + v? 0(0)} (4) Here, the regulizer term 0 should be small if the HMM system has high complexity and parameter overfitting becomes a problem; 0 should be large if the HMM-states' pdfs are shaped smoothly and system generalization works well. The constant v 2: 0 is a control parameter that adjusts the tradeoff between the pure ML solution and the smoothness of penalization. In Eqn. (4) the term Q(0) becomes larger the more data is used for training (which makes the ML estimation become more reliable) and the influence of the term v? 0 gets less important, relatively. The basic idea when constructing an expression for the regulizer 0 that favors smooth HMM systems is, that in the case of simple and smooth systems the state-dependent emission pdfs bk (.) should fall into several groups of similar pdfs. This is in contrast to the traditional state-tying that forces identical pdfs in each group. In the following, these clusters of similar emission pdfs are described by a probabilistic mixture model. Each pdf is assumed to be generated by a mixture of I different mixture components Pi (. ). In this case the probability (-density) of generating the emission pdf bk (.) is given by: I p(bkO) = L Ci . Pi(bk(?)) (5) i=l with the mixture weights Ci that are constrained to 0 ::::; Ci ::::; 1 and 1 = 2::=1 ci. The i-th mixture component Pi (.) is used to model the i-th cluster of HMM-emission pdfs. Each cluster is represented by a prototype pdf that is denoted as fJi (.) for the i-th cluster; the distance (using a suitable metric) between a HMM emission pdf bk 0 and the i-th prototype pdfis denoted as Di(bk (.)). If these distances are small for all HMM emission probabilities there are several small clusters of emission probabilities and the regulizer term 0 should be large. Now, it is assumed that the distances follow a negative exponential distribution (with a deviation parameter Ai), yielding an expression for the mixture components: p; (b.O) - (g A;,,) . exp ( - ~ A;" . D;" (bh)) ) (6) In Eqn. (6) the more general case of Z independent streams is given. Hence, the HMM emission pdfs and the cluster prototype pdfs are split up into Z different pdfs b~) (.) and fJ;Z) (.), respectively and the stream dependent distances D i,z and parameters Ai,z are used. Now, for the regulizer term 0 the log-likelihood of the mixture model in Eqn. (5) over all emission pdfs in the HMM system can be used: K 0(0) = L logp(b k=l k (?)) (7) 740 4 C. Neukirchen and G. Rigoll Regularization example: discrete HMMs As an example for parameter estimation in the regularization framework, a discrete HMM system with different VQs for each of the Z streams is considered here: Each VQ subdivides the feature space into J z different partitions (i.e. the z-th codebook size is J z) and the VQ-partition labels are denoted m)z) . If the observation subvector x (z) is in the j-th VQ-partition the VQ output is m(z) (x(z)) = m)Z). Since the discrete kind HMM output probabilities b~\m(z)) are used here, the regulizer's , prototypes are the discrete probabilities (3~z) (m (z) ). As a distance metric between the HMM emission probabilities and the prototype probabilities used in Eqn. (6) the asymmetric Kullback-Leibler divergence is applied: (8) 4.1 Estimation of HMM parameters using regularization The parameter set e of the HMM system to be estimated mainly consists of the discrete HMM emission probabilities (transition probabilities are not subject of regularization here). To get an iterative parameter estimation in the EM style, Eqn. (4) must be maximized; e.g. by setting the derivative of Eqn. (4) with respect to the HMM -parameter b~) (m )z) ) to zero and application of Lagrange multipliers with regard to the constraint 1 = EJ~ 1 biz)(m ;z)) . This leads to a quite complex solution that can be only solved numerically. The optimization problem can be simplified if the mixture in Eqn. (5) is replaced by the maximum approximation; i.e. only the maximum component in the sum is considered. The corresponding index of the maximum component is denoted i * : (9) In this simplified case the HMM parameter estimation is given by: (10) This is a weighted sum of the well known ML solution and the regulizer's prototype probability (3i~ (.) that is selected by the maximum search in Eqn. (9). The larger the value ofthe constant II, the stronger is the force that pushes the estimate of the HMM emission probability biz) (m ;z)) towards the prototype probability (3i~ (.). The situation when II tends towards infinity corresponds to the case of traditional state-tying, because all different states that fall into the same cluster i* make use of (3i~ (.) as emission probability in the z-th stream. 4.2 Estimation of regulizer parameters The parameter set ~ of the regulizer consists of the mixture weights Ci, the deviation parameters Ai,z , and of the discrete prototype probabilities (3~z) (m ;z) ) in the case of regulizing 741 Controlling the Complexity ofHMM Systems by Regularization discrete HMMs. These parameters can be set in advance by making use of prior knowledge; e.g. the prototype probabilities can be obtained from a simple HMM system that uses a small number of states. Alternatively, the regulizer's parameters can be estimated in a similar way as in (Nowlan, 1992) by maximizing Eqn. (7). Since there is no direct solution to this optimization problem, maximization must be performed in an EM-like iterative procedure that uses the HMM emission pdfs bk (.) as training data for the mixture model and by increasing the following auxiliary function in each step: R(~) K I k=1 i=1 K I k=1 i=1 L L P(ilbk(?)) ?logp(i, bk(?)) = L L P(ilb k(?)) . log (Ci . Pi(bk(?))) (11) with the posterior probability used as weighting factor given by: P(ilb k (.)) ICi . Pi(bk(')) 2:: 1=1 Cl . Pl(bk(?)) (12) Again, maximization of Eqn. (11) can be performed by setting the derivative of R(~) with respect to the regulizer's parameters to zero under consideration of the constraints ci and 1 = f3~Z)(m~Z)) by application of Lagrange multipliers. For the es1= timation of the regulizer parameters this yields: 2:::=1 o = 2:::::1 K ~ .L Ci = P(ilb k (-)) (13) 2:::=1 P(ilbk(?)) (14) k=1 ~. _ ~,z ~(z)( i 2:::=1 Di,z(b~) (.)) . P(ilbk(-)) exp ( (z)) _ mj - ~ ~exp 2:::=1 P(ilbk(')) 'IOgb~)(m)Z))) K. 2::k=1 P(zlbk(')) P(llb k (.)) 'IOgb~)(m}Z))) (2:::=1 1=1 (15) K 2::k=1 P(llb k (?)) The estimate Ci can be interpreted as the a:-erage probability that a HMM emission probability falls into the i-th mixture cluster; Ai,z is the inverse ofthe weighted average distance between the emission probabilities and the prototype probability f3;z) ( .). The estimate ~;z)(m)zl) is the average probability over all emission probabilities for the VQ-label m~zl weighted in the log-domain. If the Euclidean distance between the discrete probabilities is used instead of Eqn. (8) to measure the differences between the HMM emission probabilities and the prototypes Di 'Z (b~)(m(zl)) = Jz L (f3jz\myl) - b~z)(m;Zl)) 2 (16) j=1 the estimate of the prototype probabilities is given by the average of the HMM probabilities weighted in the original space: (17) 742 5 C. Neukirchen and G. Rigo/l Experimental results To investigate the performance of the regularization methods described above a HMM speech recognition system for the speaker-independent resource management (RM) continuous speech task is built up. For training 3990 sentences from 109 different speakers are used. Recognition results are given as word error rates averaged over the official DARPA RM test sets feb'89, oct'89, feb'91 and sep'92, consisting of 1200 sentences from 40 different speakers, totally. Recognition is done via a beam search guided Viterbi decoder using the DARPA RM word pair grammar (perplexity: 60). 'As acoustic features every 10 ms 12 MFCC coefficients and the relative signal power are extracted from the speech signal along with the dynamic ~- and ~~-features, comprising 39 features per frame. The HMM system makes use of standard 3-state discrete probability phonetic models. Four different neural networks, trained by the MMI method, that is described in in (Rigoll, 1997) and extended in (Neukirchen, 1998), are used as VQ to quantize the features into Z = 4 different streams of discrete labels. The codebook size in each stream is set to 200. A simple system with models for 47 monophones and for the most prominent 33 function words (totally 394 states) yields a word error rate of 8.6%. A system that makes use of the more detailed (but untied) word internal triphone models (totally 6921 states) yields 12.2% word error. Hence HMM overfitting because of insufficient training data is a severe problem in this case. Traditional methods to overcome the effects of overfitting like interpolating between triphones and monophones (Bahl, 1983), data driven state-clustering and decision tree clustering yield error rates of 6.5%, 8.3% and 6.4%, respectively. It must be noted that in contrast to the usual training procedure in (Rigoll, 1996) no further smoothing methods are applied to the HMM emission probabilities here. In a first series of experiments the untied triphone system is regulized by a quite simple mixture of I = 394 density components, i.e. the number of clusters in the penalty term is identical to the number of states in the monophone system. In this case the prototype probabilities are initialized by the emission probabilities of the monophone system; the mixture weights and the deviation parameters in the regulizer are set to be uniform, initially. In order to test the inluence of the tradeoff parameter v it is set to 50, 10 and 2, respectively. The corresponding word error rates are 8.4%, 6.9% and 6.3%, respectively. In the case of large vs regularization degrades to a tying of trip hone states to monophone states and ,the error rate tends towards the monophone system performance. For smaller vs there is a good tradeoff between data fitting and HMM smoothness yielding improved system performance. The initial prototype probability settings provided by the monophone system do not seem to be changed much by regulizer parameter estimation, since the system performance only changes slightly when the regulizer's parameter reestimation is not incorporated. In preliminary experiments the regularization method is also used for speaker adaptation. A speaker-independent system trained on the Wall Street Journal (WSJ) database yields an error rate of32.4% on the Nov. 93 S33>0 test set with 10 different non-native speakers. The speaker-independent HMM emission probabilities are used to initialize the prototype probabilities of the regulizer. Then, speaker-dependent systems are built up for each speaker using only 40 fast enrollment sentences for training along with regularization (v is set to 10). Now, the error rate drops to 25.7% what is better than the speaker adaptation method described in (Rottland, 1998) that yields 27.3% by a linear feature space transformation. In combination both methods achieve 23.0% word error. 6 Summary and Discussion A method to avoid parameter overfitting in HMM systems by application of a regularization term that favor smooth and simple models has been presented here. The complexity Controlling the Complexity of HMM Systems by Regularization 743 measure applied to the HMMs is based on a finite mixture of negative exponential distributions, that generates the state-dependent emission probabilities. This kind of regularization term can be interpreted as a soft state-tying, since it forces the HMM emission probabilities to form a finite set of clusters. The effect of regularization has been demonstrated on the RM task by improving overfitted trip hone models. On a WSJ non-native speaker adaption task with limited training data, regularization outperforms feature space transformations. Eqn. (4) may be also interpreted from a perspective of Bayesian inference: the term v . n plays the role of setting a prior distribution on the HMM parameters to be estimated. Hence, the use of a mixture model for n is equivalent to using a special kind of prior in the framework of MAP estimation for HMMs (Gauvain, 1994). References L.R. Bahl, F. Jelinek, L.R. Mercer, 'A Maximum Likelihood Approach to Continuous Speech Recognition', IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 5, No.2 Mar. 1983, pp. 179-190. L.R. Bahl, P.v. de Souza, P.S. Gopalakrishnan,D. Nahamoo, M.A. Picheny, (1991) Context dependent modeling of phones in continuous speech using decision trees. Proc. DARPA speech and natural language processing workshop, 264-270. E.B. Baum, D. Haussler, (1989) What size net gives valid generalization? Neural Computation, 1:151-160. Y. Ie Cun, J. Denker, S. Solla, R.E. Howard, L.D. Jackel, (1990) Optimal brain damage. Advances in Neural Information Processing Systems 2, San Mateo, CA, Morgan Kauffinan. J.L. Gauvain, C.R. Lee, (1994) Maximum a posteriori estimation for multivariate Gaussian mixture observations of markov chains. IEEE Transaction Speech and Audio Proc., Vol. 2, 2:291-298. KJ. Lang, A.H. Waibel, G.E. Hinton, (1990) A time-delay neural network architecture for isolated word recognition. Neural Networks, 3:23~3. Ch. Neukirchen, D. Willett, S. Eickeler, S. Muller, (1998) Exploiting acoustic feature correlations by joint neural vector quantizer design in a discrete HMM system. Proc. ICASSP'98,5-8. S.J. Nowlan, G.E. Hinton, (1992) Simplifying neural networks by soft weight-sharing. Neural Computation, 4:473~93. D.C. Plaut, S.J. Nowlan, G.E. Hinton, (1986) Experiments on learning by backpropagation. technical report CMU-CS-86-126, Carnegie-Mellon University, Pittsburgh, PA. G. Rigoll, Ch. Neukirchen, J. Rottland, (1996) A new hybrid system based on MMI-neural networks for the RM speech recognition task. Proc. ICASSP'96, 865-868. G. Rigoll, Ch. Neukirchen, (1997) A new approach to hybrid HMMIANN speech recognition using mutual information neural networks. Advances in Neural Information Processing Systems 9, Cambridge, MA, MIT Press, 772-778. J. Rottland, Ch. Neukirchen, G. Rigoll, (1998) Speaker adaptation for hybrid MMIconnectionist speech recognition systems. Pmc. ICASSP '98, 465~68. 'S.J. Young, (1992) The general use of tying in phoneme-based HMM speech recognizers. Proc. ICASSP '92, 569- 572. SJ. Young, P.C. Woodland (1993) The use of state tying in continuous speech recognition. Proc. Eurospeech '93, 2203-2206. 

1519 |@word stronger:1 simplifying:1 initial:1 series:1 selecting:1 outperforms:1 nowlan:4 gauvain:2 lang:2 must:5 written:1 partition:3 drop:1 v:2 intelligence:1 selected:2 quantizer:2 plaut:2 codebook:2 along:2 constructed:1 direct:1 become:1 consists:2 fitting:1 iogb:2 brain:1 decomposed:1 subvectors:1 increasing:1 becomes:4 totally:3 provided:1 underlying:1 s33:1 what:2 tying:10 kind:3 interpreted:6 transformation:2 every:1 ti:1 ofa:1 classifier:1 rm:5 control:1 zl:4 tends:2 mercator:1 mateo:1 christoph:1 hmms:8 limited:3 statistically:1 averaged:1 backpropagation:1 procedure:2 word:9 get:2 close:1 bh:1 context:2 influence:1 rottland:3 equivalent:1 map:1 demonstrated:2 baum:2 maximizing:1 assigns:1 pure:1 estimator:1 adjusts:1 haussler:1 traditionally:1 controlling:4 play:1 gerhard:2 us:2 pa:1 recognition:12 asymmetric:1 native:2 database:1 role:1 solved:2 solla:1 overfitted:2 complexity:12 neglected:1 dynamic:1 trained:2 subdivides:1 sep:1 darpa:3 joint:1 icassp:4 represented:1 train:2 forced:1 fast:1 quite:2 larger:3 grammar:1 favor:3 sequence:1 net:1 adaptation:4 flexibility:1 achieve:1 exploiting:1 cluster:13 generating:1 wsj:2 augmenting:1 auxiliary:2 c:1 guided:1 drawback:1 generalization:4 wall:1 preliminary:1 pl:1 sufficiently:2 considered:2 exp:3 viterbi:1 estimation:10 proc:6 label:4 jackel:1 weighted:4 mit:1 gaussian:2 avoid:2 ej:1 es1:1 emission:26 pdfs:16 likelihood:5 mainly:1 contrast:2 posteriori:1 inference:1 dependent:9 initially:1 hidden:1 favoring:1 comprising:1 germany:1 among:1 denoted:8 constrained:1 special:2 smoothing:1 initialize:1 mutual:1 comprise:1 f3:2 shaped:1 identical:4 report:1 ilb:3 serious:1 few:1 divergence:1 individual:1 hmmiann:1 replaced:2 argmax:2 consisting:1 investigate:1 severe:1 introduces:1 mixture:18 yielding:2 chain:1 tree:4 euclidean:1 penalizes:1 initialized:1 isolated:1 monophone:5 neukirchen:9 classify:1 soft:6 modeling:1 enrollment:1 rigo:1 logp:2 maximization:3 deviation:3 uniform:1 delay:1 successful:1 eurospeech:1 density:3 sensitivity:1 ie:2 probabilistic:2 lee:1 again:1 management:1 derivative:2 style:1 de:2 coefficient:1 caused:1 stream:10 reg:1 performed:2 capability:1 variance:1 phoneme:1 maximized:1 yield:6 ofthe:2 generalize:1 bayesian:1 mmi:2 mfcc:1 sharing:8 email:1 rdt:1 against:2 pp:1 di:3 popular:1 knowledge:3 higher:1 follow:1 improved:1 done:2 mar:1 correlation:1 eqn:12 bahl:4 biz:2 effect:3 true:1 multiplier:2 adequately:1 regularization:27 hence:4 leibler:1 during:1 speaker:13 noted:1 m:1 prominent:1 pdf:6 fj:1 consideration:1 novel:1 common:3 bko:1 numerically:1 willett:1 mellon:1 cambridge:1 ai:4 smoothness:2 language:1 recognizers:1 add:1 feb:2 triphones:1 posterior:1 multivariate:1 perspective:1 driven:2 perplexity:1 phone:1 phonetic:2 binary:1 muller:1 seen:1 morgan:1 triphone:5 determine:1 signal:2 ii:3 smooth:4 technical:1 match:1 basic:1 metric:3 expectation:1 cmu:1 sometimes:1 llb:2 beam:1 extra:1 subject:1 tend:1 seem:2 split:2 enough:1 architecture:1 reduce:3 idea:1 prototype:15 tradeoff:3 vqs:1 rigoll:9 expression:2 penalty:4 monophones:2 speech:14 woodland:1 detailed:2 amount:2 simplest:1 generate:1 estimated:4 per:1 discrete:14 carnegie:1 vol:2 group:2 four:1 duisburg:3 backward:1 sum:2 inverse:1 decision:4 nahamoo:1 constraint:2 infinity:1 untied:2 generates:1 relatively:1 department:1 waibel:1 combination:1 smaller:1 slightly:1 em:5 increasingly:1 cun:2 making:1 resource:1 vq:10 count:1 fji:1 available:1 denker:1 original:1 clustering:3 ensure:1 objective:2 xiw:1 degrades:1 damage:1 usual:1 traditional:6 gradient:1 distance:8 mapped:1 hmm:48 decoder:1 street:1 gopalakrishnan:1 index:1 relationship:1 insufficient:2 balance:2 pmc:1 negative:3 design:1 adjustable:3 observation:7 markov:2 hone:2 howard:1 finite:3 situation:1 extended:1 incorporated:1 hinton:3 frame:1 community:1 souza:1 bk:14 pair:1 required:1 subvector:2 trip:2 connection:2 optimized:1 sentence:3 acoustic:2 trans:1 able:1 pattern:2 built:2 reliable:1 power:1 suitable:1 natural:1 force:3 hybrid:3 kj:1 prior:4 relative:1 ofhmm:4 penalization:2 mercer:1 share:1 pi:5 changed:1 summary:1 fall:4 jelinek:1 regard:1 overcome:1 transition:2 avoids:1 valid:1 forward:1 commonly:2 ici:1 simplified:2 san:1 cope:1 transaction:1 picheny:1 nov:1 sj:1 uni:1 kullback:1 ml:6 overfitting:9 reestimation:1 pittsburgh:1 assumed:4 alternatively:1 continuous:7 iterative:2 search:2 timation:1 learn:1 transfer:1 mj:1 jz:1 ca:1 improving:2 quantize:1 complex:2 cl:1 constructing:3 domain:2 official:1 interpolating:1 augmented:1 exponential:3 tied:2 weighting:1 young:4 removing:1 rk:1 decay:1 workshop:1 ci:9 push:1 chn:1 smoothly:1 lagrange:2 contained:1 ch:4 corresponds:1 adaption:1 extracted:1 ma:1 oct:1 towards:4 change:1 determined:2 total:2 invariance:1 experimental:1 internal:1 incorporate:1 audio:1

436 SIMULATION AND MEASUREMENT OF THE ELECTRIC FIELDS GENERATED BY WEAKLY ELECTRIC FISH Brian Rasnow 1, Christopher Assad2, Mark E. Nelson3 and James M. Bow~ Divisions of Physics1,Elecbical Engineerini, and Biolo~ Caltech, Pasadena, 91125 ABSTRACT The weakly electric fish, Gnathonemus peters;;, explores its environment by generating pulsed elecbic fields and detecting small pertwbations in the fields resulting from nearby objects. Accordingly, the fISh detects and discriminates objects on the basis of a sequence of elecbic "images" whose temporal and spatial properties depend on the timing of the fish's electric organ discharge and its body position relative to objects in its environmenl We are interested in investigating how these fish utilize timing and body-position during exploration to aid in object discrimination. We have developed a fmite-element simulation of the fish's self-generated electric fields so as to reconstruct the electrosensory consequences of body position and electric organ discharge timing in the fish. This paper describes this finite-element simulation system and presents preliminary electric field measurements which are being used to tune the simulation. INTRODUCTION The active positioning of sensory structures (i.e. eyes, ears, whiskers, nostrils, etc.) is characteristic of the information seeking behavior of all exploratory animals. Yet, in most existing computational models and in many standard experimental paradigms, the active aspects of sensory processing are either eliminated or controlled (e.g. by stimulating fIXed groups of receptors or by stabilizing images). However, it is clear that the active positioning of receptor surfaces directly affects the content and quality of the sensory infonnation received by the nervous system. Thus. controlling the position of sensors during sensory exploration constitutes an important feature of an animals strategy for making sensory discriminations. Quantitative study of this process could very well shed light on the algorithms and internal representations used by the nervous system in discriminating peripheral objects. Studies of the active use of sensory surfaces generally can be expected to pose a number of experimental challenges. This is because, in many animals, the sensory surfaces involved are themselves structurally complicated, making it difficult to reconstruct p0sition sequences or the consequences of any repositioning. For example, while the sen- Simulation and Measurement of the Weakly Electric Fish sory systems of rats have been the subjects of a great deal of behavioral (Welker, 1964) and neurophysiological study (Gibson & Welker, 1983), it is extremely difficult to even monitor the movements of the perioral surfaces (lips, snout, whiskers) used by these animals in their exploration of the world let alone reconstruct the sensory consequences. For these reasons we have sought an experimental animal with a sensory system in which these sensory-motor interactions can be more readily quantified. The experimental animal which we have selected for studying the control of sensory surface position during exploration is a member of a family of African freshwater fish (Monniridae) that use self-generated electric fields to detect and discriminate objects in their environment (Bullock & Heiligenberg, 1986). The electrosensory system in these fish relies on an "electric organ" in their tails which produces a weak pulsed electric field in the surrounding environment (significant within 1-2 body lengths) that is then detected with an array of electrosensors that are extremely sensitive to voltage drops across the skin. These "electroreceptors" allow the fISh to respond to the perturbations in the electric field resulting from objects in the environment which differ in conductivity from the surrounding water (Fig. 1). . . conducting ? object IIID electric organ ? electroreceptors electric field lines Figure 1. The peripheral electrosensory system of Gnathonemus petersii consists of an "electric organ" current source at the base of the tail and several thousand "electroreceptor" cells distributed non uniformly over the fish's body. A conducting object near the fish causes a local increase in the current through the skin. These fISh are nocturnal, and rely more on their electric sense than on any other sensory system in perfonning a wide range of behaviors (eg. detecting and localizing objects such as food). It is also known that these fish execute exploratory movements, changing their body position actively as they attempt an electrosensory discrimination (Toerring & Belbenoit, 1979). Our objective is to understand how these movements change the distribution of the electric field on the animals skin, and to determine what, if any, relationship this has to the discrimination process. There are several clear advantages of this system for our studies. First, the electrore- 437 438 Rasnow, Assad, Nelson and Bower ceptors are in a fixed position with respect to each other on the surface of the animal. Therefore, by knowing the overall body position of the animal it is possible to know the exact spatial relationship of electroreceptors with respect to objects in the environment. Second, the physical equations governing the self-generated electric fIeld in the fish's environment are well understood. As a consequence, it is relatively straightforward to reconstruct perturbations in the electric field resulting from objects of different shape and conductance. Third, the electric potential can be readily measured, providing a direct measure of the electric field at a distance from the fish which can be used to reconstruct the potential difference across the animals skin. And finally, in the particular species of fish we have chosen to work with, Gnathonemus petersii, individual animals execute a brief (100 J.1Sec) electric organ discharge (BOD) at intervals of 30 msec to a few seconds. Modification of the firing pattern is 1cnown to be correlated with changes in the electrical environment (Lissmann, 1958). Thus, when the electric organ discharges, it is probable that the animal is interested in "taking a look" at its surroundings. In few other sensory systems is there as direct an indication of the attentional state of the subject. Having stated the advantages of this system for the study we have undertaken, it is also the case that considerable effort will still be necessary to answer the questions we have posed. For example, as described in this paper, in order to use electric field measurements made at a distance to infer the voltages across the surface of the animal's skin, it is necessary to develop a computer model of the fish and its environment. This will allow us to predict the field on the animal's skin surface given different body poSitions relative to objects in the environment. This paper describes our first steps in constructing this simulation system. Experimental Approach and Methods Simulations of Fish Electric Fields The electric potential, cll(x), generated by the EOD of a weakly electric fish in a fish tank is a solution ofPoisson's equation: Ve(pVell) = f where p(x)and f(x) are the impedance magnitude and source density at each point x inside and surrounding the fish. Our goal is to solve this equation for ell given the current source density, f, generated by the electric organ and the impedances, p, corresponding to the properties of the fish and external objects (rocks, worms, etc.). Given p and f. this equation can be solved for the potential ell using a variety of iterative approximation schemes. Iterative methods, in general, first discretize the spatial domain of the problem into a set of "node" points, and convert Poisson's equation into a set of algebraic equations with the nodal potentials as the unknown parameters. The node values, in this case, each represent an independent degree of freedom of the system and, as a consequence, there are as many equations as there are nodes. This very large system of equations can Simulation and Measurement of the Weakly Electric Fish then be solved using a variety of standard techniques, including relaxation methods, conjugate gradient minimization, domain decomposition and multi-grid methods. To simulate the electric fields generated by a fish, we currently use a 2-dimensional fmite element domain discretization (Hughes, 1987) and conjugate gradient solver. We chose the finite element method because it allows us to simulate the electric fields at much higher resolution in the area of interest close to the animal's body where the electric field is largest and where errors due to the discretization would be most severe. The fmite element method is based on minimizing a global function that corresponds to the potential energy of the electric field. To compute this energy, the domain is decomposed into a large number of elements, each with uniform impedance (see Fig. 2). The global energy is expressed as a sum of the contributions from each element, where the potential within each element is assumed to be a linear interpolation of the potentials at the nodes or vertices of each element The conjugate gradient solver determines the values of the node potentials which minimize the global energy function. 1\ IVrv'V 1\ 1\ rv:V J J 1\/1\ .J1\/ 1\11\/1\/1\11\/1\/1\ 1\11\/[\ 1\1 IV r--.. V v 7' v [7 If\ If\ '\ '\ V\ V If\ J\ 1'\ 'w l/\ V 11'\ '\ 1/ :1'\ '\ '\ '\ V '\ 1'\" '\ 11\1/\ \ '\ V Figure 2. The inner region of a fmite element grid constructed for simulating in 2-dimensions the electric field generated by an electric fish. Measurement of Fish Electric Fields Another aspect of our experimental approach involves the direct measurement of the potential generated by a fish's EOD in a fish tank using arrays of small electrodes and differential amplifiers. The electrodes and electronics have a high impedance which minimizes their influence on the electric fields they are designed to measure. The electrodes are made by pulling a 1mm glass capillary tube across a heated tungsten filament, resulting in a fine tapered tip through which a 1~ silver wire is run. The end of this wire is melted in a flame leaving a 200J,un ball below the glass insulation. Several electrodes are then mounted as an array on a microdrive attached to a modified X-Yplotter under computer control and giving better than 1mm positioning accuracy. Generated potentials are amplified by a factor of 10 - 100, and digitized at a rate of 100kHz per channel with a 12 bit AID converter using a Masscomp 5700 computer. An array processor searches this 439 440 Rasnow, Assad, Nelson and Bower continuous stream of data for EOD wavefonns. which are extracted and saved along with the position of the electrode array. Results Calibration of the Simulator In order to have confidence in the overall system, it was fD'St necessary to calibrate both the recording and the simulation procedures. To do this we set up relatively simple geometrical arrangements of sources and conductors in a fish tank for which the potential could be found analytically. The calibration source was an electronic "fake fish" circuit that generated signals resembling the discharge of Gnathonemus. Point current source A point source in a 2-dimensional box is perhaps the simplest configuration with which to initially test our electric field reconstruction system. The analytic solution for the potential from a point current source centered in a grounded. conducting 2-dimensional box is: . (.n7t). (n7tx). h (.n7ty ) 00 4>(x. y) = L n =1 sm("2 sm L sm \L ri1t n L cosh(T) Our fmite element simulation. based on a regular 80 x 80 node grid differs from the above expression by less than 1%. except in the elements adjacent to the source. where the potential change across these elements is large and is not as accurately reconstructed by a linear interpolation (Fig. 3). Smaller elements surrounding the source would improve the accuracy. however. one should note the analytic solution is infmite at the location of the "point" source whereas the measured and simulated sources (and real fish) have finite current densities. To measure the real equivalent of a point source in a 2-dimensional box. we used a linear current source (a wire) which ran the full depth of a real 3-dimensional tank. Measurements made in the midplane of the tank agree with the simulation and analytic solution to better than 5% (Fig. 3.). Uncertainty in the positions of the ClUTent source and recording sites relative to the position of the conducting walls probably accounts for much of this difference. Simulation and Measurement of the Weakly Electric Fish 1----~~--~--~----~------- o - measured x - simulated 00 2 4 6 8 10 12 14 16 dislaDce from source Figure 3. Electric potential of a point current source centered in a grounded 2-dimensional box. Measurements of Fish Fields and 2-Dimensional Simulations Calibration of our fmite element model of an electric fish requires direct measurements of the electric potential close to a discharging fish. Fig. 4 shows a recording of a single EOD sampled with 5 colinear electrodes near a restrained fish. The wavefonn is bipolar, with the fIrst phase positive if recorded near the animals head and negative if recorded near the tail (relative to a remote reference). We used the peak amplitude of the larger second phase of the wavefonn to quantify the electric potential recorded at each location. Note that the potential reverses sign at a point approximately midway along the tail. This location corresponds to the location of the null potential shown in Fig. 5. 1500 1000 $' 5500 I 0 -soo -1000 -1ro 200 ~sec Figure 4. EOD waveform sampled simultaneously from 5 electrodes. 441 442 Rasnow, Assad, Nelson and Bower Measurements of EODs from a restrained fish exhibited an extraordinarily small variance in amplitude and waveform over long periods of time. In fact, the peak-peak amplitude of the EOD varied by less than 0.4% in a sample of 40 EOD's randomly chosen during a 30 minute period. Thus we are able to directly compare waveforms sampled sequentially without renonnalizing for fluctuations in EOD amplitude. Figure 5 shows equipotential lines reconstructed from a set of 360 measurements made in the midplane of a restrained Gnathonemus. Although the observed potential resembles that from a purely dipolar source (Fig. 6), careful inspection reveals an asymmetry between the head and tail of the fISh. This asymmetry can be reproduced in our simulations by adjusting the electrical properties of the fish. Qualitatively, the measured fields can be reproduced by assigning a low impedance to the internal body cavity and a high impedance to the skin. However, in order to match the location of the null potential, the skin impedance must vary over the length of the body. We are currently quantifying these parameters, as described in the next section. !!!!m 1'!I!fl!IPf!~m !~ ...... II.. ?? 1.. . ...... 1 ??? Figure 5. Measured potentials (at peak of second phase of EOD) recorded from a restrained Gnathonemus petersii in the midplane of the fish. Equipotential lines are 20 mV apart. Inset shows relative location of fish and sampling points in the fISh tank. Figure 6. Equipotential lines from a 2-dimensional finite element simulation of a dipole using the grid of Fig. 2. The resistivity of the fish was set equal to that of the sWToundings in this simulation. Simulation and Measurement of the Weakly Electric Fish Future Directions There is still a substantial amount of work that remains to be done before we achieve our goal of being able to fully reconstruct the pattern of electroreceptor activation for any arbitrary body position in any particular environment. First. it is clear that we require more information about the electrical structure of the fISh itself. We need an accurate representation of the internal impedance distribution p(x) of the body and skin as well as of the source density f(x) of the electric organ. To some extent this can be addressed as an inverse problem, namely given the measured potential cl>(x), what choice of p(x) and f(x) best reproduces the data. Unfortunately, in the absence of further constraints, there are many equally valid solution, thus we will need to directly measure the skin and body impedance of the fish. Second, we need to extend our finite-element simulations of the fish to 3-dimensions which, although conceptually straight forward, requires substantial technical developments to be able to (a) specify and visualize the space-filling set of 3-dimensional finite-elements (eg. tetrahedrons) for arbitrary configurations, (b) compute the solution to the much larger set of equations (typically a factor of 100-1(00) in a reasonable time, and (c) visualize and analyze the resulting solutions for the 3-dimensional electrical fields. As a possible solution to (b), we are developing and testing a parallel processor implementation of the simulator. References Bullock, T. H. & Heiligenberg, W. (Eds.) (1986). "Electroreception", Wiley & Sons, New York. Gibson, J. M. & Welker. W. I. (1983). Quantitative Studies of Stimulus Coding in FirstOrder Vibrissa Afferents of Rats. 1. Receptive Field Properties and Threshold Distributions. Somatosensory Res. 1:51-67. Hughes, T. J. (1987). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, New Jersey. Lissmann. H.W. (1958). On the function and evolution of electric organs in fish. J. Exp. Bioi. 35:156-191. Toening, M. J. and Belbenoit. P. (1979). Motor Programmes and Electroreception in Monnyrid Fish. Behav. Ecol. Sociobiol. 4:369-379. Welker, W. I. (1964). Analysis of Sniffing of the Albino Rat Behaviour 22:223-244. 443 

152 |@word simulation:18 electrosensory:4 decomposition:1 electroreceptors:3 fmite:6 electronics:1 configuration:2 existing:1 current:8 discretization:2 activation:1 yet:1 assigning:1 must:1 readily:2 j1:1 midway:1 shape:1 analytic:3 motor:2 drop:1 designed:1 discrimination:4 alone:1 selected:1 nervous:2 accordingly:1 perioral:1 inspection:1 detecting:2 node:6 location:6 nodal:1 constructed:1 direct:4 differential:1 along:2 consists:1 behavioral:1 inside:1 expected:1 behavior:2 themselves:1 multi:1 simulator:2 detects:1 decomposed:1 food:1 solver:2 circuit:1 null:2 what:2 minimizes:1 developed:1 eod:9 temporal:1 quantitative:2 firstorder:1 shed:1 bipolar:1 ro:1 control:2 conductivity:1 discharging:1 positive:1 before:1 understood:1 timing:3 local:1 consequence:5 receptor:2 firing:1 interpolation:2 approximately:1 fluctuation:1 chose:1 resembles:1 quantified:1 tungsten:1 range:1 filament:1 testing:1 wavefonns:1 hughes:2 differs:1 insulation:1 procedure:1 area:1 gibson:2 confidence:1 regular:1 close:2 prentice:1 influence:1 heated:1 equivalent:1 resembling:1 straightforward:1 resolution:1 stabilizing:1 clutent:1 dipole:1 array:5 exploratory:2 discharge:5 controlling:1 exact:1 element:20 observed:1 electrical:4 solved:2 thousand:1 region:1 remote:1 movement:3 ran:1 substantial:2 discriminates:1 environment:10 dynamic:1 weakly:7 depend:1 colinear:1 purely:1 division:1 basis:1 jersey:1 surrounding:4 detected:1 extraordinarily:1 whose:1 posed:1 solve:1 larger:2 reconstruct:6 itself:1 reproduced:2 sequence:2 advantage:2 indication:1 rock:1 sen:1 reconstruction:1 interaction:1 bow:1 achieve:1 amplified:1 bod:1 electrode:7 asymmetry:2 produce:1 generating:1 silver:1 object:14 develop:1 pose:1 measured:6 received:1 electroreception:2 involves:1 somatosensory:1 revers:1 quantify:1 differ:1 direction:1 waveform:3 saved:1 exploration:4 centered:2 require:1 behaviour:1 wall:1 preliminary:1 brian:1 probable:1 mm:2 ecol:1 hall:1 exp:1 great:1 predict:1 visualize:2 sought:1 vary:1 currently:2 infonnation:1 sensitive:1 largest:1 organ:10 minimization:1 sensor:1 modified:1 voltage:2 sense:1 detect:1 glass:2 typically:1 initially:1 pasadena:1 interested:2 tank:6 overall:2 iiid:1 development:1 animal:16 spatial:3 ell:2 field:28 equal:1 having:1 eliminated:1 sampling:1 look:1 constitutes:1 filling:1 future:1 stimulus:1 few:2 surroundings:1 randomly:1 simultaneously:1 ve:1 individual:1 phase:3 attempt:1 freedom:1 conductance:1 amplifier:1 interest:1 fd:1 severe:1 light:1 electroreceptor:2 accurate:1 necessary:3 iv:1 re:1 localizing:1 calibrate:1 vertex:1 uniform:1 answer:1 repositioning:1 st:1 density:4 explores:1 freshwater:1 discriminating:1 cll:1 peak:4 tip:1 ear:1 tube:1 recorded:4 sniffing:1 external:1 actively:1 account:1 potential:23 sec:2 coding:1 mv:1 afferent:1 stream:1 analyze:1 complicated:1 parallel:1 contribution:1 minimize:1 capillary:1 accuracy:2 variance:1 characteristic:1 conducting:4 conceptually:1 weak:1 accurately:1 straight:1 processor:2 african:1 resistivity:1 ed:1 tetrahedron:1 energy:4 involved:1 james:1 nostril:1 static:1 sampled:3 adjusting:1 amplitude:4 higher:1 specify:1 execute:2 box:4 done:1 snout:1 governing:1 christopher:1 quality:1 perhaps:1 ipf:1 pulling:1 evolution:1 analytically:1 restrained:4 infmite:1 deal:1 eg:2 adjacent:1 during:4 self:3 rat:3 equipotential:3 midplane:3 heiligenberg:2 geometrical:1 image:2 physical:1 attached:1 khz:1 extend:1 tail:5 measurement:14 significant:1 grid:4 calibration:3 surface:8 etc:2 base:1 pulsed:2 apart:1 caltech:1 determine:1 paradigm:1 period:2 signal:1 ii:1 rv:1 full:1 infer:1 positioning:3 match:1 technical:1 long:1 equally:1 controlled:1 dipolar:1 poisson:1 represent:1 grounded:2 cell:1 whereas:1 fine:1 interval:1 addressed:1 source:19 leaving:1 exhibited:1 probably:1 subject:2 recording:3 member:1 near:4 variety:2 affect:1 converter:1 inner:1 knowing:1 expression:1 effort:1 peter:1 algebraic:1 york:1 cause:1 behav:1 generally:1 fake:1 clear:3 nocturnal:1 tune:1 amount:1 cosh:1 simplest:1 fish:51 sign:1 per:1 group:1 threshold:1 monitor:1 changing:1 tapered:1 undertaken:1 utilize:1 welker:4 relaxation:1 convert:1 sum:1 run:1 inverse:1 uncertainty:1 respond:1 family:1 reasonable:1 electronic:1 bit:1 fl:1 constraint:1 nearby:1 aspect:2 simulate:2 extremely:2 relatively:2 developing:1 peripheral:2 ball:1 conjugate:3 describes:2 across:5 smaller:1 son:1 wavefonn:2 bullock:2 making:2 modification:1 equation:9 agree:1 remains:1 know:1 end:1 studying:1 simulating:1 giving:1 seeking:1 skin:10 objective:1 question:1 arrangement:1 strategy:1 receptive:1 gradient:3 distance:2 attentional:1 simulated:2 sory:1 nelson:3 extent:1 reason:1 water:1 length:2 relationship:2 providing:1 minimizing:1 difficult:2 unfortunately:1 stated:1 negative:1 implementation:1 unknown:1 discretize:1 wire:3 sm:3 finite:8 head:2 digitized:1 perturbation:2 varied:1 vibrissa:1 arbitrary:2 namely:1 able:3 below:1 pattern:2 challenge:1 including:1 soo:1 rely:1 scheme:1 improve:1 brief:1 eye:1 relative:5 fully:1 whisker:2 mounted:1 degree:1 allow:2 understand:1 wide:1 taking:1 distributed:1 dimension:2 depth:1 world:1 valid:1 sensory:13 forward:1 made:4 qualitatively:1 programme:1 reconstructed:2 cavity:1 global:3 active:4 investigating:1 sequentially:1 reveals:1 reproduces:1 assumed:1 un:1 iterative:2 search:1 continuous:1 lip:1 impedance:9 channel:1 cl:1 electric:46 constructing:1 domain:4 body:14 fig:8 site:1 aid:2 wiley:1 structurally:1 position:13 msec:1 bower:3 third:1 minute:1 inset:1 rasnow:4 magnitude:1 neurophysiological:1 expressed:1 albino:1 corresponds:2 determines:1 relies:1 extracted:1 stimulating:1 bioi:1 goal:2 quantifying:1 careful:1 absence:1 content:1 change:3 considerable:1 except:1 uniformly:1 conductor:1 worm:1 specie:1 discriminate:1 experimental:6 assad:3 internal:3 mark:1 eods:1 perfonning:1 correlated:1

Exploiting generative models discriminative classifiers Tommi S. Jaakkola* MIT Artificial Intelligence Laboratorio 545 Technology Square Cambridge, MA 02139 ? In David Haussler Department of Computer Science University of California Santa Cruz, CA 95064 Abstract Generative probability models such as hidden ~larkov models provide a principled way of treating missing information and dealing with variable length sequences. On the other hand , discriminative methods such as support vector machines enable us to construct flexible decision boundaries and often result in classification performance superior to that of the model based approaches. An ideal classifier should combine these two complementary approaches. In this paper, we develop a natural way of achieving this combination by deriving kernel functions for use in discriminative methods such as support vector machines from generative probability models. We provide a theoretical justification for this combination as well as demonstrate a substantial improvement in the classification performance in the context of D~A and protein sequence analysis. 1 Introduction Speech, vision , text and biosequence data can be difficult to deal with in the context of simple statistical classification problems. Because the examples to be classified are often sequences or arrays of variable size that may have been distorted in particular ways, it is common to estimate a generative model for such data, and then use Bayes rule to obtain a classifier from this model. However. many discriminative methods, which directly estimate a posterior probability for a class label (as in Gaussian process classifiers [5]) or a discriminant function for the class label (as in support vector machines [6]) have in other areas proven to be superior to * Corresponding author. T. S. Jaakkola and D. Haussler 488 generative models for classification problems. The problem is that there has been no systematic way to extract features or metric relations between examples for use with discriminative methods in the context of difficult data types such as those listed above. Here we propose a general method for extracting these discriminatory features using a generative model. V{hile the features we propose are generally applicable, they are most naturally suited to kernel methods. 2 Kernel methods Here we provide a brief introduction to kernel methods; see, e.g., [6] [5] for more details. Suppose now that we have a training set of examples Xl and corresponding binary labels 51 (?1) . In kernel methods. as we define them. the label for a new example X is obtained from a weighted sum of the training labels. The weighting of each training label 52 consists of two parts: 1) the overall importance of the example Xl as summarized with a coefficient '\1 and 2) a measure of pairwise "similarity" between between XI and X, expressed in terms of a kernel function K(X2' X). The predicted label S for the new example X is derived from the following rule: s ~ sign ( ~ S, '\,K(X,. X) ) (1) We note that this class of kernel methods also includes probabilistic classifiers, in \vhich case the above rule refers to the label with the maximum probability. The free parameters in the classification rule are the coefficients '\1 and to some degree also the kernel function K . To pin down a particular kernel method. two things need to be clarified. First , we must define a classification loss . or equivalently, the optimization problem to solve to determine appropriate values for the coefficients '\1' Slight variations in the optimization problem can take us from support vector machines to generalized linear models. The second and the more important issue is the choice of the kernel function - the main topic of this paper. \Ve begin with a brief illustration of generalized linear models as kernel methods. 2.1 Generalized linear models For concreteness we consider here only logistic regression models. while emphasizing that the ideas are applicable to a larger class of models l . In logistic regression models , the probability of the label 5 given the example X and a parameter vector e is given by2 P(5IX. e) = (7 (5e T X) (2) where (7(z) = (1 + e- z) - l is the logistic function. To control the complexity of the model when the number of training examples is small we can assign a prior distribution p(e) over the parameters. \Ve assume here that the prior is a zero mean Gaussian with a possibly full covariance matrix L:. The maximum a posteriori (l\IAP) estimate for the parameters e given a training set of examples is found by 1 Specifically. it applies to all generalized linear models whose transfer functions are log-concave. 2Here we assume that the constant + 1 is appended to every feature vector X so that an adjustable bias term is included in the inner product T X. e Exploiting Generative Models in Discriminative Classifiers 489 maximizing the following penalized log-likelihood: I: log P(S, IX 1, B) + log P(B) where the constant c does not depend on B. It is straightforward to show, simply by taking the gradient with respect to the parameters , that the solution to this (concave) maximization problem can be written as 3 (4) Xote that the coefficients A, appear as weights on the training examples as in the definition of the kernel methods . Indeed. inserting the above solution back into the conditional probability model gives (5) By identifying !..:(X/. X) = X;'f.X and noting that the label with the maximum probability is the aile that has the same sign as the sum in the argument. this gives the decision rule (1). Through the above derivation , we have written the primal parameters B in terms of the dual coefficients A,.J. Consequently. the penalized log-likelihood function can be also written entirely in terms of A, : the resulting likelihood function specifies how the coefficients are to be optimized. This optimization problem has a unique solution and can be put into a generic form. Also , the form of the kernel function that establishes the connection between the logistic regression model and a kernel classifier is rather specific , i.e .. has the inner product form K(X,. X) = X;'f.X. However. as long as the examples here can be replaced with feature vectors derived from the examples. this form of the kernel function is the most general. \Ve discuss this further in the next section. 3 The kernel function For a general kernel fUIlction to be valid. roughly speaking it only needs to be positive semi-definite (see e.g. [7]). According to the t-Iercer 's theorem. any such valid kernel function admits a representation as a simple inner product bet\\'een suitably defined feature vectors. i.e .. !":(X,.Xj) = 0\,0.'\) . where the feature vectors come from some fixed mapping X -> ? .'\. For example. in the previous section the kernel function had the form X;'f.Xj ' which is a simple inner product for the transformed feature vector ? .'\ = 'f. 1- X. Specifying it simple inner product in the feature space defines a Euclidean metric space. Consequently. the Euclidean distances between the feature vectors are obtained directly from the kernel fUllction: with the shorthand notation K ,} = 3This corresponds to a Legendre transformation of the loss functions log a( z) . .}This is possible for all those e that could arise as solutions to the maximum penalized likelihood problem: in other words. for all relevant e. 490 T. S. Jaakkola and D. Haussler K(Xi , Xj) we get II<Px, - <PxJ W = K ti - 2Ktj + K jj . In addition to defining the metric structure in the feature space, the kernel defines a pseudo metric in the original example space through D(Xi,Xj) = II<px. - <pxJII. Thus the kernel embodies prior assumptions about the metric relations between the original examples. No systematic procedure has been proposed for finding kernel functions, let alone finding ones that naturally handle variable length examples etc. This is the topic of the next section. 4 Kernels from generative probability models: the Fisher kernel The key idea here is to derive the kernel function from a generative probability model. We arrive at the same kernel function from two different perspectives, that of enhancing the discriminative power of the model and from an attempt to find a natural comparison between examples induced by the generative model. Both of these ideas are developed in more detail in the longer version of this paper[4]. We have seen in the previous section that defining the kernel function automatically implies assumptions about metric relations between the examples. We argue that these metric relations should be defined directly from a generative probability model P(XIO). To capture the generative process in a metric between examples we use the gradient space of the generative model. The gradient of the log-likelihood with respect to a parameter describes how that parameter contributes to the process of generating a particular example 5 . This gradient space also naturally preserves all the structural assumptions that the model encodes about the generation process. To develop this idea more generally, consider a parametric class of models P(XIO) , o E e. This class of probability models defines a Riemannian manifold Ale with a local metric given by the Fisher information matrix 6 I, where I = Ex{UxU{}, Us = \1 () log P(XIB), and the expectation is over P(XIO) (see e.g. [1]). The gradient of the log-likelihood , Us , is called the Fisher score, and plays a fundamental role in our development. The local metric on lvle defines a distance between the current model P(XIO) and a nearby model P(XIO+J). This distance is given by D(O, 0+15) = ~JT 16, which also approximates the KL-divergence between the two models for a sufficiently small 6. The Fisher score Us = \l(} log P(XIB) maps an example X into a feature vector that is a point in the gradient space of the manifold Ale. We call this the Fisher score mapping. This gradient Us can be used to define the direction of steepest ascent in log P(X 10) for the example X along the manifold, i.e. , the gradient in the direction 6 that maximizes log P( X 10) while traversing the minimum distance in the manifold as defined by D(O , 0 + 6). This latter gradient is known as the natural gradient (see e.g. [1]) and is obtained from the ordinary gradient via <Ps = I - I Ux. We will call the mapping X ~ <Px the natural mapping of examples into feature vectors 7 . The natural kernel of this mapping is the inner product between these 5For the exponential family of distributions, under the natural parameterization () , these gradients, less a normalization constant that depends on () , form sufficient statistics for the example. 6For simplicity we have suppressed the dependence of I and Ux on the parameter setting (), or equivalently, on the position in the manifold . 7 Again, we have suppressed dependence on the parameter setting () here. Exploiting Generative Models in Discriminative Classifiers 491 feature vectors relative to the local Riemannian metric: (6) We call this the Fisher kernel owing to the fundamental role played by the Fisher scores in its definition. The role of the information matrix is less significant; indeed, in the context of logistic regression models, the matrix appearing in the middle of the feature vectors relates to the covariance matrix of a Gaussian prior, as show above. Thus, asymptotically, the information matrix is immaterial, and the simpler kernel KU(X i , Xj) ex Ux) provides a suitable substitute for the Fisher kernel. u.Z, We emphasize that the Fisher kernel defined above provides only the basic comparison between the examples, defining what is meant by an "inner product" between the examples when the examples are objects of various t.ypes (e .g. variable length sequences). The way such a kernel funct.ion is used in a discriminative classifier is not specified here. Using the Fisher kernel directly in a kernel classifier, for example, amounts to finding a linear separating hyper-plane in the natural gradient. (or Fisher score) feature space. The examples may not. be linearly separable in this feature space even though the natural metric st.ructure is given by t.he Fisher kernel. It may be advantageous to search in the space of quadratic (or higher order) decision boundaries, which is equivalent to transforming the Fisher kernel according to R(X t , Xj) = (1 + K(X t ? x)))m and using the resulting kernel k in the classifier. \Ve are now ready to state a few properties of the Fisher kernel function. So long as the probability model P(XIB) is suitably regular then the Fisher kernel derived from it is a) a valid kernel function and b) invariant to any invertible (and differentiable) transformation of the model parameters. The rather informally stated theorem below motivates the use of this kernel function in a classification setting. Theorem 1 Given any suitably regular probability model P(XIB) with parameters B and assuming that the classification label is included as a latent variable, the Fisher kernel K(X 1 , X)) = V~, I-I Ux] derived from this model and employed in a kernel classifier is. asymptotically. never inferior to the MAP decision rule from this model. The proofs and other related theorems are presented in the longer version of this paper [4]. To summarize, we have defined a generic procedure for obtaining kernel functions from generative probability models. Consequently the benefits of generative models are immediately available to the discriminative classifier employing this kernel function . We now turn the experimental demonstration of the effectiveness of such a combined classifier. 5 Experimental results Here we consider two relevant examples from biosequence analysis and compare the performance of the combined classifier to the best generative models used in these problems. vVe start with a DNA splice site classification problem, where the objective is to recognize true splice sites, i.e. , the boundaries between expressed regions (exons) in a gene and the intermediate regions (introns) . The dat.a set used in our experiments consisted of 9350 DNA fragments from C. elegans. Each of the T S. Jaakkola and D. Haussler 492 2029 true examples is a sequence X over the DNA alphabet {A, G, T, C} of length 25; the 7321 false examples are similar sequences that occur near but not at 5' splice sites. All recognition rates we report on this data set are averages from 7-fold cross-validation. To use the combined classifier in this setting requires us to choose a generative model for the purpose of deriving the kernel function. In order to test how much the performance of the combined classifier depends on the quality of the underlying generative model, we chose the poorest model possible. This is the model where the DKA residue in each position in the fragment is chosen independently of others, i.e., P(XIB) = P(XzIBz) and , furthermore , the parameters Bz are set such that P( Xzl OJ) = 1/4 for all i and all Xl E {A. G, T, C} . This model assigns the same probability to all examples X. We can still derive the Fisher kernel from such a model and use it in a discriminative classifier. In this case we used a logistic regression model as in (5) with a quadratic Fisher kernel K(X/. X j ) = (1 + K(Xz, Xj))2. Figure 1 shows the recognition performance of this kernel method, using the poor generative model, in comparison to the recognition performance of a naive Bayes model or a hierarchical mixture model. The comparison is summarized in ROC style curves plotting false positive errors (the errors of accepting false examples) as a function of false negative errors (the errors of missing true examples) when we vary the classification bias for the labels. The curves show that even with such a poor underlying generative model, the combined classifier is consistently better than either of the better generative models alone. n;!l In the second and more serious application of the combined classifier. we consider the well-known problem of recognizing remote homologies (evolutionary/structural similarities) between protein sequences 8 that have low residue identity. Considerable recent work has been done in refining hidden l\Iarkov models for this purpose as reviewed in [2], and such models current achieve the best performance. We use these state-of-the-art HMMs as comparison cases and also as sources for deriving the kernel function. Here we used logistic regression with the simple kernel K u (X1 ' X J)' as the number of parameters in the Hj\IMs was several thousand. The experiment was set up as follows. We picked a particular superfamily (glycosyltransferases) from the TIl'vI-barrel fold in the SCOP protein structure classification [3], and left out one of the four major families in this superfamily for testing while training the HMJlvI as well as the combined classifier on sequences corresponding to the remaining three families . The false training examples for the discriminative method came from those sequences in the same fold but not in the same superfamily. The test sequences consisted of the left-out family (true examples) and proteins outside the TIM barrel fold (false examples). The number of training examples varied around 100 depending on the left-out family. As the sequences among the four glycosyltransferase families are extremely different, this is a challenging discrimination problem. Figure lc shows the recognition performance curves for the HMM and the corresponding kernel method, averaged over the four-way cross validation. The combined classifier yields a substantial improvement in performance over the HJl..IM alone. 8These are variable length sequences thus rendering many discriminative methods inapplicable. 493 Exploiting Generative Models in Discriminative Classifiers 022 022 02 02 002' 002 0'. !!O16 ;01. .016 ~O ,. 1012 'to 12 i ~ 0' ~ a) ~ 0015 00' 0' ~008 0 08 006 006 004 004 0020 002 0 04 006 008 FaN t'leQllttve tata 0' b) 0020 000' 002 004 006 False I'Wgalllle rate 008 00 0' 0" c) 06 la1H~_rala Figure 1: a) & b) Comparison of classification performance between a kernel classifiers from t he uniform model (solid line) and a mixt ure model (dashed line) . In a) t he mixt ure model is a naive Bayes model and in b) it has t hree components in each class . c) Comparison of homology recognition performance between a hidden Mar kov model (das hed line) and t he corresponding kernel classifier (solid line). 6 Discussion The model based kernel fun ction derived in this pa per provides a generic mechanism for incorporating generative models into discriminative classifiers. For d iscrimination, the resulting combined classifier is guaranteed to be superior t o the generative model alone wit h little addi t ional computational cost . Vie not e that t he power of t he new classifier arises to a large ext.ent from the use of Fisher scores as features in place of original exa mples. It is possible to use t hese features with any classifier. e.g. a feed-forward neur al net, but kernel methods are most naturally suited for incorp orating them . F inally we note that while we have used classification t o guide the development of the kernel fun ction , t he results are directly applicable t o regression . clustering. or even interpolation problems, all of which can easily exploit metric relations among the examples defined by the Fisher kernel. References [1] S.-I. Amari . Natural gradient works efficient ly in learning. 10:251- 276, 1998. Neural Computation, [2] R. Durbin , S. Eddy, A. K rogh, a nd G . :\Iitchison . Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Aczds. Cambridge C niversity Press, 1998. [3] T. Hubba rd , A. Murzin , S. Brenner , a nd C. C hothia . seo?: a structural classification of proteins database. NA R , 25(1) :236- 9, Jan . 199 7. [4] T. S. Jaakkola a nd D . Haussler . Exploiting generative models in discriminat ive classifiers. 1998. Revised and exten ded version . \Vill be available from http : //r,l'.lY . ai . mit . edu/ rv tommi. [5] D . J. C. MacKay. Introduction to gaussia n processes. http : //wol.ra . phy . cam . ac.uk/mackay/. 1997. Availa ble from [6] V. Vapnik. The nature of statistical learning theory. Springer-Verlag. 1995. [7] G. Wahba . Spline models f or observational data. CB:\IS-NSF Regional Conference Series in Applied t>.lathematics , 1990. 

1520 |@word middle:1 version:3 advantageous:1 nd:3 suitably:3 covariance:2 solid:2 phy:1 series:1 score:6 fragment:2 current:2 must:1 written:3 cruz:1 treating:1 discrimination:1 alone:4 generative:26 intelligence:1 parameterization:1 plane:1 steepest:1 accepting:1 provides:3 clarified:1 ional:1 simpler:1 along:1 consists:1 shorthand:1 kov:1 combine:1 pairwise:1 ra:1 indeed:2 roughly:1 xz:1 automatically:1 little:1 begin:1 notation:1 underlying:2 maximizes:1 barrel:2 what:1 developed:1 finding:3 transformation:2 pseudo:1 vhich:1 every:1 ti:1 concave:2 fun:2 classifier:30 uk:1 control:1 ly:2 appear:1 positive:2 vie:1 local:3 ext:1 ure:2 interpolation:1 chose:1 specifying:1 challenging:1 hmms:1 discriminatory:1 averaged:1 unique:1 testing:1 definite:1 procedure:2 jan:1 area:1 word:1 refers:1 regular:2 protein:6 get:1 put:1 context:4 equivalent:1 map:2 missing:2 maximizing:1 murzin:1 straightforward:1 independently:1 wit:1 simplicity:1 identifying:1 immediately:1 assigns:1 rule:6 haussler:5 array:1 deriving:3 handle:1 variation:1 justification:1 suppose:1 play:1 mples:1 pa:1 recognition:5 database:1 role:3 capture:1 thousand:1 region:2 remote:1 principled:1 substantial:2 transforming:1 complexity:1 xio:5 cam:1 hese:1 immaterial:1 depend:1 hjl:1 funct:1 inapplicable:1 exon:1 easily:1 various:1 derivation:1 alphabet:1 ction:2 artificial:1 dka:1 hyper:1 outside:1 whose:1 larger:1 solve:1 ive:1 amari:1 statistic:1 addi:1 sequence:13 differentiable:1 net:1 propose:2 product:7 inserting:1 relevant:2 achieve:1 ent:1 exploiting:5 p:1 generating:1 object:1 tim:1 derive:2 develop:2 depending:1 ac:1 predicted:1 come:1 implies:1 tommi:2 direction:2 owing:1 enable:1 wol:1 observational:1 assign:1 discriminat:1 biological:1 im:1 sufficiently:1 around:1 cb:1 mapping:5 major:1 vary:1 purpose:2 applicable:3 label:12 by2:1 pxj:1 seo:1 establishes:1 weighted:1 mit:2 gaussian:3 rather:2 hj:1 bet:1 jaakkola:5 derived:5 refining:1 improvement:2 consistently:1 likelihood:6 posteriori:1 hidden:3 relation:5 transformed:1 overall:1 classification:14 flexible:1 issue:1 dual:1 among:2 development:2 art:1 mackay:2 construct:1 never:1 report:1 others:1 spline:1 serious:1 few:1 preserve:1 ve:4 divergence:1 recognize:1 replaced:1 attempt:1 hed:1 mixture:1 primal:1 biosequence:2 traversing:1 euclidean:2 theoretical:1 maximization:1 ordinary:1 cost:1 uniform:1 recognizing:1 combined:9 st:1 fundamental:2 gaussia:1 systematic:2 probabilistic:2 invertible:1 na:1 again:1 choose:1 possibly:1 exten:1 style:1 til:1 scop:1 summarized:2 includes:1 coefficient:6 depends:2 vi:1 picked:1 start:1 bayes:3 appended:1 square:1 yield:1 classified:1 definition:2 naturally:4 proof:1 riemannian:2 eddy:1 back:1 feed:1 higher:1 done:1 though:1 mar:1 furthermore:1 hand:1 defines:4 logistic:7 quality:1 consisted:2 true:4 homology:2 deal:1 inferior:1 generalized:4 demonstrate:1 mixt:2 superior:3 common:1 he:7 approximates:1 slight:1 ims:1 significant:1 cambridge:2 ai:1 rd:1 had:1 xib:5 similarity:2 longer:2 etc:1 posterior:1 recent:1 perspective:1 verlag:1 binary:1 came:1 seen:1 minimum:1 employed:1 determine:1 dashed:1 semi:1 relates:1 ale:2 ii:2 rv:1 full:1 cross:2 long:2 regression:7 basic:1 vision:1 expectation:1 enhancing:1 bz:1 metric:13 kernel:60 normalization:1 ion:1 addition:1 residue:2 source:1 regional:1 ascent:1 induced:1 thing:1 elegans:1 effectiveness:1 call:3 extracting:1 structural:3 near:1 noting:1 ideal:1 intermediate:1 rendering:1 xj:7 een:1 wahba:1 inner:7 idea:4 hile:1 fullction:1 speech:1 speaking:1 jj:1 generally:2 santa:1 listed:1 informally:1 amount:1 dna:3 http:2 specifies:1 nsf:1 sign:2 per:1 key:1 four:3 achieving:1 asymptotically:2 concreteness:1 sum:2 distorted:1 arrive:1 aile:1 family:6 place:1 decision:4 ble:1 poorest:1 entirely:1 guaranteed:1 played:1 fold:4 quadratic:2 fan:1 exa:1 durbin:1 occur:1 x2:1 encodes:1 nearby:1 argument:1 extremely:1 separable:1 px:3 department:1 according:2 neur:1 combination:2 poor:2 legendre:1 describes:1 suppressed:2 ypes:1 invariant:1 pin:1 discus:1 turn:1 mechanism:1 iap:1 available:2 hierarchical:1 appropriate:1 generic:3 appearing:1 original:3 substitute:1 hree:1 remaining:1 clustering:1 embodies:1 exploit:1 dat:1 objective:1 parametric:1 dependence:2 evolutionary:1 gradient:14 vill:1 distance:4 separating:1 hmm:1 topic:2 manifold:5 argue:1 tata:1 discriminant:1 assuming:1 length:5 illustration:1 demonstration:1 equivalently:2 difficult:2 stated:1 negative:1 motivates:1 availa:1 adjustable:1 nucleic:1 revised:1 defining:3 varied:1 david:1 kl:1 specified:1 optimized:1 connection:1 california:1 below:1 summarize:1 oj:1 power:2 suitable:1 natural:9 technology:1 brief:2 ready:1 extract:1 naive:2 text:1 prior:4 relative:1 loss:2 generation:1 proven:1 validation:2 degree:1 sufficient:1 plotting:1 penalized:3 free:1 bias:2 guide:1 taking:1 superfamily:3 benefit:1 boundary:3 curve:3 valid:3 author:1 forward:1 employing:1 emphasize:1 gene:1 dealing:1 discriminative:15 xi:3 search:1 latent:1 reviewed:1 ku:1 transfer:1 nature:1 ca:1 obtaining:1 contributes:1 da:1 main:1 linearly:1 arise:1 ded:1 complementary:1 x1:1 vve:1 site:3 roc:1 lc:1 position:2 exponential:1 xl:3 weighting:1 ix:2 splice:3 down:1 emphasizing:1 theorem:4 specific:1 jt:1 intron:1 admits:1 incorporating:1 false:7 vapnik:1 importance:1 suited:2 simply:1 expressed:2 ux:4 applies:1 springer:1 corresponds:1 ma:1 conditional:1 identity:1 consequently:3 fisher:20 considerable:1 brenner:1 included:2 specifically:1 called:1 experimental:2 support:4 latter:1 arises:1 meant:1 ex:2

SMEM Algorithm for Mixture Models N aonori U eda Ryohei Nakano {ueda, nakano }@cslab.kecl.ntt.co.jp NTT Communication Science Laboratories Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237 Japan Zoubin Ghahramani Geoffrey E. Hinton zoubin@gatsby.uc1.ac.uk g.hinton@ucl.ac.uk Gatsby Computational Neuroscience Unit, University College London 17 Queen Square, London WC1N 3AR, UK Abstract We present a split and merge EM (SMEM) algorithm to overcome the local maximum problem in parameter estimation of finite mixture models. In the case of mixture models, non-global maxima often involve having too many components of a mixture model in one part of the space and too few in another, widely separated part of the space. To escape from such configurations we repeatedly perform simultaneous split and merge operations using a new criterion for efficiently selecting the split and merge candidates. We apply the proposed algorithm to the training of Gaussian mixtures and mixtures of factor analyzers using synthetic and real data and show the effectiveness of using the split and merge operations to improve the likelihood of both the training data and of held-out test data. 1 INTRODUCTION Mixture density models, in particular normal mixtures, have been extensively used in the field of statistical pattern recognition [1]. Recently, more sophisticated mixture density models such as mixtures of latent variable models (e.g., probabilistic PCA or factor analysis) have been proposed to approximate the underlying data manifold [2]-[4]. The parameter of these mixture models can be estimated using the EM algorithm [5] based on the maximum likelihood framework [3] [4]. A common and serious problem associated with these EM algorithm is the local maxima problem. Although this problem has been pointed out by many researchers, the best way to solve it in practice is still an open question. Two of the authors have proposed the deterministic annealing EM (DAEM) algorithm [6], where a modified posterior probability parameterized by temperature is derived to avoid local maxima. However, in the case of mixture density models, local maxima arise when there are too many components of a mixture models in one part of the space and too few in another. It is not possible to move a component from the overpopulated region to the underpopulated region without passing N Ueda, R. Nakano, Z. Ghahramani and G. E. Hinton 600 through positions that give lower likelihood. We therefore introduce a discrete move that simultaneously merges two components in an overpopulated region and splits a component in an underpopulated region. The idea of split and merge operations has been successfully applied to clustering or vector quantization (e.g., [7]). To our knowledge, this is the first time that simultaneous split and merge operations have been applied to improve mixture density estimation. New criteria presented in this paper can efficiently select the split and merge candidates. Although the proposed method, unlike the DAEM algorithm, is limited to mixture models, we have experimentally comfirmed that our split and merge EM algorithm obtains better solutions than the DAEM algorithm. 2 Split and Merge EM (SMEM) Algorithm The probability density function (pdf) of a mixture of M density models is given by M p(x; 8) = L amP(xlwm;Om), where am 2: 0 and (1) m=l The p(xlwm ; Om) is a d-dimensional density model corresponding to the component The EM algorithm, as is well known, iteratively estimates the parameters 8 = {(am, Om), m = 1, ... , M} using two steps. The E-step computes the expectation of the complete data log-likelihood. Wm . Q(818(t?) = L L P(wmlx; 8(t?) logamP(xlwm;Om), X (2) m where P(wmlx; 8(t?) is the posterior probability which can be computed by P( Wm I' 8(t?) = X, M a~p(xlwm; o~;h (t) (t) Lm'=l am,p(xlwm,;Om') . (3) Next, the M-step maximizes this Q function with respect to 8 to estimate the new parameter values 8(t+1). Looking at (2) carefully, one can see that the Q function can be represented in the form of a direct sum; i.e., Q(818(t?) = L~=l qm(818(t?), where qm(818(t?) = LXEx P(wmlx ; 8(t?) logamP(xlw m; Om) and depends only on am and Om . Let 8* denote the parameter values estimated by the usual EM algorithm. Then after the EM algorithm has converged, the Q function can be rewritten as Q* = q7 + q; + qk + L q:n. (4) m,m,/i,j,k We then try to increase the first three terms of the right-hand side of (4) by merging two components Wi and Wj to produce a component Wi', and splitting the component Wk into two components Wj' and Wk" To reestimate the parameters of these new components, we have to initialize the parameters corresponding to them using 8*. The initial parameter values for the merged component Wi' can be set as a linear combination of the original ones before merge: and Oi' P(wJ ?l x,? 8*) = O*~ wX P(w ? lx?8*)+O*~ J wX Lx P(wil x ; 8*) + Lx P(wjlx; 8*) t t, (5) SMEM Algorithm for Mixture Models 601 On the other hand, as for two components Wj' and Wk', we set (6) where t is some small random perturbation vector or matrix (i.e., Iltll?IIOk 11)1. The parameter reestimation for m = i', j' and k' can be done by using EM steps, but note that the posterior probability (3) should be replaced with (7) so that this reestimation does not affect the other components. o~;h La~p(xlwm; (I am,p x m , (t) W .O(t)) m' x L m'=i,j,k P(wm'lx; 8*), m = i',j', k'. m'=i',j',k' (7) Clearly Lm'=i',j',k' P(wm, Ix; 8(t)) = Lm=i,j,k P{wmlx; 8*) always holds during the reestimation process. For convenience, we call this EM procedure the partial EM procedure. After this partial EM procedure, the usual EM steps, called the full EM procedure, are performed as a post processing. After these procedures, if Q is improved, then we accept the new estimate and repeat the above after setting the new paramters to 8*. Otherwise reject and go back to 8* and try another candidate. We summarize these procedures as follows: [SMEM Algorithm] 1. Perform the usual EM updates. Let 8* and Q* denote the estimated parameters and corresponding Q function value, respectively. 2. Sort the split and merge candidates by computing split and merge criteria (described in the next section) based on 8*. Let {i, j, k}c denote the cth candidate. 3. For c = 1, ... , C max , perform the following: After initial parameter settings based on 8 *, perform the partial EM procedure for {i, j, k }c and then perform the full EM procedure. Let 8** be the obtained parameters and Q** be the corresponding Q function value. If Q** > Q*, then set Q* f - Q**, 8* f - 8** and go to Step 2. 4. Halt with 8* as the final parameters. Note that when a certain split and merge candidate which improves the Q function value is found at Step 3, the other successive candidates are ignored. There is therefore no guarantee that the split and the merge candidates that are chosen will give the largest possible improvement in Q. This is not a major problem, however, because the split and merge operations are performed repeatedly. Strictly speaking, C max = M(M -1)(M - 2)/2, but experimentally we have confirmed that C max '" 5 may be enough. 3 Split and Merge Criteria Each of the split and merge candidates can be evaluated by its Q function value after Step 3 of the SMEM algorithm mentioned in Sec.2. However, since there are so many candidates, some reasonable criteria for ordering the split and merge candidates should be utilized to accelerate the SMEM algorithm. In general, when there are many data points each of which has almost equal posterior probabilities for any two components, it can be thought that these two components 1 In the case of mixture Gaussians, covariance matrices E i , and E k , should be positive definite. In this case, we can initialize them as E i , = E k , = det(Ek)l/d Id indtead of (6). N. Ueda. R. Nakano. Z. Ghahramani and G. E. Hinton 602 might be merged. criterion: To numerically evaluate this, we define the following merge Jmerge(i,j; 8*) = P i (8*fp j (8*), (8) where Pi(8*) = (P(wilxl; 8*), ... , P(wilxN; 8*))T E nN is the N-dimensional vector consisting of posterior probabilities for the component Wi. Clearly, two components Wi and Wj with larger Jmerge(i,j; 8*) should be merged. As a split criterion (Jsplit), we define the local Kullback divergence as: J split (k ; 8- *) = J( Pk x; 8*) - Iog Pk(X; (I 8*) e) d x, P x Wk; k (9) which is the distance between two distributions: the local data density Pk(X) around the component Wk and the density of the component Wk specified by the current parameter estimate ILk and ~k' The local data density is defined as: (10) This is a modified empirical distribution weighted by the posterior probability so that the data around the component Wk are focused. Note that when the weights are equal, i.e., P(wklx; 8*) = 11M, (10) is the usual empirical distribution, i.e., Pk(X; 8*) = (liN) E~=l 6(x - xn). Since it can be thought that the component with the largest Jspl it (k; 8*) has the worst estimate of the local density, we should try to split it. Using Jmerge and Jsp/it, we sort the split and merge candidates as follows. First, merge candidates are sorted based on Jmerge. Then, for each sorted merge ' candidate {i,j}e, split candidates excluding {i,j}e are sorted as {k}e. By combining these results and renumbering them, we obtain {i, j , k }e. 4 4.1 Experiments Gaussian mixtures First, we show the results of two-dimensional synthetic data in Fig. 1 to visually demonstrate the usefulness of the split and merge operations. Initial mean vectors and covariance matrices were set to near mean of all data and unit matrix, respectively. The usual EM algorithm converged to the local maximum solution shown in Fig. l(b), whereas the SMEM algorithm converged to the superior solution shown in Fig. led) very close to the true one. The split of the 1st Gaussian shown in Fig. l(c) seems to be redundant, but as shown in Fig. led) they are successfully merged and the original two Gaussians were improved. This indicates that the split and merge operations not only appropriately assign the number of Gaussians in a local data space, but can also improve the Gaussian parameters themselves. Next, we tested the proposed algorithm using 20-dimensional real data (facial images) where the local maxima make the optimization difficult. The data size was 103 for training and 103 for test. We ran three algorithms (EM, DAEM, and SMEM) for ten different initializations using the K-means algorithm. We set M = 5 and used a diagonal covariance for each Gaussian. As shown in Table 1, the worst solution found by the SMEM algorithm was better than the best solutions found by the other algorithms on both training and test data. 603 SMEM Algorithmfor Mixture Models :'(:0: . .~.~ . ' " .; ':.. . \.';,,: ," ...... . . :~~1 co - ...';q\' ~.t./ ?"{J?? .... O?:6 '..!' " ~:. " 2 .: " ? i,:"'..... ....:...: . ,,' :. ' (a) True Gaussians and generated data (b) Result by EM (t=72) . ,' .. (c) Example of split and merge (t=141) ? I .' ? :: . : :,?? t , ' ?? (d) Final result by SMEM (t=212) Figure 1: Gaussian mixture estimation results. Table 1: Log-likelihood I data point -145 Training data Test data initiall value EM DAEM SMEM min -159.1 1.n -157.3 -163.2 -148.2 0.24 -147.7 -148.6 -147.9 0.04 -147.8 -147.9 -145.1 0.08 -145.0 -145.2 mean std max min -168.2 2.80 -165.5 -174.2 -159.8 1.00 -158.0 -160.8 -159.7 0.37 -159.6 -159.8 -155.9 0.09 -155.9 -156.0 mean std max '~-150 ~ '0 8-'55 ? ~ g.-,,, ...J Table 2: No. of iterations mean sId max min EM DAEM SMEM 47 16 65 37 147 39 189 103 155 44 219 109 , EM -1&5 I ste~ : , , , I with split and'merge I : : I-------'--~ ,-.----~--~~ , ~ ~ .. .. ~ m ~ No. of iterations ~ ____ ~ ~ = Figure 2: Trajectories of loglikelihood. Upper (lower) corresponds to training (test) data. Figure 2 shows log-likelihood value trajectories accepted at Step 3 of the SMEM algorithm during the estimation process 2. Comparing the convergence points at Step 3 marked by the '0' symbol in Fig. 2, one can see that the successive split and merge operations improved the log-likelihood for both the training and test data, as we expected. Table 2 compares the number of iterations executed by the three algorithms. Note that in the SMEM algorithm, the EM-steps corresponding to rejected split and merge operations are not counted. The average rank of the accepted split and merge candidates was 1.8 (STD=O.9) , which indicates that the proposed split and merge criteria work very well . Therefore, the SMEM algorithm was about 155 x 1.8/47 c::: 6 times slower than the original EM algorithm. 4.2 Mixtures of factor analyzers A mixture of factor analyzers (MFA) can be thought of as a reduced dimension mixture of Gaussians [4]. That is, it can extract locally linear low-dimensional manifold underlying given high-dimensional data. A single FA model assumes that an observed D-dimensional variable x are generated as a linear transformation of some lower K-dimensionallatent variable z rv N(O, I) plus additive Gaussian noise v rv N(O, w). w is diagonal. That is, the generative model can be written as 2Dotted lines in Fig. 2 denote the starting points of Step 2. Note that it is due to the initialization at Step 3 that the log-likelihood decreases just after the split and merge. N. Ueda, R. Nakano, Z. Ghahrarnani and G. E. Hinton 604 ??~~~~--~~?-X~l"? ?~?~~~~--~'"O':',,-x-:.,. ?~~~~--~~?-x~; (a) Initial values (b) Result by EM (c) Result by SMEM Rgure 3: Extraction of 1D manifold by using a mixture of factor analyzers. x = Az + v + J-L. Here J-L is a mean vector. Then from simple calculation, we can see that x N(J-L, AAT + '11). Therefore, in the case of a M mixture of FAs, x L~=l omN(J-Lm, AmA~ + 'lim). See [4] for the details. Then, in this case, the Q function is also decomposable into M components and therefore the SMEM algorithm is straightforwardly applicable to the parameter estimation of the MFA models. t'V t'V Figure 3 shows the results of extracting a one-dimensional manifold from threedimensional data (nOisy shrinking spiral) using the EM and the SMEM algorithms 3. Although the EM algorithm converged to a poor local maxima, the SMEM algorithm successfully extracted data manifold. Table 3 compares average log-likelihood per data point over ten different initializations. The log-likelihood values were drastically improved on both training and test data by the SMEM algorithm. The MFA model is applicable to pattern recognition tasks [2][3] since once an MFA model is fitted to each class, we can compute the posterior probabilities for each data point. We tried a digit recognition task (10 digits (classes))4 using the MFA model. The computed log-likelihood averaged over ten classes and recognition accuracy for test data are given in Table 4. Clearly, the SMEM algorithm consistently improved the EM algorithm on both log-likelihood and recognition accuracy. Note that the recognition accuracy by the 3-nearest neighbor (3NN) classifier was 88.3%. It is interesting that the MFA approach by both the EM and SMEM algorithms could outperform the nearest neighbor approach when K = 3 and M = 5. This suggests that the intrinsic dimensionality of the data would be three or so. ' 3In this case, each factor loading matrix Am becomes a three dimensional column vector corresponding to each thick line in Fig. 3. More correctly, the center position and the direction of each thick line are f..Lm and Am, respectively. And the length of each thick line is 2 IIAmll. 4The data were created using the degenerate Glucksman's feature (16 dimensional data) by NTT labs.[8]. The data size was 200/class for training and 200/class for test. 605 SMEM Algorithm/or Mixture Models Table 4: Digit recognition results Table 3: Log-likelihood I data point EM Training -7.68 (0.151) Test -7.75 (0.171) -7.26 (0.017) EM SMEM EM SMEM K=3 M=5 M=10 -3.18 -3.09 -3.15 -3.05 89.0 87.5 91.3 88.7 K=8 M=5 M=10 -3.14 -3.04 -3.11 -3.01 85.3 82.5 87.3 85.1 -7.33 (0.032) O:STD 5 Log-likelihood / data point Recognition rate ("!o) SMEM Conclusion We have shown how simultaneous split and merge operations can be used to move components of a mixture model from regions of the space in which there are too many components to regions in which there are too few. Such moves cannot be accomplished by methods that continuously move components through intermediate locations because the likelihood is lower at these locations. A simultaneous split and merge can be viewed as a way of tunneling through low-likelihood barriers, thereby eliminating many non-global optima. In this respect, it has some similarities with simulated annealing but the moves that are considered are long-range and are very specific to the particular problems that arise when fitting a mixture model. Note that the SMEM algorithm is applicable to a wide variety of mixture models, as long as the decomposition (4) holds. To make the split and merge method efficient we have introduced criteria for deciding which splits and merges to consider and have shown that these criteria work well for low-dimensional synthetic datasets and for higher-dimensional real datasets. Our SMEM algorithm conSistently outperforms standard EM and therefore it would be very useful in practice. References [1] MacLachlan, G. and Basford K., "Mixture models: Inference and application to clustering," Marcel Dekker, 1988. [2] Hinton G. E., Dayan P., and Revow M., "Modeling the minifolds of images of handwritten digits," IEEE Trans. PAMI, vol.8, no.1, pp. 65-74, 1997. [3] Tipping M. E. and Bishop C. M., "Mixtures of probabilistic principal component analysers," Tech. Rep. NCRG-97-3, Aston Univ. Birmingham, UK, 1997. [4] Ghahramani Z. and Hinton G. E., "The EM algorithm for mixtures of factor analyzers," Tech. Report CRG-TR-96-1, Univ. of Toronto, 1997. [5] Dempster A. P., Laird N. M. and Rubin D. B., "Maximum likelihood from incomplete data via the EM algorithm," Journal of Royal Statistical Society B, vol. 39, pp. 1-38, 1977. [6] Ueda N. and Nakano R., "Deterministic annealing EM algorithm," Neural Networks, voLl1, no.2, pp.271-282, 1998. [7] Ueda N. and Nakano R., "A new competitive learning approach based on an equidistortion principle for designing optimal vector quantizers," Neural Networks, vol.7, no.8, pp.1211-1227, 1994. [8] Ishii K., "Design of a recognition dictionary using artificially distorted characters," Systems and computers in Japan, vol.21, no.9, pp. 669-677, 1989. 

1521 |@word eliminating:1 seems:1 loading:1 dekker:1 open:1 tried:1 covariance:3 decomposition:1 thereby:1 tr:1 xlw:1 initial:4 configuration:1 selecting:1 amp:1 outperforms:1 current:1 comparing:1 written:1 additive:1 wx:2 update:1 generative:1 location:2 lx:4 successive:2 toronto:1 direct:1 ryohei:1 fitting:1 introduce:1 expected:1 themselves:1 seika:1 becomes:1 underlying:2 maximizes:1 transformation:1 guarantee:1 qm:2 classifier:1 uk:4 unit:2 before:1 positive:1 aat:1 local:12 id:1 merge:34 pami:1 might:1 plus:1 initialization:3 suggests:1 co:2 limited:1 range:1 averaged:1 practice:2 definite:1 digit:4 procedure:8 empirical:2 reject:1 thought:3 zoubin:2 convenience:1 close:1 cannot:1 deterministic:2 center:1 go:2 starting:1 focused:1 decomposable:1 splitting:1 designing:1 recognition:9 utilized:1 std:4 q7:1 observed:1 worst:2 region:6 wj:5 ordering:1 decrease:1 ran:1 mentioned:1 dempster:1 wil:1 accelerate:1 represented:1 separated:1 univ:2 london:2 analyser:1 widely:1 solve:1 larger:1 loglikelihood:1 otherwise:1 noisy:1 laird:1 final:2 ucl:1 dimensionallatent:1 ste:1 combining:1 degenerate:1 ama:1 az:1 convergence:1 optimum:1 jsp:1 produce:1 ac:2 nearest:2 marcel:1 direction:1 thick:3 merged:4 assign:1 crg:1 strictly:1 hold:2 around:2 considered:1 normal:1 visually:1 deciding:1 lm:5 major:1 dictionary:1 estimation:5 birmingham:1 applicable:3 largest:2 successfully:3 weighted:1 clearly:3 gaussian:7 always:1 modified:2 avoid:1 derived:1 improvement:1 consistently:2 rank:1 likelihood:17 indicates:2 tech:2 ishii:1 am:7 inference:1 dayan:1 nn:2 accept:1 initialize:2 field:1 equal:2 once:1 having:1 extraction:1 report:1 escape:1 few:3 serious:1 simultaneously:1 divergence:1 replaced:1 consisting:1 mixture:33 held:1 wc1n:1 partial:3 facial:1 incomplete:1 fitted:1 column:1 modeling:1 ar:1 queen:1 usefulness:1 too:6 straightforwardly:1 synthetic:3 cho:1 st:1 density:11 probabilistic:2 continuously:1 daem:6 ek:1 japan:2 sec:1 wk:7 depends:1 performed:2 try:3 lab:1 wm:4 sort:2 competitive:1 om:7 square:1 oi:1 accuracy:3 qk:1 efficiently:2 sid:1 handwritten:1 trajectory:2 confirmed:1 researcher:1 converged:4 simultaneous:4 reestimate:1 eda:1 pp:5 associated:1 basford:1 knowledge:1 lim:1 improves:1 dimensionality:1 sophisticated:1 carefully:1 back:1 higher:1 tipping:1 improved:5 done:1 evaluated:1 rejected:1 just:1 hand:2 true:2 laboratory:1 iteratively:1 during:2 criterion:10 pdf:1 complete:1 demonstrate:1 temperature:1 image:2 recently:1 common:1 superior:1 jp:1 ncrg:1 numerically:1 pointed:1 analyzer:5 similarity:1 posterior:7 certain:1 rep:1 accomplished:1 redundant:1 rv:2 full:2 kyoto:1 ntt:3 calculation:1 renumbering:1 lin:1 long:2 post:1 halt:1 iog:1 expectation:1 iteration:3 whereas:1 annealing:3 appropriately:1 unlike:1 effectiveness:1 call:1 extracting:1 near:1 intermediate:1 split:37 enough:1 spiral:1 variety:1 affect:1 idea:1 det:1 pca:1 soraku:1 passing:1 speaking:1 repeatedly:2 algorithmfor:1 ignored:1 useful:1 involve:1 extensively:1 ten:3 locally:1 reduced:1 outperform:1 dotted:1 neuroscience:1 estimated:3 per:1 correctly:1 paramters:1 discrete:1 vol:4 sum:1 parameterized:1 distorted:1 almost:1 reasonable:1 ueda:6 tunneling:1 ilk:1 min:3 cslab:1 combination:1 poor:1 em:38 character:1 wi:5 cth:1 operation:10 rewritten:1 gaussians:5 apply:1 slower:1 original:3 assumes:1 clustering:2 nakano:7 ghahramani:4 hikaridai:1 threedimensional:1 society:1 move:6 question:1 fa:2 usual:5 diagonal:2 distance:1 simulated:1 gun:1 manifold:5 length:1 difficult:1 executed:1 quantizers:1 design:1 perform:5 upper:1 datasets:2 finite:1 hinton:7 communication:1 looking:1 excluding:1 perturbation:1 introduced:1 specified:1 merges:2 trans:1 pattern:2 fp:1 summarize:1 max:6 royal:1 mfa:6 improve:3 smem:30 aston:1 created:1 extract:1 interesting:1 geoffrey:1 rubin:1 kecl:1 principle:1 pi:1 repeat:1 drastically:1 side:1 neighbor:2 wide:1 barrier:1 overcome:1 dimension:1 xn:1 computes:1 author:1 counted:1 approximate:1 obtains:1 kullback:1 global:2 reestimation:3 latent:1 table:8 artificially:1 pk:4 noise:1 arise:2 fig:8 gatsby:2 shrinking:1 position:2 candidate:16 ix:1 specific:1 bishop:1 symbol:1 intrinsic:1 quantization:1 merging:1 led:2 corresponds:1 extracted:1 sorted:3 marked:1 viewed:1 revow:1 experimentally:2 principal:1 overpopulated:2 called:1 accepted:2 la:1 select:1 college:1 underpopulated:2 evaluate:1 tested:1 lxex:1

An entropic estimator for structure discovery Matthew Brand Mitsubishi Electric Research Laboratories, 201 Broadway, Cambridge MA 02139 brand@merl.com Abstract We introduce a novel framework for simultaneous structure and parameter learning in hidden-variable conditional probability models, based on an en tropic prior and a solution for its maximum a posteriori (MAP) estimator. The MAP estimate minimizes uncertainty in all respects: cross-entropy between model and data; entropy of the model ; entropy of the data's descriptive statistics. Iterative estimation extinguishes weakly supported parameters, compressing and sparsifying the model. Trimming operators accelerate this process by removing excess parameters and, unlike most pruning schemes, guarantee an increase in posterior probability. Entropic estimation takes a overcomplete random model and simplifies it, inducing the structure of relations between hidden and observed variables. Applied to hidden Markov models (HMMs), it finds a concise finite-state machine representing the hidden structure of a signal. We entropically model music, handwriting, and video time-series, and show that the resulting models are highly concise, structured, predictive, and interpretable: Surviving states tend to be highly correlated with meaningful partitions of the data, while surviving transitions provide a low-perplexity model of the signal dynamics. 1 . An entropic prior In entropic estimation we seek to maximize the information content of parameters. For conditional probabilities , parameters values near chance add virtually no information to the model, and are therefore wasted degrees of freedom. In contrast, parameters near the extrema {O, I} are informative because they impose strong constr?aints on the class of signals accepted by the model. In Bayesian terms, our prior should assert that parameters that do not reduce uncertainty are improbable. We can capture this intuition in a surprisingly simple form: For a model of N conditional probabilities 9 = {(h , . . . , ()N } we write (1) whence we can see that the prior measures a model's freedom from ambiguity (H(9) is an entropy measure). Applying Pe (.) to a multinomial yields the posterior p (LlI) e U W <X P(wI9)Pe(9) P(w) <X [II . z ()W;] 1 Pe(9) P(w) <X II . (){;/;+w; z (2) z where Wi is evidence for event type i. With extensive evidence this distribution converges to "fair"(ML) odds for w, but with scant evidence it skews to stronger odds. M Brand 724 1.1 MAP estimator To obtain MAP estimates we set the derivative of log-posterior to zero, using Lagrange multipliers to ensure L:i (}i = 1, W' (3) 1+ (); +IOg(}i+"\ We obtain (}i by working backward from the Lambert W function, a multi-valued inverse function satisfying W(x)eW(x) =x. Taking logarithms and setting y = logx, 0= -W(x) -logW(x) + logx - W(e Y ) -log W(e Y ) + y -1 IjW(e Y ) +logljW(e Y )+logz+y-Iogz -z zjW(e Y ) + 10gzjW(e Y ) + y -logz (4) Setting (}i = zjW(e Y ) , y = 1 +"\+logz, and z = -Wi, eqn. 4 simplifies to eqn. 3, implying (5) Equations 3 and 5 together yield a quickly converging fix-point equation for ..\ and therefore for the entropic MAP estimate. Solutions lie in the W -1 branch of Lambert's function. See [Brand, 1997] for methods we developed to calculate the little-known W function. 1.2 Interpretation The negated log-posterior is equivalent to a sum of entropies: H(O) + D(wIIO) + H(w) (6) Maximizing Pe(Olw) minimizes entropy in all respects: the parameter entropy H(O); the cross-entropy D (w "0) between the parameters 0 and the data's descriptive statistics w; and the entropy of those statistics H (w), which are calculated relative to the structure of the model. Equivalently, the MAP estimator minimizes the expected coding length, making it a maximally efficient compressor of messages consisting of the model and the data coded relative to the model. Since compression involves separating essential from accidental structure, this can be understood as a form of noise removal. Noise inflates the apparent entropy of a sampled process; this systematically biases maximum likelihood (ML) estimates toward weaker odds, more so in smaller samples. Consequently, the entropic prior is a countervailing bias toward stronger odds. 725 An Entropic Estimator for Structure Discovery 1.3 Model trimming Because the prior rewards sparse models, it is possible to remove weakly supported parameters from the model while improving its posterior probability, such that P e (O\(}iIX) > P e (OIX). This stands in contrast to most pruning schemes, which typically try to minimize damage to the posterior. Expanding via Bayes rule and taking logarithms we obtain (7) where hi((}i) is the entropy due to (}i. For small (}i, we can approximate via differentials: () . {)H(O) ~ {)(}i > () . {)logP(XIO) t {)(} i (8) By mixing the left- and right-hand sides of equations 7 and 8, we can easily identify trimmable parameters-those that contribute more to the entropy than the log-likelihood. E.g., for multinomials we set hi ((}i) = -(}i log (}i against r.h.s. eqn. 8 and simplify to obtain < exp [ - {)logP(XIO)] {)(}i (9) Parameters can be trimmed at any time during training; at convergence trimming can bump the model out of a local probability maximum, allowing further training in a lowerdimensional and possibly smoother parameter subspace. 2 Entropic HMM training and trimming In entropic estimation of HMM transition probabilities, we follow the conventional E-step, calculating the probability mass for each transition to be used as evidence w: T-l Ij,i L aj(t) Pilj Pi(Xt+1) fh(t + 1) (10) where P ilJ is the current estimate of the transition probability from state j to state i; Pi(Xt+d is the output probability of observation Xt+1 given state i, and Q, {3 are obtained from forward-backward analysis and follow the notation of Rabiner [1989]. For the Mstep, we calculate new estimates {Pi lj h = 0 by applying the MAP estimator in ?1.1 to each w = {,j ,i k That is, w is a vector of the evidence for each kind of transition out of a single state; from this evidence the MAP estimator calculates probabilities O. (In BaumWelch re-estimation, the maximum-likelihood estimator simply sets Pilj = Ij ,i/ 2:i Ij,d In iterative estimation, e.g., expectation-maximization (EM), the entropic estimator drives weakly supported parameters toward zero, skeletonizing the model and concentrating evidence on surviving parameters until their estimates converge to near the ML estimate. Trimming appears to accelerate this process by allowing slowly dying parameters to leapfrog to extinction. It also averts numerical underflow errors. For HMM transition parameters, the trimming criterion of egn. 9 becomes (11 ) where Ij (t) is the probability of state j at time t. The multinomial output distributions of a discrete-output HMM can be en tropically re-estimated and trimmed in the same manner. M. Brand 726 Entropic versus ML HMM models of Bach chorales 90 7S .-.-~-.-----,..., go r:\\~ ~~ o S 20 5 IS 25 3S .t5.....,5----+.-.,-~ , states at initialization Figure 1: Left: Sparsification, classification, and prediction superiority of entropically estimated HMMs modeling Bach chorales. Lines indicate mean performance over 10 trials; error bars are 2 standard deviations . Right: High-probability states and subgraphs of interest from an entropically estimated 35-state chorale HMM. Tones output by each state are listed in order of probability. Extraneous arcs have been removed for clarity. 3 Structure learning experiments To explore the practical utility of this framework, we will use entropically estimated HMMs as a window into the hidden structure of some human-generated time-series. Bach Chorales: We obtained a dataset of melodic lines from 100 of I.S. Bach's 371 surviving chorales from the UCI repository [Merz and Murphy, 1998], and transposed all into the key of C. We compared entropically and conventionally estimated HMMs in prediction and classification tasks, training both from identical random initial conditions and trying a variety of different initial state-counts. We trained with 90 chorales and testing with the remaining 10. In ten trials, all chorales were rotated into the test set. Figure 1 illustrates that despite substantial loss of parameters to sparsification, the entropically estimated HMMs were, on average, better predictors of notes. (Each test sequence was truncated to a random length and the HMMs were used to predict the first missing note.) They also were better at discriminating between test chorales and temporally reversed test chorales-challenging because Bach famously employed melodic reversal as a compositional device. With larger models, parameter-trimming became state-trimming: An average of 1.6 states were "pinched off" the 35-state models when all incoming transitions were deleted. While the conventionally estimated HMMs were wholly uninterpretable, in the entropically estimated HMMs one can discern several basic musical structures (figure 1, right), including self-transitioning states that output only tonic (C-E-G) or dominant (G-B-D) triads, lower- or upper-register diatonic tones (C-D-E or F-G-A-B), and mordents (A-nGA). We also found chordal state sequences (F-A-C) and states that lead to the tonic (C) via the mediant (E) or the leading tone (B). Handwriting: We used 2D Gaussian-output HMMs to analyze handwriting data. Training data, obtained from the UNIPEN web site [Reynolds, 1992], consisted of sequences of normalized pen-position coordinates taken at 5msec intervals from 10 different individuals writing the digits 0-9. The HMMs were estimated from identical data and initial conditions (random upper-diagonal transition matrices; random output parameters). The diagrams in Figure 2 depict transition graphs of two HMMs modeling the pen-strokes for the digit "5," mapped onto the data. Ellipses indicate each state's output probability iso-contours (receptive field); X s and arcs indicate state dwell and transition probabilities, respectively, by their thicknesses. Entropic estimation induces an interpretable automaton that captures essential structure and timing of the pen-strokes. 50 of the 80 original transition parameters 727 An Entropic Estimator for Structure Discovery ConlUStOn MatrIX WIth 93 0% acct.JIlIcy ~~Y", :,y . .. . eonrus.on Matnll WIth 96 0%,accuracy . '. :,"' .s -': " .,-!.:~ ,~_~~" / . -, 6 a. conventional b. en tropic c. conventional d. en tropic Figure 2: (a & b): State machines of conventionally and entropically estimated hidden Markov models of writing "S." (c & d): Confusion matrices for all digits. were trimmed. Estimation without the entropic prior results in a wholly opaque model, in which none of the original dynamical parameters were trimmed. Model concision leads to better classification-the confusion matrices show cumulative classificMion error over ten trials with random initializations. Inspection of the parameters for the model in 2b showed that all writers began in states 1 or 2. From there it is possible to follow the state diagram to reconstruct the possible sequences of pen-strokes: Some writers start with the cap (state 1) while others start with the vertical (state 2); all loop through states 3-8 and some return to the top (via state 10) to add a horizontal (state 12) or diagonal (state 11) cap. Office activity: Here we demonstrate a model of human activity learned from mediumto long-term ambient video. By activity, we mean spatio-temporal patterns in the pose, position, and movement of one's body. To make the vision tractable, we consider the activity of a single person in a relatively stable visual environment, namely, an office. We track the gross shape and position of the office occupant by segmenting each image into foreground and background pixels. Foreground pixels are identified with reference to an acquired statistical model of the background texture and camera noise. Their ensemble properties such as motion or color are modeled via adaptive multivariate Gaussian distributions, re-estimated in each frame. A single bivariate Gaussian is fitted to the foreground pixels and we record the associated ellipse parameters [mean x , meany, timean x , timean y, mass, timass, elongation, eccentricity]. Sequences of these observation vectors are used to train and test the HMMs. Approximately 30 minutes of data were taken at SHz from an SGI IndyCam. Data was collected automatically and at random over several days by a program that started recording whenever someone entered the room after it had been empty S+ minutes. Backgrounds were re-Iearned during absences to accommodate changes in lighting and room configuration. Prior to training, HMM states were initialized to tile the image with their receptive fields, and transition probabilities were initialized to prefer motion to adjoining tiles. Three sequences ranging from 1000 to 1900 frames in length were used for entropic training of 12, 16,20, 2S, and 30-state HMMs. Entropic training yielded a substantially sparsified model with an easily interpreted state machine (see figure 3). Grouping of states into activities (done only to improve readability) was done by adaptive clustering on a proximity matrix which combined Mahalonobis distance and transition probability between states. The labels are the author's description of the set of frames claimed by each state cluster during forward-backward analysis of test data. Figure 4 illustrates this analysis, showing frames from a test sequence to which specific states are strongly tuned. State S (figure 3 right) is particularly interesting-it has a very non-specific receptive field, no self-transition, and an extremely low rate of occupancy. Instead of modeling data, it serves to compress the model by summarizing transition patterns that are common to several other states. The entropic model has proven to be quite superior for segmented new video into activities and detecting anomalous behavior. M Brand 728 -~ --. ~ ...".". :~ irlitialization .. .-. tinalmochtl . ~~ . ' Figure 3: Top: The state machine found by en tropic training (left) is easily labeled and interpreted. The state machine found by conventional training (right) is not, begin fully connected. Bottom: Transition matrices after (1) initialization, (2) entropic training, (3) conventional training, and (4 & 5) entropic training from larger initializations. The top row indicates initial probabilities of each state; each subsequent row indicates the transition probabilities out of a state. Color key: 0 = 0; ? = 1. The state machines above are extracted from 2 & 3. Note that 4 & 5 show the same qualitative structure as 2, but sparser, while 3 shows no almost no structure at all. Figure 4: Some sample frames assigned high state-specific probabilities by the model. Note that some states are tuned to velocities, hence the difference between states 6 and 11. 4 Related work HMMs: The literature of structure-learning in HMMs is based almost entirely on generateand-test algorithms. These algorithms work by merging [Stokke and Omohundro, 1994] or splitting [Takami and Sagayama, 1991] states, then retraining the model to see if any advantage has been gained. Space constraints force us to summarize a recent literature review: There are now more than 20 variations and improvements on these approaches, plus some heuristic constructive algorithms (e.g., [Wolfertstetter and Ruske, 1995]). Though these efforts use a variety of heuristic techniques and priors (including MDL) to avoid detrimental model changes, much of the computation is squandered and reported run-times often range from hours to days. Entropic estimation is exact. monotonic, and orders of magnitude faster-only slightly longer than standard EM parameter estimation. MDL: Description length minimization is typically done via gradient ascent or search via model comparison; few estimators are known. Rissanen [1989] introduced an estimator for binary fractions, from which Vovk [1995] derived an approximate estimator for Bernoulli An Entropic Estimator for Structure Discovery 729 models over discrete sample spaces. It approximates a special case of our exact estimator, which handles multinomial models in continuous sample spaces. Our framework provides a unified Bayesian framework for two issues that are often treated separately in MDL: estimating the number of parameters and estimating their values. MaxEnt: Our prior has different premises and an effect opposite that of the "standard" MaxEnt prior e- aD (9i1 9 o). Nonetheless, our prior can be derived via MaxEnt reasoning from the premise that the expectation of the perplexity over all possible models is finite [Brand, 1998]. More colloquially, we almost always expect there to be learnable structure. Extensions: For simplicity of exposition (and for results that are independent of model class), we have assumed prior independence of the parameters and taken H (8) to be the combined parameter entropies of the model's component distributions. Depending on the model class, we can also provide variants of eqns. 1-8 for H (8) =conditional entropy or H (8) =entropy rate of the model. In Brand [1998] we present entropic MAP estimators for spread and covariance parameters with applications to mixtures-of-Gaussians, radial basis functions, and other popular models. In the same paper we generalize eqns. 1-8 with a temperature term, obtaining a MAP estimator that minimizes the free energy of the model. This folds deterministic annealing into EM, turning it into a quasi-global optimizer. It also provides a workaround for one known limitation of entropy minimization: It is inappropriate for learning from data that is atypical of the source process. Open questions: Our framework is currently agnostic w.r.t. two important questions: Is there an optimal trimming policy? Is there a best entropy measure? Other questions naturally arise: Can we use the entropy to estimate the peakedness of the posterior distribution, and thereby judge the appropriateness of MAP models? Can we also directly minimize the entropy of the hidden variables, thereby obtaining discriminant training? 5 Conclusion Entropic estimation is highly efficient hillclimbing procedure for simultaneously estimating model structure and parameters. It provides a clean Bayesian framework for minimizing all entropies associated with modeling, and an E-MAP algorithm that brings the structure of a randomly initialized model into alignment with hidden structures in the data via parameter extinction . The applications detailed here are three of many in which entropically estimated models have consistently outperformed maximum likelihood models in classification and prediction tasks. Most notably, it tends to produce interpretable models that shed light on the structure of relations between hidden variables and observed effects. References Brand, M. (1997). Structure discovery in conditional probability models via an entropic prior and parameter extinction. NeuraL Computation To appear; accepted 8/98. Brand, M. (1998). Pattern discovery via entropy minimization. To appear in Proc .. ArtificiaL Intelligence and Statistics #7. Merz, C. and Murphy, P. (1998). UCI repository of machine learning databases. Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2):257-286. Reynolds, D. (1992). Handwritten digit data. UNIPEN web site, hUp:llhwr.nici.kun.nl/unipen/. Donated by HP Labs, Bristol, England. Rissanen, J. (1989). Stochastic CompLexit)' and StatisticaL Inquiry. World Scientific. Stolcke, A. and Omohundro, S. (1994). Best-first model merging for hidden Markov model induction. TR-94-003, International Computer Science Institute, U.c. Berkeley. Takami, 1.-1. and Sagayama, S. (1991). Automatic generation of the hidden Markov model by successive state splitting on the contextual domain and the temporal domain. TR SP91-88, IEICE. Vovk, V. G. (1995). Minimum description length estimators under the optimal coding scheme. In Vitanyi, P., editor, Proc. ComputationaL Learning Theory / Europe, pages 237-251. Springer-Verlag. Wolfertstetter, F. and Ruske, G. (1995). Structured Markov models for speech recognition. In InternationaL Conference on Acoustics. Speech. and SignaL Processing, volume I, pages 544-7. 

1522 |@word trial:3 repository:2 compression:1 stronger:2 retraining:1 extinction:3 open:1 seek:1 mitsubishi:1 covariance:1 concise:2 thereby:2 tr:2 accommodate:1 initial:4 configuration:1 series:2 complexit:1 tuned:2 reynolds:2 current:1 com:1 contextual:1 chordal:1 numerical:1 partition:1 informative:1 subsequent:1 shape:1 mstep:1 remove:1 interpretable:3 depict:1 implying:1 intelligence:1 selected:1 device:1 tone:3 inspection:1 iso:1 record:1 detecting:1 provides:3 contribute:1 readability:1 successive:1 differential:1 qualitative:1 manner:1 introduce:1 acquired:1 notably:1 expected:1 behavior:1 multi:1 ijw:1 automatically:1 little:1 window:1 inappropriate:1 becomes:1 begin:1 estimating:3 notation:1 colloquially:1 mass:2 agnostic:1 kind:1 interpreted:2 minimizes:4 substantially:1 developed:1 dying:1 unified:1 extremum:1 sparsification:2 guarantee:1 assert:1 temporal:2 berkeley:1 donated:1 shed:1 superiority:1 appear:2 segmenting:1 understood:1 local:1 timing:1 tends:1 despite:1 approximately:1 plus:1 extinguishes:1 initialization:4 challenging:1 someone:1 hmms:15 range:1 practical:1 camera:1 testing:1 digit:4 procedure:1 logz:3 wholly:2 scant:1 radial:1 melodic:2 onto:1 operator:1 applying:2 writing:2 equivalent:1 map:12 conventional:5 missing:1 maximizing:1 deterministic:1 go:1 automaton:1 simplicity:1 splitting:2 subgraphs:1 estimator:18 rule:1 egn:1 handle:1 coordinate:1 variation:1 exact:2 iearned:1 velocity:1 satisfying:1 particularly:1 recognition:2 labeled:1 database:1 observed:2 bottom:1 capture:2 calculate:2 compressing:1 connected:1 triad:1 movement:1 removed:1 substantial:1 intuition:1 environment:1 gross:1 workaround:1 xio:2 reward:1 concision:1 dynamic:1 trained:1 weakly:3 predictive:1 writer:2 basis:1 accelerate:2 easily:3 train:1 mahalonobis:1 artificial:1 apparent:1 quite:1 larger:2 valued:1 heuristic:2 reconstruct:1 skews:1 statistic:4 descriptive:2 sequence:7 advantage:1 uci:2 loop:1 entered:1 mixing:1 description:3 inducing:1 sgi:1 convergence:1 empty:1 eccentricity:1 cluster:1 produce:1 converges:1 rotated:1 depending:1 pose:1 ij:4 strong:1 involves:1 indicate:3 judge:1 appropriateness:1 stochastic:1 human:2 premise:2 fix:1 extension:1 proximity:1 exp:1 predict:1 bump:1 matthew:1 optimizer:1 entropic:24 fh:1 estimation:11 proc:2 outperformed:1 label:1 currently:1 minimization:3 gaussian:3 always:1 avoid:1 office:3 derived:2 leapfrog:1 improvement:1 consistently:1 bernoulli:1 likelihood:4 indicates:2 contrast:2 summarizing:1 whence:1 posteriori:1 typically:2 lj:1 hidden:12 relation:2 quasi:1 i1:1 pixel:3 issue:1 classification:4 extraneous:1 special:1 field:3 elongation:1 identical:2 foreground:3 others:1 logx:2 simplify:1 few:1 inflates:1 randomly:1 simultaneously:1 individual:1 murphy:2 consisting:1 freedom:2 interest:1 trimming:9 message:1 highly:3 alignment:1 mdl:3 mixture:1 nl:1 light:1 adjoining:1 ambient:1 improbable:1 logarithm:2 initialized:3 re:4 maxent:3 overcomplete:1 fitted:1 merl:1 modeling:4 logp:2 maximization:1 deviation:1 predictor:1 reported:1 thickness:1 combined:2 person:1 international:2 discriminating:1 off:1 together:1 quickly:1 ilj:1 ambiguity:1 possibly:1 slowly:1 tile:2 derivative:1 leading:1 return:1 coding:2 register:1 ad:1 try:1 lab:1 analyze:1 start:2 bayes:1 minimize:2 accuracy:1 became:1 musical:1 ensemble:1 yield:2 identify:1 rabiner:2 generalize:1 bayesian:3 handwritten:1 lambert:2 lli:1 none:1 occupant:1 lighting:1 drive:1 bristol:1 stroke:3 simultaneous:1 inquiry:1 whenever:1 against:1 nonetheless:1 energy:1 naturally:1 associated:2 transposed:1 handwriting:3 sampled:1 dataset:1 concentrating:1 popular:1 color:2 cap:2 appears:1 day:2 follow:3 maximally:1 done:3 though:1 strongly:1 until:1 working:1 eqn:3 hand:1 web:2 horizontal:1 brings:1 aj:1 scientific:1 ieice:1 effect:2 consisted:1 multiplier:1 normalized:1 hence:1 assigned:1 laboratory:1 during:3 self:2 eqns:2 criterion:1 trying:1 omohundro:2 demonstrate:1 confusion:2 motion:2 temperature:1 reasoning:1 image:2 ranging:1 novel:1 began:1 common:1 superior:1 multinomial:4 volume:1 interpretation:1 approximates:1 cambridge:1 automatic:1 hp:1 had:1 stable:1 europe:1 longer:1 add:2 dominant:1 posterior:7 multivariate:1 showed:1 recent:1 perplexity:2 claimed:1 verlag:1 binary:1 minimum:1 lowerdimensional:1 impose:1 nici:1 employed:1 converge:1 maximize:1 signal:4 ii:2 branch:1 smoother:1 segmented:1 faster:1 england:1 cross:2 bach:5 long:1 coded:1 iog:1 ellipsis:1 calculates:1 converging:1 prediction:3 basic:1 anomalous:1 variant:1 vision:1 expectation:2 background:3 separately:1 interval:1 annealing:1 diagram:2 source:1 unlike:1 ascent:1 recording:1 tend:1 virtually:1 surviving:4 odds:4 peakedness:1 near:3 stolcke:1 indycam:1 variety:2 independence:1 identified:1 opposite:1 reduce:1 simplifies:2 utility:1 trimmed:4 effort:1 speech:3 compositional:1 detailed:1 listed:1 pinched:1 ten:2 takami:2 induces:1 tutorial:1 estimated:12 track:1 write:1 discrete:2 diatonic:1 sparsifying:1 key:2 rissanen:2 deleted:1 clarity:1 clean:1 backward:3 wasted:1 graph:1 fraction:1 sum:1 nga:1 baumwelch:1 inverse:1 run:1 uncertainty:2 opaque:1 discern:1 almost:3 prefer:1 uninterpretable:1 entirely:1 hi:2 dwell:1 vitanyi:1 accidental:1 fold:1 yielded:1 activity:6 constraint:1 extremely:1 relatively:1 structured:2 smaller:1 slightly:1 em:3 wi:2 constr:1 making:1 taken:3 equation:3 count:1 tractable:1 reversal:1 serf:1 gaussians:1 original:2 compress:1 top:3 remaining:1 ensure:1 clustering:1 iix:1 calculating:1 music:1 ellipse:1 question:3 olw:1 damage:1 receptive:3 diagonal:2 gradient:1 detrimental:1 subspace:1 reversed:1 pilj:2 mapped:1 separating:1 distance:1 hmm:7 collected:1 discriminant:1 toward:3 induction:1 length:5 modeled:1 minimizing:1 equivalently:1 kun:1 tropic:4 broadway:1 policy:1 negated:1 allowing:2 upper:2 vertical:1 observation:2 markov:6 arc:2 finite:2 truncated:1 sparsified:1 tonic:2 frame:5 introduced:1 namely:1 extensive:1 acoustic:1 learned:1 hour:1 bar:1 dynamical:1 pattern:3 summarize:1 program:1 including:2 video:3 event:1 treated:1 force:1 turning:1 representing:1 scheme:3 chorale:9 improve:1 occupancy:1 temporally:1 started:1 conventionally:3 logw:1 entropically:9 prior:14 literature:2 discovery:6 removal:1 review:1 relative:2 loss:1 fully:1 expect:1 interesting:1 limitation:1 generation:1 proven:1 versus:1 degree:1 editor:1 systematically:1 famously:1 pi:3 row:2 supported:3 surprisingly:1 free:1 bias:2 weaker:1 side:1 institute:1 taking:2 stokke:1 sparse:1 calculated:1 transition:17 stand:1 contour:1 t5:1 cumulative:1 forward:2 author:1 adaptive:2 world:1 excess:1 pruning:2 approximate:2 ml:4 global:1 incoming:1 assumed:1 spatio:1 search:1 iterative:2 pen:4 continuous:1 expanding:1 obtaining:2 improving:1 electric:1 sagayama:2 domain:2 spread:1 noise:3 arise:1 fair:1 body:1 site:2 en:5 position:3 msec:1 lie:1 pe:4 atypical:1 removing:1 minute:2 transitioning:1 xt:3 specific:3 showing:1 learnable:1 evidence:7 bivariate:1 essential:2 grouping:1 merging:2 gained:1 texture:1 magnitude:1 illustrates:2 sparser:1 entropy:21 simply:1 explore:1 visual:1 hillclimbing:1 lagrange:1 compressor:1 monotonic:1 springer:1 underflow:1 chance:1 extracted:1 ma:1 conditional:5 consequently:1 exposition:1 room:2 absence:1 content:1 change:2 vovk:2 accepted:2 oix:1 merz:2 brand:10 meaningful:1 ew:1 hup:1 constructive:1 correlated:1

Optimizing Classifiers for Imbalanced Training Sets Grigoris Karakoulas Global Analytics Group Canadian Imperial Bank of Commerce 161 Bay St., BCE-ll, Toronto ON, Canada M5J 2S8 Email: karakoulOcibc.ca John Shawe-Taylor Department of Computer Science Royal Holloway, University of London Egham, TW20 OEX England Email: jstOdcs.rhbnc.ac.uk Abstract Following recent results [9 , 8] showing the importance of the fatshattering dimension in explaining the beneficial effect of a large margin on generalization performance, the current paper investigates the implications of these results for the case of imbalanced datasets and develops two approaches to setting the threshold. The approaches are incorporated into ThetaBoost, a boosting algorithm for dealing with unequal loss functions. The performance of ThetaBoost and the two approaches are tested experimentally. Keywords: Computational Learning Theory, Generalization, fat-shattering, large margin, pac estimates, unequal loss, imbalanced datasets 1 Introduction Shawe-Taylor [8] demonstrated that the output margin can also be used as an estimate of the confidence with which a particular classification is made. In other words if a new example has an output value we]) clear of the threshold we can be more confident of the associated classification than when the output value is closer to the threshold. The current paper applies this result to the case where there are different losses associated with a false positive, than with a false negative. If a significant number of data points are misclassified we can use the criterion of minimising the empirical loss. If, however, the data is correctly classified the empirical loss is zero for all correctly separating hyperplanes. It is in this case that the approach can provide insight into how to choose the hyperplane and threshold. In summary, the paper suggests ways in which a hyperplane should be optimised for imbalanced datasets where the loss associated with misclassifying the less prevalent class is higher. 254 2 G. Karakoulas and J Shawe-Taylor Background to the Analysis Definition 2.1 [3} Let F be a set of real-valued functions. We say that a set of points X is ,-shattered by F if there are real numbers rx indexed by x E X such that for all binary vectors b indexed by X, there is a function fb E F realising dichotomy b with margin ,. The fat-shattering dimension Fat.:F of the set F is a function from the positive real numbers to the integers which maps a value, to the size of the largest ,-shattered set, if this is finite, or infinity otherwise. In general we are concerned with classifications obtained by thresholding real-valued functions. The classification values will be {-I, I} instead of the usual {O, I} in order to simplify some expressions. Hence, typically we will consider a set F of functions mapping from an input space X to the reals. For the sake of simplifying the presentation of our results we will assume that the threshold used for classification is O. The results can be extended to other thresholds without difficulty. Hence we implicitly use the classification functions H = T(F) = {T(J) : f E F}, where T(f) is the function f thresholded at O. We will say that f has , margin on the training set {(Xi, Yi) : i = 1, ... , m} , if minl<i<m{yd(Xi)} = ,. Note that a positive margin implies that T(f) is consistent. - - Definition 2.2 Given a real-valued function f : X --+ [-1, 1] used for classification by thresholding at 0, and probability distribution P on X x {-I, I}, we use er p (f) to denote the following probability erp(f) = P{(x, y) : yf(x) ::; O}. Further suppose o ::; TJ ::; 1, then we use er p(fITJ) to denote the probability er p (f ITJ) = P { ( x, y) : y f ( x) ::; 0 II f ( x ) I 2: TJ}? The probability erp(fITJ) is the probability of misclassification of a randomly chosen example given that it has a margin of TJ or more. We consider the following restriction on the set of real-valued functions. Definition 2.3 The real-valued function class F is closed under addition of constants if TJ E ~, f E F :::> f + TJ E F. Note that the linear functions (with threshold weights) used in perceptrons [9] satisfy this property as do neural networks with linear output units. Hence, this property applies to the Support Vector Machine, and the neural network examples. We now quote a result from [8]. Theorem 2.4 [8} Let F be a class of real-valued functions closed under addition of constants with fat-shattering dimension bounded by Fat.:Fh) which is continuous from the right. With probability at least 1 - t5 over the choice of a random m sample (Xi, Yi) drawn according to P the following holds. Suppose that for some f E F, TJ > 0, 1. yd(xd 2: -TJ + 2, for all (Xi, yd in the sample, 2. n = I{i: yi/(xd 2: TJ + 2,}I, 3. n 2: 3J2m(2dln(288m) log2(12em) + In(32m 2jt5)), = Let d Fat.:Fh /6). Then the probability that a new example with margin TJ misclassified is bounded by ~ (2dlog2(288m)lOg2(12em) + log2 32;n2). 1S 255 Optimizing Classijers for Imbalanced Training Sets 3 Unequal Loss Functions We consider the situation where the loss associated with an example is different for misclassification of positive and negative examples. Let Lh(x, y) be the loss associated with the classification function h on example (x, y). For the analysis considered above the loss function is taken to be Lh(X, y) = Ih(x) - YI, that is 1 if the point x is misclassified and 0 otherwise. This is also known as the discrete loss. In this paper we consider a different loss function for classification functions. Definition 3.1 The loss function L{3 is defined as L{3(x, y) h (x) f. y, and 0, otherwise. = f3y + (1 - y), if We first consider the classical approach of minimizing the empirical loss, that is the loss on the training set. Since, the loss function is no longer binary the standard theoretical results that can be applied are much weaker than for the binary case. The algorithmic implications will, however, be investigated under the assumption we are using a hyperplane parallel to the maximal margin hyperplane. The empirical risk is given by ER(h) = 2:::1 Lf3(Xi, yd, for the training set {(Xi,Yi): i = 1, ... ,m}. Assuming that the training set can be correctly classified by the hypothesis class this criterion will not be able to distinguish between consistent hypotheses, hence giving no reason not to choose the standard maximal margin choice. However, there is a natural way to introduce the different losses into the maximal margin quadratic programming procedure [1]. Here, the constraints given are specified as Yi ((w . Xi) + 0) ~ 1, i ~ 1,2, ... , m. In order to force the hyperplane away from the positive points which will incur greater loss, a natural heuristic is to set Yi = -1 for negative examples and Yi = 1/f3 for positive points, hence making them further from the decision boundary. In the case where consistent classification is possible, the effect of this will be to move the hyperplane parallel to itself so that the margin on the positive side is f3 times that on the negative side. Hence, to solve the problem we simply use the standard maximal margin algorithm [1] and then replace the threshold 0 with 1 b = 1 + f3[(w, x+) + f3(w? x-)), (1) where x+ (x-) is one of the closest positive (negative) points. The alternative approach we wish to employ is to consider other movements of the hyperplane parallel to itself while retaining consistency. Let be the margin of the maximal margin hyperplane. We consider a consistent hyperplane hI] with margin + "I to the positive examples, and "I to the negative example. The basic analytic tool is Theorem 2.4 which will be applied once for the positive examples and once for the negative examples (note that classifications are in the set {-I, I}). ,0 ,0 ,0 - Theorem 3.2 Let ho be the maximal margin hyperplane with margin ,0, while hI] is as above with "I < ,0? Set = (,0 + "I) /2 and ,- = (,0 - "I) /2. With probability at least 1 - J over the choice of a random m sample (Xi, Yi) drawn according to P the following holds. Suppose that for ho ,+ 1. no = I{i: YihO(xd ~ 2"1 + ,o}1, 2. no ~ 3J2m(dln(288m) log2(12em) + In(8/J)), Let d+ = FatF(r+ /6) and d- = FatF(r- /6). Then we can bound the expected loss by G. Karakoulas and 1. Shawe- Taylor 256 Proof: Using Theorem 2.4 we can bound the probability of error given that the correct classification is positive in terms of the expression with the fat shattering dimension d+ and n = no, while for a negative example we can bound the probability of error in terms of the expression with fat shattering dimension d- and n = m. Hence, the expected loss can be bounded by taking the maximum of the second bound with n+ in place of m together with a factor {3 in front of the second log term and the first bound multiplied by {3 ?? The bound obtained suggests a way of optimising the choice of "I, namely to minimise the expression for the fat shattering dimension of linear functions [9]. Solving for "I in terms of {a and {3 gives "I = {a (( W - 1) / ( W + 1) ) . (2) This choice of "I does not in general agree with that suggested by the choice of the threshold b in the previous section. In a later section we report on initial experiments for investigating the performance of these different choices. 4 The ThetaBoost Algorithm The above idea for adjusting the margin in the case of unequal loss function can also be applied to the AdaBoost algorithm [2] which has been shown to maximise the margin on the training examples and hence the generalization can be bounded in terms of the margin and the fat-shattering dimension of the functions that can be produced by the algorithm [6]. We will first develop a boosting algorithm for unequal loss functions and then extend it for adjustable margin. More specifically, assume: (i) a set of training examples (Xl, yd, ... , (Xrn, Yrn) where Xi E X and Y E Y = {-I, + I} j (ii) a weak learner that outputs hypotheses h : X -r {-I, + I} and (iii) the unequal loss function L(3 (y) of Definition 3.1. = We assign initial weight Dl (i) = w+ to the n+ positive examples and Dt{ i) wto the n- negative examples, where w+n+ + w- n- = 1. The values can be set so that w+ /w- = {3 or they can be adjusted using a validation set. The generalization of AdaBoost to the case of an unequal loss function is given as the AdaUBoost algorithm in Figure 1. We adapt theorem 1 in [7] for this algorithm. Theorem 4.1 Assuming the notation and algorithm of Figure 1, the following bound holds on the training error of H T w+li: H(xd #- Yi 11 + w-li: = H(xd #- Yi = -11::; IT Zt. (3) t=l The choice of w+ and w- will force uneven probabilities of misclassification on the training set, but to ensure that the weak learners concentrate on misclassified positive examples we define Z (suppressing the subscript) as (4) i Thus, to minimize training error we should seek to minimize Z with respect to Q' (the voting coefficient) on each iteration of boosting. Following [7], we introduce the notation W++, W_+, W+_ and W __ , where for Sl and S2 E {-I, +1} D(i) i :y, =31 ,h(x ,)=32 (5) 257 Optimizing Classifers for Imbalanced Training Sets By equating to zero the first derivative of (4) with respect to a, Z'(a), and using (5) we have - exp( -0'/ J3)W++/ ,6+exp(a/ ,6)W_+/,6+exp(a)W+_ -exp( -a)W__ = o. Letting Y = exp(a) we get a polynomial in Y: (6) where C 1 = -W++/,6, C 2 = W_+/,6, C3 = W+_, and C4 = -W__ . The root of this polynomial can be found numerically. Since Z" (a) > 0, Z' (a) can have at most one zero and this gives the unique minimum of Z(a). The solution for a from (6) is used (as at) when taking the distance of a training example from the standard threshold on each iteration of the AdaUBoost algorithm in Figure 1 as well as when combining the weak learners in H(x). The ThetaBoost algorithm searches for a positive and a negative support vector (SV) point such that the hyperplane separating them has the largest margin. Once these SV points are found we can then apply the formulas (1) and (2) of Sections 3.1 and 3.2 respectively to compute values for adjusting the threshold. See Figure 2 for the complete algorithm. Algorithm AdaUBoost(X, Y, (3) 1. Initialize Dt{i) as described above. 2. For t ? ? ? ? ? = 1, ... , T train weak learner using distribution Dt; get weak hypothesis h t ; choose at E lR ; update: Dt+l(i) = Dt(i) exp[-at(3iYih(xdl/Zt where (3i = 1/(3 if Yi = 1 and 1 if otherwise, and Zt is a normalization factor such that Li Dt+1(i) = 1; 3. Output the final hypothesis: H(x) = sgn (L'f=l atht(x)). Algorithm ThetaBoost(X, Y, (3, 6M 1. H(x) ) = AdaUBoost(X, Y, ,6); 2. Remove from the training dataset the false positive and borderline points; 3. Find the smallest H(x+) and mark this as the SV+; and remove any negative points with value greater than H(SV+); 4. Find the first negative point that is next in ranking to the SV+ and mark this as SV_; and compute the margin as the sum of distances, d+ and d_, of SV+ and SV_ from the standard threshold; 5. Check for candidate SV_ 's that are near to the current one and change the margin by at least 6M ; 6. Use SV+ and SV_ to compute the theta threshold from Eqn (1) and (2); 7. Output the final hypothesis: H(x) = sgn (L'f=l atht(x) - e) Figure 1: The AdaUBoost and Theta-Boost algorithms. 258 5 G. Karakoulas and J Shawe-Taylor Experiments The purpose of the experiments reported in this section is two-fold: (i) to compare the generalization performance of AdaUBoost against that of standard Adaboost on imbalanced datasetsj (ii) to examine the two formulas for choosing the threshold in ThetaBoost and evaluate their effect on generalization performance. For the evaluations in (i) and (ii) we use two performance measures: the average Li3 and the geometric mean of accuracy (g-mean) [4]. The latter is defined as 9 = Jprecision . recall, where .. preClSlOn = # positives correct # posItIves . . pre d?Icte d j _ recaII - # positives correct # true POSI.tJ.ves . The g-mean has recently been proposed as a performance measure that, in contrast to accuracy, can capture the "specificity" trade-off between false positives and true positives in imbalanced datasets [4]. It is also independent of the distribution of examples between classes. For our initial experiments we used the satimage dataset from the UCI repository [5] and used a uniform D 1 ? The dataset is about classifying neigborhoods of pixels in a satelite image. It has 36 continuous attributes and 6 classes. We picked class 4 as the goal class since it is the less prevalent one (9.73% of the dataset). The dataset comes in a training (4435 examples) and a test (2000 examples) set. Table 1 shows the performance on the test set of AdaUBoost, AdaBoost and C4.5 for different values of the beta parameter. It should be pointed out that the latter two algorithms minimize the total error assuming an equal loss function (13 = 1). In the case of equal loss AdaUBoost simply reduces to AdaBoost. As observed from the table the higher the loss parameter the bigger the improvement of AdaUBoost over the other two algorithms. This is particularly apparent in the values of g-mean. f3 values 1 2 4 8 16 AdaUBoost avgLoss g-mean 0.0545 0.773 0.0895 0.865 0.13 0.889 0.1785 0.898 0.267 0.89 AdaBoost avgLoss g-mean 0.0545 0.773 0.0831 0.773 0.1662 0.773 0.3324 0.773 0.664 0.773 C4.5 avgLoss g-mean 0.724 0.0885 0.136 0.724 0.231 0.724 0.724 0.421 0.801 0.724 Table 1: Generalization performance in the SatImage dataset. Figure 2 shows the generalization performance of ThetaBoost in terms of average loss (13 = 2) for different values of the threshold (). The latter ranges from the largest margin of negative examples that corresponds to SV_ to the smallest margin of positive examples that corresponds to SV+. This range includes the values of band TJ given by formulas (I) and (2). In this experiment,sM was set to 0.2. As depicted in the figure , the margin defined by b achieves better generalization performance than the margin defined by TJ. In particular, b is closer to the value of () that gives the minimum loss on this test set. In addition, ThetaBoost with b performs better than AdaUBoost on this test set. We should emphasise, however, that the differences are not significant and that more extensive experiments are required before the two approaches can be ranked reliably. Optimizing Classifers for Imbalanced Training Sets 259 0.2.----------.----------.,-----------, 0.18 0.16 en en .3 ~0.14 l!!Q) ~ 0.12 0.1 0.08L--------L--------'----------' -50 o 50 100 Threshold e Figure 2: Average Loss L{3 (13 = 2) on test set as a function of () 6 Discussion In the above we built a theoretical framework for optimaIly setting the margin given an unequal loss function. By applying this framework to boosting we developed AdaUBoost and ThetaBoost that generalize Adaboost, a weIl known boosting algorithm, for taking into account unequal loss functions and adjusting the margin in imbalanced datasets. Initial experiments have shown that both these factors improve the generalization performance of the boosted classifier. References [lJ Corinna Cortes and Vladimir Vapnik, Machine Learning, 20, 273- 297, 1995. [2J Yoav Freund and Robert Schapire, pages 148-156 in Proceedings of the International Conference on Machine Learning, ICML '96, 1996. [3] Michael J. Kearns and Robert E. Schapire, pages 382- 391 in Proceedings of the 31st Symposium on the Foundations of Computer Science, FOCS'90, 1990. [4J Kubat, M., Holte, R. and Matwin, S., Machine Learning, 30, 195-215, 1998. [5] Merz, C.J. and Murphy, P.M. (1997). UCI repository of machine learning databases. http://www.ics.uci.edu/ mlearn/MLRepository.html. [6] R. Schapire, Y. Freund , P. Bartlett, W. Sun Lee, pages 322- 330 in Proceedings of International Conference on Machine Learning, ICML '97, 1997. [7] Robert Schapire and Yoram Singer, in Proceedings of the Eleventh Annual Conference on Computational Learning Theory, COLT'98, 1998. [8] John Shawe-Taylor, Algorithmica, 22,157-172,1998. [9J John Shawe-Taylor, Peter Bartlett, Robert Williamson and Martin Anthony, IEEE Trans. Inf. Theory, 44 (5) 1926-1940, 1998. 

1523 |@word repository:2 polynomial:2 seek:1 simplifying:1 initial:4 suppressing:1 current:3 john:3 analytic:1 remove:2 update:1 lr:1 boosting:5 toronto:1 hyperplanes:1 beta:1 symposium:1 focs:1 eleventh:1 introduce:2 li3:1 expected:2 examine:1 bounded:4 notation:2 kubat:1 developed:1 voting:1 xd:5 fat:10 classifier:2 uk:1 unit:1 positive:21 maximise:1 before:1 optimised:1 subscript:1 yd:5 equating:1 suggests:2 analytics:1 range:2 commerce:1 unique:1 borderline:1 procedure:1 tw20:1 empirical:4 confidence:1 word:1 pre:1 specificity:1 get:2 risk:1 applying:1 xrn:1 restriction:1 www:1 map:1 demonstrated:1 insight:1 suppose:3 programming:1 hypothesis:6 particularly:1 database:1 observed:1 capture:1 sun:1 movement:1 trade:1 solving:1 incur:1 learner:4 matwin:1 train:1 london:1 dichotomy:1 choosing:1 apparent:1 heuristic:1 valued:6 solve:1 say:2 otherwise:4 itself:2 final:2 maximal:6 karakoulas:4 uci:3 combining:1 develop:1 ac:1 keywords:1 implies:1 come:1 concentrate:1 correct:3 attribute:1 sgn:2 icte:1 assign:1 generalization:10 d_:1 adjusted:1 hold:3 considered:1 ic:1 exp:6 mapping:1 algorithmic:1 achieves:1 smallest:2 fh:2 purpose:1 wto:1 quote:1 largest:3 tool:1 boosted:1 improvement:1 prevalent:2 check:1 contrast:1 shattered:2 typically:1 lj:1 misclassified:4 pixel:1 classification:12 html:1 colt:1 retaining:1 initialize:1 equal:2 once:3 f3:5 shattering:7 optimising:1 icml:2 report:1 develops:1 simplify:1 employ:1 randomly:1 bce:1 murphy:1 algorithmica:1 lf3:1 evaluation:1 tj:12 implication:2 closer:2 lh:2 indexed:2 taylor:7 oex:1 theoretical:2 yoav:1 uniform:1 front:1 reported:1 sv:8 confident:1 st:2 international:2 lee:1 off:1 j2m:2 michael:1 together:1 itj:1 posi:1 choose:3 derivative:1 li:3 account:1 w_:3 includes:1 coefficient:1 satisfy:1 ranking:1 fatshattering:1 later:1 root:1 picked:1 closed:2 parallel:3 minimize:3 accuracy:2 fitj:2 generalize:1 weak:5 produced:1 rx:1 classified:2 mlearn:1 fatf:2 email:2 definition:5 against:1 associated:5 proof:1 dataset:6 adjusting:3 recall:1 higher:2 dt:6 adaboost:7 eqn:1 yf:1 yrn:1 effect:3 true:2 hence:8 ll:1 mlrepository:1 criterion:2 complete:1 performs:1 image:1 recently:1 s8:1 extend:1 classifers:2 numerically:1 significant:2 consistency:1 pointed:1 shawe:7 longer:1 closest:1 imbalanced:10 recent:1 optimizing:4 inf:1 binary:3 yi:12 minimum:2 greater:2 holte:1 minl:1 ii:4 reduces:1 england:1 adapt:1 minimising:1 bigger:1 j3:1 basic:1 iteration:2 normalization:1 background:1 addition:3 xdl:1 integer:1 near:1 canadian:1 iii:1 concerned:1 idea:1 minimise:1 expression:4 rhbnc:1 bartlett:2 f:1 peter:1 clear:1 band:1 schapire:4 http:1 sl:1 misclassifying:1 dln:2 correctly:3 discrete:1 group:1 threshold:16 imperial:1 drawn:2 erp:2 thresholded:1 sum:1 place:1 decision:1 investigates:1 bound:7 hi:2 distinguish:1 fold:1 quadratic:1 annual:1 infinity:1 constraint:1 sake:1 martin:1 department:1 according:2 beneficial:1 em:3 making:1 taken:1 agree:1 singer:1 letting:1 multiplied:1 apply:1 away:1 egham:1 alternative:1 corinna:1 ho:2 ensure:1 log2:4 yoram:1 giving:1 classical:1 move:1 usual:1 distance:2 separating:2 reason:1 assuming:3 minimizing:1 vladimir:1 robert:4 negative:13 reliably:1 zt:3 adjustable:1 datasets:5 sm:1 finite:1 situation:1 extended:1 incorporated:1 canada:1 namely:1 required:1 specified:1 c3:1 extensive:1 c4:3 unequal:9 boost:1 trans:1 able:1 suggested:1 built:1 royal:1 misclassification:3 difficulty:1 natural:2 force:2 ranked:1 improve:1 theta:2 geometric:1 freund:2 loss:32 validation:1 foundation:1 consistent:4 thresholding:2 bank:1 classifying:1 summary:1 side:2 weaker:1 explaining:1 taking:3 emphasise:1 boundary:1 dimension:7 fb:1 t5:1 made:1 implicitly:1 dlog2:1 dealing:1 global:1 investigating:1 xi:9 continuous:2 search:1 bay:1 table:3 ca:1 williamson:1 investigated:1 anthony:1 s2:1 n2:1 atht:2 realising:1 en:2 wish:1 xl:1 candidate:1 theorem:6 formula:3 showing:1 pac:1 er:4 cortes:1 dl:1 ih:1 false:4 vapnik:1 importance:1 margin:31 depicted:1 simply:2 applies:2 corresponds:2 goal:1 presentation:1 satimage:2 replace:1 experimentally:1 change:1 specifically:1 hyperplane:11 kearns:1 total:1 merz:1 perceptrons:1 holloway:1 uneven:1 support:2 mark:2 latter:3 evaluate:1 tested:1

Batch and On-line Parameter Estimation of Gaussian Mixtures Based on the Joint Entropy Yoram Singer AT&T Labs singer@research.att.com Manfred K. Warmuth University of California, Santa Cruz manfred@cse.ucsc.edu Abstract We describe a new iterative method for parameter estimation of Gaussian mixtures. The new method is based on a framework developed by Kivinen and Warmuth for supervised on-line learning. In contrast to gradient descent and EM, which estimate the mixture's covariance matrices, the proposed method estimates the inverses of the covariance matrices. Furthennore, the new parameter estimation procedure can be applied in both on-line and batch settings. We show experimentally that it is typically faster than EM, and usually requires about half as many iterations as EM. 1 Introduction Mixture models, in particular mixtures of Gaussians, have been a popular tool for density estimation, clustering, and un-supervised learning with a wide range of applications (see for instance [5, 2] and the references therein). Mixture models are one of the most useful tools for handling incomplete data, in particular hidden variables. For Gaussian mixtures the hidden variables indicate for each data point the index of the Gaussian that generated it. Thus, the model is specified by ajoint density between the observed and hidden variables. The common technique used for estimating the parameters of a stochastic source with hidden variables is the EM algorithm. In this paper we describe a new technique for estimating the parameters of Gaussian mixtures. The new parameter estimation method is based on a framework developed by Kivinen and Warmuth [8] for supervised on-line learning. This framework was successfully used in a large number of supervised and un-supervised problems (see for instance [7, 6, 9, 1]). Our goal is to find a local minimum of a loss function which, in our case, is the negative log likelihood induced by a mixture of Gaussians. However, rather than minimizing the Parameter Estimation of Gaussian Mixtures 579 loss directly we add a tenn measuring the distance of the new parameters to the old ones. This distance is useful for iterative parameter estimation procedures. Its purpose is to keep the new parameters close to the old ones. The method for deriving iterative parameter estimation can be used in batch settings as well as on-line settings where the parameters are updated after each observation. The distance used for deriving the parameter estimation method in this paper is the relative entropy between the old and new joint density of the observed and hidden variables. For brevity we tenn the new iterative parameter estimation method the joint-entropy (JE) update. The JE update shares a common characteristic with the Expectation Maximization [4, 10] algorithm as it first calculates the same expectations. However, it replaces the maximization step with a different update of the parameters. For instance, it updates the inverse of the covariance matrix of each Gaussian in the mixture, rather than the covariance matrices themselves. We found in our experiments that the JE update often requires half as many iterations as EM. It is also straightforward to modify the proposed parameter estimation method for on-line setting where the parameters are updated after each new observation. As we demonstrate in our experiments with digit recognition, the on-line version of the JE update is especially useful in situations where the observations are generated by a nonstationary stochastic source. 2 Notation and preliminaries Let S be a sequence of training examples (Xl, X2, ..? , XN) where each Xi is a ddimensional vector in ~d. To model the distribution of the examples we use m ddimensional Gaussians. The parameters of the i-th Gaussian are denoted by 8 i and they include the mean-vector and the covariance matrix The density function of the ith Gaussian, denoted P(xI8d, is We denote the entire set of parameters of a Gaussian mixture by 8 = {8i }:: 1 = {Wi, Pi' C i }::l where w = (WI, ... , w m ) is a non-negative vector of mixture coefficients such that 2:::1 Wi = 1. We denote by P(xI8) = 2:;:1 w;P(xI8d the likelih~od of an observation x according to a Gaussian mixture with parameters_8. Let 8 i and 8 i be two Gaussian distributions. For brevity, we d~note by E; (Z) and E j (Z) the expectation of a random variable Z with respect to 8i and 8 j ? Let f be a parametric function whose parameters constitute a matrix A = (a;j). We denote by {) f / {)A the matrix of partial derivatives of f with respect to the elements in A. That is, the ij element of {) f / {)A is {) f / {)aij. Similarly, let B = (bij(x)) a matrix whose elements are functions of a scalar x. Then, we denote by dB / dx the matrix of derivatives of the elements in B with respect to x, namely, the ij element of dB / dx is dbij (x) / dx. 3 The framework for deriving updates Kivinen and Warmuth [8] introduced a general framework for deriving on-line parameter updates. In this section we describe how to apply their framework for the problem of Y. Singer and M. K. Warmuth 580 parameter estimation of Gaussian mixtures in a batch setting. We later discuss how a simple modification gives the on-line updates. Given a set of data points S in ~d and a number m, the goal is to find a set of m Gaussians that minimize the loss on the data, denoted as loss(SI8). For density estimation the natural loss function is the negative log-likelihood of the data loss(SI8) = -(I/ISI) In P(SI8) ~f -(I/ISI) L: xES In P(xI8). The best parameters which minimize the above loss cannot be found analytically. The common approach is to use iterative methods such as EM [4, 10] to find a local minimizer of the loss. In an iterative parameter estimation framework we are given the old set of parameters 8 t and we need to find a set of new parameters 8 t +1 that induce smaller loss. The framework introduced by Kivinen and Warmuth [8] deviates from the common approaches as it also requires to the new parameter vector to stay "close" to the old set of parameters which incorporates all that was learned in the previous iterations. The distance of the new parameter setting 8 t +1 from the old setting 8 t is measured by a non-negative distance function Ll(8 t +1 , 8 t ). We now search for a new set of parameters 8 t +1 that minimizes the distance summed with the loss multiplied by 17. Here 17 is a non-negative number measuring the relative importance of the distance versus the loss. This parameter 17 will become the learning rate of the update. More formally, the update is found by setting 8 t+1 = arg mineUt(8) whereU t (8) = Ll(8,8 t ) + 17loss(SI8) + A(L:::1 Wi -1). (We use a Lagrange multiplier A to enforce the constraint that the mixture coefficients sum to one.) By choosing the apropriate distance function and 17 = 1 one can show that EM becomes the above update. For most distance functions and learning rates the minimizer of the function U t (8) cannot be found analytically as both the distance function and the log-likelihood are usually non-linear in 8. Instead, we expand the log-likelihood using a first order Taylor expansion around the old parameter setting. This approximation degrades the further the new parameter values are from the old ones, which further motivates the use of the distance function Ll(8, 8 t ) (see also the discussion in [7]). We now seek a new set of parameters 8 t + 1 = argmineVt(8) where m Vt(8) = ~(8, 0 t ) + '7 (loss(510 t ) + (8 - 0 t ) . V' e l0ss(510t)) + A(L w. - 1) . (1) .=1 Here V' eloss(SI8t) denotes the gradient of the loss at 8 t . We use the above method Eq. (1) to derive the updates of this paper. For density estimation, it is natural to use the relative entropy between the new and old density as a distance. In this paper we use the joint density between the observed (data points) and hidden variables (the indices of the Gaussians). This motivates the name joint-entropy update. 4 Entropy based distance functions We first consider the relative entropy between the new and old parameter parameters of a single Gaussian. Using the notation introduced in Sec. 2, the relative entropy between two Gaussian distributions denoted by 8i , 8i is ~(8., 8i) def = [ P(xI0.) JXE~d P(xI0i) In P(xI8.) dx zI}n le.1 le.1 -z- - I IE-(( -)Te--)) z ' X -I-'i ? (X -I-'i 1-(( X + zEi - I I-'i) Te? ( x-I-'. )) Parameter Estimation of Gaussian Mixtures 581 Using standard (though tedious) algebra we can rewrite the expectations as follows: A(8-- i, 8) -i U ICil = 2"1] n -;;:;ICil -d 2 1 (C-1C-) + 2"tr i i + 2"1(J.li - J.l )T C i-1(J.li - J.li ) . (2) The relative entropy between the new and the old mixture models is the following - ~(0,0) f def - f P(xI8) = ix P(xI0) In P(xI0)dx = ix ~- L~1 w.P(xI8.) 7:: w.P(xI0.)ln ~:1 w.P(xI0./ x . (3) Ideally, we would like to use the above distance function in V t to give us an update of in terms of 8. However, there isn't a closed form expression for Eq. (3). Although the relative entropy between two Gaussians is a convex function in their parameters, the relative entropy between two Gaussian mixtures is non-convex. Thus, the loss function V t (e) may have multiple minima, making the problem of finding arg mine V t (e) difficult. e In order to sidestep this problem we use the log-sum inequality [3] to obtain an upper bound for the distance function ~(e, 8). We denote this upper bound as Li(e, 8). =L: m - W, In ;;;, w, _=1 + L: m w, j - , p(xle ,) In p(xle I ) dx = P(x e,l _=1 x L: m - W, In -;;;, w, + ,=1 L: m - - WI~(e" e,l . (4) 1=1 We call the new distance function Li(e, 8) the joint-entropy distance. Note that in this distance the parameters of Wi and Wi are "coupled" in the sense that it is a convex combination of the distances 6.(8 i , 8d. In particular, Li(8 , 8) as a function of the parameters Wi, Pi' Ci does not remain constant any more when the parameters of the individual Gaussians are permuted. Furthermore, Li (e, 8) is also is sufficiently convex so that finding the minimizer of V t is possible (see below). 5 The updates We are now ready to derive the new parameter estimation scheme. This is done by setting the partial derivatives of V t , with respect to to O. That is, our problem consists of solving the following equations e, a~(e, e) _ 1) - 151 aw, a In p(5Ie) +>- = aw, 0, a~(e, e) _ - 1) 151 aJ.L, a In P(5Ie) = aJ.L , 0, a~(e , e) ac, 1) aln p(5Ie) 151 ac, - = o. We now use the fact that Ci and thus C;l is symmetric. The derivatives of Li(e, 8), as defined by Eq. (4) and Eq. (2), with respect to Wi, Pi and C\, are In -W. w. ICd + 1 + -21 In -ICd aE(0,0) ac. 1 ( -1 - ) 1()TC-1 () + -tr 2 C Ci + -2"",,,,,,,,,,,"", (5) ll . - II . II -" t (6) alii aE(0,0) - d -2 __ 1 - (C-- 1 C-1) 2 Wi i + . . (7) Y Singer and M. K. Warmuth 582 To simplify the notation throughout the rest of the paper we define the following variables f3.(x) d ef = P(xI0i) ( d ef wi P (xI0i) P(x\0) and (X i x) = P(x\0) = P ('1t x, 0i) = wif3i(X ) . The partial derivatives of the log-likelihood are computed similarly: oln P(SI0) = OWi OC. = ~a ( ) L.; P' x X? s (8) ~ w.P(xI0.) -1 ( L.; P(xI0) C i X-I-I.) oln P(SI0) 01-1. oIn P(SI0) ~ P(xI0i) L.; P(xI0) X?S = x?s = (9) ~s _l ~ wiP(xI0.) (C:- 1 2 ~ -1 ( L.;(X.(x)C. X-I-Ii) L.; P(xI0) ? _ C:- 1 ( ? x _ .)( 1-1. x _ 1-1. )TC:-1) ? x?s -t L(X,(x)(Ci 1 - C i 1 (x-l-li)(X-I-I.f c ;-t). (10) x?s We now need to decide on an order for updating the parameter classes Wi, Pi ' and C i . We use the same order that EM uses, namely, Wi, then Pi' and finally, C i . (After doing one pass over all three groups we start again using the same order.) Using this order results in a simplified set of equations as several terms in Eq. (5) cancel out. Denote the size of the sample by N = lSI. We now need to sum the derivatives from Eq. (5) and Eq. (8) while using the fact that the Lagrange multiplier). simply assures that the new weight Wi sum to one. By setting the result to zero, we get that w. t- E:l WJ exp (-N Ex? s f3i(X?) (11) Similarly, we sum Eq. (6) and Eq. (9), set the result to zero, and get that I-li t-I-I. + ~ Lf3i(X) (x -I-Ii)' (12) x?s Finally, we do the same for C i . We sum Eq. (7) and Eq. (10) using the newly obtained Pi' Cit t- Ci 1 + ~ Lf3.(x) (Cit - C;-l(X -I-I.)(x -l-lifCi1) . x?s (13) We call the new iterative parameter estimation procedure the joint-entropy (JE) update. To summarize, the JE update is composed of the following alternating steps: We first calculate for each observation x the value !3i(X) = P(xI8;}j P(xI8) and then update the parameters as given by Eq. (11), Eq. (12), and Eq. (13). The JE update and EM differ in several aspects. First, EM uses a simple update for the mixture we!ghts w . Second, EM uses the expectations (with respect to the current parameters) of the sufficient statistics [4] for Pi and C; to find new sets of mean vectors and covariance matrices. The JE uses a (slightly different) weighted average of the observation and, in addition, it adds the old parameters. The learning rate TJ determines the proportion to be used in summing the old parameters and the newly estimated parameters. Last, EM estimates the covariance matrices Ci whereas the new update estimates the inverses, C;l, of these matrices. Thus, it is potentially be more stable numerically in cases where the covariance matrices have small condition number. To obtain an on-line procedure we need to update the parameters after each new observation at a time. That is, rather than summing over all xES, for a new observation Xt, we update Parameter Estimation of Gaussian Mixtures -3.0 I II I JE ot8_1 .9 -3.1 J . "!/ /018=1.5 .: / ot8:1 .1../ / ota=l .OS ~~ I ~ -32 --::::_________ 'i / /. _~" EM __-0':;:;--< S r'" ! -0170 - 0 171 ~ l-//' ./' ~-3.3 583 EM , , r EU ,, ._...- ~----7---!----:-,--7---:----!---' /( ,/ -3.4 -<), lo, rr" ..... BJ ............... .9'" -0' o 50 100 150 200 Number 01 iterations 250 300 10 15 ~ ._- ~ ~ ................. .......... E'" ~ ~ ~ ~ Figure 1: Left: comparison of the convergence rate of EM and the JE update with different learning rates. Right: example of a case where EM initially increases the likelihood faster than the JE update. the parameters and get a new set of parameters 8 t +1 using the current parameters 8 t ? The new parameters are then used for inducing the likelihood of the next observation Xt+ 1. The on-line parameter estimation procedure is composed of the following steps: Xj e, 1. Set: (3i (Xt ) = PP(Xj e) . 2. Parameter updates: (a) Wj f- Wj exp (-1]t(3j (xt)) / + (b) J,lj f- J,lj 1]t 1 (c) Ci f- Ci 1 I:j=1 Wj exp ( -1]t(3j (xt)) (3j (xt) (Xt - J,lj) + 1]t (3j(xt) (Cil - Ci 1 (Xt - J,lj)(Xt - J,lj)TCi1). To guarantee convergence of the on-line update one should use a diminishing learning rate, that is 1]t -t 0 as t -t 00 (for further motivation see [lID. 6 Experiments We conducted numerous experiments with the new update. Due to the lack of space we describe here only two. In the first experiment we compared the JE update and EM in batch settings. We generated data from Gaussian mixture distributions with varying number of components (m 2 to 100) and dimensions (d 2 to 20). Due to the lack of space we describe here results obtained from only one setting. In this setting the examples were generated by a mixture of 5 components with w = (0.4 , 0.3,0.2,0.05,0.05). The mean vectors were the 5 standard unit vectors in the Euclidean space 1R5 and we set all of covariances matrices to the identity matrix. We generated 1000 examples. We then run EM and the JE update with different learning rates (1] 1.9,1.5,1.1,1.05). To make sure that all the runs will end in the same local maximum we fist performed three EM iterations. The results are shown on the left hand side of Figure 1. In this setting, the JE update with high learning rates achieves much faster convergence than EM. We would like to note that this behavior is by no means esoteric - most of our experiments data yielded similar results. = = = We found a different behavior in low dimensional settings. On the right hand side of Figure 1 we show convergence rate results for a mixture containing two components each of which is a single dimension Gaussians. The mean of the two components were located Y. Singer and M. K. Warmuth 584 at 1 and -1 with the same variance of 2. Thus, there is a significant "overlap" between the two Gaussian constituting the mixture. The mixture weight vector was (0 .5,0 .5). We generated 50 examples according to this distribution and initialized the parameters as fol0.01,1-'2 -0.01, 0"1 0"2 2, WI W2 0.5 We see that initially lows: 1-'1 EM increases the likelihood much faster than the JE update. Eventually, the JE update convergences faster than EM when using a small learning rate (in the example appearing in Figure 1 we set 'rJ = 1.05). However, in this setting, the JE update diverges when learning rates larger than 'rJ 1.1 are used. This behavior underscores the advantages of both methods. EM uses a fixed learning rate and is guaranteed to converge to a local maximum of the likelihood, under conditions that typically hold for mixture of Gaussians [4, 12]. the JE update, on the other hand, encompasses a learning rate and in many settings it converges much faster than EM. However, the superior performance in high dimensional cases demands its price in low dimensional "dense" cases. Namely, a very conservative learning rate, which is hard to tune, need to be used. In these cases, EM is a better alternative, offering almost the same convergence rate without the need to tune any parameters. = = = = = = = Acknowledgments Thanks to Duncan Herring for careful proof reading and providing us with interesting data sets. References [1] E. Bauer, D. Koller, and Y. Singer. Update rules for parameter estimation in Bayesian networks. In Proc. of the 13th Annual Con! on Uncertainty in AI, pages 3-13, 1997. [2] C.M. Bishop. Neural Networks and Pattern Recognition. Oxford Univ. Press, 1995. [3] Thomas M. Cover and Joy A Thomas. Elements of Information Theory. Wiley, 1991. [4] AP. Dempster, N.M. Laird, and D.B. Rubin. Maximum-likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B39:1-38, 1977. [5] R.O. Duda and P.E. Hart. Pattern Classification and Scene Analysis. Wiley, 1973. [6] D. P. Helmbold, J. Kivinen, and M.K. Warmuth. Worst-case loss bounds for sigmoided neurons. In Advances in Neural Information Processing Systems 7, pages 309-315, 1995. [7] D.P. Helmbold, R.E. Schapire, Y.Singer, and M.K. Warmuth. A comparison of new and old algorithms for a mixture estimation problem. Machine Learning, Vol. 7, 1997. [8] J. Kivinen and M.K. Warmuth. Additive versus exponentiated gradient updates for linear prediction. Information and Computation, 132(1): 1-64, January 1997. [9] J. Kivinen and M.K. Warmuth. Relative loss bounds for multidimensional regression problems. In Advances in Neural Information Processing Systems 10, 1997. [10] R.A Redner and H.E Walker. Mixture densities, maximum likelihood and the EM algorithm. SIAM Review, 26(2), 1984. [11] D.M. Titterington, A.EM. Smith, and U.E. Makov. Statistical Analysis of Finite Mixture Distributions. Wiley, 1985. [12] C.E Wu. On the convergence properties of the EM algorithm. Annals of Stat., 11 :95103, 1983. 

1525 |@word version:1 proportion:1 duda:1 tedious:1 si8:4 seek:1 covariance:9 b39:1 tr:2 tci1:1 att:1 offering:1 current:2 com:1 od:1 dx:6 herring:1 cruz:1 additive:1 ota:1 update:38 joy:1 half:2 tenn:2 warmuth:12 ith:1 smith:1 manfred:2 cse:1 ucsc:1 become:1 consists:1 behavior:3 themselves:1 isi:2 becomes:1 estimating:2 notation:3 aln:1 minimizes:1 developed:2 titterington:1 finding:2 guarantee:1 multidimensional:1 unit:1 local:4 modify:1 oxford:1 ap:1 therein:1 range:1 acknowledgment:1 digit:1 procedure:5 induce:1 get:3 cannot:2 close:2 straightforward:1 convex:4 helmbold:2 rule:1 deriving:4 updated:2 annals:1 us:5 element:6 recognition:2 updating:1 dbij:1 located:1 observed:3 worst:1 calculate:1 wj:4 eu:1 oln:2 dempster:1 ideally:1 mine:1 rewrite:1 solving:1 algebra:1 joint:7 univ:1 describe:5 choosing:1 whose:2 larger:1 furthennore:1 statistic:1 laird:1 sequence:1 rr:1 advantage:1 xi8:7 ajoint:1 inducing:1 convergence:7 diverges:1 converges:1 derive:2 ac:3 stat:1 measured:1 ij:2 eq:14 ddimensional:2 indicate:1 differ:1 stochastic:2 preliminary:1 hold:1 around:1 sufficiently:1 exp:3 bj:1 achieves:1 purpose:1 estimation:22 proc:1 si0:3 successfully:1 tool:2 weighted:1 gaussian:20 rather:3 varying:1 likelihood:11 underscore:1 contrast:1 sense:1 typically:2 entire:1 lj:5 initially:2 hidden:6 diminishing:1 koller:1 expand:1 arg:2 classification:1 denoted:4 summed:1 f3:1 r5:1 cancel:1 simplify:1 composed:2 individual:1 lf3:1 mixture:30 tj:1 partial:3 incomplete:2 old:14 taylor:1 euclidean:1 initialized:1 instance:3 cover:1 measuring:2 maximization:2 conducted:1 aw:2 thanks:1 density:9 siam:1 ie:4 stay:1 again:1 containing:1 derivative:6 sidestep:1 li:10 makov:1 sec:1 coefficient:2 later:1 performed:1 lab:1 closed:1 doing:1 start:1 minimize:2 variance:1 characteristic:1 bayesian:1 pp:1 proof:1 con:1 newly:2 popular:1 redner:1 supervised:5 done:1 though:1 furthermore:1 hand:3 o:1 lack:2 aj:2 name:1 multiplier:2 analytically:2 alternating:1 symmetric:1 ll:4 oc:1 demonstrate:1 ef:2 common:4 superior:1 permuted:1 sigmoided:1 xi0:10 numerically:1 significant:1 ai:1 similarly:3 stable:1 add:2 inequality:1 vt:1 minimum:2 converge:1 ii:5 fist:1 multiple:1 rj:2 faster:6 hart:1 calculates:1 prediction:1 regression:1 ae:2 expectation:5 iteration:5 addition:1 whereas:1 walker:1 source:2 w2:1 rest:1 sure:1 induced:1 db:2 incorporates:1 call:2 nonstationary:1 xj:2 zi:1 f3i:1 icd:2 expression:1 constitute:1 useful:3 santa:1 tune:2 cit:2 schapire:1 lsi:1 estimated:1 vol:1 group:1 sum:6 run:2 inverse:3 uncertainty:1 throughout:1 almost:1 decide:1 wu:1 duncan:1 def:2 bound:4 guaranteed:1 replaces:1 yielded:1 annual:1 constraint:1 x2:1 scene:1 aspect:1 according:2 combination:1 smaller:1 remain:1 em:29 slightly:1 wi:15 lid:1 modification:1 making:1 handling:1 ln:1 equation:2 assures:1 discus:1 eventually:1 singer:7 end:1 gaussians:9 multiplied:1 apply:1 enforce:1 appearing:1 batch:5 alternative:1 thomas:2 denotes:1 clustering:1 include:1 yoram:1 especially:1 society:1 parametric:1 degrades:1 gradient:3 distance:19 index:2 providing:1 minimizing:1 difficult:1 potentially:1 negative:5 motivates:2 upper:2 observation:9 neuron:1 finite:1 descent:1 january:1 situation:1 introduced:3 namely:3 specified:1 xle:2 california:1 learned:1 usually:2 below:1 pattern:2 reading:1 summarize:1 encompasses:1 royal:1 overlap:1 natural:2 kivinen:7 scheme:1 numerous:1 ready:1 icil:2 coupled:1 isn:1 alii:1 deviate:1 review:1 relative:9 loss:17 zei:1 interesting:1 versus:2 sufficient:1 rubin:1 share:1 pi:7 lo:1 last:1 aij:1 side:2 exponentiated:1 wide:1 bauer:1 dimension:2 xn:1 simplified:1 constituting:1 keep:1 summing:2 xi:1 un:2 iterative:7 search:1 expansion:1 oin:1 dense:1 owi:1 motivation:1 je:18 cil:1 wiley:3 xl:1 ix:2 bij:1 xt:10 bishop:1 wip:1 x:2 importance:1 ci:9 te:2 demand:1 entropy:13 tc:2 simply:1 lagrange:2 scalar:1 minimizer:3 determines:1 goal:2 identity:1 careful:1 price:1 experimentally:1 hard:1 conservative:1 pas:1 formally:1 brevity:2 ex:1

Recurrent Cortical Amplification Produces Complex Cell Responses Frances S. Chance~ Sacha B. Nelson~ and L. F. Abbott Volen Center and Department of Biology Brandeis University Waltham, MA 02454 Abstract Cortical amplification has been proposed as a mechanism for enhancing the selectivity of neurons in the primary visual cortex. Less appreciated is the fact that the same form of amplification can also be used to de-tune or broaden selectivity. Using a network model with recurrent cortical circuitry, we propose that the spatial phase invariance of complex cell responses arises through recurrent amplification of feedforward input. Neurons in the network respond like simple cells at low gain and complex ceUs at high gain. Similar recurrent mechanisms may playa role in generating invariant representations of feedforward input elsewhere in the visual processing pathway. 1 INTRODUCTION Synaptic input to neurons in the primary visual cortex is primarily recurrent, arising from other cortical cells. The dominance of this type of connection suggests that it may play an important role in cortical information processing. Previous studies proposed that recurrent connections amplify weak feedforward input to the cortex (Douglas et aI., 1995) and selectively amplify tuning for specific stimulus characteristics, such as orientation or direction of movement (Douglas et aI., 1995; Ben-Yishai et aI., 1995; Somers et aI., 1995; Sompolinsky and Shapley, 1997). Cortical cooling and shocking experiments provide evidence that there is cortical amplification through recurrent connections, but they do not show increases in orientation or direction selectivity as a result of this amplification (Ferster et aI., 19%; Chung and Ferster, 1998). Recurrent connections can also decrease neuronal selectivity through the same form of amplification, generating responses that are insensitive to certain stimulus features. Although the ability to sharpen tuning may be an important feature in cortical processing, the capacity to broaden tuning for particular stimulus attributes is also desirable. Neurons in the primary visual cortex can be divided into two classes based on their re- Recurrent Cortical Amplification Produces Complex Cell Responses 91 sponses to visual stimuli such as counterphase and drifting sinusoidal gratings. Simple cells show tuning for orientation, spatial frequency, and spatial phase of a grating (Movshon et aI., 1978a). Complex cells exhibit orientation and spatial frequency tuning, but are insensitive to spatial phase (Movshon et aI., 1978b). A counterphase grating, s(x, t) = cos(Kx - cp) cos(wt), is one in which the spatial phase, CP,and spatial frequency, K, are held constant but the contrast, s(x, t), varies sinusoidally in time at some frequency w. In response to a counterphase grating, the activity of a simple cell oscillates at the same frequency as the stimulus, w. A complex cell response is modulated at twice the frequency, 2w. To create a drifting grating of frequency 1/, s( x, t) = cos( K x - I/t), the spatial phase and spatial frequency are held constant but the grating is moved at velocity 1// K. A simple cell response to a drifting grating is highly modulated at frequency 1/, while a complex cell response to a drifting grating is elevated but relatively unmodulated. The differences between complex and simple cell responses are a direct consequence of the complex cell spatial phase insensitivity. Previous models of complex cells generate spatial-phase invariant responses through converging sets of feedforward inputs with a wide range of spatial phase preferences but similar orientation and spatial frequency selectivities (Hubel and Wiesel, 1962; Mel et aI., 1998). These models do not incorporate recurrent connections between complex cells, which are known to be particularly strong (Toyama et al., 1981). We propose that the spatial phase invariance of complex cell responses can arise from a broadening of spatial phase tuning by cortical amplification (Chance et aI., 1998). The model neurons exhibit simple cell behavior when weakly coupled and complex cell behavior when strongly coupled, suggesting that the two classes of neurons in the primary visual cortex may arise from the same basic cortical circuit. 2 THE MODEL The activity of neuron i in the model network is characterized by a firing rate rio Each neuron sums feedforward and recurrent input and responds as described by the standard rate-model equation dri Trdi ~ = Ii + L.J Wijrj - rio Ii represents the feedforward input to cell i, W ij is the weight of the synapse from neuron j to neuron i, and Tr is a time constant. Previous studies have suggested that, for a neuron receiving many inputs, Tr is small, closer to a synaptic time constant than the membrane time constant (Ben-Yishai et al., 1995; Treves, 1993). Thus we choose Tr = 1 ms. The feedforward input describes the response of a simple cell with a Gabor receptive field Ii = [/ dxGi(x) 1 00 dt' H(t')s(x, t - t')] + ' where s(x, t) represents the contrast function of the visual stimulus and the notation [ 1+ indicates rectification. The temporal response function is (Adelson and Bergen, 1985) (at')5 (at')7) H(t') = exp(-at') ( - - - - 5! where we use a 7!' = l/ms. The spatial filter is a Gabor function, G= exp ( - 2:; ) cos(kix - <Pi)' where Ui determines the spatial extent of the receptive field, ki is the preferred spatial frequency, and <Pi is the preferred spatial phase. The values of <Pi are equally distributed F S. Chance, S. B. Nelson and L. F Abbott 92 over the interval [-180 0 , 180 0 ) . To give the neurons a realistic bandwidth, (j i is chosen such that ki(ji = 2.5 . Initially we consider a simplified case in which k i = 1 for all cells. Later we consider the spatial frequency selectivity of neurons in the network and allow the value of k i to range from 0 to 3.5 cycles/deg. In this paper we assume that the model network describes one orientation column of the primary visual cortex, and thus all neurons have the same orientation tuning. All stimuli are of the optimal orientation for the network. Spatial phase tuning is selectively broadened in the model because the strength of a recurrent connection between two neurons is independent of the spatial phase selectivities of their feedforward inputs. In the model with all k i = 1, the recurrent input is determined by 9 Wij = (N -1) ' for all i =I j. N is the number of cells in the network, and 0 ~ 9 < gmax, where gmax is the largest value of 9 for which the network remains stable. In this case gmax = 1. 3 RESULTS The steady-state solution of the rate-model equation is given by Ti = Ii + L WijTj . To solve this equation, we express the rates and feedfoward inputs in terms of a complete set of eigenvectors ~r of the recurrent weight matrix, L Wij~r = ~/.l~r for I-L = 1,2, ... , N, where ~/.l are the eigenvalues. The solution is then This equation displays the phenomenon of cortical amplification if one or more of the eigenvalues is near one. If we assume only one eigenvalue, ~1 , is close to one, the factor 1~1 in the denominator causes the I-L = 1 term to dominate and we find Ti ~ ~t L Ij~J (1 ~1) -1. The input combination L Ij~J dominates the response, determining selectivity, and this mode is amplified by a factor 1/(1 - ~1)' We refer to this amplification factor as the cortical gain. In the case where W ij = g/(N - 1) for i =I j, the largest eigenvalue is ~1 = 9 and the corresponding eigenvector has all components equal to each other. For 9 near one, the recurrent input to neuron i is then proportional to Lj [cos(~-1>j)]+ which, for large numbers of cells with uniformly placed preferred spatial phases 1>i, is approximately independent of ~, the spatial phase of the stimulus. When 9 is near zero, the network is at low gain and the response of neuron i is roughly proportional to its feedforward input, [cos(~ - 1>j)]+, and is sensitive to spatial phase. The response properties of simple and complex cells to drifting and counterphase gratings are duplicated by the model neuron, as shown in figure 1. For low gain (gain = 1, top panels of figures lA and IB), the neuron acts as a simple cell and its activity is modulated at the same frequency as the stimulus (w for counterphase gratings and v for drifting gratings). At high gain (gain = 20), the neuron responds like a complex cell, exhibiting frequency doubling in the response to a counterphase grating (bottom panel of Figure 1A) and an elevated DC response to a drifting grating (bottom panel, Figure IB). Intermediate gain (gain =5) produces intermediate behavior (middle panels). The basis of this model is that the amplified mode is independent of spatial phase. If the amplified mode depends on spatial frequency or orientation, neurons at high gain can be selective for these attributes. To show that the model can retain selectivity for other 93 Recurrent Cortical Amplification Produces Complex Cell Responses 500 1000 500 1000 500 1000 1~~1VY)!\< o 500 1000 time (ms) time (ms) Figure 1: The effects of recurrent input on the responses of a neuron in the model network. The responses of one neuron to a 2 Hz counterphase grating (A) and to a 2 Hz drifting grating (B) are shown for different levels of network gain. From top to bottom in A and B, the gain of the network is one, five, and twenty. stimulus characteristics while maintaining spatial phase insensitivity, we allowed the spatial frequency selectivity which each neuron receives from its feedforward input, ki' to vary from neuron to neuron and also modified the recurrent weight matrix so that the strength of the connection between two neurons, i and j, depends on k i - k j . The dependence is modeled as a difference of Gaussians, so the recurrent weight matrix is now 9 [ ((k i - kj W ij = (N -1) 2exp 20'~ )2) - ((k i - kj exp 20'; )2)] . Thus neurons that receive feedforward input tuned for similar spatial frequencies excite each other and neurons that receive very differently tuned feedforward input inhibit each other. This produces complex cells that are tuned to a variety of spatial frequencies, but are still insensitive to spatial phase (see figure 2). The spatial frequency tuning curve width is primarily determined by 0' c = 0.5 cycle/deg and 0' s = 1 cycle/deg. Cells within the same network do not have to exhibit the same level of gain. In previous figures, the gain of the network was determined by a parameter 9 that described the strength of all the connections between neurons. In figure 3, the recurrent input to cell i is determined by W ij = gi/(N - 1), where the values of gi are chosen randomly within the allowed range. The gain of each neuron depends on the value of gi for that neuron. As shown in figure 3, a range of complex and simple cell behaviors now coexist within the same network. 4 DISCUSSION In the recurrent model we have presented, as in Hubel and Wiesel's feedforward model, the feedforward input to a complex cell arises from simple cells. Measurements by Alonso and F S. Chance, S. B. Nelson and L. F Abbott 94 A c:~ 100 ~ 50 o a. (/) xctS E #. O-+-~r----r--~----' -180 -90 o 90 phase (deg) 180 o 1 2 3 spatial frequency (cyc/deg) Figure 2: Neurons in a high-gain network can be selective for spatial frequency while remaining insensitive to spatial phase. Both spatial phase and spatial frequency tuning are included in the feedforward input. A) The spatial phase tuning curves of three representative neurons from a high-gain network. B) The spatial frequency tuning curves of the same three neurons as in A. Martinez (1998) support this circuitry. However, direct excitatory input to complex cells arising from the LGN has also been reported (Hoffman and Stone, 1971; Singer et aI., 1975; Ferster and Lindstrom, 1983). Supporting these measurements is evidence that certain stimuli can excite complex cells without strong excitation of simple cells (Hammond and Mackay, 1975, 1977; Movshon, 1975) and also that complex cells still respond when simple cells are silenced (Malpeli, 1983; Malpeli et ai, 1986; Mignard and Malpeli, 1991). In accordance with this, the weak feedforward simple cell input in the recurrent model could probably be replaced by direct LGN input, as in the feedforward model of Mel et al. (1998). The proposed model makes definite predictions about complex cell responses. If the phaseinvariance of complex cell responses is due to recurrent interactions, manipulations that modify the balance between feedforward and recurrent drive should change the nature of the responses in a predictable manner. The model predicts that blocking local excitatory connections should turn complex cells into simple cells. Conversely, manipulations that increase cortical gain should make simple cells act more like complex cells. One way to increase cortical gain may be to block or partially block inhibition since this increases the influence of excitatory recurrent connections. Experiments along these lines have been performed, and blockade of inhibition does indeed cause simple cells to take on complex cell properties (Sillito, 1975; Shulz et aI., 1993). In a previous study, Hawken, Shapley, and Grosof (1996) noted that the temporal frequency tuning curves for complex cells are narrower for counterphase stimuli than for drifting stimuli. The recurrent model reproduces this result as long as the integration of synaptic inputs depends on temporal frequency. Such a dependence is provided, for example, by short-term synaptic depression (Chance et aI., 1998). Hubel and Wiesel's feedforward model (1962) does not reproduce this effect, even with synaptic depression at the synapses. We have presented a model of primary visual cortex in which complex cell response characteristics arise from recurrent amplification of simple cell responses. The complex cell responses in the high gain regime arise because recurrent connections selectively deamplify selectivity for spatial phase. Thus recurrent connections can act to generate invariant representation of input data. A similar mechanism could be used to produce responses that are independent of other stimulus attributes, such as size or orientation. Given the ubiquity of invariant representations in the visual pathway, this mechanism may have widespread use. 95 Recurrent Cortical Amplification Produces Complex Cell Responses 1~~~ o ~1001JV\M eft. 50 o -+-------,.--------. o 500 time (ms) 1000 500 1000 1~~~ o 500 1000 time (ms) Figure 3: Responses to a 4 Hz drifting grating of four neurons from a large network consisting of a mixture of simple and complex cells. The two traces on the left represent simple cells and the two traces on the right represent complex cells. Acknowledgements Research supported by the Sloan Center for Theoretical Neurobiology at Brandeis University, the National Science Foundation (DMS-95-03261), the W.M. Keck Foundation, the National Eye Institute (EY-11116), and the Alfred P. Sloan Foundation. References Adelson, E. H. & Bergen, J. R. Spatiotemporal energy models for the perception of motion. 1. Opt. Soc. Am. A. 2,284-299 (1985) Alonso, J-M. & Martinez, L. M. Functional connectivity between simple cells and complex cells in cat striate cortex. Nature Neuroscience 1,395-403 (1998) Ben-Yishai, R., Bar-Or, L. & Sompolinsky, H. Theory of orientation tuning in visual cortex. Proc. Natl. Acad. Sci. USA 92,3844-3848 (1995) Chance F. S., Nelson S. B. & Abbott L. F. Complex cells as cortically amplified simple cells. (submitted) Chung, S. & Ferster, D. Strength and orientation tuning of the thalamic input to simple cells revealed by electrically evoked cortical suppression. Neuron 20,1177-1189 (1998) Douglas, R. J., Koch, c., Mahowald, M., Martin , K. A. C. & Suarez, H. H. Recurrent excitation in neocortical circuits. Science 269 981-985 (1995) Ferster, D., Chung, S. & Wheat, H. Orientation selectivity ofthalamic input to simple cells of cat visual cortex. Nature 380,249-252 (1996) Ferster, D. & Lindstrom, S. An intracellular analysis of geniculo-cortical connectivity in area 17 of the cat. 1. Physiol . (Lond) 342,181-215 (1983) Hammond, P. & MacKay, D. M. Differential responses of cat visual cortical cells to textured stimuli. Exp. Brain Res. 22,427-430 (1975) 96 F S. Chance, S. B. Nelson and L. F Abbott Hammond, P. & MacKay, D. M. Differential responsiveness of simple and complex cells in cat striate cortex to visual texture. Exp. Brain Res. 30, 275-296 (1977) Hawken, M. J., Shapley, R. M. & Grosof, D. H. Temporal-frequency selectivity in monkey visual cortex. Vis. Neurosci. 13477-492 (1996) Hoffman, K. P. & Stone, J. Conduction velocity of afferents of cat visual cortex: a correlation with cortical receptive field properties. Brain Res. 32,460-466 (1971) Hubel, D. H. & Wiesel, T. N. Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. 1. Physiol.160, 106-154 (1962) Malpeli, J. G. Activity of cells in area 17 of the cat in absence of input from layer A of lateral geniculate nucleus. 1. Neurophysiol. 49,595-610 (1983) Malpeli, J. G., Lee, C., Schwark, H. D. & Weyand, T. G. Cat area 17.1. Pattern of thalamic control of cortical layers. 1. Neurophysiol. 56,1062-1073 (1986) Mel, B. W., Ruderman, D. L. & Archie, K. A. Translation-invariant orientation tuning in visual complex cells could derive from intradendritic computations. 1. Neurosci. 1843254334 (1998) Mignard, M. & Malpeli, J. G. Paths of information flow through visual cortex. Science 251, 1249-1251 (1991) Movshon, J. A. The velocity tuning of single units in cat striate cortex. 1. Physiol. 249, 445-468 (1975) Movshon, J., Thompson, I. & Tolhurst, D. Spatial summation in the receptive fields of simple cells in the cat's striate cortex. 1. Physiol. (Lond) 283, 53-77 (1978) Movshon, J., Thompson, I. & Tolhurst, D. Receptive field organization of complex cells in cat's striate cortex. 1. Physiol. (Lond) 283,79-99 (1978) Shulz, D. E., Bringuier, B. & Fregnac, Y. A complex-like structure of simple visual cortical receptive fields is masked by GABA-A intracortical inhibition. Soc .jor Neurosci . Abs. 19, 628 (1993) Sillito, A. M. The contribution of inhibitory mechanisms to the receptive field properties of neurones in the striate cortex of the cat. 1. Physiol. (Lond) 250,305-329 (1975) Singer, W., Tretter, F. & Cynader, M. Organization of cat striate cortex: a correlation of receptive-field properties with afferent and efferent connections. 1. Neurophysiol. 38, 10801098 (1975) Somers, D. C., Nelson, S. B. & Sur, M. An emergent model of orientation selectivity in cat visual cortical simple cells. 1. Neurosci. 15,5448-5465 (1995) Sompolinsky, H. & Shapley, R. New perspectives on the mechanisms for orientation selectivity. Current Opinion in Neurobiology 7, 514-522 (1997) Toyama, K., Kimura, M. & Tanaka, K. Organization of cat visual cortex as investigated by cross-correlation technique. 1. Neurophysiol. 46,202-214 (1981) Treves, A. Mean-field analysis of neuronal spike dynamics. Network 4, 259-284 (1993) 

1526 |@word middle:1 wiesel:4 tr:3 tuned:3 current:1 physiol:6 realistic:1 feedfoward:1 short:1 tolhurst:2 preference:1 kix:1 five:1 along:1 direct:3 differential:2 pathway:2 shapley:4 manner:1 indeed:1 roughly:1 behavior:4 brain:3 provided:1 notation:1 circuit:2 panel:4 eigenvector:1 monkey:1 kimura:1 temporal:4 toyama:2 ti:2 act:3 oscillates:1 control:1 unit:1 broadened:1 unmodulated:1 accordance:1 modify:1 local:1 consequence:1 acad:1 firing:1 path:1 approximately:1 twice:1 evoked:1 suggests:1 conversely:1 co:6 range:4 block:2 definite:1 area:3 gabor:2 amplify:2 close:1 coexist:1 influence:1 center:2 thompson:2 dominate:1 weyand:1 play:1 velocity:3 particularly:1 cooling:1 predicts:1 blocking:1 bottom:3 role:2 suarez:1 wheat:1 cycle:3 sompolinsky:3 movement:1 decrease:1 inhibit:1 predictable:1 ui:1 dynamic:1 weakly:1 cynader:1 basis:1 textured:1 neurophysiol:4 differently:1 emergent:1 cat:16 solve:1 ability:1 gi:3 eigenvalue:4 propose:2 interaction:2 malpeli:6 insensitivity:2 amplified:4 amplification:14 moved:1 keck:1 produce:7 generating:2 ben:3 derive:1 recurrent:31 ij:6 grating:16 soc:2 strong:2 waltham:1 exhibiting:1 direction:2 attribute:3 filter:1 opinion:1 opt:1 summation:1 koch:1 exp:6 circuitry:2 vary:1 proc:1 geniculate:1 sensitive:1 largest:2 create:1 hoffman:2 modified:1 indicates:1 contrast:2 suppression:1 am:1 rio:2 bergen:2 lj:1 initially:1 wij:2 france:1 selective:2 lgn:2 reproduce:1 orientation:16 spatial:42 integration:1 mackay:3 field:10 equal:1 biology:1 represents:2 adelson:2 stimulus:15 primarily:2 shulz:2 randomly:1 national:2 replaced:1 phase:24 consisting:1 ab:1 organization:3 highly:1 mixture:1 natl:1 yishai:3 held:2 closer:1 gmax:3 re:4 theoretical:1 column:1 sinusoidally:1 mahowald:1 masked:1 sacha:1 archie:1 reported:1 conduction:1 varies:1 spatiotemporal:1 retain:1 lee:1 receiving:1 fregnac:1 connectivity:2 choose:1 chung:3 suggesting:1 sinusoidal:1 de:1 intracortical:1 sloan:2 afferent:2 depends:4 vi:1 later:1 performed:1 thalamic:2 contribution:1 characteristic:3 weak:2 intradendritic:1 hammond:3 drive:1 submitted:1 synapsis:1 synaptic:5 energy:1 frequency:26 dm:1 efferent:1 gain:21 duplicated:1 dt:1 response:31 synapse:1 strongly:1 binocular:1 correlation:3 receives:1 ruderman:1 widespread:1 mode:3 usa:1 effect:2 blockade:1 width:1 noted:1 mel:3 steady:1 excitation:2 m:6 stone:2 complete:1 neocortical:1 cp:2 motion:1 functional:2 ji:1 insensitive:4 elevated:2 eft:1 refer:1 measurement:2 ai:13 tuning:17 sharpen:1 stable:1 cortex:21 inhibition:3 playa:1 perspective:1 manipulation:2 selectivity:15 certain:2 responsiveness:1 ey:1 ii:4 desirable:1 characterized:1 cross:1 long:1 divided:1 equally:1 converging:1 prediction:1 basic:1 denominator:1 enhancing:1 represent:2 cell:68 receive:2 interval:1 probably:1 hz:3 dri:1 flow:1 near:3 feedforward:19 intermediate:2 revealed:1 variety:1 geniculo:1 architecture:1 bandwidth:1 movshon:6 cyc:1 neurones:1 cause:2 depression:2 eigenvectors:1 tune:1 generate:2 vy:1 inhibitory:1 neuroscience:1 arising:2 alfred:1 express:1 dominance:1 four:1 jv:1 douglas:3 abbott:5 sum:1 respond:2 somers:2 hawken:2 ki:3 layer:2 display:1 activity:4 strength:4 lond:4 relatively:1 martin:1 department:1 combination:1 electrically:1 gaba:1 membrane:1 describes:2 invariant:5 rectification:1 equation:4 remains:1 turn:1 mechanism:6 singer:2 gaussians:1 ubiquity:1 drifting:10 broaden:2 top:2 remaining:1 maintaining:1 spike:1 receptive:9 primary:6 dependence:2 striate:7 responds:2 exhibit:3 sci:1 capacity:1 lateral:1 alonso:2 nelson:6 extent:1 sur:1 modeled:1 balance:1 trace:2 twenty:1 neuron:37 supporting:1 neurobiology:2 volen:1 dc:1 treves:2 connection:13 tanaka:1 suggested:1 bar:1 perception:1 pattern:1 regime:1 eye:1 coupled:2 kj:2 acknowledgement:1 determining:1 proportional:2 foundation:3 nucleus:1 pi:3 translation:1 elsewhere:1 excitatory:3 placed:1 supported:1 appreciated:1 allow:1 institute:1 wide:1 distributed:1 curve:4 cortical:24 jor:1 simplified:1 brandeis:2 preferred:3 deg:5 reproduces:1 hubel:4 excite:2 sillito:2 nature:3 broadening:1 investigated:1 complex:39 intracellular:1 neurosci:4 arise:4 martinez:2 allowed:2 neuronal:2 silenced:1 representative:1 cortically:1 ib:2 lindstrom:2 specific:1 evidence:2 dominates:1 sponses:1 texture:1 kx:1 visual:22 partially:1 doubling:1 chance:7 determines:1 ma:1 narrower:1 ferster:6 absence:1 change:1 included:1 determined:4 uniformly:1 wt:1 invariance:2 la:1 selectively:3 support:1 arises:2 modulated:3 incorporate:1 phenomenon:1

Facial Memory is Kernel Density Estimation (Almost) Matthew N. Dailey Garrison W. Cottrell Department of Computer Science and Engineering U.C. San Diego La Jolla, CA 92093-0114 {mdailey,gary}@cs.ucsd.edu Thomas A. Busey Department of Psychology Indiana University Bloomington, IN 47405 busey@indiana.edu Abstract We compare the ability of three exemplar-based memory models, each using three different face stimulus representations, to account for the probability a human subject responded "old" in an old/new facial memory experiment. The models are 1) the Generalized Context Model, 2) SimSample, a probabilistic sampling model, and 3) MMOM, a novel model related to kernel density estimation that explicitly encodes stimulus distinctiveness. The representations are 1) positions of stimuli in MDS "face space," 2) projections of test faces onto the "eigenfaces" of the study set, and 3) a representation based on response to a grid of Gabor filter jets. Of the 9 model/representation combinations, only the distinctiveness model in MDS space predicts the observed "morph familiarity inversion" effect, in which the subjects' false alarm rate for morphs between similar faces is higher than their hit rate for many of the studied faces. This evidence is consistent with the hypothesis that human memory for faces is a kernel density estimation task, with the caveat that distinctive faces require larger kernels than do typical faces. 1 Background Studying the errors subjects make during face recognition memory tasks aids our understanding of the mechanisms and representations underlying memory, face processing, and visual perception. One way of evoking such errors is by testing subjects' recognition of new faces created from studied faces that have been combined in some way (e.g. Solso and McCarthy, 1981; Reinitz, Lammers, and Cochran 1992). Busey and Tunnicliff (submitted) have recently examined the extent to which image-quality morphs between unfamiliar faces affect subjects' tendency to make recognition errors. Their experiments used facial images of bald males and morphs between these images (see ? \i<" Facial Memory Is Kernel Density Estimation (Almost) 25 .. ... " <: . . \ ,';'.>,"i'.. > " I' '.' ",.,,", .,. :- . ?... ' .'~ ,.;;.' ' .. . .'... . . Il" .. r ::;;1;f~ Figure 1: Three normalized morphs from the database. Figure 1) as stimuli. In one study, Busey (in press) had subjects rate the similarity of all pairs in a large set of faces and morphs, then performed a multidimensional scaling (MDS) of these similarity ratings to derive a 6~dimensional "face space" (Valentine and Endo, 1992). In another study, "Experiment 3" (Busey and Tunnicliff, submitted), 179 subjects studied 68 facial images, including 8 similar pairs and 8 dissimilar pairs, as determined in a pilot study. These pairs were included in order to study how morphs between similar faces and dissimilar faces evoke false alanns. We call the pair of images from which a morph are derived its "parents," and the morph itself as their "child." In the experiment's test phase, the subjects were asked to make new/old judgments in response to 8 of the 16 morphs, 20 completely new distractor faces, the 36 non-parent targets and one of the parents of each of the 8 morphs. The results were that, for many of the morphlparent pairs, subjects responded "old" to the unstudied morph more often than to its studied parent. However, this effect (a morphfamiliarity inversion) only occurred for the morphs with similar parents. It seems that the similar parents are so similar to their "child" morphs that they both contribute toward an "old" (false alann) response to the morpho Researchers have proposed many models to account for data from explicit memory experiments. Although we have applied other types of models to Busey and Tunnicliff's data with largely negative results (Dailey et al., 1998), in this paper, we limit discussion to exemplar-based models, such as the Generalized Context Model (Nosofsky, 1986) and SAM (Gillund and Shiffrin, 1984). These models rely on the assumption that subjects explicitly store representations of each of the stimuli they study. Busey and Tunnicliff applied several exemplar-based models to the Experiment 3 data, but none of these models have been able to fully account for the observed similar morph familiarity inversion without positing that the similar parents are explicitly blended in memory, producing prototypes near the morphs. We extend Busey and Tunnicliff's (submitted) work by applying two of their exemplar models to additional image-based face stimulus representations, and we propose a novel exemplar model that accounts for the similar morphs' familiarity inversion. The results are consistent with the hypothesis that facial memory is a kernel density estimation (Bishop, 1995) task, except that distinctive exemplars require larger kernels. Also, on the basis of our model, we can predict that distinctiveness with respect to the study set is the critical factor influencing kernel size, as opposed to a context-free notion of distinctiveness. We can easily test this prediction empirically. 2 Experimental Methods 2.1 Face Stimuli and Normalization The original images were 104 digitized 560x662 grayscale images of bald men, with consistent lighting and background and fairly consistent position. The subjects varied in race and extent of facial hair. We automatically located the left and right eyes on each face using a simple template correlation technique then translated, rotated, scaled and cropped each image so the eyes were aligned in each image. We then scaled each image to 114x 143 to speed up image processing. Figure 1 shows three examples of the normalized morphs (the original images are copyrighted and cannot be published) . 26 M N. Dailey, G. W Cottrell and T. A. Busey 2.2 Representations Positions in multidimensional face space Many researchers have used a multidimensional scaling approach to model various phenomena in face processing (e.g. Valentine and Endo, 1992). Busey (in press) had 343 subjects rate the similarity of pairs of faces in the test set and performed a multidimensional scaling on the similarity matrix for 100 of the faces (four non-parent target faces were dropped from this analysis). The process resulted in a 6-dimensional solution with r2 = 0.785 and a stress of 0.13. In the MDS modeling results described below, we used the 6-dimensional vector associated with each stimulus as its representation. Principal component projections "Eigenfaces," or the eigenvectors of the covariance matrix for a set of face images, are a common basis for face representations (e.g. Turk and Pentland, 1991). We performed a principal components analysis on the 68 face images used in the study set for Busey and Tunnicliff's experiment to get the 67 non-zero eigenvectors of their covariance matrix. We then projected each of the 104 test set images onto the 30 most significant eigenfaces to obtain a 30-dimensional vector representing each face. l Gabor filter responses von der Malsburg and colleagues have made effective use of banks of Gabor filters at various orientations and spatial frequencies in face recognition systems. We used one form of their wavelet (Buhmann, Lades, and von der Malsburg, 1990) at five scales and 8 orientations in an 8x8 square grid over each normalized face image as the basis for a third face stimulus representation. However, since this representation resulted in a 2560-dimensional vector for each face stimulus, we performed a principal components analysis to reduce the dimensionality to 30, keeping this representation's dimensionality the same as the eigenface representation's. Thus we obtained a 30-dimensional vector based on Gabor filter responses to represent each test set face image. 2.3 Models The Generalized Context Model (GCM) There are several different flavors to the GCM. We only consider a simple sum-similarity form that will lead directly to our distinctivenessmodulated density estimation model. Our version of GCM's predicted P(old), given a representation y of a test stimulus and representations x E X of the studied exemplars, is predy = a + {3 L e- c (d x ?y )2 xEX where a and {3linearly convert the probe's summed similarity to a probability, X is the set of representations of the study set stimuli; c is used to widen or narrow the width of the similarity function, and dx,y is either Ilx - yll, the Euclidean distance between x and y or the weighted Euclidean distance VLk Wk(Xk - Yk)2 where the "attentional weights" Wk are constants that sum to 1. Intuitively, this model simply places a Gaussian-shaped function over each of the studied exemplars, and the predicted familiarity of a test probe is simply the summed height of each of these surfaces at the probe's location. Recall that two of our representations, PC projection space and Gabor filter space, are 30-dimensional, whereas the other, MDS, is only 6-dimensional. Thus allowing adaptive weights for the MDS representation is reasonable, since the resulting model only uses 8 parameters to fit 100 points, but it is clearly unreasonable to allow adaptive weights in PC and Gabor space, where the resulting models would be fitting 32 parameters to 100 points. Thus, for all models, we report results in MDS space both with and without adaptive weights, but do not report adaptive weight results for models in PC and Gabor space. SimSample Busey and Tunnicliff (submitted) proposed SimSample in an attempt to remedy the GCM's poor predictions of the human data. It is related to both GCM, in that it 1We used 30 eigenfaces because with this number, our theoretical "distinctiveness" measure was best correlated with the same measure in MDS space. 27 Facial Memory Is Kernel Density Estimation (Almost) uses representations in MDS space, and SAM (Gillund and Shiffrin, 1984), in that it involves sampling exemplars. The idea behind the model is that when a subject is shown a test stimulus, instead of a summed comparison to all of the exemplars in memory, the test probe probabilistically samples a single exemplar in memory, and the subject responds "old" if the probe's similarity to the exemplar is above a noisy criterion. The model has a similarity scaling parameter and two parameters describing the noisy threshold function. Due to space limitations, we cannot provide the details of the model here. Busey and Tunnicliff were able to fit the human data within the SimS ample framework, but only when they introduced prototypes at the locations of the morphs in MDS space and made the probability of sampling the prototype proportional to the similarity of the parents. Here, however, we only compare with the basic version that does not blend exemplars. Mixture Model of Memory (MMOM) In this model, we assume that subjects, at study time, implicitly create a probability density surface corresponding to the training set. The subjects' probability of responding "old" to a probe are then taken to be proportional to the height of this surface at the point corresponding to the probe. The surface must be robust in the face of the variability or noise typically encountered in face recognition (lighting changes, perspective changes, etc.) yet also provide some level of discrimination support (i.e. even when the intervals of possible representations for a single face could overlap due to noise, some rational decision boundary must still be constructed). If we assume a Gaussian mixture model, in which the density surface is built from Gaussian "blobs" centered on each studied exemplar, the task is a form of kernel density estimation (Bishop, 1995). We can fonnulate the task of predicting the human subjects' P(old) in this framework, then, as optimizing the priors and widths of the kernel functions to minimize the mean squared error of the prediction. However, we also want to minimize the number of free parameters in the model - parsimonious methods for setting the priors and kernel function widths potentially lead to more useful insights into the principles underlying the human data. If the priors and widths were held constant, we would have a simple two parameter model predicting the probability a subject responds "old" to a test stimulus y: predy = L oe- I!x_~1!2 2 .. xEX where a folds together the uniform prior and normalization constants, and (7 is the standard deviation of the Gaussian kernels. If we ignore the constants, however, this model is essentially the same as the version of the GCM described above. As the results section will show, this model cannot fully account for the human familiarity data in any of our representational spaces. To improve the model, we introduce two parameters to allow the prior (kernel function height) and standard deviation (kernel function width) to vary with the distinctiveness of the studied exemplar. This modification has two intuitive motivations. First, when humans are asked which of two parent faces a 50% morph is most similar to, if one parent is distinctive and the other parent is typical, subjects tend to choose the more distinctive parent (Tanaka et aI., submitted). Second, we hypothesize that when a human is asked to study and remember a set of faces for a recognition test, faces with few neighbors will likely have more relaxed (wider) discrimination boundaries than faces with many nearby neighbors. Thus in each representation space, for each studied face x, we computed d(x), the theoretical distinctiveness of each face, as the Z-scored average distance to the five nearest studied faces. We then allowed the height and width of each kernel function to vary with d(x): predy = L xEX 0(1 + cod(x?e _ I!x_yl!2 2("(l+c .. d(x?2 As was the case for GCM and SimSample, we report the results of using a weighted Euclidean distance between y and x in MDS space only. 28 M. N Dailey. G. W. Cottrell and T. A. Busey Model " MDS space GCM 0.1633 SimS ample 0.1521 MMOM 0.1601 I MDS + weights I PC projections I Gabor jets I 0.1417 0.1404 0.1528 0.1745 0.1756 0.1992 0.1624 0.1704 0.1668 Table 1: RMSE for the three models and three representations. Quality of fit for models with adaptive attentional weights are only reported for the low-dimensional representation ("MDS + weights"). The baseline RMSE, achievable with a constant prediction, is 0.2044. 2.4 Parameter fitting and model evaluation For each of the twelve combinations of models with face representations, we searched parameter space by simple hill climbing for the parameter settings that minimized the mean squared error between the model's predicted P(old) and the actual human P(old) data. We rate each model's effectiveness with two criteria. First, we measure the models' global fit with RMSE over all test set points. A model's RMSE can be compared to the baseline performance of the "dumbest" model, which simply predicts the mean human P(old) of 0.5395, and achieves an RMSE of 0.2044. Second, we evaluate the extent to which a model predicts the mean human response for each of the six categories of test set stimuli: 1) nonparent targets, 2) non-morph distractors, 3) similar parents, 4) dissimilar parents, 5) similar morphs, and 6) dissimilar morphs. If a model correctly predicts the rank ordering of these category means, it obviously accounts for the similar morph familiarity inversion pattern in the human data. As long as models do an adequate job of fitting the human data overall, as measured by RMSE, we prefer models that predict the morph familiarity inversion effect as a natural consequence of minimizing RMSE. 3 Results Table 1 shows the global fit of each model/representation pair. The SimSample model in MDS space provides the best quantitative fit. GeM generally outperforms MMOM, indicating that for a tight quantitative fit, having parameters for a linear transformation built into the model is more important than allowing the kernel function to vary with distinctiveness. Also of note is that the PC projection representation is consistently outperformed by both the Gabor jet representation and the MDS space representation. But for our purposes, the degree to which a model predicts the mean human responses for each of the six categories of stimuli is more important, given that it is doing a reasonably good job globally. Figure 2 takes a more detailed look at how well each model predicts the human category means. Even though SimSample in MDS space has the best global fit to the human familiarity ratings, it does not predict the familiarity inversion for similar morphs. Only the mixture model in weighted MDS space correctly predicts the morph familiarity effect. All of the other models underpredict the human responses to the similar morphs. 4 Discussion The results for the mixture model are consistent with the hypothesis that facial memory is a kernel density estimation task, with the caveat that distinctive exemplars require larger kernels. Whereas true density estimation would tend to deemphasize outliers in sparse areas of the face space, the human data show that the priors and kernel function widths for outliers should actually be increased. Two potentially significant problems with the work presented here are first, we experimented with several models before finding that MMOM was able to predict the morph familiarity inversion effect, and second, we are fitting a single Facial Memory Is Kernel Density Estimation (Almost) GCMlMDS SimSamplelMDS 0.6 i ~ f ~ i 0." iI:" 0.2 f ~ T SP OM OA iE" 0.4 0.2 f 0.2 0 0 .4 iE" 0.2 f ~ T SP OM 0." 0 .2 f 0 .2 ~ T SP OM OP SM 0 SimSampleIPC 0 .6 0." iil:"0A iill:: 0." 0.2 f ~ f 0.2 SP OM ~ 0.2 GCWGabor T SP OM OP SM 0 0.6 0.6 0." iill:: 0." iit'" 0." 0.2 ~ ~ t 0.2 DP SM T SP OM 0 D T SP OM 0 0.6 0.2 0.0 0.0 0 .0 OM MMOWGabor SimSample/Gabor f SP 0.0 DP SM 0 T 0.6 0.0 T 0 MMOMIPC 0 .6 OP SM OM 0.0 OP SM 0,0 SP MMOMlMDS+wts iill:: 0 T 0.6 0." GCMlPC r DP SM 0 0.0 OP SM iE" SP OM 0.6 0.0 r 0.0 T SimSamplelMDS+wts 0.6 iit'" ~ OP SM GCMlMDS+wts ~ 0.6 0 .0 OP SM f MMOMlMDS 0.6 0 .0 iE" 29 OP SM T SP OM 0 OP SM T SP OM 0 Actual _Predicted r::::=:I Figure 2: Average actual/predicted responses to the faces in each category. Key: DP = Dissimilar parents; SM Similar morphs; T Non-parent targets; SP Similar parents; DM Dissimilar morphs; D = Distractors. = = = = experiment. The model thus must be carefully tested against new data, and its predictions empirically validated. Since a theoretical distinctiveness measure based on the sparseness of face space around an exemplar was sufficient to account for the similar morphs' familiarity inversion, we predict that distinctiveness with respect to the study set is the critical factor influencing kernel size, rather than context-free human distinctiveness judgments. We can easily test this prediction by having subjects rate the distinctiveness of the stimuli without prior exposure and then determine whether their distinctiveness ratings improve or degrade the model's fit. A somewhat disappointing (though not particularly surprising) aspect of our results is that the model requires a representation based on human similarity judgments. Ideally, we would prefer to provide an information-processing account using image-based representations like eigenface projections or Gabor filter responses. Interestingly, the efficacy of the image-based representations seems to depend on how similar they are to the MDS representations. The PC projection representation performed the worst, and distances between pairs of PC representations had a correlation of 0.388 with the distances between pairs of MDS representations. For the Gabor filter representation, which performed better, the correlation is 0.517. In future work, we plan to investigate how the MDS representation (or a representation like it) might be derived directly from the face images. 30 M N. Dailey, G. W Cottrell and T A. Busey Besides providing an infonnation-processing account of the human data, there are several other avenues for future research. These include empirical testing of our distinctiveness predictions, evaluating the applicability of the distinctiveness model in domains other than face processing, and evaluating the ability of other modeling paradigms to account for this data. Acknowledgements We thank Chris Vogt for comments on a previous draft, and other members of Gary's Unbelievable Research Unit (GURU) for earlier comments on this work. This research was supported in part by NIMH grant MH57075 to GWe. References Bishop, C. M. (1995). Neural networks for pattern recognition. Oxford University Press, Oxford. Busey, T. A. (1999). Where are morphed faces in multi-dimensional face space? Psychological Science. In press. Busey, T. A. and Tunnicliff, J. (submitted). Accounts of blending, distinctiveness and typicality in face recognition. Journal of Experimental Psychology: Learning, Memory, and Cognition. Dailey, M. N., Cottrell, G. W., and Busey, T. A. (1998). Eigenfaces for familiarity. In Proceedings of the Twentieth Annual Conference of the Cognitive Science Society, pages 273-278, Mahwah, NJ. Erlbaum. Gillund, G. and Shiffrin, R. (1984). A retrieval model for both recognition and recall. Psychological Review, 93(4):411-428. J. Buhmann, M. L. and von der Malsburg, C. (1990). Size and distortion invariant object recognition by hierarchical graph matching. In Proceedings of the IJCNN International Joint Conference on Neural Networks, volume II, pages 411-416. Nosofsky, R. M. (1986). Attention, similarity, and the identification-categorization relationship. Journal of Experimental Psychology: General, 116(1):39-57. Reinitz, M., Lammers, W., and Cochran, B. (1992). Memory-conjunction errors: Miscombination of stored stimulus features can produce illusions of memory. Memory & Cognition, 20(1):1-11. Solso, R. L. and McCarthy, J. E. (1981). Prototype formation offaces: A case of pseudomemory. British Journal of Psychology, 72(4):499-503. Tanaka, J., Giles, M., Kremen, 5., and Simon, V. (submitted). Mapping attract or fields in face space: The atypicality bias in face recognition. Turk, M. and Pentland, A. (1991). Eigenfaces for recognition. The Journal of Cognitive Neuroscience, 3:71-86. Valentine, T. and Endo, M. (1992). Towards an exemplar model of face processing: The effects of race and distinctiveness. The Quarterly Journal of Experimental Psychology, 44A(4):671-703. 

1527 |@word version:3 inversion:9 achievable:1 seems:2 vogt:1 covariance:2 efficacy:1 interestingly:1 outperforms:1 surprising:1 yet:1 dx:1 must:3 cottrell:5 hypothesize:1 xex:3 discrimination:2 xk:1 caveat:2 provides:1 draft:1 contribute:1 location:2 gillund:3 positing:1 five:2 height:4 constructed:1 fitting:4 introduce:1 distractor:1 multi:1 globally:1 automatically:1 actual:3 underlying:2 finding:1 indiana:2 transformation:1 nj:1 remember:1 quantitative:2 multidimensional:4 scaled:2 hit:1 unit:1 grant:1 evoking:1 producing:1 before:1 engineering:1 influencing:2 dropped:1 limit:1 consequence:1 oxford:2 might:1 studied:10 examined:1 testing:2 illusion:1 area:1 empirical:1 gabor:12 projection:7 matching:1 get:1 onto:2 cannot:3 context:5 applying:1 exposure:1 attention:1 typicality:1 insight:1 notion:1 diego:1 target:4 us:2 hypothesis:3 recognition:12 particularly:1 located:1 predicts:7 database:1 observed:2 vlk:1 worst:1 oe:1 ordering:1 yk:1 nimh:1 asked:3 ideally:1 depend:1 tight:1 distinctive:5 completely:1 basis:3 translated:1 easily:2 joint:1 iit:2 various:2 effective:1 cod:1 formation:1 larger:3 distortion:1 ability:2 itself:1 noisy:2 obviously:1 blob:1 propose:1 aligned:1 shiffrin:3 representational:1 intuitive:1 parent:18 produce:1 categorization:1 rotated:1 object:1 wider:1 derive:1 measured:1 exemplar:18 nearest:1 op:9 job:2 c:1 predicted:5 involves:1 filter:7 centered:1 human:22 eigenface:2 require:3 blending:1 around:1 iil:1 cognition:2 predict:5 mapping:1 matthew:1 bald:2 vary:3 achieves:1 purpose:1 estimation:11 outperformed:1 infonnation:1 create:1 weighted:3 clearly:1 gaussian:4 rather:1 probabilistically:1 conjunction:1 derived:2 validated:1 consistently:1 rank:1 baseline:2 attract:1 typically:1 overall:1 orientation:2 plan:1 spatial:1 summed:3 fairly:1 field:1 shaped:1 having:2 sampling:3 look:1 future:2 minimized:1 report:3 stimulus:18 few:1 x_:1 widen:1 resulted:2 phase:1 attempt:1 investigate:1 evaluation:1 male:1 mixture:4 pc:7 behind:1 held:1 dailey:6 wts:3 facial:10 old:13 euclidean:3 theoretical:3 psychological:2 increased:1 modeling:2 earlier:1 giles:1 blended:1 applicability:1 deviation:2 uniform:1 erlbaum:1 reported:1 stored:1 guru:1 morph:11 morphs:21 combined:1 density:13 twelve:1 international:1 ie:4 probabilistic:1 together:1 nosofsky:2 von:3 squared:2 opposed:1 choose:1 cognitive:2 account:11 wk:2 explicitly:3 race:2 fonnulate:1 performed:6 doing:1 cochran:2 simon:1 rmse:7 om:12 il:1 square:1 offaces:1 responded:2 minimize:2 largely:1 judgment:3 climbing:1 identification:1 none:1 lighting:2 researcher:2 published:1 submitted:7 against:1 colleague:1 frequency:1 turk:2 dm:1 endo:3 associated:1 rational:1 bloomington:1 pilot:1 recall:2 distractors:2 dimensionality:2 carefully:1 actually:1 higher:1 response:10 though:2 correlation:3 gcm:8 quality:2 effect:6 normalized:3 true:1 remedy:1 lades:1 during:1 width:7 criterion:2 generalized:3 stress:1 hill:1 image:21 novel:2 recently:1 common:1 empirically:2 volume:1 alann:1 occurred:1 extend:1 unfamiliar:1 significant:2 morphed:1 ai:1 grid:2 had:3 similarity:12 surface:5 etc:1 mccarthy:2 perspective:1 optimizing:1 jolla:1 disappointing:1 store:1 der:3 additional:1 relaxed:1 somewhat:1 determine:1 paradigm:1 ii:2 jet:3 morpho:1 long:1 retrieval:1 prediction:7 basic:1 hair:1 essentially:1 kernel:22 represent:1 normalization:2 background:2 cropped:1 whereas:2 want:1 interval:1 comment:2 subject:20 tend:2 ample:2 member:1 effectiveness:1 call:1 near:1 affect:1 fit:9 psychology:5 reduce:1 idea:1 prototype:4 avenue:1 whether:1 six:2 reinitz:2 adequate:1 useful:1 generally:1 detailed:1 eigenvectors:2 category:5 neuroscience:1 correctly:2 key:1 four:1 threshold:1 graph:1 sum:2 convert:1 place:1 almost:4 reasonable:1 parsimonious:1 decision:1 prefer:2 scaling:4 fold:1 encountered:1 annual:1 ijcnn:1 encodes:1 nearby:1 aspect:1 speed:1 department:2 combination:2 poor:1 sam:2 modification:1 intuitively:1 outlier:2 invariant:1 taken:1 describing:1 mechanism:1 studying:1 unreasonable:1 probe:7 quarterly:1 hierarchical:1 thomas:1 original:2 responding:1 include:1 malsburg:3 society:1 blend:1 md:21 responds:2 dp:4 distance:6 attentional:2 thank:1 oa:1 degrade:1 chris:1 extent:3 toward:1 besides:1 relationship:1 providing:1 minimizing:1 potentially:2 negative:1 allowing:2 sm:13 pentland:2 variability:1 digitized:1 ucsd:1 varied:1 rating:3 introduced:1 pair:10 yll:1 unbelievable:1 narrow:1 tanaka:2 able:3 below:1 perception:1 pattern:2 built:2 including:1 memory:20 critical:2 overlap:1 natural:1 rely:1 predicting:2 buhmann:2 representing:1 improve:2 eye:2 created:1 x8:1 prior:7 understanding:1 acknowledgement:1 review:1 fully:2 men:1 limitation:1 proportional:2 degree:1 sufficient:1 consistent:5 principle:1 bank:1 supported:1 free:3 keeping:1 bias:1 allow:2 neighbor:2 eigenfaces:6 face:57 distinctiveness:17 template:1 sparse:1 boundary:2 evaluating:2 made:2 adaptive:5 san:1 projected:1 unstudied:1 ignore:1 implicitly:1 evoke:1 global:3 gem:1 grayscale:1 iill:3 table:2 reasonably:1 robust:1 ca:1 domain:1 sp:13 linearly:1 motivation:1 noise:2 alarm:1 scored:1 mahwah:1 child:2 allowed:1 solso:2 garrison:1 aid:1 position:3 explicit:1 third:1 wavelet:1 british:1 familiarity:13 bishop:3 r2:1 experimented:1 sims:2 evidence:1 false:3 sparseness:1 flavor:1 ilx:1 simply:3 likely:1 twentieth:1 visual:1 gary:2 towards:1 change:2 included:1 typical:2 determined:1 except:1 principal:3 tendency:1 la:1 experimental:4 indicating:1 support:1 searched:1 dissimilar:6 evaluate:1 tested:1 phenomenon:1 correlated:1

Orientation, Scale, and Discontinuity as Emergent Properties of Illusory Contour Shape Karvel K. Thornber NEC Research Institute 4 Independence Way Princeton, NJ 08540 Lance R. Williams Dept. of Computer Science University of New Mexico Albuquerque, NM 87131 Abstract A recent neural model of illusory contour formation is based on a distribution of natural shapes traced by particles moving with constant speed in directions given by Brownian motions. The input to that model consists of pairs of position and direction constraints and the output consists of the distribution of contours joining all such pairs. In general, these contours will not be closed and their distribution will not be scale-invariant. In this paper, we show how to compute a scale-invariant distribution of closed contours given position constraints alone and use this result to explain a well known illusory contour effect. 1 INTRODUCTION It has been proposed by Mumford[3] that the distribution of illusory contour shapes can be modeled by particles travelling with constant speed in directions given by Brownian motions. More recently, Williams and Jacobs[7, 8] introduced the notion of a stochastic completion field, the distribution of particle trajectories joining pairs of position and direction constraints, and showed how it could be computed in a local parallel network. They argued that the mode, magnitude and variance of the completion field are related to the observed shape, salience, and sharpness of illusory contours. Unfortunately, the Williams and Jacobs model, as described, has some shortcomings. Recent psychophysics suggests that contour salience is greatly enhanced by closure[2]. Yet, in general, the distribution computed by the Williams and Jacobs model does not consist of closed contours. Nor i:; it scale-invariant-doubling the distances between the constraints does not produce a comparable completion field of K. K. Thornber and L. R. Williams 832 double the size without a corresponding doubling of the particle's speeds. However, the Williams and Jacobs model contains no intrinsic mechanism for speed selection. The speeds (like the directions) must be specified a priori. In this paper, we show how to compute a scale-invariant distribution of closed contours given position constraints alone. 2 2.1 TECHNICAL DETAILS SHAPE DISTRIBUTION Consistent with our earlier work[5, 6], in this paper we do not use the same distribution described by Mumford[3] but instead assume a distribution of completion shapes consisting of straight-line base-trajectories modified by random impulses drawn from a mixture of two limiting distributions. The first distribution consists of weak but frequently acting impulses (we call this the Gaussian-limit). The distribution of these weak impulses has zero mean and variance equal to (7~. The weak impulses act at Poisson times with rate R g . The second distribution consists of strong but infrequently acting impulses (we call this the Poisson-limit). Here, the magnitude of the random impulses is Gaussian distributed with zero mean. However, the variance is equal to (72 (where (7~ ? (7;). The strong impulses act at Poisson times with rate Rp < < Particles decay with half-life equal to a parameter T. The effect is that particles tend to travel in smooth, short paths punctuated by occasional orientation discontinuities. See [5, 6]. kg. 2.2 EIGENSOURCES Let i and j be position and velocity constraints, (xi,id and (xj,Xj). Then P(jl i) is the conditional probability that a particle beginning at i will reach j. Note that these transition probabilities are not symmetric, i.e., P(j 1 i) 1= P(i 1 j). However, by time-reversal symmetry, P(j 1 i) = P(I 1 J) where I = (Xi, -Xi) and J = (Xj, -Xj). Given only the matrix of transition probabilities, P, we would like to compute the relative number of closed contours satisfying a given position and velocity constraint. We begin by noting that, due to their randomness, only increasingly smaller and smaller fractions of contours are likely to satisfy increasing numbers of constraints. Suppose we let s~l) contours start at Xi with Xi. Then (2) _ ~ Sj - ui P( J'I '/.) (1) , Si is the relative number of contours through Xj with Xj, i.e., which satisfy two constraints. In general, (n+1) _ ~ P( 'I .) (n) Sj - ui J '/, Si Now suppose we compute the eigenvector, with largest, real positive eigenvalue, and take s~1) = Si. Then clearly si n + 1 ) = AnSi. This implies that as the number of constraints satisfied increases by one, the number of contours remaining in the sample of interest decreases by A. However, the ratios of the Si remain invariant. Letting n pass to infinity, we see that the Si are just the relative number of contours through i. To summarize, having started with all possible contours, we are now left with only those bridging pairs of constraints at all past-times. By solving AS = Ps for s we know their relative numbers. We refer to the components of s as the eigensources of the stochastic completion field. Emergent Properties of Illusory Contour Shape 2.3 833 STOCHASTIC COMPLETION FIELDS Note that the eigensources alone do not represent a distribution of closed contours. In fact, the majority of contours contributing to s will not satisfy a single additional constraint. However , the following recurrence equation gives the number of contours which begin at constraint i and end at constraint j and satisfy n - 1 intermediate constraints p(n+1) (j I i) = Lk P(j I k)p(n) (k Ii) where p( 1) (j I i) = P(j Ii). Given the above recurrence equation , we can define an expression for the relative number of contours of any length which begin and end at constraint i: Ci = lim n -+ oo p(n)(i I i)/ Lj p(n)(j I)) Using a result from the theory of positive matrices[l}, it is possible to show that the above expression is simply Ci = Si 8 d Lj Sj8j where sand s are the right and left eigenvectors of P with largest positive real eigenvalue, i.e., AS = Ps and AS = pTs. Because of the time-reversal symmetry of P, the right and left eigenvectors are related by a permutation which exchanges opposite directions, i.e. , 8i = St. Finally, given sand s, it is possible to compute the relative number of closed contours through an arbitrary position and velocity in the plane, i.e., to compute the stochastic completion field. If", = (x, x) is an arbitrary position and velocity in the plane, then C(",) = >.s~s Li P(", I i)Si . Lj P(j I",)8j gives the relative probability that a closed contour will pass through ",. Note, that this is a natural generalization of the Williams and Jacobs[7] factorization of the completion field into the product of source and sink fields. 2.4 SCALE-INVARIANCE Under the restriction that particles have constant speed, the transition probability matrix, P, becomes block-diagonal. Each block corresponds to a different possible speed, 'Y- Since the components of any given eigenvector will be confined to a single block , we can consider P to be a function of, and solve: A(r) s(r) = P(r)s(r) Let Amax (r) be the largest positive real eigenvalue of P(r) and let ,max be the speed where Amax (r) is maximized. Then Sma x (rma x), i.e., the eigenvector of P (rmax) associated with Amax (rma x), is the limiting distribution over all spatial scales. 3 3.1 EXPERIMENTS EIGHT POINT CIRCLE Given eight points spaced uniformly around the perimeter of a circle of diameter, d = 16, we would like to find the distribution of directions through each point and the corresponding completion field (Figure 1 (left)). Neither the order of traversal , directions, i.e., xdlxil, or speed , i.e. , , = IXil. are specified a priori. In all of our experiments, we sample direction at 5? intervals. Consequently, there are 72 discrete directions and 576 position-direction pairs, i.e., P(r) is of size 576 x 576. 1 lThe parameters defining the distribution of completion shapes are T = Rga~ = 0.0005 and 'T = 9.5. For simplicity, we assume the pure Gaussian-limit case described in [6] . K. K. Thornber and L. R. Williams 834 ? ? ? ? ? ? ? (two sizes) o b .- w I / " Circle ,0 ? " I Point w~ I . a / Eight ,..,,,o ?a I i Il. '0 xo og~~~~~~~~~~~~~~~ :l~ o 0 " ,0 '" 30 20 c d Figure 1: Left: (a) The eight position constraints. Neither the order of traversal, directions, or speed are specified a priori. (b) The eigenvector, Smax (,max) represents the limiting distribution over all spatial scales. (c) The product of smaxC!max) and smaxC!max). Orientations tangent to the circle dominate the distribution of closed contours. (d) The stochastic completion field, C, due to smaxC!max). Right: Plot of magnitude of maximum positive real eigenvalue, >'max, vs. logl.l (1/,) for eight point circle with d = 16.0 (solid) and d = 32.0 (dashed). ~ == ~ 11 ~ ==::J U Figure 2: Observers report that as the width of the arms increases, the shape of the illusory contour changes from a circle to a square[4]. First, we evaluated Amax b) over the velocity interval [1.1- 1 , 1.1- 3o J using standard numerical routines and plotted the magnitude of the largest, real positive eigenvalue, Amax vs. logl.l(l/,). The function reaches its maximum value at '"'(max:::::: 1.1- 2 Consequently, the eigenvector, Smax (1.1 - 2 ?) represents the limiting distribution over all spatial scales (Figure 1 (right)). ?. Next, we scaled the test Figure by a factor of two, i.e., d' = 32.0 and plotted A~axb) over the same interval (Figure 1 (right)). We observe that A~ax(1.1-x+7) :::::: Amax (1.1- X ), i.e., when plotted using a logarithmic x-axis, the functions are identical except for a translation. It follows that '"'(~ax :::::: logl.1 7 x '"'(max:::::: 2.0 x '"'(max' This confirms the scale-invariance of the system-doubling the size of the Figure results in a doubling of the selected speed. 3.2 KOFFKA CROSS The Koffka Cross stimulus (Figure 2) has two basic degrees of freedom which we call diameter (i.e. , d) and arm width (i .e., w) (Figure 3 (a)). We are interested in how Emergent Properties of Illusory Contour Shape (a) (b) r---~ o (-0 5w . O.5d) 835 ( O.5w ,O.Sd ) (--O.Sd , 05w) (e) (OSd , 05w) r----- .- -......, d U (--OSd, --05w) ( 0 Sd ,-O.5w ) (--05w. -O.5d) (O.5w. --OSd ) n u (d) ------, Figure 3: (a) Koffka Cross showing diameter, d, and width , w. (b) Orientation and position constraints in terms of d and w. The normal orientation at each endpoint is indicated by the solid lines while the dashed lines represent plus or minus one standard deviation (i.e. , 12 .8?) of the Gaussian weighting function. (c) Typically perceived as square. (d) Typically perceived as circle. The positions of the line endpoints is the same. the stochastic completion field changes as these parameters are varied. Observers report that as the width of the arms increases, the shape of the illusory contour changes from a circle to a square[4]. The endpoints of the lines comprising the Koftka Cross can be used to define a set of position and orientation constraints (Figure 3 (b)). The position constraints are specified in terms of the parameters, d and w. The orientation constraints take the form of a Gaussian weighting function which assigns higher probabilities to contours passing through the endpoints with orientations normal to the lines. 2 The prior probabilities assigned to each positiondirection pair by the Gaussian weighting function form a diagonal matrix, D: where P(r) is the transition probability matrix for the random process at scale " A(r) is an eigenvalue of Q(,), and s(r) is the corresponding eigenvector. Let Amax(r) be the largest positive real eigenvalue of Q(r) and let ,max be the scale where Amax(r) is maximized. Then smax(rmax), i.e., the eigenvector of Q(rmax) associated with Amax (rma x), is the limiting distribution over all spatial scales. First, we used a Koffka Cross where d = 2.0 and w = 0.5 and evaluated Amax (r) over the velocity interval [8.0 x 1.1- 1 , 8.0 x 1.1- 8 ?] using standard numerical routines. 3 The function reaches its maximum value at ,max::::; 8.0 X 1.1- 62 (Figure 4 (left)). Observe that the completion field due to the eigenvector, smax(8.0 x 1.1- 62 ), is dominated by contours of a predominantly circular shape (Figure 4 (right)). We then uniformly scaled the Koffka Cross Figure by a factor of two, i.e., d' = 4.0 and 20bserve that Figure 3 (c) is perceived as a square while Figure 3 (d) is perceived as a circle. Yet the positions of the line endpoints is the same. It follows that the orientations of the lines affect the percept. We have chosen to model this dependence through the use of a Gaussian weighting function which favors contours passing through the endpoints of the lines in the normal direction. It is possible to motivate this based on the statistics of natural scenes. The distribution of relative orientations at contour crossings is maximum at 90? and drops to nearly zero at 0 0 and 180 0 ? 3The parameters defining the distribution of completion shapes were: T = RgO'~ = 0.0005, T = 9.5, ?p = O'~/T = 100.0 and Rp = 1.0 X 10- 8 . As an anti-aliasing measure, the transition probabilities, P(j I i) , were averaged over initial conditions modeled as Gaussians of variance = = 0.00024 and O'J = 0.0019. See [6]. 0'; 0'; K. K. Thornber and L. R. Williams 836 Koffke ... Crosses (TWO sizes) '"'~ LIl~ _ -_=0>== , I a o -o > c 00 -. CI~ Wo -o , Ira ...a ~o a D.. ; ~/-- X0 / i / og-HTnTrnTnTrnTnTrnTnTnTrnTn~~ ~ci a 20 '0 60 .. 0 x Figure 4: Left: Plot of magnitude of maximum positive real eigenvalue, >'max, vs. logl.l (1h) for Koffka Crosses with d = 2.0 and w = 0.5 (solid) and d = 4.0 and w = 1.0 (dashed). Right: The completion field due to the eigenvector, sma x (8 .0 x 1.1- 62 ) . w' = 1.0 and plotted Anax (,) over the same interval (Figur~ 4 (left)) . Observe that A~ax (8.0 X 1.1- x+ ) :::::: Amax(8.0 x 1.1- X). As before, thls confirms the scaleinvariance of the system. Next, we studied how the relative magnitudes of the local maxima of Amax (,) change as the parameter w is varied. We begin with a Koffka Cross where d = 2.0 and w = 0.5 and observe that Amax(r) has two local maxima (Figure 5 (left)). We refer to the larger of these maxima as ,circle . As previously noted, this maximum is located at approximately 8.0 x 1.1- 62 . The second maximum is located at approximately 8.0 x 1.1 -32. When the completion field due to the eigenvector, smax(8.0 x 1.1- 32 ), is rendered, we observe that the distribution is dominated by contours of predominantly square shape (Figure 5(a)). For this reason , we refer to this local maximum as ,square. Now consider a Koffka Cross where the widths of the arms are doubled but the diameter remains the same, i.e., d' = 2.0 and w' = 1.0. We observe that A~ax (r) still has two local maxima, one at approximately 8.0 x 1.1- 63 and a second at approximately 8.0 x l.1- 29 (Figure 5 (left)). When we render the completion fields due to the eigenvectors, s~ax(8.0x 1.1- 63 ) and s~ax(8.0 x 1.1- 29 ), we find that the completion fields have the same general character as before-the contours associated with the smaller spatial scale (i.e., lower speed) are approximately circular and those associated with the larger spatial scale (Le., higher speed) are approximately square (Figure 5 (d) and (c)). Accordingly, we refer to the locations of the respective local maxima as '~ircle and ,~quar e ' However, what is most interesting is that the relative magnitudes of the local maxima have reversed. Whereas we previously observed that Amax(,circle) > Amax(rsquare), we now observe that A~ax(r~quare) > A~ax(r~ircle)' Therefore, the completion field due to the eigenvector, s~ax(r~quar e ) [not s~ax(r~ircle)!l represents the limiting distribution over all spatial scales. This is consistent with the transition from circle to square reported by human observers when the widths of the arms of the Koffka Cross are increased. 837 Emergent Properties of Illusory Contour Shape Koffke Crosses (two widths) - b -, ~:...- a c o > ( Gl o .-wo. OI~ d a b c d -o ?? (l:o o ~o o Q. 20 40 60 80 X Figure 5: Plot of magnitude of maximum positive real eigenvalue, Ama x, vs. log 1.1 (1/"'() for Koffka Crosses with d = 2.0 and w = 0.5 (solid) and d = 2.0 and w = 1.0 (dashed) . Stochastic completion fields for Koffka Cross due to (a) Sm ax ("'(.quar e ) is a local optimum for w = 0.5 (b) Sma x ("'(ci rcl e ) is the global optimum for w = 0.5 (c) s~ax("'(~quar e ) is the global optimum for w = 1.0 (d) s~a x ("'(~quar e ) is a local optimum for w = 1.0. These results are consistent with the circle-to-square transition perceived by human subjects when the width of the arms of the Koffka Cross are increased . 4 CONCLUSION We have improved upon a previous model of illusory contour formation by showing how to compute a scale-invariant distribution of closed contours given position constraints alone. We also used our model to explain a previously unexplained perceptual effect. References [1] Horn, R.A ., and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press , p. 500 , 1985. [2] Kovacs, I. and B. Julesz, A Closed Curve is Much More than an Incomplete One: Effect of Closure in Figure-Ground Segmentation, Pmc. Natl. Acad. Sci. USA, 90, pp. 7495-7497, 1993. [3] Mumford, D., Elastica and Computer Vision, Algebraic Geometry and Its Applications, Chandrajit Bajaj (ed.) , Springer-Verlag, New York, 1994. [4) Sambin , M., Angular Margins without Gradients, Italian Journal of Psychology 1, pp. 355-361, 1974. [5] Thornber, KK and L.R. Williams, Analytic Solution of Stochastic Completion Fields, Biological Cybernetics 75 , pp. 141-151, 1996. [6] Thornber , KK and L.R. Williams, Characterizing the Distribution of Completion Shapes with Corners Using a Mixture of Random Processes, Intl. Workshop on Energy Minimization Methods in Computer Vision, Venice, Italy, 1997. [7] Williams, L.R. and D.W . Jacobs, Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience, Neural Computation 9(4) , pp. 837-858, 1997. [8) Williams, L.R. and D.W. Jacobs, Local Parallel Computation of Stochastic Completion Fields, Neural Computation 9(4), pp. 859-881 , 1997. 

1528 |@word closure:2 confirms:2 jacob:7 minus:1 solid:4 initial:1 contains:1 past:1 si:8 yet:2 must:1 ixil:1 numerical:2 shape:17 analytic:1 plot:3 drop:1 v:4 alone:4 half:1 selected:1 accordingly:1 plane:2 beginning:1 short:1 location:1 consists:4 x0:1 nor:1 frequently:1 aliasing:1 increasing:1 becomes:1 begin:4 what:1 kg:1 rmax:3 eigenvector:11 nj:1 act:2 scaled:2 positive:9 before:2 local:10 sd:3 limit:3 acad:1 joining:2 id:1 path:1 approximately:6 plus:1 studied:1 suggests:1 factorization:1 averaged:1 horn:1 block:3 doubled:1 selection:1 restriction:1 williams:13 punctuated:1 sharpness:1 simplicity:1 assigns:1 pure:1 amax:15 dominate:1 notion:1 limiting:6 enhanced:1 suppose:2 pt:1 crossing:1 velocity:6 satisfying:1 infrequently:1 located:2 observed:2 decrease:1 ui:2 traversal:2 motivate:1 solving:1 upon:1 sink:1 emergent:4 univ:1 shortcoming:1 formation:2 larger:2 solve:1 favor:1 statistic:1 eigenvalue:9 product:2 elastica:1 rma:3 ama:1 double:1 p:2 optimum:4 smax:5 produce:1 intl:1 oo:1 completion:24 figur:1 quare:1 strong:2 implies:1 direction:13 stochastic:10 human:2 sand:2 argued:1 exchange:1 generalization:1 biological:1 around:1 ground:1 normal:3 sma:3 perceived:5 travel:1 unexplained:1 largest:5 minimization:1 clearly:1 gaussian:7 modified:1 og:2 ira:1 ax:12 greatly:1 lj:3 typically:2 italian:1 interested:1 comprising:1 orientation:10 priori:3 ircle:3 spatial:7 psychophysics:1 field:21 equal:3 having:1 identical:1 represents:3 nearly:1 report:2 stimulus:1 koffka:12 geometry:1 consisting:1 freedom:1 interest:1 circular:2 mixture:2 natl:1 perimeter:1 respective:1 incomplete:1 circle:13 plotted:4 increased:2 earlier:1 deviation:1 johnson:1 reported:1 st:1 nm:1 satisfied:1 corner:1 li:1 rcl:1 satisfy:4 observer:3 closed:11 start:1 parallel:2 il:1 square:9 oi:1 variance:4 percept:1 maximized:2 spaced:1 weak:3 albuquerque:1 trajectory:2 cybernetics:1 straight:1 randomness:1 explain:2 reach:3 ed:1 energy:1 pp:5 associated:4 illusory:12 lim:1 segmentation:1 routine:2 higher:2 improved:1 evaluated:2 just:1 angular:1 logl:4 mode:1 indicated:1 impulse:7 usa:1 effect:4 assigned:1 symmetric:1 width:8 recurrence:2 noted:1 motion:2 recently:1 predominantly:2 endpoint:6 jl:1 refer:4 cambridge:1 particle:8 moving:1 rgo:1 base:1 brownian:2 recent:2 showed:1 italy:1 verlag:1 life:1 additional:1 dashed:4 ii:2 smooth:1 technical:1 cross:15 basic:1 vision:2 poisson:3 represent:2 confined:1 thornber:6 whereas:1 interval:5 source:1 subject:1 tend:1 call:3 noting:1 intermediate:1 scaleinvariance:1 independence:1 xj:6 affect:1 psychology:1 opposite:1 expression:2 bridging:1 wo:2 render:1 algebraic:1 passing:2 york:1 chandrajit:1 eigenvectors:3 julesz:1 diameter:4 discrete:1 traced:1 drawn:1 neither:2 fraction:1 ansi:1 venice:1 comparable:1 constraint:22 infinity:1 scene:1 dominated:2 speed:13 rendered:1 smaller:3 remain:1 increasingly:1 character:1 invariant:6 xo:1 equation:2 previously:3 remains:1 mechanism:1 know:1 letting:1 reversal:2 end:2 travelling:1 gaussians:1 eight:5 observe:7 occasional:1 rp:2 remaining:1 mumford:3 dependence:1 diagonal:2 gradient:1 distance:1 reversed:1 sci:1 majority:1 lthe:1 reason:1 length:1 modeled:2 kk:2 ratio:1 mexico:1 pmc:1 unfortunately:1 lil:1 sm:1 anti:1 defining:2 varied:2 arbitrary:2 kovacs:1 introduced:1 pair:6 specified:4 discontinuity:2 summarize:1 max:12 lance:1 natural:3 arm:6 lk:1 started:1 axis:1 prior:1 tangent:1 contributing:1 relative:10 permutation:1 interesting:1 degree:1 consistent:3 translation:1 koftka:1 gl:1 salience:3 institute:1 characterizing:1 distributed:1 curve:1 transition:7 contour:39 osd:3 sj:2 global:2 xi:5 symmetry:2 bajaj:1 position:16 perceptual:1 weighting:4 showing:2 decay:1 consist:1 intrinsic:1 workshop:1 ci:5 nec:1 magnitude:8 margin:1 logarithmic:1 simply:1 likely:1 doubling:4 springer:1 corresponds:1 conditional:1 consequently:2 axb:1 change:4 except:1 uniformly:2 acting:2 pas:2 invariance:2 dept:1 princeton:1

Stationarity and Stability of Autoregressive Neural Network Processes Friedrich Leisch\ Adrian Trapletti 2 & Kurt Hornik l 1 Institut fur Statistik Technische UniversiUit Wien Wiedner Hauptstrafie 8-10 / 1071 A-1040 Wien, Austria firstname.lastname@ci.tuwien.ac.at 2 Institut fiir Unternehmensfiihrung Wirtschaftsuniversi tat Wien Augasse 2-6 A-lOgO Wien, Austria adrian. trapletti@wu-wien.ac.at Abstract We analyze the asymptotic behavior of autoregressive neural network (AR-NN) processes using techniques from Markov chains and non-linear time series analysis. It is shown that standard AR-NNs without shortcut connections are asymptotically stationary. If linear shortcut connections are allowed, only the shortcut weights determine whether the overall system is stationary, hence standard conditions for linear AR processes can be used. 1 Introduction In this paper we consider the popular class of nonlinear autoregressive processes driven by additive noise, which are defined by stochastic difference equations of form (1) where ft is an iid. noise process. If g( . .. , (J) is a feedforward neural network with parameter ("weight") vector (J, we call Equation 1 an autoregressive neural network process of order p, short AR-NN(p) in the following. AR-NNs are a natural generalization of the classic linear autoregressive AR(p) process (2) See, e.g., Brockwell & Davis (1987) for a comprehensive introduction into AR and ARMA (autoregressive moving average) models. F. Leisch, A. Trapletti and K. Hornik 268 One of the most central questions in linear time series theory is the stationarity of the model, i.e., whether the probabilistic structure of the series is constant over time or at least asymptotically constant (when not started in equilibrium). Surprisingly, this question has not gained much interest in the NN literature, especially there are-up to our knowledge-no results giving conditions for the stationarity of ARNN models. There are results on the stationarity of Hopfield nets (Wang & Sheng, 1996), but these nets cannot be used to estimate conditional expectations for time series prediction. The rest of this paper is organized as follows: In Section 2 we recall some results from time series analysis and Markov chain theory defining the relationship between a time series and its associated Markov chain. In Section 3 we use these results to establish that standard AR-NN models without shortcut connections are stationary. We also give conditions for AR-NN models with shortcut connections to be stationary. Section 4 examines the NN modeling of an important class of non-stationary time series, namely integrated series. All proofs are deferred to the appendix . 2 2.1 Some Time Series and Markov Chain Theory Stationarity Let ~t denote a time series generated by a (possibly nonlinear) autoregressive process as defined in (1). If lEft = 0, then 9 equals the conditional expectation 1E(~t I~t-l' ... , ~t-p) and g(~t-l' ... , ~t-p) is the best prediction for ~t in the mean square sense. If we are interested in the long term properties of the series, we may ask whether certain features such as mean or variance change over time or remain constant. The time series is called weakly stationary if lE~t = Jl and cov(~t,~t+h) = ,h, 'it, i.e., mean and covariances do not depend on the time t. A stronger criterion is that the whole distribution (and not only mean and covariance) of the process does not depend on the time, in this case the series is called strictly stationary. Strong stationarity implies weak stationarity if the second moments of the series exist. For details see standard time series textbooks such as Brockwell & Davis (1987). = If ~t is strictly stationary, then IP (~t E A) rr( A), 'it and rrO is called the stationary distribution of the series. Obviously the series can only be stationary from the beginning if it is started with the stationary distribution such that ~o '" rr. If it is not started with rr, e.g., because ~o is a constant , then we call the series asymptotically stationary if it converges to its stationary distribution: lim IP(~t E A) t-HX) 2.2 = rr(A) Time Series as Markov Chains Using the notation Xt-l (~t-l"" ,~t_p)' (3) (g(Xt-d,~t-l"" , ~t-p+d (ft,O, ... ,O)' (4) (5) we can write scalar autoregressive models of order p such as (1) or (2) as a first order vector model (6) 269 Stationarity and Stability ofAutoregressive Neural Network Processes with Xt, et E lR P (e.g., Chan & Tong, 1985). If we write = pn(x,A) p( x, A) = IP{Xt+n E Alxt pl (x, A) = x} for the probability of going from point x to set A E B in n steps, then {xd with p(x , A) forms a Markov chain with state space (lRP , B,>'), where B are the Borel sets on lR P and>' is the usual Lebesgue measure. The Markov chain {xd is called cp-irreducible, if for some IT-finite measure cp on (lR P , B, >.) 00 n=l whenever cp(A) > O. This me~ns essentially, that all parts of the state space can be reached by the Markov chain irrespective of the starting point. Another important property of Markov chains is aperiodicity, which loosely speaking means that there are no (infinitely often repeated) cycles. See, e.g., Tong (1990) for details . The Markov chain {Xt} is called geometrically ergodic, if there exists a probability measure 1I"(A) on (lR P , B, >.) and a p > 1 such that Vx E lR P : lim pnllpn(x,.) - 11"(?)11 = 0 n-+oo where II . II denotes the total variation. Then 11" satisfies the invariance equation 1I"(A) = ! p(x, A) 1I"(dx) , VA E B There is a close relationship between a time series and its associated Markov chain. If the Markov chain is geometrically ergodic, then its distribution will converge to 11" and the time series is asymptotically stationary. If the time series is started with distribution 11", i.e., Xo "" 11", then the series {~d is strictly stationary. 3 Stationarity of AR-NN Models We now apply the concepts defined in Section 2 to the case where 9 is defined by a neural network. Let x denote a p-dimensional input vector, then we consider the following standard network architectures: Single hidden layer perceptrons: g(x) = 'Yo + L,8ilT(ai + a~x) (7) where ai, ,8i and 'Yo are scalar weights, aj are p-dimensional weight vectors, and IT(') is a bounded sigmoid function such as tanh(?). Single hidden layer perceptrons with shortcut connections: (8) where c is an additional weight vector for shortcut connections between inputs and output. In this case we define the characteristic polynomial c(z) associated with the linear shortcuts as c(z) =1- ClZ - C2z2 - . .. - cP zP, ZE C. 270 F. Leisch, A. TrapleUi and K. Hornik Radial basis function networks: (9) where mj are center vectors and ?( ...) is one of the usual bounded radial basis functions such as ?(x) = exp( _x 2 ). Lemma 1 Let {xtl be defined by (6), let IEjt:tl < 00 and let the PDF of f:t be positive everywhere in JR. Then if 9 is defined by any of (7), (8) or (9), the Markov chain {Xt} is ?-irreducible and aperiodic. Lemma 1 basically says that the state space of the Markov chain, i.e., the points that can be reached, cannot be reduced depending on the starting point. An example for a reducible Markov chain would be a series that is always positive if only Xo > (and negative otherwise). This cannot happen in the AR-NN(p) case due to the unbounded additive noise term. ? Theorem 1 Let {~tl be defined by (1), {xtl by (6), further let IEktl PDF of f:t be positive everywhere in JR. Then < 00 and the 1. If 9 is a network without linear shortcuts as defined in (7) and (9), then { x tl is geometrically ergodic and {~tl is asymptotically stationary. 2. If 9 is a network with linear shortcuts as defined in (8) and additionally c(z) f 0, Vz E C : Izl ~ 1, then {xtl is geometrically ergodic and {~tl is asymptotically stationary. The time series {~t} remains stationary if we allow for more than one hidden layer (-+ multi layer perceptron, MLP) or non-linear output units, as long as the overall mapping has bounded range. An MLP with shortcut connections combines a (possibly non-stationary) linear AR(p) process with a non-linear stationary NN part. Thus, the NN part can be used to model non-linear fluctuations around a linear process like a random walk. The only part of the network that controls whether the overall process is stationary are the linear shortcut connections (if present). If there are no shortcuts, then the process is always stationary. With shortcuts, the usual test for stability of a linear system applies. 4 Integrated Models An important method in classic time series analysis is to. first transform a nonstationary series into a stationary one and then model the remainder by a stationary process. The probably most popular models of this kind are autoregressive integrated moving average (ARIMA) models, which can be transformed into stationary ARMA processes by simple differencing. Let I::!..k denote the k-th order difference operator et - ~t-l I::!..(~t - ~t-d (10) = ~t - 2~t-l + ~t-2 (11) (12) 271 Stationarity and Stability ofAutoregressive Neural Network Processes with ~ 1 = ~. E.g., a standard random walk ~t = ~t-l +ft is non-stationary because of the growing variance, but can be transformed into the iid (and hence stationary) noise process ft by taking first differences. If a time series is non-stationary, but can be transformed into a stationary series by taking k-th differences, we call the series integrated of order k. Standard MLPs or RBFs without shortcuts are asymptotically stationary. It is therefore important to take care that these networks are only used to model stationary processes. Of course the network can be trained to mimic a non-stationary process on a finite time interval, but the out-of-sample or prediction performance will be poor, because the network inherently cannot capture some important features of the process. One way to overcome this problem is to first transform the process into a stationary series (e.g., by differencing an integrated series) and train the network on the transformed series (Chng et al., 1996). As differencing is a linear operation, this transformation can also be easily incorporated into the network by choosing the shortcut connections and weights from input to hidden units accordingly. Assume we want to model an integrated series of integration order k, such that ~k~t = g(~k~t_l' . .. ' ~k~t_p) + ft where ~k~t is stationary. By (12) this is equivalent to ~t k ~(-lt-l (~)~t-n + g(~k~t_l' ... ' ~k~t_p) + ft k ~(-lt-l (~)~t-n + g(~t-l' ... ,~t-p-k) + ft which (for p > k) can be modeled by an MLP with shortcut connections as defined by (8) where the shortcut weight vector c is fixed to (~) := 0 for n >k and 9 is such that g(~t-l' ... ,~t-p-k) = g(~kXt_d. This is always possible and can basically be obtained by adding c to all weights between input and first hidden layer of g. An AR-NN(p) can model integrated series up to integration order p. If the order of integration is known , the shortcut weights can either be fixed, or the differenced series is used as input. If the order is unknown, we can also train the complete network including the shortcut connections and implicitly estimate the order of integration. After training the final model can be checked for stationarity by looking at the characteristic roots of the polynomial defined by the shortcut connections. 4.1 Fractional Integration Up to now we have only considered integrated series with positive integer order of integration, i.e., kEN. In the last years models with fractional integration order became very popular (again). Series with integration order of 0.5 < k < 1 can be shown to exhibit self-similar or fractal behavior, and have long memory. These type of processes were introduced by Mandelbrot in a series of paper modeling river flows, e.g., see Mandelbrot & Ness (1968). More recently, self-similar processes were used to model Ethernet traffic by Leland et al. (1994). Also some financial time series such as foreign exchange data series exhibit long memory and self-similarity. FLeisch. A. Trapletti and K. Hornik 272 The fractional differencing operator ~ k , k E [-1, 1] is defined by the series expansion k ~ f(-k+n) ~ ~t = ~ r(-k)f(n + 1)~t-n (13) which is obtained from the Taylor series of (1 - z)k. For k > 1 we first use Equation (12) and then the above series for the fractional remainder. For practical computation, the series (13) is of course truncated at some term n = N. An ARNN(p) model with shortcut connections can approximate the series up to the first p terms. 5 Summary We have shown that AR-NN models using standard NN architectures without shortcuts are asymptotically stationary. If linear shortcuts between inputs and outputs are included-which many popular software packages have already implementedthen only the weights of the shortcut connections determine if the overall system is stationary. It is also possible to model many integrated time series by this kind of networks . The asymptotic behavior of AR-NNs is especially important for parameter estimation, predictions over larger intervals of time, or when using the network to generate artificial time series. Limiting (normal) distributions of parameter estimates are only guaranteed for stationary series. We therefore always recommend to transform a non-stationary series to a stationary series if possible (e.g ., by differencing) before training a network on it. Another important aspect of stationarity is that a single trajectory displays the complete probability law of the process. If we have observed one long enough trajectory of the process we can (in theory) estimate all interesting quantities of the process by averaging over time. This need not be true for non-stationary processes in general, where some quantities may only be estimated by averaging over several independent trajectories. E.g., one might train the network on an available sample and then use the trained network afterwards-driven by artificial noise from a random number generator-to generate new data with similar properties than the training sample. The asymptotic stationarity guarantees that the AR-NN model cannot show "explosive" behavior or growing variance with time. We currently are working on extensions of this paper in several directions. AR-NN processes can be shown to be strong mixing (the memory of the process vanishes exponentially fast) and have autocorrelations going to zero at an exponential rate . Another question is a thorough analysis of the properties of parameter estimates (weights) and tests for the order of integration. Finally we want to extend the univariate results to the multivariate case with a special interest towards cointegrated processes. Acknowledgement This piece of research was supported by the Austrian Science Foundation (FWF) under grant SFB#OlO ('Adaptive Information Systems and Modeling in Economics and Management Science') . Stationarity and Stability ofAutoregressive Neural Network Processes 273 Appendix: Mathematical Proofs Proof of Lemma 1 It can easily be shown that {xe} is <p-irreducible if the support of the probability density function (PDF) of ?t is the whole real line, i.e., the PDF is positive everywhere in IR (Chan & Tong, 1985). In this case every non-null p-dimensional hypercube is reached in p steps with positive probability (and hence every non-null Borel set A). A necessary and sufficient condition for {Xt} to be aperiodic is that there exists a set A and positive integer n such that pn(x, A) > 0 and pn+l (x, A) > 0 for all x E A (Tong, 1990, p. 455). In our case this is true for all n due to the unbounded additive noise. Proof of Theorem 1 We use the following result from nonlinear time series theory: Theorem 2 (Chan & Tong 1985) Let {Xt} be defined by (1), (6) and let G be compact, i.e. preserve compact sets. IfG can be decomposedasG = Gh+Gd andGd(-) is of bounded range, G h(-) is continuous and homogeneous, i.e., Gh(ax) = aGh(x), the origin is a fixed point of G h and Gh is uniform asymptotically stable, IEI?tl < 00 and the PDF of ?t is positive everywhere in IR, then {Xt} is geometrically ergodic. The noise process ?t fulfills the conditions by assumption. Clearly all networks are continuous compact functions. Standard MLPs without shortcut connections and RBFs have a bounded range , hence G h == 0 and G == G d , and the series {ee} is asymptotically stationary. If we allow for linear shortcut connections between the input and the outputs, we get G h = c'x and G d = 70 l.:i (3i(T(ai aix) i.e., G h is the linear shortcut part of the network, and Gd is a standard MLP without shortcut connections. Clearly, G h is continuous, homogeneous and has the origin as a fixed point. Hence, the series {eel is asymptotically stationary if G h is asymptotically stable, i.e., when all characteristic roots of Gh have a magnitude less than unity. Obviously the same is true for RBFs with shortcut connections. Note that the model reduces to a standard linear AR(p) model if Gd == O. + + References Brockwell, P. J. & Davis, R. A. (1987). Time Series: Theory and Methods. Springer Series in Statistics. New York, USA: Springer Verlag. Chan, K. S. & Tong, H. (1985). On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Advances in Applied Probability, 17, 666-678. Chng, E . S., Chen, S., & Mulgrew, B. (1996). Gradient radial basis function networks for nonlinear and nonstationary time series prediction. IEEE Transactions on Neural Networks, 7(1), 190- 194. Husmeier, D. & Taylor, J. G . (1997). Predicting conditional probability densities of stationary stochastic time series. Neural Networks, 10(3),479-497. Jones, D. A. (1978). Nonlinear autoregressive processes. Proceedings of the Royal Society London A, 360, 71- 95. Leland, W. E., Taqqu, M. S., Willinger, W., & Wilson, D. V. (1994) . On the self-similar nature of ethernet traffic (extended version). IEEE/ACM Transactions on Networking, 2(1), 1- 15. Mandelbrot, B. B. & Ness, J . W. V. (1968). Fractional brownian motions, fractional noises and applications. SIAM Review, 10(4), 422-437. Tong, H. (1990). Non-linear time series: A dynamical system approach. New York, USA: Oxford University Press. Wang, T. & Sheng, Z. (1996). Asymptotic stationarity of discrete-time stochastic neural networks. Neural Networks, 9(6) , 957-963. 

1529 |@word version:1 polynomial:2 stronger:1 adrian:2 tat:1 covariance:2 moment:1 series:59 kurt:1 dx:1 willinger:1 additive:3 happen:1 stationary:43 accordingly:1 beginning:1 short:1 lr:5 unbounded:2 mathematical:1 mandelbrot:3 xtl:3 combine:1 leland:2 behavior:4 growing:2 multi:1 tuwien:1 notation:1 bounded:5 null:2 kind:2 textbook:1 transformation:1 guarantee:1 thorough:1 every:2 xd:2 control:1 unit:2 grant:1 positive:8 before:1 oxford:1 fluctuation:1 logo:1 might:1 range:3 practical:1 radial:3 get:1 cannot:5 close:1 operator:2 equivalent:1 deterministic:1 center:1 economics:1 starting:2 ergodic:5 agh:1 examines:1 financial:1 stability:5 classic:2 variation:1 limiting:1 homogeneous:2 origin:2 ze:1 observed:1 ft:7 reducible:1 wang:2 capture:1 cycle:1 vanishes:1 taqqu:1 trained:2 weakly:1 depend:2 basis:3 easily:2 hopfield:1 train:3 fast:1 london:1 universiuit:1 artificial:2 choosing:1 larger:1 say:1 otherwise:1 cov:1 statistic:1 transform:3 ip:3 final:1 obviously:2 rr:4 net:2 remainder:2 alxt:1 brockwell:3 mixing:1 zp:1 converges:1 oo:1 depending:1 ac:2 strong:2 implies:1 ethernet:2 lyapunov:1 direction:1 aperiodic:2 stochastic:4 vx:1 exchange:1 hx:1 generalization:1 strictly:3 pl:1 extension:1 around:1 considered:1 normal:1 exp:1 equilibrium:1 mapping:1 estimation:1 tanh:1 currently:1 vz:1 lrp:1 clearly:2 always:4 pn:3 wilson:1 ax:1 yo:2 fur:1 sense:1 nn:15 foreign:1 integrated:9 hidden:5 going:2 transformed:4 interested:1 overall:4 integration:9 ness:2 special:1 equal:1 jones:1 mimic:1 recommend:1 t_l:2 irreducible:3 preserve:1 comprehensive:1 lebesgue:1 explosive:1 stationarity:15 interest:2 mlp:4 deferred:1 chain:15 necessary:1 institut:2 loosely:1 taylor:2 walk:2 arma:2 wiedner:1 modeling:3 ar:18 technische:1 uniform:1 nns:3 gd:3 density:2 river:1 siam:1 probabilistic:1 eel:1 aix:1 again:1 central:1 management:1 possibly:2 iei:1 wien:5 piece:1 root:2 analyze:1 traffic:2 reached:3 rbfs:3 mlps:2 square:1 ir:2 aperiodicity:1 became:1 variance:3 characteristic:3 weak:1 iid:2 basically:2 trajectory:3 networking:1 whenever:1 checked:1 associated:3 proof:4 popular:4 ask:1 austria:2 knowledge:1 recall:1 lim:2 fractional:6 organized:1 ergodicity:1 sheng:2 working:1 nonlinear:5 autocorrelations:1 leisch:3 aj:1 usa:2 concept:1 true:3 hence:5 self:4 lastname:1 davis:3 criterion:1 pdf:5 complete:2 cp:4 gh:4 motion:1 recently:1 sigmoid:1 exponentially:1 jl:1 extend:1 ai:3 moving:2 stable:2 similarity:1 multivariate:1 brownian:1 chan:4 driven:2 certain:1 verlag:1 xe:1 additional:1 care:1 determine:2 converge:1 husmeier:1 ii:2 afterwards:1 reduces:1 long:5 va:1 prediction:5 austrian:1 essentially:1 expectation:2 izl:1 want:2 interval:2 rest:1 probably:1 flow:1 call:3 nonstationary:2 integer:2 fwf:1 ee:1 feedforward:1 enough:1 architecture:2 whether:4 sfb:1 speaking:1 york:2 fractal:1 ken:1 reduced:1 generate:2 exist:1 estimated:1 chng:2 write:2 discrete:1 arima:1 asymptotically:12 geometrically:5 year:1 package:1 everywhere:4 ilt:1 wu:1 appendix:2 layer:5 guaranteed:1 display:1 software:1 statistik:1 aspect:1 poor:1 jr:2 remain:1 unity:1 xo:2 equation:5 remains:1 available:1 operation:1 apply:1 denotes:1 giving:1 especially:2 establish:1 society:1 hypercube:1 question:3 already:1 quantity:2 usual:3 exhibit:2 gradient:1 me:1 modeled:1 relationship:2 differencing:5 negative:1 unknown:1 markov:15 finite:2 truncated:1 defining:1 cointegrated:1 incorporated:1 looking:1 extended:1 introduced:1 namely:1 friedrich:1 connection:18 t_p:3 differenced:1 dynamical:1 firstname:1 including:1 memory:3 royal:1 natural:1 predicting:1 started:4 irrespective:1 review:1 literature:1 acknowledgement:1 asymptotic:4 law:1 interesting:1 generator:1 foundation:1 sufficient:1 course:2 summary:1 surprisingly:1 last:1 supported:1 allow:2 perceptron:1 taking:2 overcome:1 autoregressive:10 adaptive:1 transaction:2 approximate:1 compact:3 implicitly:1 fiir:1 continuous:3 additionally:1 mj:1 nature:1 inherently:1 hornik:4 expansion:1 whole:2 noise:8 allowed:1 repeated:1 tl:6 borel:2 tong:7 n:1 exponential:1 ifg:1 theorem:3 xt:9 exists:2 adding:1 gained:1 ci:1 magnitude:1 chen:1 lt:2 univariate:1 infinitely:1 scalar:2 applies:1 springer:2 satisfies:1 acm:1 conditional:3 towards:1 shortcut:30 change:1 included:1 averaging:2 lemma:3 called:5 total:1 invariance:1 perceptrons:2 support:1 fulfills:1 olo:1

594 Range Image Restoration using Mean Field Annealing Wesley E. Snyder Griff L. Bilbro Center for Communications and Signal Processing North Carolina State University Raleigh, NC Abstract A new optimization strategy, Mean Field Annealing, is presented. Its application to MAP restoration of noisy range images is derived and experimentally verified. 1 Introduction The application which motivates this paper is image analysis; specifically the analysis of range images. We [BS86] [GS87] and others [YA85][BJ88] have found that surface curvature has the potential for providing an excellent, view-invariant feature with which to segment range images. Unfortunately, computation of curvature requires, in turn, computation of second derivatives of noisy data. We cast this task as a restoration problem: Given a measurement g(z, y), we assume that g(z, y) resulted from the addition of noise to some "ideal" image fez, y) which we must estimate from three things: 1. The measurement g(z, y). 2. The statistics of the noise, here assumed to be zero mean with variance (1'2. 3. Some a priori knowledge of the smoothness of the underlying surface(s). We will turn this restoration problem into a minimization, and solve that minimization using a strategy called Mean Field A nnealing. A neural net appears to be the ideal architecture for the reSUlting algorithm, and some work in this area has already been reported [CZVJ88]. 2 Simulated Annealing and Mean Field Anneal? Ing The strategy of SSA may be summarized as follows: Let H(f) be the objective function whose minimum we se~k, wher~ /is somt' parameter vector. A parameter T controls the algorithm. The SSA algorithm begins at a relatively high value of T which is gradually reduced. Under certain conditions, SSA will converge to a global optimum, [GGB4] [RS87] H (f) = min{ H (fie)} Vfie (1) Range Image Restoration Using Mean Field Annealing even though local minima may occur. However, SSA suffers from two drawbacks: ? It is slow, and ? there is no way to directly estimate [MMP87] a continuously-valued derivatives. I or its The algorithm presented in section 2.1 perturbs (typically) a single element of fat each iteration. In Mean Field Annealing, we perturb the entire vector f at each iteration by making a deterministic calculation which lowers a certain average of H, < H(f) >, at the current temperature. We thus perform a rather conventional non-linear minimization (e.g. gradient descent), until a minimum is found at that temperature. We will refer to the minimization condition at a given T as the equilibrium for that T. Then, T is reduced, and the previous equilibrium is used as the initial condition for another minimization. MFA thus converts a hard optimization problem into a sequence of easier problems. In the next section, we justify this approach by relating it to SSA. 2.1 Stochastic Simulated Annealing The problem to be solved is to find j where minimization with the following strategy: j minimizes H(f). SSA solves this 1. Define PT ex e- H / T . 2. Find the equilibrium conditions on PT, at the current temperature, T. By equilibrium, we mean that any statistic ofpT(f) is constant. These statistics could be derived from the Markov chain which SSA constructs: jO, p, ... , IN, ... , although in fact such statistical analysis is never done in normal running of an SSA algorithm. 3. Reduce T gradually. 4. As T --+ 0, PT(f) becomes sharply peaked at j, the minimum. 2.2 Mean Field Annealing In Mean Field Annealing, we provide an analytic mechanism for approximating the equilibrium at arbitrary T. In MFA, we define an error function, -H fe--ordl EMF(Z, T) = Tln--=-H- f eTdl + fe -Hfl T (H-Ho)dl - / j --- f e- TdJ - --. (2) which follows from Peierl's inequality [BGZ76]: F -H ~ Fo+ < H - Ho > -Hg (3) where F = -Tlnf e---r-dl and Fo = -Tlnf e T dl . The significance of EMF is as follows: the minimum of EMF determines the best approximation given the form 595 596 Bilbro and Snyder of Ho to the equilibrium statistics of the SSA-generated MRF at T. We will then anneal on T. In the next section, we choose a special form for Ho to simplify this process even further. 1. Define some Ho(f, z) which will be used to estimate H(f). 2. At temperature T, minimize EMF(Z) where EMF is a functional of Ho and H which characterizes the difference between Ho and H. The process of minimizing EMF will result in a value of the parameter z, which we will denote as ZT. 3. Define HT(f) = Ho(f, ZT) and for(f) ex e- iiT / T . 3 Image Restoration Using MFA We choose a Hamiltonian which represents both the noise in the image, and our a priori knowledge of the local shape of the image data. " -2 1 2 (Ii - gil 2 HN = " L.J ? (1' , (4) (5) where 18( represents [Bes86] the set of values of pixels neighboring pixel i (e.g. the value of I at i along with the I values at the four nearest neighbors of i); A is some scalar valued function of that set of pixels (e.g. the 5 pixel approximation to the Laplacian or the 9 pixel approximation to the quadratic variation); and (6) The noise term simply says that the image should be similar to the data, given noise of variance (1'2. The prior term drives toward solutions which are locally planar. Recently, a simpler V(z) = z2 and a similar A were successfully used to design a neural net [CZVJ88] which restores images consisting of discrete, but 256-valued pixels. Our formulation of the prior term emphasizes the importance of "point processes," as defined [WP85] by Wolberg and Pavlidis. While we accept the eventual necessity of incorporating line processes [MMP87] [Mar85] [GG84] [Gem87] into restoration, our emphasis in this paper is to provide a rigorous relationship between a point process, the prior model, and the more usual mathematical properties of surfaces. Using range imagery in this problem makes these relationships direct. By adopting this philosophy, we can exploit the results of Grimson [Gri83] as well as those of Brady and Horn [BH83] to improve on the Laplacian. The Gaussian functional form of V is chosen because it is mathematically convenient for Boltzmann statistics and beca.use it reflects the following shape properties recommended for grey level images in the literature and is especially important if Range Image Restoration Using Mean Field Annealing line processes are to be omitted: Besag [Bes86] notes that lito encourage smooth variation", V(A) "should be strictly increasing" in the absolute value of its argument and if "occasional abrupt changes" are expected, it should "quickly reach a maximum" . Rational functions with shapes similar to our V have been used in recent stochastic approaches to image processing [GM85]. In Eq. 6, T is a "soft threshold" which represents our prior knowledge of the probability of various values of \7 2 f (the Laplacian of the undegraded image). For T large, we imply that high values of the Laplacian are common - f is highly textured; for small values of T, we imply that f is generally smooth. We note that for high values of T, the prior term is insignificant, and the best estimate of the image is simply the data. We choose the Mean Field Hamiltonian to be (7) and find that the optimal ZT approximately minimizes (8) both at very high and very low T . We have found experimentally that this approximation to ZT does anneal to a satisfactory restoration. At each temperature, we use gradient descent to find ZT with the following approximation to the gradient of <H>: (9) and -b V(r?) - , - y'2;(T+T) e- .. ? 2( .. h) (lO) . Differentiating Eq. 8 with this new notation, we find (11) Since 6'+11,; is non-zero only when i 8 < H > _8 :J! . ) + v = i, :J!j (1' 2 gj we have +L L -II IT'(. ')+11 ) II and this derivative can be used to find the equilibrium condition. Algorithm (12) 597 598 Bilbro and Snyder 1. Initially, we use the high temperature assumption, which eliminates the prior term entirely, and results in Z; T =g;; for = 00. (13) This will provide the initial estimate of z. Any other estimate quickly converges to g. 2. Given an image z;, form the image ri (L ? z);, where the ? indicates convolution. = 3. Create the image V. = V' (r?) P , ..? = - -----l= _-.!'L e - :II(T~ ~T+T)T+T .. ) ? 4. Using 12, perform ordinary non-linear minimization of < H > starting from the current z. The particular strategy followed is not critical. We have successfully used steepest descent and more sophisticated conjugate gradient [PFTV88] methods. The simpler methods seem adequate fot Gaussian noise. 5. Update z to the minimizing z found in step 4. 6. Reduce T and go to 2. When T is sufficiently close to 0, the algorithm is complete. In step 6 above, T essentially defines the appropriate low-temperature stopping point. In section 5, we will elaborate on the determination of T and other such constants. 4 Performance In this section, we describe the performance of the algorithm as it is applied to several range images. We will use range images, in which the data is of the form z = z(z, y). 4.1 (14) Images With High Levels of Noise Figure 1 illustrates a range image consisting of three objects, a wedge (upper left), a cylinder with rounded end and hole (right), and a trapezoidal block viewed from the top. The noise in this region is measured at (1' = 3units out of a total range of about 100 units. Unsophisticated smoothing will not estimate second derivatives of such data without blurring. Following the surface interpolation literature, [Gri83] [BB83] we use the quadratic variation as the argument of the penalty function for the prior term to (15) and performing the derivative in a manner analogous to Eq. 11 and 12. The Laplacian of the restoration is shown in Figure 2. Figure 3 shows a cross-section taken as indicated by the red line on Figure 2. Fig. 1 Original rallge image I n J~~ l\~lll Fig. 2 Laplacian of the restored image 4.2 Fig. 3 Cross section Through Laplacian along Red Line Comparison With Existing Techniques Accurate computation of surface derivatives requires extremely good smoothing of surface noise, while segmentation requires preservat.ion of edges. One suc.h adapt.ive smoothing technique,[Ter87] iterative Gaussian smoothing (IGS) has been successfully applied to range imagery. [PB87] Following this strategy, step edges are first detected, and smoothing is then applied using a small center-weighted kernel. At edges, an even smaller kernel, called a "molecule", is used to smooth right up to the edge without blurring the edge. The smoothing is then iterated. 600 Bilbro and Snyder The results, restoration and Laplacian, of IGS are not nearly as sharp as those shown in Figure 2. Determining the Parameters 5 Although the optimization strategy described in section 3 has no hard thresholds, several parameters exist either explicitly in Eq. 8 or implicitly in the iteration. Good estimates of these parameters will result in improved performance, faster convergence, or both. The parameters are: (1' the standard deviation of the noise b the relative magnitude of the prior term 11 = T + T the initial temperature and T the final temperature. The decrement in T which defines the annealing schedule could also be considered a parameter. However, we have observed that 10% or less per step is always good enough. We find that for depth images of polyhedral scenes, T = 0 so that only one parameter is problem dependent: (1'. For the more realistic case of images which also contain curved surfaces, however, see our technical report [BS88], which also describes the MFA derivation in much more detail. The standard deviation of the noise must be determined independently for each problem class. It is straightforward to estimate (1' to within 50%, and we have observed experimentally that performance of the algorithm is not sensitive to this order of error. We can analytically show that annealing occurs in the region T:::::: IL12(1'2 and choose TJ 2ILI2(1'2. Here, ILI2 is the squared norm of the operator Land ILI2 20 for the usual Laplacian and ILI2 12.5 for the quadratic variation. Further analysis shows that b .J2;ILI(1' is a good choice for the coefficient of the prior term. = = = = References [Bes86] [BGZ76] [BH83] [BJ88] J. Besag. On the statistical analysis of dirty pictures. Journal of the Royal Stati6ticCJl Society, B 48(3), 1986. E. Brezin, J. C. Le Guillon, and J. Zinn-Justin. Field theoretical approach to critical phenomena. In C. Domb and M.S. Green, editors, PhCJ6e Tran6ition6 and Critical Phenomena, chapter 3, Academic Press, New York, 1976. M. Brady and B.K.P Horn. Symm~tric operators for surface interpolation. CVGIP, 22, 1983. P.J. Besl and R.C. Jain. Segmentation through variable-order surface fitting. IEEE PAMI, 10(2), 1988. Range Image Restoration Using Mean Field Annealing G. Bilbro and W. Snyder. A Linear Time Theory for Recognizing Surface6 in 3-D. Technical Report CCSP-NCSU TR-86/8, Center for Communications and Signal Processing, North Carolina State University, 1986. [BS88] G. L. Bilbro and W. E. Snyder. Range Image Re6toration U6ing Mean Field Annealing. Technical Report NETR 88-19, Available from the Center for Communications and Signal Processing, North Carolina State University, 1988. [CZVJ88] R. Chellappa, Y.-T. Zhou, A. Vaid, and B. K. Jenkins. Image restoration using a neural network. IEEE Tran6action6 on ASSP, 36(7):1141-1151, July 1988. [Gem87] D. Geman. Stochastic model for boundary detection. Vi6ion and Image Computing, 5(2):61-65, 1987. [GG84] D. Geman and S. Geman. Stochastic relaxation, Gibbs Distributions, and the Bayesian restoration of images. IEEE Tran6action6 on PAMI, PAMI-6(6):721-741, November 1984. [GM85] S. Geman and D.E. McClure. Bayesian image analysis: an application to single photon emission tomography. Proceeding6 of the American Stati6tical A uociation, Stati6tical Computing Section, 12-18, 1985. rGri83] W.E.L. Grimson. An implementation of computational theory of visual surface interpolation. CVGIP, 22, 1983. [GS87] B. R. Groshong and W. E. Snyder. Range image segmentation for object recognition. In 18th Pitt6burgh Conference on Modeling and Simulation, Pittsburgh, PA, April 1987. [Mar85] J .L. Marroquin. Probabili6tic Solution to Inver6e Problem6. PhD thesis, M.LT, Cambridge, MA, September 1985. [MMP87] J. Marroquin, S. Mitter, and T. Poggio. Probabilistic solution of illposed problems in computational vision. Journal of American Stati6tical Auociation, 82(397):76-89, March 1987. T. Ponce and M. Brady. Toward a surface primal sketch. In T. Kanade, [PB87] editor, Three Dimen6ional Machine Vi6ion, Kluwer Press, 1987. [PFTV88] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipe6 in C. Cambridge University Press, 1988. [RS87] F. Romeo and A. Sangiovanni-Vencentelli. Probabalistic hill climbing algorithms: properties and applications. In Chapel Hill Conference on VLSI, Computer Science Press, Chapel Hill, NC, 1987. [Ter87] D. Terzopoulos. The role of constraints and discontiuities in visiblesurface reconstruction. In Proc. of 7th International Conf. on AI, pages 1073-1077, 1987. [WP85] G. Wolberg and T. Pavlidis. Restoration of binary images using stochastic relaxation with annealing. Pattern Recognition Letter6, 3(6):375-388, December 1985. M. Brady A. Yiulle and H. Asada. Describing surfaces. CVGIP, August [YA85] 1985. [BS86] 601 

153 |@word norm:1 grey:1 simulation:1 carolina:3 tr:1 necessity:1 initial:3 existing:1 current:3 z2:1 must:2 numerical:1 realistic:1 shape:3 analytic:1 update:1 steepest:1 hamiltonian:2 simpler:2 mathematical:1 along:2 direct:1 fitting:1 polyhedral:1 manner:1 expected:1 fez:1 lll:1 increasing:1 becomes:1 begin:1 underlying:1 notation:1 proceeding6:1 minimizes:2 brady:4 fat:1 control:1 unit:2 local:2 interpolation:3 approximately:1 pami:3 emphasis:1 range:15 pavlidis:2 bilbro:6 horn:2 fot:1 unsophisticated:1 block:1 illposed:1 area:1 convenient:1 beca:1 close:1 operator:2 conventional:1 map:1 deterministic:1 center:4 go:1 straightforward:1 starting:1 independently:1 abrupt:1 chapel:2 variation:4 analogous:1 pt:3 pa:1 element:1 recognition:2 trapezoidal:1 geman:4 observed:2 role:1 solved:1 region:2 sangiovanni:1 wher:1 grimson:2 segment:1 blurring:2 textured:1 iit:1 various:1 chapter:1 derivation:1 jain:1 describe:1 chellappa:1 detected:1 whose:1 solve:1 valued:3 say:1 ive:1 besl:1 statistic:5 noisy:2 final:1 sequence:1 net:2 reconstruction:1 neighboring:1 j2:1 convergence:1 optimum:1 converges:1 object:2 measured:1 nearest:1 eq:4 solves:1 wedge:1 drawback:1 stochastic:5 tln:1 mathematically:1 strictly:1 sufficiently:1 considered:1 normal:1 equilibrium:7 omitted:1 proc:1 sensitive:1 create:1 successfully:3 reflects:1 weighted:1 minimization:7 gaussian:3 always:1 rather:1 zhou:1 derived:2 emission:1 romeo:1 ponce:1 indicates:1 besag:2 rigorous:1 dependent:1 stopping:1 vetterling:1 typically:1 entire:1 accept:1 initially:1 vlsi:1 pixel:6 priori:2 restores:1 special:1 smoothing:6 field:13 construct:1 never:1 represents:3 nearly:1 peaked:1 others:1 report:3 simplify:1 resulted:1 consisting:2 cylinder:1 detection:1 highly:1 primal:1 tj:1 hg:1 chain:1 accurate:1 edge:5 encourage:1 poggio:1 theoretical:1 soft:1 modeling:1 restoration:15 ordinary:1 deviation:2 recognizing:1 reported:1 international:1 probabilistic:1 rounded:1 continuously:1 quickly:2 jo:1 imagery:2 squared:1 thesis:1 choose:4 hn:1 conf:1 american:2 derivative:6 potential:1 photon:1 domb:1 summarized:1 north:3 coefficient:1 explicitly:1 view:1 characterizes:1 red:2 minimize:1 variance:2 climbing:1 bayesian:2 iterated:1 emphasizes:1 drive:1 fo:2 suffers:1 reach:1 rational:1 gg84:2 knowledge:3 segmentation:3 schedule:1 sophisticated:1 marroquin:2 appears:1 wesley:1 planar:1 improved:1 april:1 formulation:1 done:1 though:1 until:1 sketch:1 defines:2 indicated:1 contain:1 analytically:1 satisfactory:1 hill:3 complete:1 temperature:9 image:36 recently:1 common:1 functional:2 relating:1 kluwer:1 measurement:2 refer:1 cambridge:2 gibbs:1 ai:1 smoothness:1 surface:12 gj:1 curvature:2 recent:1 certain:2 inequality:1 binary:1 minimum:5 converge:1 recommended:1 signal:3 ii:4 july:1 ing:1 technical:3 smooth:3 academic:1 fie:1 determination:1 calculation:1 cross:2 mcclure:1 adapt:1 faster:1 laplacian:9 mrf:1 emf:6 essentially:1 vision:1 symm:1 iteration:3 kernel:2 adopting:1 ion:1 hfl:1 addition:1 annealing:14 eliminates:1 thing:1 december:1 seem:1 ideal:2 enough:1 architecture:1 reduce:2 penalty:1 york:1 wolberg:2 adequate:1 generally:1 se:1 locally:1 tomography:1 reduced:2 exist:1 gil:1 per:1 discrete:1 snyder:7 four:1 threshold:2 verified:1 ht:1 relaxation:2 convert:1 zinn:1 nnealing:1 entirely:1 followed:1 quadratic:3 occur:1 constraint:1 sharply:1 ri:1 scene:1 ncsu:1 argument:2 min:1 extremely:1 performing:1 relatively:1 march:1 conjugate:1 smaller:1 describes:1 making:1 invariant:1 gradually:2 taken:1 turn:2 describing:1 mechanism:1 end:1 available:1 jenkins:1 occasional:1 appropriate:1 netr:1 ho:8 original:1 top:1 running:1 dirty:1 exploit:1 cvgip:3 perturb:1 especially:1 approximating:1 society:1 objective:1 already:1 occurs:1 restored:1 strategy:7 usual:2 september:1 gradient:4 perturbs:1 simulated:2 toward:2 relationship:2 providing:1 minimizing:2 nc:2 unfortunately:1 fe:2 design:1 implementation:1 motivates:1 zt:5 boltzmann:1 perform:2 upper:1 convolution:1 markov:1 descent:3 curved:1 november:1 communication:3 assp:1 arbitrary:1 sharp:1 august:1 cast:1 justin:1 pattern:1 royal:1 green:1 mfa:4 critical:3 improve:1 imply:2 picture:1 prior:9 literature:2 determining:1 relative:1 editor:2 land:1 lo:1 raleigh:1 terzopoulos:1 neighbor:1 differentiating:1 absolute:1 boundary:1 depth:1 ig:2 implicitly:1 global:1 pittsburgh:1 assumed:1 iterative:1 kanade:1 molecule:1 probabalistic:1 excellent:1 anneal:3 suc:1 significance:1 decrement:1 noise:11 fig:3 mitter:1 elaborate:1 slow:1 insignificant:1 ssa:9 dl:3 incorporating:1 importance:1 phd:1 magnitude:1 illustrates:1 hole:1 easier:1 flannery:1 lt:1 simply:2 visual:1 lito:1 scalar:1 determines:1 teukolsky:1 ma:1 viewed:1 eventual:1 experimentally:3 hard:2 change:1 specifically:1 determined:1 asada:1 justify:1 called:2 total:1 ili:1 u6ing:1 philosophy:1 griff:1 phenomenon:2 ex:2

A Precise Characterization of the Class of Languages Recognized by Neural Nets under Gaussian and other Common Noise Distributions Wolfgang Maass* Inst. for Theoretical Computer Science, Technische Universitat Graz Klosterwiesgasse 3212, A-80lO Graz, Austria email: maass@igi.tu-graz.ac.at Eduardo D. Sontag Oep. of Mathematics Rutgers University New Brunswick, NJ 08903, USA email: sontag@hilbert.rutgers.edu Abstract We consider recurrent analog neural nets where each gate is subject to Gaussian noise, or any other common noise distribution whose probability density function is nonzero on a large set. We show that many regular languages cannot be recognized by networks of this type, for example the language {w E {O, I} * I w begins with O}, and we give a precise characterization of those languages which can be recognized. This result implies severe constraints on possibilities for constructing recurrent analog neural nets that are robust against realistic types of analog noise. On the other hand we present a method for constructing feedforward analog neural nets that are robust with regard to analog noise of this type. 1 Introduction A fairly large literature (see [Omlin, Giles, 1996] and the references therein) is devoted to the construction of analog neural nets that recognize regular languages. Any physical realization of the analog computational units of an analog neural net in technological or biological systems is bound to encounter some form of "imprecision" or analog noise at its analog computational units. We show in this article that this effect has serious consequences for the computational power of recurrent analog neural nets. We show that any analog neural net whose computational units are subject to Gaussian or other common noise distributions cannot recognize arbitrary regular languages. For example, such analog neural net cannot recognize the regular language {w E {O, I} * I w begins with O}. ? Partially supported by the Fonds zur F6rderung der wissenschaftlichen Forschnung (FWF), Austria, project P12153. W Maass and E. D. Sontag 282 A precise characterization of those regular languages which can be recognized by such analog neural nets is given in Theorem 1.1. In section 3 we introduce a simple technique for making feedforward neural nets robust with regard to the same types of analog noise. This method is employed to prove the positive part of Theorem 1.1. The main difficulty in proving Theorem 1.1 is its negative part, for which adequate theoretical tools are introduced in section 2. Before we can give the exact statement of Theorem 1.1 and discuss related preceding work we have to give a precise definition of computations in noisy neural networks. From the conceptual point of view this definition is basically the same as for computations in noisy boolean circuits (see [Pippenger, 1985] and [Pippenger, 1990]). However it is technically more involved since we have to deal here with an infinite state space. We will first illustrate this definition for a concrete case, a recurrent sigmoidal neural net with Gaussian noise, and then indicate the full generality of our result, which makes it applicable to a very large class of other types of analog computational systems with analog noise. Consider a recurrent sigmoidal neural net N consisting of n units, that receives at each time step t an input Ut from some finite alphabet U (for example U = {O, I}). The internal state of N at the end of step t is described by a vector Xt E [-1, l]n, which consists of the outputs of the n sigmoidal units at the end of step t. A computation step of the network N is described by Xt+1 = a(Wxt + h + UtC + Vi) where W E IRnxn and c, h E IRn represent weight matrix and vectors, a is a sigmoidal activation function (e.g., a(y) = 1/(1 + e- Y applied to each vector component, and VI, V2 , ? ?? is a sequence of n-vectors drawn independently from some Gaussian distribution. In analogy to the case of noisy boolean circuits [Pippenger, 1990] one says that this network N recognizes a language L ~ U* with reliability c (where c E (O,~] is some given constant) if immediately after reading an arbitrary word w E U* the network N is with probability 2: ~ + c in an accepting state in case that w E L, and with probability c in an accepting state in case that w rf. LI. ? : :; ! - We will show in this article that even if the parameters of the Gaussian noise distribution for each sigmoidal unit can be determined by the designer of the neural net, it is impossible to find a size n, weight matrix W, vectors h, C and a reliability c E (0, so that the resulting recurrent sigmoidal neural net with Gaussian noise accepts the simple regular language {w E {0,1}*1 w begins with O} with reliability c. This result exhibits a fundamental limitation for making a recurrent analog neural net noise robust, even in a case where the noise distribution is known and of a rather benign type. This quite startling negative result should be contrasted with the large number of known techniques for making a feedforward boolean circuit robust against noise, see [Pippenger, 1990]. !] Our negative result turns out to be of a very general nature, that holds for virtually all related definitions of noisy analog neural nets and also for completely different models for analog computation in the presence of Gaussian or similar noise. Instead of the state set [-1, l]n one can take any compact set n ~ IRn , and instead of the map (x, u) t-+ W x + h + uc one can consider an arbitrary map I : n x U ~ 0 for a compact set 0 ~ IRn where f (', u) is Borel measurable for each fixed U E U. Instead of a sigmoidal activation function a and a Gaussian distributed noise vector V it suffices to assume that a : IRn ~ n is some arbitrary Borel measurable function and V is some IRn -valued random variable with a density ?(.) that has a wide support2 ? In order to define a computation in such system we consider for 1 According to this definition a network N that is after reading some w E U? in an accepting state with probability strictly between c and + c does not recognize any language L ~ U?. 2More precisely: We assume that there exists a subset no of n and some constant Co > 0 such t- t 283 Analog Neural Nets with Gaussian Noise each U E U the stochastic kernel Ku defined by Ku(x, A) := Prob [a(f(x, u) + V) E A] for x E n and A S;;; n. For each (signed, Borel) measure /-l on n, and each U E U, we let lKu/-l be the (signed, Borel) measure defined on n by (lKu/-l)(A) := Ku(x , A)d/-l(x) . Note that lKu /-l is a probability measure whenever /-l is. For any sequence of inputs W = U1 , .?. ,U r , we consider the composition of the evolution operators lKu. : J (1) If the probability distribution of states at any given instant is given by the measure /-l, then the distribution of states after a single computation step on input U E U is given by lKu /-l, and after r computation steps on inputs W = U1,"" U r , the new distribution is IKw /-l, where we are using the notation (1). In particular, if the system starts at a particular initial state ~, then the distribution of states after r computation steps on W is IKwbe, where be is the probability measure concentrated on {O. That is to say, for each measurable subset Fen Prob [Xr+1 E F I Xl =~, input = w] = (lKwbe)(F). We fix an initial state ~ E n, a set of "accepting" or "final" states F, and a "reliability" level E > 0, and say that the resulting noisy analog computational system M recognizes the language L S;;; U* if for all w E U* : 1 wEL ~ (lKwbe)(F) ;::: 2 + E w(j.L (lKwbe)(F) :::; 1 2- E . In general a neural network that simulates a DFA will carry out not just one, but a fixed number k of computation steps (=state transitions) of the form x' = a(W x + h + uc + V) for each input symbol U E U that it reads (see the constructions described in [Omlin, Giles, 1996], and in section 3 of this article). This can easily be reflected in our model by formally replacing any input sequence w = UI, U2 , . . . , U r from U* by a padded sequence W = UI , bk - I , U2 , bk - I , ... ,U r , bk - I from (U U {b})*, where b is a blank symbol not in U, and bk - I denotes a sequence of k - 1 copies of b (for some arbitrarily fixed k ;::: 1). This completes our definition of language recognition by a noisy analog computational system M with discrete time. This definition essentially agrees with that given in [Maass, Orponen, 1997]. We employ the following common notations from formal language theory: We write WI w2 for the concatenation of two strings WI and W2, for the set of all concatenations of r strings from U, U* for the set of all concatenations of any finite number of strings from U, and UV for the set of all strings WI W2 with WI E U and W2 E V . The main result of this article is the following: ur Theorem 1.1 Assume that U is some arbitrary finite alphabet. A language L S;;; U* can be recognized by a noisy analog computational system of the previously specified type if and only if L E1 UU* E2 for two finite subsets E1 and E2 of U* . = A corresponding version of Theorem 1.1 for discrete computational systems was previously shown in [Rabin, 1963]. More precisely, Rabin had shown that probabilistic automata with strictly positive matrices can recognize exactly the same class of languages L that occur in our Theorem 1.1. Rabin referred to these languages as definite languages. Language recognition by analog computational systems with analog noise has previously been investigated in [Casey, 1996] for the special case of bounded noise and perfect reliability n that the following two properties hold: <jJ(v) :::: Co for all v E Q := a-I (no) (that is, Q is the set consisting of all possible differences z - y, with a(z) E no and yEn) and a-I (no) has finite and nonzero Lebesgue measuremo =). (a- 1 (no)). W Maass and E. D. Sontag 284 (i.e. ~ l vll:S;71 ?>(v)dv = 1 for some small TJ > 0 and c = 1/2 in our terminology), and in [Maass, Orponen, 1997] for the general case. It was shown in [Maass, Orponen, 1997] that any such system can only recognize regular languages. Furthermore it was shown there that if ~lvll:S;71 ?>(v)dv = 1 for some small TJ > 0 then all regular languages can be recognized by such systems. In the present paper we focus on the complementary case where the condition "~lvll:S;71 ?>(v)dv = 1 for some small '" > 0" is not satisfied, i.e. analog noise may move states over larger distances in the state space. We show that even if the probability of such event is arbitrarily small, the neural net will no longer be able to recognize arbitrary regular languages. 2 A Constraint on Language Recognition We prove in this section the following result for arbitrary noisy computational systems M as specified at the end of section 1: Theorem 2.1 Assume that U is some arbitrary alphabet. If a language L ~ U* is recognized by M, then there are subsets E1 and E2 of u:S;r, for some integer r, such that L = E1 U U* E 2. In other words: whether a string w E U* belongs to the language L can be decided by just inspecting the first r and the last r symbols of w. 2.1 A General Fact about Stochastic Kernels Let (5, S) be a measure space, and let K be a stochastic kernel 3 . As in the special case of the Ku's above, for each (signed) measure f-t on (5, S), we let II<?t be the (signed) measure defined on S by (II<?t)(A) := J K(x, A)df-t(x) . Observe that II<?t is a probability measure whenever f-t is. Let c > 0 be arbitrary. We say that K satisfies Doeblin's condition (with constant c) if there is some probability measure p on (5, S) so that K(x, A) ~ cp(A) for all x E 5, A E S. (2) (Necessarily c ::; 1, as is seen by considering the special case A = 5.) This condition is due to [Doeblin, 1937]. We denote by Ilf-til the total variation of the (signed) measure f-t. Recall that Ilf-til is defined as follows. One may decompose 5 into a disjoint union of two sets A and B, in such a manner that f-t is nonnegative on A and nonpositive on B. Letting the restrictions of f-t to A and B be "f-t+" and "-f-t-" respectively (and zero on B and A respectively), we may decompose f-t as a difference of nonnegative measures with disjoint supports, f-t = f-t+ - f-t- . Then, Ilf-til = f-t+ (A) + f-t- (B). The following Lemma is a "folk" fact ([Papinicolaou, 1978]). Lemma 2.2 Assume that K satisfies Doeblin's condition with constant c. Let f-t be any (signed) measure such that f-t(5) = o. Then 111I<?t11 ::; (1 - c) 1If-t1l. ? 2.2 Proof of Theorem 2.1 Lemma 2.3 There is a constant c constantc,foreveryu E U. > 0 such that Ku satisfies Doeblin's condition with Proof Let no, co, and 0 < rno < 1 be as in the second footnote, and introduce the following (Borel) probability measure on no: Ao(A) := ~A (0'-1 (A)) rno . 3That is to say, K(x,?) is a probability distribution for each x, and K(-, A) is a measurable function for each Borel measurable set A. 285 Analog Neural Nets with Gaussian Noise Pick any measurable A ~ no and any yEn. Then, Z(y, A) Prob [O"(y = + V) (?(v) dv JAy ~ E A] = Prob [y +V E 0"-1 (A)] coA(Ay) = CoA (0"-1 (A?) = comoAo(A) , where Ay := 0"-1 (A) - {y} ~ Q. We conclude that Z(y, A) ~ cAo(A) for all y, A, where c = como. Finally, we extend the measure AO to all of n by assigning zero measure to the complement of no, that is, p(A) := Ao(A no) for all measurable subsets A of n . Pick u E U; we will show that Ku satisfies Doeblin's condition with the above constant c (and using p as the "comparison" measure in the definition). Consider any x E nand measurable A ~ n. Then, n Ku(x, A) = Z(f(x, u), A) ~ Z(f(x, u), A n no) ~ cAo(A n no) = cp(A) , ? as required. For every two probability measures 1-'1,1-'2 on n, applying Lemma 2.2 to I-' := 1-'1 -1-'2, we know that 111Ku1-'1 -1Ku1-'211 ::; (1 - c) 111-'1 - 1-'211 for each u E U . Recursively, then, we conclude: (3) IllKwl-'l -lKw1-'211 ::; (1 111-'1 - 1-'211::; 2(1 - ct for all words w of length ~ ct r. Now pick any integer r such that (1 - ct < 2c. From Equation (3), we have that for all w of length ~ r and any two probability measures 1-'1,1-'2. In particular, this means that, for each measurable set A, (4) for all such w. (Because, for any two probability measures set A, 2Ivl(A) - v2(A)1 ::; Ilvl - v211 ?) VI and V2, and any measurable Lemma 2.4 Pick any v E U* and wE ur. Then w E L {:::::::} vw E L . Proof Assume that w E L, that is, (lKw t5e)(F) ~ ~+E. Applying inequality (4) to the measures 1-'1 := t5e and 1-'2 := lKvt5e and A = F, we have that l(lKwt5e)(F) - (lKvw t5e)(F) I < 2E,andthisimpliesthat(lKvwt5e)(F) > ~-E,i.e. , vw E L. (Since ~-E < (lKvwt5e)(F) < ~ + E is ruled out.) If w ~ L, the argument is similar. ? We have proved that So, n where El := L u~r and E2 := L proof of Theorem 2.1. nur are both included in u~r. This completes the ? 286 3 W. Maass and E. D. Sontag Construction of Noise Robust Analog Neural Nets In this section we exhibit a method for making feedforward analog neural nets robust with regard to arbitrary analog noise of the type considered in the preceding sections. This method will be used to prove in Corollary 3.2 the missing positive part of the claim of the main result (Theorem 1.1) of this article. Theorem 3.1 Let C be any (noiseless) feedfOlward threshold circuit, and let u : ~ -+ [-1, 1] be some arbitrary function with u( u) -+ 1 for u -+ 00 and u( u) -+ -1 for u -+ -00. Furthermore assume that 8, p E (0, 1) are some arbitrary given parameters. Then one can transform for any given analog noise of the type considered in section 1 the noiseless threshold circuit C into an analog neural net Nc with the same number of gates, whose gates employ the given function u as activation function, so that for any circuit input ~ E {-I, l}m the output of the noisy analog neural net Nc differs with probability ~ 1- 8 by at most p from the output ofC. Idea of the proof Let k be the maximal fan-in of a gate in C, and let w be the maximal absolute value of a weight in C. We choose R > so large that the density function ?>(.) of the noise vector V satisfies for each gate with n inputs in C ? { JIVil~R ?>(v)dv:c:; ? 28 for i= 1, ... ,n. n Furthermore we choose Uo > so large that u(u) ~ 1 - p/(wk) for u ~ Uo and u(u) :c:; -1 + p/(wk) for u :c:; -Uo . Finally we choose a factor "/ > so large that ,,/(1- p) - R ~ Uo. LetNc be the analog neural net that results from C through multiplication of all weights and thresholds with "/ and through replacement of the Heaviside activation functions of the gates in C by the given activation function u. ? ? The following Corollary provides the proof of the positive part of our main result Theorem 1.1. It holds for any u considered in Theorem 3.1. Corollary 3.2 Assume that U is some arbitrary finite alphabet, and language L ~ U* is of the form L = El U U* E2 for two arbitrary finite subsets El and E2 of U*. Then the language L can be recognized by a noisy analog neural net N with any desired reliability E E (0, ~), in spite of arbitrary analog noise of the type considered in section 1. Proof. We first construct a feed forward threshold circuit C for recognizing L, that receives each input symbol from U in the form of a bitstring u E {a, 1}' (for some fixed I ~ log2 \U\), that is encoded as the binary states of l input units of the boolean circuit C. Via a tapped delay line of fixed length d (which can easily be implemented in a feedforward threshold circuit by d layers, each consisting of l gates that compute the identity function on a single binary input from the preceding layer) one can achieve that the feed forward circuit C computes any given boolean function of the last d sequences from {O, 1}1 that were presented to the circuit. On the other hand for any language of the form L = El U U* E2 with E 1 , E2 finite there exists some dEN such that for each w E U* one can decide whether w E L by just inspecting the last d characters of w. Therefore a feedforward threshold circuit C with a tapped delay line of the type described above can decide whether wE L. = t). We apply Theorem 3.1 to this circuit C for 8 p = min(~ - E, We define the set F of accepting states for the resulting analog neural net Nc as the set of those states where the computation is completed and the output gate of Nc assumes a value ~ 3/4. Then according to Theorem 3.1 the analog neural net Nc recognizes L with reliability E. To be formally precise, one has to apply Theorem 3.1 to a threshold circuit C that receives its Analog Neural Nets with Gaussian NOise 287 input not in a single batch, but through a sequence of d batches. The proof of Theorem 3.1 readily extends to this case. ? 4 Conclusions We have exhibited a fundamental limitation of analog neural nets with Gaussian or other common noise distributions whose probability density function is nonzero on a large set: They cannot accept the very simple regular language {w E {O, 1 }*I w begins with O}. This holds even if the designer of the neural net is allowed to choose the parameters of the Gaussian noise distribution and the architecture and parameters of the neural net. The proof of this result introduces new mathematical arguments into the investigation of neural computation, which can also be applied to other stochastic analog computational systems. We also have presented a method for makingfeedfOlward analog neural nets robust against the same type of noise. This implies that certain regular languages, such as for example {w E {O, 1 }*I wends with O} can be recognized by a recurrent analog neural net with Gaussian noise. In combination with our negative result this yields a precise characterization of all regular languages that can be recognized by recurrent analog neural nets with Gaussian noise, or with any other noise distribution that has a large support. References [Casey, 1996] Casey, M., "The dynamics of discrete-time computation, with application to recurrent neural networks and finite state machine extraction", Neural Computation 8,1135-1178,1996. [Doeblin, 1937] Doeblin, W., "Sur Ie proprietes asymtotiques de mouvement regis par certain types de chaInes simples", Bull. Math. Soc. Roumaine Sci. 39(1): 57-115; (2) 3-61,1937. [Maass, Orponen, 1997] Maass, W., and Orponen, P. "On the effect of analog noise on discrete-time analog computations", Advances in Neural Information Processing Systems 9, 1997, 218-224; journal version: Neural Computation 10(5), 1071-1095, 1998. [amlin, Giles, 1996] amlin, C. W., Giles, C. L. "Constructing deterministic finite-state automata in recurrent neural networks", J. Assoc. Comput. Mach. 43 (1996), 937972. [Papinicolaou, 1978] Papinicolaou, G., "Asymptotic Analysis of Stochastic Equations", in Studies in Probability Theory, MAA Studies in Mathematics, vol. 18, 111-179, edited by M. Rosenblatt, Math. Assoc. of America, 1978. [Pippenger, 1985] Pippenger, N., "On networks of noisy gates", IEEE Sympos. on Foundations of Computer Science, vol. 26, IEEE Press, New York, 30-38, 1985. [Pippenger, 1989] Pippenger, N., ':Invariance of complexity measures for networks with unreliable gates", J. of the ACM, vol. 36, 531-539,1989. [Pippenger, 1990] Pippenger, N., "Developments in 'The Synthesis of Reliable Organisms from Unreliable Components' ", Proc. of Symposia in Pure Mathematics, vol. 50, 311-324,1990. [Rabin, 1963] Rabin, M., "Probabilistic automata", Information and Control, vol. 6, 230245, 1963. 

1530 |@word version:2 rno:2 pick:4 recursively:1 doeblin:7 carry:1 initial:2 irnxn:1 orponen:5 blank:1 activation:5 assigning:1 readily:1 realistic:1 benign:1 nur:1 accepting:5 v211:1 characterization:4 provides:1 math:2 sigmoidal:7 mathematical:1 lku:5 symposium:1 prove:3 consists:1 manner:1 introduce:2 coa:2 utc:1 considering:1 begin:4 project:1 notation:2 bounded:1 circuit:14 string:5 nj:1 eduardo:1 every:1 exactly:1 assoc:2 control:1 unit:7 uo:4 positive:4 before:1 consequence:1 mach:1 signed:6 therein:1 co:3 decided:1 union:1 definite:1 differs:1 xr:1 word:3 regular:12 spite:1 cannot:4 operator:1 impossible:1 applying:2 ilf:3 restriction:1 measurable:10 deterministic:1 map:2 missing:1 independently:1 automaton:3 immediately:1 pure:1 proving:1 variation:1 vll:1 construction:3 exact:1 tapped:2 recognition:3 graz:3 technological:1 edited:1 ui:2 complexity:1 dynamic:1 support2:1 technically:1 klosterwiesgasse:1 completely:1 easily:2 america:1 alphabet:4 sympos:1 whose:4 quite:1 larger:1 valued:1 encoded:1 say:5 transform:1 noisy:11 final:1 sequence:7 net:35 maximal:2 tu:1 cao:2 realization:1 achieve:1 perfect:1 illustrate:1 recurrent:11 ac:1 soc:1 implemented:1 implies:2 indicate:1 uu:1 stochastic:5 suffices:1 fix:1 ao:3 decompose:2 investigation:1 biological:1 inspecting:2 strictly:2 hold:4 considered:4 claim:1 proc:1 applicable:1 bitstring:1 agrees:1 ofc:1 tool:1 gaussian:16 rather:1 corollary:3 focus:1 casey:3 ivl:1 regis:1 inst:1 el:4 nand:1 accept:1 irn:5 development:1 special:3 fairly:1 uc:2 construct:1 extraction:1 serious:1 employ:2 recognize:7 consisting:3 lebesgue:1 replacement:1 possibility:1 severe:1 introduces:1 tj:2 devoted:1 folk:1 ruled:1 desired:1 theoretical:2 rabin:5 giles:4 boolean:5 bull:1 technische:1 subset:6 recognizing:1 delay:2 universitat:1 density:4 fundamental:2 ie:1 probabilistic:2 synthesis:1 concrete:1 satisfied:1 choose:4 til:3 li:1 ku1:2 de:2 wk:2 igi:1 vi:3 view:1 wolfgang:1 start:1 yen:2 yield:1 basically:1 startling:1 footnote:1 whenever:2 email:2 definition:8 against:3 involved:1 e2:8 proof:9 nonpositive:1 proved:1 austria:2 recall:1 ut:1 hilbert:1 feed:2 reflected:1 generality:1 furthermore:3 just:3 hand:2 receives:3 replacing:1 usa:1 effect:2 evolution:1 imprecision:1 read:1 nonzero:3 maass:10 deal:1 ay:2 cp:2 common:5 physical:1 analog:47 extend:1 organism:1 composition:1 uv:1 mathematics:3 language:31 had:1 reliability:7 longer:1 wxt:1 belongs:1 certain:2 inequality:1 binary:2 arbitrarily:2 der:1 fen:1 seen:1 preceding:3 employed:1 recognized:10 ii:3 full:1 e1:4 essentially:1 noiseless:2 rutgers:2 df:1 represent:1 kernel:3 zur:1 completes:2 w2:4 exhibited:1 subject:2 virtually:1 simulates:1 fwf:1 integer:2 vw:2 presence:1 feedforward:6 architecture:1 idea:1 whether:3 sontag:5 york:1 wissenschaftlichen:1 jj:1 adequate:1 dfa:1 concentrated:1 designer:2 disjoint:2 rosenblatt:1 discrete:4 write:1 vol:5 terminology:1 threshold:7 amlin:2 drawn:1 padded:1 prob:4 extends:1 decide:2 bound:1 wel:1 ct:3 layer:2 fan:1 nonnegative:2 occur:1 constraint:2 precisely:2 u1:2 lkw:1 argument:2 min:1 according:2 combination:1 character:1 ur:2 wi:4 making:4 dv:5 den:1 equation:2 previously:3 discus:1 turn:1 know:1 letting:1 end:3 apply:2 observe:1 v2:3 batch:2 encounter:1 gate:10 denotes:1 assumes:1 t11:1 recognizes:3 completed:1 log2:1 instant:1 ikw:1 move:1 exhibit:2 distance:1 t1l:1 sci:1 concatenation:3 length:3 sur:1 nc:5 statement:1 negative:4 finite:10 precise:6 arbitrary:15 introduced:1 bk:4 complement:1 required:1 specified:2 accepts:1 able:1 reading:2 rf:1 reliable:1 power:1 omlin:2 event:1 difficulty:1 literature:1 multiplication:1 asymptotic:1 par:1 limitation:2 analogy:1 foundation:1 article:5 lo:1 supported:1 last:3 copy:1 formal:1 wide:1 absolute:1 distributed:1 regard:3 transition:1 computes:1 forward:2 compact:2 unreliable:2 conceptual:1 conclude:2 nature:1 ku:7 robust:8 investigated:1 necessarily:1 constructing:3 main:4 noise:35 allowed:1 complementary:1 referred:1 borel:6 xl:1 comput:1 jay:1 theorem:18 xt:2 symbol:4 exists:2 fonds:1 partially:1 u2:2 maa:1 satisfies:5 acm:1 identity:1 pippenger:10 included:1 infinite:1 determined:1 contrasted:1 lemma:5 total:1 invariance:1 formally:2 internal:1 support:2 brunswick:1 heaviside:1

Finite-Sample Convergence Rates for Q-Learning and Indirect Algorithms Michael Kearns and Satinder Singh AT&T Labs 180 Park Avenue Florham Park , NJ 07932 {mkearns,baveja}@research.att.com Abstract In this paper, we address two issues of long-standing interest in the reinforcement learning literature. First, what kinds of performance guarantees can be made for Q-learning after only a finite number of actions? Second, what quantitative comparisons can be made between Q-learning and model-based (indirect) approaches, which use experience to estimate next-state distributions for off-line value iteration? We first show that both Q-learning and the indirect approach enjoy rather rapid convergence to the optimal policy as a function of the number of state transitions observed. In particular, on the order of only (Nlog(1/c)/c 2 )(log(N) + loglog(l/c)) transitions are sufficient for both algorithms to come within c of the optimal policy, in an idealized model that assumes the observed transitions are "well-mixed" throughout an N-state MDP. Thus, the two approaches have roughly the same sample complexity. Perhaps surprisingly, this sample complexity is far less than what is required for the model-based approach to actually construct a good approximation to the next-state distribution. The result also shows that the amount of memory required by the model-based approach is closer to N than to N 2 ? For either approach, to remove the assumption that the observed transitions are well-mixed, we consider a model in which the transitions are determined by a fixed, arbitrary exploration policy. Bounds on the number of transitions required in order to achieve a desired level of performance are then related to the stationary distribution and mixing time of this policy. 1 Introduction There are at least two different approaches to learning in Markov decision processes: indirect approaches, which use control experience (observed transitions and payoffs) to estimate a model, and then apply dynamic programming to compute policies from the estimated model; and direct approaches such as Q-Iearning [2], which use control Convergence Rates for Q-Leaming and Indirect Algorithms 997 experience to directly learn policies (through value functions) without ever explicitly estimating a model. Both are known to converge asymptotically to the optimal policy [1, 3] . However, little is known about the performance of these two approaches after only a finite amount of experience . A common argument offered by proponents of direct methods is that it may require much more experience to learn an accurate model than to simply learn a good policy. This argument is predicated on the seemingly reasonable assumption that an indirect method must first learn an accurate model in order to compute a good policy. On the other hand, proponents of indirect methods argue that such methods can do unlimited off-line computation on the estimated model, which may give an advantage over direct methods, at least if the model is accurate. Learning a good model may also be useful across tasks, permitting the computation of good policies for multiple reward functions [4]. To date, these arguments have lacked a formal framework for analysis and verification. In this paper, we provide such a framework, and use it to derive the first finite-time convergence rates (sample size bounds) for both Q-learning and the standard indirect algorithm. An important aspect of our analysis is that we separate the quality of the policy generating experience from the quality of the two learning algorithms. In addition to demonstrating that both methods enjoy rather rapid convergence to the optimal policy as a function of the amount of control experience, the convergence rates have a number of specific and perhaps surprising implications for the hypothetical differences between the two approaches outlined above. Some of these implications, as well as the rates of convergence we derive, were briefly mentioned in the abstract; in the interests of brevity, we will not repeat them here, but instead proceed directly into the technical material. 2 MDP Basics Let M be an unknown N-state MDP with A actions . We use PM(ij) to denote the probability of going to state j, given that we are in state i and execute action a; and RM(i) to denote the reward received for executing a from i (which we assume is fixed and bounded between 0 and 1 without loss of generality). A policy 1r assigns an action to each state. The value of state i under policy 1r, VM(i), is the expected discounted sum of rewards received upon starting in state i and executing 1r forever : VM(i) = E7r[rl + ,r2 + ,2r3 + ...], where rt is the reward received at time step t under a random walk governed by 1r from start state i, and 0 ~ , < 1 is the discount factor . It is also convenient to define values for state-action pairs (i, a): QM(i, a) = RM (i) Lj PM (ij) VM(j) . The goal of learning is to approximate the optimal policy 1r* that maximizes the value at every state; the optimal value function is denoted QM. Given QM' we can compute the optimal policy as 1r*(i) = argmaxa{QM(i,a)}. +, If M is given , value iteration can be used to compute a good approximation to the optimal value function. Setting our initial guess as Qo(i, a) = 0 for all (i, a), we iterate as follows: RM(i) +, 2)PM(ij)Ve(j)] (1) j where we define \Il(j) = maxv{Qe(j, b)}. It can be shown that after I! iterations, max(i,aj{IQe(i, a) - QM(i , a)1} ~ Given any approximation Q to QM we can compute the greedy approximation 1r to the optimal policy 1r* as 1r(i) = argmaxa{Q(i, a)}. ,e. 998 3 M Kearns and S. Singh The Parallel Sampling Model In reinforcement learning, the transition probabilities PM(ij) are not given, and a good policy must be learn ed on the basis of observed experience (transitions) in M . Classical convergence results for algorithms such as Q-Iearning [1] implicitly assume that the observed experience is generated by an arbitrary "exploration policy" 7r, and then proceed to prove convergence to the optimal policy if 7r meets certain minimal conditions - namely, 7r must try every state-action pair infinitely often, with probability 1. This approach conflates two distinct issues : the quality of the exploration policy 7r, and the quality ofreinforcement learning algorithms using experience generated by 7r. In contrast, we choose to separate these issues. If the exploration policy never or only very rarely visits some state-action pair, we would like to have this reflected as a factor in our bounds that depends only on 7r; a separate factor depending only on the learning algorithm will in turn reflect how efficiently a particular learning algorithm uses the experience generated by 7r . Thus, for a fixed 7r, all learning algorithms are placed on equal footing, and can be directly compared. There are probably various ways in which this separation can be accomplished; we now introduce one that is particularly clean and simple. We would like a model of the ideal exploration policy - one that produces experiences that are "well-mixed", in the sense that every state-action pair is tried with equal frequency. Thus, let us define a parallel sampling subroutine PS(M) that behaves as follows: a single call to PS( M) returns, for every state-action pair (i, a), a random next state j distributed according to PM (ij). Thus, every state-action pair is executed simultaneously, and the resulting N x A next states are reported. A single call to PS(M) is therefore really simulating N x A transitions in M, and we must be careful to multiply the number of calls to PS(M) by this factor if we wish to count the total number of transitions witnessed. What is PS(M) modeling? It is modeling the idealized exploration policy that manages to visit every state-action pair in succession, without duplication, and without fail. It should be intuitively obvious that such an exploration policy would be optimal, from the viewpoint of gathering experience everywhere as rapidly as possible. We shall first provide an analysis, in Section 5, of both direct and indirect reinforcement learning algorithms, in a setting in which the observed experience is generated by calls to PS(M). Of course, in any given MDP M , there may not be any exploration policy that meets the ideal captured by PS(M) - for instance, there may simply be some states that are very difficult for any policy to reach, and thus the experience generated by any policy will certainly not be equally mixed around the entire MDP. (Indeed, a call to PS(M) will typically return a set of transitions that does not even correspond to a trajectory in M.) Furthermore, even if PS(M) could be simulated by some exploration policy, we would like to provide more general results that express the amount of experience required for reinforcement learning algorithms under any exploration policy (where the amount of experience will , of course, depend on properties of the exploration policy). Thus, in Section 6, we sketch how one can bound the amount of experience required under any 7r in order to simulate calls to PS(M) . (More detail will be provided in a longer version of this paper.) The bound depends on natural properties of 7r, such as its stationary distribution and mixing time. Combined with the results of Section 5, we get the desired two-factor bounds discussed above: for both the direct and indirect approaches, a bound on the total number of transitions required, consisting of one factor that depends only on the algorithm, and another factor that depends only on the exploration policy. Convergence Rates for Q-Learning and Indirect Algorithms 4 999 The Learning Algorithms We now explicitly state the two reinforcement learning algorithms we shall analyze and compare. In keeping with the separation between algorithms and exploration policies already discussed, we will phrase these algorithms in the parallel sampling framework, and Section 6 indicates how they generalize to the case of arbitrary exploration policies. We begin with the direct approach. Rather than directly studying standard Q-Iearning, we will here instead examine a variant that is slightly easier to analyze, and is called phased Q-Iearning. However, we emphasize that all of our resuits can be generalized to apply to standard Q-learning (with learning rate a(i, a) = t(i~a)' where t(i, a) is the number oftrials of (i, a) so far) . Basically, rather than updating the value function with every observed transition from (i , a), phased Q-Iearning estimates the expected value of the next state from (i, a) on the basis of many transitions, and only then makes an update. The memory requirements for phased Q-learning are essentially the same as those for standard Q-Iearning. Direct Algorithm - Phased Q-Learning: As suggested by the name , the algorithm operates in phases. In each phase, the algorithm will make mD calls to PS(M) (where mD will be determined by the analysis), thus gathering mD trials of every state-action pair (i, a) . At the fth phase, the algorithm updates the estimated value function as follows: for every (i , a), Ql+d i , a) = RM(i) + ,_1_ ~ Oeu?) (2) mD k=l where jf, ... , j~ are the m D next states observed from (i, a) on the m D calls to PS(M) during t~e fth phase. The policy computed by the algorithm is then the greedy policy determined by the final value function. Note that phased Q-learning is quite like standard Q-Iearning, except that we gather statistics (the summation in Equation (2)) before making an update. We now proceed to describe the standard indirect approach . Indirect Algorithm: The algorithm first makes m[ calls to PS(M) to obtain m[ next state samples for each (i, a) . It then builds an empirical model of the transition probabilities as follows: PM(ij) = #(~aj) , where #(i -+a j) is the number of times state j was reached on the m[ trials of (i, a). The algorithm then does value iteration (as described in Section 2) on the fixed model PM(ij) for f[ phases. Again , the policy computed by the algorithm is the greedy policy dictated by the final value function . Thus , in phased Q-Iearning, the algorithm runs for some number fD phases, and each phase requires mD calls to PS(M), for a total number of transitions fD x mD x N x A . The direct algorithm first makes m j calls to PS(M) , and then runs f[ phases of value iteration (which requires no additional data) , for a total number of transitions m[ x N x A. The question we now address is: how large must mD, m[, fD' f[ be so that, with probability at least 1 - 6, the resulting policies have expected return within f. of the optimal policy in M? The answers we give yield perhaps surprisingly similar bounds on the total number of transitions required for the two approaches in the parallel sampling model. 5 Bounds on the Number of Transitions We now state our main result. M Kearns and S. Singh 1000 Theorem 1 For any MDP M: ? For an appropriate choice of the parameters mJ and and fJ, the total number of calls to PS(M) required by the indirect algorithm in order to ensure that, with probability at least 1 - 6, the expected return of the resulting policy will be within f of the optimal policy, is O((I/f 2 )(log(N/6) + loglog(l/f)). (3) ? For an appropriate choice of the parameters mD and fD, the total number of calls to PS(M) required by phased Q-learning in order to ensure that, with probability at least 1 - 6, the expected return of the resulting policy will be within f of the optimal policy, is O((log(1/f)/f 2 )(log(N/6) + log log(l/f)). (4) The bound for phased Q-learning is thus only O(log(l/f)) larger than that for the indirect algorithm. Bounds on the total number of transitions witnessed in either case are obtained by multiplying the given bounds by N x A . Before sketching some of the ideas behind the proof of this result, we first discuss some of its implications for the debate on direct versus indirect approaches. First of all, for both approaches, convergence is rather fast: with a total number of transitions only on the order of N log(N) (fixing f and 6 for simplicity), near-optimal policies are obtained. This represents a considerable advance over the classical asymptotic results: instead of saying that an infinite number of visits to every state-action pair are required to converge to the optimal policy, we are claiming that a rather small number of visits are required to get close to the optimal policy. Second, by our analysis, the two approaches have similar complexities, with the number of transitions required differing by only a log(l/f) factor in favor of the indirect algorithm. Third - and perhaps surprisingly - note that since only O(log(N)) calls are being made to PS(M) (again fixing f and 6), and since the number of trials per state-action pair is exactly the number of calls to PS(M), the total number of non-zero entries in the model PM (ij) built by the indirect approach is in fact only O(log( N)). In other words , PM (ij) will be extremely sparse - and thus, a terrible approximation to the true transition probabilities - yet still good enough to derive a near-optimal policy! Clever representation of PM(ij) will thus result in total memory requirements that are only O(N log(N)) rather than O(N2). Fourth, although we do not have space to provide any details, if instead of a single reward function, we are provided with L reward functions (where the L reward functions are given in aqvance of observing any experience), then for both algorithms, the number of transitions required to compute near-optimal policies for all L reward functions simultaneously is only a factor of O(log(L)) greater than the bounds given above. Our own view of the result and its implications is: ? Both algorithms enjoy rapid convergence to the optimal policy as a function of the amount of experience. ? In general, neither approach enjoys a significant advantage in convergence rate, memory requirements, or handling multiple reward functions. Both are quite efficient on all counts. We do not have space to provide a detailed proof of Theorem 1, but instead provide some highlights of the main ideas. The proofs for both the indirect algorithm and phased Q-Iearning are actually quite similar, and have at their heart two slightly /001 Convergence Rates for Q-Learning and Indirect Algorithms different uniform convergence lemmas. For phased Q-Iearning, it is possible to show that, for any bound fD on the number of phases to be executed, and for any T > 0, we can choose mD so that mD (l/mD)LVtU?)- LPijVtU) < T (5) j k=l will hold simultaneously for every (i, a) and for every phase f = 1, . . . , fD. In other words, at the end of every phase, the empirical estimate of the expected next-state value for every (i, a) will be close to the true expectation, where here the expectation is with respect to the current estimated value function Vt. For the indirect algorithm, a slightly more subtle uniform convergence argument is required. Here we show that it is possible to choose, for~any bound fI on the number of iterations of value iteration to be executed on the PM(ij), and for any T > 0, a value mI such that (6) j j for every (i,a) and every phase f = 1, . . . ,fI, where the VtU) are the value functions resulting from performing true value iteration (that is, on the PM (ij)). Equation (6) essentially says that expectations of the true value functions are quite similar under either the true or estimated model, even though the indirect algorithm never has access to the true value functions . In either case, the uniform convergence results allow us to argue that the corresponding algorithms still achieve successive contractions, as in the classical proof of value iteration. For instance, in the case of phased Q-Iearning, if we define b..l max(i ,a){IQe(i, a) - Ql(i , a)l}, we can derive a recurrence relation for b..l+ 1 as follows : = m ,(l/m) L VtU?) -, L Pij VtU) j k=l "E'I',~x,} { < 7 < ,T + ,b..l . (7) (y P;j v,(j) +") - y P;j V, (j) }S) (9) ~ Here we have made use of Equation (5). Since b.. o = 0 (Qo = Qo) , this recurrence gives b..l :::; Tb/(l--,)) for any f. From this it is not hard to show that for any (i,a) IQdi , a) - Q*(i, a)1 :::; Tb/(l -,)) + ,l . (10) From this it can be shown that the regret in expected return suffered by the policy computed by phased Q-Learning after f phases is at most (T, /(1-,) +,l )(2/(1-,)). The proof proceeds by setting this regret smaller than the desired f, solving for f and T, and obtaining the resulting bound on m D. The derivation of bounds for the indirect algorithm is similar. 6 Handling General Exploration Policies As promised, we conclude our technical results by briefly sketching how we can translate the bounds obtained in Section 5 under the idealized parallel sampling model into 1002 M Kearns and S. Singh bounds applicable when any fixed policy 1r is guiding the exploration. Such bounds must, of course, depend on properties of 1r. Due to space limitations, we can only outline the main ideas; the formal statements and proofs are deferred to a longer version of the paper. Let us assume for simplicity that 1r (which may be a stochastic policy) defines an ergodic Markov process in the MDP M. Thus, 1r induces a unique stationary distribution PM,1[(i, a) over state-action pairs - intuitively, PM ,1[(i, a) is the frequency of executing action a from state i during an infinite random walk in M according to 1r. Furthermore, we can introduce the standard notion of the mixing time of 1r to its stationary distribution - informally, this is the number T1[ of steps required such that the distribution induced on state-action pairs by T1[-step walks according to 1r will be "very close" to PM,1[ 1. Finally, let us define P1[ = min(i,a){PM,1[(i, an. Armed with these notions, it is not difficult to show that the number of steps we must take under 1r in order to simulate, with high probability, a call to the oracle PS(M) , is polynomial in the quantity T1[ / P1[. The intuition is straightforward: at most every T1[ steps, we obtain an "almost independent" draw from PM,1[(i, a); and with each independent draw, we have at least probability p of drawing any particular (i, a) pair. Once we have sampled every (i, a) pair, we have simulated a call to PS(M). The formalization of these intuitions leads to a version of Theorem 1 applicable to any 1r, in which the bound is multiplied by a factor polynomial in T1[ / P1[, as desired. However, a better result is possible . In cases where P1[ may be small or even 0 (which would occur when 1r simply does not ever execute some action from some state), the factor T1[ / P1[ is large or infinite and our bounds become weak or vacuous. In such cases, it is better to define the sub-MDP M1[(O'), which is obtained from M by simply deleting any (i, a) for which PM,1[(i, a) < a, where a> 0 is a parameter of our choosing . In M1[ (a), P1[ > a by construction, and we may now obtain convergence rates to the optimal policy in M1[ (a) for both Q-Iearning and the indirect approach like those given in Theorem 1, multiplied by a factor polynomial in T1[/O'. (Technically, we must slightly alter the algorithms to have an initial phase that detects and eliminates small-probability state-action pairs, but this is a minor detail.) By allowing a to become smaller as the amount of experience we receive from 1r grows, we can obtain an "anytime" result, since the sub-MDP M1[(O') approaches the full MDP M as 0'-+0. References [1] Jaakkola, T., Jordan, M. I., Singh, S. On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6(6), 1185-1201, 1994. [2] C. J. C. H. Watkins. Learning from Delayed Rewards. Ph.D. thesis, Cambridge University, 1989. [3] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, 1998. [4] S. Mahadevan. Enhancing Transfer in Reinforcement Learning by Building Stochastic Models of Robot Actions. In Machine Learning: Proceedings of the Ninth International Conference, 1992. 1 Formally, the degree of closeness is measured by the distance between the transient and stationary distributions. For brevity here we will simply assume this parameter is set to a very small, constant value. 

1531 |@word trial:3 version:3 briefly:2 polynomial:3 tried:1 contraction:1 initial:2 mkearns:1 att:1 current:1 com:1 surprising:1 yet:1 must:8 remove:1 update:3 maxv:1 stationary:5 greedy:3 guess:1 footing:1 successive:1 direct:9 become:2 prove:1 fth:2 introduce:2 indeed:1 expected:7 rapid:3 p1:6 examine:1 roughly:1 discounted:1 detects:1 little:1 armed:1 provided:2 estimating:1 bounded:1 begin:1 maximizes:1 what:4 kind:1 differing:1 nj:1 guarantee:1 quantitative:1 every:18 hypothetical:1 iearning:12 exactly:1 rm:4 qm:6 control:3 enjoy:3 before:2 t1:7 sutton:1 meet:2 phased:12 unique:1 regret:2 empirical:2 convenient:1 word:2 argmaxa:2 get:2 close:3 clever:1 straightforward:1 starting:1 ergodic:1 simplicity:2 assigns:1 notion:2 construction:1 programming:2 us:1 particularly:1 updating:1 observed:9 mentioned:1 intuition:2 complexity:3 reward:10 dynamic:2 singh:5 depend:2 solving:1 technically:1 upon:1 basis:2 indirect:24 various:1 derivation:1 distinct:1 fast:1 describe:1 choosing:1 quite:4 larger:1 say:1 drawing:1 florham:1 statistic:1 favor:1 final:2 seemingly:1 advantage:2 nlog:1 date:1 rapidly:1 mixing:3 translate:1 achieve:2 convergence:19 p:21 requirement:3 produce:1 generating:1 executing:3 derive:4 depending:1 fixing:2 measured:1 ij:12 minor:1 received:3 come:1 stochastic:3 exploration:16 transient:1 material:1 require:1 really:1 summation:1 ofreinforcement:1 hold:1 around:1 proponent:2 applicable:2 mit:1 rather:7 barto:1 jaakkola:1 indicates:1 contrast:1 e7r:1 sense:1 lj:1 entire:1 typically:1 relation:1 going:1 subroutine:1 issue:3 denoted:1 equal:2 construct:1 never:2 once:1 sampling:5 represents:1 park:2 alter:1 simultaneously:3 ve:1 delayed:1 phase:14 consisting:1 vtu:3 interest:2 fd:6 multiply:1 certainly:1 deferred:1 behind:1 implication:4 accurate:3 closer:1 experience:21 walk:3 desired:4 minimal:1 witnessed:2 instance:2 modeling:2 phrase:1 entry:1 uniform:3 reported:1 answer:1 combined:1 international:1 standing:1 off:2 vm:3 michael:1 sketching:2 again:2 reflect:1 thesis:1 choose:3 return:6 explicitly:2 idealized:3 depends:4 try:1 view:1 lab:1 analyze:2 observing:1 reached:1 start:1 parallel:5 il:1 efficiently:1 succession:1 correspond:1 yield:1 generalize:1 weak:1 manages:1 basically:1 trajectory:1 multiplying:1 reach:1 ed:1 frequency:2 obvious:1 proof:6 mi:1 sampled:1 anytime:1 subtle:1 actually:2 reflected:1 execute:2 though:1 generality:1 furthermore:2 predicated:1 hand:1 sketch:1 qo:3 defines:1 quality:4 perhaps:4 aj:2 mdp:10 grows:1 building:1 name:1 true:6 during:2 recurrence:2 qe:1 generalized:1 outline:1 fj:1 fi:2 common:1 behaves:1 rl:1 discussed:2 m1:4 significant:1 cambridge:1 outlined:1 pm:18 baveja:1 access:1 robot:1 longer:2 own:1 dictated:1 certain:1 vt:1 accomplished:1 captured:1 additional:1 greater:1 converge:2 multiple:2 full:1 technical:2 long:1 equally:1 visit:4 permitting:1 variant:1 basic:1 essentially:2 expectation:3 enhancing:1 iteration:9 receive:1 addition:1 suffered:1 eliminates:1 probably:1 duplication:1 induced:1 jordan:1 call:17 near:3 ideal:2 mahadevan:1 enough:1 iterate:1 idea:3 avenue:1 proceed:3 action:20 useful:1 detailed:1 informally:1 amount:8 discount:1 ph:1 induces:1 terrible:1 estimated:5 per:1 shall:2 express:1 demonstrating:1 promised:1 neither:1 clean:1 asymptotically:1 sum:1 run:2 everywhere:1 fourth:1 throughout:1 reasonable:1 saying:1 almost:1 separation:2 draw:2 decision:1 bound:22 oracle:1 occur:1 unlimited:1 aspect:1 simulate:2 argument:4 extremely:1 min:1 performing:1 according:3 across:1 slightly:4 smaller:2 making:1 intuitively:2 gathering:2 heart:1 equation:3 turn:1 r3:1 count:2 fail:1 discus:1 end:1 studying:1 lacked:1 multiplied:2 apply:2 appropriate:2 simulating:1 assumes:1 ensure:2 build:1 classical:3 already:1 question:1 quantity:1 rt:1 md:11 distance:1 separate:3 simulated:2 argue:2 difficult:2 executed:3 ql:2 statement:1 debate:1 claiming:1 policy:55 unknown:1 allowing:1 markov:2 finite:4 payoff:1 ever:2 ninth:1 arbitrary:3 conflates:1 vacuous:1 pair:15 required:15 namely:1 address:2 suggested:1 proceeds:1 tb:2 built:1 max:2 memory:4 deleting:1 natural:1 literature:1 asymptotic:1 loss:1 highlight:1 mixed:4 limitation:1 versus:1 degree:1 offered:1 sufficient:1 verification:1 gather:1 pij:1 viewpoint:1 course:3 surprisingly:3 repeat:1 placed:1 keeping:1 enjoys:1 formal:2 allow:1 sparse:1 distributed:1 transition:25 made:4 reinforcement:7 far:2 approximate:1 emphasize:1 forever:1 implicitly:1 satinder:1 conclude:1 iterative:1 learn:5 mj:1 transfer:1 obtaining:1 main:3 n2:1 formalization:1 sub:2 guiding:1 wish:1 governed:1 watkins:1 loglog:2 third:1 theorem:4 specific:1 r2:1 closeness:1 easier:1 simply:5 infinitely:1 goal:1 careful:1 leaming:1 jf:1 considerable:1 hard:1 determined:3 except:1 operates:1 infinite:3 kearns:4 lemma:1 total:11 called:1 rarely:1 formally:1 brevity:2 handling:2

Mean field methods for classification with Gaussian processes Manfred Opper Neural Computing Research Group Division of Electronic Engineering and Computer Science Aston University Birmingham B4 7ET, UK. opperm~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, S6lvegatan 14 A S-223 62 Lund, Sweden CONNECT, The Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, 2100 Copenhagen 0, Denmark winther~thep.lu.se Abstract We discuss the application of TAP mean field methods known from the Statistical Mechanics of disordered systems to Bayesian classification models with Gaussian processes. In contrast to previous approaches, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given. 1 Modeling with Gaussian Processes Bayesian models which are based on Gaussian prior distributions on function spaces are promising non-parametric statistical tools. They have been recently introduced into the Neural Computation community (Neal 1996, Williams & Rasmussen 1996, Mackay 1997). To give their basic definition, we assume that the likelihood of the output or target variable T for a given input s E RN can be written in the form p(Tlh(s)) where h : RN --+ R is a priori assumed to be a Gaussian random field. If we assume fields with zero prior mean, the statistics of h is entirely defined by the second order correlations C(s, S') == E[h(s)h(S')], where E denotes expectations M Opper and 0. Winther 310 with respect to the prior. Interesting examples are C(s, s') (1) C(s, s') (2) The choice (1) can be motivated as a limit of a two-layered neural network with infinitely many hidden units with factorizable input-hidden weight priors (Williams 1997). Wi are hyperparameters determining the relevant prior lengthscales of h(s). The simplest choice C(s, s') = 2::i WiSiS~ corresponds to a single layer percept ron with independent Gaussian weight priors. In this Bayesian framework, one can make predictions on a novel input s after having received a set Dm of m training examples (TJ.1., sJ.1.), J.L = 1, ... , m by using the posterior distribution of the field at the test point s which is given by p(h(s)IDm) = J p(h(s)l{hV}) p({hV}IDm) II dhJ.1.. (3) J.1. p(h(s)1 {hV}) is a conditional Gaussian distribution and p({hV}IDm) = ~P({hV}) II p(TJ.1.IhJ.1.). (4) J.1. is the posterior distribution of the field variables at the training points. Z is a normalizing partition function and (5) is the prior distribution of the fields at the training points. Here, we have introduced the abbreviations hJ.1. = h(sJ.1.) and CJ.1.V == C(sJ.1., SV). The major technical problem of this approach comes from the difficulty in performing the high dimensional integrations. Non-Gaussian likelihoods can be only treated by approximations, where e.g. Monte Carlo sampling (Neal 1997), Laplace integration (Barber & Williams 1997) or bounds on the likelihood (Gibbs & Mackay 1997) have been used so far. In this paper, we introduce a further approach, which is based on a mean field method known in the Statistical Physics of disordered systems (Mezard, Parisi & Virasoro 1987). We specialize on the case of a binary classification problem, where a binary class label T = ?1 must be predicted using a training set corrupted by i.i.d label noise. The likelihood for this problem is taken as where I\, is the probability that the true classification label is corrupted, i. e. flipped and the step function, 0(x) is defined as 0(x) = 1 for x > 0 and 0 otherwise. For such a case, we expect that (by the non-smoothness of the model), e.g. Laplace's method and the bounds introduced in (Gibbs & Mackay 1997) are not directly applicable. 31J Mean Field Methods for Classification with Gaussian Processes 2 Exact posterior averages In order to make a prediction on an input s, ideally the label with maximum posterior probability should be chosen, i.e. TBayes = argmax r p( TIDm), where the predictive probability is given by P(TIDm) = dhp(Tlh) p(hIDm). For the binary case the Bayes classifier becomes TBayes = sign(signh(s)), where we throughout the paper let brackets (... ) denote posterior averages. Here, we use a somewhat simpler approach by using the prediction T = sign((h(s))) . J This would reduce to the ideal prediction, when the posterior distribution of h(s) is symmetric around its mean (h(s)). The goal of our mean field approach will be to provide a set of equations for approximately determining (h( s)) . The starting point of our analysis is the partition function Z = JII dX;:ihJ.L J.L IIp(TJ.LlhJ.L)e~ LI' ,u cI'UxI'X U- (6) L I' hl'xl' , J.L where the new auxiliary variables x/l (integrated along the imaginary axis) have been introduced in order to get rid of C- l in (5). It is not hard to show from (6) that the posterior averages of the fields at the m training inputs and at a new test point s are given by (7) l/ l/ We have thus reduced our problem to the calculation of the "microscopic orderparameters" (x/l). 1 Averages in Statistical Physics can be calculated from derivatives of -In Z with respect to small external fields, which are then set to zero, An equivalent formulation uses the Legendre transform of -In Z as a function of the expectations , which in our case is given by G( {(XJ.L) , ((XJ.L)2)}) = -In Z(, /l, A) + L(xJ.L)'yJ.L + ~ L J.L with Z( bJ.L, A/l}) = JII dX;:ihJ.L IIp(TJ.LlhJ.L)e~ /l J.L AJ.L((XJ.L)2) . (8) J.L LI',JAI'Ol'u+Cl' u)x l'x u + LI' xl'(-yl' - hl'). (9) The additional averages ((XJ.L)2) have been introduced , because the dynamical variables xJ.L (unlike Ising spins) do not have fixed length. The external fields ,J.L , AJ.L must be eliminated from t~ = = 0 and the true expectation values of xJ.L and 8G ) -- 0 , (x J.L)2 are th ose wh'ICh sat'IS fy 8 ? 8G xl' )2) -- 8(xl' t; 3 Naive mean field theory So far, this description does not give anything new. Usually G cannot be calculated exactly for the non-Gaussian likelihood models of interest. Nevertheless, based on mean field theory (MFT) it is possible to guess an approximate form for G. 1 Although the integrations are over the imaginary axis, these expectations come out positive. This is due to the fact that the integration "measure" is complex as well. 312 M Opper and 0. Winther Mean field methods have found interesting applications in Neural Computing within the framework of ensemble learning, where the the exact posterior distribution is approximated by a simpler one using product distributions in a variational treatment. Such a "standard" mean field method for the posterior of the hf.L (for the case of Gaussian process classification) is in preparation and will be discussed somewhere else. In this paper, we suggest a different route, which introduces nontrivial corrections to a simple or "naive" MFT for the variables xl-'. Besides the variational method (which would be purely formal because the distribution of the xf.L is complex and does not define a probability), there are other ways to define the simple MFT. E.g., by truncating a perturbation expansion with respect to the "interactions" Cf.LV in G after the first order (Plefka 1982). These approaches yield the result G ~ Gnaive = Go - ~ :LCI-'f.L((XI-')2) - ~ I-' :L CI-'v(xl-')(XV). (10) 1-' , v, wl-f.L Go is the contribution to G for a model without any interactions i.e. when CI-'v = 0 in (9), i.e. it is the Legendre transform of - In Zo = l: In [~+ (1 - 2~) <I> (TI-' ; ; ) ] , I-' where <I>(z) = J~oo .:}f;e- t2 / 2 is an error function. For simple models in Statistical Physics, where all interactions CI-'V are positive and equal, it is easy to show that Gnaive will become exact in the limit of an infinite number of variables xl-'. Hence, for systems with a large number of nonzero interactions having the same orders of magnitude, one may expect that the approximation is not too bad. 4 The TAP approach Nevertheless, when the interactions Cf.LV can be both positive and negative (as one would expect e.g. when inputs have zero mean), even in the thermodynamic limit and for nice distributions of inputs, an additional contribution tlG must be added to the "naive" mean field theory (10). Such a correction (often called an Onsager reaction term) has been introduced for a spin glass model by (Thouless, Anderson & Palmer 1977) (TAP). It was later applied to the statistical mechanics of single layer perceptrons by (Mezard 1989) and then generalized to the Bayesian framework by (Opper & Winther 1996, 1997). For an application to multilayer networks, see (Wong 1995). In the thermodynamic limit of infinitely large dimension of the input space, and for nice input distributions, the results can be shown coincide with the results of the replica framework. The drawback of the previous derivations of the TAP MFT for neural networks was the fact that special assumptions on the input distribution had been made and certain fluctuating terms have been replaced by their averages over the distribution of random data, which in practice would not be available. In this paper, we will use the approach of (Parisi & Potters 1995), which allows to circumvent this problem. They concluded (applied to the case of a spin model with random interactions of a specific type), that the functional form of tlG should not depend on the type of the "single particle" contribution Go. Hence, one may use any model in Go, for which G can be calculated exactly (e.g. the Gaussian regression model) and subtract the naive mean field contribution to obtain the 313 Mean Field Methods for Classification with Gaussian Processes desired I:1G. For the sake of simplicity, we have chosen the even simpler model p( TI-'l hi-') '"" 6 (hi-') without changing the final result. A lengthy but straightforward calculation for this problem leads to the result (11) with RI-' == ((xl-')2) - (Xi-')2. The Ai-' must be eliminated using to the equation t j( = 0, which leads I' (12) Note, that with this choice, the TAP mean field theory becomes exact for Gaussian likelihoods , i. e. for standard regression problems. Finally, setting the derivatives of GT AP = Gnaive + I:1G with respect to the 4 variables (xl-'), ((xl-')2) ,rl-" AI-' equal to zero, we obtain the equat ions (13) v where D( z ) = e- z 2 /2 /..,j2; is the Gaussian measure. These eqs . have to be solved numerically together with (12). In contrast, for the naive MFT , the simpler result AI-' = C 1-'1-' is found. 5 Simulations Solving the nonlinear system of equations (12,13) by iteration turns out to be quite straightforward. For some data sets to get convergence, one has to add a diagonal term v to the covariance matrix C: Cij -+ Cij +6ijV. It may be shown that this term corresponds to learning with Gaussian noise (with variance v ) added the Gaussian random field. Here, we present simulation results for a single data set, the Sonar - Mines versus Rocks using the same training/test set split as in the original study by (Gorman & Sejnowski 1988). The input data were pre-processed by linear rescaling such that over the training set each input variable has zero mean and unit variance. In some cases the mean field equations failed to converge using the raw data. A further important feature of TAP MFT is the fact that the method also gives an approximate leave-one-out estimator for the generalization error , C]oo expressed in terms of the solution to the mean field equations (see (Opper & Winther 1996, 1997) for more details) . It is also possible to derive a leave-one-out estimator for the naive MFT (Opper & Winther to be published). Since we so far haven't dealt with the problem of automatically estimating the hyperparameters, their number was drastically reduced by setting Wi = (TiN in the covariances (1) and (2). The remaining hyperparameters, a 2 , K, and v were chosen M. Opper and 0. Winther 314 Table 1: The result for the Sonar data. Covariance Function Algorithm ?test TAP Mean Field (1) 0.183 (2) 0.077 Naive Mean Field (1) 0.154 (2) 0.077 Simple Percept ron 0.269(?0.048) Back-Prop Best 21ayer - 12 Hidden 0.096(?0.018) as to minimize ?Ioo . It turned out that the lowest without noise: K, = v = O. ?Ioo ?exact 100 0.260 0.212 0.269 0.221 ?Joo 0.260 0.212 0.269 0.221 was found from modeling The simulation results are shown in table 1. The comparisons for back-propagation is taken from (Gorman & Sejnowski 1988). The solution found by the algorithm turned out to be unique, i.e. different order presentation of the examples and different initial values for the (XIL) converged to the same solution. In table 1, we have also compared the estimate given by the algorithm with the exact leave-one-out estimate ?i~~ct obtained by going through the training set and keeping an example out for testing and running the mean field algorithm on the rest. The estimate and exact value are in complete agreement. Comparing with the test error we see that the training set is 'hard' and the test set is 'easy'. The small difference for test error between the naive and full mean field algorithms also indicate that the mean field scheme is quite robust with respect to choice of AIL ' 6 Discussion More work has to be done to make the TAP approach a practical tool for Bayesian modeling. One has to find better methods for solving the equations. A conversion into a direct minimization problem for a free energy maybe helpful. To achieve this, one may probably work with the real field variables hJ.l. instead of the imaginary XIL . A further problem is the determination of the hyperparameters of the covariance functions. Two ways seem to be interesting here. One may use the approximate free energy G, which is essentially the negative logarithm of the Bayesian evidence to estimate the most probable values of the hyperparameters. However, an estimate on the errors made in the TAP approach would be necessary. Second, one may use the built-in leave-one-out estimate to estimate the generalization error. Again an estimate on the validity of the approximation is necessary. It will further be interesting to apply our way of deriving the TAP equations to other models (Boltzmann machines, belief nets, combinatorial optimization problems), for which standard mean field theories have been applied successfully. Acknowledgments This research is supported by the Swedish Foundation for Strategic Research and by the Danish Research Councils for the Natural and Technical Sciences through the Danish Computational Neural Network Center (CONNECT). Mean Field Methods for Classification with Gaussian Processes 315 References D. Barber and C. K. I. Williams, Gaussian Processes for Bayesian Classification via Hybrid Monte Carlo, in Neural Information Processing Systems 9, M . C. Mozer, M. I. Jordan and T. Petsche, eds., 340-346. MIT Press (1997). M. N. Gibbs and D. J. C . Mackay, Variational Gaussian Process Classifiers, Preprint Cambridge University (1997). R. P. Gorman and T. J. Sejnowski, Analysis of Hidden Units in a Layered Network Trained to Classify Sonar Targets, Neural Networks 1, 75 (1988) . D. J. C. Mackay, Gaussian Processes, A Replacement for Neural Networks , NIPS tutorial 1997, May be obtained from http://wol.ra.phy.cam.ac . uk/pub/mackay/. M. Mezard, The Space of interactions in Neural Networks: Gardner's Computation with the Cavity Method, J. Phys. A 22, 2181 (1989). M. Mezard and G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond, Lecture Notes in Physics, 9, World Scientific (1987). R. Neal, Bayesian Learning for Neural Networks, Lecture Notes in Statistics, Springer (1996). R. M. Neal, Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification, Technical Report CRG-TR-97-2, Dept. of Computer Science, University of Toronto (1997). M. Opper and O. Winther, A Mean Field Approach to Bayes Learning in FeedForward Neural Networks, Phys. Rev. Lett. 76, 1964 (1996). M. Opper and O. Winther, A Mean Field Algorithm for Bayes Learning in Large Feed-Forward Neural Networks, in Neural Information Processing Systems 9, M. C. Mozer, M. I. Jordan and T. Petsche, eds., 225-231. MIT Press (1997). G. Parisi and M. Potters, Mean-Field Equations for Spin Models with Orthogonal Interaction Matrices, J . Phys. A: Math. Gen . 28 5267 (1995). T. Plefka, Convergence Condition of the TAP Equation for the Infinite-Range Ising Spin Glass, J. Phys. A 15, 1971 (1982). D. J. Thouless, P. W. Anderson and R. G. Palmer , Solution of a 'Solvable Model of a Spin Glass ', Phil. Mag. 35, 593 (1977). C. K. I. Williams, Computing with Infinite Networks, in Neural Information Processing Systems 9, M. C. Mozer, M. I. Jordan and T. Petsche, eds., 295-301. MIT Press (1997). C. K. I. Williams and C. E. Rasmussen, Gaussian Processes for Regression, in Neural Information Processing Systems 8, D. S. Touretzky, M. C. Mozer and M. E. Hasselmo eds., 514-520, MIT Press (1996). K. Y. M. Wong, Microscopic Equations and Stability Conditions in Optimal Neural Networks, Europhys. Lett. 30, 245 (1995). 

1532 |@word simulation:4 covariance:4 tr:1 initial:1 phy:1 pub:1 mag:1 imaginary:3 reaction:1 comparing:1 dx:2 written:1 must:4 partition:2 guess:1 manfred:1 math:1 toronto:1 ron:2 simpler:4 along:1 direct:1 become:1 specialize:1 introduce:1 ra:1 mechanic:2 ol:1 automatically:1 becomes:2 estimating:1 lowest:1 ail:1 onsager:1 ti:2 exactly:2 classifier:2 uk:3 unit:3 positive:3 engineering:1 xv:1 limit:4 approximately:1 ap:1 palmer:2 range:1 unique:1 practical:1 acknowledgment:1 yj:1 testing:1 practice:1 pre:1 suggest:1 get:2 cannot:1 layered:2 wong:2 equivalent:1 center:1 phil:1 williams:6 go:4 starting:1 truncating:1 straightforward:2 simplicity:1 estimator:2 deriving:1 stability:1 laplace:2 target:2 exact:7 us:1 agreement:1 approximated:1 ising:2 preprint:1 solved:1 hv:5 equat:1 mozer:4 ideally:1 mine:1 cam:1 trained:1 depend:1 solving:2 predictive:1 purely:1 division:1 derivation:1 zo:1 monte:3 ole:1 sejnowski:3 lengthscales:1 europhys:1 quite:2 otherwise:1 statistic:2 transform:2 final:1 parisi:4 net:1 rock:1 interaction:8 product:1 j2:1 relevant:1 turned:2 gen:1 achieve:1 opperm:1 description:1 convergence:2 xil:2 leave:4 oo:2 derive:1 ac:2 received:1 eq:1 auxiliary:1 predicted:1 come:2 indicate:1 drawback:1 disordered:2 wol:1 ijv:1 generalization:2 probable:1 crg:1 correction:2 around:1 bj:1 major:1 niels:1 birmingham:1 applicable:1 label:4 combinatorial:1 council:1 hasselmo:1 wl:1 successfully:1 tool:2 minimization:1 mit:4 gaussian:23 hj:2 likelihood:6 contrast:2 glass:4 helpful:1 integrated:1 hidden:4 going:1 classification:10 priori:1 integration:4 mackay:6 special:1 field:35 equal:2 having:2 sampling:1 eliminated:2 flipped:1 t2:1 report:1 haven:1 thouless:2 replaced:1 argmax:1 replacement:1 interest:1 introduces:1 bracket:1 tj:4 ihj:3 bohr:1 necessary:2 sweden:1 orthogonal:1 logarithm:1 desired:1 theoretical:1 virasoro:2 classify:1 modeling:3 strategic:1 plefka:2 too:1 connect:2 corrupted:2 sv:1 winther:10 physic:5 yl:1 together:1 again:1 iip:2 external:2 derivative:2 rescaling:1 li:3 jii:2 later:1 bayes:3 hf:1 contribution:4 minimize:1 spin:7 variance:2 percept:2 ensemble:1 yield:1 dealt:1 bayesian:9 raw:1 lu:1 carlo:3 lci:1 published:1 converged:1 dhj:1 phys:4 touretzky:1 ed:4 lengthy:1 definition:1 danish:2 energy:2 dm:1 treatment:1 wh:1 knowledge:1 cj:1 back:2 feed:1 ayer:1 swedish:1 formulation:1 done:1 anderson:2 correlation:1 nonlinear:1 propagation:1 aj:2 scientific:1 validity:1 true:2 hence:2 symmetric:1 nonzero:1 neal:4 anything:1 generalized:1 complete:1 variational:3 novel:1 recently:1 functional:1 rl:1 b4:1 discussed:1 numerically:1 mft:7 cambridge:1 gibbs:3 ai:3 smoothness:1 particle:1 had:1 joo:1 tlg:2 gt:1 add:1 posterior:9 route:1 certain:1 binary:3 additional:2 somewhat:1 converge:1 ii:3 thermodynamic:2 full:1 technical:3 xf:1 determination:1 calculation:2 prediction:4 basic:1 regression:4 multilayer:1 essentially:1 expectation:4 iteration:1 ion:1 else:1 concluded:1 rest:1 unlike:1 probably:1 seem:1 jordan:3 ideal:1 feedforward:1 split:1 easy:2 xj:7 reduce:1 motivated:1 se:1 maybe:1 processed:1 simplest:1 reduced:2 http:1 tutorial:1 sign:2 group:1 nevertheless:2 changing:1 replica:1 tlh:2 ich:1 throughout:1 electronic:1 entirely:1 layer:2 bound:2 hi:2 ct:1 nontrivial:1 ri:1 sake:1 performing:1 legendre:2 wi:2 rev:1 hl:2 taken:2 equation:10 discus:1 turn:1 needed:1 available:1 apply:1 fluctuating:1 petsche:3 original:1 denotes:1 remaining:1 cf:2 running:1 somewhere:1 uxi:1 added:2 parametric:1 diagonal:1 microscopic:2 blegdamsvej:1 barber:2 fy:1 idm:3 denmark:1 potter:2 length:1 besides:1 dhp:1 cij:2 negative:2 implementation:1 boltzmann:1 conversion:1 rn:2 perturbation:1 community:1 introduced:6 copenhagen:2 tap:11 nip:1 beyond:1 dynamical:1 usually:1 lund:2 built:1 belief:1 difficulty:1 treated:1 circumvent:1 natural:1 hybrid:1 solvable:1 scheme:1 aston:2 gardner:1 axis:2 naive:8 jai:1 prior:7 nice:2 determining:2 expect:3 lecture:2 interesting:4 versus:1 lv:2 foundation:1 supported:1 rasmussen:2 keeping:1 free:2 drastically:1 formal:1 institute:1 opper:9 calculated:3 dimension:1 world:1 lett:2 forward:1 made:2 coincide:1 far:3 sj:3 approximate:3 cavity:1 rid:1 sat:1 assumed:1 xi:2 thep:1 sonar:4 table:3 promising:1 robust:1 ioo:2 expansion:1 cl:1 complex:2 factorizable:1 noise:3 hyperparameters:5 ose:1 mezard:4 xl:10 tin:1 bad:1 specific:1 normalizing:1 evidence:1 ci:4 magnitude:1 gorman:3 subtract:1 infinitely:2 failed:1 expressed:1 springer:1 corresponds:2 prop:1 conditional:1 abbreviation:1 goal:1 presentation:1 hard:2 infinite:3 called:1 perceptrons:1 preparation:1 dept:1

Modeling Surround Suppression in VI Neurons with a Statistically-Derived Normalization Model Eero P. Simoncelli Center for Neural Science, and Courant Institute of Mathematical Sciences New York University eero.simoncelli@nyu.edu Odelia Schwartz Center for Neural Science New York University odelia@cns.nyu.edu Abstract We examine the statistics of natural monochromatic images decomposed using a multi-scale wavelet basis. Although the coefficients of this representation are nearly decorrelated, they exhibit important higher-order statistical dependencies that cannot be eliminated with purely linear proc~ssing. In particular, rectified coefficients corresponding to basis functions at neighboring spatial positions, orientations and scales are highly correlated. A method of removing these dependencies is to divide each coefficient by a weighted combination of its rectified neighbors. Several successful models of the steady -state behavior of neurons in primary visual cortex are based on such "divisive normalization" computations, and thus our analysis provides a theoretical justification for these models. Perhaps more importantly, the statistical measurements explicitly specify the weights that should be used in computing the normalization signal. We demonstrate that this weighting is qualitatively consistent with recent physiological experiments that characterize the suppressive effect of stimuli presented outside of the classical receptive field. Our observations thus provide evidence for the hypothesis that early visual neural processing is well matched to these statistical properties of images. An appealing hypothesis for neural processing states that sensory systems develop in response to the statistical properties of the signals to which they are exposed [e.g., 1, 2]. This has led many researchers to look for a means of deriving a model of cortical processing purely from a statistical characterization of sensory signals. In particular, many such attempts are based on the notion that neural responses should be statistically independent. The pixels of digitized natural images are highly redundant, but one can always find a linear decomposition (i.e., principal component analysis) that eliminates second-order corResearch supported by an Alfred P. Sloan Fellowship to EPS, and by the Sloan Center for Theoretical Neurobiology at NYU. 154 E. P Simoncelli and 0. Schwartz relation. A number of researchers have used such concepts to derive linear receptive fields similar to those determined from physiological measurements [e.g., 16,20]. The principal components decomposition is, however, not unique. Because of this, these early attempts required additional constraints, such as spatial locality and/or symmetry, in order to achieve functions approximating cortical receptive fields. More recently, a number of authors have shown that one may use higher-order statistical measurements to uniquely constrain the choice of linear decomposition [e.g., 7, 9]. This is commonly known as independent components analysis. Vision researchers have demonstrated that the resulting basis functions are similar to cortical receptive fields, in that they are localized in spatial position, orientation and scale [e.g., 17, 3]. The associated coefficients of such decompositions are (second-order) decorrelated, highly kurtotic, and generally more independent than principal components. But the response properties of neurons in primary visual cortex are not adequately described by linear processes. Even if one chooses to describe only the mean firing rate of such neurons, one must at a minimum include a rectifying, saturating nonlinearity. A number of authors have shown that a gain control mechanism, known as divisive normalization, can explain a wide variety of the nonlinear behaviors of these neurons [18, 4, II, 12,6]. In most instantiations of normalization, the response of each linear basis function is rectified (and typically squared) and then divided by a uniformly weighted sum of the rectified responses of all other neurons. PhYSiologically, this is hypothesized to occur via feedback shunting inhibitory mechanisms [e.g., 13, 5]. Ruderman and Bialek [19] have discussed divisive normalization as a means of increasing entropy. In this paper, we examine the joint statistics of coefficients of an orthonormal wavelet image decomposition that approximates the independent components of natural images. We show that the coefficients are second-order decorrelated, but not independent. In particular, pairs of rectified responses are highly correlated. These pairwise dependencies may be eliminated by dividing each coefficient by a weighted combination of the rectified responses of other neurons, with the weighting determined from image statistics. We show that the resulting model, with all parameters determined from the statistics of a set of images, can account for recent physiological observations regarding suppression of cortical responses by stimuli presented outside the classical receptive field. These concepts have been previously presented in [21, 25]. 1 Joint Statistics of Orthonormal Wavelet Coefficients Multi-scale linear transforms such as wavelets have become popular for image representation. 'TYpically, the basis functions of these representations are localized in spatial position, orientation, and spatial frequency (scale). The coefficients resulting from projection of natural images onto these functions are essentially uncorrelated. In addition, a number of authors have noted that wavelet coefficients have significantly non-Gaussian marginal statistics [e.g., 10,14]. Because of these properties, we believe that wavelet bases provide a close approximation to the independent components decomposition for natural images. For the purposes of this paper, we utilize a typical separable decomposition, based on symmetric quadrature mirror filters taken from [23]. The decomposition is constructed by splitting an image into four subbands (lowpass, vertical, horizontal, diagonal), and then recursively splitting the lowpass subband. Despite the decorrelation properties of the wavelet decomposition, it is quite evident that wavelet coefficients are not statistically independent [26, 22]. Large-magnitude coefficients (either positive or negative) tend to lie along ridges with orientation matching that of the subband. Large-magnitude coefficients also tend to occur at the same relative spatiallocations in subbands at adjacent scales, and orientations. To make these statistical relationships 

1533 |@word dividing:1 effect:1 hypothesized:1 concept:2 approximating:1 classical:2 adequately:1 symmetric:1 filter:1 receptive:5 primary:2 decomposition:9 adjacent:1 diagonal:1 bialek:1 exhibit:1 uniquely:1 noted:1 steady:1 recursively:1 evident:1 ridge:1 demonstrate:1 image:11 relationship:1 recently:1 must:1 early:2 negative:1 purpose:1 discussed:1 proc:1 approximates:1 eps:1 measurement:3 vertical:1 neuron:7 surround:1 observation:2 weighted:3 provides:1 characterization:1 nonlinearity:1 always:1 gaussian:1 neurobiology:1 digitized:1 cortex:2 mathematical:1 along:1 constructed:1 base:1 become:1 derived:1 recent:2 pair:1 required:1 pairwise:1 suppression:2 behavior:2 examine:2 multi:2 minimum:1 additional:1 decomposed:1 typically:2 relation:1 increasing:1 redundant:1 signal:3 ii:1 matched:1 pixel:1 simoncelli:3 orientation:5 decorrelation:1 natural:5 spatial:5 marginal:1 field:5 divided:1 shunting:1 eliminated:2 look:1 nearly:1 vision:1 essentially:1 stimulus:2 schwartz:2 control:1 normalization:6 addition:1 positive:1 fellowship:1 relative:1 despite:1 cns:1 suppressive:1 eliminates:1 attempt:2 localized:2 firing:1 tend:2 highly:4 monochromatic:1 consistent:1 uncorrelated:1 statistically:3 variety:1 unique:1 supported:1 regarding:1 institute:1 neighbor:1 wide:1 divide:1 significantly:1 theoretical:2 projection:1 matching:1 feedback:1 cortical:4 modeling:1 sensory:2 author:3 kurtotic:1 qualitatively:1 cannot:1 onto:1 close:1 york:2 commonly:1 generally:1 demonstrated:1 center:3 successful:1 transforms:1 instantiation:1 characterize:1 dependency:3 eero:2 splitting:2 chooses:1 inhibitory:1 physiologically:1 importantly:1 deriving:1 orthonormal:2 alfred:1 notion:1 correlated:2 justification:1 symmetry:1 four:1 squared:1 hypothesis:2 utilize:1 account:1 sum:1 quadrature:1 coefficient:13 explicitly:1 sloan:2 vi:1 ssing:1 position:3 lie:1 weighting:2 wavelet:8 removing:1 rectifying:1 occur:2 exposed:1 constraint:1 purely:2 constrain:1 nyu:3 physiological:3 basis:5 evidence:1 joint:2 lowpass:2 mirror:1 separable:1 magnitude:2 rectified:6 researcher:3 describe:1 explain:1 combination:2 locality:1 entropy:1 outside:2 decorrelated:3 led:1 quite:1 visual:3 appealing:1 frequency:1 saturating:1 statistic:6 associated:1 gain:1 taken:1 popular:1 previously:1 mechanism:2 neighboring:1 higher:2 courant:1 determined:3 typical:1 uniformly:1 specify:1 achieve:1 response:8 subbands:2 principal:3 divisive:3 horizontal:1 ruderman:1 nonlinear:1 include:1 odelia:2 derive:1 develop:1 perhaps:1 believe:1 subband:2

Analog VLSI Cellular Implementation of the Boundary Contour System Gert Cauwenberghs and James Waskiewicz Department of Electrical and Computer Engineering Johns Hopkins University 3400 North Charles Street Baltimore, MD 21218-2686 E-mail: {gert, davros }@bach. ece. jhu. edu Abstract We present an analog VLSI cellular architecture implementing a simpli.fied version of the Boundary Contour System (BCS) for real-time image processing. Inspired by neuromorphic models across several layers of visual cortex, the design integrates in each pixel the functions of simple cells, complex cells, hyper-complex cells, and bipole cells, in three orientations interconnected on a hexagonal grid. Analog current-mode CMOS circuits are used throughout to perform edge detection, local inhibition, directionally selective long-range diffusive kernels, and renormalizing global gain control. Experimental results from a fabricated 12 x 10 pixel prototype in 1.2 J-tm CMOS technology demonstrate the robustness of the architecture in selecting image contours in a cluttered and noisy background. 1 Introduction The Boundary Contour System (BCS) and Feature Contour System (FCS) combine models for processes of image segmentation, feature filling, and surface reconstruction in biological vision systems [1 ],[2]. They provide a powerful technique to recognize patterns and restore image quality under excessive fixed pattern noise, such as in SAR images [3]. A related model with similar functional and structural properties is presented in [4]. The motivation for implementing a relatively complex model such as BCS and FCS on the focal-plane is dual. First, as argued in [5], complex neuromorphic active pixel designs become viable engineering solutions as the feature size of the VLSI technology shrinks significantly below the optical diffraction limit, and more transistors can be stuffed in each pixel. The pixel design that we present contains 88 transistors, likely the most complex G. Cauwenberghs and J. Waskiewicz 658 Bipole Cells (long-range orientational cooperaHon) ...... Diffusive Network ...... Local itlhibiticnt Input Image (locally normalized and contrast enhanced; diffused) BCS Focal-Plane Receptors; ...... Ri1ndom-Access Inputs FCS Figure 1: Diagram of BCSIFCS model for image segmentation, feature filling, and surface reconstruction. Three layers represent simple, complex and bipole cells. active pixel imager ever put on silicon. Second, our motivation is to extend the functionality of previous work on analog VLSI neuromorphic image processors for image boundary segmentation, e.g. [6, 7, 5, 8,9] which are based on simplified physical models that do not include directional selectivity and/or long-range signal aggregation for boundary fonnation in the presence of significant noise and clutter. The analog VLSI implementation of BCS reported here is a first step towards this goal, with the additional objectives of real-time, low-power operation as required for demanding target recognition applications. As an alternative to focal-plane optical input, the image can be loaded electronically through random-access pixel addressing. The BCS model encompasses visual processing at different levels, including several layers of cells interacting through shunting inhibition, long-range cooperative excitation, and renonnalization. The implementation architecture, shown schematically in Figure 1, partitions the BCS model into three levels: simple cells, complex and hypercomplex cells, and bipole cells. Simple cells compute unidirectional gradients of nonnalized intensity obtained from the photoreceptors. Complex (hyper-complex) cells perfonn spatial and directional competition (inhibition) for edge fonnation. Bipole cells perfonn long-range cooperation for boundary contour enhancement, and exert positive feedback (excitation) onto the hypercomplex cells. Our present implementation does not include the FCS model, which completes and fills features through diffusive spatial filtering of the image blocked by the edges fonned in BCS. 2 Modified BeS Algorithm and Implementation We adopted the BCS algorithm for analog continuous-time implementation on a hexagonal grid, extending in three directions u, v and w on the focal plane as indicated schematically in Figure 2. For notational convenience, let subscript 0 denote the center pixel and ?u, ?v and ?w its six neighbors. Components of each complex cell "vector" C i at grid location i, along three directions of edge selectivity, are indicated with superscript indices u, v and w. In the implemented circuit model, a pixel unit consists of a photosensor (or random-access analog memory) sourcing a current indicating light intensity, gradient computation and rectification circuits implementing simple cells in three directions, and one complex (hyper- 659 Analog VLSI Cellular Implementation of the Boundary Contour System Figure 2: Hexagonal arrangement of Bes pixels, at the level of simple and complex cells, extending in three directions u, v and w in the focal plane. complex) cell and one bipole cell for each of the three directions. The photosensors generate a current Ii that is proportional to intensity. Through current mirrors, the currents Ii propagate in the three directions u, v, and w as noted in Figure 2. Rectified finite-difference gradient estimates of Ii are obtained for each of the three hexagonal directions. These gradients excite the complex cells cl. Lateral inhibition among spatially (i) and directionally (j) adjacent complex cells implement the function of hypercomplex cells for edge enhancement and noise reduction. The complex output (Cl) is inhibited by local complex cell outputs in the two competing directions of j. Co is additionally inhibited by the complex cells of the four nearest neighbors in competing locations i with parallel orientation. Bf, A directionally selective interconnected diffusive network of bipole cells interacting provides long range cooperative feedback, and enhances smooth with the complex cells edge contours while reducing spurious edges due to image clutter. is excited by bipole on the line crossing i in the same direction j. interaction received from the bipole cell cl, Bf cl The operation of the (hyper-)complex cells in the hexagonal arrangement is summarized in the following equation, for one of the three directions u: where: 1. 1~(Iv + Iw) - 101 represents the rectified gradient input as approximated on the hexagonal grid; + Co) is the inhibition from locally opposing directions; o:'(C;: + c::; + c~v + C~w) is inhibition from non-aligned neighbors in the same 2. 0:(C8 3. direction; and 4. f3B8 is the excitation through long-range cooperation from the bipole cell. 660 G. Cauwenberghs and J Waskiewicz Figure 3: Network of bipole cells, implemented on a hexagonal resistive grid using orientationally tuned diffusors extending in three directions. glat! gvert determines the spatial extent of the dipole, whereas glat! gcross sets the directional selectivity. The bipole cell resistive grid (Figure 3) implements a three-fold cross-coupled, directionally polarized, long-range diffusive kernel, formulated as follows: (2) where K::, K::, and K~ represent spatial convolutional kernels implementing bipole fields symmetrically polarized in the u, v and w directions. Diffusive kernels can be efficiently implemented with a distributed representation using resistive diffusive elements [7, 10]. Three linear networks of diffusor elements are used, complemented with cross-links of adjustable strength, to control the degree of direction selectivity and the spatial spread of the kernel. Finally, the result (2) is locally normalized, before it is fed back onto the complex cells. 3 Analog VLSI Implementation The simplified circuit diagram of the BCS cell, including simple, complex and bipole cell functions on a hexagonal grid, is shown in Figure 4. The image is acquired either optically from phototransistors on the focal-plane, or in direct electronic format through random-access pixel addressing, Figure 4 (a). The simple cell portion in Figure 4 (b) combines the local intensity 10 with intensities Iv and Iw received from neighboring cells to compute the rectified gradient in (l), using distributed current mirrors and an absolute value circuit. A pMOS load converts the complex cell output into a voltage representation C8 for distribution to neighboring nodes and complementary orientations: local inhibition for spatial and directional competition in Figure 4 (c), and longrange cooperation through the bipole layer in Figure 4 (d). The linear diffusive kernel is implemented in current-mode using ladder structures of subthreshold MOS transistors [7], three families extending in each direction with cross-links for directional dispersion as indicated in Figure 3. Voltage biases control the spatial extent and directional selectivity of the interactions, as 661 Analog VLSI Cellular Implementation of the Boundary Contour System Va Vo Cov 4 Va' Cow ~ ~_WU C+~ CoU PHOTO II~YVin T (C) Vnorm Cou (a) Bou Bov Cou V+v4 B+uu BoW Vve~ ~IBoU v~ (b) (d) VbM~1Mm Vnorm Vthresh4 (e) Figure 4: Simplified circuit schematic of one BCS cell in the hexagonal array, showing only one of three directions, the other directions being symmetrical in implementation. (a) Photosensor and random-access input selection circuit. (b) Simple cell rectified gradient calculation. (c) Complex cell spatial and orientational inhibition. (d) Bipole cell directionallong range cooperation. (e) Bipole global gain and threshold control. well as the relative strength of inhibition and excitation, and the level of renormalization, for the complex and bipole cells. The values for gvert. glat and gcross controlling the bipole kernel are set externally by applying gate bias voltages Vvert. Vlat and Vcross, respectively. Likewise, the constants a, a' and /3 in (1) are set independently by the applied source voltages Va, Va' and Vt1. Global normalization and thresholding of the bipo1e response for improved stability of edge formation is achieved through an additional diffusive network that acts as a localized Gilbert-type current normalizer (only partially shown in Figure 4 (e?. 4 Experimental Results A prototype 12 x 10 pixel array has been fabricated and tested. The pixel unit, illustrated in Figure 5 (a), has been designed for testability, and has not been optimized for density. The pixel contains 88 transistors including a phototransistor, a large sample-and-hold capacitor, and three networks of interconnections in each of the three directions, requiring a fanin/fan-out of 18 node voltages across the interface of each pixel unit. A micrograph of the Tiny 2.2 x 2.2 sq. mm chip, fabricated through MOSIS in 1.2 J.Lm CMOS technology, is shown in Figure 5 (b). We have tested the BCS chip both under focal-plane optical inputs, and random-access direct electronic inputs. Input currents from optical input under ambient room lighting conditions are around 30 nA. The experimental results reported here are obtained by feeding test inputs electronically. The response of the BCS chip to two test images of interest are shown in Figures 6 and 7. G. Cauwenberghs and J Waskiewicz 662 (a) (b) Figure 5: BCS processor. (a) Pixel layout. (b) Chip micrograph. . , ~~~~~-ir\~"':";-4;? - .. - .. -,,- -\ 'I \ I II-~ -->';- :";- ':""'l){ ' -\-~ X \ \ , \ \ .. . -- -\-*~ *-\-*+ , .:.,..-';- .' \ \ - \ \ \ X\ \ / \ \ (b) . .. . " --:- ,***\ / \ \-" \ '. *- ~ , '. -~---~ -\-+ ':"""T ... \ ){ + +~~ --.:.,.. , - \ 1 \ -:- -:-- ' \ + :..;* :..; .:,, \- ..,.+ , ...\,.. ..... -*-\-+-r '\: " . -- ~~~ -\- -\-,""""':"--+-4;- .... ->.;-* :..;-.;- :..; -I- (a) - \ \ \ l \ . -'.;... - '* * \ '........ -'o- ~ -'o- -\- -\-- -\-....,.* \- \ \ \ Y \ \ \ \ j j \ . \ \ (c) Figure 6: Experimental response of the BCS chip to a curved edge. (a) Reconstructed input image. (b) Complex field. (c) Bipolefield. The thickness of the bars on the grid represent the measured components in the three directions. Figure 6 illustrates the interpolating directional response to a curved edge in the input. varying in direction between two of the principal axes (u and w in the example). Interpolation between quantized directions is important since implementing more axes on the grid incurs a quadratic cost in complexity. The second example image contains a bar with two gaps of different diameter. for the purpose of testing BCS's capacity to extend contour boundaries across clutter. The response in Figure 7 illustrates a characteristic of bipole operation. in which short-range discontinuities are bridged but large ones are preserved. 5 Conclusions An analog VLSI cellular architecture implementing the Boundary Contour System (BCS) on the focal plane has been presented. A diffusive kernel with distributed resistive networks has been used to implement long-range interactions of bipole cells without the need of excessive global interconnects across the array of pixels. The cellular model is fairly easy to implement. and succeeds in selecting boundary contours in images with significant clutter. Analog VLSI Cellular Implementation of the Boundary Contour System -T~~+-\-~ ---:--...!.r-\-~ \.\ \ ''' \ ~~ -\-~ ~ ___ -+---Io;-*"'\ \ \\\\ I \ I , \ \ I \ I X ---~---~-- \ ; \ 1" \ . X \ +-*~",-"'"-~? ----1 .. ---- \ -~ \ ,\ \ \ \ \ I \ ~ \ \ \ 1 \ ~ -T++ ~ + - +++* ~ . '\ \ \ ,v \, . . (b) ? , '. ? ----+~~--~ j x x Y x \ X/ \ ~ ~ \, \ \ \ -f--'-*-'I""-\-~+-'-*...:_ \ \ \ ~ \ 1 '. \ X \ '\ ,; '. :,' , 1. \ \ '/ \ \ \ . . ~+~...!.::-+ .. ~ -\- * - - - -I- - -\- + -\? (a) 663 \\ .:\\ \ \\\\ "\ -.\\\\\ \~~ '~\ \\\\\\\ (c) Figure 7: Experimental response of the BCS chip to a bar with two gaps of different size. (a) Reconstructed input image. (b) Complex field. (c) Bipolefield. Experimental results from a 12 x 10 pixel prototype demonstrate expected BCS operation on simple examples. While this size is small for practical applications, the analog cellular architecture is fully scalable towards higher resolutions. Based on the current design, a 10, OOO-pixel array in 0.5 J.tm CMOS technology would fit a 1 cm 2 die. Acknowledgments This research was supported by DARPA and ONR under MURI grant NOO0l4-95-1-0409. Chip fabrication was provided through the MOSIS service. References [1] S. Grossberg, "Neural Networks for Visual Perception in Variable Illumination," Optics News, pp. 5-10, August 1988. [2] S. Grossberg, "A Solution of the Figure-Ground Problem for Biological Vision," Neural Networks, vol. 6, pp. 463-482, 1993. [3] S. Grossberg, E. Mingolla, and J. Williamson, "Synthetic Aperture Radar Processing by a Multiple Scale Neural System for Boundary and Surface Representation," Neural Networks, vol. 9 (1), January 1996. [4] Z.P. Li, "A Neural Model of Contour Integration in the Primary Visual Cortex," Neural Computation, vol. 10 (4), pp. 903-940, 1998. [5] K.A. Boahen, "A Retinomorphic Vision System," IEEE Micro, vol. 16 (5), pp. 30-39, Oct. 1996. [6] lG. Harris, C. Koch, and J. Luo, "A Two-Dimensional Analog VLSI Circuit for Detecting Discontinuities in Early Vision," Science, vol. 248, pp. 1209-1211, June 1990. [7] A.G. Andreou, K.A. Boahen, P.O. Pouliquen, A. Pavasovic, R.E. Jenkins, and K. Strohbehn, "Current-Mode Subthreshold MOS Circuits for Analog VLSI Neural Systems," IEEE Transactions on Neural Networks, vol. 2 (2), pp 205-213, 1991. [8] L. Dron McIlrath, "A CCD/CMOS Focal-Plane Array Edge Detection Processor Implementing the Multiscale Veto Algorithm," IEEE 1. Solid State Circuits, vol. 31 (9), pp 1239-1248, 1996. [9] P. Venier, A. Mortara. X. Arreguit and E.A. Vittoz, "An Integrated Cortical Layer for Orientation Enhancement," IEEE 1. Solid State Circuits, vol. 32 (2), pp 177-186, Febr. 1997. [10] E. Fragniere, A. van Schaik and E. Vittoz, "Reactive Components for Pseudo-Resistive Networks," Electronic Letters, vol. 33 (23), pp 1913-1914, Nov. 1997. 

1534 |@word version:1 bf:2 propagate:1 excited:1 incurs:1 solid:2 reduction:1 contains:3 selecting:2 optically:1 tuned:1 current:11 luo:1 john:1 partition:1 designed:1 plane:9 short:1 schaik:1 provides:1 quantized:1 node:2 location:2 detecting:1 along:1 direct:2 become:1 viable:1 consists:1 resistive:5 combine:2 acquired:1 expected:1 inspired:1 provided:1 circuit:11 cm:1 fabricated:3 perfonn:2 pseudo:1 act:1 control:4 imager:1 unit:3 grant:1 positive:1 before:1 engineering:2 local:5 service:1 limit:1 io:1 receptor:1 subscript:1 interpolation:1 exert:1 co:2 range:11 grossberg:3 practical:1 acknowledgment:1 testing:1 implement:4 sq:1 jhu:1 significantly:1 bipole:21 onto:2 convenience:1 selection:1 put:1 applying:1 phototransistor:1 gilbert:1 center:1 layout:1 cluttered:1 bou:1 independently:1 resolution:1 dipole:1 array:5 fill:1 stability:1 gert:2 sar:1 enhanced:1 target:1 controlling:1 crossing:1 element:2 recognition:1 approximated:1 muri:1 cooperative:2 electrical:1 news:1 boahen:2 complexity:1 testability:1 radar:1 darpa:1 chip:7 vt1:1 hyper:4 formation:1 pmos:1 interconnection:1 cov:1 noisy:1 superscript:1 directionally:4 transistor:4 reconstruction:2 interconnected:2 interaction:3 neighboring:2 aligned:1 bow:1 competition:2 enhancement:3 extending:4 renormalizing:1 cmos:5 measured:1 nearest:1 received:2 implemented:4 uu:1 vittoz:2 direction:22 functionality:1 implementing:7 argued:1 venier:1 feeding:1 biological:2 mm:2 hold:1 around:1 koch:1 ground:1 fragniere:1 mo:2 lm:1 early:1 purpose:1 integrates:1 iw:2 modified:1 varying:1 voltage:5 ax:2 june:1 notational:1 contrast:1 normalizer:1 integrated:1 spurious:1 vlsi:12 selective:2 pixel:19 dual:1 orientation:4 among:1 retinomorphic:1 spatial:8 integration:1 fairly:1 field:3 represents:1 filling:2 excessive:2 inhibited:2 micro:1 recognize:1 opposing:1 detection:2 interest:1 light:1 ambient:1 edge:11 mortara:1 iv:2 neuromorphic:3 cost:1 addressing:2 fabrication:1 reported:2 thickness:1 synthetic:1 density:1 v4:1 hopkins:1 na:1 li:1 summarized:1 north:1 cauwenberghs:4 portion:1 aggregation:1 parallel:1 unidirectional:1 ir:1 convolutional:1 loaded:1 characteristic:1 efficiently:1 likewise:1 subthreshold:2 directional:7 lighting:1 rectified:4 processor:3 pp:9 james:1 gain:2 bridged:1 segmentation:3 orientational:2 back:1 higher:1 response:6 improved:1 ooo:1 photosensors:1 shrink:1 multiscale:1 mode:3 quality:1 indicated:3 normalized:2 requiring:1 spatially:1 illustrated:1 adjacent:1 noted:1 excitation:4 die:1 demonstrate:2 vo:1 interface:1 image:18 charles:1 functional:1 physical:1 analog:15 extend:2 silicon:1 significant:2 blocked:1 grid:9 focal:9 arreguit:1 access:6 cortex:2 surface:3 inhibition:9 selectivity:5 onr:1 additional:2 simpli:1 signal:1 ii:5 multiple:1 bcs:19 fonnation:2 smooth:1 calculation:1 bach:1 long:9 cross:3 shunting:1 va:4 schematic:1 scalable:1 vision:4 pavasovic:1 kernel:8 represent:3 normalization:1 achieved:1 cell:43 diffusive:10 preserved:1 background:1 whereas:1 schematically:2 baltimore:1 diagram:2 completes:1 source:1 veto:1 photosensor:2 capacitor:1 structural:1 presence:1 symmetrically:1 easy:1 fit:1 architecture:5 competing:2 cow:1 prototype:3 tm:2 six:1 mingolla:1 clutter:4 locally:3 diameter:1 generate:1 vol:9 four:1 threshold:1 micrograph:2 stuffed:1 mosis:2 convert:1 letter:1 powerful:1 throughout:1 family:1 electronic:3 wu:1 polarized:2 diffraction:1 layer:5 fold:1 fan:1 quadratic:1 strength:2 optic:1 c8:2 hexagonal:9 optical:4 relatively:1 format:1 department:1 across:4 rectification:1 equation:1 pouliquen:1 fed:1 photo:1 adopted:1 operation:4 jenkins:1 alternative:1 robustness:1 gate:1 include:2 ccd:1 diffused:1 objective:1 arrangement:2 primary:1 md:1 enhances:1 gradient:7 link:2 lateral:1 capacity:1 street:1 mail:1 extent:2 cellular:8 index:1 lg:1 implementation:11 design:4 adjustable:1 perform:1 dispersion:1 finite:1 curved:2 january:1 ever:1 nonnalized:1 diffusor:2 interacting:2 august:1 vbm:1 intensity:5 required:1 optimized:1 andreou:1 discontinuity:2 renonnalization:1 bar:3 below:1 pattern:2 perception:1 encompasses:1 bov:1 including:3 memory:1 power:1 demanding:1 restore:1 interconnects:1 technology:4 ladder:1 coupled:1 relative:1 fully:1 filtering:1 proportional:1 localized:1 degree:1 thresholding:1 tiny:1 cooperation:4 sourcing:1 supported:1 electronically:2 bias:2 neighbor:3 absolute:1 distributed:3 van:1 boundary:13 feedback:2 cortical:1 contour:14 fonned:1 simplified:3 hypercomplex:3 transaction:1 reconstructed:2 nov:1 longrange:1 aperture:1 global:4 active:2 photoreceptors:1 symmetrical:1 excite:1 continuous:1 additionally:1 williamson:1 complex:27 cl:4 interpolating:1 spread:1 motivation:2 noise:3 complementary:1 fied:1 vve:1 renormalization:1 externally:1 load:1 showing:1 phototransistors:1 mirror:2 illumination:1 illustrates:2 gap:2 fanin:1 fcs:4 strohbehn:1 likely:1 visual:4 partially:1 determines:1 complemented:1 harris:1 oct:1 goal:1 formulated:1 towards:2 room:1 reducing:1 principal:1 ece:1 experimental:6 succeeds:1 indicating:1 reactive:1 tested:2

DTs: Dynamic Trees Christopher K. I. Williams Nicholas J. Adams Institute for Adaptive and Neural Computation Division of Informatics, 5 Forrest Hill Edinburgh, EHI 2QL, UK. http://www.anc.ed.ac . uk/ ckiw~dai.ed.ac.uk nicka~dai.ed.ac.uk Abstract In this paper we introduce a new class of image models, which we call dynamic trees or DTs. A dynamic tree model specifies a prior over a large number of trees, each one of which is a tree-structured belief net (TSBN). Experiments show that DTs are capable of generating images that are less blocky, and the models have better translation invariance properties than a fixed, "balanced" TSBN. We also show that Simulated Annealing is effective at finding trees which have high posterior probability. 1 Introduction In this paper we introduce a new class of image models, which we call dynamic trees or DTs. A dynamic tree model specifies a prior over a large number of trees, each one of which is a tree-structured belief net (TSBN) . Our aim is to retain the advantages of tree-structured belief networks, namely the hierarchical structure of the model and (in part) the efficient inference algorithms, while avoiding the "blocky" artifacts that derive from a single, fixed TSBN structure. One use for DTs is as prior models over labellings for image segmentation problems. Section 2 of the paper gives the theory of DTs, and experiments are described in section 3. 2 Theory There are two essential components that make up a dynamic tree network (i) the tree architecture and (ii) the nodes and conditional probability tables (CPTs) in the given tree. We consider the architecture question first. DTs: Dynamic Trees 635 o o o o o o 0 0 o 0 0 o 000 10000000000000000 (a) (c) (d) Figure 1: (a) "Naked" nodes, (b) the "balanced" tree architecture, (c) a sample from the prior over Z, (d) data generated from the tree in (c). Consider a number of nodes arranged into layers, as in Figure lea). We wish to construct a tree structure so that any child node in a particular layer will be connected to a parent in the layer above. We also allow there to be a null parent for each layer, so that any child connected to it will become a new root. (Technically we are constructing a forest rather than a tree.) An example of a structure generated using this method is shown in Figure 1 (c). There are a number of ways of specifying a prior over trees. If we denote by Zi the indicator vector which shows to which parent node i belongs, then the tree structure is specified by a matrix Z whose columns are the individual Zi vectors (one for each node). The scheme that we have investigated so far is to set P(Z) = It P(Zi). In our work we have specified P(Zi) as follows. Each child node is considered to have a "natural" parent-its parent in the balanced structure shown in Figure l(b). Each node in the parent layer is assigned an "affinity" for each child node, and the "natural" parent has the highest affinity. Denote the affinity of node k in the parent layer by ak. Then we choose P(Zi = ek) = e!3a/e / EjEPai e!3 a j , where (3 is some positive constant and ek is the unit vector with a 1 in position k. Note that the "null" parent is included in the sum, and has affinity anull associated with it, which affects the relative probability of "orphans". We have named this prior the "full-time-node-employment" prior as all the nodes participate in the creation of the tree structure to some degree. Having specified the prior over architectures, we now need to translate this into a TSBN. The units in the tree are taken to be C-class multinomial random variables. Each layer of the structure has associated with it a prior probability vector 7f1 and CPT MI. Given a particular Z matrix which specifies a forest structure, the probability of a particular instantiation of all of the random variables is simply the product of the probabilities of all of the trees, where the appropriate root probabilities and CPTs are picked up from the 7fIS and MIS. A sample generated from the tree structure in Figure l(c) is shown in Figure led). C. K. I. Williams and N. 1. Adams 636 Our intuition as to why DTs may be useful image models is based on the idea that most pixels in an image are derived from a single object. We think of an object as being described by a root of a tree, with the scale of the object being determined by the level in the tree at which the root occurs. In this interpretation the ePTs will have most of their probability mass on the diagonal. Given some data at the bottom layer of units, we can form a posterior over the tree structures and node instantiations of the layers above. This is rather like obtaining a set of parses for a number of sentences using a context-free grammar l . In the DT model as described above different examples are explained by different trees. This is an important difference with the usual priors over belief networks as used, e.g. in Bayesian averaging over model structures. Also, in the usual case of model averaging, there is normally no restriction to TSBN structures, or to tying the parameters (1rIS and MIS) between different structures. 2.1 Inference in DTs We now consider the problem of inference in DTs, i.e. obtaining the posterior P(Z, XhlXv) where Z denotes the tree-structure, Xv the visible units (the image clamped on the lowest level) and X h the hidden units. In fact, we shall concentrate on obtaining the posterior marginal P(ZIXv), as we can obtain samples from P(XhIXv, Z) using standard techniques for TSBNs. There are a very large number of possible structures; in fact for a set of nodes created from a balanced tree with branching factor b and depth D (with the top level indexed by 1) there are IT~=2(b(d-2) + l)b(d-l) possible forest structures. Our objective will be to obtain the maximum a posteriori (MAP) state from the posterior P(ZIXv) ex P(Z)P(XvIZ) using Simulated Annealing.2 This is possible because two components P(Z) and P(XvIZ) are readily evaluated. P(XvIZ) can be computed from ITr (Exr A(X r )'7r(xr)), where A(Xr) and 7r(xr) are the Pearl-style vectors of each root r of the forest. An alternative to sampling from the posterior P(Z, XhlXv ) is to use approximate inference. One possibility is to use a mean-field-type approximation to the posterior of the form QZ(Z)Qh(Xh) (Zoubin Ghahramani, personal communication, 1998). 2.2 Comparing DTs to other image models Fixed-structure TSBNs have been used by a number of authors as models of images (Bouman and Shapiro, 1994), (Luettgen and Willsky, 1995). They have an attractive multi-scale structure, but suffer from problems due to the fixed tree structure, which can lead to very "blocky" segmentations. Markov Random Field (MRF) models are also popular image models; however, one of their main limitations is that inference in a MRF is NP-hard. Also, they lack an hierarchical structure. On the other hand, stationarity of the process they define can be easily ensured, which lCFGs have a O(n 3 ) algorithm to infer the MAP parse; however, this algorithm depends crucially on the one-dimensional ordering of the inputs. We believe that the possibility of crossed links in the DT architecture means that this kind of algorithm is not applicable to the DT case. Also, the DT model can be applied to 2-d images, where the O(n 3 ) algorithm is not applicable. 2It is also possible to sample from the posterior using, e.g. Gibbs Sampling. 637 DTs: Dynamic Trees is not the case for fixed-structure TSBNs. One strategy to overcome the fixed structure of TSBNs is to break away from the tree structure, and use belief networks with cross connections e.g. (Dayan et ai., 1995). However, this means losing the linear-time belief-propagation algorithms that can be used in trees (Pearl, 1988) and using approximate algorithms. While it is true that inference over DTs is also NP-hard, we do retain a"clean" semantics based on the fact that we expect that each pixel should belong to one object, which may lead to useful approximation schemes. 3 Experiments In this section we describe two experiments conducted on the DT models. The first has been designed to compare the translation performance of DTs with that of the balanced TSBN structure and is described in section 3.1. In section 3.2 we generate 2-d images from the DT model, find the MAP Dynamic Tree for these images, and contrast their performance in relative to the balanced TSBN. 3.1 Comparing DTs with the balanced TSBN We consider a 5-1ayer binary tree with 16 leaf nodes, as shown in Figure 1. Each node in the tree is a binary variable, taking on values of white/black. The 7r1'S, M,'s and affinities were set to be equal in each layer. The values used were 7r = (0.75,0.25) with 0.75 referring to white, and M had values 0.99 on the diagonal and 0.01 offdiagonal. The affinities 3 were set as 1 for the natural parent, 0 for the nearest neighbour(s) of the natural parent, -00 for non-nearest neighbours and anull = 0, with f3 = 1.25. " !\I~ ~i . .. , ., .., /" I r \ 1\ " \,' , \ ... - -~,~~~~.~.~,~.~~~~. (a) 5 black nodes I \ r \1 , I I' ~,/ " \ I r \ \ . ?? ".. ?" " '. " \,,: - . .... . : ~ \' '/ \ I I I \~ (b) 4 black nodes Figure 2: Plots of the unnormalised log posterior vs position of the input pattern for (a) the 5-black-nodes pattern and (b) 4-black-nodes pattern. To illustrate the effects of translation, we have taken a stimulus made up of a bar of five black pixels, and moved it across the image. The unnormalised log posterior for a particular Z configuration is logP(Z) + logP(XvIZ). This is computed for the balanced TSBN architecture, and compared to the highest value that can be found by conducting a search over Z. These results are plotted in Figure 2(a). The x-axis denotes the position of the left hand end of the bar (running from 1 to 3The affinities are defined up to the addition of an arbitrary constant. 638 C. K. I. Williams and N. 1. Adams 12), and the y-axis shows the posterior probability. Note that due to symmetries there are in reality fewer than 12 distinct configurations. Figure 2(a) shows clearly that the balanced TSBN is a poor model for this stimulus, and that much better interpretations can be found using DTs, even though the "natural parent" idea ensures that the logP(Z) is always larger for the balanced tree. Notice also how the balanced TSBN displays greater sensitivity of the log posterior with respect to position than the DT model. Figure 2 shows both the "optimal" log posterior (found "by hand", using intuitions as to the best trees), and the those of the MAP models discovered by Simulated Annealing. Annealing was conducted from a starting temperature of 1.0 and exponentially decreased by a factor of 0.9. At each temperature up to 2000 proposals could be made, although transition to the next temperature would occur after 200 accepted steps. The run was deemed to have converged after five successive temperature steps were made without accepting a single step. We also show the log posterior of trees found by Gibbs sampling from which we report the best configuration found from four separate runs (with different random starting positions), each of which was run for 25,000 sweeps through all of the nodes . In Figure 2(b) we have shown the log posterior for a stimulus made up of four black nodes 4 . In this case the balanced TSBN is even more sensitive to the stimulus location, as the four black nodes fit exactly under one sub-tree when they are in positions 1, 5, 9 or 13. By contrast, the dynamic tree is less sensitive to the alignment, although it does retain a preference for the configuration most favoured by the balanced TSBN. This is due to the concept of a "natural" parent built into the (current) architecture (but see Section 4 for further discussion) . Clearly these results are somewhat sensitive to settings of the parameters. One of the most important parameters is the diagonal entry in the CPT. This controls the relative desirability of having a disconnection against a transition in the tree that involves a colour change. For example, if the diagonal entry in the CPT is reduced to 0.95, the gap between the optimal and balanced trees in Figure 2(b) is decreased. We have experimented with CPT entries of 0.90,0.95 and 0.99, but otherwise have . not needed to explore the parameter space to obtain the results shown. 3.2 Generating from the prior and finding the MAP Tree in 2-d We now turn our attention to 2-d images. Considering a 5 layer quad-tree node arrangement gives a total of 256 leaf nodes or a 16x16 pixel image. A structural plot of such a tree generated from the prior is shown in figure 3. Each sub-plot is a slice through the tree showing the nodes on successive levels. The boxes represent a single node on the current level and their shading indicates the tree to which they belong. Nodes in the parent layer above are superimposed as circles and the lines emanating from them shows their connectivity. Black circles with a smaller white circle inside are used to indicate root nodes. Thus in the example above we see that the forest consists of five trees, four of whose roots lie at level 3 (which between them account for most of the black in the image, Figure 3(f?, while the root node at level 1 is responsible for the background. 4The parameters are the same as above, except that encourage disconnections at this level. anull in level 3 was set to 10.0 to 639 DTs: Dynamic Trees (a) (b) (c) (e) (d) (f) Figure 3: Plot of the MAP Dynamic Tree of the accompanying image (f). Broadly speaking the parameters for the 2-d DTs were set to be similar to the I-d trees of the previous section, except that the disconnection affinities were set to favour disconnections higher up the tree, and to values for the leaf level such that leaf disconnection probabilities tend to zero. In practice this resulted in all leaves being connected to parent nodes (which is desirable as we believe that single-pixel objects are unlikely). The (3 values increase with tree depth so that lower levels nodes choose parents from a tighter neighbourhood. The 7ft and M t values were unchanged, and again we consider binary valued nodes. A suite of 600 images were created by sampling DTs from the above prior and then generating 5 images from each. Figure 3(f) shows an example of an image generated by the DT and it can be seen that the "blockiness" exhibited by balanced TSBNs is not present . . ':. . (a) (b) Figure 4: (a) Comparison of the MAP DT log posterior against that of the quad-tree for 600 images, (b) tree generated from the "part-time-node-employment" prior. 640 C. K. I. Williams and N. J Adams The MAP Dynamic Tree for each of these images was found by Simulated Annealing using the same exponential strategy described earlier, and their log posteriors are compared with those of the balanced TSBN in the plot 4(a). The line denotes the boundary of equal log posterior and the location of all the points above this clearly shows that in every case the MAP tree found has a higher posterior. 4 Discussion Above we have demonstrated that DT models have greater translation invariance and do not exhibit the blockiness of the balanced TSBN model. We also see that Simulated Annealing methods are successful at finding trees that have high posterior probability. We now discuss some extensions to the model. In the work above we have kept the balanced tree arrangement of nodes. However, this could be relaxed, giving rise to roughly equal numbers of nodes at the various levels (cf stationary wavelets). This would be useful (a) for providing better translation invariance and (b) to avoid slight shortages of hidden units that can occur when patterns that are "misaligned" wrt the balanced tree are presented. In this case the prior over Z would need to be adjusted to ensure a high proportion of tree-like structures, by generating the z's and x's in layers, so that the z's can be contingent on the states of the units in the layer above. We have devised a prior of this nature and called it the "part-timeemployment" prior as the nodes can decide whether or not they wish to be employed in the tree structure or remain redundant and inactive. An example tree generated from this prior is shown in figure 4(b); we plan to explore this direction further in on-going research. Other research directions include the learning of parameters in the networks (e.g. using EM), and the introduction of additional information at the nodes; for example one might use real-valued variables in addition to the multinomial variables considered above. These additional variables might be used to encode information such as that concerning the instantiation parameters of objects. Acknowledgements This work stems from a conversation between CW and Zoubin Gharahmani at the Isaac Newton Institute in October 1997. We thank Zoubin Ghahramani, Geoff Hinton and Peter Dayan for helpful conversations, and the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) for hospitality during the "Neural Networks and Machine Learning" programme. NJA is supported by an EPSRC research studentship, and the work of CW is partially supported by EPSRC grant GR/L03088, Combining Spatially Distributed Predictions From Neural Networks. References Bouman, C. A. and M. Shapiro (1994). A Multiscale Random Field Model for Bayesian Image Segmentation. IEEE Transactions on Image Processing 3(2),162-177. Dayan, P., G. E. Hinton, R. M. Neal, and R. S. Zemel (1995). The Helmholtz Machine. Neural Computation 7(5)r 889-904. Luettgen, M. R. and A. S. Willsky (1995). Likelihood Calculation for a Class of Multiscale Stocahstic Models, with Application to Texture Discrimination. IEEE Trans. Image Processing 4(2?, 194-207. Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo, CA: Morgan Kaufmann. 

1535 |@word proportion:1 crucially:1 shading:1 configuration:4 current:2 comparing:2 readily:1 visible:1 designed:1 plot:5 v:1 stationary:1 discrimination:1 leaf:5 fewer:1 accepting:1 node:38 location:2 successive:2 preference:1 five:3 mathematical:1 become:1 consists:1 inside:1 introduce:2 roughly:1 multi:1 tsbn:16 quad:2 considering:1 mass:1 null:2 lowest:1 tying:1 kind:1 finding:3 suite:1 every:1 exactly:1 ensured:1 uk:5 control:1 unit:7 normally:1 grant:1 positive:1 xv:1 ak:1 black:10 might:2 mateo:1 specifying:1 misaligned:1 responsible:1 practice:1 xr:3 zoubin:3 context:1 www:1 restriction:1 map:9 demonstrated:1 williams:4 attention:1 starting:2 qh:1 exr:1 losing:1 helmholtz:1 bottom:1 ft:1 epsrc:2 ensures:1 connected:3 ordering:1 highest:2 balanced:19 intuition:2 dynamic:13 personal:1 employment:2 creation:1 technically:1 division:1 easily:1 geoff:1 various:1 distinct:1 effective:1 describe:1 emanating:1 zemel:1 whose:2 larger:1 valued:2 plausible:1 otherwise:1 grammar:1 think:1 advantage:1 net:2 product:1 combining:1 translate:1 moved:1 parent:16 r1:1 generating:4 adam:4 object:6 derive:1 illustrate:1 ac:3 nearest:2 involves:1 indicate:1 concentrate:1 direction:2 f1:1 tighter:1 adjusted:1 extension:1 accompanying:1 considered:2 applicable:2 sensitive:3 clearly:3 hospitality:1 always:1 aim:1 desirability:1 rather:2 avoid:1 tsbns:5 derived:1 encode:1 indicates:1 superimposed:1 likelihood:1 contrast:2 posteriori:1 inference:7 helpful:1 dayan:3 unlikely:1 fis:1 hidden:2 going:1 semantics:1 pixel:5 plan:1 marginal:1 field:3 construct:1 equal:3 having:2 f3:1 sampling:4 np:2 stimulus:4 report:1 intelligent:1 neighbour:2 resulted:1 individual:1 stationarity:1 possibility:2 alignment:1 blocky:3 capable:1 encourage:1 tree:66 indexed:1 circle:3 plotted:1 bouman:2 column:1 earlier:1 logp:3 entry:3 successful:1 conducted:2 gr:1 referring:1 ehi:1 sensitivity:1 retain:3 probabilistic:1 informatics:1 connectivity:1 again:1 luettgen:2 choose:2 ek:2 style:1 account:1 depends:1 crossed:1 cpts:2 root:8 picked:1 break:1 offdiagonal:1 kaufmann:1 conducting:1 bayesian:2 converged:1 ed:3 against:2 blockiness:2 isaac:2 associated:2 mi:3 popular:1 conversation:2 segmentation:3 higher:2 dt:10 ayer:1 arranged:1 evaluated:1 though:1 box:1 hand:3 parse:1 christopher:1 multiscale:2 lack:1 propagation:1 artifact:1 believe:2 effect:1 concept:1 true:1 assigned:1 spatially:1 neal:1 white:3 attractive:1 during:1 branching:1 hill:1 temperature:4 reasoning:1 image:26 multinomial:2 exponentially:1 belong:2 interpretation:2 slight:1 cambridge:1 gibbs:2 ai:1 had:1 posterior:20 belongs:1 binary:3 seen:1 morgan:1 dai:2 greater:2 somewhat:1 relaxed:1 employed:1 additional:2 contingent:1 redundant:1 ii:1 full:1 desirable:1 infer:1 stem:1 calculation:1 cross:1 devised:1 concerning:1 prediction:1 mrf:2 represent:1 lea:1 proposal:1 addition:2 background:1 annealing:6 decreased:2 exhibited:1 tend:1 call:2 structural:1 affect:1 fit:1 zi:5 architecture:7 idea:2 itr:1 favour:1 whether:1 inactive:1 colour:1 suffer:1 peter:1 speaking:1 cpt:4 useful:3 shortage:1 reduced:1 http:1 specifies:3 shapiro:2 generate:1 notice:1 broadly:1 shall:1 four:4 clean:1 kept:1 sum:1 run:3 named:1 decide:1 forrest:1 layer:14 display:1 occur:2 ri:1 structured:3 poor:1 remain:1 across:1 smaller:1 em:1 labellings:1 explained:1 taken:2 turn:1 discus:1 needed:1 wrt:1 end:1 ckiw:1 hierarchical:2 away:1 appropriate:1 nicholas:1 neighbourhood:1 alternative:1 denotes:3 top:1 running:1 cf:1 ensure:1 include:1 newton:2 giving:1 ghahramani:2 unchanged:1 sweep:1 objective:1 question:1 arrangement:2 occurs:1 strategy:2 usual:2 diagonal:4 exhibit:1 affinity:8 cw:2 link:1 separate:1 simulated:5 thank:1 participate:1 willsky:2 providing:1 ql:1 october:1 rise:1 markov:1 hinton:2 communication:1 discovered:1 arbitrary:1 namely:1 specified:3 sentence:1 connection:1 dts:19 pearl:3 trans:1 bar:2 pattern:4 built:1 belief:6 natural:6 indicator:1 scheme:2 axis:2 created:2 deemed:1 unnormalised:2 prior:18 acknowledgement:1 relative:3 par:1 expect:1 limitation:1 degree:1 translation:5 naked:1 supported:2 free:1 allow:1 disconnection:5 institute:3 taking:1 edinburgh:1 slice:1 overcome:1 depth:2 boundary:1 transition:2 studentship:1 distributed:1 author:1 made:4 adaptive:1 san:1 programme:1 far:1 transaction:1 approximate:2 instantiation:3 search:1 why:1 table:1 reality:1 qz:1 nature:1 nja:1 ca:1 obtaining:3 symmetry:1 forest:5 orphan:1 anc:1 investigated:1 constructing:1 main:1 child:4 x16:1 favoured:1 sub:2 position:6 wish:2 xh:1 exponential:1 lie:1 clamped:1 wavelet:1 showing:1 experimented:1 essential:1 texture:1 gap:1 led:1 simply:1 explore:2 partially:1 conditional:1 hard:2 change:1 included:1 determined:1 except:2 averaging:2 total:1 called:1 invariance:3 accepted:1 avoiding:1 ex:1

Divisive Normalization, Line Attractor Networks and Ideal Observers Sophie Deneve l Alexandre Pougetl, and P.E. Latham 2 Institute for Computational and Cognitive Sciences, Georgetown University, Washington, DC 20007-2197 2Dpt of Neurobiology, UCLA, Los Angeles, CA 90095-1763, U.S.A. 1 Georgetown Abstract Gain control by divisive inhibition, a.k.a. divisive normalization, has been proposed to be a general mechanism throughout the visual cortex. We explore in this study the statistical properties of this normalization in the presence of noise. Using simulations, we show that divisive normalization is a close approximation to a maximum likelihood estimator, which, in the context of population coding, is the same as an ideal observer. We also demonstrate analytically that this is a general property of a large class of nonlinear recurrent networks with line attractors. Our work suggests that divisive normalization plays a critical role in noise filtering, and that every cortical layer may be an ideal observer of the activity in the preceding layer. Information processing in the cortex is often formalized as a sequence of a linear stages followed by a nonlinearity. In the visual cortex, the nonlinearity is best described by squaring combined with a divisive pooling of local activities. The divisive part of the nonlinearity has been extensively studied by Heeger and colleagues [1], and several authors have explored the role of this normalization in the computation of high order visual features such as orientation of edges or first and second order motion[ 4]. We show in this paper that divisive normalization can also playa role in noise filtering. More specifically, we demonstrate through simulations that networks implementing this normalization come close to performing maximum likelihood estimation. We then demonstrate analytically that the ability to perform maximum likelihood estimation, and thus efficiently extract information from a population of noisy neurons, is a property exhibited by a large class of networks. Maximum likelihood estimation is a framework commonly used in the theory of ideal observers. A recent example comes from the work of Itti et al., 1998, who have shown that it is possible to account for the behavior of human subjects in simple discrimination tasks. Their model comprised two distinct stages: 1) a network 105 Divisive Normalization. Line Attractor Networks and Ideal Observers which models the noisy response of neurons with tuning curves to orientation and spatial frequency combined with divisive normalization, and 2) an ideal observer (a maximum likelihood estimator) to read out the population activity of the network. Our work suggests that there is no need to distinguish between these two stages, since, as we will show, divisive normalization comes close to providing a maximum likelihood estimation. More generally, we propose that there may not be any part of the cortex that acts as an ideal observer for patterns of activity in sensory areas but, instead , that each cortical layer acts as an ideal observer of the activity in the preceding layer. 1 The network Our network is a simplified model of a cortical hypercolumn for spatial frequency and orientation. It consists of a two dimensional array of units in which each unit is indexed by its preferred orientation, 8i , and spatial frequency, >'j. 1.1 LGN model Units in the cortical layer are assumed to receive direct inputs from the lateral geniculate nucleus (LG N). Here we do not model explicitly the LG N, but focus instead on the pooled LGN input onto each cortical unit . The input to each unit is denoted aij' We distinguish between the mean pooled LGN input, fij(8, >'), as a function of orientation, 8, and spatial frequency, >., and the noise distribution around this mean, P(aijI8, >.). In response to a stimulus of orientation, 8, spatial frequency, >., and contrast, G, the mean LGN input onto unit ij is a circular Gaussian with a small amount of spontaneous activity, 1/: J'J,,(8 ,/\') - KG - exp (COS(>. - >'j) 2 ~A 1+ cos(8 - 8i ) 2 ~o - 1) + 1/, (1) where K is a constant. Note that spatial frequency is treated as a periodic variable; this was done for convenience only and should have negligible effects on our results as long as we keep>. far from 27m, n an integer. On any given trial the LGN input to cortical unit ij, aij, is sampled from a Gaussian noise distribution with variance ~;j: (2) In our simulations, the variance of the noise was either kept fixed (~'fj = ~2) or set to the mean activity (~t = Jij(8, >.)). The latter is more consistent with the noise that has been measured experimentally in the cortex. We show in figure I-A an example of a noisy LGN pattern of activity. 1.2 Cortical Model: Divisive Normalization Activities in the cortical layer are updated over time according to: S. Deneve, A. Pouget and P. E. Latham 106 A. CORTEX :::::::: r:::L:: t ::;-:::;::::: - r- , _ :- :::~ '~ DO 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.1 D.' 1 Contrast Figure 1: A- LGN input (bottom) and stable hill in the cortical network after relaxation (top). The position of the stable hill can be used to estimate orientation (0) and spatial frequency (5.). B- Inverse of the variance of the network estimate for orientation using Gaussian noise with variance equal to the mean as a function of contrast and number of iterations (0, dashed; 1, diamond; 2, circle; and 3, square). The continuous curve corresponds to the theoretical upper bound on the inverse of the variance (i.e. an ideal observer). C- Gain curve for contrast for the cortical units after 1, 2 and 3 iterations. (3) where {Wij,kt} are the filtering weights, Oij(t) is the activity of unit ij at time t, S is a constant, and J1. is what we call the divisive inhibition weight. The filtering weights implement a two dimensional Gaussian filter: Wij,kl = Wi-k,j - l = Kwexp (COS[27!'(i -2 k )/P] ~w~ where Kw is a constant, there are p 2 units. ~w~ and ~WA -1 + cos[27!'(j ~ l)/Pl-1) (4) ~WA control the width of the filtering weights, and On each iteration the activity is filtered by the weights, squared, and then normalized by the total local activity. Divisive normalization per se only involves the squaring and division by local activity. We have added the filtering weights to obtain a local pooling of activity between cells with similar preferred orientations and spatial frequencies. This pooling can easily be implemented with cortical lateral connections and it is reasonable to think that such a pooling takes place in the cortex. Divisive Normalization, Line Attractor Networks and Ideal Observers 2 107 Simulation Results Our simulations consist of iterating equation 3 with initial conditions determined by the presentation orientation and spatial frequency. The initial conditions are chosen as follows: For a given presentation angle, (}o, and spatial frequency, Ao, determine the mean cortical activity, /ij((}o, AO), via equation 1. Then generate the actual cortical activity, {aij}, by sampling from the distribution given in equation 2. This serves as our set of initial conditions: Oij (t = 0) = aij' Iterating equation 3 with the above initial conditions, we found that for very low contrast the activity of all cortical units decayed to zero. Above some contrast threshold, however, the activities converged to a smooth stable hill (see figure I-A for an example with parameters (Jw(} = (Jw).. = (J(} = (J).. = I/V8, K = 74, C = 1, J.L = 0.01). The width of the hill is controlled by the width of the filtering weights. Its peak, on the other hand, depends on the orientation and spatial frequency of the LGN input, (}o and Ao. The peak can thus be used to estimate these quantities (see figure I-A). To compute the position of the final hill, we used a population vector estimator [3] although any unbiased estimator would work as well. In all cases we looked at, the network produced an unbiased estimate of (}o and Ao. In our simulations we adjusted (Jw(} and (Jw).. so that the stable hill had the same profile as the mean LGN input (equation 1). As a result, the tuning curves of the cortical units match the tuning curves specified by the pooled LGN input. For this case, we found that the estimate obtained from the network has a variance close to the theoretical minimum, known as the Cramer-Rao bound [3]. For Gaussian noise of fixed variance, the variance of the estimate was 16.6% above this bound, compared to 3833% for the population vector applied directly to the LGN input. In a ID network (orientation alone), these numbers go to 12.9% for the network versus 613% for population vector. For Gaussian noise with variance proportional to the mean, the network was 8.8% above the bound, compared to 722% for the population vector applied directly to the input. These numbers are respectively 9% and 108% for the I-D network. The network is therefore a close approximation to a maximum likelihood estimator, i.e., it is close to being an ideal observer of the LGN activity with respect to orientation and spatial frequency. As long as the contrast, C, was superthreshold, large variations in contrast did not affect our results (figure I-B). However, the tuning of the network units to contrast after reaching the stable state was found to follow a step function whereas, for real neurons, the curves are better described by a sigmoid [2]. Improved agreement with experiment was achieved by taking only 2-3 iterations, at which point the performance of the network is close to optimal (figure I-B) and the tuning curves to contrast are more realistic and closer to sigmoids (figure I-C). Therefore, reaching a stable state is not required for optimal performance, and in fact leads to contrast tuning curves that are inconsistent with experiment. 3 Mathematical Analysis We first prove that line attractor networks with sufficiently small noise are close approximations to a maximum likelihood estimator. We then show how this result applies to our simulations with divisive normalization. S. Deneve, A. Pouget and P. E. Latham J08 3.1 General Case: Line Attractor Networks Let On be the activity vector (denoted by bold type) at discrete time, n, for a set of P interconnected units. We consider a one dimensional network, i.e., only one feature is encoded; generalization to multidimensional networks is straightforward. A generic mapping for this network may be written (5) where H is a nonlinear function. We assume that this mapping admits a line attractor, which we denote G(O), for which G(O) = H(G(O)) where 0 is a continuous variable. 1 Let the initial state of the network be a function of the presentation parameter, 00 , plus noise, 00 = F(Oo) +N (6) where F(Oo) is the function used to generate the data (in our simulations this would correspond to the mean LGN input, equation 1). Iterating the mapping, equation 5, leads eventually to a point on the line attractor. Consequently, as n -+ 00 , On -+ G(O) . The parameter 0 provides an estimate of 00 . To determine how well the network does we need to find fJO :::: 0 - 00 as a function of the noise, N, then average over the noise to compute the mean and variance of fJO. Because the mapping, equation 5, is nonlinear, this cannot be done exactly. For small noise, however, we can take a perturbative approach and expand around a point on the attractor. For line at tractors there is no general method for choosing which point on the attractor to expand around. Our approach will be to expand around an arbitrary point, G( 0), and choose 0 by requiring that the quadratic terms be finite. Keeping terms up to quadratic order, equation 6 may be written G(O) In . + fJo n . fJo o + (7) n-l I.: (Jm . fJo ~ o) . H" . (J m . fJo o ) , (8) m=O where J(O) == [8G (o)H(G(0))f is the Jacobian (the subscript T means transpose), H" is the Hessian of H evaluated at G(O) and a "." represents the standard dot product. Because the mapping, equation 5, admits a line attractor , J has one eigenvalue equal to 1 and all others less than 1. Denote the eigenvector with eigenvalue 1 as y and its adjoint v t : J . v = v and JT . v t = yt. It is not hard to show that y = 8oG(0), up to a multiplicative constant. Since J has an eigenvalue equal to 1, to avoid the quadratic term in Eq. 8 approaching infinity as n -+ 00 we require that lim I n-too n . fJo o = O. (9) IThe line attractor is, in fact , an idealization ; for P units the attractors associated with equation 5 consists of P isolated points. However, for P large, the attractors are spaced closely enough that they may be considered a line. Divisive Normalization. Line Attractor Networks and Ideal Observers 109 This equations has an important consequence: it implies that, to linear order, limn-too 60 n = 0 (see equation 8), which in turn implies that 0 00 = G(O) which, ~nally, implies that 0 = O. Consequently we can find the network estimator of 00 , 0, by computing O. We now turn to that task. It is straightforward to show that JOO = vv t . Combining this expression for J with equation 9, using equation 7 to express 600 in terms of 00 and G(O), and, finally using equation 6 to express 00 in terms of the initial mean activity, F(Oo), and the noise, N, we find that v t (0) . [F(Oo) - G(O) Using 00 =0- + N] = O. (10) 60 and expanding F(Oo) to first order in 60 then yields 60 = vt(O) . [N + F(O) - G(O)] vt(O) . F'(O) . (11) As long as v t is orthogonal to F(O) - G(O), (60) = 0 and the estimator is unbiased. This must be checked on a case by case basis, but for the circularly symmetric networks we considered orthogonality is satisfied. We can now calculate the variance of the network estimate, (60)2. Assuming v t . [F(O) - G(O)] = 0, equation 11 implies that 2 vt.R?v t (60) = [v t . F'F ' (12) where a prime denotes a derivative with respect to 0 and R is the covariance matrix of the noise, R = (NN). The network is equivalent to maximum likelihood when this variance is equal to the Cramer-Rao bound [3], (60)bR. If the noise, N, is Gaussian with a covariance matrix independent of 0, this bound is equal to: 2 (60)CR = 1 F'. R - l . F' (13) For independent Gaussian noise of fixed variance, (T2, and zero covariance, the variance of the network estimate, equation 12, becomes (T2 1(IF'1 2 cos2 f-L) where f-L is the angle between v t and F'. The Cramer-Rao bound, on the other hand, is equal to (T2 IIF'1 2 . These expressions differ only by cos 2 J1., which is 1 if F ex v t . In addition, it is close to 1 for networks that have identical input and output tunin curves, F(O) = G(O), and the Jacobian, J, is nearly symmetric, so that v ::::: v (recall that v = G'). If these last two conditions are satisfied, the network comes close to being a maximum likelihood estimator. 1 3.2 Application to Divisive Normalization Divisive normalization is a particular example of the general case considered above. For simplicity, in our simulations we chose the input and output tuning curves to be equal (F = G in the above notation), which lead to a value of 0.87 for cos2 f-L (evaluated numerically). This predicted a variance 15% above the Cramer-Rao S. Deneve, A. Pouget and P E. Latham 110 bound for independent Gaussian noise with fixed variance, consistent with the 16% we obtained in our simulations. The network also handles fairly well other noise distributions, such as Gaussian noise with variance proportional to the mean, as illustrated by our simulations. 4 Conclusions We have recently shown that a subclass of line attractor networks can be used as maximum likelihood estimators[3]. This paper extend this conclusion to a much wider class of networks, namely, any network that admits a line (or, by straightforward extension of the above analysis, a higher dimensional) attractor. This is true in particular for networks using divisive normalization, a normalization which is thought to match quite closely the nonlinearity found in the primary visual cortex and MT. Although our analysis relies on the existence of an attractor, this is not a requirement for obtaining near optimal noise filtering. As we have seen, 2-3 iterations are enough to achieve asymptotic performance (except at contrasts barely above threshold). What matters most is that our network implement a sequence of low pass filtering to filter out the noise, followed by a square nonlinearity to compensate for the widening of the tuning curve due to the low pass filter, and a normalization to weaken contrast dependence. It is likely that this process would still clean up noise efficiently in the first 2-3 iterations even if activity decayed to zero eventually, that is to say, even if the hills of activity were not stable states. This would allow us to apply our approach to other types of networks, including those lacking circular symmetry and networks with continuously clamped inputs. To conclude, we propose that each cortical layer may read out the activity in the preceding layer in an optimal way thanks to the nonlinear pooling properties of divisive normalization, and, as a result, may behave like an ideal observer. It is therefore possible that the ability to read out neuronal codes in the sensory cortices in an optimal way may not be confined to a few areas like the parietal or frontal cortex, but may instead be a general property of every cortical layer. References [1] D. Heeger. Normalization of cell responses in cat striate cortex. Visual Neuroscience, 9:181- 197,1992. [2] L. Itti, C. Koch, and J. Braun. A quantitative model for human spatial vision threshold on the basis of non-linear interactions among spatial filters. In R. Lippman, J. Moody, and D. Touretzky, editors, Advances in Neural Information Processing Systems, volume 11. Morgan-Kaufmann, San Mateo, 1998. [3] A. Pouget, K. Zhang, S. Deneve, and P. Latham. Statistically efficient estimation using population coding. Neural Computation, 10:373- 401, 1998. [4] E. Simoncelli and D. Heeger. A model of neuronal responses in visual area MT. Vision Research, 38(5):743- 761 , 1998. 

1536 |@word trial:1 cos2:2 simulation:11 covariance:3 initial:6 nally:1 perturbative:1 written:2 must:1 realistic:1 j1:2 discrimination:1 alone:1 filtered:1 provides:1 zhang:1 mathematical:1 direct:1 consists:2 prove:1 behavior:1 actual:1 jm:1 becomes:1 notation:1 what:2 kg:1 eigenvector:1 quantitative:1 every:2 multidimensional:1 act:2 subclass:1 braun:1 exactly:1 control:2 unit:15 negligible:1 local:4 consequence:1 id:1 subscript:1 plus:1 chose:1 studied:1 mateo:1 suggests:2 co:5 statistically:1 implement:2 lippman:1 area:3 thought:1 onto:2 close:10 convenience:1 cannot:1 context:1 equivalent:1 yt:1 go:1 straightforward:3 formalized:1 simplicity:1 pouget:4 estimator:10 array:1 population:8 handle:1 variation:1 updated:1 spontaneous:1 play:1 agreement:1 fjo:7 bottom:1 role:3 calculate:1 ithe:1 division:1 basis:2 easily:1 cat:1 distinct:1 choosing:1 quite:1 encoded:1 say:1 ability:2 think:1 noisy:3 final:1 sequence:2 eigenvalue:3 propose:2 interconnected:1 jij:1 product:1 interaction:1 combining:1 achieve:1 adjoint:1 los:1 requirement:1 wider:1 oo:5 recurrent:1 measured:1 ij:4 eq:1 implemented:1 predicted:1 involves:1 come:4 implies:4 differ:1 fij:1 closely:2 filter:4 human:2 implementing:1 require:1 ao:4 generalization:1 adjusted:1 extension:1 pl:1 around:4 cramer:4 sufficiently:1 considered:3 exp:1 koch:1 mapping:5 estimation:5 geniculate:1 gaussian:10 reaching:2 avoid:1 cr:1 og:1 focus:1 likelihood:11 contrast:13 squaring:2 nn:1 expand:3 wij:2 lgn:13 among:1 orientation:13 denoted:2 spatial:14 fairly:1 equal:7 washington:1 sampling:1 iif:1 kw:1 represents:1 identical:1 nearly:1 others:1 stimulus:1 t2:3 few:1 attractor:18 circular:2 kt:1 edge:1 closer:1 orthogonal:1 indexed:1 circle:1 isolated:1 theoretical:2 weaken:1 rao:4 comprised:1 too:2 periodic:1 combined:2 thanks:1 decayed:2 peak:2 continuously:1 moody:1 squared:1 satisfied:2 choose:1 cognitive:1 derivative:1 itti:2 account:1 coding:2 pooled:3 bold:1 matter:1 explicitly:1 depends:1 multiplicative:1 observer:13 square:2 variance:17 who:1 efficiently:2 kaufmann:1 correspond:1 spaced:1 yield:1 produced:1 converged:1 touretzky:1 checked:1 colleague:1 frequency:12 associated:1 gain:2 sampled:1 recall:1 lim:1 alexandre:1 higher:1 follow:1 response:4 improved:1 jw:4 done:2 evaluated:2 stage:3 hand:2 nonlinear:4 effect:1 normalized:1 unbiased:3 true:1 requiring:1 analytically:2 read:3 symmetric:2 illustrated:1 width:3 hill:7 demonstrate:3 latham:5 motion:1 fj:1 recently:1 sigmoid:1 mt:2 volume:1 extend:1 numerically:1 tuning:8 nonlinearity:5 had:1 dot:1 joo:1 stable:7 cortex:11 inhibition:2 playa:1 recent:1 prime:1 vt:3 seen:1 minimum:1 morgan:1 preceding:3 determine:2 dashed:1 simoncelli:1 smooth:1 match:2 long:3 compensate:1 controlled:1 vision:2 iteration:6 normalization:23 achieved:1 cell:2 confined:1 receive:1 whereas:1 addition:1 limn:1 exhibited:1 pooling:5 subject:1 inconsistent:1 integer:1 call:1 j08:1 near:1 presence:1 ideal:13 enough:2 affect:1 approaching:1 br:1 angeles:1 expression:2 hessian:1 v8:1 generally:1 iterating:3 se:1 amount:1 extensively:1 generate:2 neuroscience:1 per:1 discrete:1 express:2 threshold:3 clean:1 kept:1 deneve:5 relaxation:1 idealization:1 inverse:2 angle:2 place:1 throughout:1 reasonable:1 layer:9 bound:8 followed:2 distinguish:2 quadratic:3 activity:24 orthogonality:1 infinity:1 ucla:1 performing:1 according:1 wi:1 equation:18 turn:2 eventually:2 mechanism:1 serf:1 apply:1 generic:1 existence:1 dpt:1 top:1 denotes:1 added:1 quantity:1 looked:1 primary:1 dependence:1 striate:1 lateral:2 barely:1 assuming:1 code:1 providing:1 lg:2 perform:1 diamond:1 upper:1 neuron:3 finite:1 behave:1 parietal:1 neurobiology:1 dc:1 arbitrary:1 namely:1 hypercolumn:1 kl:1 specified:1 connection:1 required:1 pattern:2 including:1 critical:1 treated:1 widening:1 oij:2 extract:1 georgetown:2 tractor:1 asymptotic:1 lacking:1 filtering:9 proportional:2 versus:1 nucleus:1 consistent:2 editor:1 last:1 keeping:1 transpose:1 aij:4 allow:1 vv:1 institute:1 taking:1 curve:11 cortical:17 sensory:2 author:1 commonly:1 san:1 simplified:1 far:1 preferred:2 keep:1 assumed:1 conclude:1 continuous:2 ca:1 expanding:1 obtaining:1 symmetry:1 did:1 noise:25 profile:1 neuronal:2 position:2 heeger:3 clamped:1 jacobian:2 jt:1 explored:1 admits:3 consist:1 circularly:1 sigmoids:1 explore:1 likely:1 visual:6 applies:1 corresponds:1 relies:1 presentation:3 consequently:2 experimentally:1 hard:1 specifically:1 determined:1 except:1 sophie:1 total:1 pas:2 divisive:21 latter:1 frontal:1 ex:1

Maximum Conditional Likelihood via Bound Maximization and the CEM Algorithm Tony J ebara and Alex Pentland Vision and Modeling, MIT Media Laboratory, Cambridge MA http://www.rnedia.rnit.edu/ ~ jebara { jebara,sandy }~rnedia.rnit.edu Abstract We present the CEM (Conditional Expectation Maximi::ation) algorithm as an extension of the EM (Expectation M aximi::ation) algorithm to conditional density estimation under missing data. A bounding and maximization process is given to specifically optimize conditional likelihood instead of the usual joint likelihood. We apply the method to conditioned mixture models and use bounding techniques to derive the model's update rules . Monotonic convergence, computational efficiency and regression results superior to EM are demonstrated. 1 Introduction Conditional densities have played an important role in statistics and their merits over joint density models have been debated. Advantages in feature selection , robustness and limited resource allocation have been studied. Ultimately, tasks such as regression and classification reduce to the evaluation of a conditional density. However, popularity of maximumjoint likelihood and EM techniques remains strong in part due to their elegance and convergence properties . Thus , many conditional problems are solved by first estimating joint models then conditioning them . This results in concise solutions such as the N adarya- Watson estimator [2], Xu's mixture of experts [7], and Amari's em-neural networks [1]. However, direct conditional density approaches [2, 4] can offer solutions with higher conditional likelihood on test data than their joint counter-parts. 495 Maximum Conditional Likelihood via Bound Maximization and CEM 1~~1 01 '" (a) La = -4.2 L; !i. = -2.4 15 (b) Lb = -5 .2 Lb 20 = -1.8 Figure 1: Average Joint (x, y) vs. Conditional (ylx) Likelihood Visualization Pop at [6] describes a simple visualization example where 4 clusters must be fit with 2 Gaussian models as in Figure 1. Here, the model in (a) has a superior joint likelihood (La> Lb) and hence a better p(x, y) solution. However, when the models are conditioned to estimate p(ylx), model (b) is superior (Lb > L~). Model (a) yields a poor unimodal conditional density in y and (b) yields a bi-modal conditional density. It is therefore of interest to directly optimize conditional models using conditionallikelihood. We introduce the CEM (Conditional Expectation Maximization) algorithm for this purpose and apply it to the case of Gaussian mixture models. 2 EM and Conditional Likelihood For joint densities, the tried and true EM algorithm [3] maximizes joint likelihood over data. However , EM is not as useful when applied to conditional density estimation and maximum conditional likelihood problems. Here, one typically resorts to other local optimization techniques such as gradient descent or second order Hessian methods [2]. We therefore introduce CEM, a variant of EM , which targets conditional likelihood while maintaining desirable convergence properties. The CEM algorithm operates by directly bounding and decoupling conditional likelihood and simplifies M-step calculations. In EM, a complex density optimization is broken down into a two-step iteration using the notion of missing data. The unknown data components are estimated via the E-step and a simplified maximization over complete data is done in the M-step. In more practical terms, EM is a bound maximization : the E-step finds a lower bound for the likelihood and the M-step maximizes the bound. M P(Xi, Yi18) =L p(m, Xi, Yi1 8 ) (1) m=l Consider a complex joint density p(Xi , Yi 18) which is best described by a discrete (or continuous) summation of simpler models (Equation 1) . Summation is over the 'missing components' m . L:~llog(P(Xi' Yi 18 t ?) -Iog(p(xi ' YiI 8t - 1 )) l:!.l > ",\,N L...i=l h I p(m ,X. ,Y.le t ) L...m=l im og p(m,X"Y.let I) ",\,M h h p(m,x. ,y.le t- I ) were im = ",\,M L...,,=I p(n ,x"y .Ie t- I ) By appealing to Jensen's inequality, EM obtains a lower bound for the incremental log-likelihood over a data set (Equation 2) . Jensen's inequality bounds the logarithm of the sum and the result is that the logarithm is applied to each simple (2) T. Jebara and A. Pentland 496 model p(m, Xi , yd8) individually. It then becomes straightforward t.o compute the derivatives with respect to e and set to zero for maximization (M-step) . AJ ,\"M " I 8) = =c""-7 L...m=IP(m,xi,y;j8) p( Yi IXi, 8) - = L..: p( m , Y i Xi , M+------'-----'- (3) Lm=IP(m,XiI 8 ) m=l However, the elegance of EM is compromised when we consider a conditioned density as in Equation :3. The corresponding incremental conditional log-likelihood, L:l.lc, is shown in Equation 4. L~llog(p(Yilxi' 8 t )) -log(p(ydxi, 8 t I L.... og !=1 ,\"N LMm_I P(m ,X. ,Y.10)t H L;"=I p(m ,X. ,Y.10 t - l ) -I og 1) LMn_IP(n ,X, I0 t) (4) M Ln=1 p(n ,X.10 t - l ) The above is a difference between a ratio of joints and a ratio of marginals. If Jensen's inequality is applied to the second term in Equation 4 it yields an upper bound since the term is subtracted (this would compromise convergence). Thus, only the first ratio can be lower bounded with Jensen (Equation 5). L:l.jC>~~h' I p(m,xi,YiI 8t ) -I - L..: L..: 2m og ( 18 t - 1) og i=1 m=1 p m, Xi, Yi - L~~lp(n,xiI8t) M Ln=1 p(n, XiI8 t - 1) (5) Note the lingering logarithm of a sum which prevents a simple M-Step. At this point, one would resort to a Generalized EM (GEM) approach which requires gradient or second-order ascent techniques for the M-step. For example, Jordan et al. overcome the difficult M-step caused by EM with an Iteratively Re- Weighted Least Squares algorithm in the mixtures of experts architecture [4]. 3 Conditional Expectation Maximization The EM algorithm can be extended by substituting Jensen's inequality for a different bound. Consider the upper variational bound of a logarithm x-I 2: log(x) (which becomes a lower bound on the negative log). The proposed logarithm's bound satisfies a number of desiderata: (1) it makes contact at the current operating point 1, (2) it is tangential to the logarithm, (3) it is a tight bound, (4) it is simple and (5) it is the variational dual of the logarithm. Substituting this linear bound into the incremental conditional log-likelihood maintains a true lower bounding function Q (Equation 6). The Mixture of Experts formalism [4J offers a graceful representation of a conditional density using experts (conditional sub-models) and gates (marginal sub-models). The Q function adopts this form in Equation 7. 1 The current operating point is 1 since the previous iteration's value e t - 1 . e t model in the ratio is held fixed at the Maximum Conditional Likelihood via Bound Maximization and CEM L~l L~=1 {him(logp(Yilm,Xi,e t ) +logp(m,xile t ) where Zim = log(p(m,xi,Yile t -- 1 )) 497 - Zim) - riP(m , xile) + and ri = (L~=1p(n'Xilet-1) )-1 ir} (7) Computing this Q function forms the CE-step in the Conditional Expectation Maximization algorithm and it results in a simplified M-step. Note the absence of the logarithm of a sum and the decoupled models. The form here allows a more straightforward computation of derivatives with respect to e t and a more tractable M-Step. For continuous missing data, a similar derivation holds. At this point , without loss of generality, we specifically attend to the case of a conditioned Gaussian mixture model and derive the corresponding M-Step calculations. This serves as an implementation example for comparison purposes. 4 CEM and Bound Maxinlization for Gaussian Mixtures In deriving an efficient M-step for the mixture of Gaussians, we call upon more bounding techniques that follow the CE-step and provide a monotonically convergent learning algori thm . The form ofthe condi tional model we will train is obtained by conditioning a joint mixture of Gaussians. We write the conditional density in a experts-gates form as in Equation 8. We use unnormalized Gaussian gates N(x; p,~) = exp( - ~(x - p)T~-1 (x - p? since conditional models do not require true marginal densities over x (i .e. that necessarily integrate to 1). Also, note that the parameters of the gates (0:' , px , :E xx ) are independent of the parameters of the experts (vm,rm,om). Both gates and experts are optimized independently and have no variables in common. An update is performed over the experts and then over the gates. If each of those causes an increase, we converge to a local maximum of conditional loglikelihood (as in Expectation Conditional Maximization [5]). p(Ylx,8) (8) To update the experts , we hold the gates fixed and merely take derivatives of the Q function with respect to the expert parameters (<l>m = {v m , rm, am} ) and set them to O. Each expert is effectively decoupled from other terms (gates, other experts , etc .). The solution reduces to maximizing the log of a single conditioned Gaussian and is analytically straightforward. 8Q(e t ,e(t-l?) 8<1>'" (9) Similarly, to update the gate mixing proportions, derivatives of the Q function are taken with respect to O:'m and set to O. By holding the other parameters fixed , the update equation for the mixing proportions is numerically evaluated (Equation 10). N N O:'m := LriN(xi;P~,:E~x) le(l-I) {Lhim}-l i=l i=l (10) T. Jebara and A. Pentland 498 , c, O~ 01 I ~ r 01 I i \ - ' ., i \, ',_ 00 _1 ( a) ... : ~, ,>/~ \ \ ,f at //---~ ',,---, ,1 \ f Function -2 ..!..: _I ':li :~ c' ----- 00 I I?Ji '--... 01' I ~ ''''\ I ~ jr' --...-.;==i -J j--- - - ' - ,- - 10 Ii " (b) Bound on f..L ( c) g Function (d) Bound on I: xx Figure 2: Bound Width Computation and Example Bounds 4.1 Bounding Gate Means Taking derivatives of Q and setting to a is not as straightforward for the case of the gate means (even though they are decoupled). What is desired is a simple update rule (i.e. computing an empirical mean). Therefore, we further bound the Q function for the M-step. The Q function is actually a summation of sub-elements Qim and we bound it instead by a summation of quadratic functions on the means (Equation 11). N Q(e t , e(t-1)) = M N LL i=l Q(e t , e(t-1))im > m=l M LL kim - Wimllf..L~ - ciml1 2 (11) i=l m=l Each quadratic bound has a location parameter cim (a centroid), a scale parameter Wim (narrowness), and a peak value at kim. The sum of quadratic bounds makes e contact with the Q function at the old values of the model t - 1 where the gate mean was originally f..L':* and the covariance is I:':x*' To facilitate the derivation, one may assume that the previous mean was zero and the covariance was identity if the data is appropriately whitened with respect to a given gate. The parameters of each quadratic bound are solved by ensuring that it contacts the corresponding Qim function at t - 1 and they have equal derivatives at contact (i .e. tangential contact) . Sol ving these constraints yields quadratic parameters for each gate m and data point i in Equation 12 (kim is omitted for brevity) . e (12) > The tightest quadratic bound occurs when Wim is minimal (without violating the inequality). The expression for Wim reduces to finding the minimal value, wim, as in Equation 13 (here p2 = xT xd. The f function is computed numerically only once and stored as a lookup table (see Figure 2(a)). We thus immediately compute the optimal wim and the rest of the quadratic bound's parameters obtaining bounds as in Figure 2(b) where a Qim is lower bounded. 1 * -_ Wim . rlCl'm max C { e- 1 -p 2 2 2 e- 2 C e CP 2 c cp - 1 } ' h?1m _ +- - 2 , . r l Cl'm e 1 - -p 2 2 h?1m f( p) + - (13) 2 The gate means f..L~ are solved by maximizing the sum of the M x N parabolas which bound Q. The update is f..L': = (2: wimCim) (2: wim)-l. This mean is subsequently unwhitened to undo earlier data transformations. Maximum Conditional Likelihood via Bound Maximization and CEM 499 [ ,~, :l -1 " ~ 0'-, ?. i . ..... : - J ... . (a) Data (b) CEM p(ylx) (c) CEM IC (d) EM fit (e) EM p(ylx) (f) EM I C Figure 3: Conditional Density Estimation for CEM and EM 4.2 Bounding Gate Covariances Having derived the update equation for gate means, we now turn our attention to the gate covariances. We bound the Q function with logarithms of Gaussians. Maximizing this bound (a sum of log-Gaussians) reduces to the maximum-likelihood estimation of a covariance matrix . The bound for a Qim sub-component is shown in Equation 14. Once again, we assume the data has been appropriately whitened with respect to the gate's previous parameters (the gate's previous mean is 0 and previous covariance is identity). Equation 15 solves for the log-Gaussian parameters (again p2 = XTXi). Q( Dt,D(t-1));m > Iog (N) = k im QQ. _ - ",m T -1 Cim WimCimL..xx Wim Iog I",m L.. xx I (14) (15) > The computation for the minimal Wim simplifies to wim = riQ:mg(p) . The 9 function is derived and plotted in Figure 2(c). An example of a log-Gaussian bound is shown in Figure 2( d) a sub-component of the Q function. Each sub-component corresponds to a single data point as we vary one gate 's covariance. All M x N log-Gaussian bounds are computed (one for each data point and gate combination) and are summed to bound the Q function in its entirety. To obtain a final answer for the update of the gate covariances E~ we simply maximize the sum of log Gaussians (parametrized by wim, kim, Cim). The update is E~x = (2: WimCimCim T) (2: wim)-l. This covariance is subsequently unwhitened , inverting the whitening transform applied to the data. 5 ] '~ - -- -:----" !' ' 1 Results The CEM algorithm updates the conditioned mixture of Gaussians by computing him and rim in the CE steps and interlaces these with updates on the experts, mixing proportions, gate means and gate covariances. For the mixture of Gaussians , each CEM update has a computation time that is comparable with that of an EM update (even for high dimensions). However, conditional likelihood (not joint) is monotonically increased . Consider the 4-cluster (x , y) data in Figure 3(a). The data is modeled with a conditional density p(ylx) using only 2 Gaussian models . Estimating the density with CEM yields the p(ylx) shown in Figure 3(b). CEM exhibits monotonic conditional likelihood growth (Figure 3(c)) and obtains a more conditionally likely model. In T. Jebara and A. Pentland 500 Algorithm Abalone Table 1: Test Results. Class label regression accuracy data. (CNNO=cascadecorrelation, hidden units, CCN5=5 hidden LD=linear discriminant). a the EM case, a joint p(x, y) clusters the data as in Figure 3(d) . Conditioning it yields the p(ylx) in Figure 3(e) . Figure 3(f) depicts EM's non-monotonic evolution of conditional log-likelihood. EM produces a superior joint likelihood but an inferior conditional likelihood. Note how the CEM algorithm utilized limited resources to capture the multimodal nature of the distribution in y and ignored spurious bimodal clustering in the x feature space. These properties are critical for a good conditional density p(ylx). For comparison , standard databases were used from DCI 2. Mixture models were trained with EM and CEM , maximizingjoint and conditional likelihood respectively. Regression results are shown in Table 1. CEM exhibited , monotonic conditional loglikelihood growth and out-performed other methods including EM with the same 2-Gaussian model (EM2 and CEM2). 6 Discussion We have demonstrated a variant of EM called CEM which optimizes conditional likelihood efficiently and monotonically. The application of CEM and bound maximization to a mixture of Gaussians exhibited promising results and better regression than EM . In other work , a MAP framework with various priors and a deterministic annealing approach have been formulated. Applications of the CEM algorithm to non-linear regressor experts and hidden Markov models are currently being investigated . Nevertheless , many applications CEM remain to be explored and hopefully others will be motivated to extend the initial results . Acknowledgements Many thanks to Michael Jordan and Kris Popat for insightful discussions. References [1] S. Amari. Information geometry of em and em algorithms for neural networks. Neural Networks , 8(9), 1995 . [2 ] C. Bishop. Neural Networks Jor Pattern Recognition. Oxford Press, 1996. 3 [ ] A. Dempster, N. Laird, and D. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal oj the Royal Statistical Society, B39, 1977. [4] M. Jordan and R. Jacobs. Hierarchical mixtures of experts and the em algorithm . Neural Computation, 6:181 - 214, 1994. [5] X. Meng and D. Rubin. Maximum likelihood estimation via the ecm algorithm : A general framework. Biometrika , 80(2), 1993. [6] A . Popat. Conjoint probabilistic subband modeling (phd. thesis). Technical Report 461, M.LT. Media Laboratory, 1997. [7] 1. Xu , M. Jordan , and G. Hinton . An alternative model for mixtures of experts . In Neural InJormation Processing Systems 7, 1995. 2http://www.ics.uci.edu/'''-'mlearn/MLRepository.html 

1537 |@word proportion:3 tried:1 covariance:10 b39:1 jacob:1 concise:1 ld:1 initial:1 current:2 must:1 update:14 v:1 yi1:1 location:1 simpler:1 direct:1 introduce:2 becomes:2 estimating:2 bounded:2 xx:4 maximizes:2 medium:2 what:1 finding:1 transformation:1 xd:1 growth:2 biometrika:1 rm:2 unit:1 attend:1 local:2 oxford:1 meng:1 studied:1 limited:2 bi:1 practical:1 empirical:1 ving:1 selection:1 www:2 optimize:2 map:1 demonstrated:2 missing:4 maximizing:3 deterministic:1 straightforward:4 attention:1 independently:1 immediately:1 rule:2 estimator:1 deriving:1 notion:1 qq:1 target:1 rip:1 ixi:1 element:1 recognition:1 utilized:1 parabola:1 database:1 role:1 solved:3 capture:1 counter:1 sol:1 xile:2 dempster:1 broken:1 ultimately:1 trained:1 tight:1 compromise:1 upon:1 efficiency:1 multimodal:1 joint:14 various:1 derivation:2 train:1 loglikelihood:2 qim:4 amari:2 statistic:1 transform:1 laird:1 ip:2 final:1 injormation:1 advantage:1 mg:1 yii:2 uci:1 mixing:3 convergence:4 cluster:3 produce:1 incremental:3 derive:2 p2:2 solves:1 strong:1 entirety:1 subsequently:2 require:1 summation:4 im:4 extension:1 hold:2 ic:2 exp:1 lm:1 substituting:2 yilxi:1 vary:1 sandy:1 omitted:1 purpose:2 estimation:5 label:1 currently:1 wim:12 individually:1 him:2 weighted:1 mit:1 gaussian:11 ecm:1 og:5 derived:2 zim:2 likelihood:29 centroid:1 kim:4 am:1 tional:1 i0:1 typically:1 hidden:3 spurious:1 classification:1 dual:1 html:1 summed:1 marginal:2 equal:1 once:2 having:1 others:1 report:1 tangential:2 geometry:1 interest:1 evaluation:1 mixture:15 held:1 decoupled:3 incomplete:1 old:1 logarithm:9 re:1 desired:1 plotted:1 minimal:3 increased:1 formalism:1 modeling:2 earlier:1 logp:2 maximization:13 stored:1 answer:1 thanks:1 density:19 peak:1 ie:1 probabilistic:1 vm:1 regressor:1 michael:1 again:2 rlcl:1 algori:1 thesis:1 expert:16 resort:2 derivative:6 li:1 lookup:1 jc:1 caused:1 maximumjoint:1 performed:2 maintains:1 om:1 square:1 ir:1 accuracy:1 efficiently:1 yield:6 ofthe:1 kris:1 mlearn:1 elegance:2 rim:1 actually:1 higher:1 originally:1 violating:1 follow:1 dt:1 modal:1 done:1 evaluated:1 though:1 generality:1 hopefully:1 aj:1 facilitate:1 true:3 evolution:1 hence:1 analytically:1 laboratory:2 iteratively:1 conditionally:1 ll:2 width:1 inferior:1 unnormalized:1 abalone:1 mlrepository:1 generalized:1 complete:1 cp:2 variational:2 superior:4 common:1 ji:1 rnit:2 conditioning:3 extend:1 marginals:1 numerically:2 cambridge:1 similarly:1 operating:2 whitening:1 etc:1 ebara:1 optimizes:1 inequality:5 watson:1 yi:4 converge:1 maximize:1 monotonically:3 ii:1 unimodal:1 desirable:1 reduces:3 technical:1 calculation:2 offer:2 iog:3 ensuring:1 variant:2 regression:5 desideratum:1 whitened:2 vision:1 expectation:6 iteration:2 bimodal:1 condi:1 annealing:1 appropriately:2 rest:1 exhibited:2 ascent:1 undo:1 jordan:4 call:1 fit:2 architecture:1 reduce:1 simplifies:2 expression:1 motivated:1 hessian:1 cause:1 ignored:1 useful:1 ylx:9 http:2 estimated:1 popularity:1 xii:1 discrete:1 write:1 nevertheless:1 ce:3 merely:1 sum:7 ydxi:1 comparable:1 bound:37 played:1 convergent:1 quadratic:7 constraint:1 alex:1 ri:1 graceful:1 px:1 combination:1 poor:1 jr:1 describes:1 cascadecorrelation:1 em:31 remain:1 appealing:1 lp:1 taken:1 ln:2 resource:2 visualization:2 remains:1 equation:17 turn:1 merit:1 tractable:1 serf:1 gaussians:8 tightest:1 apply:2 hierarchical:1 subtracted:1 alternative:1 robustness:1 gate:25 clustering:1 tony:1 maintaining:1 subband:1 society:1 contact:5 occurs:1 usual:1 exhibit:1 gradient:2 parametrized:1 discriminant:1 modeled:1 ratio:4 difficult:1 holding:1 dci:1 negative:1 implementation:1 unknown:1 upper:2 markov:1 descent:1 pentland:4 extended:1 hinton:1 lb:4 thm:1 jebara:5 xtxi:1 inverting:1 optimized:1 pop:1 pattern:1 max:1 including:1 oj:1 royal:1 critical:1 ation:2 cim:3 prior:1 acknowledgement:1 loss:1 j8:1 allocation:1 conjoint:1 integrate:1 rubin:2 taking:1 overcome:1 dimension:1 jor:1 adopts:1 simplified:2 obtains:2 em2:1 cem:23 gem:1 xi:13 continuous:2 compromised:1 table:3 promising:1 nature:1 decoupling:1 obtaining:1 investigated:1 complex:2 necessarily:1 cl:1 bounding:7 xu:2 depicts:1 lc:1 sub:6 debated:1 down:1 xt:1 bishop:1 insightful:1 jensen:5 popat:2 explored:1 effectively:1 phd:1 conditioned:6 unwhitened:2 lt:1 simply:1 likely:1 prevents:1 monotonic:4 corresponds:1 satisfies:1 ma:1 conditional:42 identity:2 formulated:1 absence:1 specifically:2 operates:1 llog:2 called:1 la:2 brevity:1

Robot Docking using Mixtures of Gaussians Matthew Williamson* Roderick Murray-Smith t Volker Hansen t Abstract This paper applies the Mixture of Gaussians probabilistic model, combined with Expectation Maximization optimization to the task of summarizing three dimensional range data for a mobile robot. This provides a flexible way of dealing with uncertainties in sensor information, and allows the introduction of prior knowledge into low-level perception modules. Problems with the basic approach were solved in several ways: the mixture of Gaussians was reparameterized to reflect the types of objects expected in the scene, and priors on model parameters were included in the optimization process. Both approaches force the optimization to find 'interesting' objects, given the sensor and object characteristics. A higher level classifier was used to interpret the results provided by the model, and to reject spurious solutions. 1 Introduction This paper concerns an application of the Mixture of Gaussians (MoG) probabilistic model (Titterington et aI., 1985) for a robot docking application. We use the ExpectationMaximization (EM) approach (Dempster et aI., 1977) to fit Gaussian sub-models to a sparse 3d representation of the robot's environment, finding walls, boxes, etc .. We have modified the MoG formulation in three ways to incorporate prior knowledge about the task, and the sensor characteristics: the parameters of the Gaussians are recast to constrain how they fit the data, priors on these parameters are calculated and incorporated into the EM algorithm, and a higher level processing stage is included which interprets the fit of the Gaussians on the data, detects misclassifications, and providing prior information to guide the modelfitting. The robot is equipped with a LIDAR 3d laser range-finder (PIAP, 1995) which it uses to identify possible docking objects. The range-finder calculates the time of flight for a light pulse reflected off objects in the scene. The particular LIDAR used is not very powerful, making objects with poor reflectance (e.g., dark, shiny, or surfaces not perpendicular to the *Corresponding author: MIT AI Lab, Cambridge, MA, USA. rna t t@ai . rni t . edu tDept. of Mathematical Modelling, Technical University of Denmark. rod@imm. dtu. dk tDaimlerChrysler, Alt-Moabit 96a, Berlin, Germany. hansen@dbag.bIn. dairnierbenz . com 946 M M Williamson, R. Murray-Smith and V. Hansen laser beam) invisible. The scan pattern is also very sparse, especially in the vertical direction, as shown in the scan of a wall in Figure 1. However, if an object is detected, the range returned is accurate (?1-2cm). When the range data is plotted in Cartesian space it forms a number of sparse clusters, leading naturally to the use of MoG clustering algorithms to make sense of the scene. While the Gaussian assumption is not an ideal model of the data, the generality of MoG, and its ease of implementation and analysis motivated its use over a more specialized approach. The sparse nature of the data inspired the modifications to the MoG formulation described in this paper. Model-based object recognition from dense range images has been widely reported (see (Arman and Aggarwal, 1993) for a review), but is not relevant in this case given the sparseness of the data. Denser range images could be collected by combining multiple scans, but the poor visibility of the sensor hampers the application of these techniques. The advantage of the MoG technique is that the segmentation is "soft", and perception proceeds iteratively during learning. This is especially useful for mobile robots where evidence accumulates over time, and the allocation of attention is time and state-dependent. The EM algorithm is useful since it is guaranteed to converge to a local maximum. The following sections of the paper describe the re-parameterization of the Gaussians to model plane-like clusters, the formulation of the priors, and the higher level processing which interprets the clustered data in order to both move the robot and provide prior information to the model-fitting algorithm. -<>.2 e -0. ~ -0.6 ~-08 -, -,.. 2 Figure 1: Plot showing data from a LIDAR scan of a wall, plotted in Cartesian space. The robot is located at the origin, with the y axis pointing forward, x to the right, and z up. The sparse scan pattern is visible, as well as the visibility constraint: the wall extends beyond where the scan ends, but is invisible to the LIDAR due to the orientation of the wall 2 Mixture of Gaussians model The range-finder returns a set of data, each of which is a position in Cartesian space Xi = (Xi, Yi, Zi). The complete set of data D = {Xl ... XN} is modeled as being generated by a mixture density M P(xn) =L P(xn Ii, JLi, E i , 1l'i)P( i), i=l where we use a Gaussian as the sub-model, with mean JLi, variance Ei and weight 1l'i' which makes the probability of a particular data point: M P(xnIJL, E, 1l') = ~ (21l')3/:jE I1/2 exp ( -~(Xn i JLi)TE;l(xn - JLi)) Robot Docking Using Mixtures of Gaussians 947 Given a set of data D, the most likely set of parameters is found using the EM algorithm. This algorithm has a number of advantages, such as guaranteed convergence to a local minimum, and efficient computational performance. In 3D Cartesian space, the Gaussian sub-models form ellipsoids, where the size and orientation are determined by the covariance matrix ~~. In the general case, the EM algorithm can be used to learn all the parameters of ~i. The sparseness of the LIDAR data makes this parameterization inappropriate, as various odd collections of points could be clustered together. By changing the parameterization of ~~ to better model plane-like structures, the system can be improved. The reparameterization is most readily expressed in terms of the eigenvalues Ai and eigenvectors ~ of the covariance matrix ~i = ~Ai ~ -I. The covariance matrix of a normal approximation to a plane-like vertical structure will have a large eigenvalue in the z direction, and in the x-y plane one large and one small I = ~T = ~, eigenvalue. Since ~i is symmetrical, the eigenvectors are orthogonal, and ~i can be written: v:- o where Oi is the angle of orientation of the ith sub-model in the x-y plane, ai scales the cluster in the x and y directions, and bi scales in the z direction. The constant, controls the aspect ratio of the ellipsoid in the x-y plane. I The optimal values of these parameters (a, b) are found using EM, first calculating the probability that data point Xn is modeled by Gaussian i, (h tn ) for every data point Xn and every Gaussian i, hin = 7ril~il-1/2 exp (-~(Xn - fli)T~il(Xn - fli)) --~M~--------~~~--------~--------~-- Li==1 7ril~~I - 1/2exp (-~(Xn - fldT~il(Xn - fli))' This "responsibility" is then used as a weighting for the updates to the other parameters, {) _ ~ t -I ( 2 Ln htn(Xnl - flil)(X n 2 - fli2) ) t 2 an Ln htn[(Xnl - fl~I)2 - (Xn2 - fli2)2] (r - l)((xnl - flid sin 0 + (Xn2 - fl~2) COSO)2 + (Xnl - flid 2 + (Xn2 - fli2)2 Ln hinxn Ln h tn ' fli Ln hin( 2, Ln hin ' b_ t - Ln h in (Xn3 - fln3)2 Ln h tn ' where Xnl is the first element of Xn etc. and ( corresponds to the projection of the data into the plane ofthe cluster. It is im~ortant to update the means fli first, and use the new values to update the other parameters. Figure 2 shows a typical model response on real LIDAR data. 2.1 Practicalities of application, and results Starting values for the model parameters are important, as EM is only guaranteed to find a local optimum. The Gaussian mixture components are initialized with a large covariance, allowing them to pick up data and move to the correct positions. We found that initializing the means fli to random data points, rather than randomly in the input space, tended to 1By experimentation, a value of'Y of 0.01 was found to be reasonable for this application. 2Intuition for the Oi update can be obtained by considering that (Xnl - fltl) is the x component of the distance between Xn and /.Li, which is IXn - /.Ld cos and similarly (Xn2 - /.Li2) is IXn - /.Li Isin so tan 2() = sin 20 = 2 sin 0 cos 0 = 2(xn1 -1'.1 )(xn 2 -1'.2) . e, cos 20 cos 2 0-sin 2 0 (X n 1-l'i1 )2 -(Xn2 -1'.2)2 e, M. M. Williamson, R. Murray-Smith and V. Hansen 948 O '+ ~~ 1 ;Ui?h? " ----..-~ ... + ? ? Figure 2: Example of clustering of the 3d data points. The left hand graph shows the view from above (the x-y plane), and the right graph shows the view from the side (the y-z plane), with the robot positioned at the origin. The scene shows a box at an oblique angle, with a wall behind. The extent of the plane-like Gaussian sub-models is illustrated using the ellipses, which are drawn at a probability of 0.5. work better, especially given the sensor characteristics-if the LIDAR returned a range measurement, it was likely to be part of an interesting object. Despite the accuracy of measurement, there are still outlying data points, and it is impossible to fully segment the data into separate objects. One simple solution we found was to define a "junk" Gaussian. This is a sub-model placed in the center of the data, with a large covariance ~. This Gaussian then becomes responsible for the outliers in the data (i.e. sparsely distributed data over the whole scene, none of which are associated with a specific object), allowing the object-modeling Gaussians to work undistracted. The use of EM with the a, b, e parameterization found and represented plane-like data clusters better than models where all the elements of the covariance matrix were free to adapt. It also tended to converge faster, probably due to the reduced numbers of parameters in the covariance matrix (3 as opposed to 6). Although the algorithm is constrained to find planes, the parameterization was flexible enough to model other objects such as thin vertical lines (say from a table leg). The only problem with the algorithm was that it occasionally found poor local minimum solutions, such as illustrated in Figure 3. This is a common problem with least squares based clustering methods (Duda and Hart, 1973) . O. o. O. OB 07 07 06 06 os os 04 04 03 ?? 02 ? -, 01 0 -o.s os .. 03 0.2 01 I ..%.6 -04 -02 02 04 06 08 Figure 3: Two examples of 'undesirable' local minimum solutions found by EM. Both graphs show the top view of a scene of a box in front of a wall. The algorithm has incorrectly clustered the box with the left hand side of the wall. 949 Robot Docking Using Mixtures ofGaussians 3 Incorporating prior information As well as reformulating the Gaussian models to suit our application, we also incorporated prior knowledge on the parameters of the sub-models. Sensor characteristics are often well-defined, and it makes sense to use these as early as possible in perception, rather than dealing with their side-effects at higher levels of reasoning. Here, e.g., the visibility constraint, by which only planes which are almost perpendicular to the lidar rays are visible, could be included by writing P(x n ) = I:~~l P(xnli, f3t)P(i)P(visiblelf3i), the updates could be recalculated, and the feature immediately brought into the modeling process. In addition, prior knowledge about the locations and sizes of objects, maybe from other sensors, can be used to influence the modeling procedure. This allows the sensor to make better use of the sparse data. For a model with parameters f3 and data D, Bayes rule gives: P(f3) P(,8ID) = P(D) II P(xn lf3)? Normally the logarithm of this is taken, to give the log-likelihood, which in the case of mixtures of Gaussians is L(DIf3) = log(p({/-li, 7ri,ai,bi ,6Q)) -log(p(D)) + LlogLp(xnli,/-li,7ri,ai,bi,Oi) n To include the parameter priors in the EM algorithm, distributions for the different parameters are chosen, then the log-likelihood is differentiated as usual to find the updates to the parameters (McMichael, 1995). The calculations are simplified if the priors on all the parameters are assumed to be independent, p( {/-li, 7rt , ai, bt , Od) = It p(/-ldp( 7ri)P( ai)p(bdp( Od? The exact form of the prior distributions varies for different parameters, both to capture different behavior and for ease of implementation. For the element means (/-li), a flat distribution over the data is used, specifying that the means should be among the data points. For the element weights, a multinomial Dirichlet prior can be used, p(7ri la) = n::~1J n~l 7rf. When the hyperparameter a > 0, the algorithm favours weights around 1/NI, and when -1 < a < 0, weights close to 0 or 1. 3 The expected value of ai (written as ai) can be encoded using a truncated inverse exponential prior (McMichael, 1995), setting p(ailai) = Kexp(-at/(2ai)), where K is a normalizing factor. 4 The prior for bi has the same form. Priors for Ot were not used, but could be useful to capture the visibility constraint. Given these distributions, the updates to the parameters become I:n h in + a /-li I: nI:j h jn + a o'i I:n h in(/, + a; 2 I: nh in bt = I:n h in (X n3 - /-ln3) I:n h in 2 - + bt . The update for /-li is the same as before, the prior having no effect. The update for at and bt forces them to be near ai and bi , and the update for 7ri is affected by the hyperparameter a. The priors on ai and bi had noticeable effects on the models obtained. Figure 4 shows the results from two fits, starting from identical initial conditions. By adjusting the size of the prior, the algorithm can be guided into finding different sized clusters. Large values of the prior are shown here to demonstrate its effect. 3In this paper we make little use of the Q priors, but introducing separate Q;'S for each object could be a useful next step for scenes with varying object sizes. 4To deal with the case when a, = 0, the prior is truncated, setting p(a;!a,) = 0 when a, < Perit . 950 .. M M Williamson. R. Murray-Smith and V. Hansen ~ .' . " ..... \....t , ~ 1, 4' ~. ., . .~ ? f f.1J ~ ; . ~ ~, \....t - ~ ) ~ 6JiiZC3!' . .. ~.' ., ., ~ ? ~ ~'. . ~ f , f:> ~ ; . ... . @ ,. .. ) ., .. '.' Figure 4: Example of the action of the priors on ai and bi . The photograph shows a visual image of the scene: a box in front of a wall, and the priors were chosen to prefer a distribution matching the wall. The two left hand graphs show the top and side view of the scene clustered without priors, while the two right hand graphs use priors on ai and bi . The priors give a preference for large values of ai and bi , so biasing the optimization to find a mixture component matching the whole wall as opposed to just the top of it. 4 Classification and diagnosis FEATURES SENSOR MODEL FITIING DATA EM ALGORITHM PRIOR HIGHER LEVEL MOVE COMMAND PROCESSING FOR ROBOT INFORMATION Figure 5: Schematic of system This section describes how higher-level processing can be used to not only interpret the clusters fitted by the EM algorithm, but also affect the model-fitting using prior information. The processes of model-fitting and analysis are thus coupled, and not sequential. The results of the model fitting are primarily processed to steer the robot. Once the cluster has been recognized as a boxlwaIVetc., the location and orientation are used to calculate a move command. To perform the object-recognition, we used a simple classifier on a feature vector extracted from the clustered data. The labels used were specific to docking, and commonly clustered objects - boxes, walls, thin vertical lines. but also included labels for clustering errors (like those shown in Figure 3). The features used were the values of the parameters ai, bi , giving the size of the clusters, but also measures of the visibility of the clusters, and the skewness of the within-cluster data. The classification used simple models of the probability distributions of the features fi' given the objects OJ (i.e. P(hIOj)), using a set of training data. In addition to moving the robot, the classifier can modify the behavior of the model fitting algorithm. If a poor clustering solution is found, EM can be re-run with slightly different initial conditions. If the probable locations or sizes of objects are known from previous scans, or indeed from other sensors, then these can constrain the clustering through priors, or provide initial means. Robot Docking Using Mixtures ofGaussians 951 5 Summary This paper shows that the Mixture of Gaussians architecture combined with EM optimization and the use of parameter priors can be used to segment and analyze real data from the 3D range-finder of a mobile robot. The approach was successfully used to guide a mobile robot towards a docking object, using only its range-finder for perception. For the learning community this provides more than an example of the application of a probabilistic model to a real task. We have shown how the usual Mixture of Gaussians model can be parameterized to include expectations about the environment in a way which can be readily extended. We have included prior knowledge at three different levels: 1. The use of problem-specific parameterization of the covariance matrix to find expected patterns (e.g. planes at particular angles). 2. The use of problem-specific parameter priors to automatically rule-out unlikely objects at the lowest level of perception. 3. The results of the clustering process were post-processed by higher-level classification algorithms which interpreted the parameters of the mixture components, diagnosed typical misclassification, provided new priors for future perception, and gave the robot control system new targets. It is expected that the basic approach can be fruitfully applied to other sensors, to problems which track dynamically changing scenes, or to problems which require relationships between objects in the scene to be accounted for and interpreted. A problem common to all modeling approaches is that it is not trivial to determine the number and types of clusters needed to represent a given scene. Recent work with Markov-Chain Monte-Carlo approaches has been successfully applied to mixtures of Gaussians (Richardson and Green, 1997), allowing a Bayesian solution to this problem, which could provide control systems with even richer probabilistic information (a series of models conditioned on number of clusters). Acknowledgements All authors were employed by Daimler-Benz AG during stages of the work. R. MurraySmith gratefully acknowledges the support of Marie Curie TMR grant FMBICT96 I 369. References Arman, F. and Aggarwal, J. K. (1993). Model-based object recognition in dense-range images-a review. ACM Computing Surveys, 25 (1), 5-43. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39, 1-38. Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis. New York, Wiley. McMichael, D. W. (1995). Bayesian growing and pruning strategies for MAP-optimal estimation of gaussian mixture models. In 4th lEE International Con! on Artificial Neural Networks, pp. 364-368. PIAP (1995) . PIAP impact report on TRC lidar performance. Technical Report 1, Industrial Research Institute for Automation and Measure ments, 02-486 Warszawa, AI. Jerozolimskie 202, Poland. Richardson, S. and Green, P. J. (1997). On Bayesian anaysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society B, 50 (4), 700-792. Titterington, D., Smith, A., and Makov, U. (1985). Statistical Analysis of Finite Mixture Distributions. Chichester, John Wiley & Sons. 

1538 |@word duda:2 pulse:1 covariance:8 pick:1 ld:1 initial:3 series:2 com:1 od:2 trc:1 written:2 readily:2 john:1 visible:2 visibility:5 plot:1 update:10 parameterization:6 plane:14 ith:1 smith:5 oblique:1 provides:2 location:3 preference:1 mathematical:1 shiny:1 become:1 fitting:5 ray:1 indeed:1 expected:4 behavior:2 growing:1 inspired:1 detects:1 automatically:1 little:1 equipped:1 inappropriate:1 considering:1 becomes:1 provided:2 lowest:1 cm:1 interpreted:2 skewness:1 titterington:2 finding:2 ag:1 every:2 classifier:3 control:3 normally:1 grant:1 before:1 local:5 modify:1 despite:1 accumulates:1 id:1 dynamically:1 specifying:1 co:4 ease:2 range:12 perpendicular:2 bi:10 responsible:1 procedure:1 reject:1 projection:1 matching:2 undesirable:1 close:1 impossible:1 writing:1 influence:1 map:1 center:1 attention:1 starting:2 survey:1 immediately:1 rule:2 reparameterization:1 jli:4 target:1 tan:1 exact:1 us:1 origin:2 element:4 recognition:3 located:1 coso:1 sparsely:1 module:1 solved:1 initializing:1 capture:2 calculate:1 intuition:1 roderick:1 dempster:2 environment:2 ui:1 segment:2 various:1 represented:1 laser:2 describe:1 monte:1 detected:1 artificial:1 encoded:1 widely:1 richer:1 denser:1 say:1 richardson:2 laird:1 advantage:2 eigenvalue:3 relevant:1 combining:1 convergence:1 cluster:13 optimum:1 object:24 odd:1 noticeable:1 expectationmaximization:1 direction:4 guided:1 correct:1 bin:1 require:1 clustered:6 wall:12 probable:1 im:1 around:1 normal:1 exp:3 recalculated:1 matthew:1 pointing:1 early:1 estimation:1 label:2 hansen:5 successfully:2 mit:1 brought:1 sensor:11 gaussian:12 rna:1 modified:1 rather:2 volker:1 mobile:4 varying:1 command:2 modelling:1 likelihood:3 industrial:1 summarizing:1 sense:2 dependent:1 bt:4 unlikely:1 spurious:1 i1:2 germany:1 ldp:1 among:1 flexible:2 orientation:4 classification:4 constrained:1 once:1 f3:2 having:1 identical:1 thin:2 future:1 report:2 primarily:1 randomly:1 hamper:1 suit:1 lf3:1 chichester:1 mixture:19 light:1 behind:1 chain:1 accurate:1 ln3:1 orthogonal:1 incomplete:1 logarithm:1 initialized:1 re:2 plotted:2 fitted:1 soft:1 modeling:4 steer:1 maximization:1 introducing:1 f3t:1 fruitfully:1 front:2 reported:1 varies:1 combined:2 density:1 international:1 probabilistic:4 off:1 lee:1 together:1 tmr:1 reflect:1 opposed:2 leading:1 return:1 li:9 makov:1 automation:1 view:4 lab:1 responsibility:1 analyze:1 bayes:1 curie:1 square:1 ni:2 oi:3 il:3 variance:1 characteristic:4 accuracy:1 identify:1 ofthe:1 bayesian:3 none:1 carlo:1 tended:2 pp:1 naturally:1 associated:1 xn1:1 junk:1 con:1 adjusting:1 knowledge:5 segmentation:1 positioned:1 higher:7 reflected:1 response:1 improved:1 formulation:3 box:6 diagnosed:1 generality:1 just:1 stage:2 flight:1 hand:4 ei:1 o:3 usa:1 effect:4 ril:2 reformulating:1 iteratively:1 illustrated:2 deal:1 sin:4 during:2 complete:1 demonstrate:1 invisible:2 tn:3 hin:3 reasoning:1 image:4 fi:1 common:2 specialized:1 multinomial:1 flid:2 nh:1 interpret:2 measurement:2 cambridge:1 ai:21 ofgaussians:2 similarly:1 gratefully:1 had:1 moving:1 robot:18 surface:1 etc:2 recent:1 occasionally:1 yi:1 minimum:3 employed:1 recognized:1 converge:2 determine:1 ii:2 xn2:5 multiple:1 aggarwal:2 technical:2 faster:1 adapt:1 calculation:1 hart:2 post:1 finder:5 ellipsis:1 calculates:1 schematic:1 impact:1 basic:2 hioj:1 expectation:2 mog:6 represent:1 beam:1 addition:2 htn:2 ot:1 probably:1 near:1 ideal:1 enough:1 affect:1 fit:4 zi:1 misclassifications:1 architecture:1 gave:1 interprets:2 rod:1 favour:1 motivated:1 returned:2 york:1 action:1 useful:4 eigenvectors:2 maybe:1 dark:1 processed:2 daimler:1 reduced:1 track:1 li2:1 diagnosis:1 hyperparameter:2 affected:1 drawn:1 changing:2 marie:1 isin:1 graph:5 run:1 angle:3 inverse:1 uncertainty:1 powerful:1 parameterized:1 extends:1 almost:1 reasonable:1 ob:1 prefer:1 fl:2 guaranteed:3 kexp:1 constraint:3 constrain:2 scene:13 ri:5 flat:1 n3:1 aspect:1 poor:4 describes:1 slightly:1 em:15 son:1 making:1 modification:1 leg:1 outlier:1 taken:1 ln:8 benz:1 needed:1 end:1 b_:1 gaussians:14 experimentation:1 murraysmith:1 differentiated:1 jn:1 top:3 clustering:7 include:2 dirichlet:1 calculating:1 giving:1 reflectance:1 practicality:1 murray:4 especially:3 society:2 move:4 strategy:1 rt:1 usual:2 distance:1 separate:2 berlin:1 collected:1 extent:1 trivial:1 denmark:1 modeled:2 relationship:1 ellipsoid:2 providing:1 ratio:1 implementation:2 unknown:1 perform:1 allowing:3 vertical:4 markov:1 finite:1 incorrectly:1 reparameterized:1 truncated:2 extended:1 incorporated:2 community:1 beyond:1 proceeds:1 perception:6 pattern:4 biasing:1 recast:1 rf:1 oj:1 green:2 royal:2 misclassification:1 force:2 ixn:2 dtu:1 axis:1 acknowledges:1 coupled:1 docking:8 prior:35 review:2 acknowledgement:1 poland:1 fully:1 interesting:2 allocation:1 xnl:6 rni:1 rubin:1 bdp:1 summary:1 accounted:1 placed:1 free:1 guide:2 side:4 institute:1 sparse:6 distributed:1 calculated:1 xn:15 author:2 forward:1 collection:1 commonly:1 simplified:1 outlying:1 pruning:1 dealing:2 imm:1 symmetrical:1 assumed:1 xi:2 table:1 nature:1 learn:1 williamson:4 fli:6 dense:2 whole:2 je:1 wiley:2 sub:7 position:2 exponential:1 xl:1 weighting:1 specific:4 showing:1 dk:1 alt:1 ments:1 concern:1 evidence:1 incorporating:1 normalizing:1 sequential:1 te:1 conditioned:1 cartesian:4 sparseness:2 photograph:1 likely:2 visual:1 expressed:1 applies:1 corresponds:1 extracted:1 ma:1 acm:1 sized:1 towards:1 lidar:9 included:5 determined:1 typical:2 la:1 support:1 scan:7 incorporate:1

Fisher Scoring and a Mixture of Modes Approach for Approximate Inference and Learning in Nonlinear State Space Models Thomas Briegel and Volker Tresp Siemens AG, Corporate Technology Dept. Information and Communications Otto-Hahn-Ring 6,81730 Munich, Germany {Thomas.Briegel, Volker.Tresp} @mchp.siemens.de Abstract We present Monte-Carlo generalized EM equations for learning in nonlinear state space models. The difficulties lie in the Monte-Carlo E-step which consists of sampling from the posterior distribution of the hidden variables given the observations. The new idea presented in this paper is to generate samples from a Gaussian approximation to the true posterior from which it is easy to obtain independent samples. The parameters of the Gaussian approximation are either derived from the extended Kalman filter or the Fisher scoring algorithm. In case the posterior density is multimodal we propose to approximate the posterior by a sum of Gaussians (mixture of modes approach). We show that sampling from the approximate posterior densities obtained by the above algorithms leads to better models than using point estimates for the hidden states. In our experiment, the Fisher scoring algorithm obtained a better approximation of the posterior mode than the EKF. For a multimodal distribution, the mixture of modes approach gave superior results. 1 INTRODUCTION Nonlinear state space models (NSSM) are a general framework for representing nonlinear time series. In particular, any NARMAX model (nonlinear auto-regressive moving average model with external inputs) can be translated into an equivalent NSSM. Mathematically, a NSSM is described by the system equation (1) where Xt denotes a hidden state variable, (t denotes zero-mean uncorrelated Gaussian noise with covariance Qt and Ut is an exogenous (deterministic) input vector. The time-series measurements Yt are related to the unobserved hidden states Xt through the observation equation (2) where Vt is uncorrelated Gaussian noise with covariance lit. In the following we assume that the nonlinear mappings fw{.) and gv{.) are neural networks with weight vectors w and v, respectively. The initial state Xo is assumed to be Gaussian distributed with mean ao and covariance Qo. All variables are in general multidimensional. The two challenges 404 T Briegel and V. Tresp in NSSMs are the interrelated tasks of inference and learning. In inference we try to estimate the states of unknown variables Xs given some measurements Yb ... , Yt (typically the states of past (s < t), present (s = t) or future (s > t) values of Xt) and in learning we want to adapt some unknown parameters in the model (i.e. neural network weight vectors wand v) given a set of measurements. 1 In the special case of linear state space models with Gaussian noise, efficient algorithms for inference and maximum likelihood learning exist. The latter can be implemented using EM update equations in which the E-step is implemented using forward-backward Kalman filtering (Shumway & Stoffer, 1982). If the system is nonlinear, however, the problem of inference and learning leads to complex integrals which are usually considered intractable (Anderson & Moore, 1979). A useful approximation is presented in section 3 where we show how the learning equations for NSSMs can be implemented using two steps which are repeated until convergence. First in the (Monte-Carlo) E-step, random samples are generated from the unknown variables (e.g. the hidden variables Xt) given the measurements. In the second step (a generalized M-step) those samples are treated as real data and are used to adapt Iw (.) and gv (.) using some version of the backpropagation algorithm. The problem lies in the first step, since it is difficult to generate independent samples from a general multidimensional distribution. Since it is difficult to generate samples from the proper distribution the next best thing might be to generate samples using an approximation to the proper distribution which is the idea pursued in this paper. The first thing which might come to mind is to approximate the posterior distribution of the hidden variables by a multidimensional Gaussian distribution since generating samples from such a distribution is simple. In the first approach we use the extended Kalman filter and smoother to obtain mode and covariance ofthis Gaussian. 2 Alternatively, we estimate the mode and the covariance of the posterior distribution using an efficient implementation of Fisher scoring derived by Fahrmeir and Kaufmann (1991) and use those as parameters of the Gaussian. In some cases the approximation of the posterior mode by a single Gaussian might be considered too crude. Therefore, as a third solution, we approximate the posterior distribution by a sum of Gaussians (mixture of modes approach). Modes and covariances of those Gaussians are obtained using the Fisher scoring algorithm. The weights of the Gaussians are derived from the likelihood of the observed data given the individual Gaussian. In the following section we derive the gradient of the log-likelihood with respect to the weights in I w (.) and gv (.). In section 3, we show that the network weights can be updated using a Monte-Carlo E-step and a generalized M-step. Furthermore, we derive the different Gaussian approximations to the posterior distribution and introduce the mixture of modes approach. In section 4 we validate our algorithms using a standard nonlinear stochastic time-series model. In section 5 we present conclusions. 2 THE GRADIENTS FOR NONLINEAR STATE SPACE MODELS Given our assumptions we can write the joint probability of the complete data for t 1, ... , T as 3 p(Xr, Yr, Ur) = p(Ur) p(xo) r r t=1 t=l II p(Xt IXt-l. ut} II p(Yt IXt, ut} (3) 1 In this paper we focus on the case s :::; t (smoothing and offline learning, respectively). 2Independently from our work, a single Gaussian approximation to the E-step using the EKFS has been proposed by Ghahramani & Roweis (1998) for the special case of a RBF network. They show that one obtains a closed form M-step when just adapting the linear parameters by holding the nonlinear parameters fixed. Although avoiding sampling, the computational load of their M-step seems to be significant. 3In the following, each probability density is conditioned on the current model. For notational convenience, we do not indicate this fact explicitly. Fisher Scoring and Mixture of Modes for Inference and Learning in NSSM 405 = where UT {Ul,"" UT} is a set of known inputs which means that p( UT) is irrelevant in the following. Since only YT = {Yl,"" YT} and UT are observed, the log-likelihood of the model is log L = log J p(XT, YTIUT)p(UT) dXT ex log J p(XT, YTIUT ) dXT (4) with XT = {xo, ... , XT}. By inserting the Gaussian noise assumptions we obtain the gradients of the log-likelihood with respect to the neural network weight vectors wand v, respectively (Tresp & Hofmann, 1995) T 810gL 8w ex 810gL 8v ex ~J!8fw(Xt-l,Ut)( ) I L.J 8w Xt -fw(Xt-llUd p(Xt,Xt-l YT,UT)dxt-ldxt t=1 ~J8gv(Xt'Ut)( ) L.J 8v Yt-gv(Xt,ut} p(Xt!YT,UT)dxt. (5) t=1 3 3.1 APPROXIMATIONS TO THE E-STEP Monte-Carlo Generalized EM Learning The integrals in the previous equations can be solved using Monte-Carlo integration which leads to the following learning algorithm. xr 1. Generate S samples {xo, ... , };=1 from P(XT \YT , UT ) assuming the current model is correct (Monte-Carlo E-Step). 2. Treat those samples as real data and update w new = wold void + 1]&I~~L with stepsize 1] and T aIogL 8w aIogL 8 v ex S t=1 s=1 ex = 5 ~2:2:8fw(Xt-l,udl T + 1] &~! Land v new 8w (x:-fW(X:_l,ud) (6) Xt-l=t:_ 1 5 ~~~8gv (Xt,Ut)1 ( _ ("S )) SL.JL.J 8 Yt gv Xt,Ut t=1 s=1 V Xt=i:; (7) (generalized M-step). Go back to step one. The second step is simply a stochastic gradient step. The computational difficulties lie in the first step. Methods which produce samples from multivariate distributions such as Gibbs sampling and other Markov chain Monte-Carlo methods have (at least) two problems. First, the sampling process has to "forget" its initial condition which means that the first samples have to be discarded and there are no simple analytical tools available to determine how many samples must be discarded . Secondly, subsequent samples are highly correlated which means that many samples have to be generated before a sufficient amount of independent samples is available. Since it is so difficult to sample from the correct posterior distribution p(XT !YT, UT) the idea in this paper is to generate samples from an approximate distribution from which it is easy to draw samples. In the next sections we present approximations using a multivariate Gaussian and a mixture of Gaussians. 3.2 Approximate Mode Estimation Using the Extended Kalman Filter Whereas the Kalman filter is an optimal state estimator for linear state space models the extended Kalman filter is a suboptimal state estimator for NSSMs based on locallinearizations of the nonlinearities. 4 The extended Kalman filter and smoother (EKFS) algorithm is 4 Note that we do not include the parameters in the NSSM as additional states to be estimated as done by other authors, e.g. Puskorius & Feldkamp (1994). T. Briegel and V. Tresp 406 a forward-backward algorithm and can be derived as an approximation to posterior mode estimation for Gaussian error sequences (Sage & Melsa, 1971). Its application to our framework amounts to approximating x~ode ~ x~KFS where x~KFS is the smoothed estimate of Xt obtained from forward-backward extended Kalman filtering over the set of measurements YT and x~ode is the mode of the posterior distribution p( Xt IYT , UT). We use x~KFS as the center of the approximating Gaussian. The EKFS also provides an estimate of the error covariance of the state vector at each time step t which can be used to form the covariance matrix of the approximating Gaussian. The EKFS equations can be found in Anderson & Moore (1979). To generate samples we recursively apply the following algorithm. Given xLI is a sample from the Gaussian approximation of p(xt-IIYT, UT) at time t - 1 draw from p(XtIXt-1 = X:_I' YT, UT). The last conditional density is Gaussian a sample with mean and covariance calculated from the EKFS approximation and the lag-one error covariances derived in Shumway & Stoffer (1982), respectively. xt 3.3 Exact Mode Estimation Using the Fisher Scoring Algorithm If the system is highly nonlinear, however, the EKFS can perform badly in finding the posterior mode due to the fact that it uses a first order Taylor series expansion of the nonlinearities fw (.) and gv(.) (for an illustration, see Figure 1). A u!:>cful- and computationally tractable - alternative to the EKFS is to compute the "exact" posterior mode by maximizing logp(XT IYrr, UT) with respect to XT. A suitable way to determine a stationary point of the log posterior, or equivalently, of P(XT, YTIUT) (derived from (3) by dropping P(UT)) FS old . to app Iy rlS D' h . W'It h the current estImate . XT . IS er scormg. ' we get a better estImate X~s,new = X;S,old + 1] J for the unknown state sequence XT where J is the solution of (8) with the score function s(XT S(XT) ) = alogp(::~YTIUT) and the expected information matrix = E[_a210~1X;{fIUT'J.5 T T By extending the arguments given in Fahrmeir & Kaufmann (1991) to nonlinear state space models it turns out that solving equation (8) e.g. to compute the inverse of the expected information matrix - can be performed by Cholesky decomposition in one forward and backward pass. 6 The forward-backward steps can be implemented as a fast EKFS-Iike algorithm which has to be iterated to obtain the maximum posterior estimates x~ode x;S (see Appendix). Figure 1 shows the estimate obtained by the Fisher scoring procedure for a bimodal posterior density. Fisher scoring is successful in finding the "exact" mode, the EKFS algorithm is not. Samples of the approximating Gaussian are generated in the same way as in the last section. = 3.4 The Mixture of Modes Approach The previous two approaches to posterior mode smoothing can be viewed as single Gaussian approximations of the mode of p(XTIYT, UT). In some cases the approximation of the posterior density by a single Gaussian might be considered too crude, in particular if the posterior distribution is multimodal. In this section we approximate the posterior by a weighted sum of m Gaussians p(XT IYT, UT) ~ :E~I okp(XT Ik) where p(XT Ik) is the k-th Gaussian. If the individual Gaussians model the different modes we are able to model multimodal posterior distributions accurately. The approximations of the individual modes are local maxima of the Fisher scoring algorithm which are f~)Und by starting the algorithm using different initial conditions. Given the different Gaussians, the optimal weighting facp(YTlk)p(k)jp(YT) where p(YTlk) jp(YTIXT)P(XTlk)dXT is the tors are ok = = SNote that the difference between the Fisher scoring and the Gauss-Newton update is that in the fonner we take the expectation of the information matrix. 6The expected information matrix is a positive definite blOCk-tridiagonal matrix. Fisher Scoring and Mixture of Modes for Inference and Learning in NSSM 407 likelihood of the data given mode k. If we approximate that integral by inserting the Fisher scoring solutions x;S,k for each time step t and linearize the nonlinearity gv (.) about the Fisher scoring solutions, we obtain a closed form solution for computing the ok (see Appendix). The resulting estimator is a weighted sum of the m single Fisher scoring estimates x~M L::=l ok x;s,k. The mixture of modes algorithm can be found in the Appendix. For the learning task samples of the mixture of Gaussians are based on samples of each of the m single Gaussians which are obtained the same way as in subsection 3.2. = 4 EXPERIMENTAL RESULTS In the first experiment we want to test how well the different approaches can approximate the posterior distribution of a nonlinear time series (inference). As a time-series model we chose 1 2 = 0.5 Xt-l + 25 1 +Xt-l 2 + 8eas ( 1.2(t -I},) g(xt} = 20Xt, (9) xt_ 1 the covariances Qt = 10, lit = 1 and initial conditions ao = 0 and Qo = 5 which is f(Xt-l, Ut} considered a hard inference problem (Kitagawa, 1987). At each time step we calculate the expected value of the hidden variables Xt, t = 1, ... , 400 based on a set of measurements Y400 = {Yl, ... , Y400} (which is the optimal estimator in the mean squared sense) and based on the different approximations presented in the last section. Note that for the single mode approximation, x~ode is the best estimate of Xt based on the approximating Gaussian. For the mixture of modes approach, the best estimate is L:~l ok x;S,k where x;S,k is the mode of the k-th Gaussian in the dimension of Xt. Figure 2 (left) shows the mean squared error (MSE) of the smoothed estimates using the different approaches. The Fisher scoring (FS) algorithm is significantly better than the EKFS approach. In this experiment, the mixture of modes (MM) approach is significantly better than both the EKFS and Fisher scoring. The reason is that the posterior probability is multimodal as shown in Figure 1. In the second experiment we used the same time-series model and trained a neural network to approximate fw (.) where all covariances were assumed to be fixed and known. For adaptation we used the learning rules of section 3 using the various approximations to the posterior distribution of XT . Figure 2 (right) shows the results. The experiments show that truly sampling from the approximating Gaussians gives significantly better results than using the expected value as a point estimate. Furthermore, using the mixture of modes approach in conjunction with sampling gave significantly better results than the approximations using a single Gaussian . When used for inference, the network trained using the mixture of modes approach was not significantly worse than the true model (5% significance level, based on 20 experiments). 5 CONCLUSIONS In our paper we presented novel approaches for inference and learning in NSSMs. The application of Fisher scoring and the mixture of modes approach to nonlinear models as presented in our paper is new. Also the idea of sampling from an approximation to the posterior distribution of the hidden variables is presented here for the first time. Our results indicate that the Fisher scoring algorithm gives better estimates of the expected value of the hidden variable than the EKFS based approximations. Note that the Fisher scoring algorithm is more complex in requiring typically 5 forward-backward passes instead of only one forward-backward pass for the EKFS approach. Our experiments also showed that if the posterior distribution is multi modal, the mixture of modes approach gives significantly better estimates if compared to the approaches based on a single Gaussian approximation. Our learning experiments show that it is important to sample from the approximate distributions and that it is not sufficient to simply substitute point estimates. Based on the T. Briegel and V. Tresp 408 0.4,---- - - - - - - - - - - - - - - ---, 0 . 2 , - - -- -- - - - - - - - - - - - - - - - . 0 . 18 0.16 , , , , 0.14 0 . 35 .. 0.3 1t 0.25 .8 0.2 1.0.15 0.05 , ,,- .... , o - 'o---~~~-=-=--~o--~--~~-~ t =-= 2 9 5 Figure 1: Approximations to the posterior distribution p( x t iY400, U400) for t = 294 and t = 295. The continuous line shows the posterior distribution based on Gibbs sampling using 1000 samples and can be considered a close approximation to the true posterior. The EKFS approximation (dotted) does not converge to a mode. The Fisher scoring solution (dashdotted) finds the largest mode. The mixture of modes approach with 50 modes (dashed) correctly finds the two modes. sampling approach it is also possible to estimate hyperparameters (e.g. the covariance matrices) which was not done in this paper. The approaches can also be extended towards online learning and estimation in various ways (e.g. missing data problems). Appendix: Mixture of Modes Algorithm The mixture of modes estimate x~M is derived as a weighted sum of k = 1, . .. ,m individual Fisher scoring (mode) estimates x;S ,k. For m = 1 we obtain the Fisher scoring algorithm of subsection 3.3. First, one performs the set of forward recursions (t = 1, ... , T) for each single mode estimator k. ",k ""'tit-I = Btk p(~FS , k)"'k tXt_I ",k ""'t-llt-I ""'t-Ilt-I pT( , FS ,k)+Q tXt_I t pT('FS ,k)(",k )-1 tXt_I ""'tit-I (10) (11 ) (12) ,FS ,k) St ( x t k 'Yt with the initialization :E~lo = Qo, 'Yo ing recursions (t = T , .. . , 1) k T + Bt k (13) 'Yt-l = So (X~S , k). Then, one performs the set of backward smooth:E kt - l l t - l (Dkt-I )-1 - Bk:Ek Bkt T t tit-I :E~_1 (D~_d-l 0:_1 ( Dtk 1)-1 'Yt-l k = + B;:E~ B; T + Bkok t t (14) (15) (16) = 8fw(Xt_l ' U') I G ( Z ) -- &YdXt,Utll ( ) - &logp(Xr,YrIUT) I a d &Xt_l Xt_l=Z' &Xt Xt=Z, St Z &Xt Xt=Z n initialization o} = :E}'Y}. The k individual mode estimates x;S ,k are obtained by iterative application of the update rule X~S , k := '7 Ok + X~S , k with stepsize '7 where X~S , k = {X~S , k, .. . ,X~S,k} with Pt(z) t and Ok = {o~ , ... , o} }. After convergence we obtain the mixture of modes estimate as the weighted . h'ttng coe ffi Clents . k := 0'0k were h k( T -, 1 "' , 0) sum X,MM = ",m 6k=1 0' k~FS x t ' k WIt. h welg 0' O't t = t are computed recursively starting with a uniform prior O'} = (N(xlp,:E) stands for a Gaussian with center p and covariance :E evaluated at x): .k k O't = O'~+IN(Ytlgv(xfs,k, ur), nn (17) (18) Fisher Scoring and Mixture of Modes for Inference and Learning in NSSM 0.8 e 0.7 8 o.s 7 f: ';"4 ~3 2 409 ~ 0.5 ~O.4 ~ 0.3 0.2 0 . "1 0 0 Figure 2: Left (inference): The heights of the bars indicate the mean squared error between the true Xt (which we know since we simulated the system) and the estimates using the various approximations. The error bars show the standard deviation derived from 20 repetitions of the experiment. Based on the paired t-test, Fisher scoring is significantly better than the EKFS and all mixture of modes approaches are significantly better than both EKFS and Fisher scoring based on a 1% rejection region. The mixture of modes approximation with 50 modes (MM 50) is significantly better than the approximation using 20 modes. The improvement of the approximation using 20 modes (MM 20) is not significantly better than the approximation with 10 (MM 10) modes using a 5% rejection region. Right (learning): The heights of the bars indicate the mean squared error between the true fw (.) (which is known) and the approximations using a multi-layer perceptron with 3 hidden units and T = 200. Shown are results using the EKFS approximation, (left) the Fisher scoring approximation (center) and the mixture of modes approximation (right). There are two bars for each experiment: The left bars show results where the expected value of x t calculated using the approximating Gaussians are used as (single) samples for the generalized M-step - in other words - we use a point estimate for Xt. Using the point estimates, the results of all three approximations are not significantly different based on a 5% significance level. The right bars shows the result where S = 50 samples are generated for approximating the gradient using the Gaussian approximations. The results using sampling are all significantly better than the results using point estimates (l % significance level). The sampling approach using the mixture of modes approximation is significantly better than the other two sampling-based approaches (l % significance level). If compared to the inference results of the experiments shown on the left, we achieved a mean squared error of 6.02 for the mixture of modes approach with 10 modes which is not significantly worse than the results the with the true model of 5.87 (5% significance level). References Anderson, B. and Moore, J. (1979) Optimal Filtering, Prentice-Hall, New Jersey. Fahnneir, L. and Kaufmann, H. (1991) On Kalman Filtering. Posterior Mode Estimation and Fisher Scoring in Dynamic Exponential Family Regression, Metrika, 38, pp. 37-60. Ghahramani, Z. and Roweis, S. (1999) Learning Nonlinear Stochastic Dynamics using the Generalized EM ALgorithm, Advances in Neural Infonnation Processing Systems 11, eps. M. Keams, S. Solla, D. Cohn, MIT Press, Cambridge, MA. Kitagawa, G. (1987) Non-Gaussian State Space Modeling of Nonstationary Time Series (with Comments), JASA 82, pp. 1032-1063. Puskorius, G. and Feldkamp, L. (1994) NeurocontroL of Nonlinear Dynamical Systems with KaLman Filter Trained Recurrent Networks, IEEE Transactions on Neural Networks, 5:2, pp. 279-297. Sage, A. and Melsa, J. (1971) Estimation Theory with Applications to Communications and Control, McGraw-Hill, New York. Shumway, R. and Stoffer, D. (1982) Time Series Smoothing and Forecasting Using the EM Algorithm, Technical Report No. 27, Division of Statistics, UC Davis. Tresp, V. and Hofmann, R. (1995) Missing and Noisy Data in NonLinear Time-Series Prediction, Neural Networks for Signal Processing 5, IEEE Sig. Proc. Soc., pp. 1-10. 

1539 |@word dtk:1 version:1 seems:1 covariance:14 decomposition:1 recursively:2 initial:4 series:10 score:1 past:1 current:3 must:1 subsequent:1 hofmann:2 gv:8 update:4 stationary:1 pursued:1 yr:1 metrika:1 regressive:1 provides:1 height:2 welg:1 ik:2 consists:1 introduce:1 xt_l:3 expected:7 multi:2 feldkamp:2 ag:1 unobserved:1 finding:2 multidimensional:3 control:1 unit:1 before:1 positive:1 local:1 treat:1 might:4 chose:1 initialization:2 block:1 definite:1 backpropagation:1 xr:3 procedure:1 adapting:1 significantly:14 word:1 get:1 convenience:1 close:1 prentice:1 equivalent:1 deterministic:1 yt:17 center:3 maximizing:1 go:1 missing:2 starting:2 independently:1 wit:1 estimator:5 rule:2 updated:1 pt:3 exact:3 us:1 sig:1 observed:2 solved:1 calculate:1 region:2 solla:1 und:1 dynamic:2 trained:3 solving:1 xtixt:1 tit:3 division:1 translated:1 multimodal:5 joint:1 various:3 jersey:1 fast:1 dkt:1 monte:8 lag:1 otto:1 statistic:1 noisy:1 online:1 sequence:2 analytical:1 propose:1 adaptation:1 inserting:2 roweis:2 validate:1 convergence:2 extending:1 produce:1 generating:1 ring:1 derive:2 linearize:1 recurrent:1 qt:2 ex:5 soc:1 implemented:4 come:1 indicate:4 correct:2 filter:7 stochastic:3 ao:2 secondly:1 mathematically:1 kitagawa:2 mm:5 considered:5 hall:1 mapping:1 tor:1 estimation:6 proc:1 iw:1 infonnation:1 largest:1 repetition:1 tool:1 weighted:4 mit:1 gaussian:31 ekf:1 volker:2 conjunction:1 derived:8 focus:1 yo:1 notational:1 improvement:1 likelihood:6 sense:1 inference:14 nn:1 typically:2 bt:1 hidden:10 keams:1 germany:1 udl:1 smoothing:3 integration:1 special:2 uc:1 sampling:13 lit:2 rls:1 future:1 report:1 individual:5 highly:2 stoffer:3 mixture:27 truly:1 puskorius:2 chain:1 kt:1 integral:3 kfs:3 taylor:1 old:2 iiyt:1 modeling:1 logp:2 deviation:1 uniform:1 successful:1 tridiagonal:1 too:2 ixt:2 st:2 density:6 yl:2 iy:1 squared:5 worse:2 external:1 ek:1 nonlinearities:2 de:1 explicitly:1 performed:1 try:1 closed:2 exogenous:1 iike:1 kaufmann:3 xli:1 iterated:1 accurately:1 carlo:8 app:1 llt:1 pp:4 subsection:2 ut:25 ea:1 back:1 ok:6 modal:1 yb:1 done:2 wold:1 evaluated:1 anderson:3 furthermore:2 just:1 until:1 qo:3 cohn:1 nonlinear:17 mode:57 requiring:1 true:6 moore:3 davis:1 generalized:7 hill:1 complete:1 performs:2 novel:1 superior:1 jp:2 jl:1 eps:1 measurement:6 significant:1 cambridge:1 gibbs:2 nonlinearity:1 moving:1 okp:1 posterior:33 multivariate:2 showed:1 irrelevant:1 vt:1 scoring:28 additional:1 determine:2 converge:1 ud:1 dashed:1 ii:2 smoother:2 signal:1 corporate:1 ing:1 smooth:1 technical:1 adapt:2 paired:1 prediction:1 regression:1 expectation:1 bimodal:1 achieved:1 whereas:1 want:2 ode:4 void:1 pass:1 comment:1 briegel:5 thing:2 nonstationary:1 easy:2 gave:2 suboptimal:1 idea:4 ul:1 forecasting:1 f:7 york:1 useful:1 amount:2 generate:7 sl:1 exist:1 dotted:1 estimated:1 correctly:1 write:1 dropping:1 backward:8 sum:6 wand:2 inverse:1 ilt:1 family:1 draw:2 appendix:4 layer:1 alogp:1 badly:1 btk:1 argument:1 munich:1 em:5 ur:3 xo:4 computationally:1 equation:8 turn:1 ffi:1 mind:1 know:1 tractable:1 available:2 gaussians:12 apply:1 stepsize:2 alternative:1 thomas:2 substitute:1 denotes:2 include:1 newton:1 coe:1 bkt:1 ghahramani:2 hahn:1 approximating:8 xlp:1 gradient:5 simulated:1 reason:1 assuming:1 kalman:10 illustration:1 iyt:2 equivalently:1 difficult:3 holding:1 sage:2 implementation:1 proper:2 unknown:4 perform:1 observation:2 markov:1 discarded:2 extended:7 communication:2 smoothed:2 bk:1 able:1 bar:6 usually:1 dynamical:1 challenge:1 suitable:1 difficulty:2 treated:1 xtiyt:1 recursion:2 representing:1 technology:1 auto:1 tresp:7 prior:1 shumway:3 dxt:5 filtering:4 xtlk:1 jasa:1 sufficient:2 uncorrelated:2 land:1 lo:1 dashdotted:1 neurocontrol:1 gl:2 last:3 offline:1 perceptron:1 fahrmeir:2 mchp:1 distributed:1 calculated:2 dimension:1 stand:1 forward:8 author:1 transaction:1 approximate:12 obtains:1 mcgraw:1 assumed:2 alternatively:1 continuous:1 iterative:1 expansion:1 mse:1 complex:2 significance:5 noise:4 hyperparameters:1 repeated:1 xt_:1 exponential:1 lie:3 crude:2 third:1 weighting:1 load:1 xt:53 er:1 x:1 intractable:1 ofthis:1 ekfs:17 conditioned:1 rejection:2 forget:1 interrelated:1 simply:2 ma:1 conditional:1 viewed:1 rbf:1 towards:1 fisher:28 fw:9 hard:1 pas:2 gauss:1 experimental:1 siemens:2 cholesky:1 latter:1 dept:1 avoiding:1 correlated:1

81 WHAT SIZE NET GIVES VALID GENERALIZATION?* Eric B. Baum Department of Physics Princeton University Princeton NJ 08540 David Haussler Computer and Information Science University of California Santa Cruz, CA 95064 ABSTRACT We address the question of when a network can be expected to generalize from m random training examples chosen from some arbitrary probability distribution, assuming that future test examples are drawn from the same distribution. Among our results are the following bounds on appropriate sample vs. network size. Assume o < ? $ 1/8. We show that if m > O( ~log~) random examples can be loaded on a feedforward network of linear threshold functions with N nodes and W weights, so that at least a fraction 1 - t of the examples are correctly classified, then one has confidence approaching certainty that the network will correctly classify a fraction 1 - ? of future test examples drawn from the same distribution. Conversely, for fully-connected feedforward nets with one hidden layer, any learning algorithm using fewer than O( random training examples will, for some distributions of examples consistent with an appropriate weight choice, fail at least some fixed fraction of the time to find a weight choice that will correctly classify more than a 1 - ? fraction of the future test examples. '!') INTRODUCTION In the last few years, many diverse real-world problems have been attacked by back propagation. For example "expert systems" have been produced for mapping text to phonemes [sr87], for determining the secondary structure of proteins [qs88], and for playing backgammon [ts88]. In such problems, one starts with a training database, chooses (by making an educated guess) a network, and then uses back propagation to load as many of the training examples as possible onto the network. The hope is that the network so designed will generalize to predict correctly on future examples of the same problem. This hope is not always realized. * This paper will appear in the January 1989 issue of Neural Computation. For completeness, we reprint this full version here, with the kind permission of MIT Press. ? 1989, MIT Press 82 Baum and Haussler We address the question of when valid generalization can be expected. Given a training database of m examples, what size net should we attempt to load these on? We will assume that the examples are drawn from some fixed but arbitrary probability distribution, that the learner is given some accuracy parameter E, and that his goal is to produce with high probability a feedforward neural network that predicts correctly at least a fraction 1 - E of future examples drawn from the same distribution. These reasonable assumptions are suggested by the protocol proposed by Valiant for learning from examples [val84]. However, here we do not assume the existence of any "target function"; indeed the underlying process generating the examples may classify them in a stochastic manner, as in e.g. [dh73]. Our treatment of the problem of valid generalization will be quite general in that the results we give will hold for arbitrary learning algorithms and not just for back propagation. The results are based on the notion of capacity introduced by Cover [cov65] and developed by Vapnik and Chervonenkis [vc7l], [vap82]. Recent overviews of this theory are given in [dev88], [behw87b] and [poI84], from the various perspectives of pattern recognition, Valiant's computational learning theory, and pure probability theory, respectively. This theory generalizes the simpler counting arguments based on cardinality and entropy used in [behw87a] and [dswshhj87], in the latter case specifically to study the question of generalization in feedforward nets (see [vap82] or [behw87b]). The particular measures of capacity we use here are the maximum number of dichotomies that can be induced on m inputs, and the Vapnik-CheMlonenki. (Ve) Dimen.ion, defined below. We give upper and lower bounds on these measures for classes of networks obtained by varying the weights in a fixed feedforward architecture. These results show that the VC dimension is closely related to the number of weights in the architecture, in analogy with the number of coefficients or "degrees of freedom" in regression models. One particular result, of some interest independent of its implications for learning, is a construction of a near minimal size net architecture capable of implementing all dichotomies on a randomly chosen set of points on the n-hypercube with high probability. Applying these results, we address the question of when a network can be expected to generalize from m random training examples chosen from some arbitrary probability distribution, assuming that future test examples are drawn from the same distribution. Assume 0 < E < 1/8. We show that ifm ~ O(~log.r:) random examples can be loaded on a feedforward network of linear threshold functions with N nodes and W weights, so that at least a fraction 1 - j of the examples are correctly classified, then one has confidence approaching certainty that the network will correctly classify a fraction 1 - E of future test examples drawn from the same distribution. Conversely, for fully-connected feedforward nets with one hidden layer, any learning algorithm using fewer than O( ~) random training examples will, for some distributions of examples consistent with an appropriate weight choice, fail at least some fixed fraction of the time to find a weight choice that will correctly classify more than a 1 - E fraction of the future test examples. What Size Net Gives Valid Generalization? Ignoring the constant and logarithmic factors, these results suggest that the appropriate number of training examples is approximately the number of weights times the inversel of the accuracy parameter E. Thus, for example, if we desire an accuracy level of 90%, corresponding to E 0.1, we might guess that we would need about 10 times as many training examples as we have weights in the network. This is in fact the rule of thumb suggested by Widrow [wid87], and appears to work fairly well in practice. At the end of Section 3, we briefly discuss why learning algorithms that try to minimize the number of non-zero weights in the network [rum87] [hin87] may need fewer training examples. = DEFINITIONS We use In to denote the natural logarithm and log to denote the logarithm base 2. We define an ezample as a pair (i, a), i E ~n, a E {-I, +1}. We define a random sample as a sequence of examples drawn independently at random from some distribution D on ~n X {-1, +1}. Let I be a function from ~n into {-1, +1}. We define the error of I, with respect to D, as the probability a;/; I(i) for (i,a) a random example. Let F be a class of {-1, +l}-valued functions on ~n and let S be a set of m points in ~n . A dichotomy of S induced by I E F is a partition of S into two disjoint subsets S+ and S- such that I(i) +1 for i E S+ and I(i) -1 for i E S-. By .6. F (S) we denote the number of distinct dichotomies of S induced by functions I E F, and by .6.F(m) we denote the maximum of .6.F(S) over all S C ~n of cardinality m. We say S is shattered by F if .6.F(S) = 2151 , i.e. all dichotomies of S can be induced by functions in F. The Vapnik-CheMlonenkis (VC) dimension of F, denoted VCdim(F), is the cardinality of the largest S C ~n that is shattered by F, i.e. the largest m such that .6.F ( m) 2m ? = = = A feedforward net with input from ~n is a directed acyclic graph G with an ordered sequence ofn source nodes (called inputs) and one sink (called the output). Nodes of G that are not source nodes are called computation nodes, nodes that are neither source nor sink nodes are called hidden nodes. With each computation node n. there is associated a function" : ~inde't'ee(n,) ~ {-I, +1}, where indeg7'ee(n.) is the number of incoming edges for node n,. With the net itself there is associated a function I : ~n ~ {-I, +1} defined by composing the I,'s in the obvious way, assuming that component i of the input i is placed at the it" input node. A Jeedlorward architecture is a class of feedforward nets all of which share the same underlying graph. Given a graph G we define a feedforward architecture by a class of functions F, from ~'nde't'ee(n,) associating to each computation node n, 1 It should be noted that our bounds differ significantly from those given in [dev88] in that the latter exhibit a dependence on the inverse of e2 ? This is because we derive our results from Vapnik's theorem on the uniform relative deviation of frequencies from their probabilities ([vap82], see Appendix A3 of [behw87b]), giving sharper bounds as E approaches o. 83 84 Baum and Haussler to {-I, +1}. The resulting architecture consists of all feedforward nets obtained by choosing a particular function" from F, for each computation node ft,. We will identify an architecture with the class offunctions computed by the individual nets within the architecture when no confusion will arise. CONDITIONS SUFFICIENT FOR VALID GENERALIZATION Theorem 1: Let F be a feedforward architecture generated by an underlying graph G with N > 2 computation nodes and F, be the class of functions associated with computation node ft, of G, 1 < i < N. Let d = E~l VCdim(Fl). Then AF(m) < n~lAF,(m)::; (Nem/d)d for m > d, where e is the base of the natural logarithm. Proof: Assume G has n input nodes and that the computation nodes of G are ordered so that node receives inputs only from input nodes and from computation nodes nj, 1 < j ::; i - I . Let S be a set of m points in ~n. The dichotomy induced on S by the function in node nl can be chosen in at most AFI (m) ways. This choice determines the input to node nz for each of the m points in S. The dichotomy induced on these m inputs by the function in node nz can be chosen in at most AF:a(m) ways, etc. Any dichotomy of S induced by the whole network can be obtained by choosing dichotomies for each of the ni's in this manner, hence AF(m) < nf:l AF,(m). n, By a theorem of Sauer [sau72], whenever VCdim(F) = Ie < 00, AF(m) < (em/Ie)l for all m > Ie (see also [behw87b]). Let ~ = VCdim(Fi), 1 < i < N. Thus d Ef:l~. Then n~l AF,(m) < n~l(em/~)'" for m > d. Using the fact that E~l -ailogai < logN whenever a. > 0, 1 < i < N, and E~l ai = I, and setting ai ~/d, it is easily verified that n~l ~d. > (d/N)d. Hence n~l(em/di)d. < (Nem/d)d. = = Corollary 2: Let F be the class of all functions computed by feedforward nets defined on a fixed underlying graph G with E edges and N > 2 computation nodes, each of which computes a linear threshold function. Let W E + N (the total number of weights in the network, including one weight per edge and one threshold per computation node). Then AF(m) < (Nem/W)W for all m > Wand VCdim(F) < 2Wlog(eN). = Proof: The first inequality follows from directly from Theorem 1 using the fact that VCdim(F) = Ie + 1 when F is the class of all linear threshold functions on ~l (see e.g. [wd81]). For the second inequality, it is easily verified that for N > 2 and m 2Wlog(eN), (N em/W)W < 2m. Hence this is an upper bound on VCdim(F). = Using VC dimension bounds given in [wd81], related corollaries can be obtained for nets that use spherical and other types of polynomial threshold functions. These bounds can be used in the following. What Size Net Gives Valid Generalization? Theorem 3 [vapS2} (see [behw87b), Theorem A3.3): Let F be a class offunctions2 on ~n, 0 < l' < 1,0 < ?,6 < 1. Let S be a random sequence of m examples drawn independently according to the distribution D. The probability that there exists a function in F that disagrees with at most a fraction (1 - 1')? of the examples in S and yet has error greater than ? (w.r.t. D) is less than From Corollary 2 and Theorem 3, we get: Corollary 4: Given a fixed graph G with E edges and N linear threshold units (i.e. W = E + N weights), fixed 0 < ? < 1/2, and m random training examples, where 32W 1 32N m>-n-, ? ? if one can find a choice of weights so that at least a fraction 1- ?/2 of the m training examples are correctly loaded, then one has confidence at least 1 - Se- 1?5W that the net will correctly classify all but a fraction ? of future examples drawn from the same distribution. For m 64W I 64N > --;- n--;-, the confidence is at least 1 - Se-em/S2. Proof: Let l' = 1/2 and apply Theorem 3, using the bound on aF(m) given in Corollary 2. This shows that the probability that there exists a choice of the weights that defines a function with error greater than ? that is consistent with at least a fraction 1 - ?/2 of the training examples is at most When m = !ll!.ln!!K this is S(2e 3fN' E In!!K)W which is less than Se- 1. 5W for N -> e e' E ' 2 and ? < 1/2. When m > 84EW In 8~N, (2N em/W) W < e Em / S2 , so S(2N em/W) W e- Em / 16 < Se-em/S2. The constant 32 is undoubtably an overestimate. No serious attempt has been made to minimize it. Further, we do not know if the log term is unavoidable. Nevertheless, even without these terms, for nets with many weights this may represent a considerable number of examples. Such nets are common in cases where the complexity of the rule being learned is not known in advance, so a large architecture is chosen 2 We assume some measurability conditions on the class F. See [poI84], [behwS7b1 for details. 85 86 Baum and Haussler in order to increase the chances that the rule can be represented. To counteract the concomitant increase in the size of the training sample needed, one method that has been explored is the use of learning algorithms that try to use as little of the architecture as possible to load the examples, e.g. by setting as many weights to zero as possible, and by removing as many nodes as possible (a node can be removed if all its incoming weights are zero.) [rumS7] [hin87]. The following shows that the VC dimension of such a "reduced" architecture is not much larger than what one would get if one knew a priori what nodes and edges could be deleted. Corollary 5: Let F be the class of all functions computed by linear threshold feedforward nets defined on a fixed underlying graph G with N' > 2 computation nodes and E' ~ N' edges, such that at most E > 2 edges have non-zero weights and at most N ~ 2 nodes have at least one incoming edge with a non-zero weight. Let W = E + N. Then the conclusion of Corollary 4 holds for sample size 32W 32NE' l m>-n--f f Prool sketch: We can bound dF( m) by considering the number of ways the N nodes and E edges that remain can be chosen from among those in the initial network. A crude upper bound is (N')N (E')E. Applying Corollary 2 to the remaining network gives dF(m) ~ (N')N(E')E(Nem/W)w. This is at most (N E'em/W)w. The rest of the analysis is similar to that in Corollary 4. This iridicates that minimizing non-zero weights may be a fruitful approach. Similar approaches in other learning contexts are discussed in [hauSS] and [litSS]. CONDITIONS NECESSARY FOR VALID GENERALIZATION The following general theorem gives a lower bound on the number of examples needed for distribution-free learning, regardless of the algorithm used. Theorem 6 [ehkvS7] (see also [behw87b]): Let F be a class of {-I, +1}-valued functions on ~n. with VCdim(F) > 2. Let A be any learning algorithm that takes as input a sequence of {-I, +1}-labeled examples over ~n. and produces as output a function from ~n. into {-I, +1}. Then for any 0 < f ~ l/S, 0 < 0 ~ l~ and m 1- fl 1 VCdim(F) -1] 3 2e ' e n7' v < maz [ - there exists (1) a function I E F and (2) a distribution D on ~n X {-I, +1} for which Prob((E, a) : a f. I(E)) = 0, such that given a random sample of size m chosen according to D, with probability at least 0, A produces a function with error greater than e. What Size Net Gives Valid Generalization? This theorem can be used to obtain a lower bound on the number of examples needed to train a net, assuming that the examples are drawn from the worst-case distribution that is consistent with some function realizable on that net. We need only obtain lower bounds on the VC dimension of the associated architecture. In this section we will specialize by considering only fully-connected networks of linear threshold units that have only one hidden layer. Thus each hidden node will have an incoming edge from each input node and an outgoing edge to the output node, and no other edges will be present. In [b88] a slicing construction is given that shows that a one hidden layer net of threshold units with n inputs and 2j hidden units can shatter an arbitrary set of 2jn vectors in general position in ~". A corollary of this result is: Theorem 7: The class of one hidden layer linear threshold nets taking input from ~" with k hidden units has VC dimension at least 2L~Jn. Note that for large k and n, 2 L~ Jn is approximately equal to the total number W of weights in the network. A special case of considerable interest occurs when the domain is restricted to the hypercube: {+1,-1}". Lemma 6 of [lit88] shows that the class of Boolean? functions on {+1, _I}" represented by disjunctive normal form expressions with k terms, k < 0(2,,/2/Vn) , where each term is the conjunction of n/2 literals, has VC dimension at least kn/4. Since these functions can be represented on a linear threshold net with one hidden layer of k units, this provides a lower bound on the VC dimension of this architecture. We also can use the slicing construction of [b88] to give a lower bound approaching kn/2. The actual result is somewhat stronger in that it shows that for large n a randomly chosen set of approximately kn/2 vectors is shattered with high probability. Theorem 8: With probability approaching 1 exponentially in n, a set S of m < 2,,/3 vectors chosen randomly and uniformly from {+1, _I}" can be shattered by the one hidden layer architecture with 2rm/l(n(1 - 1~0,,))J1linear threshold units in its hidden layer. Prool,ketch: With probability approaching 1 exponentially in n no pair of vectors in S are negations of each other. Assume n > eto. Let l' = In(l- I~O,,)J. Divide S at random into m/1'1 disjoint subsets S1I ... , Srm/t'l each containing no more than l' vectors. We will describe a set T of ?1 vectors as Iliceable if the vectors in T are linearly independent and the subspace they span over the reals does not contain any ?l vector other than the vectors in T and their negations. In [od188] it is shown, for large n, that any random set of l' vectors has probability P 4(;)(~)" +0(( 110)") of not being sliceable. Thus the probability that some S. is not sliceable is 0(mn2(~)"), which is exponentially small for m < 2,,/3. Hence with probability approaching 1 exponentially in n, each S, is sliceable, 1 ~ i $ m/ 1'1. r = Consider any Boolean function I on S and let S: = {i E S, : f(i) r = +1}, 87 88 Baum and Haussler r 1 < i < m/7'1. If Si is sliceable and no pair of vectors in S are negations of each other then we may pass a plane through the points in that doesn't contain any other points in S. Shifting this plane parallel to itself slightly we can construct two half spaces whose intersection forms a slice of~" containing and no other points in S. Using threshold units at the hidden layer recognizing these two half spaces, with weights to the output unit +1 and -1 appropriately, the output unit receives input +2 for any point in the slice and 0 for any point not in the slice. Doing this for each and thresholding at 1 implements the function f. st st S: We can now apply Theorem 6 to show that any neural net learning algorithm using too few examples will be fooled by some reasonable distributions. Corollary 9: For any learning algorithm training Ii net with k linear threshold functions in its hidden layer, and 0 < l ~ 1/8, if the algorithm uses (a) fewer than 2l lc/;'f,,-1 examples to learn a function from ~" to {-I, +1}, or (b) fewer than l"lll/2J(mQ,:I:(1/!~~-10/(ln n?)J-1 examples to learn a function from {-I, +1}" to {-I, +1}, for k ~ O(2 n / 3 ), then there exist distributions D for which (i) there exists a choice of weights such that the network exactly classifies its inputs according to D, but (ii) the learning algorithm will have probability at least .01 of finding a choice of weights which in fact has error greater than E. CONCLUSION We have given theoretical lower and upper bounds on the sample size vs. net size needed such that valid generalization can be expected. The exact constants we have given in these formulae are still quite crudej it may be expected that the actual values are closer to 1. The logarithmic factor in Corollary 4 may also not be needed, at least for the types of distributions and architectures seen in practice. Widrow's experience supports this conjecture [wid87]. However, closing the theoretical gap lower bound on the worst case between the O( ': log ~) upper bound and the (2 ( sample size for architectures with one hidden layer of threshold units remains an interesting open problem. Also, apart from our upper bound, the case of multiple hidden layers is largely open. Finally, our bounds are obtained under the assumption that the node functions are linear threshold functions (or at least Boolean valued). We conjecture that similar bounds also hold for classes of real valued functions such as sigmoid functions, and hope shortly to establish this. 1f) Acknowledgements We would like to thank Ron Rivest for suggestions on improving the bounds given in Corollaries 4 and 5 in an earlier draft of this paper, and Nick Littlestone for many helpful comments. The research of E. Baum was performed by the Jet Propulsion Laboratory, California Institute of Technology, as part of its Innovative Space What Size Net Gives Valid Generalization? Technology Center, which is sponsored by the Strategic Defense Initiative Organization/Innovative Science and Technology through an agreement with the National Aeronautics and Space Administration (NASA). D. Haussler gratefully acknowledges the support of ONR grant NOOOI4-86-K-0454. Part of this work was done while E. Baum was visiting UC Santa Cruz. References [b88]BAUM, E. B., (1988) On the capabilities of multilayer perceptrons, J. of Complexity, 4, 1988, ppI93-215. [behw87a]BLUMER, A., EHRENFEUCHT, A. HAUSSLER, D., WARMUTH, M., (1987), Occam's Razor, Int: Proc. Let., 24, 1987, pp377-380. [behw87b]BLUMER, A., EHRENFEUCHT, A. HAUSSLER, D., WARMUTH, M., (1987), Learnability and the Vapnik-Chervonenkis dimension, UC Santa Cruz Tech. Rep. UCSC-CRL-87-20 (revised Oct., 1988) and J. ACM, to appear. [cov65]COVER, T., (1965), Geometrical and statistical properties of systems of linear inequalities with applications to pattern recognition, IEEE Trans. Elect. Comp., V14, pp326-334. [dev88]DEVROYE, L., (1988), Automatic pattern recognition, a study of the probability of error, IEEE Trans. P AMI, V10, N4, pp530-543. [dswshhj87]DENKER J., SCHWARTZ D., WITTNER B., SOLLA S., HOP FIELD J., HOWARD R., JACKEL L., (1987), Automatic learning, rule extraction, and generalization, Complex Systems 1 pp877-922. [dh73]DUDA, R., HART, P., (1973), Pattern clallification and scene analysis, Wiley, New York. [ehkv87]EHRENFEUCHT, A., llAUSSLER, D., KEARNS, M., VALIANT, L., (1987), A general lower bound on the number of examples needed for learning, UC Santa Cruz Tech. Rep. UCSC-CRL-87-26 and Information and Computation, to appear. [hau88]HAUSSLER, D., (1988), Quantifying inductive bias: AI learning algorithms and Valiant's learning framework, Artificial Intelligence, 36, 1988, pp177-221. [hin87]HINTON, G., (1987), Connectionist learning procedures, Artificial Intelligence, to appear. [lit88]LITTLESTONE, N., (1988) Learning quickly when irrelevant attributes abound: a new linear threshold algorithm, Machine Learning, V2, pp285-318. [odI88]ODLYZKO, A., (1988), On subspaces spanned by random selections of ?1 vectors, J. Comb. Th. A, V47, Nt, pp124-133. [poI84]POLLARD, D., (1984), Convergence 0/ stochastic procelles, Springer-Verlag, New York. ' 89 90 Baum and Haussler [qs88]QUIAN, N., SEJNOWSKI, T. J., (1988), Predicting the secondary structure of globular protein using neural nets, Bull. Math. Biophys. 5, 115-137. [rum87]RUMELHART, D., (1987), personal communication. [sau72]SAUER, N., (1972), On the density of families of sets, J. Comb. Th. A, V13, 145-147. [sr87]SEJNOWSKI, T.J., ROSENBERG, C. R., (1987), NET Talk: a parallel network that learns to read aloud, Complex Systems, vi pp145-168. [ts88]TESAURO G., SEJNOWSKI, T. J.,(1988), A 'neural' network that learns to play backgammon, in Neural Information Procelling Sy,tem" ed. D.Z. Anderson, AlP, NY, pp794-803. [val84]VALIANT, L. G., (1984), A theory of the learnable, Comm. ACM V27, Nil pp1l34-1142. [vc71]VAPNIK, V.N., Chervonenkis, A. Ya., (1971), On the uniform convergence of relative frequencies of events to their probabilities, Th. Probe and its Applications, V17, N2, pp264-280. [vap82]VAPNIK, V.N., (1982), E,timation of Dependence, Ba,ed on Empirical Data, Springer Verlag, NY. [wd81]WENOCUR, R. S., DUDLEY, R. M., (1981) Some special Vapnik-Chervonenkis classes, Discrete Math., V33, pp313-318. [wid87]WIDROW, B, (1987) ADALINE and MADALINE - 1963, Plenary Speech, Vol I, Proc. IEEE 1st Int. Conf. on Neural Networks, San Diego, CA, pp143-158. 

154 |@word briefly:1 version:1 polynomial:1 stronger:1 maz:1 duda:1 open:2 initial:1 chervonenkis:4 ketch:1 nt:1 si:1 yet:1 cruz:4 fn:1 partition:1 offunctions:1 designed:1 sponsored:1 v:2 half:2 fewer:5 guess:2 intelligence:2 warmuth:2 plane:2 completeness:1 provides:1 node:36 ron:1 draft:1 math:2 simpler:1 shatter:1 ucsc:2 initiative:1 consists:1 specialize:1 dimen:1 comb:2 manner:2 indeed:1 expected:5 nor:1 ifm:1 spherical:1 little:1 actual:2 lll:1 cardinality:3 considering:2 abound:1 classifies:1 underlying:5 rivest:1 what:8 kind:1 developed:1 finding:1 nj:2 certainty:2 nf:1 exactly:1 rm:1 schwartz:1 unit:11 val84:2 grant:1 appear:4 overestimate:1 educated:1 approximately:3 might:1 nz:2 conversely:2 directed:1 v27:1 practice:2 implement:1 procedure:1 empirical:1 significantly:1 confidence:4 protein:2 suggest:1 get:2 onto:1 selection:1 context:1 applying:2 fruitful:1 center:1 baum:9 regardless:1 independently:2 pure:1 slicing:2 rule:4 haussler:10 spanned:1 his:1 mq:1 notion:1 target:1 construction:3 play:1 diego:1 exact:1 us:2 agreement:1 rumelhart:1 recognition:3 predicts:1 database:2 labeled:1 ft:2 disjunctive:1 worst:2 s1i:1 v13:1 connected:3 solla:1 removed:1 comm:1 complexity:2 pp1l34:1 personal:1 eric:1 learner:1 sink:2 easily:2 various:1 represented:3 talk:1 train:1 distinct:1 mn2:1 describe:1 sejnowski:3 artificial:2 dichotomy:9 choosing:2 aloud:1 quite:2 whose:1 larger:1 valued:4 say:1 itself:2 sequence:4 net:31 adaline:1 eto:1 convergence:2 produce:3 generating:1 prool:2 derive:1 widrow:3 v10:1 differ:1 closely:1 attribute:1 stochastic:2 vc:8 alp:1 implementing:1 globular:1 vcdim:9 generalization:12 hold:3 normal:1 mapping:1 predict:1 proc:2 jackel:1 largest:2 hope:3 mit:2 always:1 varying:1 rosenberg:1 conjunction:1 corollary:13 backgammon:2 fooled:1 tech:2 realizable:1 helpful:1 shattered:4 hidden:16 issue:1 among:2 denoted:1 logn:1 priori:1 special:2 fairly:1 uc:3 equal:1 construct:1 field:1 extraction:1 hop:1 lit:1 tem:1 future:9 connectionist:1 serious:1 few:2 randomly:3 ve:1 national:1 individual:1 undoubtably:1 negation:3 attempt:2 freedom:1 organization:1 interest:2 nl:1 implication:1 edge:12 capable:1 closer:1 necessary:1 experience:1 sauer:2 afi:1 divide:1 logarithm:3 littlestone:2 theoretical:2 minimal:1 plenary:1 classify:6 earlier:1 boolean:3 cover:2 bull:1 strategic:1 deviation:1 subset:2 uniform:2 srm:1 recognizing:1 too:1 learnability:1 kn:3 chooses:1 st:3 density:1 ie:4 physic:1 quickly:1 unavoidable:1 containing:2 literal:1 conf:1 expert:1 coefficient:1 int:2 vi:1 performed:1 try:2 doing:1 start:1 parallel:2 capability:1 minimize:2 ni:1 accuracy:3 loaded:3 phoneme:1 largely:1 sy:1 identify:1 generalize:3 thumb:1 produced:1 comp:1 classified:2 whenever:2 ed:2 definition:1 frequency:2 obvious:1 e2:1 associated:4 proof:3 di:1 treatment:1 noooi4:1 ezample:1 back:3 nasa:1 appears:1 dh73:2 done:1 anderson:1 just:1 sketch:1 receives:2 propagation:3 defines:1 measurability:1 contain:2 inductive:1 hence:4 read:1 laboratory:1 ehrenfeucht:3 ll:1 razor:1 noted:1 elect:1 confusion:1 nde:1 geometrical:1 ef:1 fi:1 common:1 sigmoid:1 overview:1 exponentially:4 discussed:1 v14:1 ai:3 automatic:2 closing:1 gratefully:1 etc:1 base:2 aeronautics:1 recent:1 perspective:1 irrelevant:1 apart:1 tesauro:1 verlag:2 inequality:3 onr:1 rep:2 seen:1 greater:4 somewhat:1 ii:2 full:1 multiple:1 jet:1 af:8 wittner:1 hart:1 regression:1 multilayer:1 df:2 represent:1 ion:1 source:3 appropriately:1 rest:1 comment:1 induced:7 n7:1 ee:3 near:1 counting:1 feedforward:14 architecture:17 approaching:6 associating:1 administration:1 expression:1 quian:1 defense:1 pollard:1 york:2 speech:1 santa:4 se:4 reduced:1 exist:1 disjoint:2 correctly:10 per:2 diverse:1 discrete:1 vol:1 threshold:18 nevertheless:1 drawn:10 deleted:1 neither:1 verified:2 graph:7 fraction:13 year:1 wand:1 counteract:1 inverse:1 prob:1 family:1 reasonable:2 vn:1 appendix:1 bound:23 layer:12 fl:2 scene:1 argument:1 span:1 innovative:2 conjecture:2 department:1 according:3 remain:1 slightly:1 em:11 n4:1 making:1 restricted:1 ln:2 remains:1 discus:1 fail:2 needed:6 know:1 end:1 generalizes:1 apply:2 denker:1 probe:1 v2:1 appropriate:4 dudley:1 permission:1 shortly:1 existence:1 jn:3 remaining:1 vc71:1 giving:1 establish:1 hypercube:2 question:4 realized:1 occurs:1 dependence:2 visiting:1 exhibit:1 subspace:2 thank:1 capacity:2 propulsion:1 assuming:4 devroye:1 procelling:1 concomitant:1 minimizing:1 madaline:1 sharper:1 ba:1 upper:6 revised:1 howard:1 attacked:1 january:1 hinton:1 communication:1 arbitrary:5 david:1 introduced:1 pair:3 nick:1 california:2 learned:1 vap82:4 trans:2 address:3 suggested:2 below:1 pattern:4 including:1 shifting:1 event:1 natural:2 predicting:1 technology:3 ne:1 reprint:1 acknowledges:1 text:1 disagrees:1 acknowledgement:1 determining:1 relative:2 fully:3 inde:1 interesting:1 suggestion:1 analogy:1 acyclic:1 lit88:2 degree:1 sufficient:1 consistent:4 thresholding:1 playing:1 share:1 occam:1 placed:1 last:1 free:1 bias:1 institute:1 taking:1 slice:3 dimension:9 valid:10 world:1 computes:1 doesn:1 made:1 san:1 odlyzko:1 incoming:4 knew:1 timation:1 why:1 learn:2 ca:2 composing:1 ignoring:1 improving:1 complex:2 protocol:1 domain:1 linearly:1 whole:1 s2:3 arise:1 n2:1 en:2 ny:2 wiley:1 wlog:2 lc:1 position:1 crude:1 learns:2 theorem:14 ofn:1 removing:1 load:3 formula:1 learnable:1 explored:1 a3:2 exists:4 vapnik:8 valiant:5 biophys:1 gap:1 b88:3 entropy:1 intersection:1 logarithmic:2 desire:1 ordered:2 springer:2 determines:1 chance:1 acm:2 oct:1 nem:4 goal:1 blumer:2 quantifying:1 crl:2 considerable:2 specifically:1 uniformly:1 ami:1 lemma:1 kearns:1 called:4 total:2 secondary:2 pas:1 nil:1 ya:1 ew:1 perceptrons:1 support:2 latter:2 outgoing:1 princeton:2

General-purpose localization of textured ? ? Image regions Rutb Rosenboltz? XeroxPARC 3333 Coyote Hill Rd. Palo Alto, CA 94304 Abstract We suggest a working definition of texture: Texture is stuff that is more compactly represented by its statistics than by specifying the configuration of its parts. This definition suggests that to fmd texture we look for outliers to the local statistics, and label as texture the regions with no outliers. We present a method, based upon this idea, for labeling points in natural scenes as belonging to texture regions, while simultaneously allowing us to label lowlevel, bottom-up cues for visual attention. This method is based upon recent psychophysics results on processing of texture and popout. 1 WHAT IS TEXTURE, AND WHY DO WE WANT TO FIND IT? In a number of problems in computer VlSlon and image processing, one must distinguish between image regions that correspond to objects and those which correspond to texture, and perform different processing depending upon the type of region. Current computer vision algorithms assume one magically knows this region labeling. But what is texture? We have the notion that texture involves a pattern that is somehow homogeneous, or in which signal changes are "too complex" to describe, so that aggregate properties must be used instead (Saund, 1998). There is by no means a firm division between texture and objects; rather, the characterization often depends upon the scale of interest (Saund, 1998). ? Email: rruth@parc.xerox.com 818 R. Rosenholtz Ideally the defmition of texture should probably depend upon the application. We investigate a definition that we believe will be of fairly general utility: Texture is stuff that seems to belong to the local statistics. We propose extracting several texture features, at several different scales, and labeling as texture those regions whose feature values are likely to have come from the local distribution. Outliers to the local statistics tend to draw our attention (Rosenholtz, 1997, 1998). The phenomenon is often referred to as "popout." Thus while labeling (locally) statistically homogeneous regions as texture, we can simultaneously highlight salient outliers to the local statistics. Our revised defmition is that texture is the absence of popout. In Section 2, we discuss previous work in both human perception and in fmding texture and regions of interest in an image. In Section 3, we describe our method. We present and discuss results on a number of real images in Section 4. 2 PREVIOUS WORK See (Wolfe, 1998) for a review of the visual search literature. Popout is typically studied using simple displays, in which an experimental subject searches for the unusual, target item, among the other, distractor items. One typically attempts to judge the "saliency," or degree to which the target pops out, by studying the efficiency of search for that item. Typically popout is modeled by a relatively lowlevel operator, which operates independently on a number of basic features of the image, including orientation, contrast/color, depth, and motion. In this paper, we look only at the features of contrast and orientation. Within the image-processing field, much of the work in fmding texture has defmed as texture any region with a high luminance variance, e.g. Vaisey & Gersho (1992). Unfortunately, the luminance variance in a region containing an edge can be as high as that in a textured region. Won & Park (1997) use model fitting to detect image blocks containing an edge, and then label blocks with high variance as containing texture. Recently, several computer vision researchers have also tackled this problem. Leung & Malik (1996) found regions of completely deterministic texture. Other researchers have used the defmition that if the luminance goes up and then down again (or vice versa) it's texture (Forsyth et aI, 1996). However, this method will treat lines as if they were texture. Also, with no notion of similarity within a texture (also lacking in the image-processing work), one would mark a "fault" in a texture as belonging to that texture. This would be unacceptable for a texture synthesis application, in which a routine that tried to synthesize such a texture would most likely fail to reproduce the (highly visible) fault. More recently, Shi and Malik (1998) presented a method for segmenting images based upon texture features. Their method performs extremely well at the segmentation task, dividing an image into regions with internal similarity that is high compared to the similarity across regions. However, it is difficult to compare with their results, since they do not explicitly label a subset of the resulting regions as texture. Furthermore, this method may also tend to mark a "fault" in a texture as belonging to that texture. This is both because the method is biased against separating out small regions, and because the grouping of a patch with one region depends as much upon the difference between that patch and other regions as it does upon the similarity between the patch and the given region. Very little computer vision work has been done on attentional cues. Milanese et al (1993) found salient image regions using both top-down information and a bottomup "conspicuity" operator, which marks a local region as more salient the greater the General-Purpose Localization o/Textured Image Regions 819 difference between a local feature value and the mean feature value in the surrounding region. However. for the same difference in means. a local region is less salient when there is a greater variance in the feature values in the surrounding region (Duncan & Humphreys. 1989; Rosenholtz. 1997). We use as our saliency measure a test for outliers to the local distribution. This captures. in many cases. the dependence of saliency on difference between a given feature value and the local mean. relative to the local standard deviation. We will discuss our saliency measure in greater detail in the following section. 3 FINDING TEXTURE AND REGIONS OF INTEREST We compute multiresolution feature maps for orientation and contrast. and then look for outliers in the local orientation and contrast statistics. We do this by fast creating a 3-level Gaussian pyramid representation of the image. To extract contrast. we filter the pyramid with a difference of circularly symmetric Gaussians. The response of these filters will oscillate. even in a region with constant-contrast texture (e.g. a sinewave pattern). We approximate a computation of the maximum response of these filters over a small region by fast squaring the filter responses. and then filtering the contrast energy with an appropriate Gaussian. Finally. we threshold the contrast to eliminate low-contrast regions ("flat" texture). These thresholds (one for each scale) were set by examining the visibility of sinewave patterns of various spatial frequencies. We compute orientation in a simple and biologically plausible way. using Bergen & Landy's (1991) "back pocket model" for low-level computations: 1. Filter the pyramid with horizontal. vertical. and ?45? oriented Gaussian second derivatives. 2. Compute opponent energy by squaring the filter outputs. pooling them over a region 4 times the scale of the second derivative filters. and subtracting the vertical from the horizontal response and the +45 0 from the _45 0 response. 3. Normalize the opponent energy at each scale by dividing by the total energy in the 4 orientation energy bands at that scale. The result is two images at each scale of the pyramid. To a good approximation. in regions which are strongly oriented. these images represent kcos(26) and ksin(26). where 6 is the local orientation at that scale. and k is a value between 0 and 1 which is related to the local orientation specificity. Orientation estimates from points with low specificity tend to be very noisy. In images of white noise. 80% of the estimates of k fall below 0.5. therefore with 80% confidence. an orientation specificity of k>0.5 did not occur due to chance. We use this value to threshold out orientation estimates with low "orientedness.?? We then estimate D, the local feature distribution, for each feature and scale, using the method of Parzen windows. The blurring of the distribution estimate by the Parzen window mimics uncertainty in estimates of feature values by the visual system. We collect statistics over a local integration region. For texture processing. the size of this region is ind.ependent of viewing distance, and is roughly lOS in diameter, where S is the support of the Gaussian 2nd derivative filters used to extract the texture features (Kingdom & Keeble, 1997; Kingdom et ai, 1995). We next compute a non-parametric measure of saliency: ( saliency = -IO~ P(v ID) maxP(x ID) % , ) (1) R. Rosenholtz 820 Note that if D were Gaussian N(~,a2), this simplifies to (X_tt)2 (2) '1a 2 which should be compared to the standard parametric test for outliers, which uses the measure (x - tt)/a. Our saliency measure is essentially a more general, nonparametric form of this measure (i.e. it does not assume a Gaussian distribution). Points with saliency less than 0.5 are labeled as candidate texture points. If D were Gaussian, this would correspond to feature estimates within one standard deviation of the mean. Points with saliency greater than 3.1 are labeled as candidates for bottom-up attentional cues. If D were Gaussian, this would correspond to feature estimates more than 2.50 from the mean, a standard parametric test for outliers. One could, of course, keep the raw saliency values, as a measure of the likelihood that a region contained texture, rather than setting a hard threshold. We use a hard threshold in our examples to better display the results. Both the texture images and the region of interest images are median-filtered to remove extraneous points. 4 EXPERIMENTAL RESULTS Figure 3 shows several example images. Figures 2, 3, and 4 show texture found at each scale of processing. The striped and checkered patterns represent oriented and homogeneous contrast texture, respectively. The absence of an image in any of these figures means that no texture of the given type was found in that image at the given scale. Note that we perform no segmentation of one texture from another. For the building image, the algorithm labeled bricks and window panes as fme-scale texture, and windows and shutters as coarser-scale texture. The leopard skin and low-frequency stripes in the lower right comer of the leopard image were correctly labeled as texture. In the desk image, the "wood" texture was correctly identified. The regular pattern of windows were marked as texture in the hotel image. In the house image, the wood siding, trees, and part of the grass were labeled as texture (much of the grass was low contrast and labeled as "flat" texture). One of the bushes is correctly identified as having coarser texture than the other has. In the lighthouse image, the house sans window, fence, and tower were marked, as well as a low-frequency oriented pattern in the clouds. Figure 5 shows the regions of interest that were found (the striped and plaid patterns here have no meaning but were chosen for maximum visibility). Most complex natural scenes had few interesting low-level attentional areas. In the lighthouse image, the life preserver is marked. In the hotel, curved or unusual angular windows are identified as attentional cues, as well as the top of the building. Both of these results are in agreement with psychophysical results showing that observers quickly identify curved or bent lines among straight lines (reviewed in Wolfe, 1998). The simpler desk scene yields more intuitive results, with each of the 3 objects labeled, as well as the phone cord. Bottom-up attentional cues are outliers to the local distribution of features, and we have suggested that texture is the absence of such outliers. This definition captures some of the intuition that texture is homogeneous and statistical in nature. We presented a method for fmding contrast and orientation outliers, and results both on localizing texture and on finding popout in natural images. For the simple desk image, the algorithm highlights salient regions that correspond to our notions of the important objects in the scene. On complicated natural scenes, its results are less intuitive; suggesting that search in natural scenes makes use of higher-level General-Purpose Localization o/Textured Image Regions 821 processing such as grouping into objects. This result should not be terribly surprising, but serves as a useful check on simple low-level models of visual attention. The algorithm does a good job of identifying textured regions at a number of different scales, with the results perhaps more intuitive at finer scales. Acknowledgments This work was partially supported by an NRC postdoctoral award at NASA Ames. Many thanks to David Marimont and Eric Saund for useful discussions. References J. R. Bergen and M. S. Landy (1991), "Computational modeling of visual texture segmentation," Computational Models of Visual Processing, Landy and Movshon (eds.), pp. 252-271, MIT Press, Cambridge, MA. J. Duncan and G. Humphreys (1989), "Visual search and stimulus similarity," Psych. Review 96, pp. 433-458. D. Forsyth, J. Malik, M. Fleck, H. Greenspan, T. Leung, S. Belongie, C. Carson, and C. Bregler (1996), "Finding pictures of objects in collections of images," ECCV Workshop on Object Representation, Cambridge. F. A. A. Kingdom, D. Keeble, D., and B. Moulden (1995), "Sensitivity to orientation modulation in micropattern-based textures," Vis. Res. 35, 1, pp. 79-91. F. A. A. Kingdom and D. Keeble (1997), "The mechanism for scale invariance in orientation-defined textures." Invest. Ophthal. and Vis . Sci. (Suppl.) 38, 4, p. 636. T. K. Leung and J. Malik (1996), "Detecting, localizing, and grouping repeated scene elements from an image," Proc. 4th European Con! On Computer Vision, 1064, 1, pp. 546-555, Springer-Verlag, Cambridge. R. Milanese, H. Wechsler, S. Gil, J. -M. Bost, and T. Pun (1993), "Integration of bottom-up and top-down cues for visual attention using non-linear relaxation," Proc. IEEE CVPR, pp. 781-785, IEEE Computer Society Press, Seattle. R. Rosenholtz (1997), "Basic signal detection theory model does not explain search among heterogeneous distractors." Invest. Ophthal. and Vis. Sci. (Suppl.) 38, 4, p. 687. R. Rosenholtz (1998), "A simple saliency model explains a number of motion popout phenomena." Invest. Ophthal. and Vis. Sci. (Suppl.) 39,4, p. 629. E. Saund (1998), "Scale and the ShapelTexture Continuum," Xerox Internal Technical Memorandum. J. Shi and J. Malik (1998), "SelfInducing Relational Distance and its Application to Image Segmentation," Proc. jth European Con! on Computer Vision, Burkhardt and Neumann (eds.), 1406, 1, pp. 528-543, Springer, Freiburg. J. Vaisey and A. Gersho (1992), "Image compression with variable block size segmentation." IEEE Trans. Signal Processing 40,8, pp. 2040-2060. J. M. Wolfe (1998), "Visual search: a review," Attention, H. Pashler (ed.), pp. 1374, Psychology Press Ltd., Hove, East Sussex, UK. C. S. Won and D. K. Park (1997), "Image block classification and variable block size segmentation using a model-fitting criterion," Opt. Eng. 36, 8, pp. 2204-2209. 822 R. Rosenholtz Figure I: Original images. (a) (b) Figure 2: Fine-scale texture. (a) oriented texture, (b) homogeneous contrast texture. -. ~" . ~ - & (a) .... I. ... , ..' 1". .~ ?, ? ? - .. ' .. ? (b) Figure 3: Medium-scale texture. (a) oriented texture, (b) homogeneous contrast texture. General-Purpose Localization of Textured Image Regions 823 (a) (b) Figure 4: Coarse-scale texture. (a) oriented texture, (b) homogeneous contrast texture . .. . , I, ? ? Figure 5: Regions of interest. 

1540 |@word compression:1 seems:1 nd:1 tried:1 eng:1 configuration:1 current:1 com:1 surprising:1 must:2 visible:1 visibility:2 remove:1 grass:2 cue:6 item:3 filtered:1 detecting:1 coarse:1 characterization:1 ames:1 simpler:1 pun:1 unacceptable:1 fitting:2 roughly:1 distractor:1 little:1 window:7 alto:1 medium:1 what:2 psych:1 finding:3 stuff:2 uk:1 segmenting:1 local:17 treat:1 io:1 id:2 modulation:1 studied:1 specifying:1 suggests:1 collect:1 statistically:1 acknowledgment:1 block:5 area:1 confidence:1 regular:1 specificity:3 suggest:1 operator:2 pashler:1 deterministic:1 map:1 shi:2 go:1 attention:5 lowlevel:2 independently:1 moulden:1 identifying:1 notion:3 memorandum:1 target:2 homogeneous:7 us:1 agreement:1 wolfe:3 synthesize:1 element:1 stripe:1 coarser:2 labeled:7 bottom:4 cloud:1 capture:2 region:42 cord:1 intuition:1 ideally:1 depend:1 parc:1 localization:4 upon:8 division:1 efficiency:1 textured:6 completely:1 compactly:1 blurring:1 comer:1 eric:1 represented:1 various:1 surrounding:2 fast:2 describe:2 labeling:4 aggregate:1 firm:1 whose:1 plausible:1 cvpr:1 maxp:1 statistic:7 noisy:1 propose:1 subtracting:1 multiresolution:1 intuitive:3 normalize:1 los:1 invest:3 seattle:1 neumann:1 object:7 depending:1 job:1 dividing:2 involves:1 come:1 judge:1 plaid:1 filter:8 human:1 terribly:1 viewing:1 explains:1 opt:1 bregler:1 leopard:2 sans:1 milanese:2 continuum:1 a2:1 purpose:4 proc:3 label:4 palo:1 vice:1 mit:1 gaussian:8 rather:2 greenspan:1 fence:1 likelihood:1 check:1 contrast:15 detect:1 bergen:2 squaring:2 leung:3 typically:3 eliminate:1 reproduce:1 among:3 orientation:14 classification:1 extraneous:1 spatial:1 integration:2 psychophysics:1 fairly:1 field:1 having:1 park:2 look:3 mimic:1 stimulus:1 few:1 oriented:7 simultaneously:2 attempt:1 detection:1 interest:6 investigate:1 highly:1 edge:2 tree:1 re:1 brick:1 modeling:1 localizing:2 deviation:2 subset:1 coyote:1 examining:1 too:1 thanks:1 sensitivity:1 synthesis:1 parzen:2 quickly:1 again:1 containing:3 creating:1 derivative:3 suggesting:1 forsyth:2 explicitly:1 depends:2 vi:4 saund:4 observer:1 complicated:1 fme:1 variance:4 correspond:5 saliency:11 identify:1 yield:1 raw:1 researcher:2 finer:1 straight:1 explain:1 ed:3 email:1 definition:4 against:1 energy:5 hotel:2 frequency:3 pp:9 con:2 color:1 distractors:1 segmentation:6 pocket:1 routine:1 back:1 nasa:1 higher:1 response:5 done:1 strongly:1 furthermore:1 angular:1 working:1 horizontal:2 somehow:1 perhaps:1 believe:1 building:2 symmetric:1 white:1 ind:1 defmed:1 sussex:1 won:2 criterion:1 carson:1 hill:1 freiburg:1 tt:1 performs:1 motion:2 image:39 meaning:1 recently:2 belong:1 defmition:3 versa:1 cambridge:3 ai:2 rd:1 had:1 similarity:5 recent:1 phone:1 lighthouse:2 verlag:1 fault:3 life:1 greater:4 signal:3 technical:1 bent:1 award:1 basic:2 heterogeneous:1 vision:5 essentially:1 popout:7 represent:2 pyramid:4 suppl:3 want:1 fine:1 median:1 biased:1 probably:1 subject:1 tend:3 pooling:1 marimont:1 extracting:1 psychology:1 identified:3 idea:1 simplifies:1 shutter:1 utility:1 ltd:1 movshon:1 oscillate:1 useful:2 burkhardt:1 nonparametric:1 desk:3 locally:1 band:1 diameter:1 conspicuity:1 rosenholtz:7 gil:1 correctly:3 salient:5 threshold:5 luminance:3 relaxation:1 wood:2 uncertainty:1 patch:3 draw:1 duncan:2 distinguish:1 display:2 tackled:1 occur:1 striped:2 scene:7 flat:2 extremely:1 pane:1 relatively:1 xerox:2 belonging:3 magically:1 across:1 biologically:1 outlier:11 discus:3 fail:1 mechanism:1 know:1 gersho:2 serf:1 unusual:2 studying:1 gaussians:1 opponent:2 appropriate:1 original:1 top:3 landy:3 wechsler:1 society:1 psychophysical:1 malik:5 skin:1 parametric:3 dependence:1 distance:2 attentional:5 separating:1 sci:3 tower:1 bost:1 modeled:1 fmding:3 difficult:1 unfortunately:1 kingdom:4 perform:2 allowing:1 vertical:2 revised:1 curved:2 relational:1 david:1 pop:1 trans:1 suggested:1 below:1 pattern:7 perception:1 including:1 natural:5 picture:1 extract:2 review:3 literature:1 relative:1 lacking:1 highlight:2 interesting:1 filtering:1 degree:1 eccv:1 course:1 supported:1 jth:1 fall:1 depth:1 collection:1 approximate:1 checkered:1 keep:1 belongie:1 bottomup:1 postdoctoral:1 search:7 why:1 reviewed:1 nature:1 ca:1 complex:2 european:2 did:1 nrc:1 noise:1 repeated:1 fmd:1 referred:1 candidate:2 house:2 humphreys:2 down:3 sinewave:2 showing:1 grouping:3 workshop:1 circularly:1 texture:68 likely:2 visual:9 contained:1 partially:1 springer:2 chance:1 ma:1 marked:3 absence:3 change:1 hard:2 operates:1 total:1 invariance:1 experimental:2 east:1 fleck:1 internal:2 mark:3 support:1 bush:1 phenomenon:2

Active Noise Canceling using Analog NeuroChip with On-Chip Learning Capability Jung-Wook Cho and Soo-Young Lee Computation and Neural Systems Laboratory Department of Electrical Engineering Korea Advanced Institute of Science and Technology 373-1 Kusong-dong, Yusong-gu, Taejon 305-701, Korea sylee@ee.kaist.ac.kr Abstract A modular analogue neuro-chip set with on-chip learning capability is developed for active noise canceling. The analogue neuro-chip set incorporates the error backpropagation learning rule for practical applications, and allows pin-to-pin interconnections for multi-chip boards. The developed neuro-board demonstrated active noise canceling without any digital signal processor. Multi-path fading of acoustic channels, random noise, and nonlinear distortion of the loud speaker are compensated by the adaptive learning circuits of the neuro-chips. Experimental results are reported for cancellation of car noise in real time. 1 INTRODUCTION Both analog and digital implementations of neural networks have been reported. Digital neuro-chips can be designed and fabricated with the help of well-established CAD tools and digital VLSI fabrication technology [1]. Although analogue neurochips have potential advantages on integration density and speed over digital chips[2], they suffer from non-ideal characteristics of the fabricated chips such as offset and nonlinearity, and the fabricated chips are not flexible enough to be used for many different applications. Also, much careful design is required, and the fabricated chip characteristics are fairly dependent upon fabrication processes. For the implementation of analog neuro-chips, there exist two different approaches, i.e., with and without on-chip learning capability [3,4], Currently the majority of analog neuro-chips does not have learning capability, while many practical applications require on-line adaptation to continuously changing environments, and must have online adaptation learning capability. Therefore neuro-chips with on-chip learning capability are essential for such practical applications. Modular architecture is also 665 Active Noise Canceling with Analog On-Chip Learning Neuro-Chip advantageous to provide flexibility of implementing many large complex systems from same chips. Although many applications have been studied for analog neuro-chips, it is very important to find proper problems where analog neuro-chips may have potential advantages over popular DSPs. We believe applications with analog input/output signals and high computational requirements are those good problems. For example, active noise controls [5] and adaptive equalizers [6,7] are good applications for analog neuro-chips. In this paper we report a demonstration of the active noise canceling, which may have many applications in real world. A modular analog neuro-chip set is developed with on-chip learning capability, and a neuro-board is fabricated from multiple chips with PC interfaces for input and output measurements. Unlike our previous implementations for adaptive equalizers with binary outputs [7], both input and output values are analogue in this noise canceling. ..-.---1-1.... 0 II' . xl, ~t---(~.-~-.lI'"'"'"iI--- (~ Figure 1. Block diagram of a synapse cell 2 ~ ._._.- i') Figure 2. Block diagram of a neuron cell ANALOG NEURO-CHIP WITH ON-CHIP LEARNING We had developed analog neuro-chips with error backpropagation learning capability. With the modular architecture the developed analog neuro-chip set consists of a synapse chip and a neuron chip.[8] The basic cell of the synapse chip is shown in Figure 1. Each synapse cell receives two inputs, i.e., pre-synaptic neural activation x and error correction term 8, and generates two outputs, i.e., feed-forward signal wx and back-propagated error w8. Also it updates a stored weight w by the amount of x8. Therefore, a synapse cell consists of three multiplier circuits and one analogue storage for the synaptic weight. Figure 2 shows the basic cell in the neuron chip, which collects signals from synapses in the previous layer and distributes to synapses in the following layer. Each neuron body receives two inputs, i.e., post-synaptic neural activation 0 and back-propagated error 8 from the following layer, and generates two outputs, i.e., Sigmoid-squashed neural activation 0 and a new backpropagated error 8 multiplied by a bell-shaped Sigmoid-derivative. The backpropagated error may be input to the synapse cells in the previous layer. To provide easy connectivity with other chips, the two inputs of the synapse cell are represented as voltage, while the two outputs are as currents for simple current summation. On the other hand the inputs and outputs of the neuron cell are represented as currents and voltages, respectively. For simple pin-to-pin connections between chips, one package pin is maintained to each input and output of the chip. No time- J.-W Cho and s.-Y. Lee 666 multiplexing is introduced, and no other control is required for multi-chip and multilayer systems. However, it makes the number of package pins the main limiting factor for the number of synapse and neuron cells in the developed chip sets. Although many simplified multipliers had been reported for high-density integration, their performance is limited in linearity, resolution, and speed. For on-chip learning, it is desirable to have high precision, and a faithful implementation of the 4-quadranr Gilbert multipliers is used. Especially, the mUltiplier for weight updates in the synapse cell requires high precision.[9] The synaptic weight is stored on a capacitor, and an MaS switch is used to allow current flow from the multiplier to the capacitor during a short time interval for weight adaptation. For applications like active noise controls [5] and telecommunications [6,7], tapped analog delay lines are also designed and integrated in the synapse chip. To reduce offset accumulation, a parallel analog delay line is adopted. Same offset voltage is introduced for operational amplifiers at all nodes [10] . Diffusion capacitors with 2.2 pF are used for the storage of the tapped analog delay line. In a synapse chip 250 synapse cells are integrated in a 25xl0 array with a 25-tap analog delay line. Inputs may be applied either from the analog delay line or from external pins in parallel. To select a capacitor in the cell for refresh, decoders are placed in columns and rows. The actual size of the synapse cell is 14111m x 17911m, and the size of the synapse chip is 5.05mm x 5.05mm. The chip is fabricated in a 0.811m single-poly CMOS process. On the other hand, the neuron chip has a very simple structure, which consists of 20 neuron cells without additional circuits. The Sigmoid circuit [3] in the neuron cell uses a differential pair, and the slope and amplitude are controlled by a voltage-controlled resistor [II]. Sigmoid-derivative circuit is also using differential pair with min-select circuit. The size of the neuron cell is 177.2I1m x 62.4l1m. Synapse Chip PC Neuron PC Chip , ~ _ ' Target I N t-.,.' ----t~ I I DSP TMS320C51 N I Output : -:-r!1"iJ-[B~h:---'-:_._.-.____;--..; r:1-r:"1' L!..r~ ..._~Q_-I_""',_ _ I Input - L -_ _ _--I ANN Board Figure 3: Block diagram of the analog neuro-board GDAB tv.c : 32ch ArIC : 48<:h D1 : 16bitll DO : 48bitll Active Noise Canceling with Analog On-Chip Learning Neuro-Chip 667 Using these chip sets, an analog neuro-system is constructed. Figure 3 shows a brief block diagram of the analog neuro-system, where an analogue neuro-board is interfaced to a host computer through a GDAB (General Data Acquisition Board). The GDAB board is specially designed for the data interface with the analogue neuro-chips. The neuro-board has 6 synapse chips and 2 neuron chips with the 2-layer Perceptron architecture. For test and development purposes, a DSP, ADC and DAC are installed on the neuro-board to refresh and adjust weights. Forward propagation time of the 2 layers Perceptron is measured as about 30 f..lsec. Therefore the computation speed of the neuro-board is about 266 MCPS (Mega Connections Per Second) for recall and about 200 MCUPS (Mega Connections Updates Per Second) for error backpropagation learning. To achieve this speed with a DSP, about 400 MIPS is required for recall and at least 600 MIPS for error-back propagation learning. C 1 (z) Channel Error Signal Noise Source Adaptive Filter or Multilayer Perceptron Figure 4: Structure of a feedforward active noise canceling 3 ACTIVE NOISE CANCELING USING NEURO-CHIP Basic architecture of the feed forward active noise canceling is shown in Figure 4. An area near the microphone is called "quiet zone," which actually means noise should be small in this area. Noise propagates from a source to the quiet zone through a dispersive medium, of which characteristics are modeled as a finite impulse response (FIR) filter with additional random noise. An active noise canceller should generate electric signals for a loud speaker, which creates acoustic signals to cancel the noise at the quiet zone. In general the electric-to-acoustic signal transfer characteristics of the loud speaker is nonlinear, and the overall active noise canceling (ANC) system also becomes nonlinear. Therefore, multilayer Perceptron has a potential advantage over popular transversal adaptive filters based on linear-mean.-square (LMS) error minimization. Experiments had been conducted for car noise canceling. The reference signal for the noise source was extracted from an engine room, while a compact car was running at 60 kmlhour. The difference of the two acoustic channels, i.e., H(z) = C1 (z) / C2 ( z) , addition noise n, and nonlinear characteristics of the loud speaker need be compensated. Two different acoustic channels are used for the experiments. The first channel Hl (z) = 0.894 + 0.447z- 1 is a minimum phase channel, while the second non- J.-W Cho and S-Y. Lee 668 minimum phase channel H2 (z) = 0.174 + 0.6z -I + 0.6z -2 + 0.174z -3 characterizes frequency-selective multipath fading with a deep spectral amplitude null. A simple cubic distortion model was used for the characteristics of the loud speaker.[12] To compare performance of the neuro-chip with digital processors, computer simulation was first conducted with error backpropagation algorithm for a single hidden-layer Perceptron as well as the LMS algorithm for a transversal adaptive filter. Then, the same experimental data were provided to the developed neuro-board by a personal computer through the GDAB. o ro a:: -5 c o :.;:; g -10 . "0 CI.> 0::: CI.> .~ o -15 z - 20 ~-----~------~----~------~--=-~ o 5 10 15 20 25 Signal-to-Distortion Ratio (a) o 15 -5 0::: c o ....... ~ -10 "0 CI.> 0::: CI.> .~ o -15 z -20~----~----~----~----~----~ o 5 10 15 20 Signal-to-Distortion Ratio 25 (b) Figure 5: Noise Reduction Ratio (dB) versus Signal-to-Distortion Ratio (dB) for (a) a simple acoustic channel HI (z) and (b) a multi-path fading acoustic channel H2 (z) _ Here, '+', '*', 'x', and '0' denote results ofLMS algorithm, neural networks simulation, neural network simulation with 8-bit input quantization, and neuro-chips, respectively_ Active Noise Canceling with Analog On-Chip Learning Neuro-Chip 669 Results for the channels HI ( z) and H2 (z) are shown in Figures 5(a) and 5(b), respectively. Each point in these figures denotes the result of one experiment with different parameters. The horizontal axes represent Signal-to-Distortion Ratio (SDR) of the speaker nonlinear characteristics. The vertical axes represent Noise Reduction Ratio (NRR) of the active noise canceling systems. As expected, severe nonlinear distortion of the loud speaker resulted in poor noise canceling for the LMS canceller. However, the performance degradation was greatly reduced by neural network canceller. With the neuro-chips the performance was worse than that of computer simulation. Although the neuro-chip demonstrated active noise canceling and worked better than LMS cancellers for very small SDRs, i.e. , very high nonlinear distortions, its performance became saturated to -8 dB and -5 dB NRRs, respectively. The performance saturation was more severe for the harder problem with the complicated H 2 (z ) channel. The performance degradation with neuro-chips may come from inherent limitations of analogue chips such as limited dynamic ranges of synaptic weights and signals, unwanted offsets and nonlinearity, and limited resolution of the learning rate and sigmoid slope. [9] However, other side effects of the GDAB board, i.e., fixed resolution of AID converters and D/A converters for data 110, also contributed to the performance degradation. The input and output resolutions of the GDAB were J6 bit and 8 bit, respectively. Unlike actual real-world systems the input values of the experimental analogue neuro-chips are these 8-bit quantized values. As shown in Figures 5, results of the computer simulation with 8-bit quantized target values showed much degraded performance compared to the floating-point simulations. Therefore, a significant portion of the poor performance in the experimental analogue system may be contributed from the AID converters, and the analogue system may work better in real world systems. Actual acoustic signals are plotted in Figure 6. The top, middle, and bottom signals denote noise , negated speaker signal, and residual noise at the quiet zone, respectively. Figure 6: Examples of noise, negated loud-speaker canceling signal, and residual error J.-w. Cho and s.-Y. Lee 670 4 CONCLUSION In this paper we report an experimental results of active noise canceling using analogue neuro-chips with on-chip learning capability. Although the its performance is limited due to nonideal characteristics of analogue chip itself and also peripheral devices, it clearly demonstrates feasibility of analogue chips for real world applications. Acknowledgements This research was supported by Korean Ministry of Information and Telecommunications. References [1] T. Watanabe, K. Kimura, M. Aold, T. Sakata & K. Ito (1993) A Single 1.5-V Digital Chip for a 106 Synapse Neural Network, IEEE Trans. Neural Network, VolA, No.3, pp.387-393. [2J T. Morie and Y. Amemiya (1994) An All-Analog Expandable Neural Network LSI with On-Chip Backpropagation Learning, IEEE Journal of Solid State Circuits, vo1.29, No.9, pp.1086-1093. [3J J.-W. Cho, Y. K. Choi, S.-Y. Lee (1996) Modular Neuro-Chip with On-Chip Learning and Adjustable Learning Parameters, Neural Processing Letters, VolA, No.1. [4J J. Alspector, A. Jayakumar, S. Luna (1992) Experimental evaluation of learning in neural microsystem, Advances in Neural Information Processing Systems 4, pp. 871-878 . [5 J B. Widrow, et al. (1975) Adative Noise Cancelling: Principles and Applications, Proceeding of IEEE, Vo1.63, No.12, pp.1692-1716. [6] J. Choi, S.H. Bang, BJ. Sheu (1993) A Programmable Analog VLSI Neural Network Processor for Communication Receivers, IEEE Transaction on Neural Network, VolA, No.3, ppA84-495. [7J J.-W. Cho and S.-Y. Lee (1998) Analog neuro-chips with on-chip learning capability for adaptive nonlinear equalizer, Proc. lJCNN, pp. 581-586, May 4-9, Anchorage, USA. [8J J. Van der Spiegel, C. Donham, R. Etienne-Cummings, S. Fernando (1994) Large scale analog neural computer with programmable architecture and programmable time constants for temporal pattern analysis, Proc. ICNN, pp. 1830-1835. [9J Y.K. Choi, K.H. Ahn, and S.Y. Lee (1996) Effects of multiplier offsets on onchip learning for analog neuro-chip, Neural Processing Letters, vol. 4, No.1, 1-8. [1OJ T. Enomoto, T. Ishihara and M. Yasumoto (1982) Integrated tapped MaS analogue delay line using switched-capacitor technique, Electronics Lertters, Vo1.l8, pp.193-194. [11 J P.B. Allen, D.R. Holberg (1987) CMOS Analog Circuit Design, Holt, Douglas Rinehart and Winston. [12J F. Gao and W.M. Snelgrove (1991) Adaptive linearization of a loudspeaker, Proc. International Conference on Acoustics, Speech and Signal processing, pp. 3589-3592. 

1541 |@word middle:1 advantageous:1 donham:1 simulation:6 solid:1 harder:1 reduction:2 electronics:1 nrr:1 current:4 cad:1 activation:3 must:1 refresh:2 wx:1 designed:3 update:3 device:1 short:1 quantized:2 node:1 anchorage:1 constructed:1 c2:1 differential:2 consists:3 expected:1 alspector:1 multi:4 actual:3 pf:1 becomes:1 provided:1 linearity:1 circuit:8 medium:1 null:1 developed:7 adc:1 fabricated:6 kimura:1 temporal:1 w8:1 unwanted:1 ro:1 demonstrates:1 control:3 engineering:1 installed:1 path:2 onchip:1 studied:1 mcups:1 collect:1 limited:4 range:1 practical:3 faithful:1 block:4 backpropagation:5 area:2 bell:1 pre:1 holt:1 storage:2 microsystem:1 gilbert:1 accumulation:1 demonstrated:2 compensated:2 resolution:4 rinehart:1 rule:1 array:1 limiting:1 target:2 us:1 tapped:3 bottom:1 electrical:1 environment:1 dynamic:1 personal:1 upon:1 creates:1 gu:1 chip:73 represented:2 modular:5 kaist:1 distortion:8 interconnection:1 sakata:1 spiegel:1 itself:1 online:1 advantage:3 adaptation:3 cancelling:1 flexibility:1 achieve:1 requirement:1 cmos:2 help:1 widrow:1 ac:1 measured:1 ij:1 come:1 filter:4 implementing:1 require:1 l1m:1 icnn:1 summation:1 correction:1 mm:2 bj:1 lm:4 purpose:1 proc:3 currently:1 tool:1 minimization:1 clearly:1 voltage:4 ax:2 dsp:3 greatly:1 dependent:1 integrated:3 hidden:1 vlsi:2 i1m:1 selective:1 overall:1 flexible:1 development:1 transversal:2 integration:2 fairly:1 shaped:1 cancel:1 report:2 inherent:1 resulted:1 floating:1 phase:2 sdr:1 amplifier:1 evaluation:1 adjust:1 severe:2 saturated:1 pc:3 korea:2 morie:1 plotted:1 column:1 delay:6 fabrication:2 conducted:2 reported:3 stored:2 cho:6 density:2 international:1 lee:7 dong:1 continuously:1 connectivity:1 fir:1 worse:1 external:1 luna:1 jayakumar:1 derivative:2 li:1 potential:3 characterizes:1 portion:1 capability:10 parallel:2 complicated:1 slope:2 square:1 degraded:1 became:1 characteristic:8 interfaced:1 expandable:1 j6:1 processor:3 synapsis:2 canceling:18 synaptic:5 acquisition:1 frequency:1 pp:8 propagated:2 popular:2 recall:2 car:3 dispersive:1 amplitude:2 actually:1 back:3 feed:2 cummings:1 response:1 synapse:17 hand:2 receives:2 horizontal:1 dac:1 nonlinear:8 propagation:2 impulse:1 believe:1 usa:1 effect:2 multiplier:6 yusong:1 laboratory:1 during:1 maintained:1 speaker:9 q_:1 allen:1 interface:2 sigmoid:5 sheu:1 analog:29 measurement:1 significant:1 cancellation:1 nonlinearity:2 had:3 ahn:1 showed:1 binary:1 der:1 minimum:2 additional:2 ministry:1 fernando:1 signal:19 ii:3 multiple:1 desirable:1 host:1 post:1 controlled:2 feasibility:1 neuro:39 basic:3 multilayer:3 represent:2 cell:17 c1:1 addition:1 interval:1 diagram:4 source:3 unlike:2 specially:1 db:4 flow:1 incorporates:1 capacitor:5 ee:1 near:1 ideal:1 feedforward:1 enough:1 easy:1 mips:2 switch:1 architecture:5 converter:3 reduce:1 suffer:1 speech:1 programmable:3 deep:1 amount:1 backpropagated:2 reduced:1 generate:1 exist:1 lsi:1 per:2 mega:2 vol:1 changing:1 douglas:1 diffusion:1 equalizer:3 package:2 letter:2 telecommunication:2 taejon:1 bit:5 layer:7 hi:2 winston:1 fading:3 worked:1 multiplexing:1 generates:2 lsec:1 speed:4 min:1 loudspeaker:1 department:1 tv:1 peripheral:1 poor:2 hl:1 aold:1 mcps:1 pin:7 adopted:1 multiplied:1 multipath:1 spectral:1 denotes:1 running:1 top:1 etienne:1 especially:1 squashed:1 quiet:4 majority:1 decoder:1 sylee:1 modeled:1 ratio:6 demonstration:1 korean:1 adative:1 implementation:4 design:2 proper:1 adjustable:1 contributed:2 negated:2 i_:1 vertical:1 neuron:12 finite:1 communication:1 dsps:1 introduced:2 pair:2 required:3 connection:3 tap:1 acoustic:9 engine:1 established:1 trans:1 pattern:1 saturation:1 oj:1 soo:1 analogue:15 residual:2 advanced:1 technology:2 brief:1 x8:1 acknowledgement:1 limitation:1 versus:1 digital:7 h2:3 ljcnn:1 switched:1 propagates:1 principle:1 row:1 l8:1 jung:1 placed:1 supported:1 side:1 allow:1 perceptron:5 institute:1 van:1 world:4 forward:3 adaptive:8 simplified:1 transaction:1 compact:1 active:17 receiver:1 channel:11 transfer:1 operational:1 anc:1 complex:1 poly:1 electric:2 main:1 noise:35 body:1 board:13 cubic:1 aid:2 precision:2 watanabe:1 resistor:1 xl:1 ito:1 young:1 choi:3 offset:5 essential:1 quantization:1 kr:1 ci:4 nonideal:1 linearization:1 gao:1 ch:1 loud:7 extracted:1 ma:2 ann:1 careful:1 bang:1 room:1 vo1:3 distributes:1 degradation:3 microphone:1 called:1 experimental:6 zone:4 select:2 xl0:1 d1:1

Bayesian modeling of human concept learning Joshua B. Tenenbaum Department of Brain and Cognitive Sciences Massachusetts Institute of Technology, Cambridge, MA 02139 jbt@psyche.mit.edu Abstract I consider the problem of learning concepts from small numbers of positive examples, a feat which humans perform routinely but which computers are rarely capable of. Bridging machine learning and cognitive science perspectives, I present both theoretical analysis and an empirical study with human subjects for the simple task oflearning concepts corresponding to axis-aligned rectangles in a multidimensional feature space. Existing learning models, when applied to this task, cannot explain how subjects generalize from only a few examples of the concept. I propose a principled Bayesian model based on the assumption that the examples are a random sample from the concept to be learned. The model gives precise fits to human behavior on this simple task and provides qualitati ve insights into more complex, realistic cases of concept learning. 1 Introduction The ability to learn concepts from examples is one of the core capacities of human cognition. From a computational point of view, human concept learning is remarkable for the fact that very successful generalizations are often produced after experience with only a small number of positive examples of a concept (Feldman, 1997). While negative examples are no doubt useful to human learners in refining the boundaries of concepts, they are not necessary in order to make reasonable generalizations of word meanings, perceptual categories, and other natural concepts. In contrast, most machine learning algorithms require examples of both positive and negative instances of a concept in order to generalize at all, and many examples of both kinds in order to generalize successfully (Mitchell, 1997). This paper attempts to close the gap between human and machine concept learning by developing a rigorous theory for concept learning from limited positive evidence and testing it against real behavioral data. I focus on a simple abstract task of interest to both cognitive science and machine learning: learning axis-parallel rectangles in ?R m . We assume that each object x in our world can be described by its values (XI, ... , xm) on m real-valued observable dimensions, and that each concept C to be learned corresponds to a conjunction of independent intervals (mini (C) ~ Xi ~ maXi (C? along each dimension 60 1. B. Tenenbaum (b) (a) (e) \ - ...... - . ~ r-------------. I - I I I I + :C + I ~ " ...... ~ . " - >" ~ ? ! I + + I t.. ... ____ ... __ ...... ___, Figure 1: (a) A rectangle concept C. (b-c) The size principle in Bayesian concept learning: of the man y hypotheses consistent wi th the observed posi ti ve examples, the smallest rapidly become more likely (indicated by darker lines) as more examples are observed. i. For example, the objects might be people, the dimensions might be "cholesterol level" and "insulin level", and the concept might be "healthy levels". Suppose that "healthy levels" applies to any individual whose cholesterol and insulin levels are each greater than some minimum healthy level and less than some maximum healthy level. Then the concept "healthy levels" corresponds to a rectangle in the two-dimensional cholesterol/insulin space. The problem of generalization in this setting is to infer, given a set of positive (+) and negative (-) examples of a concept C, which other points belong inside the rectangle corresponding to C (Fig. 1a.). This paper considers the question most relevant for cognitive modeling: how to generalize from just a few positive examples? In machine learning, the problem of learning rectangles is a common textbook example used to illustrate models of concept learning (Mitchell, 1997). It is also the focus of stateof-the-art theoretical work and applications (Dietterich et aI., 1997). The rectangle learning task is not well known in cognitive psychology, but many studies have investigated human learning in similar tasks using simple concepts defined over two perceptually separable dimensions such as size and color (Shepard, 1987). Such impoverished tasks are worth our attention because they isolate the essential inductive challenge of concept learning in a form that is analytically tractable and amenable to empirical study in human subjects. This paper consists of two main contributions. I first present a new theoretical analysis of the rectangle learning problem based on Bayesian inference and contrast this model's predictions with standard learning frameworks (Section 2). I then describe an experiment with human subjects on the rectangle task and show that, of the models considered, the Bayesian approach provides by far the best description of how people actually generalize on this task when given only limited positive evidence (Section 3). These results suggest an explanation for some aspects of the ubiquotous human ability to learn concepts from just a few positive examples. 2 Theoretical analysis Computational approaches to concept learning. Depending on how they model a concept, different approaches to concept learning differ in their ability to generalize meaningfully from only limited positive evidence. Discriminative approaches embody no explicit model of a concept, but only a procedure for discriminating category members from members of mutually exclusive contrast categories. Most backprop-style neural networks and exemplar-based techniques (e.g. K -nearest neighbor classification) fall into this group, along with hybrid models like ALCOVE (Kruschke, 1992). These approaches are ruled out by definition; they cannot learn to discriminate positive and negative instances ifthey have seen only positive examples. Distributional approaches model a concept as a probability distribution over some feature space and classify new instances x as members of C if their Bayesian Modeling ofHuman Concept Learning 61 estimated probability p(xIG) exceeds a threshold (J. This group includes "novelty detection" techniques based on Bayesian nets (Jaakkola et al., 1996) and, loosely, autoencoder networks (Japkowicz et al., 1995). While p(xIG) can be estimated from only positive examples, novelty detection also requires negative examples for principled generalization, in order to set an appropriate threshold (J which may vary over many orders of magnitude for different concepts. For learning from positive evidence only, our best hope are algorithms that treat a new concept G as an unknown subset of the universe of objects and decide how to generalize G by finding "good" subsets in a hypothesis space H of possible concepts. The Bayesian framework. For this task, the natural hypothesis space H corresponds to all rectangles in the plane. The central challenge in generalizing using the subset approach is that any small set of examples will typically be consistent with many hypotheses (Fig. Ib). This problem is not unique to learning rectangles, but is a universal dilemna when trying to generalize concepts from only limited positive data. The Bayesian solution is to embed the hypothesis space in a probabilistic model of our observations, which allows us to weight different consistent hypotheses as more or less likely to be the true concept based on the particular examples observed. Specifically, we assume that the examples are generated by random sampling from the true concept. This leads to the size principle: smaller hypotheses become more likely than larger hypotheses (Fig. Ib - darker rectangles are more likely), and they become exponentially more likely as the number of consistent examples increases (Fig. lc). The size principle is the key to understanding how we can learn concepts from only a few positive examples. = Formal treatment. We observe n positive examples X {xCI), ... , x Cn )} of concept G and want to compute the generalization/unction p(y E GIX), i.e. the probability that some new object y belongs to G given the observations X. Let each rectangle hypothesis h be denoted by a quadruple (11,/ 2,81,82), where Ii E [-00,00] is the location of h's lower-left comer and 8i E [0,00] is the size of h along dimension i. Our probabilistic model consists of a prior density p( h) and a likelihood function p( X Ih) for each hypothesis h E H. The likelihood is determined by our 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 C. For n examples we then have: p(Xlh) (1) o otherwise, where Ihl denotes the size of h. For rectangle (11,/2,81,82), Ihl is simply 8182 . Note that because each hypothesis must distribute one unit mass oflikelihood over its volume for each example h p(xlh)dh 1), the probability density for smaller consistent hypotheses is greater than for larger hypotheses, and exponentially greater as a function of n. Figs. Ib,c illustrate this size principle for scoring hypotheses (darker rectang!es are more likely). cJx = The appropriate choice of p( h) depends on our background knowledge. If we have no a priori reason to prefer any rectangle hypothesis over any other, we can choose the scaleand location-invariant uninformative prior, p( h) = P(ll, 12, 81 ,82) = 1/(81,82), In any realistic application, however, we will have some prior information. For example, we may know the expected size O'i of rectangle concepts along dimension i in our domain, and then use the associated maximum entropy prior P(ll, 12, 81,82) = exp{ -( 81/0'1 + 82/ 0'2)}. The generalization function p(y E GIX) is computed by integrating the predictions of all hypotheses, weighted by their posterior probabilities p( h IX): p(y E GIX) = r p(y E Glh) p(hIX) dh, (2) lhEH where from Bayes' theorem p(hIX) ex: p(Xlh)p(h) (normalized such that fhEH p(hIX)dh = 1), and p(y E Clh) = 1 if y E hand 0 otherwise. Under the J. B. Tenenbaum 62 uninformative prior, this becomes: (3) Here ri is the maximum distance between the examples in X along dimension i, and di equals 0 if y falls inside the range of values spanned by X along dimension i, and otherwise equals the distance from y to the nearest example in X along dimension i. Under the expected-size prior, p(y E GIX) has no closed form solution valid for all n. However, except for very small values of n (e.g. < 3) and ri (e.g. < 0'i/1O), the following approximation holds to within 10% (and usually much less) error: (4) Fig. 2 (left column) illustrates the Bayesian learner's contours of equal probability of generalization (at p = 0.1 intervals), for different values of nand ri. The bold curve 0.5, a natural boundary for generalizing the concept. corresponds to p(y E GIX) Integrating over all hypotheses weighted by their size-based probabilities yields a broad gradient of generalization for small n (row 1) that rapidly sharpens up to the smallest consistent hypothesis as n increases (rows 2-3), and that extends further along the dimension with a broader range ri of observations. This figure reflects an expected-size prior with 0'1 = 0'2 = axiLwidthl2; using an uninformative prior produces a qualitatively similar plot. = Related work: MIN and Weak Bayes. Two existing subset approaches to concept learning can be seen as variants of this Bayesian framework. The classic MIN algorithm generalizes no further than the smallest hypothesis in H that includes all the positive examples (Bruner et al., 1956; Feldman, 1997). MIN is a PAC learning algorithm for the rectangles task, and also corresponds to the maximum likelihood estimate in the Bayesian framework (Mitchell, 1997). However, while it converges to the true concept as n becomes large (Fig. 2, row 3), it appears extremely conservative in generalizing from very limited data (Fig. 2, row 1). An earlier approach to Bayesian concept learning, developed independently in cognitive psychology (Shepard, 1987) and machine learning (Haussler et al., 1994; Mitchell, 1997), was an important inspiration for the framework of this paper. I call the earlier approach weak Bayes, because it embodies a different generative model that leads to a much weaker likelihood function than Eq. 1. While Eq. 1 came from assuming examples sampled randomly from the true concept, weak Bayes assumes the examples are generated by an arbitrary process independent of the true concept. As a result, the size principle for scoring hypotheses does not apply; all hypotheses consistent with the examples receive a likelihood of 1, instead of the factor of 1/lhln in Eq. 1. The extent of generalization is then determined solely by the prior; for example, under the expected-size prior, (5) Weak Bayes, unlike MIN, generalizes reasonably from just a few examples (Fig. 2, row 1). However, because Eq. 5 is independent of n or ri, weak Bayes does not converge to the true concept as the number of examples increases (Fig. 2, rows 2-3), nor does it generalize further along axes of greater variability. While weak Bayes is a natural model when the examples really are generated independently of the concept (e.g. when the learner himself or a random process chooses objects to be labeled "positive" or "negative" by a teacher), it is clearly limited as a model oflearning from deliberately provided positive examples. In sum, previous subset approaches each appear to capture a different aspect of how humans generalize concepts from positive examples. The broad similarity gradients that emerge Bayesian Modeling ofHuman Concept Learning 63 from weak Bayes seem most applicable when only a few broadly spaced examples have been observed (Fig. 2, row 1), while the sharp boundaries of the MIN rule appear more reasonable as the number of examples increases or their range narrows (Fig. 2, rows 2-3). In contrast, the Bayesian framework guided by the size principle automatically interpolates between these two regimes of similarity-based and rule-based generalization, offering the best hope for a complete model of human concept learning. 3 Experimental data from human subjects This section presents empirical evidence that our Bayesian model - but neither MIN nor weak Bayes - can explain human behavior on the simple rectangle learning task. Subjects were given the task of guessing 2-dimensional rectangular concepts from positive examples only, under the cover story of learning about the range of healthy levels of insulin and cholesterol, as described in Section 1. On each trial of the experiment, several dots appeared on a blank computer screen. Subjects were told that these dots were randomly chosen examples from some arbitrary rectangle of "healthy levels," and their job was to guess that rectangle as nearly as possible by clicking on-screen with the mouse. The dots were in fact randomly generated on each trial, subject to the constraints ofthree independent variables that were systematically varied across trials in a (6 x 6 x 6) factorial design. The three independent variables were the horizontal range spanned by the dots (.25, .5, 1, 2, 4, 8 units in a 24-unit-wide window), vertical range spanned by the dots (same), and number of dots (2,3,4,6, 10,50). Subjects thus completed 216 trials in random order. To ensure that subjects understood the task, they first completed 24 practice trials in which they were shown, after entering their guess, the "true" rectangle that the dots were drawn from. I The data from 6 subjects is shown in Fig. 3a, averaged across subjects and across the two directions (horizontal and vertical). The extent d of subjects' rectangles beyond r, the range spanned by the observed examples, is plotted as a function of rand n, the number of examples. Two patterns of generalization are apparent. First, d increases monotonically with r and decreases with n. Second, the rate of increase of d as a function of r is much slower for larger values of n. Fig. 3b shows that neither MIN nor weak Bayes can explain these patterns. MIN always predicts zero generalization beyond the examples - a horizontal line at d = 0 - for all values of rand n. The predictions of weak Bayes are also independent of rand n: d 0" log 2, assuming subjects give the tightest rectangle enclosing all points y with p(y E G\X) > 0.5. = Under the same assumption, Figs. 3c,d show our Bayesian model's predicted bounds on generalization using uninformative and expected-size priors, respectively. Both versions of the model capture the qualitative dependence of d on rand n, confirming the importance of the size principle in guiding generalization independent of the choice of prior. However, the uninformative prior misses the nonlinear dependence on r for small n, because it assumes an ideal scale invariance that clearly does not hold in this experiment (due to the fixed size of the computer window in which the rectangles appeared). In contrast, the expected-size prior naturally embodies prior knowledge about typical scale in its one free parameter 0". A reasonable value of 0" =5 units (out of the 24-unit-wide window) yields an excellent fit to subjects' average generalization behavior on this task. 4 Conclusions In developing a model of concept learning that is at once computationally principled and able to fit human behavior precisely, I hope to have shed some light on how people are able I Because dots were drawn randomly, the "true" rectangles that subjects saw during practice were quite variable and were rarely the "correct" response according to any theory considered here. Thus it is unlikely that this short practice was responsible for any consistent trends in subjects' behavior. 64 1. B. Tenenbaum to infer the correct extent of a concept from only a few positive examples. The Bayesian model has two key components: (1) a generalization function that results from integrating the predictions of all hypotheses weighted by their posterior probability; (2) the assumption that examples are sampled from the concept to be learned, and not independently of the concept as previous weak Bayes models have assumed. Integrating predictions over the whole hypothesis space explains why either broad gradients of generalization (Fig. 2, row 1) or sharp, rule-based generalization (Fig. 2, row 3) may emerge, depending on how peaked the posterior is. Assuming examples drawn randomly from the concept explains why learners do not weight all consistent hypotheses equally, but instead weight more specific hypotheses higher than more general ones by a factor that increases exponentially with the number of examples observed (the size principle). This work is being extended in a number of directions. Negative instances, when encountered, are easily accomodated by assigning zero likelihood to any hypotheses containing them. The Bayesian formulation applies not only to learning rectangles, but to learning concepts in any measurable hypothesis space - wherever the size principle for scoring hypotheses may be applied. In Tenenbaum (1999), I show that the same principles enable learning number concepts and words for kinds of objects from only a few positive examples. 2 I also show how the size principle supports much more powerful inferences than this short paper could demonstrate: automatically detecting incorrectly labeled examples, selecting relevant features, and determining the complexity of the hypothesis space. Such inferences are likely to be necessary for learning in the complex natural settings we are ultimately interested in. Acknowledgments Thanks to M. Bernstein, W. Freeman, S. Ghaznavi, W. Richards, R Shepard, and Y. Weiss for helpful discussions. The author was a Howard Hughes Medical Institute Predoctoral Fellow. References Bruner, J. A., Goodnow,J. S., & Austin, G. J. (1956). A study of thinking. New York: Wiley. Dietterich, T, Lathrop, R, & Lozano-Perez, T (1997). Solving the multiple-instance problem with axis-parallel rectangles. ArtificiaL Intelligence 89(1-2), 31-71. Feldman, J. (1997). The structure of perceptual categories. J. Math. Psych. 41, 145-170. Haussler, D., Keams, M., & Schapire, R (1994). Bounds on the sample complexity of Bayesian learning using infonnation theory and the VC-dimension. Machine Learning 14, 83-113. Jaakkola, T., Saul, L., & Jordan, M. (1996) Fast learning by bounding likelihoods in sigmoid type belief networks. Advances in NeuraL Information Processing Systems 8. Japkowicz, N., Myers, C., & Gluck, M. (1995). A novelty detection approach to classification. Proceedings of the 14th InternationaL Joint Conference on AritificaL InteLLigence. Kruschke, J. (1992). ALCOVE: An exemplar-based connectionist model of category learning. Psych. Rev. 99,22-44. Mitchell, T (1997). Machine Learning. McGraw-Hill. Muggleton, S. (preprint). Learning from positive data. Submitted to Machine Learning. Shepard, R (1987). Towards a universal law of generalization for psychological science. Science 237,1317-1323. Thnenbaum, J. B. (1999). A Bayesian Frameworkfor Concept Learning. Ph. D. Thesis, MIT Department of Brain and Cognitive Sciences. 2In the framework of inductive logic programming, Muggleton (preprint) has independently proposed that similar principles may allow linguistic grammars to be learned from positive data only. Bayesian Modeling ofHuman Concept Learning 65 MIN Bayes weak Bayes n=6 n= 12 Figure 2: Performance of three concept learning algorithms on the rectangle task. (b) MIN and weak Bayes models (a) Average data from 6 subjects 52.5 ie 2.5 2 2 ~ 1.5 1.5 '0 1 C ~ 0.5 0.5 weak Bayes (0 :: 2) "In & ?? ~ 0 weak Bayes (0:: 1) ~--~~--~----~------ o 2 4 6 0 8 n examples (c) Bayesian model (uninformative prior) 0 "In MIN "In 2 4 6 8 r: Range spanned by 2.5 (d) Bayesian model (expected-size prior) 2.5 2 2 n::2 1.5 1.5 n::3 n=4 n=6 n", 10 n= 50 o 2 4 6 8 2 4 6 8 Figure 3: Data from human subjects and model predictions for the rectangle task. PART II NEUROSCIENCE 

1542 |@word trial:5 version:1 sharpens:1 selecting:1 offering:1 existing:2 glh:1 blank:1 unction:1 assigning:1 must:1 realistic:2 confirming:1 dilemna:1 plot:1 generative:1 intelligence:2 guess:2 plane:1 core:1 short:2 cjx:1 provides:2 detecting:1 math:1 location:2 along:9 become:3 qualitative:1 consists:2 behavioral:1 inside:2 expected:7 embody:1 behavior:5 nor:3 brain:2 freeman:1 automatically:2 window:3 becomes:2 provided:1 mass:1 kind:2 psych:2 textbook:1 developed:1 finding:1 fellow:1 multidimensional:1 ti:1 shed:1 unit:5 medical:1 appear:2 positive:26 understood:1 xig:2 treat:1 quadruple:1 solely:1 might:3 limited:6 range:8 averaged:1 unique:1 responsible:1 acknowledgment:1 testing:1 practice:3 hughes:1 procedure:1 universal:2 empirical:3 word:2 integrating:4 suggest:1 cannot:2 close:1 measurable:1 xci:1 attention:1 kruschke:2 independently:5 rectangular:1 insight:1 haussler:2 rule:3 cholesterol:4 spanned:5 classic:1 suppose:1 programming:1 hypothesis:29 trend:1 richards:1 distributional:1 labeled:2 predicts:1 observed:6 preprint:2 capture:2 decrease:1 principled:3 complexity:2 ultimately:1 solving:1 learner:4 comer:1 easily:1 joint:1 routinely:1 fast:1 describe:1 artificial:1 whose:1 apparent:1 larger:3 valued:1 quite:1 otherwise:3 grammar:1 ability:3 insulin:4 myers:1 net:1 propose:1 aligned:1 relevant:2 rapidly:2 description:1 produce:1 converges:1 object:6 illustrate:2 depending:2 exemplar:2 nearest:2 job:1 eq:4 predicted:1 differ:1 direction:2 guided:1 correct:2 vc:1 human:18 enable:1 backprop:1 require:1 explains:2 generalization:19 really:1 hold:2 considered:2 exp:1 cognition:1 vary:1 smallest:3 applicable:1 infonnation:1 healthy:7 saw:1 successfully:1 alcove:2 weighted:3 reflects:1 hope:3 mit:2 clearly:2 always:1 jaakkola:2 broader:1 conjunction:1 linguistic:1 ax:1 focus:2 refining:1 likelihood:7 contrast:5 rigorous:1 helpful:1 inference:3 typically:1 unlikely:1 nand:1 keams:1 japkowicz:2 interested:1 classification:2 bruner:2 accomodated:1 stateof:1 denoted:1 priori:1 art:1 equal:3 once:1 sampling:1 broad:3 nearly:1 thinking:1 peaked:1 connectionist:1 few:8 randomly:6 ve:2 individual:1 attempt:1 detection:3 interest:1 light:1 perez:1 amenable:1 capable:1 necessary:2 experience:1 loosely:1 ruled:1 plotted:1 theoretical:4 psychological:1 instance:5 classify:1 modeling:5 column:1 earlier:2 cover:1 oflearning:2 subset:5 uniform:1 successful:1 teacher:1 chooses:1 thanks:1 density:3 international:1 gix:5 discriminating:1 ie:1 probabilistic:2 told:1 mouse:1 posi:1 thesis:1 central:1 containing:1 choose:1 xlh:3 cognitive:7 style:1 doubt:1 distribute:1 bold:1 includes:2 goodnow:1 depends:1 view:1 closed:1 bayes:17 parallel:2 contribution:1 yield:2 spaced:1 clh:1 generalize:10 weak:15 bayesian:24 produced:1 worth:1 submitted:1 explain:3 definition:1 against:1 naturally:1 associated:1 di:1 sampled:4 treatment:1 massachusetts:1 ihl:2 mitchell:5 color:1 knowledge:2 impoverished:1 actually:1 appears:1 higher:1 response:1 wei:1 rand:4 formulation:1 just:3 hand:1 horizontal:3 nonlinear:1 indicated:1 dietterich:2 concept:62 true:8 normalized:1 deliberately:1 inductive:2 analytically:1 inspiration:1 lozano:1 entering:1 jbt:1 ll:2 during:1 trying:1 hill:1 complete:1 demonstrate:1 meaning:1 common:1 sigmoid:1 shepard:4 exponentially:3 volume:1 belong:1 cambridge:1 feldman:3 ai:1 dot:8 similarity:2 posterior:3 perspective:1 belongs:1 came:1 joshua:1 scoring:3 seen:2 minimum:1 greater:4 novelty:3 converge:1 monotonically:1 ii:2 multiple:1 infer:2 exceeds:1 frameworkfor:1 muggleton:2 equally:1 prediction:6 variant:1 himself:1 receive:1 background:1 want:1 uninformative:6 interval:2 unlike:1 subject:19 isolate:1 meaningfully:1 member:3 seem:1 jordan:1 call:1 ideal:1 bernstein:1 fit:3 psychology:2 cn:1 bridging:1 interpolates:1 york:1 useful:1 factorial:1 tenenbaum:5 ph:1 category:5 simplest:1 schapire:1 ofthree:1 estimated:2 neuroscience:1 broadly:1 group:2 key:2 threshold:2 drawn:3 neither:2 rectangle:29 sum:1 powerful:1 extends:1 reasonable:3 decide:1 prefer:1 bound:2 encountered:1 constraint:1 precisely:1 ri:5 aspect:2 min:11 extremely:1 separable:1 department:2 developing:2 according:1 smaller:2 across:3 psyche:1 wi:1 rev:1 wherever:1 invariant:1 computationally:1 mutually:1 know:1 tractable:1 generalizes:2 tightest:1 apply:1 observe:1 appropriate:2 slower:1 denotes:1 assumes:2 ensure:1 completed:2 embodies:2 question:1 exclusive:1 dependence:2 guessing:1 gradient:3 distance:2 capacity:1 considers:1 extent:3 reason:1 assuming:3 mini:1 negative:7 design:1 enclosing:1 unknown:1 perform:1 predoctoral:1 vertical:2 observation:3 howard:1 incorrectly:1 extended:1 variability:1 precise:1 varied:1 arbitrary:2 sharp:2 learned:4 narrow:1 beyond:2 able:2 usually:1 pattern:2 xm:1 regime:1 appeared:2 challenge:2 explanation:1 belief:1 natural:5 hybrid:1 technology:1 axis:3 hix:3 autoencoder:1 prior:17 understanding:1 determining:1 law:1 remarkable:1 consistent:9 principle:12 story:1 systematically:1 row:10 austin:1 free:1 formal:1 weaker:1 allow:1 institute:2 neighbor:1 fall:2 wide:2 saul:1 emerge:2 boundary:3 dimension:11 curve:1 world:1 valid:1 contour:1 author:1 qualitatively:1 far:1 observable:1 mcgraw:1 feat:1 logic:1 assumed:2 xi:2 discriminative:1 why:2 learn:4 reasonably:1 investigated:1 complex:2 excellent:1 domain:1 main:1 universe:1 whole:1 bounding:1 fig:17 screen:2 darker:3 wiley:1 lc:1 guiding:1 explicit:1 clicking:1 perceptual:2 ib:3 ix:1 theorem:1 embed:1 specific:1 pac:1 maxi:1 evidence:5 essential:1 ih:1 importance:1 magnitude:1 perceptually:1 illustrates:1 gap:1 aritifical:1 gluck:1 entropy:1 generalizing:3 simply:1 likely:7 applies:2 corresponds:5 dh:3 ma:1 towards:1 man:1 specifically:1 determined:2 except:1 typical:1 miss:1 conservative:1 discriminate:1 invariance:1 e:1 experimental:1 lathrop:1 rarely:2 people:3 support:1 ex:1

Learning Mixture Hierarchies Nuno Vasconcelos Andrew Lippman MIT Media Laboratory, 20 Ames St, EI5-320M, Cambridge, MA 02139, {nuno,lip} @media.mit.edu, http://www.media.mit.edwnuno Abstract The hierarchical representation of data has various applications in domains such as data mining, machine vision, or information retrieval. In this paper we introduce an extension of the Expectation-Maximization (EM) algorithm that learns mixture hierarchies in a computationally efficient manner. Efficiency is achieved by progressing in a bottom-up fashion, i.e. by clustering the mixture components of a given level in the hierarchy to obtain those of the level above. This cl ustering requires onl y knowledge of the mixture parameters, there being no need to resort to intermediate samples. In addition to practical applications, the algorithm allows a new interpretation of EM that makes clear the relationship with non-parametric kernel-based estimation methods, provides explicit control over the trade-off between the bias and variance of EM estimates, and offers new insights about the behavior of deterministic annealing methods commonly used with EM to escape local minima of the likelihood. 1 Introduction There are many practical applications of statistical learning where it is useful to characterize data hierarchically. Such characterization can be done according to either top-down or bottom-up strategies. While the former start by generating a coarse model that roughly describes the entire space, and then successively refine the description by partitioning the space and generating sub-models for each of the regions in the partition; the later start from a fine description, and successively agglomerate sub-models to generate the coarser descriptions at the higher levels in the hierarchy. Bottom-up strategies are particularly useful when not all the data is available at once, or when the dataset is so big that processing it as whole is computationally infeasible. This is the case of machine vision tasks such as object recognition, or the indexing of video databases. In object recognition, it is many times convenient to determine not only which object is present in the scene but also its pose [2], a goal that can be attained by a hierarchical, description where at the lowest level a model is learned for each object pose and all pose models are then combined into a generic model at the top level of the hierarchy. Similarly, Learning Mixture Hierarchies 607 for video indexing, one may be interested in learning a description for each frame and then combine these into shot descriptions or descriptions for some other sort of high level temporal unit [6). In this paper we present an extension of the EM algorithm [I) for the estimation of hierarchical mixture models in a bottom-up fashion. It turns out that the attainment of this goal has far more reaching consequences than the practical applications above. In particular, because a kernel density estimate can be seen as a limiting case ofa mixture model (where a mixture component is superimposed on each sample), this extension establishes a direct connection between so-called parametric and non-parametric density estimation methods making it possible to exploit results from the vast non-parametric smoothing literature [4) to improve the accuracy of parametric estimates. Furthennore, the original EM algorithm becomes a particular case of the one now presented, and a new intuitive interpretation becomes available for an important variation of EM (known as deterministic annealing) that had previously been derived from statistical physics. With regards to practical applications, the algorithm leads to computationally efficient methods for estimating density hierarchies capable of describing data at different resolutions. 2 Hierarchical mixture density estimation Our model consists of a hierarchy of mixture densities, where the data at a given level is described by cl P(X) = L 1I"~p(Xlz~ = I , Md, (I) k= 1 where 1 is the level in the hierarchy (l = 0 providing the coarsest characterization of the data), MI the mixture model at this level, C l the number of mixture components that compose it, 11"~ the prior probability of the kth component, and z~ a binary variable that takes the value 1 if and only if the sample X was drawn from this component. The only restriction on the model is that if node j of levell + 1 is a child of node i of levell, then 11"1+1 J = 11"1+111"1 jlk k' (2) where k is the parent of j in the hierarchy of hidden variables. The basic problem is to compute the mixture parameters of the description at levell given the knowledge of the parameters at level 1 + 1. This can also be seen as a problem of clustering mixture components. A straightforward solution would be to draw a sample from the mixture density at levell + 1 and simply run EM with the number of classes of the level 1 to estimate the corresponding parameters. Such a solution would have at least two major limitations. First, there would be no guarantee that the constraint of equation (2) would be enforced, i.e. there would be no guarantee of structure in the resulting mixture hierarchy, and second it would be computationally expensive, as all the models in the hierarchy would have to be learned from a large sample. In the next section, we show that this is really not necessary. 3 Estimating mixture hierarchies The basic idea behind our approach is, instead of generating a real sample from the mixture model at level L + 1, to consider a virtual sample generated from the same model, use EM to find the expressions for the parameters of the mixture model of levell that best explain this virtual sample, and establish a closed-fonn relationship between these parameters and those of the model at level 1 + I. For this, we start by considering a virtual sample X = {XI, .. . , X C l+ l } from ; \.11+1, where each of the Xi is a virtual sample from one of N. Vasconcelos and A. Lippman 608 the C ' + 1 components of this model, with size Mi = virtual points. 11'! N, where N is the total number of We next establish the likelihood for the virtual sample under the model M" For this, as is usual in the EM literature, we assume that samples from different blocks are independent, Le. C 1+ 1 P(XIM,) = II P(XiIM,), (3) i=1 but, to ensure that the constraint of equation (2) is enforced, samples within the same block are assigned to the same component of M,. Assuming further that, given the knowledge of the assignment the samples are drawn independently from the corresponding mixture component, the likelihood of each block is given by P(XiIMd c1 c1 j = 1 j= 1 = 2: lI'~P(Xilzij = I,M,) = 2: 11'} Mi II p(XrlZij = I,M,), (4) m= 1 where Zij = Z!+I z; is a binary variable with value one if and only if the block Xi is assigned is the mth data point in Xi. Combining equations (3) to the jth component of M" and and (4) we obtain the incomplete data likelihood, under M" for the whole sample xr C 1+ 1 P(XIM,) = c1 M. II 2: 11'; II p(XrlZij = I,M,). i= 1 j = 1 (5) m= 1 This equation is similar to the incomplete data likelihood of standard EM, the main difference being that instead of having an hidden variable for each sample point, we now have one for each sample block. The likelihood of the complete data is given by C 1+ 1 P(X, ZIM,) = c1 II II [lI'~P(Xilzij = 1, M,)f?i , (6) i= 1 j = 1 where Z is a vector containing all the Zij, and the log-likelihood becomes C 1+ 1 log P(X, ZIM,) c1 = 2: 2: Zij 10g(1I';P(Xilzij = 1, M,). (7) i= 1 j = 1 Relying on EM to estimate the parameters of M, leads to the the following E-step _ _ _ P(Xi IZij = I, M,)lI'; E[zijIXi,M,]-P(zij-lIXi , M,)-~ P(X .I. - I lA) I ' (8) L..k , Zzk - , l VI/ lI'k The key quantity to compute is therefore P (Xi IZij = I, M,). Taking its logarithm h ij 10gP(Xilzij = I,M,) 1 M. Mi[M . 2:logP(xrlzij = I,M,)] , i= 1 MiE M 1+ [log P(XIZij = 1, M,)], 1 ., (9) where we have used the law of large numbers, and EM 1+ 1? ? [x] is the expected value of x according the ith mixture component of M'+ 1 (the one from which Xi was drawn). This is an easy computation for most densities commonly used in mixture modeling. It can be shown [5] that for the Gaussian case it leads to .L { ~l _1~ l +I}]M. I 1 [ g(J1.i+ ,/L~ , E~)e-~trace (~J)~' 7r~ (10) 609 Learning Mixture Hierarchies where 9(x, J1., E) is the expression for a Gaussian with mean J1. and covariance E. The M-step consists of maximizing C I +1 Q= cl L L hij 10g(1T~P(Xilzij = 1, Md) (II) i= 1 j = 1 subject to the constraint E j 7r~ = I. Once again, this is a relatively simple task for common mixture models and in [5] we show that for the Gaussian case it leads to the following parameter update equations (12) Ei hij MiJ1.~+ 1 Ei h ij Mi (3) E.~ .. M . [LhijMiE~+1 + LhijMi(J1.~+I-J1.;)(J1.!+1 -J1.;f] .(4) , 'J , i i Notice that neither equation (10) nor equations (12) to (14) depend explicitly on the underlying sample Xi and can be computed directly from the parameters of Ml+l. The algorithm is thus very efficient from a computational standpoint as the number of mixture components in Ml+ 1 is typically much smaller than the size of the sample at the bottom of the hierarchy. 4 Relationships with standard EM There are interesting relationships between the algorithm derived above and the standard EM procedure. The first thing to notice is that by making Mi = I and E~+ 1 = 0, the E and M-steps become those obtained by applying standard EM to the sample composed of the points J1.~+1 . Thus, standard EM can be seen as a particular case of the new algorithm, that learns a two level mixture hierarchy. An initial estimate is first obtained at the bottom of this hierarchy by placing a Gaussian with zero covariance on top of each data point, the model at the second level being then computed from this estimate. The fact that the estimate at the bottom level is nothing more than a kernel estimate with zero bandwidth suggests that other choices of the kernel bandwidth may lead to better overall EM estimates. Under this interpretation, the E~+I become free parameters that can be used to control the smoothness of the density estimates and the whole procedure is equivalent to the composition of three steps: I) find the kernel density estimate that best fits the sample under analysis, 2) draw a larger virtual sample from that density, and 3) compute EM estimates from this larger sample. In section 5, we show that this can leave to significant improvements in estimation accuracy, particularly when the initial sample is small, the free parameters allowing explicit control over the trade-off between the bias and variance of the estimator. Another interesting relationship between the hierarchical method and standard EM can be derived by investigating the role of the size of the underlying virtual sample (which determines M i ) on the estimates. Assuming Mi constant, Mi = M, Vi, it factors out of all summations in equations (12) to (14), the contributions of numerator and denominator canceling each other. In this case, the only significance of the choice of M is its impact on the E-step. Assuming, as before, that E~+I = 0 we once again have the EM algorithm, but where the class-conditional likelihoods of the E-step are now raised to the Mth power. If 610 N Vasconcelos and A. Lippman M is seen as the inverse of temperature, both the E and M steps become those of standard EM under deterministic annealing (DA) I [3] . The DA process is therefore naturally derived from our hierarchical formulation, which gives it a new interpretation that is significantly simpler and more intuitive than those derived from statistical physics. At the start of the process M is set to zero, i.e. no virtual samples are drawn from the Gaussian superimposed on the real dataset, and there is no virtual data. Thus, the assignments hij of the E-step simply become the prior mixing proportions 11"; and the M-step simply sets the parameters of all Gaussians in the model to the sample mean and sample covariance of the real sample. As M increases, the number of virtual points drawn from each Gaussian also increases and for M = 1 we have a single point that coincides with the point on the real training sample. We therefore obtain the standard EM equations. Increasing M further will make the E-step assignments harder (in the limit of M = 00 each point is assigned to a single mixture component) because a larger virtual probability mass is attached to each real point leading to much higher certainty with regards to the reliability of the assignment. Overall, while in the beginning of the process the reduced size of the virtual sample allows the points in the real sample to switch from mixture to mixture easily, as M is increased the switching becomes much less likely. The "exploratory" nature of the initial iterations drives the process towards solutions that are globally good, therefore allowing it to escape local minima. 5 Experimental results In this section, we present experimental results that illustrate the properties of the hierarchical EM algorithm now proposed. We start by a simple example that illustrates how the algorithm can be used to estimate hierarchical mixtures. .. . ~.,...:.:~ ..: , :... :~~y :.~.:: ~,~~,;,i 1!IO - 100 -so 0 50 100 '?.i~ i ' ;~ '~'Tl;~:. 150 Figure I: Mixture hierarchy derived from the model shown in the left. The plot relative to each level of the hierarchy is superimposed on a sample drawn from this model. Only the one-standard deviation contours are shown for each Gaussian. The plot on the left of Figure 1 presents a Gaussian mixture with 16 uniformly weighted components. A sample with 1000 points was drawn from this model, and the algorithm used to find the best descriptions for it at three resolutions (mixtures with 16, 4, and 2 Gaussian). These descriptions are shown in the figure. Notice how the mixture hierarchy naturally captures the various levels of structure exhibited by the data. This example suggests how the algorithm could be useful for applications such as object recognition or image retrieval. Suppose that each of the Gaussians in the leftmost plot of IDA is a technique drawn from analogies with statistical physics that avoids local maxima of the likelihood function (in which standard EM can get trapped) by perfonning a succession of optimizations at various temperatures [31. 611 Learning Mixture Hierarchies ( I'? j '~~~~--H~~W~~~~ o-.I? ...,r'Itol Figure 2: Object recognition task. Left: 8 of the 100 objects in the database. Right: computational savings achieved with hierarchical recognition vs full search. the figure describes how a given pose of a given object populates a 2-D feature space on which object recognition is to be perfonned. In this case, higher levels in the hierarchical representation provide a more generic description of the object. E.g. each of the Gaussians in the model shown in the middle of the figure might provide a description for all the poses in which the camera is on the same quadrant of the viewing sphere, while those in the model shown in the right might represent views from the same hemisphere. The advantage, for recognition or retrieval, of relying on a hierarchal structure is that the search can be perfonned first at the highest resolution, where it is much less expensive, only the best matches being considered at the subsequent levels. Figure 2 illustrates the application of hierarchical mixture modeling to a real object recognition task. Shown on the left side of the figure are 8 objects from the 100 contained in the Columbia object database [2]. The database consists of 72 views (obtained by positioning the camera in 5? intervals along a circle on the viewing sphere), which were evenly separated into a training and a test set. A set of features was computed for each image, and a hierarchical model was then learned for each object in the resulting feature space. While the process could be extended to any number of levels, here we only report on the case of a two-level hierarchy: at the bottom each image is described by a mixture of 8 Gaussians, and at the top each mixture (also with 8 Gaussians) describes 3 consecutive views. Thus, the entire training set is described by 3600 mixtures at the bottom resolution and 1200 at the top. Given an image of an object to recognize, recognition takes place by computing its projection into the feature space, measuring the likelihood of the resulting sample according to each of the models in the database, and choosing the most likely. The complexity of the process is proportional to the database size. The plot on the left of Figure 2 presents the recognition accuracy achieved with the hierarchical representation vs the corresponding complexity, shown as a percent of the complexity required by full search. The full-search accuracy is in this case 90%, and is also shown as a straight line in the graph. As can be seen from the figure, the hierarchical search achieves the full search accuracy with less than 40% of its complexity. We are now repeating this experiments with deeper trees, where we expect the gains to be even more impressive. We finalize by reporting on the impact of smoothing on the quality of EM estimates. For this, we conducted the following Monte Carlo experiment: I) draw 200 datasets Si, i = 1, ... ,200 from the model shown on the left of Figure 1, 2) fit each dataset with EM, 3) measure the correlation coefficient Pi, i = 1, ... ,200 between each of the EM fits and the original model, and 4) compute the sample mean pand variance p. The correlation coefficient is defined by Pi = f f(x)h(x)dxIU f(x)dxf ii(X)dx), where f(x) is the a N. Vasconcelos and A. Lippman 612 8lC 10~ OD -so , 08 - -HI) / -so - - - - 100 ? 200 ? 4CXl - - - 300 J(I) 500 - - 1000 <00 -500 -- - 1000 - o .o'-----'--~, 0--':15:-----:2' O :---25 ~--:'c30--= 36:--..... 40: :---' 45 - - - - - - - - -:: ..-- ----10 15 20 - 2S ~ . " -, -' 30 36 Figure 3: Results of the Monte Carlo experiment described on the text. Left: p as a function 17k. Right: Up as a function of 17k. The various curves in each graph correspond to to different sample sizes. true model and fi(X) the ith estimate, and can be computed in closed form for Gaussian mixtures. The experiment was repeated with various dataset sizes and various degrees of smoothing (by setting the bandwidth of the underlying Gaussian kernel to for various values of O'k). oil Figure 3 presents the results of this experiment. It is clear, from the graph on the left, that smoothing can have a significant impact on the quality of the EM estimates. This impact is largest for small samples, where smoothing can provide up to a two fold improvement estimation accuracy, but can be found even for large samples. The kernel bandwidth allows control over the trade-off between the bias and variance of the estimates. When O'k is zero (standard EM), bias is small but variance can be large, as illustrated by the graph on the right of the figure. As O'k is increased, variance decreases at the cost of an increase in bias (the reason why for large O'k aU lines in the graph of the left meet at the same point regardless ofthe sample size). The point where p is the highest is the point at which the bias-variance trade off is optimal. Operating at this point leads to a much smaller dependence of the accuracy of the estimates on the sample size or, conversely, the need for much smaller samples to achieve a given degree of accuracy. References [I] A. Dempster, N. Laird, and D. Rubin. Maximum-likelihood from Incomplete Data via the EM Algorithm. J. of the Royal Statistical Society, B-39, 1977. [2] H. Murase and S. Nayar. Visual Learning and Recognition of 3-D Objects from Appearence. International Journal of Computer Vision, 14:5-24, 1995. [3] K. Rose, E. Gurewitz, and G. Fox. Vector Quantization by Determinisc Annealing. IEEE Trans. on Information Theory, Vol. 38, July 1992. [4] J. Simonoff. Smoothing Methods in Statistics. Springer-Verlag, 1996. [5] N. Vasconcelos and A. Lippman. Learning Mixture Hierarchies. Technical report, MIT Media Laboratory, 1998. Available from ftp:l/ftp.media.mit.eduJpub/nunolHierMix.ps.gz . [6] N. Vasconcelos and A. Lippman. Content-based Pre-Indexed Video. In Proc. Int. Con! Image Processing, Santa Barbara, California, 1997. 

1543 |@word middle:1 proportion:1 covariance:3 xilzij:5 fonn:1 harder:1 shot:1 initial:3 zij:4 ida:1 od:1 si:1 dx:1 subsequent:1 partition:1 j1:8 plot:4 update:1 v:2 beginning:1 ith:2 provides:1 characterization:2 coarse:1 ames:1 node:2 simpler:1 along:1 direct:1 become:4 consists:3 combine:1 compose:1 introduce:1 manner:1 expected:1 roughly:1 behavior:1 nor:1 relying:2 globally:1 considering:1 increasing:1 becomes:4 dxf:1 estimating:2 underlying:3 medium:5 lixi:1 lowest:1 mass:1 guarantee:2 temporal:1 certainty:1 ofa:1 control:4 partitioning:1 unit:1 before:1 local:3 limit:1 consequence:1 switching:1 io:1 meet:1 might:2 au:1 suggests:2 conversely:1 practical:4 camera:2 block:5 xr:1 lippman:6 procedure:2 significantly:1 convenient:1 projection:1 pre:1 quadrant:1 get:1 applying:1 www:1 restriction:1 deterministic:3 xlz:1 equivalent:1 maximizing:1 appearence:1 straightforward:1 regardless:1 independently:1 resolution:4 insight:1 estimator:1 exploratory:1 variation:1 limiting:1 hierarchy:23 suppose:1 recognition:11 particularly:2 expensive:2 coarser:1 database:6 bottom:9 role:1 capture:1 region:1 trade:4 highest:2 decrease:1 rose:1 dempster:1 complexity:4 depend:1 efficiency:1 easily:1 various:7 separated:1 monte:2 choosing:1 larger:3 furthennore:1 statistic:1 gp:1 laird:1 advantage:1 combining:1 mixing:1 achieve:1 description:12 intuitive:2 parent:1 xim:2 p:1 generating:3 leave:1 object:16 ftp:2 illustrate:1 andrew:1 pose:5 ij:2 murase:1 mie:1 viewing:2 virtual:13 really:1 summation:1 extension:3 considered:1 major:1 achieves:1 consecutive:1 estimation:6 proc:1 largest:1 establishes:1 weighted:1 mit:5 gaussian:11 reaching:1 derived:6 zim:2 improvement:2 likelihood:11 superimposed:3 progressing:1 entire:2 typically:1 hidden:2 mth:2 interested:1 overall:2 smoothing:6 raised:1 once:3 saving:1 vasconcelos:6 having:1 placing:1 report:2 escape:2 composed:1 recognize:1 mining:1 mixture:41 behind:1 capable:1 necessary:1 fox:1 tree:1 incomplete:3 indexed:1 logarithm:1 circle:1 increased:2 modeling:2 measuring:1 logp:1 assignment:4 maximization:1 cost:1 deviation:1 conducted:1 characterize:1 combined:1 st:1 density:10 international:1 off:4 physic:3 again:2 successively:2 containing:1 resort:1 leading:1 li:4 coefficient:2 int:1 explicitly:1 vi:2 later:1 view:3 closed:2 start:5 sort:1 contribution:1 accuracy:8 pand:1 variance:7 succession:1 correspond:1 ofthe:1 carlo:2 finalize:1 drive:1 straight:1 explain:1 canceling:1 nuno:2 naturally:2 mi:8 con:1 gain:1 dataset:4 knowledge:3 higher:3 attained:1 formulation:1 done:1 correlation:2 ei:2 quality:2 oil:1 true:1 former:1 assigned:3 laboratory:2 illustrated:1 numerator:1 coincides:1 leftmost:1 agglomerate:1 complete:1 temperature:2 percent:1 image:5 fi:1 common:1 attached:1 interpretation:4 significant:2 composition:1 cambridge:1 smoothness:1 similarly:1 had:1 reliability:1 impressive:1 operating:1 hemisphere:1 barbara:1 hierarchal:1 verlag:1 binary:2 seen:5 minimum:2 determine:1 july:1 ii:8 full:4 levell:5 match:1 positioning:1 technical:1 offer:1 sphere:2 retrieval:3 impact:4 basic:2 denominator:1 vision:3 expectation:1 iteration:1 kernel:7 represent:1 achieved:3 c1:5 addition:1 fine:1 annealing:4 interval:1 standpoint:1 exhibited:1 subject:1 thing:1 intermediate:1 easy:1 switch:1 fit:3 bandwidth:4 idea:1 expression:2 useful:3 clear:2 santa:1 repeating:1 reduced:1 http:1 generate:1 notice:3 trapped:1 vol:1 key:1 drawn:8 neither:1 vast:1 graph:5 enforced:2 run:1 inverse:1 place:1 reporting:1 draw:3 hi:1 fold:1 refine:1 constraint:3 scene:1 coarsest:1 relatively:1 according:3 describes:3 smaller:3 em:31 making:2 indexing:2 jlk:1 computationally:4 equation:9 previously:1 turn:1 describing:1 available:3 gaussians:5 hierarchical:14 generic:2 original:2 top:5 clustering:2 ensure:1 exploit:1 establish:2 society:1 quantity:1 parametric:5 strategy:2 dependence:1 md:2 usual:1 kth:1 evenly:1 reason:1 assuming:3 relationship:5 providing:1 onl:1 hij:3 trace:1 allowing:2 datasets:1 extended:1 frame:1 populates:1 required:1 connection:1 california:1 learned:3 trans:1 royal:1 video:3 power:1 perfonned:2 improve:1 gz:1 columbia:1 gurewitz:1 text:1 prior:2 literature:2 relative:1 law:1 expect:1 interesting:2 limitation:1 proportional:1 analogy:1 c30:1 degree:2 rubin:1 pi:2 free:2 infeasible:1 jth:1 bias:6 side:1 deeper:1 taking:1 regard:2 curve:1 avoids:1 contour:1 simonoff:1 commonly:2 far:1 ml:2 investigating:1 xi:8 search:6 cxl:1 why:1 lip:1 nature:1 attainment:1 cl:3 domain:1 da:2 significance:1 hierarchically:1 main:1 big:1 whole:3 nothing:1 child:1 repeated:1 tl:1 fashion:2 lc:1 sub:2 explicit:2 learns:2 down:1 quantization:1 illustrates:2 simply:3 likely:2 visual:1 contained:1 springer:1 determines:1 ma:1 conditional:1 goal:2 towards:1 content:1 uniformly:1 called:1 total:1 experimental:2 la:1 perfonning:1 izij:2 nayar:1

Exploratory Data Analysis Using Radial Basis Function Latent Variable Models Alan D. Marrs and Andrew R. Webb DERA St Andrews Road, Malvern Worcestershire U.K. WR14 3PS {marrs,webb}@signal.dera.gov.uk @British Crown Copyright 1998 Abstract Two developments of nonlinear latent variable models based on radial basis functions are discussed: in the first, the use of priors or constraints on allowable models is considered as a means of preserving data structure in low-dimensional representations for visualisation purposes. Also, a resampling approach is introduced which makes more effective use of the latent samples in evaluating the likelihood. 1 INTRODUCTION Radial basis functions (RBF) have been extensively used for problems in discrimination and regression. Here we consider their application for obtaining low-dimensional representations of high-dimensional data as part of the exploratory data analysis process. There has been a great deal of research over the years into linear and nonlinear techniques for dimensionality reduction. The technique most commonly used is principal components analysis (PCA) and there have been several nonlinear generalisations, each taking a particular definition of PCA and generalising it to the nonlinear situation. One approach is to find surfaces of closest fit (as a generalisation of the PCA definition due to the work of Pearson (1901) for finding lines and planes of closest fit). This has been explored by Hastie and Stuetzle (1989), Tibshirani (1992) (and further by LeBlanc and Tibshirani, 1994) and various authors using a neural network approach (for example, Kramer, 1991). Another approach is one of variance maximisation subject to constraints on the transformation (Hotelling, 1933). This has been investigated by Webb (1996), using a transformation modelled as an RBF network, and in a supervised context in Webb (1998). An alternative strategy also using RBFs, based on metric multidimensional scaling, is described by Webb (1995) and Lowe and Tipping (1996). Here, an optimisation criterion, A. D. Marrs and A. R. Webb 530 termed stress, is defined in the transformed space and the weights in an RBF model determined by minimising the stress. The above methods use a radial basis function to model a transformation from the highdimensional data space to a low-dimensional representation space. A complementary approach is provided by Bishop et al (1998) in which the structure of the data is modelled as a function of hidden or latent variables. Termed generative topographic mapping (GTM), the model may be regarded as a nonlinear generalisation of factor analysis in which the mapping from latent space to data space is characterised by an RBF. Such generative models are relevant to a wide range of applications including radar target modelling, speech recognition and handwritten character recognition. However, one of the problems with GTM that limits its practical use for visualising data on manifolds in high dimensional space arises from distortions in the structure that it imposes. This is acknowledged in Bishop et al (1997) where 'magnification factors' are introduced to correct for the GTM's deficiency as a means of data visualisation. This paper considers two developments: constraints on the permissible models and resampiing of the latent space. Section 2 presents the background to latent variable models; Model constraints are discussed in Section 3. Section 4 describes a re-sampling approach to estimation of the posterior pdf on the latent samples. An illustration is provided in Section 5. 2 BACKGROUND Briefly, we shall re-state the basic GTM model, retaining the notation of Bishop et al (1998). Let Ui, i = 1, ... , N}, ti E RP represent measurements on the data space variables; Z E R. represent the latent variables. Let t be normally-distributed with mean y(z; W) and covariance matrix {3-1 I; y(z; W) is a nonlinear transformation that depends on a set of parameters W. Specifically, we shall assume a basis function model M L Wi<Pi(Z) y(z; W) = i=1 where the vectors Wi E R.D are to be determined through optimisation and {<Pi, i = 1, . . . , M} is a set of basis functions defined on the latent space. The data distribution may be written p(tIW,{3) = ! (1) p(tlz; W , {3)p(z)dz where, under the assumptions of normality, {3 ) p(tlz; W,{3) = ( 271" D /2 exp {{3 -"2 ly (z; W) - tll 2 } Approximating the integral by a finite sum (assuming the functions p(z) and y(z) do not vary too greatly compared with the sample spacing), we have K p(tIW,{3) = LPiP(tlzi; W,{3) i =1 which may be regarded as a function of the parameters W and {3 that characterise y . (2) Exploratory Data Analysis Using Radial Basis Function Latent Variable Models Given the data set {ti' i 531 = 1, ... ,N}, the log likelihood is given by N L(W,J3) = I)n[p(t;IW,J3)] j=l which may be maximised using a standard EM approach (Bishop et ai, 1998). In this case, we have Pj 1 = N N L~n (3) n=l as the re-estimate of the mixture component weights, Pj, at the (m ~ n _ p~m)p(tnIZi; w(m), J3(m?) - EiP~m)p(tnlzi;W(m),J3(m?) + 1) step, where (4) and (.) (m) denotes values at the mth step. Note that Bishop et al (1998) do not re-estimate P;; all values are taken to be equal. The number of P; terms to be re-estimated is K, the number of terms used to approximate the integral (1). We might expect that the density is smoothly varying and governed by a much fewer number of parameters (not dependent on K). The re-estimation equation for the D x M matrix W = [w 11 ... IWM] is = TT RT +[+TG+]-l W(m+l) (5) where G is the K x K diagonal matrix with N Gjj = LRjn n=l = and TT = [tIl .. . ltN], +T [tfJ(Zl)I .. . ltfJ(ZK)]. The term l/J3(m) = 1/(ND) E~l E~l Rjilti - w(m+l)tfJ(Zj) 12. J3 is re-estimated as Once we have determined parameters of the transformation, we may invert the model by asking for the distribution of Z given a measurement ti. That is, we require p(Zlti) = p(tilz)p(z) f p(tilz)p(z)dz (6) For example, we may plot the position of the peak of the distribution p( Ziti) for each data sample ti. 3 APPLYING A CONSTRAINT One way to retain structure is to impose a condition that ensures that a unit step in the latent space corresponds to a unit step in the data space (more or less). For a single latent variable, Xl, we may impose the constraints that l ay 12 = 1 aXl which may be written, in terms of W as ifwTWil 1 532 A. D. Marrs and A. R. Webb 8c/J. where;l = ~ The derivative of the data space variable with respect to the latent variable has unit magnitude. The derivative is of course a function of Xl and imposing such a condition at each sample point in latent space would not be possible owing to the smoothness of the RBF model. However, we may average over the latent space, where ( .) denotes average over the latent space. In general, for L latent variables we may impose a constraint JTWTW J = 1 L leading to the penalty term Tr {A(JTWTW J - IL)} where J is an M x L matrix with jth column 8?/8xj and A is a symmetric matrix of Lagrange multipliers. This is very similar to regularisation terms. It is a condition on the norm of W; it incorporates the Jacobian matrix J and a symmetric L x L matrix of Lagrange multipliers, A. The re-estimation solution for W may be written (7) with A chosen so that the constraint JT W T W J = 1 L is satisfied. We may also use the derivatives of the transformation to define a distortion measure or magnification factor, M(Zj W) = IIJTWTW J - 1112 which is a function of the latent variables and the model parameters. A value of zero shows that there is no distortion 1? An alternative to the constraint approach above is to introduce a prior on the allowable transformations using the magnification factor; for example, P(W) ~ exp(-AM(zj W)) (8) where A is a regularisation parameter. This leads to a modification to the M-step reestimation equation for W, providing a maximum a posteriori estimate. Equation (8) provides a natural generalisation of PCA since for the special case of a linear transformation (Pi = Xi, M = L), the solution for W is the PCA space as A ~ 00. 4 RESAMPLING THE LATENT SPACE Having obtained a mapping from latent space to data space using the above constraint, we seek a better estimate to the posterior pdf of the latent samples. Current versions of GTM require the latent samples to be uniformly distributed in the latent space which leads to distortions when the data of interest are projected into the latent space for visualisation. Since the responsibility matrix R can be used to determine a weight for each of the latent samples it is possible to update these samples using a resampling scheme. We propose to use a resampling scheme based upon adaptive kernel density estimation. The basic procedure places a Gaussian kernel on each latent sample. This results in a Gaussian 1Note that this differs from the measure in the paper by Bishop et aI, where a rati()-()f-areas criterion is used, a factor which is unity for zero distortion, but may also be unity for some distortions. Exploratory Data Analysis Using Radial Basis Function Latent Variable Models 533 mixture representation of the pdf of the latent samples p( x It), K p(xlt) = ~PiN(lLi' E i ), (9) i=l where each mixture component is weighted according to the latent sample weight Pi. Initially, the Ei'S are all equal, taking their value from the standard formula of Silverman (1986), Ei = hLy, (10) where matrix Y is an estimate of the covariance of p( x )and, (11) If the kernels are centered exactly on the latent samples, this model artificially inflates the variance of the latent samples. Following West (1993) we perform kernel shrinkage by making the lLi take the values (12) where jL is the mean of the latent samples. This ensures that there is no artificial inflation of the variance. To reduce the redundancy in our initially large number of mixture components, we propose a kernel reduction scheme in a similar manner to West. However, the scheme used here differs from that of West and follows a scheme proposed by Salmond (1990). Essentially, we chose the component with the smallest weight and its nearest neighbour, denoting these with subscripts 1 and 2 respectively. These components are then combined into a single component denoted with subscript c as follows, Ec = Pl[El + (lLc Pc = Pl + P2 (13) PllLl + P21L2 IL = --=---= c Pc (14) -lLl)(lL c -lLl)T] + P2[E 2 + (lLc -1L2)(lLc -1L2)T]. (15) Pc This procedure is repeated until some stopping criterion is met. The stopping criterion could be a simple limit upon the number of mixture components ie; smaller than K but sufficiently large to model the data structure. Alternatively, the average kernel covariance and between kernel covariance can be monitored and the reduction stopped before some multiple (eg. 10) of the average kernel covariance exceeds the between kernel covariance. Once a final mixture density estimate is obtained, a new set of equally weighted latent samples can be drawn from it. The new latent samples represent a better estimate of the posterior pdf of the latent samples and can be used, along with the existing RBF mapping, to calculate a new responsibility matrix R. This procedure can be repeated to obtain a further improved estimate of the posterior pdf which, after only a couple of iterations can lead to good estimates of the posterior pdf which further iterations fail to improve upon. 5 RESULTS A latent variable model based oil a spherically-symmetric Gaussian RBF has been implemented. The weights and the centres of the RBF were initialised so that the solution best approximated the zer<rdistortion principal components solution for tw<rdimensional projection. A. D. Marrs and A. R. Webb 534 For our example we chose to construct a simulated data set with easily identifiable structure. Four hundred points lying on the letters "NIPS" were sampled and projected onto a sphere of radius 50 such that the points lay between 25 0 and 175 0 longitude and 750 and 125 0 latitude with Gaussian noise of variance 4.0 on the radius of each point. The resulting data are shown in figure 1. ToY dataset Figure 1: Simulated data. I ..,:.' .:\ ~.. ~. "I: .i,~ ? .> I .~.,...... I :: I I r...1 I :., ...... ?~: -to.,. -u l' Figure 2: Results for standard GTM model. Figure 3: Results for regularisedlresampled model. Figure 2 shows results for the standard GTM (uniform grid of latent samples) projection of the data to two dimensions. The central figure shows the projection onto the latent space, exhibiting significant distortion. The left figure shows the projection of the regular grid of latent samples (red points) into the data space. Distortion of this grid can be easily seen. The right figure is a plot of the magnification factor as defined in section 3, with mean value of 4.577. For this data set most stretching occurs at the edges of the latent variable space. Figure 3 shows results for the regularisedlresampled version of the latent variable model for A = 1.0. Again the central figure shows the projection onto the latent space after 2 iterations of the resampling procedure. The left-hand figure shows the projection of the initial regular grid of latent samples into the data space. The effect of regularisation is evident by the lack of severe distortions. Finally the magnification factors can be seen in the right-hand figure to be lower, with a mean value of 0.976. Exploratory Data Analysis Using Radial Basis Function Latent Variable Models 535 6 DISCUSSION We have considered two developments of the GTM latent variable model: the incorporation of priors on the allowable model and a resampling approach to the maximum likelihood parameter estimation. Results have been presented for this regularisedlresampling approach and magnification factors lower than the standard model achieved, using the same RBF model. However, further reduction in magnification factor is possible with different RBF models, but the example illustrates that resampling offers a more robust approach. Current work is aimed at assessing the approach on realistic data sets. References Bishop, C.M. and Svensen, M. and Williams, C.K.1. (1997). Magnification factors for the GTM algorithm. lEE International Conference on Artificial Neural Networks, 465-471. Bishop, C.M. and Svensen, M. and Williams, C.K.1. (1998). GTM: the generative topographic mapping. Neural Computation, 10,215-234. Hastie, T. and Stuetzle, W. (1989). Principal curves, Journal of the American Statistical Association, 84, 502-516. Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417-441,498-520. Kramer, M.A. (1991). Nonlinear principal component analysis using autoassociative neural networks. American Institute of Chemical Engineers Journal, 37(2),233-243. LeBlanc, M. and Tibshirani, R. (1994). Adaptive principal surfaces. Journal of the American Statistical Association, 89(425), 53-664. Lowe, D. and Tipping, M. (1996). Feed-forward neural networks and topographic mappings for exploratory data analysis. Neural Computing and Applications, 4, 83-95. Pearson, K. (1901). On lines and planes of closest fit. Philosophical Magazine, 6, 559-572. Salmond, D.J. (1990). Mixture reduction algorithms for target tracking in clutter. Signal & Data processing of small targets, edited by O. Drummond, SPlE, 1305. Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall,1986. Tibshirani, R. (1992). Principal curves revisited. Statistics and Computing, 2(4), 183-190. Webb, A.R. (1995). Multidimensional scaling by iterative majorisation using radial basis functions. Pattern Recognition, 28(5), 753-759. Webb, A.R. (1996). An approach to nonlinear principal components analysis using radially-symmetric kernel functions. Statistics and Computing, 6, 159-168. Webb, A.R. (1997). Radial basis functions for exploratory data analysis: an iterative majorisation approach for Minkowski distances based on multidimensional scaling. Journal of Classification, 14(2),249-267. Webb, A.R. (1998). Supervised nonlinear principal components analysis. (submitted for publication ). West, M. (1993). Approximating posterior distributions by mixtures. J. R. Statist. Soc B, 55(2), 409-422. 

1544 |@word version:2 briefly:1 norm:1 nd:1 seek:1 covariance:6 tiw:2 tr:1 reduction:5 initial:1 denoting:1 existing:1 current:2 written:3 realistic:1 plot:2 update:1 resampling:7 discrimination:1 generative:3 fewer:1 plane:2 maximised:1 provides:1 revisited:1 along:1 manner:1 introduce:1 gov:1 lll:2 provided:2 notation:1 finding:1 transformation:8 multidimensional:3 ti:4 axl:1 exactly:1 uk:1 zl:1 normally:1 ly:1 unit:3 before:1 limit:2 subscript:2 might:1 chose:2 range:1 practical:1 maximisation:1 differs:2 silverman:2 procedure:4 stuetzle:2 area:1 projection:6 radial:9 road:1 regular:2 onto:3 context:1 applying:1 dz:2 williams:2 educational:1 regarded:2 exploratory:7 target:3 magazine:1 recognition:3 magnification:8 approximated:1 lay:1 majorisation:2 visualising:1 calculate:1 ensures:2 edited:1 ui:1 radar:1 upon:3 basis:11 easily:2 various:1 gtm:10 effective:1 artificial:2 pearson:2 distortion:9 statistic:3 topographic:3 final:1 xlt:1 leblanc:2 propose:2 relevant:1 zer:1 drummond:1 p:1 assessing:1 andrew:2 svensen:2 nearest:1 p2:2 longitude:1 implemented:1 soc:1 met:1 exhibiting:1 radius:2 correct:1 owing:1 centered:1 tlz:2 require:2 pl:2 lying:1 inflation:1 considered:2 sufficiently:1 hall:1 exp:2 great:1 mapping:6 vary:1 smallest:1 purpose:1 estimation:6 iw:1 weighted:2 gaussian:4 shrinkage:1 varying:1 tll:1 publication:1 modelling:1 likelihood:3 greatly:1 am:1 posteriori:1 dependent:1 el:1 stopping:2 initially:2 hidden:1 visualisation:3 mth:1 transformed:1 classification:1 denoted:1 retaining:1 development:3 special:1 equal:2 once:2 construct:1 having:1 sampling:1 chapman:1 inflates:1 neighbour:1 interest:1 severe:1 mixture:8 pc:3 copyright:1 integral:2 edge:1 re:8 stopped:1 column:1 asking:1 tg:1 wr14:1 hundred:1 uniform:1 too:1 combined:1 st:1 density:4 peak:1 international:1 ie:1 ltn:1 retain:1 lee:1 again:1 central:2 satisfied:1 american:3 derivative:3 til:1 leading:1 toy:1 depends:1 lowe:2 responsibility:2 red:1 hly:1 rbfs:1 il:2 variance:4 eip:1 stretching:1 modelled:2 handwritten:1 lli:2 submitted:1 definition:2 initialised:1 monitored:1 couple:1 sampled:1 dataset:1 radially:1 dimensionality:1 feed:1 tipping:2 supervised:2 improved:1 until:1 hand:2 gjj:1 ei:2 nonlinear:9 lack:1 oil:1 effect:1 multiplier:2 chemical:1 symmetric:4 spherically:1 deal:1 eg:1 ll:1 criterion:4 allowable:3 stress:2 pdf:6 tt:2 ay:1 evident:1 crown:1 jl:1 discussed:2 association:2 measurement:2 significant:1 imposing:1 ai:2 smoothness:1 grid:4 centre:1 surface:2 closest:3 posterior:6 termed:2 preserving:1 seen:2 impose:3 determine:1 signal:2 multiple:1 alan:1 exceeds:1 minimising:1 sphere:1 offer:1 equally:1 j3:6 regression:1 basic:2 optimisation:2 metric:1 essentially:1 iteration:3 represent:3 kernel:10 invert:1 achieved:1 background:2 spacing:1 permissible:1 subject:1 incorporates:1 xj:1 fit:3 psychology:1 hastie:2 reduce:1 pca:5 penalty:1 speech:1 autoassociative:1 aimed:1 characterise:1 clutter:1 extensively:1 statist:1 zj:3 estimated:2 tibshirani:4 shall:2 redundancy:1 four:1 acknowledged:1 drawn:1 pj:2 year:1 sum:1 letter:1 place:1 scaling:3 identifiable:1 constraint:10 deficiency:1 incorporation:1 minkowski:1 according:1 describes:1 smaller:1 em:1 character:1 unity:2 wi:2 tw:1 modification:1 making:1 taken:1 equation:3 pin:1 fail:1 hotelling:2 alternative:2 rp:1 dera:2 denotes:2 approximating:2 occurs:1 strategy:1 marrs:5 rt:1 diagonal:1 distance:1 simulated:2 manifold:1 considers:1 assuming:1 illustration:1 providing:1 webb:12 perform:1 finite:1 situation:1 introduced:2 philosophical:1 nip:1 pattern:1 latitude:1 including:1 natural:1 normality:1 scheme:5 improve:1 prior:3 l2:2 regularisation:3 expect:1 imposes:1 pi:4 course:1 jth:1 salmond:2 institute:1 wide:1 taking:2 distributed:2 curve:2 dimension:1 llc:3 evaluating:1 author:1 commonly:1 adaptive:2 projected:2 forward:1 ec:1 approximate:1 reestimation:1 generalising:1 xi:1 alternatively:1 latent:46 iterative:2 zk:1 robust:1 obtaining:1 investigated:1 complex:1 artificially:1 noise:1 repeated:2 complementary:1 malvern:1 west:4 position:1 xl:2 governed:1 jacobian:1 british:1 formula:1 bishop:8 jt:1 explored:1 magnitude:1 illustrates:1 smoothly:1 tfj:2 lagrange:2 tracking:1 corresponds:1 kramer:2 rbf:10 generalisation:4 determined:3 characterised:1 specifically:1 uniformly:1 principal:9 engineer:1 highdimensional:1 arises:1

Independent Component Analysis of Intracellular Calcium Spike Data Klaus Prank, Julia Borger, Alexander von zur Miihlen, Georg Brabant, Christof Schoil Department of Clinical Endocrinology Medical School Hannover D-30625 Hannover Germany Abstract Calcium (Ca2 +)is an ubiquitous intracellular messenger which regulates cellular processes, such as secretion, contraction, and cell proliferation. A number of different cell types respond to hormonal stimuli with periodic oscillations of the intracellular free calcium concentration ([Ca 2 +]i). These Ca2+ signals are often organized in complex temporal and spatial patterns even under conditions of sustained stimulation. Here we study the spatio-temporal aspects of intracellular calcium ([Ca 2+]i) oscillations in clonal J3-cells (hamster insulin secreting cells, HIT) under pharmacological stimulation (Schofi et al., 1996). We use a novel fast fixed-point algorithm (Hyvarinen and Oja, 1997) for Independent Component Analysis (ICA) to blind source separation of the spatio-temporal dynamics of [Ca 2 +]i in a HIT-cell. Using this approach we find two significant independent components out of five differently mixed input signals: one [Ca2+]i signal with a mean oscillatory period of 68s and a high frequency signal with a broadband power spectrum with considerable spectral density. This results is in good agreement with a study on high-frequency [Ca 2+]j oscillations (Palus et al., 1998) Further theoretical and experimental studies have to be performed to resolve the question on the functional impact of intracellular signaling of these independent [Ca 2 +]i signals. K. Prank et al. 932 1 INTRODUCTION Independent component analysis (ICA) (Comon, 1994; Jutten and Herault, 1991) has recently received much attention as a signal processing method which has been successfully applied to blind source separation and feature extraction. The goal of ICA is to find independent sources in an unknown linear mixture of measured sensory data. This goal is obtained by reducing 2nd-order and higher order statistical dependencies to make the signals as independent as possible. Mainly three different approaches for ICA exist. The first approach is based on batch computations minimizing or maximizing some relevant criterion functions (Cardoso, 1992; Comon, 1994). The second category contains adaptive algorithms often based on stochastic gradient methods, which may have implementations in neural networks (Amari et al., 1996; Bell and Sejnowski, 1995; Delfosse and Loubaton, 1995; Hyvarinen and Oja, 1996; Jutten and Herault, 1991; Moreau and Macchi, 1993; Oja and Karhunen, 1995). The third class of algorithms is based on a fixed-point iteration scheme for finding the local extrema of the kurtosis of a linear combination of the observed variables which is equivalent to estimating the non-Gaussian independent companents (Hyvarinen and Oja 1997). Here we use the fast fixed-point algorithm for independent component analysis proposed by Hyvarinen and Oja (1997) to analyze the spatia-temporal dynamics of intracellular free calcium ([Ca2+]i) in a hamster insulin secreting cell (HIT). Oscillations of [Ca 2+]i have been reported in a number of electrically excitable and non-excitable cells and the hypotheses of frequency coding were proposed a decade ago (Berridge and Galione, 1988). Recent experimental results clearly demonstrate that [Ca 2 +]i oscillations and their frequency can be specific for gene activation concerning the efficiency as well as the selectivity (Dolmetsch et al., 1998). Cells are highly compartmentalized structures which can not be regarded as homogenous entities. Thus, [Ca 2+]i oscillations do not occur uniformly throughout the cell but are initiated at specific sites which are distributed in a functional and nonunifortm manner. These [Ca 2 +]i oscillations spread across individual cells in the form of Ca2+ waves. [Ca2+]i gradients within cells have been proposed to initiate cell migration, exocytosis, lymphocyte, killer cell activity, acid secretion, transcellular ion transport, neurotransmitter release, gap junction regulation, and numerous other functions (Tsien and Tsien, 1990). Due to this fact it is of major importance to study the spatia-temporal aspects of [Ca 2 +]i signaling in small sub compartments using calcium-specific fluorescent reporter dyes and digital videomicroscopy rather than studying the cell as a uniform entity. The aim of this study was to define the independent components of the spatia-temporal [Ca 2 +]i signal. 2 2.1 METHODS FAST FIXED-POINT ALGORITHM USING KURTOSIS FOR INDEPENDENT COMPONENT ANALYSIS In Independent Component Analysis (ICA) the original independent sources are unknown. In this study we have recorded the [Ca 2 +]i signal in single HIT-cells under pharmacological stimulation at different subcellular regions (m = 5) in parallel. The [Ca 2+]i signals (mixtures of sources) are denoted as Xl, X2, ?? ? , X m . Each Xi is expressed as the weighted sum of n unknown statistically independent compa- Independent Component Analysis ofIntracellular Calcium Spike Data 933 nents (ICs), denoted as SI, S2, ?.? , Sn. The components are assumed to be mutually statistically independent and zero-mean. The measured signals Xi as well as the independent component variables can be arranged into vectors x = (Xl, X2, ?.. ,XIIl ) and 8 = (81,82, ... , 8 n ) respectively. The linear relationship is given by: X=A8 (I) Here A is a constant mixing matrix whose elements aij are the unknown coefficients of the mixtures. The basic problem of ICA is to estimate both the mixing matrix A and the realizations of the Si using only observations of the mixtures X j. In order to perform ICA, it is necessary to have at least as many mixtures as there are independent sources (m 2: n). The assumption of zero mean of the ICs is no restriction, as this can always be accomplished by subtracting the mean from the random vector x. The ICs and the columns of A can only be estimated up to a mUltiplicative constant, because any constant multiplying an IC in eq. 1 could be cancelled by dividing the corresponding column of the mixing matrix A by the same constant. For mathematical convenience, the ICs are defined to have unit variance making the (non-Gaussian) ICs unique, up to their signs (Comon, 1994). Here we use a novel fixed-point algorithm for ICA estimation which is based on 'contrast' functions whose extrema are closely connected to the estimation of ICs (Hyvarinen and OJ a, 1997). This method denoted as fast fixed-point algorithm has a number of desirable properties. First, it is easy to use, since there are no user-defined parameters. Furthermore, the convergence is fast, conventionally in less than 15 steps and for an appropriate contrast function, the fixed-point algorithm is much more robust against outliers than most ICA algorithms. Most solutions to the ICA problem use the fourth-order cumulant or kurtosis of the signals, defined for a zero-mean random variable x as: (2) where E{ x} denotes the mathematical expectation of x. The kurtosis is negative for source signals whose amplitude has sub-Gaussian probability densitites (distribution flatter than Gaussian, positive for super Gaussian) sharper than Gaussian, and zero for Gausssian densities. Kurtosis is a contrast function for ICA in the following sense. Consider a linear combination of the measured mixtures x, say wTx, where the vector w is constrained so that E{(w T X}2} = 1. When w T x = ?Si, for some i, i.e. when the linear combination equals, up to the sign, one of the ICs, the kurtosis of w T x is locally minimized or maximized. This property is widely used in ICA algorithms and forms the basis of the fixed-point algorithm used in this study which finds the relevant extrema of kurtosis also for non-whitened data. Based on this fact, Hyvarinen and Oja (1997) introduced a very simple and highly efficient fixed-point algorithm for computing ICA, calculated over sphered zero-mean vectors x, that is able to find the rows of the separation matrix (denoted as w) and so identify one independent source at a time. The algorithm which computes a gradient descent over the kurtosis is defined as follows: 1. Take a random initial vector Wo of unit norm. Let 1 = 1. 2. Let WI = E{V(Wf-1V}3} - 3WI-l. The expectation can be estimated using a large sample of Vk vectors. K. Prank et al. 934 3. Divide WI by its norm (e.g. the Euclidean norm II W 11= .J~i wn? 4. If 1WfWI-l 1is not close enough to 1, let 1 = 1 + 1 and go back to step 2. Otherwise, output the vector WI. To calculate more than one solution, the algorithm may be run as many times as required. It is nevertheless, necessary to remove the information contained in the solutions already found, to estimate each time a different independent component. This can be achieved, after the fourth step of the algorithm, by simply subtracting the estimated solution 8 = w T v from the unsphered data x. In the first step of analysis we determined the eigenvalues of the covariance matrix of the measured [Ca 2+]i signals Si to reduce the dimensionality of the system. Then the fast fixed-point algorithm was run using the experimental [Ca 2 +]i data to determine the lOs. The resulting lOs were analyzed in respect to their frequency content by computing the Fourier power spectrum. 2.2 MEASUREMENT OF INTRACELLULAR CALCIUM IN HIT-CELLS To measure [Ca2+]i' HIT (hamster insulin secreting tumor)-cells were loaded with the fluorescent indicator Fura-2/ AM and Fura-2 fluorescence was recorded at five different subcellular regions in parallel using a dual excitation spectrofluorometer videoimaging system. The emission wavelength was 510 nm and the excitation wavelengths were 340 nm and 380 nm respectively. The ration between the excitation wavelength (F340nm/ F38onm) which correlates to [Ca2+]i was sampled at a rate of 1 Hz over 360 s. [Ca 2 +]i spikes in this cell were induced by the administration of 1 nM arginine vasopressin (AVP). 3 RESULTS From the five experimental [Ca 2 +]i signals (Fig. 1) we determined two significant eigenvalues of the covariance matrix. The fixed-point algorithm converged in less than 15 steps and yielded two different lOs, one slowly oscillating component with a mean period of 68 s and one component with fast irregular oscillations with a flat broadband power spectrum (Fig. 2). The spectral density of the second component was considerably larger than that for the high-frequency content of the first slowly oscillating component. 4 CONCLUSIONS Ohanges in [Ca 2 +]i associated with Ca 2+ oscillations generally do not occur uniformly throughout the cell but are initiated at specific sites and are able to spread across individual cells in the form of intracellular Ca2+ waves. Furthermore, Ca 2+ signaling is not limited to single cells but occurs between adjacent cells in the form of intercellular Ca 2 + waves. The reasons for these spatio-temporal patterns of [Ca2+]i are not yet fully understood. It has been suggested that information is encoded in the frequency, rather than the amplitude, of Ca2+ oscillations, which has the advantage of avoiding prolonged exposures to high [Ca2+]i. Another advantage of 935 Independent Component Analysis ofIntracellular Calcium Spike Data =}0~~J~~~j j~~~~2i o 50 100 150 200 250 300 -4 {k~~g LL 0 50 100 150 o 50 100 150 -~ -4 =~~: o 50 100 200 250 300 : : : :~j 150 200 250 300 200 250 300 rim. (5) Figure 1: Experimental time series of [Ca2+]i in a ,B-cell (insulin secreting cell from a hamster, HIT-cell) determined in five subcellular regions. The data are given as the ratio between both exciation wavelengths of 340 nm and 380 nm respectively which correspond to [Ca2+k [Ca2+]i can be calculated from this ratio. The plotted time series are whitened. frequency modulated signaling is its high signal-to-noise ratio. In the spatial domain, the spreading of a Ca 2+ oscillation as a Ca2+ wave provides a mechanism by which the regulatory signal can be distributed throughout the cell. The extension of Ca2+ waves to adjacent cells by intercellular communication provides one mechanism by which multicellular systems can effect coordinated and cooperative cell responses to localized stimuli. In this study we demonstrated that the [Ca2+]i signal in clonal ,B-cells (HIT cells) is composed of two independent components using spatio-temporal [Ca 2 +]i data for analysis. One component can be described as large amplitude slow frequency oscillations whereas the other one is a high frequency component which exhibits a broadband power spectrum. These results are in good agreement with a previous study where only the temporal dynamics of [Ca 2 +]i in HIT cells has been studied. Using coarse-grained entropy rates computed from information-theoretic functionals we could demonstrate in that study that a fast oscillatory component of the [Ca2+]i signal can be modulated pharmacologically suggesting deterministic structure in the temporal dynamics (Palu8 et al., 1998). Since Ca2+ is central to the stimulation of insulin secretion from pancreatic ,B-cells future experimental and theoretical studies should evaluate the impact of the different oscillatory components of [Ca 2+]i onto the secretory process as well as gene transcription. One possibility to resolve that question is to use a recently proposed mathematical model which allows for the on-line decoding of the [Ca 2+]i into the cellular response represented by the activation (phospho- K. Prank et al. 936 -0 '----- - -- - -----' a 100 200 sao 100 'D. C 'd. .00 0 10- 6 L - -_ 10 -?0?'------:0,,'-0::-": ,2""' 0,':---:" 0,-:0 --=-' 0 .? frequency (Hz) 200 limets ) rime (S) a _ 0.1 _ 0 .2 _ _ _- - - ' O. S 0 .4 0 .5 frequency (Hz) Figure 2: Results from the independent component analysis by the fast fixed-point algorithm. Two independent components of [Ca 2+]i were found. A: slowlyoscillating [Ca 2+]i signal, B: fast oscillating [Ca 2 +]i signal. Fourier power spectra of the independent components. C: the major [Ca 2+]i oscillatory period is 68 s, D: flat broadband power spectrum. rylation) of target proteins (Prank et al., 1998). Very recent experimental data clearly demonstrate that specificty is encoded in the frequency of [Ca2+]i oscillations. Rapid oscillations of [Ca2+]j are able to stimulate a set of transcription factors in T-Iymphocytes whereas slow oscillations activate only one transcription factor (Dolmetsch et al., 1998). Frequency-dependent gene expression is likely to be a widespread phenomenon and oscillations of [Ca 2+]i can occur with periods of seconds to hours. The technique of independent component analyis should be able to extract the spatio-temporal features of the [Ca2+]i signal in a variety of cells and should help to understand the differential regulation of [Ca2+]i-dependent intracellular processes such as gene transcription or secretion. Acknowledgements This study was supported by Deutsche Forschungsgemeinschaft under grants Scho 466/1-3 and Br 915/4-4. Independent Component Analysis ofIntracellular Calcium Spike Data 937 References Amari, S., Cichocki, A. & Yang, H. (1996) A new learning algorithm for blind source separation. In Touretzky, D.S., Mozer, M. C. & Hasselmo, M. E., (eds.), Advances in Neural Information Processing 8, pp. 757-763. Cambridge, MA: MIT Press. Bell, A. & Sejnowski, T. (1995) An information-maximization approach to blind separation and blind deconvolution. Neural Computation 7:1129-1159. Berridge, M. & Galione, A. (1988) Cytosolic calcium oscillators. FASEB 2:30743082. Cardoso, J. F. (1992) Iterative techniques for blind source separation using only fourth-order cumulants. In Proc. EUSIPCO (pp. 739-742) . Brussels. Comon, P. (1994) Independent component analysis - a new concept? Signal Procesing 36:287-314. Delfosse, N. & Loubaton, P. (1995) Adaptive blind separation of independent sources: a deflation approach. Signal Processing 45:59-83. Dolmetsch, R E., Xu, K. & Lewis, R S. (1998) Calcium oscillations increase the efficiency and specificity of gene expression. Nature 392:933-936. Hyvarinen, A. & Oja, E. (1996) A neuron that learns to separate one independent component from linear mixtures. In Proc. IEEE Int. Conf. on Neural Networks, pp. 62-67, Washington, D.C. Hyvarinen, A. & Oja, E. (1997) A fast fixed-point algorithm for independent component analysis. Neural Computation 9:1483-1492. Jutten, C. & Herault, J. (1991) Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture. Signal Processing 24:1-10. Moureau, E., & Macchi, O. (1993) New self-adaptive algorithms for source separation based on contrast functions. In Proc. IEEE Signal Processing Workshop on Higher Order Statistics, pp. 215-219, Lake Tahoe, USA. OJ a, E. & Karhunen, J. (1995) Signal separation by nonlinear hebbian learning. In Palaniswami, M., Attikiouzel, Y., Marks, R, Fogel, D. & Fukuda, T. (eds.) Computational Intelligence - a Dynamic System Perspective pp. 83-97. IEEE Press, New York. Palus, M., Schafl, C., von zur Miihlen, A., Brabant, G. & Prank, K. (1998) Coarsegrained entropy rates quantify fast Ca 2 + dynamics modulated by pharmacological stimulation. Pacific Symposium on Biocomputing 1998:645-656. Prank, K., Laer, L., Wagner, M., von zur Miihlen, A., Brabant, G. & Schafl, C. (1998) Decoding of intracellular calcium spike trains. Europhys. Lett. 42:143-147. Schafl, C., ROssig, L., Leitolf, H., Mader, T., von zur Miihlen, A. & Brabant, G. (1996) Generation of repetitive Ca 2+ transients by bombesin requires intracellular release and influx of C a2+ through voltage-dependent and voltage independent channels in single HIT cells. Cell Calcium 19(6):485-493. Tsien, R W . & Tsien, R. Y. (1990) Calcium channels, stores, and oscillations. Annu. Rev. Cell BioI. 6:715-760. 

1545 |@word norm:3 nd:1 pancreatic:1 contraction:1 covariance:2 initial:1 contains:1 series:2 activation:2 si:4 yet:1 remove:1 intelligence:1 coarse:1 provides:2 tahoe:1 five:4 mathematical:3 differential:1 symposium:1 sustained:1 manner:1 pharmacologically:1 ica:13 rapid:1 secretory:1 proliferation:1 resolve:2 prolonged:1 estimating:1 deutsche:1 killer:1 finding:1 extremum:3 temporal:11 hit:10 unit:2 medical:1 grant:1 christof:1 positive:1 understood:1 local:1 eusipco:1 initiated:2 studied:1 limited:1 statistically:2 unique:1 intercellular:2 signaling:4 bell:2 specificity:1 protein:1 convenience:1 close:1 onto:1 restriction:1 equivalent:1 deterministic:1 demonstrated:1 maximizing:1 go:1 attention:1 exposure:1 regarded:1 target:1 user:1 hypothesis:1 agreement:2 element:1 cooperative:1 observed:1 calculate:1 region:3 connected:1 mozer:1 ration:1 dynamic:6 efficiency:2 basis:1 differently:1 represented:1 neurotransmitter:1 train:1 fast:12 activate:1 sejnowski:2 klaus:1 europhys:1 whose:3 encoded:2 widely:1 larger:1 say:1 amari:2 otherwise:1 hormonal:1 statistic:1 insulin:5 advantage:2 eigenvalue:2 kurtosis:8 subtracting:2 relevant:2 realization:1 mixing:3 subcellular:3 los:3 convergence:1 oscillating:3 unsphered:1 help:1 measured:4 school:1 received:1 wtx:1 eq:1 dividing:1 quantify:1 closely:1 stochastic:1 transient:1 rime:1 extension:1 ic:8 major:2 nents:1 a2:1 delfosse:2 estimation:2 proc:3 spreading:1 fluorescence:1 hasselmo:1 successfully:1 weighted:1 mit:1 clearly:2 gaussian:6 always:1 aim:1 super:1 hannover:2 rather:2 voltage:2 release:2 emission:1 vk:1 mainly:1 contrast:4 sense:1 wf:1 am:1 dependent:3 germany:1 dual:1 denoted:4 herault:3 spatial:2 constrained:1 homogenous:1 equal:1 extraction:1 washington:1 future:1 minimized:1 stimulus:2 loubaton:2 oja:8 composed:1 individual:2 hamster:4 highly:2 possibility:1 mixture:7 analyzed:1 secretion:4 palus:2 necessary:2 divide:1 euclidean:1 plotted:1 theoretical:2 column:2 cumulants:1 maximization:1 lymphocyte:1 uniform:1 reported:1 dependency:1 periodic:1 considerably:1 migration:1 density:3 decoding:2 von:4 central:1 recorded:2 nm:6 slowly:2 berridge:2 conf:1 suggesting:1 coding:1 flatter:1 coefficient:1 int:1 coordinated:1 blind:8 performed:1 multiplicative:1 analyze:1 wave:5 parallel:2 palaniswami:1 compartment:1 acid:1 variance:1 loaded:1 maximized:1 correspond:1 identify:1 multiplying:1 ago:1 converged:1 oscillatory:4 messenger:1 touretzky:1 ed:2 against:1 frequency:14 pp:5 associated:1 sampled:1 dimensionality:1 ubiquitous:1 organized:1 amplitude:3 rim:1 back:1 higher:2 compartmentalized:1 response:2 arranged:1 furthermore:2 transport:1 nonlinear:1 widespread:1 jutten:3 stimulate:1 usa:1 effect:1 concept:1 adjacent:2 pharmacological:3 ll:1 self:1 excitation:3 criterion:1 theoretic:1 demonstrate:3 julia:1 novel:2 recently:2 scho:1 stimulation:5 functional:2 regulates:1 significant:2 measurement:1 compa:1 cambridge:1 spatia:3 recent:2 dye:1 perspective:1 store:1 selectivity:1 accomplished:1 determine:1 period:4 signal:27 ii:1 desirable:1 hebbian:1 clinical:1 concerning:1 impact:2 j3:1 basic:1 whitened:2 expectation:2 iteration:1 repetitive:1 achieved:1 cell:36 zur:4 irregular:1 whereas:2 ion:1 source:13 hz:3 induced:1 yang:1 forschungsgemeinschaft:1 easy:1 wn:1 enough:1 variety:1 architecture:1 reduce:1 br:1 administration:1 expression:2 wo:1 york:1 generally:1 cardoso:2 locally:1 category:1 exist:1 sign:2 estimated:3 georg:1 nevertheless:1 sum:1 run:2 fourth:3 ca2:24 respond:1 throughout:3 separation:10 oscillation:18 lake:1 yielded:1 activity:1 occur:3 x2:2 flat:2 influx:1 aspect:2 fourier:2 department:1 pacific:1 brussels:1 combination:3 electrically:1 across:2 wi:4 rev:1 making:1 comon:4 outlier:1 macchi:2 mutually:1 mechanism:2 deflation:1 initiate:1 neuromimetic:1 studying:1 junction:1 coarsegrained:1 spectral:2 appropriate:1 cancelled:1 batch:1 original:1 denotes:1 fukuda:1 question:2 already:1 spike:6 occurs:1 concentration:1 exhibit:1 gradient:3 separate:1 reporter:1 entity:2 procesing:1 cellular:2 reason:1 relationship:1 ratio:3 minimizing:1 regulation:2 sharper:1 negative:1 implementation:1 calcium:15 unknown:4 perform:1 observation:1 neuron:1 descent:1 prank:7 communication:1 introduced:1 required:1 fogel:1 hour:1 able:4 suggested:1 pattern:2 oj:2 power:6 indicator:1 scheme:1 numerous:1 conventionally:1 excitable:2 extract:1 cichocki:1 sn:1 acknowledgement:1 fully:1 mixed:1 generation:1 fluorescent:2 localized:1 digital:1 sao:1 row:1 supported:1 free:2 aij:1 understand:1 wagner:1 moreau:1 distributed:2 calculated:2 lett:1 computes:1 sensory:1 adaptive:4 hyvarinen:8 correlate:1 functionals:1 arginine:1 transcription:4 gene:5 assumed:1 spatio:5 xi:2 spectrum:6 regulatory:1 iterative:1 decade:1 nature:1 channel:2 robust:1 ca:33 complex:1 domain:1 spread:2 intracellular:11 s2:1 noise:1 xu:1 site:2 broadband:4 fig:2 slow:2 sub:2 xl:2 third:1 learns:1 grained:1 annu:1 specific:4 deconvolution:1 workshop:1 importance:1 multicellular:1 faseb:1 karhunen:2 gap:1 endocrinology:1 entropy:2 tsien:4 simply:1 wavelength:4 likely:1 gausssian:1 expressed:1 contained:1 a8:1 lewis:1 ma:1 bioi:1 goal:2 oscillator:1 considerable:1 content:2 determined:3 reducing:1 uniformly:2 tumor:1 experimental:7 mark:1 sphered:1 modulated:3 alexander:1 cumulant:1 biocomputing:1 avoiding:1 evaluate:1 phenomenon:1

Vertex Identification in High Energy Physics Experiments Gideon Dror* Department of Computer Science The Academic College of Tel-Aviv-Yaffo, Tel Aviv 64044 , Israel Halina Abramowicz t David Horn t School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University, Tel Aviv 69978 , Israel Abstract In High Energy Physics experiments one has to sort through a high flux of events , at a rate of tens of MHz, and select the few that are of interest. One of the key factors in making this decision is the location of the vertex where the interaction , that led to the event , took place. Here we present a novel solution to the problem of finding the location of the vertex , based on two feedforward neural networks with fixed architectures , whose parameters are chosen so as to obtain a high accuracy. The system is tested on simulated data sets , and is shown to perform better than conventional algorithms. 1 Introduction An event in High Energy Physics (HEP) is the experimental result of an interaction during the collision of particles in an accelerator . The result of this interaction is the production of tens of particles, each of which is ejected in a different direction and energy. Due to the quantum mechanical effects involved, the events differ from one another in the number of particles produced , the types of particles, and their energies. The trajectories of produced particles are detected by a very large and sophisticated detector . ? gideon@server.mta.ac.il thalina@Dost.tau.ac.i1 *hom@n;uron.tau.ac.il 869 Vertex Identification in High Energy Physics Experiments Events are typically produced at a rate of 10 MHz, in conjunction with a data volume of up to 500 kBytes per event. The signal is very small, and is selected from the background by multilevel triggers that perform filtering either through hardware or software. In the present paper we confront one problem that is of interest in these experiments and is part of the triggering consideration. This is the location of the vertex of the interaction. To be specific we will use a simulation of data collected by the central tracking detector [1] of the ZEUS experiment [2] at the HEP laboratory DESY in Hamburg, Germany. This detector, placed in a magnetic field , surrounds the interaction point and is sensitive to the path of charged particles. It has a cylindrical shape around the axis, z, where the interaction between the incoming particles takes place. The challenge is to find an efficient and fast method to extract the exact location of the vertex along this axis. 2 The Input Data An example of an event, projected onto the z = 0 plane, is shown in Figure 1. Only the information relevant to triggering is used and displayed. The relevant points, which denote hits by the outgoing particles on wires in the detector , form five rings due to the concentric structure of the detector. Several slightly curved particle tracks emanating from the origin, which is marked with a + sign, and crossing all five rings, can easily be seen. Each track is made of 30-40 data points. All tracks appear in this projection as arcs, and indeed, when viewed in 3 dimensions, every particle follows a helical trajectory due to the solenoidal magnetic field in the detector. . "1:.- 60 40 20 Eo .? -20 -40 -60 -60 -40 -20 0 x[cml 20 40 60 Figure 1: A typical event projected onto the z = 0 plane. The dots, or hits , have a two-fold ambiguity in the determination of the xy coordinates through which the particle has moved. The correct solutions lie on curved tracks that emanate from the origin. Each physical hit is represented twice in Fig. 1 due to an inherent two-fold ambiguity in the determination of its xy coordinates. The correct solutions form curved tracks emanating from the origin. Some of those can be readily seen in the data. Due to the limited time available for decision making at the trigger level, the z coordinate is obtained from the difference in arrival times of a pulse at both ends of the CTD and is available for only a fraction of these points. The hit resolution in xy is '" 230 J.lm , while that of z-by-timing is ::: 4 cm. The quality of the z coordinate G. Dror. H. Abramowicz and D. Hom 870 information is exemplified in figure 2. Figure 2(a) shows points forming a track of a single particle on the z = 0 projection. Since the corresponding track forms a helix with small curvature, one expects a linear dependence of the z coordinate of the hits on their radial position, r = J x 2 + y2. Figure 2(b) compares the values of r with the measured z values for these points. The scatter of the data around the linear regression fit is considerable. 10~~-~-,..--~-~-~~-, 35,--,--,-,1---,1-...,--,..-----.1--1.----, 301- a) 90 :-r. - - 25f- ~20f- - >- b) 80 70 E ~60 N _ ...; 101- - 50 40 30 - 51I I I 10 20 30 40 50 x [cm) 'I 60 70 80 20 1~5 20 25 30 35 40 r[cm) 45 50 55 Figure 2: A typical example of uncertainties in the measured z values: (a) a single track taken from the event shown in figure 1, (b) the z coordinate vs r Jx 2 + y2 the distance from the z axis for the data points shown in (a). The full line is a linear regression fit. = 3 The Network Our network is based on step-wise changes in the representation of the data, moving from the input points, to local line segments and to global arcs. The nature of the data and the problem suggest it is best to separate the treatment of the xy coordinates from that of the z coordinate. Two parallel networks which perform entirely different computations, form our final system. The first network, which handles the xy information is responsible for constructing arcs that correctly identify some of the particle tracks in the event. The second network uses this information to evaluate the z location of the point where all tracks meet . 3.1 Arc Identification Network The arc identification network processes information in a fashion akin to the method visual information is processed by the primary visual system [3]. The input layer for this network is made of a large number of neurons (several tens of thousands) and corresponds to the function of the retina. Each input neuron has its distinct receptive field. The sum of all fields covers completely the relevant domain in the xy plane. This domain has 5 concentric rings, which show up in figure 1. The total area of the rings is about 5000 cm 2 , and covering it with 100000 input neurons leads to satisfactory resolution. A neuron in the input level fires when a hit is present in its receptive field. We shall label each input neuron by the (xy) coordinates of the center of its receptive field. Neurons of the second layer are line segment detectors. Each second layer neuron is labeled by (XY a), where (X, Y) are the coordinates of the center of the segment Vertex Identification in High Energy Physics Experiments and 0' 871 denotes its orientation. The activation of second layer neurons is given by VXYa = g(2:: J XY a ,xy V xy - ( 2) , (1) xy where lxY a ,ry ={ ~1 ifr.L < O.5cmArll < 2cm ifO.5cm< r.L < 1cmArii < 2cm otherwise (2) and g( x) is the standard Heaviside step function . rll and r.L are the parallel and perpendicular distances between (X , Y) and (x, y) with respect to the axis of the line segment , defined by 0' . It is important to note that at this level , values of the threshold 82 which are slightly lower than optimum are preferable, taking the risk of obtaining superfluous line segments in order to reduce the probability of missing one . Superfluous line segments are filtered out very efficiently in higher layers. Figure 3 represents the output of the second layer neurons for the input illustrated by the event of figure 1. An active second layer neuron (XY 0') is represented in this figure by a line segment centered at the point (X , Y) making an angle 0' with the x axis. The length of the line segments is immaterial and was chosen only for the purpose of visual clarity. 60 ~ 40 ... ... "Z ~ 20 E ~ 0 >- -20 -40 -60 #- .>~~ 1'1( -t!- s ~ ~ , . ~.::.. ~ ~ . "\0 ~ ~. ~ ~1t ..,. ~, . ~ ' "'" " i-!. '" ;J I' l-J.. -60 -40 -20 .~ 0 xfcml 20 40 60 Figure 3: Representation of the activity of second layer neurons XY 0' for the input of figure 1 taken by plotting the appropriate line segments in the xy plane . At some XY locations several line segments with different directions occur due to the rather low threshold parameter used , 82 = 4. Neurons of the third layer transform the representation of local line segments into local arc segments. An arc which passes through the origin is uniquely defined by its radius of curvature R and its slope at the origin . Thus , each third layer neuron is labeled by '" 8 i , where 1"'1 = 1/ R is the curvature and the sign of '" determines the orient ation of the arc . 1 < i < 5 is an index which relates each arc segment to the ring it belongs to. -The mapping between second and third layers is based on a winner-take-all mechanism. Namely, for a given local arc segment, we take the arc segment which is closest to being tangent to the local arc segment. Denoting the average radius of the ring i ( i=1 ,2, ...5) by rj and using f3i = sin -1 (y) G. Dror. H. Abramowicz and D. Horn 872 the final expression for the activation of the third layer neurons is V",lIi _0 = maxe 0<3 2 2 cos (() - 2f3i - 0:), (3) where 6 = 6(X , Y, "', (), i) = J(X - ri cos((} - f3d)2 + (Y - ri sin(() - f3d)2 is simply the distance of the center of the receptive field of the (XY 0:) neuron to the ("'(}) arc. The fourth layer is the last one in the arc identification network . Neurons belonging to this layer are global arc detectors. In other words, they detect projected tracks on the z 0 plane. A fourth level neuron is denoted by "'(} , where", and () have the previous meaning , now describing global arcs. Fourth layer neurons are connected to third layer neurons in a simple fashion , = Vd = g( L 6""",,611 ,11' V""II'i - (}4) . (4) ",'II' i Figure 4 represents the activity of fourth layer neurons . Each active neuron "'(} is equivalent in the xy plane to one arc appearing in the figure . . ~60 40 20 E ~o >- ~ -20 -40 -60 -60 -40 -20 x f<em] 20 40 60 Figure 4: Representation of the activity of fourth layer neurons "'(} for the input of figure 1 taken by plotting the appropriate arcs in t he xy plane. The arcs are not precisely congruent to the activity of the input layer which is also shown , due to the finite widths which were used, il", = 0.004 and il(} = 7r/20. This figure was produced with (}4 = 3. 3.2 z Location Network The architecture of the second network has a structure which is identical to the first one, although its computational task is different. We will use an identical labeling system for its neurons , but denote their activities by v xy . The latter will assume continuous values in this network. A first layer neuron of the z-location network receives its input from the same receptive field as its corresponding neuron in the first network. Its value , v xy , is the mean value of the z values of the points within its receptive field . If no z values are available for these points , a null value is assigned to it. The second layer neurons compute the mean value vXY a = (v xy ) of the z coordinate of the first layer neurons in their receptive field , averaging over all neurons within 873 Vertex Identification in High Energy Physics Experiments the section II(x - X) sina - (y - Y) cosal < 0.5cm/\ (x - X)2 + (y - y)2 < 4cm2} , which corresponds to the excitatory part of the synaptic connections of equation (2). If null values appear within that section they are disregarded by the averaging procedure. If all values are null , VXYa is assigned a null value too . This Z averaging procedure is similarly propagated to the third layer neurons. {xy The fourth layer neurons evaluate the Z value of the origin of each arc identified by the first network. This is performed by a simple linear extrapolation. The final z estimate of the vertex, Znet , which should be the common origin of all arcs , IS calculated by averaging the outputs of all active fourth layer neurons. 4 Results In order to test the network , we ran it over a set of 1000 events generated by a Monte-Carlo simulator as well as over a sample of physical events taken from the ZEUS experiment at the HEP laboratory DESY in Hamburg . For the former set we compared the estimate of the net Znet with the nominal location of the vertex z , whereas for the real events in the latter set , we compared it with an estimate Zrec obtained by full reconstruction algorithm , which runs off-line and uses all available data. Results of the two tests can be compared since it is well established that the result of the full reconstruction algorithm is within 1 mm from the exact location of the vertex. z z Network <Az>=-2.7?O.2 (1 = 6.1 ?O.2 140 140 120 120 100 100 80 80 60 60 40 40 20 0 -40 J ~~ -20 0 20 Histogrom <Az>= 1.9?O.3 (1 = 8.4?O.3 20 40 0 Az [em] Figure 5: Distribution of ~ z = Ze8timate - Zexact values for two types of estimates, (a) the one proposed in this paper and (b) the one based on a commonly used histogram method . We also compared our results with those of an algorithmic method used for triggering at ZEUS [4]. We shall refer to this method as the 'histogram method '. The performance of the two methods was compared on a sample of 1000 Monte-Carlo events. The network was unable to get an estimate for 16 events from the set , as compared with 15 for the histogram method (15 of those events were common G. Dror, H. Abramowicz and D. Horn 874 failures). In Figure 5 we compare the distributions of ~z = Znet - Zexact and ~Z = Zhist - Zexact for the sample of Monte-Carlo events , where Zexact is the generated location of the vertex. Both methods lead to small biases, -2.7 cm for Znet and 1.9 cm for Zhist . The resolution, as obtained from a Gaussian fit , was found to be better for the network approach (06.1 cm) as compared to the histogram method (08.4cm). In addition, it should be noted that the histogram method yields discrete results, with a step of 10 cm, whereas the current method gives continuous values. This can be of great advantage for further processing. Note that off-line, after using the whole CTD information, the resolution is better than 1 mm. = 5 = Discussion We have described a feed forward double neural network that performs a task of pattern identification by thresholding and selecting subsets of data on which a simple computation can lead to the final answer. The network uses a fixed architecture, which allows for its implementation in hardware, crucial for fast triggering purposes. The basic idea of using a fixed architecture that is inspired by the way our brain processes visual information, is similar to the the raison d 'etre of the orientation selective neural network employed by [5]. The latter was based on orientation selective cells only, which were sufficient to select linear tracks that are of interest in HEP experiments. Here we develop an arc identification method, following similar steps. Both methods can also be viewed as generalizations of the Hough transform [6] that was originally proposed for straight line identification and may be regarded as a basic element of pattern recognition problems [7]. Neither [5] nor our present proposal were considered by previous neural network analyses of HEP data [8] . The results that we have obtained are very promising. We hope that they open the possibility for a new type of neural network implementation in triggering devices of HEP experiments. Acknowledgments We are indebted to the ZEUS Collaboration whose data were used for this study. This research was partially supported by the Israel National Science Foundation . References [1] B. Foster et al. , Nuclear Instrum. and Methods in Phys. Res. A338 (1994) 254. [2] ZEUS Collab., The ZEUS Detector, Status Report 1993, DESY 1993; M. Derrick et al. , Phys. Lett. B 293 (1992) 465 . [3] D. H. Hubel and T . N. Wiesel, J. Physiol. 195 (1968) 215. [4] A. Quadt , MSc thesis, University of Oxford (1997) . [5] H. Abramowicz , D. Horn , U. Naftaly and C . Sahar-Pikielny, Nuclear Instrum. and Methods in Phys. Res. A378 (1996) 305; Advances in Neural Information Processing Systems 9, eds. M. C . Mozer , M. J. Jordan and T. Petsche, MIT Press 1997, pp. 925- 931. [6] P. V. Hough , "Methods and means to recognize complex patterns", U.S. patent 3.069.654. [7] R. O. Duda and P. E. Hart , "Pattern classification and scene analysis" , Wiley, New York , 1973. [8] B. Denby, Neural Computation, 5 (1993) 505. 

1546 |@word cylindrical:1 faculty:1 wiesel:1 duda:1 open:1 cm2:1 simulation:1 cml:1 pulse:1 denby:1 selecting:1 denoting:1 current:1 activation:2 scatter:1 readily:1 physiol:1 shape:1 v:1 selected:1 device:1 plane:7 kbytes:1 filtered:1 location:11 five:2 along:1 abramowicz:5 indeed:1 nor:1 ry:1 simulator:1 brain:1 inspired:1 null:4 israel:3 cm:13 dror:4 astronomy:1 finding:1 every:1 preferable:1 hit:6 appear:2 timing:1 local:5 oxford:1 meet:1 path:1 twice:1 co:2 limited:1 perpendicular:1 horn:4 responsible:1 acknowledgment:1 procedure:2 emanate:1 area:1 projection:2 word:1 radial:1 suggest:1 get:1 onto:2 risk:1 conventional:1 equivalent:1 charged:1 center:3 rll:1 missing:1 resolution:4 regarded:1 nuclear:2 handle:1 coordinate:11 trigger:2 nominal:1 exact:3 us:3 origin:7 crossing:1 element:1 recognition:1 labeled:2 thousand:1 connected:1 ran:1 mozer:1 immaterial:1 segment:16 completely:1 easily:1 represented:2 distinct:1 fast:2 monte:3 detected:1 emanating:2 labeling:1 whose:2 otherwise:1 transform:2 final:4 advantage:1 net:1 took:1 reconstruction:2 interaction:6 relevant:3 helical:1 moved:1 az:3 double:1 optimum:1 congruent:1 ring:6 develop:1 ac:3 measured:2 school:1 differ:1 direction:2 radius:2 correct:2 centered:1 multilevel:1 generalization:1 mm:2 around:2 considered:1 great:1 mapping:1 algorithmic:1 lm:1 jx:1 purpose:2 label:1 sensitive:1 hope:1 mit:1 gaussian:1 rather:1 conjunction:1 detect:1 typically:1 selective:2 i1:1 germany:1 classification:1 orientation:3 denoted:1 field:10 identical:2 represents:2 report:1 inherent:1 few:1 retina:1 national:1 recognize:1 fire:1 interest:3 possibility:1 superfluous:2 xy:23 hough:2 re:2 hep:6 cover:1 mhz:2 vertex:12 subset:1 expects:1 too:1 answer:1 physic:7 off:2 thesis:1 central:1 ambiguity:2 lii:1 ifo:1 performed:1 extrapolation:1 sort:1 parallel:2 slope:1 il:4 accuracy:1 beverly:1 efficiently:1 yield:1 identify:1 identification:10 produced:4 carlo:3 trajectory:2 straight:1 indebted:1 detector:9 phys:3 synaptic:1 ed:1 failure:1 energy:8 pp:1 involved:1 propagated:1 derrick:1 treatment:1 zeus:6 sophisticated:1 feed:1 higher:1 originally:1 msc:1 outgoing:1 receives:1 quality:1 aviv:4 effect:1 y2:2 lxy:1 former:1 assigned:2 laboratory:2 satisfactory:1 illustrated:1 sin:2 during:1 width:1 uniquely:1 covering:1 noted:1 performs:1 meaning:1 wise:1 consideration:1 novel:1 common:2 physical:2 patent:1 winner:1 volume:1 yaffo:1 he:1 refer:1 surround:1 vxy:1 similarly:1 particle:13 dot:1 moving:1 curvature:3 closest:1 belongs:1 hamburg:2 collab:1 server:1 pikielny:1 seen:2 eo:1 employed:1 signal:1 ii:3 relates:1 full:3 rj:1 academic:1 determination:2 hart:1 regression:2 basic:2 confront:1 histogram:5 cell:1 proposal:1 background:1 whereas:2 addition:1 crucial:1 pass:1 jordan:1 feedforward:1 fit:3 architecture:4 identified:1 triggering:5 f3i:2 reduce:1 idea:1 expression:1 akin:1 york:1 collision:1 ten:3 hardware:2 processed:1 ctd:2 sign:2 per:1 track:12 correctly:1 discrete:1 shall:2 key:1 threshold:2 clarity:1 neither:1 fraction:1 sum:1 orient:1 angle:1 run:1 uncertainty:1 fourth:7 place:2 decision:2 entirely:1 layer:25 hom:2 sackler:1 fold:2 activity:5 occur:1 precisely:1 ri:2 software:1 scene:1 department:1 mta:1 belonging:1 slightly:2 em:2 naftaly:1 making:3 taken:4 equation:1 describing:1 mechanism:1 end:1 available:4 appropriate:2 petsche:1 magnetic:2 appearing:1 denotes:1 ifr:1 receptive:7 primary:1 dependence:1 distance:3 separate:1 unable:1 simulated:1 vd:1 collected:1 length:1 index:1 znet:4 implementation:2 perform:3 wire:1 neuron:31 arc:22 finite:1 curved:3 displayed:1 concentric:2 david:1 namely:1 mechanical:1 connection:1 established:1 exemplified:1 pattern:4 gideon:2 challenge:1 tau:2 event:18 ation:1 axis:5 extract:1 raymond:1 tangent:1 f3d:2 accelerator:1 sahar:1 filtering:1 foundation:1 sufficient:1 plotting:2 thresholding:1 foster:1 helix:1 collaboration:1 production:1 excitatory:1 placed:1 last:1 supported:1 bias:1 taking:1 dimension:1 calculated:1 lett:1 quantum:1 forward:1 made:2 commonly:1 projected:3 flux:1 status:1 global:3 active:3 incoming:1 hubel:1 continuous:2 promising:1 nature:1 tel:4 obtaining:1 complex:1 constructing:1 domain:2 whole:1 arrival:1 fig:1 fashion:2 wiley:1 position:1 lie:1 uron:1 third:6 specific:1 disregarded:1 led:1 desy:3 simply:1 forming:1 visual:4 tracking:1 partially:1 corresponds:2 determines:1 marked:1 viewed:2 considerable:1 change:1 instrum:2 typical:2 averaging:4 total:1 experimental:1 maxe:1 select:2 college:1 latter:3 sina:1 evaluate:2 heaviside:1 tested:1

Attentional Modulation of Human Pattern Discrimination Psychophysics Reproduced by a Quantitative Model Laurent Itti, Jochen Braun, Dale K. Lee and Christof Koch {itti, achim, jjwen, koch}Oklab.caltech.edu Computation & Neural Systems, MSC 139-74 California Institute of Technology, Pasadena, CA 91125, U.S.A. Abstract We previously proposed a quantitative model of early visual processing in primates, based on non-linearly interacting visual filters and statistically efficient decision. We now use this model to interpret the observed modulation of a range of human psychophysical thresholds with and without focal visual attention. Our model calibrated by an automatic fitting procedure - simultaneously reproduces thresholds for four classical pattern discrimination tasks, performed while attention was engaged by another concurrent task. Our model then predicts that the seemingly complex improvements of certain thresholds, which we observed when attention was fully available for the discrimination tasks, can best be explained by a strengthening of competition among early visual filters. 1 INTRODUCTION What happens when we voluntarily focus our attention to a restricted part of our visual field? Focal attention is often thought as a gating mechanism, which selectively allows a certain spatial location and and certain types of visual features to reach higher visual processes. We here investigate the possibility that attention might have a specific computational modulatory effect on early visual processing. We and others have observed that focal visual attention can modulate human psychophysical thresholds for simple pattern discrimination tasks [7, 8, 5] When attention is drawn away from a task, for example by "cueing" [12] to another location of the display, or by a second, concurrent task [1, 7, 8], an apparently complex pattern of performance degradation is observed: For some tasks, attention has little or no effect on performance (e.g., detection of luminance increments), while for 790 L. ltti, J. Braun, D. K. Lee and C. Koch other tasks, attention dramatically improves performance (e.g., discrimination of orientation). Our specific findings with dual-task psychophysics are detailed below. These observations have been paralleled by electrophysiological studies of attention. In the awake macaque, neuronal responses to attended stimuli can be 20% to 100% higher than to otherwise identical unattended stimuli. This has been demonstrated in visual cortical areas VI, V2, and V4 [16, 11, 10,9] when the animal discriminates stimulus orientation, and in areas MT and MST when the animal discriminates the speed of stimulus motion [17]. Even spontaneous firing rates are 40% larger when attention is directed at a neuron's receptive field [9]. Whether neuronal responses to attended stimuli are merely enhanced [17] or whether they are also more sharply tuned for certain stimulus dimensions [16] remains controversial. Very recently, fMRI studies have shown similar enhancement (as measured with BOLD contrast) in area VI of humans, specifically at the retinotopic location where subjects had been instructed to focus their attention to [2, 14]. All of these observations directly address the issue of the "top-down" computational effect of attentional focusing onto early visual processing stages. This issue should be distinguished from that of the "bottom-up" control of visual attention [6], which studies which visual features are likely to attract the attention focusing mechanism (e.g., pop-out phenomena and studies of visual search). Top-down attentional modulation happens after attention has been focused to a location of the visual field, and most probably involves the massive feedback circuits which anatomically project from higher cortical areas back to early visual processing areas. In the present study, we quantify the modulatory effect of attention observed in human psychophysics using a model of early visual processing. The model is based on non-linearly interacting visual filters and statistically efficient decision [4, 5]. Although attention could modulate virtually any visual processing stage (e.g ., the decision stage, which compares internal responses from different stimuli), our basic hypothesis here - supported by electrophysiology and fMRI [16,11,10,17,9,2, 14]is that this modulation might happen very early in the visual processing hierarchy. Given this basic hypothesis, we investigate how attention should affect early visual processing in order to quantitatively reproduce the psychophysical results. 2 PSYCHOPHYSICAL EXPERIMENTS We measured attentional modulation of spatial vision thresholds using a Central task: dual-task paradigm [15, 7]: At the center of the visual field, a letter discrimination task is presented, while a pattern discrimination task is simultaneously presented at a random peripheral location (4 0 eccentricity). The central task consists of discriminating between five letters "T" or four "T" and one "L". It has been shown to efficiently engage attention [7]. The peripheral task is chosen from a battery of a classical pattern threshold measurement discrimination tasks, and is the task of interest for this study. Psychophysical thresholds are measured for two distinct conditions: In the "fully attended" condition, observers are asked to devote their entire attention to the peripheral 791 Quantitative Modeling ofAttentional Modulation task, and to ignore the central task (while still fixating the center of the screen). In the "poorly attended" condition, observers are asked to pay full attention to the central task (and the blocks of trials for which performance for the central task falls below a certain cut-off are discarded). Four classical pattern discrimination tasks were investigated, each with two volunteer subjects (average shown in Figure 1), similarly to our previous experiments [7, 8]. Screen luminance resolution was 0.2%. Screen luminance varied from 1 to 90cd/m 2 (mean 45cd/m 2 ), room illumination was 5cd/m 2 and viewing distance 80cm. The Yes/No (present/absent) paradigm was used (one stimulus presentation per trial). Threshold (75% correct peformance) was reached using a staircase procedure , and computed through a maximum-likelihood fit of a Weibull function with two degrees of freedom to the psychometric curves. Exp. 2: Orientation discrimination f 20 J 15 ., 10 c: ,g S 5 ~ ~~~~~ 0.6 0.8 Contrast Mask contrast 0.4 :2 :2 .B 0 I: e II) II> ?; -6 !0.2 i.to 0.2 c: c: 8 0 () ?? ? Figure 1: Psychophysical data and model fits using the parameters from Table 1 (P=poorly and F=fully attended). Gray curves: Model predictions for fully attended data, using the poorly attended parameters, except for -y = 2.9 and {) = 2.1 (see Results). Expo 1 measured increment contrast discrimination threshold: The observer discriminates between a 4cpd (cycles per degree) stochastic oriented mask [7] at fixed contrast , and the same mask plus a low-contrast sixth-derivative-of-Gaussian (D6G) bar; threshold is measured for bar contrast [8]. Expo 2 measured orientation discrimination thresholds: The observer discriminates between a vertical and tilted grating at 4cpd; threshold for the angle difference is measured. In addition, two contrast masking tasks were investigated for their sensitivity to non-linearities in visual processing. A 4cpd stochastic mask (50% contrast) was always present, and threshold was measured for the contrast of a vertical superimposed D6G bar. In Expo 3, the orientation of the masker was varied and its spatial frequency fixed (4cpd), while in Expo 4 the spatial period of the masker was varied and its orientation vertical. Our aim was to investigate very dissimilar tasks, in particular with respect to the decision strategy used by the observer. Using the dual-task paradigm, we found mixed attentional effects on psychophysical thresholds, including the appearance of a more pronounced contrast discrimination L. ltti, J. Braun, D. K. Lee and C. Koch 792 "dipper" in Exp. 1, substantial improvement of orientation thresholds in Exp. 2, and reduced contrast elevations due to masking in Exps. 3-4 (also see [7, 8]). 3 MODEL The model consists of three successive stages [4, 5]. In the first stage, a bank of Gabor-like linear filters analyzes a fixed location of the visual scene. Here, a singlescale model composed of 12 pairs of filters in quadrature phase, tuned for orientations o E e evenly spanning 1800 , was sufficient to account for the data (although a multiscale model may account for a wider range of psychophysical thresholds). The linear filters take values between 0.0 and 100.0, then multiplied by a gain factor A (one of the ten free parameters of the model), and to which a small background activity f. is added. In the second stage, filters non-linearly interact as follows: (1) Each unit receives non-linear self-excitation, and (2) each unit receives non-linear divisive inhibition from a pool of similarly-tuned units: With E8 being the linear response from a unit tuned for orientation 0, the pooled response R8 is given by: where W8(O') =e- (/1'_/1)2 2E~ is a Gaussian weighting function centered around 0, and 1J a positive constant to account for background activity in the pooling stage. This stage is inspired from Heeger's model of gain control in cat VI [3, 4]. Our formulation, in which none of the parameters is given a particular value, however allows for multiple outcomes, to be determined by fitting the model to our psychophysical data: A sigmoidal (S > 0, I > d') as well as simple power-law (S = 0) or even linear (! = 1, d' = 0) contrast response characteristic could emerge, the responses could be saturating d') or not (, i= d'), and the inhibitory pool size (~8) could be broad or narrow. Because striate neurons are noisy, physiological noise is assumed in the model at the outputs of the second stage. The noise level is chosen close to what is typically observed in cortical pyramidal cells, and modeled by Gaussian noise with variance equal to mean taken to some power a determined by fitting. (, = Because the decision stage - which quantitatively relates activity in the population of pooled noisy units to behavioral discrimination performance - is not fully characterized in humans, we are not in a position to model it in any detail. Instead, we trained our subjects (for 2-3 hours on each task), and assume that they perform close to an "optimal detector". Such optimal detector may be characterized in a formal manner, using Statistical Estimation Theory [4, 5]. We assume that a brain mechanism exists, which, for a given stimulus presentation, builds an internal estimate of some stimulus attribute ( (e.g., contrast, orientation, period). The central assumption of our decision stage is that this brain mechanism will perform close to an unbiased efficient statistic T, which is the best possible estimator of ( Quantitative Modeling ofAttentional Modulation 793 given the noisy population response from the second stage. The accuracy (variance) with which T estimates ( can be computed formally, and is the inverse of the Fisher Information with respect to ( [13, 4]. Simply put, this means that, from the first two stages of the model alone, we have a means of computing the best possible estimation performance for (, and consequently, the best possible discrimination performance between two stimuli with parameters (1 and (2 [4, 5]. Such statistically efficient decision stage is implementable as a neural network [13]. This decision stage provides a unified framework for optimal discrimination in any behavioral situation, and eliminates the need for task-dependent assumptions about the strategy used by the observers to perform the task in a near optimal manner. Our model allows for a quantitative prediction of human psychophysical thresholds, based on a crude simulation of the physiology of primary visual cortex (area VI). 4 RESULTS All parameters in the model were automatically adjusted in order to best fit the psychophysical data from all experiments. A multidimensional downhill simplex with simulated annealing overhead was used to minimize the root-mean-square distance between the quantitative predictions of the model and the human data [4]. The best-fit parameters obtained independently for the "fully attended" and "poorly attended" conditions are reported in Table 1. The model's simultaneous fits to our entire dataset are plotted in Figure 1 for both conditions. After convergence of the fitting procedure, a measure of how well constrained each parameter was by the data was computed as follows: Each parameter was systematically varied around its best-fit value, in 0.5% steps, and the fitting error was recomputed; the amplitude by which each parameter could be varied before the fitting error increased by more than 10% of its optimum is noted as a standard deviation in Table 1. A lower deviation indicates that the parameter is more strongly constrained by the dataset. Table 1. Model parameters for both attentional conditions. Name Symbol fully attended poorly attended Linear gaint A l.7 ? 0.2 8.2 ? 0.9 Activity-independent inhibition t S 14.1 ? 2.3 10l.5 ? 16.6 Excitatory exponent 'Y 3.36 ? 0.02 2.09 ? 0.01 Inhibitory exponent 6 2.48 ? 0.02 l.51 ? 0.02 Noise exponent a l.34 ? 0.07 1.39 ? 0.08 Background activity, linear stage f l.13 ? 0.35 1.25 ? 0.60 Background activity, pooling stage 7] 0.18 ? 0.05 0.77 ? 0.11 Spatial period tuning width X (r>. 0.85 ? 0.06 oct. 0.85 ? 0.09 oct . Orientation tuning width X (r8 26? ? 2.4? 38? ? 5.5? Orientation pooling width X ~8 48? ? 25? 50? ? 26? t Dynamic range of linear filters is [? ... 100.0 X A + 4 x For clarity, FWHM values are given rather than 17 values (FWHM = 2I7J2ln(2?. Although no human bias was introduced during the fitting procedure, interestingly, all of the model's internal parameters reached physiologically plausible best-fit values, such as, for example, slightly supra-Poisson noise level (a ~ 1.35), ~ 30? orientation tuning FWHM (full-width at half-maximum), and ~ 0.85 octave spatial period tuning FWHM. Some of the internal characteristics of the model which more closely relate to the putative underlying physiological mechanisms are shown in Figure 2. 794 a L. ltti, J. Braun, D. K. Lee and C. Koch b Transducer function Orientation tuning C 0.8 p 0.4 0.8 0.8 5.c: o.s ..c: en Ci5 ~0.4 ~o.s e 0.4 0.2 0.8 Contrast Orientation pooling F 0 -40 -20 0 20 40 Orientation (deg) Figure 2: Internals of the model. (a) The response function of individual units to contrast was sigmoidal under full (F) and almost linear under poor (P) attention. (b) Native linear orientation tuning was broader under poor (NP) than full (NF) attention, but it was sharpened in both cases by pooling (PP=pooled poor, and PF=pooled full attention). (c) There was no difference in orientation pooling width under poor (P) or full (F) attention. Using poorly attended parameters, except for -y = 2.9 and ~ = 2.1 (grey curves), yielded steep non-linear contrast response, and intermediary tuning (same width as NF). In Table 1, attention had the following significant effects on the model's parameters: 1) Both pooling exponents (-y, d) were higher; 2) the tuning width (0"/1) was narrower; 3) the linear gain (A) and associated activity-independent inhibition (5) were lower; and 4) the background activity of the pooling stage was lower. This yielded increased competition between filters: The network behaved more like a winner-take-all under full attention, and more like a linear network of independent units under poor attention. While the attentional modulation of "d and 0"/1 are easy to interpret, its effect on the A, 5 and 'fJ is more difficult to understand. Consequently, we conducted a further automatic fit, which, starting from the "poorly attended" parameters, was only allowed to alter, and d to fit the "fully attended" data. The motivation for not varying 0"/1 was that we observed significant sharpening of the tuning induced by higher exponents "d (Figure 2) . Also, slight changes in the difference , - d can easily produce large changes in the overall gain of the system, hence compensating for changes in A, 5 and 'fJ . (We however do not imply here that 0"/1, A, 5 and 'fJ are redundant parameters; there is only a small range around the best-fit point over which, and d can compensate for variations in the other parameters, without dramatically impairing the quality of fit) . Although the new fit was not as accurate as that obtained with all parameters allowed to vary, it appeared that a simple modification of the pooling exponents well captured the effect of attention (Figure 1). Hence, the "poorly attended" parameters of Table 1 well described the "poorly attended" data, and the same parameters except for, = 2.9 and d = 2.1 well described the "fully attended" data. A variety of other simple parameter modifications were also tested, but none except for the pooling exponents (-y,o) could fully account for the attentional modulation. These modifications include: Changes in gain (obtained by modifying A only, , only, or d only), in tuning (0"/1), in the extent ofthe inhibitory pool (E/I), and in the noise level (a). A more systematic study, in which all possible parameter subsets are successively examined, is currently in progress in our laboratory. 5 DISCUSSION and CONCLUSION At the basis of our results is the hypothesis that attention might modulate the earlier rather than the later stages of visual processing. We found that a very Quantitative Modeling ofAttentional Modulation 795 simple, prototypical, task-independent enhancement of the amount of competition between early visual filters accounts well for the human data. This enhancement resulted from increases in parameters 'Y and 5 in the model, and was paralleled by an increase in contrast gain and a sharpening in orientation tuning. Although it is not possible from our data to rule out any attentional modulation at later stages, our hypothesis has recently received experimental support that attention indeed modulates early visual processing in humans [2, 14]. More psychophysical experiments are needed to investigate attentional modulation at later processing stages. For example, it might be possible to study the effect of attention on the decision stage by manipulating attention during experiments involving decision uncertainty. In the absence of such results, we have attempted in our experiments to minimize the possible impact of attention on later stages, by using only simple stimulus patterns devoid of conceptual or emotional meaning, such as to involve as little as possible the more cognitive stages of visual processing. Our finding that attention may increase the amount of competition between early visual filters is accompanied by an enhancement of the gain and sensitivity of the filters, and by a sharpening of their tuning properties. The existence of two such processing states - one, more sensitive and selective inside the focus of attention, and the other, more broadly-tuned and non-specific outside - can be justified by at least two observations: First, the higher level of activity in attended neurons consumes more energy, which may not be desirable over the entire extent of visual cortices. Second, although less efficient for fine discriminations, the broadly-tuned and non-specific state may have greater ability at catching unexpected, non-specific visual events. In this perspective, this state would be desirable as an input to bottom-up, visual alerting mechanisms, which monitor the rest of our visual world while we are focusing on a specific task requiring high focal accuracy. Acknowledgements This research was supported by ONR and NSF (Caltech ERG). References [1] Bonnel AM, Stein JF, Bertucci P. Q J Exp Psychol fA} 1992;44(4):601-26 [2] Gandhi SP, Heeger DJ, Boynton GM. Inv Opht Vis Sci (ARVO'98) 1998;39(4):5194 [3] Heeger DJ. Vis Neurosci 1992;9:181-97 [4] Itti L, Braun J, Lee DK, Koch C. Proc NIPS *97 {in press) [5] Itti L, Koch C, Braun J. Inv Opht Vis Sci (Proc ARVO'98) 1998;39(4):2934 [6] Koch C, Ullman S. Hum NeurobioI1985;4:219-27 [7] Lee DK, Koch C, Braun J. Vis Res 1997:37(17):2409-18 [8] Lee DK, Koch C, Itti L, Braun J . Inv Opht Vis Sci (Proc ARVO'98) 1998;39(4):2938 [9] Luck SJ, Chelazzi L, Hillyard SA, Desimone R. J Neurophysio/1997;77{I):24-42 [10] Maunsell JH. Science 1995;270(5237)764-9 [11] Motter BC. J NeurophysioiI993;70(3):909-19 [12] Nakayama K, Mackeben M. Vis Res 1989;29(11):1631-47 [13] Pouget A, Zhang K, Deneve S, Latham PE. Neur Comp 1998;10:373-401 [14] Somers DC, et al. lnv Opht Vis Sci (Proc ARVO'98) 1998;39(4):5192 [15] Sperling G, Melchner MJ. Science 1978;202:315-8 [16] Spitzer H, Desimone R, Moran J. Science 1988;240(4850):338-40 [17] Treue S, Maunsell JH. Nature 1996;382(6591):539-41 

1547 |@word trial:2 grey:1 simulation:1 attended:18 tuned:6 bc:1 interestingly:1 tilted:1 mst:1 happen:1 discrimination:17 alone:1 half:1 provides:1 location:6 successive:1 sigmoidal:2 zhang:1 five:1 transducer:1 consists:2 fitting:7 overhead:1 behavioral:2 inside:1 manner:2 mask:4 indeed:1 brain:2 compensating:1 inspired:1 automatically:1 little:2 pf:1 project:1 retinotopic:1 linearity:1 underlying:1 circuit:1 spitzer:1 what:2 cm:1 weibull:1 unified:1 finding:2 sharpening:3 quantitative:7 w8:1 multidimensional:1 nf:2 braun:8 control:2 unit:7 maunsell:2 christof:1 positive:1 before:1 laurent:1 firing:1 modulation:12 might:4 plus:1 examined:1 range:4 statistically:3 fwhm:4 directed:1 internals:1 block:1 procedure:4 area:6 thought:1 gabor:1 physiology:1 onto:1 close:3 put:1 unattended:1 demonstrated:1 center:2 attention:37 starting:1 independently:1 focused:1 resolution:1 pouget:1 estimator:1 rule:1 erg:1 population:2 variation:1 increment:2 hierarchy:1 spontaneous:1 enhanced:1 massive:1 engage:1 gandhi:1 gm:1 hypothesis:4 cut:1 predicts:1 native:1 observed:7 bottom:2 cycle:1 e8:1 luck:1 consumes:1 voluntarily:1 discriminates:4 substantial:1 asked:2 battery:1 exps:1 dynamic:1 trained:1 basis:1 easily:1 cat:1 distinct:1 outcome:1 outside:1 larger:1 plausible:1 arvo:4 otherwise:1 ability:1 statistic:1 noisy:3 reproduced:1 seemingly:1 strengthening:1 poorly:9 pronounced:1 competition:4 convergence:1 enhancement:4 eccentricity:1 optimum:1 supra:1 produce:1 wider:1 measured:8 received:1 progress:1 sa:1 grating:1 involves:1 quantify:1 closely:1 correct:1 attribute:1 filter:12 stochastic:2 modifying:1 centered:1 human:11 viewing:1 elevation:1 adjusted:1 koch:10 around:3 exp:4 vary:1 early:11 estimation:2 proc:4 intermediary:1 currently:1 sensitive:1 concurrent:2 gaussian:3 always:1 aim:1 rather:2 varying:1 broader:1 treue:1 focus:3 improvement:2 likelihood:1 superimposed:1 indicates:1 contrast:18 am:1 dependent:1 attract:1 entire:3 typically:1 achim:1 pasadena:1 manipulating:1 reproduce:1 selective:1 issue:2 among:1 orientation:19 dual:3 overall:1 exponent:7 animal:2 spatial:6 constrained:2 psychophysics:3 singlescale:1 field:4 equal:1 identical:1 broad:1 alter:1 jochen:1 fmri:2 simplex:1 others:1 stimulus:12 quantitatively:2 cpd:4 np:1 oriented:1 melchner:1 composed:1 simultaneously:2 resulted:1 individual:1 phase:1 freedom:1 detection:1 interest:1 investigate:4 possibility:1 accurate:1 desimone:2 re:2 plotted:1 catching:1 increased:2 modeling:3 earlier:1 deviation:2 subset:1 conducted:1 reported:1 calibrated:1 devoid:1 sensitivity:2 discriminating:1 lee:7 v4:1 off:1 systematic:1 pool:3 central:6 sharpened:1 successively:1 bonnel:1 cognitive:1 derivative:1 itti:5 ullman:1 fixating:1 account:5 accompanied:1 bold:1 pooled:4 vi:11 performed:1 root:1 observer:6 later:4 apparently:1 reached:2 masking:2 minimize:2 square:1 accuracy:2 variance:2 characteristic:2 efficiently:1 ofthe:1 yes:1 none:2 comp:1 detector:2 simultaneous:1 reach:1 sixth:1 energy:1 frequency:1 pp:1 associated:1 gain:7 cueing:1 dataset:2 improves:1 electrophysiological:1 amplitude:1 back:1 focusing:3 higher:6 response:10 formulation:1 strongly:1 stage:24 msc:1 receives:2 multiscale:1 quality:1 gray:1 behaved:1 name:1 effect:9 staircase:1 unbiased:1 requiring:1 hence:2 laboratory:1 during:2 self:1 width:7 noted:1 excitation:1 octave:1 latham:1 motion:1 fj:3 meaning:1 recently:2 mt:1 winner:1 slight:1 interpret:2 measurement:1 significant:2 automatic:2 tuning:12 focal:4 similarly:2 dj:2 had:2 hillyard:1 cortex:2 inhibition:3 perspective:1 certain:5 onr:1 caltech:2 captured:1 analyzes:1 greater:1 paradigm:3 period:4 redundant:1 ii:2 relates:1 full:7 multiple:1 desirable:2 characterized:2 compensate:1 impact:1 prediction:3 involving:1 basic:2 vision:1 poisson:1 volunteer:1 cell:1 justified:1 addition:1 background:5 fine:1 annealing:1 pyramidal:1 eliminates:1 rest:1 probably:1 subject:3 dipper:1 virtually:1 pooling:10 induced:1 near:1 easy:1 peformance:1 variety:1 affect:1 fit:12 absent:1 whether:2 dramatically:2 modulatory:2 detailed:1 involve:1 amount:2 stein:1 ten:1 reduced:1 nsf:1 inhibitory:3 per:2 broadly:2 motter:1 recomputed:1 four:3 threshold:17 monitor:1 drawn:1 clarity:1 luminance:3 deneve:1 merely:1 angle:1 letter:2 inverse:1 uncertainty:1 somers:1 almost:1 putative:1 decision:10 pay:1 display:1 chelazzi:1 yielded:2 activity:9 sharply:1 awake:1 expo:4 scene:1 speed:1 neur:1 peripheral:3 poor:5 slightly:1 primate:1 happens:2 modification:3 explained:1 restricted:1 anatomically:1 taken:1 previously:1 remains:1 sperling:1 mechanism:6 needed:1 available:1 multiplied:1 away:1 v2:1 distinguished:1 existence:1 top:2 include:1 emotional:1 build:1 classical:3 psychophysical:12 added:1 hum:1 receptive:1 strategy:2 primary:1 striate:1 fa:1 devote:1 distance:2 attentional:10 simulated:1 sci:4 evenly:1 extent:2 spanning:1 modeled:1 difficult:1 steep:1 relate:1 perform:3 vertical:3 observation:3 neuron:3 discarded:1 implementable:1 situation:1 dc:1 interacting:2 varied:5 inv:3 introduced:1 pair:1 alerting:1 california:1 narrow:1 pop:1 hour:1 macaque:1 nip:1 address:1 bar:3 below:2 pattern:8 appeared:1 including:1 power:2 event:1 technology:1 imply:1 psychol:1 acknowledgement:1 law:1 fully:10 mixed:1 prototypical:1 degree:2 controversial:1 sufficient:1 boynton:1 bank:1 systematically:1 cd:3 excitatory:1 supported:2 free:1 formal:1 bias:1 understand:1 jh:2 institute:1 fall:1 emerge:1 feedback:1 dimension:1 cortical:3 curve:3 world:1 dale:1 instructed:1 sj:1 ignore:1 deg:1 reproduces:1 conceptual:1 assumed:1 masker:2 search:1 physiologically:1 table:6 mj:1 nature:1 ca:1 nakayama:1 interact:1 investigated:2 complex:2 sp:1 linearly:3 neurosci:1 motivation:1 noise:6 allowed:2 quadrature:1 neuronal:2 psychometric:1 en:1 screen:3 position:1 downhill:1 heeger:3 crude:1 pe:1 weighting:1 down:2 specific:6 gating:1 r8:2 symbol:1 dk:3 physiological:2 moran:1 exists:1 modulates:1 illumination:1 electrophysiology:1 simply:1 likely:1 appearance:1 visual:34 unexpected:1 saturating:1 oct:2 modulate:3 presentation:2 narrower:1 consequently:2 room:1 jf:1 fisher:1 absence:1 change:4 specifically:1 except:4 determined:2 degradation:1 engaged:1 divisive:1 experimental:1 attempted:1 selectively:1 formally:1 internal:4 support:1 paralleled:2 dissimilar:1 tested:1 phenomenon:1

Familiarity Discrimination of Radar Pulses Eric Grangerl, Stephen Grossberg 2 Mark A. RUbin2 , William W. Streilein 2 1 Department of Electrical and Computer Engineering Ecole Polytechnique de Montreal Montreal, Qc. H3C 3A 7 CAN ADA 2Department of Cognitive and Neural Systems, Boston University Boston, MA 02215 USA Abstract The ARTMAP-FD neural network performs both identification (placing test patterns in classes encountered during training) and familiarity discrimination (judging whether a test pattern belongs to any of the classes encountered during training). The performance of ARTMAP-FD is tested on radar pulse data obtained in the field, and compared to that of the nearest-neighbor-based NEN algorithm and to a k > 1 extension of NEN. 1 Introduction The recognition process involves both identification and familiarity discrimination. Consider, for example, a neural network designed to identify aircraft based on their radar reflections and trained on sample reflections from ten types of aircraft A . . . J. After training, the network should correctly classify radar reflections belonging to the familiar classes A . .. J, but it should also abstain from making a meaningless guess when presented with a radar reflection from an object belonging to a different, unfamiliar class. Familiarity discrimination is also referred to as "novelty detection," a "reject option," and "recognition in partially exposed environments." ARTMAP-FD, an extension of fuzzy ARTMAP that performs familiarity discrimination, has shown its effectiveness on datasets consisting of simulated radar range profiles from aircraft targets [1, 2]. In the present paper we examine the performance of ARTMAP-FD on radar pulse data obtained in the field , and compare it 876 E. Granger, S. Grossberg, M. A. Rubin and W. W. Streilein to that of NEN, a nearest-neighbor-based familiarity discrimination algorithm, and to a k > 1 extension of NEN. 2 Fuzzy ARTMAP Fuzzy ARTMAP [3] is a self-organizing neural network for learning, recognition, and prediction. Each input a learns to predict an output class K. During training, the network creates internal recognition categories, with the number of categories determined on-line by predictive success. Components of the vector a are scaled so that each ai E [0,1] (i = 1 ... M). Complement coding [4] doubles the number of components in the input vector, which becomes A (a, a C ) , where the ith component of a C is ai (I-ad. With fast learning, the weight vector w) records the largest and smallest component values of input vectors placed in the /h category. The 2M-dimensional vector Wj may be visualized as the hyperbox R j that just encloses all the vectors a that selected category j during training. = = Activation of the coding field F2 is determined by the Weber law choice function Tj(A) =1 A 1\ Wj 1 /(0:+ 1 Wj I), where (P 1\ Q)i = min(Pi , Qj) and 1 P 1= L;~ 1 Pi I? With winner-take-all coding, the F2 node J that receives the largest Fl -+ F2 input T j becomes active. Node J remains active if it satisfies the matching criterion: 1Al\wj 1/ 1A 1= 1Al\wj 1/M > p, where p E [0,1] is the dimensionless vigilance parameter. Otherwise, the network resets the active F2 node and searches until J satisfies the matching criterion. If node J then makes an incorrect class prediction, a match tracking signal raises vigilance just enough to induce a search, which continues until either some F2 node becomes active for the first time, in which case J learns the correct output class label k( J) = K; or a node J that has previously learned to predict K becomes active. During testing, a pattern a that activates node J is predicted to belong to the class K = k( J). 3 ARTMAP-FD Familiarity measure. During testing, an input pattern a is defined as familiar when a familiarity function ?(A) is greater than a decision threshold T Once a category choice has been made by the winner-take-all rule, fuzzy ARTMAP ignores the size of the input TJ. In contrast, ARTMAP-FD uses TJ to define familiarity, taking ?(A) = TJ(A) = 1 A 1\ WJ 1 (1) TjlAX 1 WJ 1 ' where TjlAX =1 WJ 1/(0:+ 1WJ I)? This maximal value of T J is attained by each input a that lies in the hyperbox RJ, since 1 A 1\ W J 1= 1 W J 1 for these points. An input that chooses category J during testing is then assigned the maximum familiarity value 1 if and only if a lies within RJ. Familiarity discrimination algorithm. ARTMAP-FD is identical to fuzzy ARTMAP during training. During testing, ?(A) is computed after fuzzy ARTMAP has yielded a winning node J and a predicted class K = k(J). If ?(A) > I, ARTMAP-FD predicts class K for the input a. If ?(A) ::; I, a is regarded as belonging to an unfamiliar class and the network makes no prediction. Note that fuzzy ARTMAP can also abstain from classification, when the baseline vigilance parameter 15 is greater than zero during testing. Typically 15 = during training, to maximize code compression. In radar range profile simulations such ? Familiarity Discrimination of Radar Pulses 877 as those described below, fuzzy ARTMAP can perform familiarity discrimination when p > 0 during both training and testing. However, accurate discrimination requires that p be close to 1, which causes category proliferation during training. Range profile simulations have also set p = 0 during both training and testing, but with the familiarity measure set equal to the fuzzy ARTMAP match function: (2) This approach is essentially equivalent to taking p = 0 during training and p > 0 during testing, with p However, for a test set input a E RJ, the function defined by (2) sets ?(A) =1 w J 1/ M, which may be large or small although a is familiar. Thus this function does not provide as good familiarity discrimination as the one defined by (1), which always sets ?(A) = 1 when a E RJ. Except as noted , all the simulations below employ the function (1), with p = O. =,. Sequential evidence accumulation. ART-EMAP (Stage 3) [5] identifies a test set object's class after exposure to a sequence of input patterns, such as differing views, all identified with that one object. Training is identical to that of fuzzy ART MAP, with winner-take-all coding at F2 . ART-EMAP generally employs distributed F2 coding during testing. With winner-take-all coding during testing as well as training, ART-EMAP predicts the object's class to be the one selected by the largest number of inputs in the sequence. Extending this approach, ARTMAP-FD accumulates familiarity measures for each predicted class K as the test set sequence is presented. Once the winning class is determined, the object's familiarity is defined as the average accumulated familiarity measure of the predicted class during the test sequence. 4 Familiarity discrimination simulations Since familiarity discrimination involves placing an input into one of two sets, familiar and unfamiliar, the receiver operating characteristic (ROC) formalism can be used to evaluate the effectiveness of ARTMAP-FD on this task. The hit rate H is the fraction of familiar targets the network correctly identifies as familiar and the false alarm rate F is the fraction of unfamiliar targets the network incorrectly identifies as familiar. An ROC curve is a plot of H vs. F, parameterized by the threshold'Y (i.e., it is equivalent to the two curves Fh) and Hh)) . The area under the ROC curve is the c-index, a measure of predictive accuracy that is independent of both the fraction of positive (familiar) cases in the test set and the positive-case decision threshold 'Y. An ARTMAP-FD network was trained on simulated radar range profiles from 18 targets out of a 36-target set (Fig. la). Simulations tested sequential evidence accumulation performance for 1, 3, and 100 observations, corresponding to 0.05, 0.15, and 5.0 sec. of observation (smooth curves, Fig. Ib) . As in the case of identification [6], a combination of multiwavelength range profiles and sequential evidence accumulation produces good familiarity discrimination, with the c-index approaching 1 as the number of sequential observations grows. Fig. Ib also demonstrates the importance of the proper choice of familiarity measure. The jagged ROC curve was produced by a familiarity discrimination simulation identical to that which resulted in the IOO-sequential-view smooth curve, but using the match function (2) instead of ? as given by (1). E. Granger, S. Grossberg, M A. Rubin and W. W. Streilein 878 IO , - - - - - ----r I 'F ~_~~~II ?""'\"MA '-"- .. o o 0 .2 0.4 0.6 F (b) 08 'Y T. (c) Figure l:(a) 36 simulation targets with 6 wing positions and 6 wing lengths, and 100 scattering centers per target. Boxes indicate randomly selected familiar targets. (b) ROC curves from ARTMAP-FD simulations, with multiwavelength range profiles having 40 center frequencies. Sequential evidence accumulation for 1, 3 and 100 views uses familiarity measure (1) (smooth curves); and for 100 views uses the match function (2) (jagged curve). (c) Training and test curves of miss rate M = (1- H) and false alarm rate F vs threshold 1', for 36 targets and one view, Training curves intersect at the point where "y = r p (predicted); and test curves intersect near the point where l' = ra (optimal). The training curves are based on data from the first training epoch, the test curves on data from 3 training epochs. 5 Familiarity threshold selection When a system is placed in operation, one particular decision threshold 'Y = r must be chosen . In a given application, selection of r depends upon the relative cost of errors due to missed targets and false alarms. The optimal r corresponds to a point on the parameterized ROC curve that is typically close to the upper left-hand corner of the unit square, to maximize correct selection of familiar targets (H) while minimizing incorrect selection of unfamiliar tar gets (F) . Validation set method. To determine a predicted threshold r p , the training data is partitioned into a training subset and a validation subset. The network is trained on the training subset, and an ROC curve (F(r) , H(r)) is calculated for the validation subset. r p is then taken to be the point on the curve that maximizes [H(r) - F(r)]. (For ease of computation the symmetry point on the curve, where 1 - H('y) = F(r), can yield a good approximation.) For a familiarity discrimination task the validation set must include examples of classes not present in the training set. Once rp is determined , the training subset and validation subset should be recombined and the network retrained on the complete training set. The retrained network and the predicted threshold r p are then employed for familiarity discrimination on the test set. On-line threshold determination. During ARTMAP-FD training, category nodes compete for new patterns as they are presented. When a node J wins the competition, learning expands the category hyperbox RJ enough to enclose the training pattern a. The familiarity measure ? for each training set input then becomes equal to 1. However, before this learning takes place, ? can be less than 1, and the degree to which this initial value of ? is less than 1 reflects the distance from the training pattern to RJ. An event of this type- a training pattern successfully coded by a category node-is taken to be representative of familiar test-set patterns. The corresponding initial values of ? are thus used to generate a training Familiarity Discrimination of Radar Pulses 879 hit rate curve, where H("() equals the fraction of training inputs with cp > ,. What about false alarms? By definition, all patterns presented during training are familiar. However, a reset event during training (Sec. 2) resembles the arrival of an unfamiliar pattern during testing. Recall that a reset occurs when a category node that predicts class K wins the competition for a pattern that actually belongs to a different class k. The corresponding values of cp for these events can thus be used to generate a training false-alarm rate curve, where F("() equals the fraction of match-tracking inputs with initial cp > "(. Predictive accuracy is improved by use of a reduced set of cp values in the trainingset ROC curve construction process. Namely, training patterns that fall inside RJ, where cp = I, are not used because these exemplars tend to distort the miss rate curve. In addition, the first incorrect response to a training input is the best predictor of the network's response to an unfamiliar testing input, since sequential search will not be available during testing. Finally, giving more weight to events occurring later in the training process improves accuracy. This can be accomplished by first computing training curves H("() and F("() and a preliminary predicted threshold r p using the reduced training set; then recomputing the curves and r p from data presented only after the system has activated the final category node of the training process (Fig. Ic). The final predicted threshold r p averages these values. This calculation can still be made on-line, by taking the "final" node to be the last one activated. Table I shows that applying on-line threshold determination to simulated radar range profile data gives good predictions for the actual hit and false alarm rates, H A and FA. Furthermore, the HA and FA so obtained are close to optimal, particularly when the ROC curve has a c-index close to one. The method is effective even when testing involves sequential evidence accumulation, despite the fact that the training curves use only single views of each target. 6 NEN Near-enough-neighbor (NEN) [7, 8] is a familiarity discrimination algorithm based on the single nearest neighbor classifier. For each familiar class K, the familiarity threshold t:l.K is the largest distance between any training pattern of class K and its nearest neighbor also of class K. During testing, a test pattern is declared unfamiliar if the distance to its nearest neighbor is greater than the threshold t:l.K corresponding to the class K of that nearest neighbor. We have extended NEN to k > I by retaining the above definition of the t:l.K's, while taking the comparison during testing to be between t:l.K and the distance between the test pattern and the closest of its k nearest neighbors which is of the class K to which the test pattern is deemed to belong. 7 Radar pulse data Identifying the type of emitter from which a radar signal was transmitted is an important task for radar electronic support measures (ESM) systems. Familiarity discrimination is a key component of this task, particularly as the continual proliferation of new emitters outstrips the ability of emitter libraries to document every sort of emitter which may be encountered. The data analyzed here, gathered by Defense Research Establishment Ottawa, con- 880 E. Granger, S. Grossberg, M. A. Rubin and W W Streilein hit rate false alarm rate accuracy actual 0.81 0.11 0.95 3x3 optimal 0.86 0.14 1.00 actual 0.77 0.24 0.93 6x6 optimal 0.77 0.23 1.00 actual 0.99 0.06 1.00 6x6* optimal 0.98 0.02 1.00 Table 1: Familiarity discrimination, using ARTMAP-FD with on-line threshold prediction, of simulated radar range profile data. Training on half the target classes (boxed "aircraft" in Fig. la) , testing on all target classes. (In 3x3 case, 4 classes out of 9 total used for training.) Accuracy equals the fraction of correctly-classified targets out of familiar targets selected by the network as familiar. The results for the 6x6' dataset involve sequential evidence accumulation, with 100 observations (5 sec.) per test target. Radar range profile simulations use 40 center frequencies evenly spaced between 18GHz and 22GHz, and wp x wl simulated targets, where wp =number of wing positions and wi =number of wing lengths. The number of range bins (2/3 m. per bin) is 60 , so each pattern vector has (60 range bins) x (40 center frequencies) = 2400 components. Training patterns are at 21 evenly spaced aspects in a 10? angular range and, for each viewing angle, at 15 downrange shifts evenly spaced within a single bin width. Testing patterns are at random aspects and downrange shifts within the angular range and half the total range profile extent of (60 bins) x (2/3 m.) =40 m. method ARTMAP-FD hit rate f. a. rate accuracy [memory 0.95 0.02 1.00 21 [I II NEN city-block metric Euclidean metric k-l k-5 k - 25 k-l k-5 k - 25 0.94 0.94 0.93 0.94 0.93 0.92 0.02 0.02 0.04 0.14 0.05 0.13 1.00 0.99 1.00 1.00 1.00 1.00 446 Table 2: Familiarity discrimination of radar pulse data set, using ARTMAP-FD and NEN with different metrics and values of k. Figure given for memory is twice number of F2 nodes (due to complement coding) for ARTMAP-FD, number of training patterns for NEN. Training (single epoch) on first three quarters of data in classes 1-9, testing on other quarter of data in classes 1-9 and all data in classes 10-12. (Values given are averages over four cyclic permutations of the the 12 classes.) ARTMAP-FD familiarity threshold determined by validation-set method with retraining. sist of radar pulses from 12 ship borne navigation radars [9]. Fifty pulses were collected from each radar, with the exception of radars #7 (100 pulses) and #8 (200 pulses). The pulses were preprocessed to yield 800 I5-component vectors. with the components taking values between a and l. 8 Results From Table 2, ARTMAP-FD is seen to perform effective familiarity discrimination on the radar pulse data. NEN (k = 1) performs comparatively poorly. Extensions of NEN to k > 1 perform well. During fielded operation these would incur the cost of the additional computation required to find the k nearest neighbors of the current test pattern , as well as the cost of higher memory requirements] relative to ARTMAP-FD. The combination of low hit rate with low false alarm rate obtained by NEN on the simulated radar range profile datasets (Table 3) suggests that the algorithm performs poorly here because it selects a familiarity threshold which is 1The memory requirements of kNN pattern classifiers can be reduced by editing techniques[8], but how the use of these methods affects performance of kNN-based familiarity discrimination methods is an open question. 881 Familiarity Discrimination ofRadar Pulses method I dataset hit rate false alarm rate accuracy memory II__ II II ARTMAP -FD 3x3 0.81 0.11 0.95 I 12 I 6x6 0.77 0.24 0.93 88 Ill-rk -----.-1.,.....,_...-,--,-N_E......N....,..-...--.-_..---r-.----..-i __ k - 5 k - 99 k - 1 I k - 5 II 3x3 6x6 0.11 0.00 1.00 II 0.11 0.00 1.00 1260 0.11 0.00 1.00 0.14 0.14 0.00 0.00 1.00 1.00 5670 Table 3: Familiarity discrimination of simulated radar range profiles using ARTMAP-FD and NEN with different values of k. Training and testing as in Table 1. ARTMAP-FD familiarity threshold determined by on-line method. City-block metric used with NEN; results with Euclidean metric were slighlty poorer. too high. ARTMAP-FD on-line threshold selection, on the other hand, yields a value for the familiarity threshold which balances the desiderata of high hit rate and low false alarm rate. This research was supported in part by grants from the Office of Naval Research, ONR NOOOI4-95-1-0657 (S . G.) and ONR NOOOI4-96-1-0659 (M. A. R ., W. W. S.) , and by a grant from the Defense Advanced Research Projects Agency and the Office of Naval Research, ONR NOOOI4-95-1-0409 (S. G. , M. A. R. , W W. S.). E. G. was supported in part by the Defense Research Establishment Ottawa and the Natural Sciences and Engineering Research Council of Canada. References [1] Carpenter, G. A., Rubin, M. A. , & Streilein, W . W ., ARTMAP-FD: Familiarity discrimination applied to radar target recognition , in ICNN'97: Proceedings of the IEEE International Conference on Neural N etworks, Houston, June 1997; [2] Carpenter, G. A., Rubin, M. A., & Streilein, W. W ., Threshold Determination for ARTMAP-FD Familiarity Discrimination, in C . H. Dagli et al., eds., Intelligent Engineering Systems Through Artificial Neural Networks, 1, 23-28, ASME, New York, 1997. [3] Carpenter, G . A., Grossberg, S. , Markuzon, N., Reynolds, J . H., & Rosen, D. E ., Fuzzy ARTMAP: A neural network architecture for incremental supervised learning of analog multidimensional maps, IEEE Transactions on N eural Networks, 3, 698-713, 1992. [4] Carpenter, G. A., Grossberg, S., & Rosen . D . B. , Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system, Neural Networks, 4,759-771, 1991. [5] Carpenter, G. A., & Ross, W . D. , ART-EMAP : A neural network architecture for object recognition by evidence accumulation , IEEE Transactions on Neural Networks, 6, 805-818, 1995. [6] Rubin, M. A., Application of fuzzy ARTMAP and ART-EMAP to automatic target recognition using radar range profiles, Neural Networks , 8, 1109-1116, 1995. [7] Dasarathy, E. V.,.Is your nearest neighbor near enough a neighbor?, in Lainious, D. G. and Tzannes, N. S., eds. Applications and Research in Informations Systems and Sciences, 1, 114-117, Hemisphere Publishing Corp. , Washington, 1977. [8] Dasarathy, B. V., ed., Nearest Neighbor(NN) Norm : NN Pattern Classification Techniques, IEEE Computer Society Press, Los Alamitos, CA, 1991. [9] Granger, E. , Savaria, Y, Lavoie, P., & Cantin, M.-A ., A comparison of self-organizing neural networks for fast clustering of radar pulses, Signal Processing , 64, 249-269, 1998. 

1548 |@word aircraft:4 compression:1 norm:1 retraining:1 open:1 pulse:15 simulation:9 initial:3 cyclic:1 ecole:1 document:1 reynolds:1 current:1 activation:1 must:2 designed:1 plot:1 discrimination:28 v:2 half:2 selected:4 guess:1 ith:1 record:1 node:15 incorrect:3 inside:1 ra:1 proliferation:2 examine:1 actual:4 becomes:5 project:1 maximizes:1 what:1 sist:1 fuzzy:13 differing:1 every:1 multidimensional:1 expands:1 continual:1 scaled:1 hit:8 demonstrates:1 classifier:2 unit:1 grant:2 positive:2 before:1 engineering:3 io:1 despite:1 accumulates:1 dasarathy:2 twice:1 resembles:1 suggests:1 ease:1 emitter:4 range:17 grossberg:6 testing:20 block:2 x3:4 area:1 intersect:2 reject:1 matching:2 induce:1 get:1 close:4 encloses:1 selection:5 dimensionless:1 applying:1 accumulation:7 equivalent:2 map:2 center:4 exposure:1 artmap:37 qc:1 identifying:1 rule:1 regarded:1 target:20 construction:1 us:3 recognition:7 particularly:2 continues:1 predicts:3 electrical:1 wj:9 environment:1 agency:1 radar:28 trained:3 raise:1 exposed:1 predictive:3 incur:1 creates:1 upon:1 eric:1 f2:8 fast:3 effective:2 artificial:1 otherwise:1 ability:1 knn:2 h3c:1 final:3 sequence:4 maximal:1 reset:3 organizing:2 poorly:2 competition:2 los:1 double:1 requirement:2 extending:1 produce:1 categorization:1 incremental:1 object:6 montreal:2 exemplar:1 nearest:10 predicted:9 involves:3 indicate:1 enclose:1 slighlty:1 correct:2 viewing:1 bin:5 icnn:1 preliminary:1 recombined:1 extension:4 ic:1 predict:2 smallest:1 fh:1 label:1 ross:1 council:1 largest:4 wl:1 successfully:1 city:2 reflects:1 activates:1 always:1 establishment:2 tar:1 office:2 june:1 naval:2 contrast:1 baseline:1 accumulated:1 nn:2 typically:2 selects:1 classification:2 ill:1 retaining:1 resonance:1 art:7 field:3 once:3 equal:5 having:1 washington:1 identical:3 placing:2 rosen:2 intelligent:1 employ:2 randomly:1 resulted:1 familiar:15 consisting:1 william:1 detection:1 fd:26 analyzed:1 navigation:1 activated:2 tj:4 accurate:1 poorer:1 euclidean:2 recomputing:1 classify:1 formalism:1 ada:1 cost:3 ottawa:2 subset:6 predictor:1 too:1 chooses:1 international:1 outstrips:1 vigilance:3 borne:1 cognitive:1 corner:1 wing:4 de:1 coding:7 sec:3 jagged:2 ad:1 depends:1 later:1 view:6 sort:1 option:1 square:1 accuracy:7 characteristic:1 yield:3 identify:1 gathered:1 spaced:3 identification:3 produced:1 classified:1 ed:3 definition:2 distort:1 frequency:3 con:1 dataset:2 noooi4:3 recall:1 improves:1 actually:1 scattering:1 attained:1 higher:1 supervised:1 x6:5 response:2 improved:1 editing:1 box:1 cantin:1 furthermore:1 just:2 stage:1 angular:2 until:2 hand:2 receives:1 grows:1 usa:1 assigned:1 wp:2 during:27 self:2 width:1 noted:1 criterion:2 asme:1 complete:1 polytechnique:1 performs:4 cp:5 reflection:4 weber:1 abstain:2 quarter:2 winner:4 belong:2 analog:2 unfamiliar:8 ai:2 automatic:1 stable:1 operating:1 closest:1 belongs:2 hemisphere:1 ship:1 corp:1 onr:3 success:1 accomplished:1 transmitted:1 seen:1 greater:3 additional:1 houston:1 employed:1 novelty:1 maximize:2 determine:1 signal:3 stephen:1 ii:6 rj:7 smooth:3 match:5 determination:3 calculation:1 coded:1 prediction:5 desideratum:1 essentially:1 metric:5 addition:1 fifty:1 meaningless:1 tend:1 effectiveness:2 near:3 enough:4 trainingset:1 affect:1 architecture:2 identified:1 approaching:1 shift:2 qj:1 whether:1 defense:3 york:1 cause:1 generally:1 involve:1 ten:1 visualized:1 category:12 reduced:3 generate:2 judging:1 correctly:3 per:3 key:1 four:1 threshold:21 preprocessed:1 lavoie:1 fraction:6 compete:1 angle:1 parameterized:2 i5:1 place:1 electronic:1 missed:1 decision:3 fl:1 encountered:3 yielded:1 your:1 declared:1 aspect:2 min:1 department:2 combination:2 belonging:3 partitioned:1 wi:1 making:1 taken:2 remains:1 previously:1 etworks:1 granger:4 hh:1 available:1 operation:2 nen:15 esm:1 rp:1 clustering:1 include:1 publishing:1 giving:1 society:1 comparatively:1 question:1 alamitos:1 occurs:1 fa:2 win:2 distance:4 simulated:7 evenly:3 extent:1 collected:1 code:1 length:2 index:3 minimizing:1 balance:1 proper:1 perform:3 upper:1 observation:4 datasets:2 incorrectly:1 extended:1 retrained:2 canada:1 complement:2 namely:1 required:1 fielded:1 learned:1 below:2 pattern:26 memory:5 event:4 natural:1 advanced:1 library:1 identifies:3 deemed:1 epoch:3 relative:2 law:1 permutation:1 validation:6 degree:1 rubin:6 pi:2 placed:2 last:1 supported:2 neighbor:12 fall:1 taking:5 distributed:1 ghz:2 curve:26 calculated:1 emap:5 ignores:1 made:2 adaptive:1 transaction:2 active:5 receiver:1 search:3 table:7 ca:1 symmetry:1 ioo:1 boxed:1 alarm:10 profile:13 arrival:1 carpenter:5 eural:1 fig:5 referred:1 representative:1 roc:9 position:2 winning:2 lie:2 ib:2 learns:2 rk:1 familiarity:44 evidence:7 false:10 sequential:9 importance:1 occurring:1 boston:2 tracking:2 partially:1 corresponds:1 satisfies:2 ma:2 determined:6 except:1 miss:2 total:2 la:2 exception:1 internal:1 mark:1 support:1 evaluate:1 tested:2