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2D Observers for Human 3D Object Recognition? Zili Liu NEC Research Institute Daniel Kersten University of Minnesota . Abstract Converging evidence has shown that human object recognition depends on familiarity with the images of an object. Further, the greater the similarity between objects, the stronger is the dependence on object appearance, and the more important twodimensional (2D) image information becomes. These findings, however, do not rule out the use of 3D structural information in recognition, and the degree to which 3D information is used in visual memory is an important issue. Liu, Knill, & Kersten (1995) showed that any model that is restricted to rotations in the image plane of independent 2D templates could not account for human performance in discriminating novel object views. We now present results from models of generalized radial basis functions (GRBF), 2D nearest neighbor matching that allows 2D affine transformations, and a Bayesian statistical estimator that integrates over all possible 2D affine transformations. The performance of the human observers relative to each of the models is better for the novel views than for the familiar template views, suggesting that humans generalize better to novel views from template views. The Bayesian estimator yields the optimal performance with 2D affine transformations and independent 2D templates. Therefore, models of 2D affine matching operations with independent 2D templates are unlikely to account for human recognition performance. 1 Introduction Object recognition is one of the most important functions in human vision. To understand human object recognition, it is essential to understand how objects are represented in human visual memory. A central component in object recognition is the matching of the stored object representation with that derived from the image input. But the nature of the object representation has to be inferred from recognition performance, by taking into account the contribution from the image information. When evaluating human performance, how can one separate the con- 830 Z Liu and D. Kersten tributions to performance of the image information from the representation? Ideal observer analysis provides a precise computational tool to answer this question. An ideal observer's recognition performance is restricted only by the available image information and is otherwise optimal, in the sense of statistical decision theory, irrespective of how the model is implemented. A comparison of human to ideal performance (often in terms of efficiency) serves to normalize performance with respect to the image information for the task. We consider the problem of viewpoint dependence in human recognition. A recent debate in human object recognition has focused on the dependence of recognition performance on viewpoint [1 , 6]. Depending on the experimental conditions, an observer's ability to recognize a familiar object from novel viewpoints is impaired to varying degrees. A central assumption in the debate is the equivalence in viewpoint dependence and recognition performance. In other words, the assumption is that viewpoint dependent performance implies a viewpoint dependent representation, and that viewpoint independent performance implies a viewpoint independent representation. However, given that any recognition performance depends on the input image information, which is necessarily viewpoint dependent, the viewpoint dependence of the performance is neither necessary nor sufficient for the viewpoint dependence of the representation. Image information has to be factored out first, and the ideal observer provides the means to do this. The second aspect of an ideal observer is that it is implementation free. Consider the GRBF model [5], as compared with human object recognition (see below). The model stores a number of 2D templates {Ti} of a 3D object 0, and reco~nizes or rejects a stimulus image S by the following similarity measure ~iCi exp UITi - SI1 2 j2( 2 ), where Ci and a are constants. The model's performance as a function of viewpoint parallels that of human observers. This observation has led to the conclusion that the human visual system may indeed, as does the model, use 2D stored views with GRBF interpolation to recognize 3D objects [2]. Such a conclusion, however, overlooks implementational constraints in the model, because the model's performance also depends on its implementations. Conceivably, a model with some 3D information of the objects can also mimic human performance, so long as it is appropriately implemented. There are typically too many possible models that can produce the same pattern of results. In contrast, an ideal observer computes the optimal performance that is only limited by the stimulus information and the task. We can define constrained ideals that are also limited by explicitly specified assumptions (e.g., a class of matching operations). Such a model observer therefore yields the best possible performance among the class of models with the same stimulus input and assumptions. In this paper, we are particularly interested in constrained ideal observers that are restricted in functionally Significant aspects (e.g., a 2D ideal observer that stores independent 2D templates and has access only to 2D affine transformations) . The key idea is that a constrained ideal observer is the best in its class. So if humans outperform this ideal observer, they must have used more than what is available to the ideal. The conclusion that follows is strong: not only does the constrained ideal fail to account for human performance, but the whole class of its implementations are also falsified. A crucial question in object recognition is the extent to which human observers model the geometric variation in images due to the projection of a 3D object onto a 2D image. At one extreme, we have shown that any model that compares the image to independent views (even if we allow for 2D rigid transformations of the input image) is insufficient to account for human performance. At the other extreme, it is unlikely that variation is modeled in terms of rigid transformation of a 3D object 2D Observers/or Hwnan 3D Object Recognition? 831 template in memory. A possible intermediate solution is to match the input image to stored views, subject to 2D affine deformations. This is reasonable because 2D affine transformations approximate 3D variation over a limited range of viewpoint change. In this study, we test whether any model limited to the independent comparison of 2D views, but with 2D affine flexibility, is sufficient to account for viewpoint dependence in human recognition. In the following section, we first define our experimental task, in which the computational models yield the provably best possible performance under their specified conditions. We then review the 2D ideal observer and GRBF model derived in [4], and the 2D affine nearest neighbor model in [8]. Our principal theoretical result is a closed-form solution of a Bayesian 2D affine ideal observer. We then compare human performance with the 2D affine ideal model, as well as the other three models. In particular, if humans can classify novel views of an object better than the 2D affine ideal, then our human observers must have used more information than that embodied by that ideal. 2 The observers Let us first define the task. An observer looks at the 2D images of a 3D wire frame object from a number of viewpoints. These images will be called templates {Td. Then two distorted copies of the original 3D object are displayed. They are obtained by adding 3D Gaussian positional noise (i.i.d.) to the vertices of the original object. One distorted object is called the target, whose Gaussian noise has a constant variance. The other is the distract or , whose noise has a larger variance that can be adjusted to achieve a criterion level of performance. The two objects are displayed from the same viewpoint in parallel projection, which is either from one of the template views, or a novel view due to 3D rotation. The task is to choose the one that is more similar to the original object. The observer's performance is measured by the variance (threshold) that gives rise to 75% correct performance. The optimal strategy is to choose the stimulus S with a larger probability p (OIS). From Bayes' rule, this is to choose the larger of p (SIO). Assume that the models are restricted to 2D transformations of the image, and cannot reconstruct the 3D structure of the object from its independent templates {Ti}. Assume also that the prior probability p(Td is constant. Let us represent S and Ti by their (x, y) vertex coordinates: (X Y )T, where X = (Xl, x2, ... , x n ), y = (yl, y2 , ... , yn). We assume that the correspondence between S and T i is solved up to a reflection ambiguity, which is equivalent to an additional template: Ti = (xr yr )T, where X r = (x n , ... ,x2,xl ), yr = (yn, ... ,y2,yl). We still denote the template set as {Td. Therefore, (1) In what follows, we will compute p(SITi)p(T i ), with the assumption that S = F (Ti) + N (0, crI 2n ), where N is the Gaussian distribution, 12n the 2n x 2n identity matrix, and :F a 2D transformation. For the 2D ideal observer, :F is a rigid 2D rotation. For the GRBF model, F assigns a linear coefficient to each template T i , in addition to a 2D rotation. For the 2D affine nearest neighbor model, :F represents the 2D affine transformation that minimizes liS - Ti11 2 , after Sand Ti are normalized in size. For the 2D affine ideal observer, :F represents all possible 2D affine transformations applicable to T i. Z Liu and D. Kersten 832 2.1 The 2D ideal observer The templates are the original 2D images, their mirror reflections, and 2D rotations (in angle ?) in the image plane. Assume that the stimulus S is generated by adding Gaussian noise to a template, the probability p(SIO) is an integration over all templates and their reflections and rotations. The detailed derivation for the 2D ideal and the GRBF model can be found in [4]. Ep(SITi)p(Ti) ex: E J d?exp (-liS - Ti(?)112 /2( 2 ) ? (2) 2.2 The GRBF model The model has the same template set as the 2D ideal observer does. Its training requires that EiJ;7r d?Ci(?)N(IITj - Ti(?)II,a) = 1, j = 1,2, ... , with which {cd can be obtained optimally using singular value decomposition. When a pair of new stimuli is} are presented, the optimal decision is to choose the one that is closer to the learned prototype, in other words, the one with a smaller value of 111- E 127r d?ci(?)exp (_liS - 2:~(?)1I2) II. (3) 2.3 The 2D affine nearest neighbor model It has been proved in [8] that the smallest Euclidean distance D(S, T) between S and T is, when T is allowed a 2D affine transformation, S ~ S/IISII, T ~ T/IITII, D2(S, T) = 1 - tr(S+S . TTT)/IITII2, (4) where tr strands for trace, and S+ = ST(SST)-l. The optimal strategy, therefore, is to choose the S that gives rise to the larger of E exp (_D2(S, Ti)/2a 2) , or the smaller of ED2(S, Ti). (Since no probability is defined in this model, both measures will be used and the results from the better one will be reported.) 2.4 The 2D affine ideal observer We now calculate the Bayesian probability by assuming that the prior probability distribution of the 2D affine transformation, which is applied to the template T i , AT + Tr (~ ~) Ti + (~: ::: ~:), = obeys a Gaussian distribution N(X o ,,,,/1 6 ), where Xo is the identity transformation (1,0,0,1,0,0). We have Ep(SIT i ) i: xl' = (a,b,c,d,tx,t y) = dX exp (-IIATi + Tr - SII 2/2( 2) = E = EC(n, a, ",/)deC 1 (QD exp (tr (KfQi(QD-1QiKi) /2(12), (6) where C(n, a, ",/) is a function of n, a, "'/; Q' = Q Q _ ( XT . X T Y T ?XT (5) + ",/-212, and X T ? Y T ) QK _ ( XT? Xs YT ?YT ' X T ?Ys Y T . Xs) -21 Y T .Ys +"'/ 2? (7) The free parameters are "'/ and the number of 2D rotated copies for each T i (since a 2D affine transformation implicitly includes 2D rotations, and since a specific prior probability distribution N(Xo, ",/1) is assumed, both free parameters should be explored together to search for the optimal results). 2D Observers for Hwnan 3D Object Recognition? 833 ? ? ? ? ? ? Figure 1: Stimulus classes with increasing structural regularity: Balls, Irregular, Symmetric, and V-Shaped. There were three objects in each class in the experiment. 2.5 The human observers Three naive subjects were tested with four classes of objects: Balls, Irregular, Symmetric, and V-Shaped (Fig. 1). There were three objects in each class. For each object, 11 template views were learned by rotating the object 60? /step, around the X- and Y-axis, respectively. The 2D images were generated by orthographic projection, and viewed monocularly. The viewing distance was 1.5 m. During the test, the standard deviation of the Gaussian noise added to the target object was (J"t = 0.254 cm. No feedback was provided. Because the image information available to the humans was more than what was available to the models (shading and occlusion in addition to the (x, y) positions of the vertices), both learned and novel views were tested in a randomly interleaved fashion. Therefore, the strategy that humans used in the task for the learned and novel views should be the same. The number of self-occlusions, which in principle provided relative depth information, was counted and was about equal in both learned and novel view conditions. The shading information was also likely to be equal for the learned and novel views. Therefore, this additional information was about equal for the learned and novel views, and should not affect the comparison of the performance (humans relative to a model) between learned and novel views. We predict that if the humans used a 2D affine strategy, then their performance relative to the 2D affine ideal observer should not be higher for the novel views than for the learned views. One reason to use the four classes of objects with increasing structural regularity is that structural regularity is a 3D property (e.g., 3D Symmetric vs. Irregular), which the 2D models cannot capture. The exception is the planar V-Shaped objects, for which the 2D affine models completely capture 3D rotations, and are therefore the "correct" models. The V-Shaped objects were used in the 2D affine case as a benchmark. If human performance increases with increasing structural regularity of the objects, this would lend support to the hypothesis that humans have used 3D information in the task. 2.6 Measuring performance A stair-case procedure [7] was used to track the observers' performance at 75% correct level for the learned and novel views, respectively. There were 120 trials for the humans, and 2000 trials for each of the models. For the GRBF model, the standard deviation of the Gaussian function was also sampled to search for the best result for the novel views for each of the 12 objects, and the result for the learned views was obtained accordingly. This resulted in a conservative test of the hypothesis of a GRBF model for human vision for the following reasons: (1) Since no feedback was provided in the human experiment and the learned and novel views were randomly intermixed, it is not straightforward for the model to find the best standard deviation for the novel views, particularly because the best standard deviation for the novel views was not the same as that for the learned Z Liu and D. Kersten 834 ones. The performance for the novel views is therefore the upper limit of the model's performance. (2) The subjects' performance relative to the model will be defined as statistical efficiency (see below). The above method will yield the lowest possible efficiency for the novel views, and a higher efficiency for the learned views, since the best standard deviation for the novel views is different from that for the learned views. Because our hypothesis depends on a higher statistical efficiency for the novel views than for the learned views, this method will make such a putative difference even smaller. Likewise, for the 2D affine ideal, the number of 2D rotated copies of each template Ti and the value I were both extensively sampled, and the best performance for the novel views was selected accordingly. The result for the learned views corresponding to the same parameters was selected. This choice also makes it a conservative hypothesis test. 3 Results Learned Views 25 IJ O e- O 20 Affine Nearest NtMghbor :g 0 Human 20 Ideal GRBF rn 20 Affine kIoai .?. ~ ? 1.5 Novel Views ? Human EJ 20 Ideal o o e- .?. :!2 0 ~ GRBF 20 Affine Nearesl N.tghbor ~ 2DAfllna~ 1.5 ~ 81 l! ~ l- I- 0.5 Object Type Object Type Figure 2: The threshold standard deviation of the Gaussian noise, added to the distractor in the test pair, that keeps an observer's performance at the 75% correct level, for the learned and novel views, respectively. The dotted line is the standard deviation of the Gaussian noise added to the target in the test pair. Fig. 2 shows the threshold performance. We use statistical efficiency E to compare human to model performance. E is defined as the information used by humans relative to the ideal observer [3] : E = (d~uman/d~deal)2, where d' is the discrimination index. We have shown in [4] that, in our task, E = ((a~1!f;actor)2 - (CTtarget)2) / ((CT~~~~~tor)2 - (CTtarget)2) , where CT is the threshold. Fig. 3 shows the statistical efficiency of the human observers relative to each of the four models. We note in Fig. 3 that the efficiency for the novel views is higher than those for the learned views (several of them even exceeded 100%), except for the planar V-Shaped objects. We are particularly interested in the Irregular and Symmetric objects in the 2D affine ideal case, in which the pairwise comparison between the learned and novel views across the six objects and three observers yielded a significant difference (binomial, p < 0.05). This suggests that the 2D affine ideal observer cannot account for the human performance, because if the humans used a 2D affine template matching strategy, their relative performance for the novel views cannot be better than for the learned views. We suggest therefore that 3D information was used by the human observers (e.g., 3D symmetry). This is supported in addition by the increasing efficiencies as the structural regularity increased from the Balls, Irregular, to Symmetric objects (except for the V-Shaped objects with 2D affine models). 2D Observers for Hwnan 3D Object Recognition? 300 .. l .." ..! " $: "" 20 Ideal 250 o Learned 200 ? Novel 250 l f '50 '" ~ "" w "- II! .'" N ~ I .Noval 0 l&arnedl ~ -------------- " '" Q l>j GRBF Modol --- t ! ObjoctTypo 300 20 Aftlne Nearest 250 200 o Learned ? Novel Q N Ighbor l,.. j" " ~ 150 300 20 Affine Ideal 250 o Learned 200 ? Novel '50 j i I ~ ObJect Type 835 0 Object Type ObjOGtType Figure 3: Statistical efficiencies of human observers relative to the 2D ideal observer, the GRBF model, the 2D affine nearest neighbor model, and the 2D affine ideal observer_ 4 Conclusions Computational models of visual cognition are subject to information theoretic as well as implementational constraints. When a model's performance mimics that of human observers, it is difficult to interpret which aspects of the model characterize the human visual system. For example, human object recognition could be simulated by both a GRBF model and a model with partial 3D information of the object. The approach we advocate here is that, instead of trying to mimic human performance by a computational model, one designs an implementation-free model for a specific recognition task that yields the best possible performance under explicitly specified computational constraints. This model provides a well-defined benchmark for performance, and if human observers outperform it, we can conclude firmly that the humans must have used better computational strategies than the model. We showed that models of independent 2D templates with 2D linear operations cannot account for human performance. This suggests that our human observers may have used the templates to reconstruct a representation of the object with some (possibly crude) 3D structural information. References [1] Biederman I and Gerhardstein P C. Viewpoint dependent mechanisms in visual object recognition: a critical analysis. J. Exp. Psych.: HPP, 21: 1506-1514, 1995. [2] Biilthoff H H and Edelman S. Psychophysical support for a 2D view interpolation theory of object recognition. Proc. Natl. Acad. Sci. , 89:60-64, 1992. [3] Fisher R A. Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh, 1925. [4] Liu Z, Knill D C, and Kersten D. Object classification for human and ideal observers. Vision Research, 35:549-568, 1995. [5] Poggio T and Edelman S. A network that learns to recognize three-dimensional objects. Nature, 343:263-266, 1990. [6] Tarr M J and Biilthoff H H. Is human object recognition better described by geon-structural-descriptions or by multiple-views? J. Exp. Psych.: HPP, 21:1494-1505,1995. [7] Watson A B and Pelli D G. QUEST: A Bayesian adaptive psychometric method. Perception and Psychophysics, 33:113-120, 1983. [8] Werman M and Weinshall D. Similarity and affine invariant distances between 2D point sets. IEEE PAMI, 17:810-814,1995. Toward a Single-Cell Account for Binocular Disparity Tuning: An Energy Model May be Hiding in Your Dendrites Bartlett W. Mel Department of Biomedical Engineering University of Southern California, MC 1451 Los Angeles, CA 90089 mel@quake.usc.edu Daniel L. Ruderman The Salk Institute 10010 N. Torrey Pines Road La Jolla, CA 92037 ruderman@salk.edu Kevin A. Archie Neuroscience Program University of Southern California Los Angeles, CA 90089 karchie@quake.usc.edu Abstract Hubel and Wiesel (1962) proposed that complex cells in visual cortex are driven by a pool of simple cells with the same preferred orientation but different spatial phases. However, a wide variety of experimental results over the past two decades have challenged the pure hierarchical model, primarily by demonstrating that many complex cells receive monosynaptic input from unoriented LGN cells, or do not depend on simple cell input. We recently showed using a detailed biophysical model that nonlinear interactions among synaptic inputs to an excitable dendritic tree could provide the nonlinear subunit computations that underlie complex cell responses (Mel, Ruderman, & Archie, 1997). This work extends the result to the case of complex cell binocular disparity tuning, by demonstrating in an isolated model pyramidal cell (1) disparity tuning at a resolution much finer than the the overall dimensions of the cell's receptive field, and (2) systematically shifted optimal disparity values for rivalrous pairs of light and dark bars-both in good agreement with published reports (Ohzawa, DeAngelis, & Freeman, 1997). Our results reemphasize the potential importance of intradendritic computation for binocular visual processing in particular, and for cortical neurophysiology in general. A Single-Cell Accountfor Binocular Disparity Tuning 1 209 Introduction Binocular disparity is a powerful cue for depth in vision. The neurophysiological basis for binocular disparity processing has been of interest for decades, spawned by the early studies of Rubel and Wiesel (1962) showing neurons in primary visual cortex which could be driven by both eyes. Early qualitative models for disparity tuning held that a binocularly driven neuron could represent a particular disparity (zero, near, or far) via a relative shift of receptive field (RF) centers in the right and left eyes. According to this model, a binocular cell fires maximally when an optimal stimulus, e.g. an edge of a particular orientation, is simultaneously centered in the left and right eye receptive fields, corresponding to a stimulus at a specific depth relative to the fixation point. An account of this kind is most relevant to the case of a cortical "simple" cell, whose phase-sensitivity enforces a preference for a particular absolute location and contrast polarity of a stimulus within its monocular receptive fields. This global receptive field shift account leads to a conceptual puzzle, however, when binocular complex cell receptive fields are considered instead, since a complex cell can respond to an oriented feature nearly independent of position within its monocular receptive field. Since complex cell receptive field diameters in the cat lie in the range of 1-3 degrees, the excessive "play" in their monocular receptive fields would seem to render complex cells incapable of signaling disparity on the much finer scale needed for depth perception (measured in minutes). Intriguingly, various authors have reported that a substantial fraction of complex cells in cat visual cortex are in fact tuned to left-right disparities much finer than that suggested by the size of the monocular RF's. For such cells, a stimulus delivered at the proper disparity, regardless of absolute position in either eye, produces a neural response in excess of that predicted by the sum of the monocular responses (Pettigrew, Nikara, & Bishop, 1968; Ohzawa, DeAngelis, & Freeman, 1990; Ohzawa et al., 1997). Binocular responses of this type suggest that for these cells, the left and right RF's are combined via a correlation operation rather than a simple sum (Nishihara & Poggio, 1984; Koch & Poggio, 1987). This computation has also been formalized in terms of an "energy" model (Ohzawa et al., 1990, 1997), building on the earlier use of energy models to account for complex cell orientation tuning (Pollen & Ronner, 1983) and direction selectivity (Adelson & Bergen, 1985). In an energy model for binocular disparity tuning, sums of linear Gabor filter outputs representing left and right receptive fields are squared to produce the crucial multiplicative cross terms (Ohzawa et al., 1990, 1997). Our previous biophysical modeling work has shown that the dendritic tree of a cortical pyramidal cells is well suited to support an approximative high-dimensional quadratic input-output relation, where the second-order multiplicative cross terms arise from local interactions among synaptic inputs carried out in quasi-isolated dendritic "subunits" (Mel, 1992b, 1992a, 1993). We recently applied these ideas to show that the position-invariant orientation tuning of a monocular complex cell could be computed within the dendrites of a single cortical cell, based exclusively upon excitatory inputs from a uniform, overlapping population of unoriented ON and OFF cells (Mel et al., 1997). Given the similarity of the "energy" formulations previously proposed to account for orientation tuning and binocu~ar disparity tuning, we hypothesized that a similar type of dendritic subunit computation could underlie disparity tuning in a binocularly driven complex cell. B. W. Mel, D. L Ruderman and K A. Archie 210 Parameter Rm Ra Value IOkO cm:l 2000cm em 1.0ILF/cm~ Vrest -70 mV 615 0.20,0.12 S/cm:l 0.05,0.03 S/cm:t. 0- 100 Hz 0.027 nS - 0.295 nS 0.5 ms, 3 ms 0.27 nS - 2.95 nS 0.5 ms, 50 ms OmV Compartments Somatic !iNa, YnR Dendritic !iNa, YnR Input frequency gAMPA TAMPA (on, of f) gNMDA 7'NMDA (on, off) Esyn Table 1: Biophysical simulation parameters. Details of HH channel implementation are given elsewhere (Mel, 1993); original HH channel implementation courtesy Ojvind Bernander and Rodney Douglas. In order that local EPSP size be held approximately constant across the dendritic arbor, peak synaptic conductance at dendritic location x was approximately scaled to the local input resistance (inversely), given by 9syn(X) = C/Rin(X), where c was a constant, and Rin(X) = max(Rin(X),200MO). Input resistance Rin(X) was measured for a passive cell. Thus 9syn was identical for all dendritic sites with input resistance below 200MO, and was given by the larger conductance value shown; roughly 50% of the tree fell within a factor of 2 of this value. Peak conductances at the finest distal tips were smaller by roughly a factor of 10 (smaller number shown). Somatic input resistance was near 24MO. The peak synaptic conductance values used were such that the ratio of steady state current injection through NMDA vs. AMPA channels was 1.2?0.4. Both AMPA and NMDA-type synaptic conductances were modeled using the kinetic scheme of Destexhe et al. (1994); synaptic activation and inactivation time constants are shown for each. 2 Methods Compartmental simulations of a pyramidal cell from cat visual cortex (morphology courtesy of Rodney Douglas and Kevan Martin) were carried out in NEURON (Hines, 1989); simulation parameters are summarized in Table 1. The soma and dendritic membrane contained Hodgkin-Huxley-type (HH) voltage-dependent sodium and potassium channels. Following evidence for higher spike thresholds and decremental propagation in dendrites (Stuart & Sakmann, 1994), HH channel density was set to a uniform, 4-fold lower value in the dendritic membrane relative to that of the cell body. Excitatory synapses from LGN cells included both NMDA and AMPAtype synaptic conductances. Since the cell was considered to be isolated from the cortical network, inhibitory input was not modeled. Cortical cell responses were reported as average spike rate recorded at the cell body over the 500 ms stimulus period, excluding the 50 ms initial transient. The binocular LGN consisted of two copies of the monocular LGN model used previously (Mel et al., 1997), each consisting of a superimposed pair of 64x64 ON and OFF subfields. LGN cells were modeled as linear, half-rectified center-surround filters with centers 7 pixels in width. We randomly subsampled the left and right LGN arrays by a factor of 16 to yield 1,024 total LGN inputs to the pyramidal cell. A Single-Cell Account for Binocular Disparity Tuning 211 A developmental principle was used to determine the spatial arrangement of these 1,024 synaptic contacts onto the dendritic branches of the cortical cell, as follows. A virtual stimulus ensemble was defined for the cell, consisting of the complete set of single vertical light or dark bars presented binocularly at zero-disparity within the cell's receptive field. Within this ensemble, strong pairwise correlations existed among cells falling into vertically aligned groups of the same (ON or OFF) type, and cells in the vertical column at zero horizontal disparity in the other eye. These binocular cohorts of highly correlated LGN cells were labeled mutual "friends". Progressing through the dendritic tree in depth first order, a randomly chosen LG N cell was assigned to the first dendritic site. A randomly chosen "friend" of hers was assigned to the second site, the third site was assigned to a friend of the site 2 input, etc., until all friends in the available subsample were assigned (4 from each eye, on average). If the friends of the connection at site i were exhausted, a new LGN cell was chosen at random for site i + 1. In earlier work, this type of synaptic arrangement was shown to be the outcome of a Hebb-type correlational learning rule, in which random, activity independent formation of synaptic contacts acted to slowly randomize the axo-dendritic interface, shaped by Hebbian stabilization of synaptic contacts based on their short-range correlations with other synapses. 3 Results Model pyramidal cells configured in this way exhibited prominent phase-invariant orientation tuning, the hallmark response property of the visual complex cell. Multiple orientation tuning curves are shown, for example, for a monocular complex cell, giving rise to strong tuning for light and dark bars across the receptive field (fig. 1). The bold curve shows the average of all tuning curves for this cell; the half-width at half max is 25?, in the normal range for complex cells in cat visual cortex (Orban, 1984). When the spatial arrangement of LGN synaptic contacts onto the pyramidal cell dendrites was randomly scrambled, leaving all other model parameters unchanged, orientation tuning was abolished in this cell (right frame), confirming the crucial role of spatially-mediated nonlinear synaptic interactions (average curve from left frame is reproduced for comparison). Disparity-tuning in an orientation-tuned binocular model cell is shown in fig. 2, compared to data from a complex cell in cat visual cortex (adapted from Ohzawa et al. (1997)). Responses to contrast matched (light-light) and contrast non-matched (light-dark) bar pairs were subtracted to produce these plots. The strong diagonal structure indicates that both the model and real cells responded most vigorously when contrast-matched bars were presented at the same horizontal position in the left and right-eye RF's (Le. at zero-disparity), whereas peak responses to contrastnon-matched bars occured at symmetric near and far, non-zero disparities. 4 Discussion The response pattern illustrated in fig. 2A is highly similar to the response generated by an analytical binocular energy model for a complex cell (Ohzawa et al., 1997): {exp (-kXi) cos (271' f XL) {exp (-kxiJ sin (271' f XL) + exp (-kX'kJ cos (271' f XR)}2 + + exp (-kXh) sin (271' f XR)}2, (1) where XL and X R are the horizontal bar positions to the two eyes, k is the factor B. W. Mel, D_ L Ruderman and K. A. Archie 212 Ordered vs. Scrambled Orientation Tuning 70 60 '0 Ql ~ Ql "''5." .e Ql (/) 50 40 +- 50 -?I- 45 -+- U ,,*- Ql .... - (/) Us -ll- (/) Ql 35 .e 30 "''5." ~ (/) 20 (/) Ql c: 8. ex: 10 0 -90 25 20 Ql ex: ordered scrambled -+- 40 Ql 30 c: 8. - 55 average lightO dark 4 light 8 light16 dark 16 15 /'--- "' + 10 -60 -30 0 30 Orientabon (degrees) 60 90 5 -90 '+---- / -60 / + , ~, I ' +_+- ~ -30 0 30 Orientation (degrees) 60 90 Figure 1: Orientation tuning curves are shown in the left frame for light and dark bars at 3 arbitrary positions_ Essentially similar responses were seen at other receptive field positions, and for other complex cells_ Bold trace indicates average of tuning curves at positions 0, 1, 2, 4, 8, and 16 for light and dark bars. Similar form of 6 curves shown reflects the translation-invariance of the cell's response to oriented stimuli, and symmetry with respect to ON and OFF input. Orientation tuning is eliminated when the spatial arrangement of LGN synapses onto the model cell dendrites is randomly scrambled (right frame). Complex Cell Model Complex Cell in Cat VI Ohzawa, Deangelis, & Freeman, 1997 Right eye position Right eye position Figure 2: Comparison of disparity tuning in model complex cell to that of a binocular complex cell from cat visual cortex. Light or dark bars were presented simultaneously to the left and right eyes. Bars could be of same polarity in both eyes (light, light) or different polarity (light, dark); cell responses for these two cases were subtracted to produce plot shown in left frame. Right frame shows data similarly displayed for a binocular complex cell in cat visual cortex (adapted from Ohzawa et al. (1997)). A Single-Cell Account for Binocular Disparity Tuning that determines the width of the subunit RF's, and 213 f is the spatial frequency. In lieu of literal simple cell "subunits" , the present results indicate that the subunit computations associated with the terms of an energy model could derive largely from synaptic interactions within the dendrites of the individual cortical cell, driven exclusively by excitatory inputs from unoriented, monocular ON and OFF cells drawn from a uniform overlapping spatial distribution. While lateral inhibition and excitation play numerous important roles in cortical computation, the present results suggest they are not essential for the basic features of the nonlinear disparity tuned responses of cortical complex cells. Further, these results address the paradox as to how inputs from both unoriented LGN cells and oriented simple cells can coexist without conflict within the dendrites of a single complex cell. A number of controls from previous work suggest that this type of subunit processing is very robustly computed in the dendrites of an individual neuron, with little sensitivity to biophysical parameters and modeling assumptions, including details of the algorithm used to spatially organize the genicula-cortical projection, specifics of cell morphology, synaptic activation density across the dendritic tree, passive membrane and cytoplasmic parameters, and details of the kinetics, voltage-dependence, or spatial distribution of the voltage-dependent dendritic channels. One important difference between a standard energy model and the intradendritic responses generated in the present simulation experiments is that the energy model has oriented RF structure at the linear (simple-cell-like) stage, giving rise to oriented, antagonistic ON-OFF subregions (Movshon, Thompson, & Tolhurst, 1978), whereas the linear stage in our model gives rise to center-surround antagonism only within individual LGN receptive fields. Put another way, the LGN-derived subunits in the present model cannot provide all the negative cross-terms that appear in the energy model equations, specifically for pairs of pixels that fall outside the range of a single LG N receptive field. While the present simulations involve numerous simplifications relative to the full complexity of the cortical microcircuit, the results nonetheless emphasize the potential importance of intradendritic computation in visual cortex. Acknowledgements Thanks to Ken Miller, Allan Dobbins, and Christof Koch for many helpful comments on this work. This work was funded by the National Science Foundation and the Office of Naval Research, and by a Slo~n Foundation Fellowship (D.R.). References Adelson, E., & Bergen, J. (1985). Spatiotemporal energy models for the perception of motion. J. Opt. Soc. Amer., A 2, 284-299. Rines, M. (1989). A program for simulation of nerve equations with branching geometries. Int. J. Biomed. Comput., 24, 55-68. Rubel, D., & Wiesel, T . (1962) . Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. J. Physiol., 160, 106- 154. Koch, C., & Poggio, T . (1987) . Biophysics of computation: Neurons, synapses, and membranes. In Edelman, G., Gall, W., & Cowan, W. (Eds.), Synaptic junction, pp. 637-697. Wiley, New York. Mel, B. (1992a). The clusteron: Toward a simple abstraction for a complex neuron. In Moody, J., Hanson, S., & Lippmann, R. (Eds.), Advances in Neural 214 B. W. Mel, D. L Ruderman and K. A Archie Information Processing Systems, vol. 4, pp. 35-42. Morgan Kaufmann, San Mateo, CA. Mel, B. (1992b). NMDA-based pattern discrimination in a modeled cortical neuron. Neural Computation, 4, 502-516. Mel, B. (1993). Synaptic integration in an excitable dendritic tree. J. Neurophysiol., 70(3), 1086-110l. Mel, B., Ruderman, D., & Archie, K. (1997). Complex-cell responses derived from center-surround inputs: the surprising power of intradendritic computation. In Mozer, M., Jordan, M., & Petsche, T. (Eds.), Advances in Neural Information Processing Systems, Vol. 9, pp. 83-89. MIT Press, Cambridge, MA. Movshon, J., Thompson, I., & Tolhurst, D. (1978). Receptive field organization of complex cells in the cat's striate cortex. J. Physiol., 283, 79-99. Nishihara, H., & Poggio, T. (1984). Stereo vision for robotics. In Brady, & Paul (Eds.), Proceedings of the First International Symposium of Robotics Research, pp. 489-505. MIT Press, Cambridge, MA. Ohzawa, I., DeAngelis, G., & Freeman, R. (1990). Stereoscopic depth discrimination in the visual cortex: Neurons ideally suited as disparity detectors. Science, 249, 1037- 104l. Ohzawa, I., DeAngelis, G. , & Freeman, R. (1997). Encoding of binocular disparity by complex cells in the cat's visual cortex. J. Neurophysiol., June. Orban, G. (1984). Neuronal operations in the visual cortex. Springer Verlag, New York. Pettigrew, J., Nikara, T., & Bishop, P. (1968). Responses to moving slits by single units in cat striate cortex. Exp. Brain Res., 6, 373-390. Pollen, D., & Ronner, S. (1983). Visual cortical neurons as localized spatial frequency filters. IEEE Trans. Sys. Man Cybero., 13, 907-916. Stuart, G., & Sakmann, B. (1994). Active propagation of somatic action potentials into neocortical pyramidal cell dendrites. Nature, 367, 69-72. 

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Blind Separation of Radio Signals Fading Channels ? In Kari Torkkola Motorola, Phoenix Corporate Research Labs, 2100 E. Elliot Rd, MD EL508, Tempe, AZ 85284, USA email: A540AA(Qemail.mot.com Abstract We apply information maximization / maximum likelihood blind source separation [2, 6) to complex valued signals mixed with complex valued nonstationary matrices. This case arises in radio communications with baseband signals. We incorporate known source signal distributions in the adaptation, thus making the algorithms less "blind". This results in drastic reduction of the amount of data needed for successful convergence. Adaptation to rapidly changing signal mixing conditions, such as to fading in mobile communications, becomes now feasible as demonstrated by simulations. 1 Introduction In SDMA (spatial division multiple access) the purpose is to separate radio signals of interfering users (either intentional or accidental) from each others on the basis of the spatial characteristics of the signals using smart antennas, array processing, and beamforming [5, 8). Supervised methods typically use a variant of LMS (least mean squares), either gradient based, or algebraic, to adapt the coefficients that describe the channels or their inverses. This is usually a robust way of estimating the channel but a part of the signal is wasted as predetermined training data, and the methods might not be fast enough for rapidly varying fading channels. Unsupervised methods either rely on information about the antenna array manifold, or properties of the signals. Former approaches might require calibrated antenna arrays or special array geometries. Less restrictive methods use signal properties only, such as constant modulus, finite alphabet, spectral self-coherence, or cyclostationarity. Blind source separation (BSS) techniques typically rely only on source signal independence and non-Gaussianity assumptions. Our aim is to separate simultaneous radio signals occupying the same frequency band, more specifically, radio signals that carry digital information. Since linear mixtures of antenna signals end up being linear mixtures of (complex) baseband signals due to the linearity of the downconversion process, we will apply BSS at the baseband stage of the receiver. The main contribution of this paper is to show that by making better use of the known signal properties, it is possible to devise algorithms that adapt much faster than algorithms that rely only on weak assumptions, such as source signal independence. We will first discuss how the probability density functions (pdf) of baseband DPSK signals could be modelled in' a way that can efficiently be used in blind separation algorithms. We will incorporate those models into information maximization and Blind Separation of Radio Signals in Fading Channels 757 into maximum likelihood approaches [2, 6). We will then continue with the maximum likelihood approach and other modulation techniques, such as QAM. Finally, we will show in simulations, how this approach results in an adaptation process that is fast enough for fading channels. 2 Models of baseband signal distributions In digital communications the binary (or n-ary) information is transmitted as discrete combinations of the amplitude and/or the phase of the carrier signal. After downconversion to baseband the instantaneous amplitude of the carrier can be observed as the length of a complex valued sample of the baseband signal, and the phase of the carrier is discernible as the phase angle of the same sample. Possible combinations that depend on the modulation method employed, are called symbol constellations. N-QAM (quadrature amplitude modulation) utilizes both the amplitude and the phase, whereby the baseband signals can only take one of N possible locations on a grid on the complex plane. In N-PSK (phase shift keying) the amplitude of the baseband signal stays constant, but the phase can take any of N discrete values. In DPSK (differential phase shift keying) the information is encoded as the difference between phases of two consecutive transmitted symbols. The phase can thus take any value, and since the amplitude remains constant, the baseband signal distribution is a circle on the complex plane. Information maximization BSS requires a nonlinear function that models the cumulative density function (cdf) of the data. This function and its derivative need to be differentiable. In the case of a circular complex distribution with uniformly distributed phase, there is only one important direction of deviation, the radial direction. A smooth cdf G for a circular distribution at the unit circle can be constructed using the hyperbolic tangent function as G(z) = tanh(w(lzl - 1)) (1) and the pdf, differentiated in the radial direction, that is, with respect to Izl is 8 g(z) = Bizi tanh(w(lzl - 1)) = w(l - tanh2(w(lzl - 1))) (2) where z = x + iy is a complex valued variable, and the parameter w controls the steepness of the slope of the tanh function. Note that this is in contrast to more commonly used coordinate axis directions to differentiate and to integrate to get the pdf from the cdf and vice versa. These functions are plotted in Fig. 1. a) CDF b) PDF Figure 1: Radial tanh with w=2.0 (equations 1 and 2) . Note that we have not been worrying about the pdf integrating to unity. Thus we could leave the first multiplicative constant w out of the definition of g. Scaling will not be important for our purposes of using these functions as the nonlinearities in the information maximization BSS. Note also that when the steepness w approaches infinity, the densities approach the ideal density of a DPSK source, the unit circle. Many other equally good choices are possible where the ideal density is reached as a limit of a parameter value. For example, the radial section of the circular "ridge" of the pdf could be a Gaussian. 758 3 K. Torkkola The information maximization adaptation equation The information maximization adaptation equation to learn the unmixing matrix W using the natural gradient is [2] AWex (guT + I)W where ~. - 8 !!Jli.. (3) YJ - 8Yi 8Ui Vector U = W x denotes a time sample of the separated sources, x denotes the corresponding time sample of the observed mixtures, and Yj is the nonlinear function approximating the cdf of the data, which is applied to each component of the u. Now we can insert (1) into Yj. Making use of {)lzI/{)z ~ Yj = zllzl this yields for 'OJ: () -() () tanh(w (I Uj I-1 ) ) = -2WYj-1 Uj-I = -() Yj Uj Uj (4) When (4) is inserted into (3) we get <lWex (I - 2 (Wjtanh(WI~~~jl-l?"j) j "H) W (5) where (.)j denotes a vector with elements of varying j. Here, we have replaced the transpose operator by the hermitian operator H, since we will be processing complex data. We have also added a subscript to W as these parameters can be learned, too. We will not show the adaptation equations due to lack of space. 4 Connection to the maximum likelihood approach Pearlmutter and Parra have shown that (3) can be derived from the maximum likelihood approach to density estimation [6]. The same fact has also been pointed out by others, for example, by Cardoso [3]. We will not repeat their straightforward derivation, but the final adaptation equation is of the following form: AWex - dO WTW = ((fj(Uj; Wj)) u T + dW Ii (Uj; Wj) j I) W. (6) where U = Wx are the sources separated from mixtures x, and fj(uj;wj) is the pdf of source j parametrized by Wj. This is exactly the form of Bell and Sejnowski when Ii is taken to be the derivative of the necessary nonlinearity gj, which was assumed to be "close" to the true cdf of the source. Thus the information maximization approach makes implicit assumptions about the cdf's of the sources in the form of the nonlinear squashing function, and does implicit density estimation, whereas in the ML approach the density assumptions are made explicit. This fact makes it more intuitive and lucid to derive the adaptation for other forms of densities, and also to extend it to complex valued variables. Now, we can use the circular pdf's (2) depicted in Fig. 1 as the densities Ii (omitting 1- tanh 2 (wj(lujl- 1)). where the steepness Wj acts as the scaling) fj(uj;wj) single parameter of the density. Now we need to compute its derivative = fj(uj;wj) = (){) Ii(uj;wj) = -2tanh(wj(IUjl-l))Ii(Uj;Wj)WjIUjl (7) Uj Uj Inserting this into (6) and changing transpose operators into hermitians yields <l Wex (I _2 (Wjtanh(WI~~~jl- 1?,,; ) ; "H) W, (8) which is exactly the information maximization rule (5). Notice that at this time we did not have to ponder what would be an appropriate way to construct the cdf from the pdf for complex valued distributions. 759 Blind Separation of Radio Signals in Fading Channels 5 Modifications for QAM and other signal constellations So far we have only looked at signals that lie on the unit circle, or that have a constant modulus. Now we will take a look at other modulation techniques, in which the alphabet is constructed as discrete points on the complex plane. An example is the QAM (quadrature amplitude modulation), in which the signal alphabet is a regular grid. For example, in 4-QAM, the alphabet could be A4 = {I+i, -I+i, -I-i, l-i}, or any scaled version of A4. In the ideal pdf of 4-QAM, each symbol is represented just as a point. Again, we can construct a smoothed version of the ideal pdf as the sum of "bumps" over all of the alphabet where the ideal pdf will be approached by increasing w. g(U) = 2:)1 - tanh2(wkl u - Uk!)) (9) k Now the density for each source j will be !i(Uj; Wj) = 2:)1 - tanh 2 (wkl u j - (10) Uk!)) k where Wj is now a vector of parameters Wk. In practice each which case a single parameter W will suffice. Wk would be equal in This density function could now be inserted into (6) resulting in the weight update equation. However, since !i(Uj; Wj) is a sum of multiple components, f' / f will not have a particularly simple form. In essence, for each sample to be processed, we would need to evaluate all the components of the pdf model of the constellation. This can be avoided by evaluating only the component of the pdf corresponding to that symbol of the alphabet U c which is nearest to the current separated sample u. This is a very good approximation when W is large. But the approximation does not even have to be a good one when W is small, since the whole purpose of using "wide" pdf components is to be able to evaluate the gradients on the whole complex plane. Figure 2 depicts examples of this approximation with two different values of w. The discontinuities are visible at the real and imaginary axes for the smaller w. a) w = 1.0 b) w = 5.0 Figure 2: A piecewise continuous PDF for a 4-QAM source using the tanh function. Thus for the 4-QAM, the complex plane will be divided into 4 quadrants, each having its own adaptation rule corresponding to the single pdf component in that quadrant. Evaluating (6) for each component of the sum gives Ll.. W'" (I _2 (w. tanh(WI~~j - '"I) Uj \ "H ) W, (11) for each symbol k of the alphabet or for the corresponding location Uk on the complex plane. This equation can be applied as such when the baseband signal is sampled at the symbol rate. With oversampling, it may be necessary to include in the pdf model the transition paths between the symbols, too. 6 Practical simplifications To be able to better vectorize the algorithm, it is practical to accumulate ~ W from a number of samples before updating the W. This amounts to computing an expectation of ~W over a number, say, 10-500 samples of the mixtures. Looking at the DPSK case, (5) or (8) the expectation of IUil in the denominator equals one "near" convergence since we assume baseband signals that are distributed on the unit circle. Also, near the solution we can assume that the separated outputs Uj are close to true distributions, the exact unit circle, which can be derived from h by increasing its steepness. At the limit the tanh will equal the sign function, when the whole adaptation, ignoring scaling, is (12) However, this simplification can only be used when the W is not too far off from the correct solution. This is especially true when the number of available samples of the mixtures is small. The smooth tanh is needed in the beginning of the adaptation to give the correct direction to the gradient in the algorithm since the pdfs of the outputs Uj are far from the ideal ones in the beginning. 7 Performance with static and fading signals We have tested the performance of the proposed algorithm both with static and dynamic (changing) mixing conditions. In the static case with four DPSK signals (8 x oversampled) mixed with random matrices the algorithm needs only about 80 sample points (corresponding to 10 symbols) of the mixtures to converge to a separating solution, whereas a more general algorithm, such as [4], needs about 800-1200 samples for convergence. We attribute this improvement to making much better use of the baseband signal distributions. In mobile communications the signals are subject to fading. If there is no direct line of sight from the transmitter to the receiver, only multiple reflected and diffracted signal components reach the receiver. When either the receiver or the transmitter is moving, for example, in an urban environment, these components are changing very rapidly. If the phases of the carrier signals in these components are aligned the components add constructively at the receiver. If the phases of carriers are 180 degrees off the components add destructively. Note that a half of a wavelength difference in the lengths of the paths of the received components corresponds to a 180 degree phase shift. This is only about 0.17 m at 900 MHz. Since this small a spatial difference can cause the signal to change from constructive interference to a null received signal, the result is that both the amplitude and the phase of the received signal vary seemingly randomly at a rate that is proportional to relative speeds of the transmitter and the receiver. The amplitude of the received signal follows a Rayleigh distribution, hence the name Rayleigh fading. As an example, Figure 3 depicts a 0.1 second fragment of the amplitude of a fading channel. 10 r o _lo F \\. '-'\ / I V -20 //--\" \1-\/\(\/'\\//-~-'V i Ii -w o 'V' \f \ , om 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.()9 0.1 Figure 3: Amplitude (in dB) of a fading radio channel corresponding to a vehicle speed of 60 mph, when the carrier is 900 Mhz. Horizontal axis is time in seconds. 761 Blind Separation of Radio Signals in Fading Channels With fading sources, the problem is to be able to adapt to changing conditions, keeping up with the fading rate. In the signal of Fig. 3 it takes less than 5 milliseconds to move from a peak of the amplitude into a deep fade. Assuming a symbol rate of 20000 symbols/second, this corresponds to a mere 100 symbols during this change. We simulated again DPSK sources oversampling by 8 relative to the symbol rate. The received sampled mixtures are (13) j where 8j[n] are the source signals, fij[n] represents the fading channel from transmitter j to receiver i, and ni[n] represents the noise observed by receiver i. In our experiments, we used a sliding window of 80 samples centered at the current sample. The weight matrix update (the gradient) was calculated using all the samples of the window, the weight matrix was updated, the window was slid one sample forward, and the same was repeated. Using this technique we were able to keep up with the fading rate corresponding to 60 mph relative speed of the transmitter and the receiver. Figure 4 depicts how the algorithm tracks the fading channels in the case of three simultaneous source signals. 4 l~ .. '/,\" .~. ~ .... ' . .)~' .... 1 :~ o 2000 4000 6000 8000 10000 12000 14000 16000 +'------'--~____'__'__~----'-----': _:E+Ff!'-------'----.L..fc .?.. ?-'-------"'-'.t o 2000 4000 6000 8000 10000 12000 14000 16000 5r-----~----~----~--~r-----~----~----~--__, 10 O~ ____ ~-L _ _L -_ _ _ _L __ _ ~L_ ____ ~ ____ ~~ __ ~ __ ~ o 2000 4000 6000 8000 10000 12000 14000 16000 Figure 4: Separation of three signals subject to fading channels. Top graph: The real parts 16 independent fading channels. 2nd graph: The inverse of the instantaneous fading conditions (only the real part is depicted) . This is one example of an ideal separation solution. 3rd graph: The separation solution tracked by the algorithm. (only the real part is depicted). Bottom graph: The resulting signal/interference (S/I) ratio in dB for each of the four separated source signals. Horizontal axis is samples. 16000 samples (8 x oversampled) corresponds to 0.1 seconds. On the average, the S/I to start with is zero. The average output S/I is 20 dB for the worst of the three separated signals. Since the mixing is now dynamic the instantaneous mixing matrix, as determined by the instantaneous fades, can occasionally be singular and cannot be inverted. Thus the signals at this instance cannot be separated. In our 0.1 second test signal this occurred four times in the three source signal case (9 independent fading paths), at which instances the output 762 K. Torkkola SII bounced to or near zero momentarily for one or more of the separated signals. Durations of these instances are short, lasting about 15 symbols, and covering about 3 per cent of the total signal time. 8 Related work and discussion Although the whole field of blind source separation has started around 1985, rather surprisingly, no application to radio communications has yet emerged. Most of the source separation algorithms are based on higher-order statistics, and these should be relatively straightforward to generalize for complex valued baseband data. Perhaps the main reason is that all theoretical work has concentrated in the case of static mixing, not in the dynamic case. Many communications channels are dynamic in nature, and thus rapidly adapting methods are necessary. Making use of all available knowledge of the sources, in this case the pdf's of the source signals, allows successful adaptation based on a very small number of samples, much smaller than by just incorporating the coarse shapes of the pdf's into the algorithm. It is not unreasonable to presume this knowledge, on the contrary, the modulation method of a communications system must certainly be known. To our knowledge, no successful blind separation of signals subject to rapidly varying mixing conditions, such as fading, has been reported in the literature. Different techniques applied to separation of various simulated radio signals under static mixing conditions have been described, for example, in [9, 4]. The maximum likelihood method reported recently by Yellin and Friedlander [9] seems to be the closest to our approach, but they only apply it to simulated baseband radio signals with static mixing conditions. It must also be noted that channel time dispersion is not taken into account in our current simulations. This is valid only in cases where the delay spread is short compared to the inverse of the signal bandwidths. If this is not a valid assumption, separation techniques for convolutive mixtures, such as in [7] or [1], need to be combined with the methods developed in this paper. References [1] S. Amari, S. Douglas, A. Cichocki, and H. H. Yang. Multichannel blind deconvolution and equalization using the natural gradient. In Proc. 1st IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications, pages 101104, Paris, France, April 16-18 1997. [2] A. Bell and T. Sejnowski. An information-maximisation approach to blind separation and blind deconvolution. Neural Computation, 7(6):1129-1159, 1995. [3] J.-F. Cardoso. Infomax and maximum likelihood for source separation. IEEE Letters on Signal Processing, 4(4):112-114, April 1997. [4] J.-F. Cardoso and B. Laheld. Equivariant adaptive source separation. IEEE 1'ransactions on Signal Processing, 44(12):3017-3030, December 1996. [5] A. Paulraj and C. B. Papadias. Array processing in mobile communications. In Handbook of Signal Processing. CRC Press, 1997. [6] B. A. Pearlmutter and L. C. Parra. A context-sensitive generalization of ICA. In International Conference on Neural Information Processing, Hong Kong, Sept. 24-27 1996. Springer. [7] K. Torkkola. Blind separation of convolved sources based on information maximization. In IEEE Workshop on Neural Networks for Signal Processing, pages 423-432, Kyoto, Japan, September 4-6 1996. [8] A.-J. van der Veen and A. Paulraj. An analytical constant modulus algorithm. IEEE 1hmsactions on Signal Processing, 44(5), May 1996. [9] D. Yellin and B. Friedlander. A maximum likelihood approach to blind separation of narrowband digital communication signals. In Proc. 30th Asilomar Conf. on Signals, Systems, and Computers, 1996. 

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An Analog VLSI Model of the Fly Elementary Motion Detector Reid R. Harrison and Christof Koch Computation and Neural Systems Program, 139-74 California Institute of Technology Pasadena, CA 91125 [harrison,koch]@klab.caltech.edu Abstract Flies are capable of rapidly detecting and integrating visual motion information in behaviorly-relevant ways. The first stage of visual motion processing in flies is a retinotopic array of functional units known as elementary motion detectors (EMDs). Several decades ago, Reichardt and colleagues developed a correlation-based model of motion detection that described the behavior of these neural circuits. We have implemented a variant of this model in a 2.0-JLm analog CMOS VLSI process. The result is a low-power, continuous-time analog circuit with integrated photoreceptors that responds to motion in real time. The responses of the circuit to drifting sinusoidal gratings qualitatively resemble the temporal frequency response, spatial frequency response, and direction selectivity of motion-sensitive neurons observed in insects. In addition to its possible engineering applications, the circuit could potentially be used as a building block for constructing hardware models of higher-level insect motion integration. 1 INTRODUCTION Flies rely heavily on visual motion information to survive. In the fly, motion information is known to underlie many important behaviors including stabilization during flight, orienting towards small, rapidly-moving objects (Egelhaaf and Borst 1993), and estimating time-tocontact for safe landings (Borst and Bahde 1988). Some motion-related tasks like extending the legs for landing can be excecuted less than 70 milliseconds after stimulus presentation. The computational machinery performing this sensory processing is fast, small, low-power, and robust. There is good evidence that motion information is first extracted by local elementary motion detectors (see Egelhaaf et al. 1988 and references therein). These EMDs are ar- An Analog VLSI Model of the Fly Elementary Motion Detector a) 881 EMD Architecture photoreceptors high-pass filters 250 , 300 350 150 200 time[ms) 250 300 350 , low-pass filters , 150 200 time[msJ ! 40 <' .?30 multipliers E-20 8 subtraction ~ 10 __ Ul EMD output o 50 100 Figure 1: Elementary Motion Detector. a) A simplified version of our EMD circuit architecture. In the actual circuit implementation, there are separate ON and OFF channels that operate in parallel. These two channels are summed after the mulipIication. b) The measured response of the EMD test circuit to a drifting sinusoidal grating. Notice that the output is phase dependent, but has a positive mean response. If the grating was drifting in the opposite direction, the circuit would give a negative mean response. ranged retinotopically, and receive input from adjacent photoreceptors. The properties of these motion-sensitive units have been studied extensively during the past 30 years. Direct recording from individual EMDs is difficult due to the small size of the cells, but much work has been done recording from large tangential cells that integrate the outputs of many EMDs over large portions of the visual field. From these studies, the behavior of individual EMDs has been inferred. If we wish to study models of motion integration in the fly, we first need a model of the EMD. Since many motion integration neurons in the fly are only a few synapses away from muscles, it may be possible in the near future to contruct models that complete the sensorimotor loop. If we wish to include the real world in the loop, we need a mobile system that works in real time. In the pursuit of such a system, we follow the neuromorphic engineering approach pioneered by Mead (Mead 1989) and implement a hardware model of the fly EMD in a commercially available 2.0-J-tm CMOS VLSI process. All data presented in this paper are from one such chip. 2 ALGORITHM AND ARCHITECTURE Figure la shows a simplified version of the motion detector. This is an elaborated version of the correlation-based motion detector first proposed by Reichardt and colleagues (see Reichardt 1987 and references therein). The Reichardt motion detector works by correlating (by means of a multiplication) the response of one photoreceptor to the delayed response of an adjacent photoreceptor. Our model uses the phase lag inherent in a low-pass filter to supply the delay. The outputs from two mirror-symmetric correlators are subtracted to remove any response to full-field flicker (ws = 0, Wt > 0). Correlation-based EMDs are not pure velocity sensors. Their response is strongly affected by the contrast and the spatial frequency components of the stimulating pattern. They can best be described as direction-selective spatiotemporal filters. The mean steady-state response R of the motion detector shown in Figure I a to a sinusoidal grating drifting in one direction can be expressed as a separable function of stimulus amplitude (D.I), temporal R. R. Harrison and C. Koch 882 c) b) a) dV I .. component 0fCTt poSl\lve t v + negative component of C ~; + dIout t dV . cTt 1= it t + lout = lin = CUr It Figure 2: EMD Subcircuits. a) Temporal derivative circuit. In combination with the firstorder low-pass filter inherent in the photoreceptor, this forms the high-pass filter with time constant TH. The feedback amplifier enforces V = Yin, and the output is the current needed for the nFET or pFET source follower to charge or discharge the capacitor C. b) Current-mode low-pass filter. The time constant TL is determined by the bias current IT (which is set by a bias voltage supplied from off-chip), the capacitance C, and the thermal voltage UT = kT /q. c) Current-mode one-quadrant multiplier. The devices shown are floating-gate nFETs. Two control gates capacitively couple to the floating node, forming a capacitive divider. frequency (Wt = 21r It), and spatial frequency (ws = 21r is): (1) (2) where D.cp is the angular separation of the photoreceptors, TH is the time constant of the high-pass filter, and TL is the time constant of the low-pass filter (see Figure 1a). (Note that this holds only for motion in a particular direction. Motion detectors are not linearly separable overall, but the single-direction analysis is useful for making comparisons.) 3 CIRCUIT DESCRIPTION In addition to the basic Reichardt model described above, we include a high-pass filter in series with the photoreceptor. This amplifies transient responses and removes the DC component of the photoreceptor signal. We primarily use the high-pass filter as a convenient circuit to switch from a voltage-mode to a current-mode representation (see Figure 2a). For the photoreceptor, we use an adaptive circuit developed by Delbruck (Delbruck and Mead 1996) that produces an output voltage proportional to log intensity. We bias the photoreceptor very weakly to attenuate high temporal frequencies. This is directly followed by a temporal derivative circuit (Mead 1989) (see Figure 2a), the result being a high-pass filter with the dominant pole TH being set by the photoreceptor cutoff frequency. The outputs of the temporal derivati?,re circuit are two unidirectional currents that represent the positive and negative components of a high-pass filtered version of the photoreceptor output. This resembles the ON and OFF channels found in many biological visual systems. Some studies suggest ON and OFF channels are present in the fly (Franceschini et al. 1989) but the evidence is mixed (Egelhaaf and Borst 1992). This two-channel representation is useful for current-mode circuits, since the following translinear circuits work only with unidirectional An Analog VLSI Model of the Fly Elementary Motion Detector 883 0.8 0.2 0.1 1 10 Temporal Frequency J, [Hz] 100 Figure 3: Temporal Frequency Response. Circuit data was taken with is = 0.05 cyclesldeg and 86% contrast. Theory trace is Rt(Wt) from Equation 2, where TH = 360 ms and TL = 25 ms were directly measured in separate experiments - these terms were not fit to the data. Insect data was taken from a wide-field motion neuron in the blowfly Calliphora erythrocephala (O ' Carroll et al. 1996). All three curves were normalized by their peak response. currents. It should be noted that the use of ON and OFF channels introduces nonlinearities into the circuit that are not accounted for in the simple model described by Equation 2. The current-mode low-pass filter is shown in Figure 2b. The time constant TL is set by the bias current 11'. This is a log-domain filter that takes advantage of the exponential behavior of field-effect transistors (FETs) in the subthreshold region of operation (Minch, personal communication). The current-mode multiplier is shown in Figure 2c. This circuit is also translinear, using a diode-connected FET to convert the input currents into log-encoded voltages. A weighted sum of the voltages is computed with a capacitive divider, and the resulting voltage is exponentiated by the output FET into the output current. The capacitive divider creates a floating node, and the charge on all these nodes must be equalized to ensure matching across independent multipliers. This is easily accomplished by exposing the chip to UV light for several minutes. This circuit represents one of a family of floating-gate MaS translinear circuits developed by Minch that are capable of computing arbitrary power laws in current mode (Minch et al. 1996). After the multiplication stage, the currents from the ON and OFF channels are summed, and the final subtraction of the left and right channels is done off-chip. There is a gain mismatch of approximately 2.5 between the left and right channels that is now compensated for manually. This mismatch must be lowered before large on-chip arrays of EMDs are practical. A new circuit designed to lessen this gain mismatch is currently being tested. It is interesting to note that there is no significant offset error in the output currents from each channel. This is a consequence of using translinear circuits which typically have gain errors due to transistor mismatch, but no fixed offset errors. 4 EXPERIMENTS As we showed in Equation 2, the motion detector's response to a drifting sinusoidal grating of a particular direction should be a separable function of Ill, temporal frequency, and R. R. Harrison and C. Koch 884 ~con 08 ..... .ID.!'ory __ ._i~ect ,, ,, ~ 0.6 , ...,, 2- !1 ~ c e 0,4 \ \ ::.: .~ ...i}. \ ~ 0.2 ... ?..... \ /'c'????? ~"'~/ ~, 2 0,001 , \ 0,01 0,1 Spatial FrequencyIs [cycleoJdeg] Figure 4: Spatial Frequency Response. Circuit data was taken with it = 4 Hz and 86% contrast. Theory trace is Rs(w s ) from Equation 2 multiplied by exp( -w s 2/ K2) to account for blurring in the optics, The photoreceptor spacing ~rp = 1.9 0 was directly measured in an separate experiment. Only K and the overall magnitude were varied to fit the data. Insect data was taken from a wide-field motion neuron in the hoverfly Volucella pelluscens (O'Carroll et al. 1996). Circuit and insect data were nonnalized by their peak response. spatial frequency, We tested the circuit along these axes using printed sinusoidal gratings mounted on a rotating drum. A lens with an 8-mm focal length was mounted over the chip. Each stimulus pattern had a fixed contrast ~I /21 and spatial frequency fs. The temporal frequency was set by the pattern's angular velocity vas seen by the chip, where it = fsv. The response of the circuit to a drifting sine wave grating is phase dependent (see Figure I b). In flies, this phase dependency is removed by integrating over large numbers of EMDs (spatial integration). In order to evaluate the perfonnance of our circuit, we measured the mean response over time. Figure 3 shows the temporal frequency response ofthe circuit as compared to theory, and to a wide-field motion neuron in the fly. The circuit exhibits temporal frequency tuning. The point of peak response is largely detennined by n, and can be changed by altering the lowpass filter bias current. The deviation of the circuit behavior from theory at low frequencies is thought to be a consequence of crossover distortion in the temporal derivative circuit. At high temporal frequencies, parasitic capacitances in current mirrors are a likely candidate for the discrepancy. The temporal frequency response of the blowfly Calliphora is broader than both the theory and circuit curves. This might be a result of time-constant adaptation found in blowfly motion-sensitive neurons (de Ruyter van Steveninck et at. 1986). Figure 4 shows the spatial frequency response of the circuit. The response goes toward zero as Ws approaches zero, indicating that the circuit greatly attenuates full-field flicker. The circuit begins aliasing at Ws = 1/2~rp, giving a response in the wrong direction. Spatial aliasing has also been observed in flies (Gotz 1965). The optics used in the experiment act as an antialiasing filter, so aliasing could be avoided by defocusing the lens slightly. Figure 5 shows the directional tuning of the circuit. It can be shown that as long as the spatial wavelength is large compared to ~rp, the directional sensitivity of a correlationbased motion detector should approximate a cosine function (Zanker 1990). The circuit's perfonnance matches this quite well. Motion sensitive neurons in the fly show cosine-like direction selectivity. Figure 6 shows the contrast response of the circuit. Insect EMDs show a saturating contrast An Analog VLSI Model of the Fly Elementary Motion Detector ~con 0.8 ~ .... ..t!J~ory __ .J!!~ect 0.6 \ ,\ ''. ~\, '. 9. 0.4 . ? ,, ''. ' ~ 02 , '. co:: ~ i a ~ri"')I.' -0.2 ~-o.4 885 lO-. .t" _ ... -0.6 " -0.8 -1 -150 -100 -50 a 50 100 150 Direction of MolIOn [deg) Figure 5: Directional Response. Circuit data was taken with it = 6 Hz, is = 0.05 cycles/deg and 86% contrast. Theory trace is cos 0:, where 0: is the direction of motion relative to the axis along the two photoreceptors. Insect data was taken from the HI neuron in the blowfly Calliphora erythrocephala (van Hateren 1990). HI is a spiking neuron with a low spontaneous firing rate. The flattened negative responses visible in the graph are a result of the cell's limited dynamic range in this region. All three curves were normalized by their peak response. response curve, which can be accounted for by introducing saturating nonlinearities before the multiplication stage (Egelhaaf and Borst 1989). We did not attempt to model contrast saturation in our circuit, though it could be added in future versions. 5 CONCLUSIONS We implemented and tested an analog VLSI model of the fly elementary motion detector. The circuit's spatiotemporal frequency response and directional selectivity is qualitatively similar to the responses of motion-sensitive neurons in the fly. This circuit could be a useful building block for constructing analog VLSI models of motion integration in flies. As an integrated, low-power, real-time sensory processor, the circuit may also have engineering applications. Acknowledgements This work was supported by the Center for Neuromorphic Systems Engineering as a part of NSF's Engineering Research Center program, and by ONR. Reid Harrison is supported by an NDSEG fellowship from ONR. We thank Bradley Minch, Ho1ger Krapp, and Rainer Deutschmann for invaluable discussions. References A. Borst and S. Bahde (1988) Visual information processing in the fly's landing system. 1. Compo Physiol. A 163: 167-173. T. Delbruck and C. Mead (1996) Analog VLSI phototransduction by continuous-time, adaptive, logarithmic photoreceptor circuits. CNS Memo No . 30, Caltech. M. Egelhaaf, K. Hausen, W. Reichardt, and C. Wehrhahn (1988) Visual course control in 886 R. R. Harrison and C. Koch / ,'JI. ~con .... -- ....?.t!'!'ory 08 ,' __.j~~ct "i'! 8- ;l ~06 >" I ? I I <.> ::; .~O.4 I ,- /' I U I '" 02 .,/ I I I I ..... i ........ I -....- 10 20 30 40 50 60 70 80 90 100 Contrast &112[[%] Figure 6: Contrast Response. Circuit data was taken with It = 6 Hz and is = 0.1 cycles/deg. Theory trace is Rj(D.I) from Equation 2 with its magnitude scaled to fit the circuit data. Insect data was taken from the HS neuron in the blowfly Calliphora erythrocephala (Egelhaaf and Borst 1989). Circuit and insect data were normalized by their peak response. flies relies on neuronal computation of object and background motion. TINS 11: 351-3 5 8. M. Egelbaaf and A. Borst (1989) Transient and steady-state response properties of movement detectors. J. Opt. Soc. Am. A 6: 116-127. M. Egelbaaf and A. Borst (1992) Are there separate ON and OFF channels in fly motion vision? Visual Neuroscience 8: 151-164. M. Egelbaaf and A. Borst (1993) A look into the cockpit of the fly: Visual orientation, algorithms, and identified neurons. J. Neurosci. 13: 4563-4574. N. Franceschini, A. Riehle, and A. Ie Nestour (1989) Directionally selective motion detection by insect neurons. In StavengaIHardie (eds.), Facets o/Vision, Berlin: Springer-Verlag. K.G. Gotz (1965) Die optischen Ubertragungseigenschaften der Komplexaugen von Drosophila. Kybemetik2: 215-221. J.H. van Hateren (1990) Directional tuning curves, elementary movement detectors, and the estimation of the direction of visual movement. Vision Res. 30: 603-614. C. Mead (1989) Analog VLSI and Neural Systems. Reading, Mass.: Addison-Wesley. B.A. Minch, C. Diorio, P. Hasler, and C. Mead (1996) Translinear circuits using subthreshold floating-gate MaS transistors. Analog Int. Circuits and Signal Processing 9: 167-179. D.C. O'Carroll, N.J. Bidwell, S.B. Laughlin, and EJ. Warrant (1996) Insect motion detectors matched to visual ecology. Nature 382: 63-66. W. Reichardt (1987) Evaluation of optical motion information by movement detectors. J. CompoPhys. A 161: 533-547. R.R. de Ruyter van Steveninck, W.H. Zaagman, and H.A.K. Mastebroek (1986) Adaptation of transient responses of a movement-sensitive neuron in the visual system of the blowfly Calliphora erythrocephala. Bioi. Cybern. 54: 223-236. J.M. Zanker (1990) On the directional sensitivity of motion detectors. Bioi. Cybern. 62: 177-183. 

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Computing with Action Potentials John J. Hopfield* Carlos D. Brody t Sam Roweis t Abstract Most computational engineering based loosely on biology uses continuous variables to represent neural activity. Yet most neurons communicate with action potentials. The engineering view is equivalent to using a rate-code for representing information and for computing. An increasing number of examples are being discovered in which biology may not be using rate codes. Information can be represented using the timing of action potentials, and efficiently computed with in this representation. The "analog match" problem of odour identification is a simple problem which can be efficiently solved using action potential timing and an underlying rhythm. By using adapting units to effect a fundamental change of representation of a problem, we map the recognition of words (having uniform time-warp) in connected speech into the same analog match problem. We describe the architecture and preliminary results of such a recognition system. Using the fast events of biology in conjunction with an underlying rhythm is one way to overcome the limits of an eventdriven view of computation. When the intrinsic hardware is much faster than the time scale of change of inputs, this approach can greatly increase the effective computation per unit time on a given quantity of hardware. 1 Spike timing Most neurons communicate using action potentials - stereotyped pulses of activity that are propagated along axons without change of shape over long distances by active regenerative processes. They provide a pulse-coded way of sending information. Individual action potentials last about 2 ros. Typical active nerve cells generate 5-100 action potentials/sec. Most biologically inspired engineering of neural networks represent the activity of a nerve cell by a continuous variable which can be interpreted as the short-time average rate of generating action potentials. Most traditional discussions by neurobiologists concerning how information is represented and processed in the brain have similarly relied on using "short term mean firing rate" as the carrier of information and the basis for computation. But this is often an ineffective way to compute and represent information in neurobiology. t *Dept. of Molecular Biology, Princeton University. jhopfield@watson.princeton. edu Computation & Neural Systems, California Institute of Technology. Computing with Action Potentials 167 To define "short term mean firing rate" with reasonable accuracy, it is necessary to either wait for several action potentials to arrive from a single neuron, or to average over many roughly equivalent cells. One of these necessitates slow processing; the other requires redundant "wetware". Since action potentials are short events with sharp rise times, action potential timing is another way that information can be represented and computed with ([Hopfield, 1995]). Action potential timing seems to be the basis for some neural computations, such as the determination of a sharp response time to an ultrasonic pulse generated by the moustache bat. In this system, the bat generates a 10 ms pulse during which the frequency changes monotonically with time (a "chirp"). In the cochlea and cochlear nucleus, cells which are responsive to different frequencies will be sequentially driven, each producing zero or one action potentials during the time when the frequency is in their responsive band. These action potentials converge onto a target cell. However, while the times of initiation of the action potentials from the different frequency bands are different, the length and propagation speed of the various axons have been coordinated to result in all the action potentials arriving at the target cell at the same time, thus recognizing the "chirped" pulse as a whole, while discriminating against random sounds of the same overall duration. Taking this hint from biology, we next investigate the use of action potential timing to represent information and compute with in one of the fundamental computational problems relevant to olfaction, noting why the elementary "neural net" engineering solution is poor, and showing why computing with action potentials lacks the deficiencies of the conventional elementary solution. 2 Analog match The simplest computational problem of odors is merely to identify a known odor when a single odor dominates the olfactory scene. Most natural odors consist of mixtures of several molecular species. At some particular strength a complex odor b can be described by the concentrations Nt of its constitutive molecular of species i. If the stimulus intensity changes, each component increases (or decreases) by the same multiplicative factor. It is convenient to describe the stimulus as a product of two factors, an intensity .A and normalized components n~ as: .A = ~jNJ => b _ Nbj \ i A ni - or Nbi -Ani _ \ b (1) The n~ are normalized, or relative concentrations of different molecules, and .A describes the overall odor intensity. Ideally, a given odor quality is described by the pattern of n~ , which does not change when the odor intensity .A changes. When a stimulus s described by a set {NJ} is presented, an ideal odor quality detector answers "yes" to the question "is odor b present?" if and only if for some value of .A: N~~.An~ J 3 't/J' (2) This general computation has been called analog match. 1 The elementary "new?at net" way to solve analog match and recognize a single odor independent of intensity would bt: to use a single "grandmother unit" of the following type. I The analog match problem of olfaction is actually viewed through olfactory receptor cells. Studies of vertebrate sensory cells have shown that each molecular species stimulates many different sensory cells, and each cell is excited by many different molecular species. The pattern of relative excitation across the population of sensory cell classes determines the odor quality in the generalist olfactory system. There are about 1000 broadly responsive cell types; thus, the olfactory systems of higher animals apparently solve an analog match problem of the type described by (2), except that the indices refer to cell types, and the actual dimension is no more than 1000. 168 1. J. Hopfield, C. D. Brody and S. Roweis Call the unknown odor vector I, and the weight vector W. The input to the unit will then be I . W. If W = n / lin II and I is pre-normalized by dividing by the Euclidean magnitude 11111, recognition can be identified by I . W > .95 , or whatever threshold describes the degree of precision in identification which the task requires. This solution has four major weaknesses. 1. Euclidean normalization is used; not a trivial calculation for real neural hardware. 2. The size of input components Ik and their importance is confounded. If a weak component has particular importance, or a strong one is not reliable, there is no way to represent this. W describes only the size of the target odor components. 3. There is no natural composition if the problem is to be broken into a hierarchy by breaking the inputs into several parts, solving independently, and feeding these results on to a higher level unit for a final recognition. This is best seen by analogy to vision. If I recognize in a picture grandmother's nose at one scale, her mouth at another, and her right eye at a third scale, then it is assuredly not grandmother. Separate normalization is a disaster for creating hierarchies. 4. A substantial number of inputs may be missing or giving grossly wrong information. The "dot-product-and-threshold" solution cannot contend with this problem. For example, in olfaction, two of the common sources of noise are the adaptation of a subset of sensors due to previous strong odors, and receptors stuck "on" due to the retention of strongly bound molecules from previous odors. All four problems are removed when the information is encoded and computed with in an action potential representation, as illustrated below.. The three channels of analog input Ia'/b, Ie are illustrated on the left. They are converted to a spike timing representation by the position of action potentials with respect to a fiducial time T. The interval between T and the time of an action potential in a channel j is equal to log Ij. Each channel is connected to an output unit through a delay line of length t,.j = logn~, where n b is the target vector to be identified. When the analog match criterion is satisAed, the pulses on all three channels will arrive at the target unit at the same time, driving it strongly. If all inputs are scaled by a, then the times of the action potentials will all be changed by log a. The three action potentials will arrive at the recognition unit simultaneously, but a a time shifted by loga. Thus a pattern can be recognized (or not) on the basis of its relative components. Scale information is retained in the time at which the recognition unit is driven. The system clearly "composes", and difficulty (3) is sunnounted. No normalization is required, eliminating difficulty (1). Each pathway has two parameters describing it, a delay (which contains the information about the pattern to be recognized) and a synaptic strength (which describes the weight of the action potential at the recognition unit). Scale and importance are separately represented. The central computational motif is very similar to that used in bat sonar, using relative timing to represent information and time delays to represent target patterns. 1\ ~--I a t ilOg b ~ log C ?? ) log Imin Imax 1 1 o :j m~a ~ ~ ---t>@recognitionunitsums EPSPs then thresholds t - -.....-+~ ._ T'; Delays set prototype pattern 0 T ~ Weights set relath'c feature importance Computing with Action Potentials 169 This system also tolerates errors due to missing or grossly inaccurate information. The figure below illustrates this fact for the case of three inputs, and contrasts the receptive fields of a system computing with action potentials with those of a conventional grandmother cell. (The only relevant variables are the projections of the input vector on the surface of the unit sphere, as illustrated.) When the thresholds are set high, both schemes recognize a small, roughly circular region around the target pattern (here chosen as 111). Lowering the recognition threshold in the action-potential based scheme results in a star-shaped region being recognized; this region can be characterized as "recognize if any two components are in the correct ratio, independent of the size of the third component." Pattern 110 is thus recognized as being similar to 111 while still rejecting most of the space as not resembling the target. In contrast, to recognize 110 with the conventional unit requires such threshold lowering that almost any vector would be recognized. Spike Timing thresh =0.7 0.6 Normalize, Dot Product 0.4 thresh =0.99 0.95 0.90 This method of representation and computation using action potential timing requires a fiducial time available to all neurons participating in stimulus encoding. Fiducial times might be externally generated by salient events, as they are in the case of moustache bat sonar. Or they could be internally generated, sporadically or periodically. In the case of the olfactory system, the first processing area of all animals has an oscillatory behavior. A large piece of the biophysics of neurons can be represented by the idea that neurons are leaky integrators, and that when their internal potential is pushed above a critical value, they produce an action potential, and their internal potential is reset a fixed distance below threshold. When a sub-threshold input having a frequency j is combined with a steady analog current I, the system generates action potentials at frequency j, but whose phase with respect to the underlying oscillation is a monotone function of I. Thus the system encodes I into a phase (or time) of an action potential with respect to the underlying rhythm. Interestingly, in mammals, the second stage of the olfactory system, the prepiriform cortex, has slow axons propagating signals across it. The propagation time delays are comparable to 1/ j. The system has the capability of encoding and analyzing information in action potential timing. 3 Time warp and speech Recognizing syllables or words independent of a uniform stretch (''uniform time warp") can in principle be cast as an analog match problem and transformed into neural variables [Hopfield, 1996]. We next describe this approach in relationship to a previous "neural network" way of recognizing words in connected speech [Hopfield and Tank, 1987, Unnikrishnan et aI., 1991, Unnikrishnan et aI., 1992] (URT for short). A block diagram below shows the UHT neural network for recognizing a small vocabulary of words in connected speech. The speech signal is passed through a bank of band-pass filters, and an elementary neural feature detector then examines whether each frequency is a local maximum of the short-term power spectrum. If so, it propagates a "1" down a delay line from that feature detector, thus converting the pattern of features in time into a pattern in space. The recognition unit for a particular word is then connected to these delay lines by a pattern of weights which are trained on a large data base. 1. 1. Hopfield, C. D. Brody and S. Roweis 170 Time De\ays UHT diagram Time De\ays Time-warp diagram The conceptual strength of this circuit is that it requires no indication of the boundaries between words. Indeed, there is no such concept in the circuit. The conceptual weakness of this "neural network" is that the recognition process for a particular word is equivalent to sliding a rigid template across the feature pattern. Unfortunately, even a single speaker has great variation in the duration of a given word under different circumstances, as illustrated in the two spectrograms below. Clearly no single template will fit these both of these utterances of "one" very well. This general problem is known as time-warp. A time-warp invariant recognizer would have considerable advantage. Two instances of "one" [~~Ci 00 0.3 Smoothed spectrogram Thresholding 00 0.3 Time since feature ~~t~ 00 0.3 00 0.3 The UHT approach represents a sequence by the presence of a signal on feature signal lines A, B, C, as shown on the left of the figure below. Suppose the end of the word occurs at some particular time as indicated. Then the feature starts and stops can be described as an analog vector of times, whose components are shown by the arrows as indicated. In this representation, a word which is spoken more slowly simply has all its vector components multiplied by a common factor. The problem of recognizing words within a uniform time warp is thus isomorphic with the analog match problem, and can be readily solved by using action potential timing and an underlying rhythm, as described above. In our present modeling, the rhythm has a frequency of 50 Hz, significantly faster than the rate at which new features appear in speech. This frequency corresponds to the clock rate at which speech features are effectively "sampled". In the UHT circuit this rate was set by the response timescale of the recognition units. But where each template in the UHT circuit attempted only a single match with the feature vector per sample, this circuit allows the attempted match of many possible time-warps with the feature vector per sample. (The range of time-warps allowed is determined by the oscillation frequency and the temporal resolution of the spike timing system.) A: 000111111000001 B: 00111100000000 C: 00000001111000 A: new., B: rep C: start feature stop feature The block diagram of the neural circuit necessary to recognize words in connected speech with uniform time warp is sketched above. It looks superficially similar to the UHT circuit beside it. except for the insertion of a ramp generator and a phase encoder between the Computing with Action Potentials 171 feature detectors and the delay system. Recognizing a feature activates a ramp generator whose output decays. This becomes the input to a "neuron" which has an additional oscillatory input at frequency f. If the ramp decay and oscillation shapes are properly matched, the logarithm of the time since the occurrence of a feature is encoded in action potential timing as above. Following this encoding system there is a set of tapped delay lines of the same style which would have been necessary to solve the olfactory decoding problem. The total the amount of hardware is similar to the UHT approach because the connections and delay lines dominate the resource requirements. The operation of the present circuit is, however, entirely different. What the present circuit does is to "remember" recent features by using ramp generators, encode the logarithms of times since features into action potential timing, and recognize the pattern with a timedelay circuit. The time delays in the present circuit have an entirely different meaning from those of the UHT circuit, since they are dimensionally not physical time, but instead are a representation of the logarithm of feature times. The time delays are only on the scale of 1/f rather than the duration of a word. There are simple biological implementations of these ideas. For example, when a neuron responds, as many do, to a step in its input by generating a train of action potentials with gradually falling firing frequency (adaptation), the temporal spacing between the action potentials is an implicit representation of the time since the "step" occurred (see [Hopfield, 1996]). For our initial engineering investigations, we used very simple features. The power within each frequency band is merely thresholded. An upward crossing of that threshold represents a "start" feature for that band, and a downward crossing a "end" feature. A pattern of such features is identified above beside the spectrograms. Although the pattern of feature vectors for the two examples of "one" do not match well because of time warp, when the logarithms of the patterns are taken, the difference between the two patterns is chiefly a shift, i.e. the dominant difference between the patterns is merely uniform time warp. To recognize the spoken digit "one", for example, the appropriate delay for each channel was chosen so as to minimize the variance of the post-delay spike times (thus aligning the spikes produced by all features), averaged over the different exemplars which contained that feature. All channels with a feature present were given a unity weight connection at that delay value; inactive channels were given weight zero. The figure below shows, on the left, the spike input to the recognition unit (top) and the sum of the EPSPs caused by these inputs (bottom). The examples of "one" produced maximum outputs in different cycles of the oscillation, corresponding to the actual "end times" at which the words should be viewed as recognized. Only the maximum cycle for each utterance is shown here. Within their maximum cycle, different examples of the utterances produced maximal outputs at different phases of the cycle, corresponding to the fact that the different utterances were recognized as having different time warp factors. The panels on the right show the result of playing spoken "four"s into the same recognition unit. Three "ones"s III Q) Several "four"s .....;... ... .< ,? ~.::~~ :, ~ ? ,,_ ......P": " ?. - ... . .... . . ::J iii _ Q) f~.{ ~----~'~"'~ " --------- .~ > ~ :1:: c::: threshold ::J "5 a. S~---'-"~~------ o time within cycle time within cycle 172 J 1. Hopfield, C. D. Brody and S. Roweis There is no difficulty in distinguishing "ones" from other digits. When, however, the possibility of adjusting the time-warp is turned off, resulting in a "rigid" template it was not possible to discriminate between "one" and other digits. (Disabling time-warp effectively forces recognition to take place at the same "time" during each oscillation. Imagine drawing a vertical line in the figure and notice that it cannot pass through all the peaks of output unit activities.) We have described the beginning of a research project to use action potentials and timing to solve a real speech problem in a "neural" fashion. Very unsophisticated features were used, and no competitive learning was employed in setting the connection weights. Even so, the system appears to function in a word-spotting mode, and displays a facility of matching patterns with time warp. Its intrinsic design makes it insensitive to burst noise and to frequency-band noise. How is computation being done? After features are detected, rates of change are slow, and little additional information is accumulated during say a 50 ms. interval. If we let "time be its own representation", as Carver Mead used to say, we let the information source be the effective clock, and the effective clock rate is only about 20 Hz. Instead, by adding a rhythm, we can interleave many calculations (in this particular case about the possibility of different time warps) while the basic inputs are changing very little. Using an oscillation frequency of 50 Hz and a resolving time of I ms in the speech example we describe increases the effective clock rate by more than a factor of 10 compared to the effective clock rate of the UHT computation. We believe that "time as its own representation" is a loser for processing information when the computation desired is complex but the data is slowly changing. No computer scientist would use a computer with a 24 Hz clock to analyze a movie because the movie is viewed at 24 frames a second. Biology will surely have found its way out of this "paced by the environment" dilemma. Finally, because problems are easy or hard according to how algorithms fit on hardware and according to the representation of information, the differences in operation between the system we have described and conventional ANN suggest the utility of thinking about other problems in a timing representation. Acknowledgements The authors thank Sanjoy Mahajan and Erik Winfree for comments and help with preparation of the manuscript. This work was supported in part by the Center for Neuromorphic Systems Engineering as a part of the National Science Foundation Engineering Research Center Program under grant EEC-9402726. Roweis is supported by the Natural Sciences and Engineering Research Council of Canada under an NSERC 1967 Award. References [Hopfield, 1995] Hopfield, J. (1995). Pattern recognition computation using action potential timing for stimulus representation. Nature, 376:3J---36. [Hopfield, 1996] Hopfield, J. (1996). Transforming neural computations and representing time. Proceedings o/the National Academy o/Sciences, 93:15440-15444. [Hopfield and Tank, 1987] Hopfield, J. and Tank, D. (1987). Neural computation by concentrating information in time. Proceedings 0/ the National Academy 0/ Sciences, 84: 1896-1900. [Unnikrishnan et aI., 1991] Unnikrishnan, K., Hopfield, J., and Tank, D. (1991). Connected digit speaker-dependent speech recognition using a neural network with time-delayed connections. IEEE Transactions on Signal ProceSSing, 39:698-713. [Unnikrishnan et aI., 1992] Unnikrishnan, K., Hopfield, J., and Tank, D. (1992). Speakerindependent digit recognition using a neural network with time-delayed connections. Neural Computation, 4:108-119. 

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Learning Continuous Attractors in Recurrent Networks H. Sebastian Seung Bell Labs, Lucent Technologies Murray Hill, NJ 07974 seung~bell-labs.com Abstract One approach to invariant object recognition employs a recurrent neural network as an associative memory. In the standard depiction of the network's state space, memories of objects are stored as attractive fixed points of the dynamics. I argue for a modification of this picture: if an object has a continuous family of instantiations, it should be represented by a continuous attractor. This idea is illustrated with a network that learns to complete patterns. To perform the task of filling in missing information, the network develops a continuous attractor that models the manifold from which the patterns are drawn. From a statistical viewpoint, the pattern completion task allows a formulation of unsupervised learning in terms of regression rather than density estimation . A classic approach to invariant object recognition is to use a recurrent neural network as an associative memory[l]. In spite of the intuitive appeal and biological plausibility of this approach, it has largely been abandoned in practical applications. This paper introduces two new concepts that could help resurrect it: object representation by continuous attractors, and learning attractors by pattern completion. In most models of associative memory, memories are stored as attractive fixed points at discrete locations in state space[l]. Discrete attractors may not be appropriate for patterns with continuous variability, like the images of a three-dimensional object from different viewpoints. When the instantiations of an object lie on a continuous pattern manifold, it is more appropriate to represent objects by attractive manifolds of fixed points, or continuous attractors. To make this idea practical, it is important to find methods for learning attractors from examples. A naive method is to train the network to retain examples in shortterm memory. This method is deficient because it does not prevent the network from storing spurious fixed points that are unrelated to the examples. A superior method is to train the network to restore examples that have been corrupted, so that it learns to complete patterns by filling in missing information. Learning Continuous Attractors in Recurrent Networks (a) 655 (b) Figure 1: Representing objects by dynamical attractors. (a) Discrete attractors. (b) Continuous attractors. Learning by pattern completion can be understood from both dynamical and statistical perspectives. Since the completion task requires a large basin of attraction around each memory, spurious fixed points are suppressed. The completion task also leads to a formulation of unsupervised learning as the regression problem of estimating functional dependences between variables in the sensory input. Density estimation, rather than regression, is the dominant formulation of unsupervised learning in stochastic neural networks like the Boltzmann machine[2] . Density estimation has the virtue of suppressing spurious fixed points automatically, but it also has the serious drawback of being intractable for many network architectures. Regression is a more tractable, but nonetheless powerful, alternative to density estimation. In a number of recent neurobiological models, continuous attractors have been used to represent continuous quantities like eye position-[3], direction of reaching[4], head direction[5], and orientation of a visual stimulus[6]. Along with these models, the present work is part of a new paradigm for neural computation based on continuous attractors. 1 DISCRETE VERSUS CONTINUOUS ATTRACTORS Figure 1 depicts two ways of representing objects as attractors of a recurrent neural network dynamics. The standard way is to represent each object by an attractive fixed point[l], as in Figure 1a. Recall of a memory is triggered by a sensory input, which sets the initial conditions. The network dynamics converges to a fixed point, thus retrieving a memory. If different instantiations of one object lie in the same basin of attraction, they all trigger retrieval of the same memory, resulting in the many-to-one map required for invariant recognition. In Figure 1b, each object is represented by a continuous manifold of fixed points. A one-dimensional manifold is shown, but generally the attractor should be multidimensional, and is parametrized by the instantiation or pose parameters of the object . For example, in visual object recognition, the coordinates would include the viewpoint from which the object is seen. The reader should be cautioned that the term "continuous attractor" is an idealization and should not be taken too literally. In real networks, a continuous attractor is only approximated by a manifold in state space along which drift is very slow. This is illustrated by a simple example, a descent dynamics on a trough-shaped energy landscape[3]. If the bottom of the trough is perfectly level, it is a line of fixed points and an ideal continuous attract or of the dynamics. However, any slight imperfections cause slow drift along the line. This sort of approximate continuous attract or is what is found in real networks, including those trained by the learning 656 H S. Seung (a) hidden layer (b) ~ visible layer Figure 2: (a) Recurrent network. (b) Feedforward autoencoder. algorithms to be discussed below. 2 DYNAMICS OF MEMORY RETRIEVAL The preceding discussion has motivated the idea of representing pattern manifolds by continuous attractors. This idea will be further developed with the simple network shown in Figure 2a, which consists of a visible layer Xl E Rnl and a hidden layer X2 E Rn2. The architecture is recurrent, containing both bottom-up connections (the n2 x nl matrix W2d and top-down connections (the nl x n2 matrix WI2). The vectors bl and b2 represent the biases ofthe neurons. The neurons have a rectification nonlinearity [x]+ = max{x, O}, which acts on vectors component by component. There are many variants of recurrent network dynamics: a convenient choice is the following discrete-time version, in which updates of the hidden and visible layers alternate in time. After the visible layer is initialized with the input vector Xl (0), the dynamics evolves as X2(t) Xl (t) = = [b 2 + W2IXI(t -1)]+ , [b l + W12X2(t)]+ . (1) If memories are stored as attractors, iteration of this dynamics can be regarded as memory retrieval. Activity circulates around the feedback loop between the two layers. One iteration of this loop is the map Xl(t - 1) ~ X2(t) ~ Xl(t). This single iteration is equivalent to the feedforward architecture of Figure 2b. In the case where the hidden layer is smaller than the visible layers, this architecture is known as an auto encoder network[7]. Therefore the recurrent network dynamics (1) is equivalent to repeated iterations of the feedforward autoencoder. This is just the standard trick of unfolding the dynamics of a recurrent network in time, to yield an equivalent feedforward network with many layers[7]. Because of the close relationship between the recurrent network of Figure 2a and the autoencoder of Figure 2b, it should not be surprising that learning algorithms for these two networks are also related, as will be explained below. 3 LEARNING TO RETAIN PATTERNS Little trace of an arbitrary input vector Xl (0) remains after a few time steps of the dynamics (1). However, the network can retain some input vectors in short-term memory as "reverberating" patterns of activity. These correspond to fixed points of the dynamics (1); they are patterns that do not change as activity circulates around the feedback loop. Learning Continuous Attraclors in Recurrent Networlcs 657 This suggests a formulation of learning as the optimization of the network's ability to retain examples in short-term memory. Then a suitable cost function is the squared difference IXI (T) - Xl (0)12 between the example pattern Xl (0) and the network's short-term memory Xl (T) of it after T time steps. Gradient descent on this cost function can be done via backpropagation through time[7]. If the network is trained with patterns drawn from a continuous family, then it can learn to perform the short-term memory task oy developing a continuous attractor that lies near the examples it is trained on. When the hidden layer is smaller than the visible layer, the dimensionality of the attractor is limited by the size of the hidden layer. For the case of a single time step (T = 1), training the recurrent network of Figure 2a to retain patterns is equivalent to training the autoencoder of Figure 2b by minimizing the squared difference between its input and output layers, averaged over the examples[8]. From the information theoretic perspective, the small hidden layer in Figure 2b acts as a bottleneck between the input and output layers, forcing the autoencoder to learn an efficient encoding of the input. For the special case of a linear network, the nature of the learned encoding is understood completely. Then the input and output vectors are related by a simple matrix multiplication. The rank of the matrix is equal to the number of hidden units. The average distortion is minimized when this matrix becomes a projection operator onto the subspace spanned by the principal components of the examples[9]. From the dynamical perspective, the principal subspace is a continuous attractor of the dynamics (1). The linear network dynamics converges to this attractor in a single iteration, starting from any initial condition. Therefore we can interpret principal component analysis and its variants as methods of learning continuous attractors[lO]. 4 LEARNING TO COMPLETE PATTERNS Learning to retain patterns in short-term memory only works properly for architectures with a small hidden layer. The problem with a large hidden layer is evident when the hidden and visible layers are the same size, and the neurons are linear. Then the cost function for learning can be minimized by setting the weight matrices equal to the identity, W 21 = W l2 = I. For this trivial minimum, every input vector is a fixed point of the recurrent network (Figure 2a), and the equivalent feedforward network (Figure 2b) exactly realizes the identity map. Clearly these networks have not learned anything. Therefore in the case of a large hidden layer, learning to retain patterns is inadequate. Without the bottleneck in the architecture, there is no pressure on the feedforward network to learn an efficient encoding. Without constraints on the dimension of the attractor, the recurrent network develops spurious fixed points that have nothing to do with the examples. These problems can be solved by a different formulation of learning based on the task of pattern completion. In the completion task of Figure 3a, the network is initialized with a corrupted version of an example. Learning is done by minimizing the completion error, which is the squared difference IXI (T) - dl 2 between the uncorrupted pattern d and the final visible vector Xl (T). Gradient descent on completion error can be done with backpropagation through time[ll]. This new formulation of learning eliminates the trivial identity map solution men- H. S. Seung 658 (a) ~1 L _ retention. ~1 .. _ 1 ~ completio~ ~ It ~ It ___ (b) topographic feature map 9x9 patch missing sensory Input retrieved memory Figure 3: (a) Pattern retention versus completion. (b) Dynamics of pattern completion. (b) 5x5 receptive fields Figure 4: (a) Locally connected architecture. (b) Receptive fields of hidden neurons. tioned above: while the identity network can retain any example, it cannot restore corrupted examples to their pristine form. The completion task forces the network to enlarge the basins of attraction of the stored memories, which suppresses spurious fixed points. It also forces the network to learn associations between variables in the sensory input. 5 LOCALLY CONNECTED ARCHITECTURE Experiments were conducted with images of handwritten digits from the USPS database described in [12]. The example images were 16 x 16, with a gray scale ranging from a to 1. The network was trained on a specific digit class, with the goal of learning a single pattern manifold. Both the network architecture and the nature of the completion task were chosen to suit the topographic structure present in visual images. The network architecture was given a topographic organization by constraining the synaptic connectivity to be local, as shown in Figure 4a. Both the visible and hidden layers of the network were 16 x 16. The visible layer represented an image, while the hidden layer was a topographic feature map. Each neuron had 5 x 5 receptive and projective fields, except for neurons near the edges, which had more restricted connectivity. In the pattern completion task, example images were corrupted by zeroing the pixels inside a 9 x 9 patch chosen at a random location, as shown in Figure 3a. The location of the patch was randomized for each presentation of an example. The size of the patch was a substantial fraction of the 16 x 16 image, and much larger than the 5 x 5 receptive field size. This method of corrupting the examples gave the completion task a topographic nature, because it involved a set of spatially contiguous pixels. This topographic nature would have been lacking if the examples had been corrupted by, for example, the addition of spatially uncorrelated noise. Figure 3b illustrates the dynamics of pattern completion performed by a network Learning Continuous Attractors in Recurrent Networks 659 trained on examples of the digit class "two." The network is initialized with a corrupted example of a "two." After the first itex:ation of the dynamics, the image is partially restored. The second iteration leads to superior restoration, with further sharpening of the image. The "filling in" phenomenon is also evident in the hidden layer. The network was first trained on a retrieval dynamics of one iteration. The resulting biases and synaptic weights were then used as initial conditions for training on a retrieval dynamics of two iterations. The hidden layer developed into a topographic feature map suitable for representing images of the digit "two." Figure 4b depicts the bottom-up receptive fields of the 256 hidden neurons. The top-down projective fields of these neurons were similar, but are not shown. This feature map is distinct from others[13) because of its use of top-down and bottom-up connections in a feedback loop. The bottom-up connections analyze images into their constituent features, while the top-down connections synthesize images by composing features. The features in the top-down connections can be regarded as a "vocabulary" for synthesis of images. Since not all combinations of features are proper patterns, there must be some "grammatical" constraints on their combination. The network's ability to complete patterns suggests that some of these constraints are embedded in the dynamical equations of the network. Therefore the relaxation dynamics (1) can be regarded as a process of massively parallel constraint satisfaction. 6 CONCLUSION I have argued that continuous attractors are a natural representation for pattern manifolds. One method of learning attractors is to train the network to retain examples in short-term memory. This method is equivalent to autoencoder learning, and does not work if the number of hidden units is large. A better method is to train the network to complete patterns. For a locally connected network, this method was demonstrated to learn a topographic feature map. The trained network is able to complete patterns, indicating that syntactic constraints on the combination of features are embedded in the network dynamics. Empirical evidence that the network has indeed learned a continuous attractor is obtained by local linearization of the network (1). The linearized dynamics has many eigenvalues close to unity, indicating the existence of an approximate continuous attractor. Learning with an increased number of iterations in the retrieval dynamics should improve the quality of the approximation. There is only one aspect of the learning algorithm that is specifically tailored for continuous attractors. This aspect is the limitation of the retrieval dynamics (1) to a few iterations, rather than iterating it all the way to a true fixed point. As mentioned earlier, a continuous attractor is only an idealization; in a real network it does not consist of true fixed points, but is just a manifold to which relaxation is fast and along which drift is slow. Adjusting the shape of this manifold is the goal of learning; the exact locations of the true fixed points are not relevant. The use of a fast retrieval dynamics removes one long-standing objection to attractor neural networks, which is that true convergence to a fixed point takes too long. If all that is desired is fast relaxation to an approximate continuous attractor, attractor neural networks are not much slower than feedforward networks. In the experiments discussed here, learning was done with backpropagation through time. Contrastive Hebbian learning[14] is a simpler alternative. Part of the image 660 H S. Seung is held clamped, the missing values are filled in by convergence to a fixed point, and an anti-Hebbian update is made. Then the missing values are clamped at their correct values, the network converges to a new fixed point, and a Hebbian update is made. This procedure has the disadvantage of requiring true convergence to a fixed point, which can take many iterations. It also requires symmetric connections, which may be a representational handicap. This paper addressed only the learning of a single attractor to represent a single pattern manifold. The problem of learning multiple attractors to represent mUltiple pattern classes will be discussed elsewhere, along with the extension to network architectures with many layers. Acknowledgments This work was supported by Bell Laboratories. I thank J. J. Hopfield, D. D. Lee, L. K. Saul, N. D. Socci, H. Sompolinsky, and D. W. Tank for helpful discussions. References [1] J. J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. USA, 79:2554-2558, 1982. [2] D. H. Ackley, G. E. Hinton, and T. J. Sejnowski. A learning algorithm for Boltzmann machines. Cognitive Science, 9:147-169, 1985. [3] H. S. Seung. How the brain keeps the eyes still. Proc . Natl. Acad. Sci. USA,93:1333913344, 1996. [4] A. P. Georgopoulos, M. Taira, and A. Lukashin. Cognitive neurophysiology of the motor cortex. Science, 260:47-52, 1993. [5] K Zhang. Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. J. Neurosci., 16:2112-2126, 1996. [6] R . Ben-Yishai, R. L. Bar-Or, and H. Sompolinsky. Theory of orientation tuning in visual cortex. Proc. Nat. Acad. Sci. USA, 92:3844-3848, 1995 . [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, chapter 8, pages 318-362. MIT Press, Cambridge, 1986. [8] G. W . Cottrell, P. Munro, and D. Zipser. Image compression by back propagation: an example of extensional programming. In N. E. Sharkey, editor, Models of cognition: a review of cognitive science. Ablex, Norwood, NJ, 1989. [9] P. Baldi and K Hornik. Neural networks and principal component analysis: Learning from examples without local minima. Neural Networks, 2:53-58, 1989. [10] H. S. Seung. Pattern analysis and synthesis in attractor neural networks. In K-Y. M. Wong, 1. King, and D.-y' Yeung, editors, Theoretical Aspects of Neural Computation : A Multidisciplinary Perspective, Singapore, 1997. Springer-Verlag. [11] F.-S. Tsung and G. W . Cottrell. Phase-space learning. Adv. Neural Info . Proc. Syst., 7:481-488, 1995. [12] Y. LeCun et al. Learning algorithms for classification: a comparison on handwritten digit recognition. In J.-H. Oh, C. Kwon, and S. Cho, editors, Neural networks: the statistical mechanics perspective, pages 261-276, Singapore, 1995. World Scientific. [13] T . Kohonen. The self-organizing map. Proc. IEEE, 78:1464-1480, 1990. [14] J. J . Hopfield, D. I. Feinstein, and R. G. Palmer. "Unlearning" has a stabilizing effect in collective memories. Nature, 304:158-159, 1983. 

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720 AN ELECTRONIC PHOTORECEPTOR SENSITIVE TO SMALL CHANGES IN INTENSITY T. Delbriick and C. A. Mead 256-80 Computer Science California Institute of Technology Pasadena, CA 91125 ABSTRACT We describe an electronic photoreceptor circuit that is sensitive to small changes in incident light intensity. The sensitivity to change8 in the intensity is achieved by feeding back to the input a filtered version of the output. The feedback loop includes a hysteretic element. The circuit behaves in a manner reminiscent of the gain control properties and temporal responses of a variety of retinal cells, particularly retinal bipolar cells. We compare the thresholds for detection of intensity increments by a human and by the circuit. Both obey Weber's law and for both the temporal contrast sensitivities are nearly identical. We previously described an electronic photoreceptor that outputs a voltage that is logarithmic in the light intensity (Mead, 1985). This report describes an extension of this circuit which was based on a suggestion by Frank Werblin that biological retinas may achieve greater sensitivity to change8 in the illumination by feeding back a filtered version of the output. OPERATION OF THE CIRCUIT The circuit (Figure 1) consists of a phototransistor (P), exponential feedback to P (Ql, Q2, and Q3), a transconductance amplifier (A), and the hysteretic element (Q4 and Qs). In general terms the operation of the circuit consists of two stages of amplification with hysteresis in the feedback loop. The light falls on the parasitic bipolar transistor P. (The rest of the circuit is shielded by metal.) P's collector is the substrate and the base is an isolated well. P and Ql form the first stage of amplification. The light produces a base current Is for P. The emitter current IE is PIs, neglecting collector resistance for now. P is typically a few hundred. The feedback current IQl is set by the gate voltage on QdQ2' which is set by the current through Q3, which is set by the feedback voltage Vjb. In equilibrium Vjb will be such that IQl = IE and some voltage Vp will be the output of the first stage. The An Electronic Photoreceptor Sensitive to Small Changes negative feedback through the transconductance amplifier A will make Vp ~ V/ b ? This voltage is logarithmic in the light intensity, since in subthreshold operation the currents through Q2 and Q3 are exponential in their gate to source voltages. The DC output of the circuit will be Vout ~ V/b = Vdd - (2kT /q) log IE, neglecting the back-gate effect for Q2. Figure Sa (DC output) shows that the assumption of subthreshold operation is valid over about 4 orders of magnitude. G Vout T O Figure 1. The photoreceptor circuit. Now what happens when the intensity increases a bit? Figure 2a shows the current through P and Ql as a function of the voltage Vp. Both P and Ql act like current sources in parallel with a resistance, where the value of the current is set, respectively, by the light intensity and by the feedback voltage V/b. When the intensity increases a bit the immediate result is that the curve labeled IE in Figure 2a will shift upwards a little to the curve labeled I~. But IQl won't change right away it is set by the delayed feedback. The effect on Vp will be that Vp will drop by the amount of the shift in the intersection of the curves in Figure 2a, to Vj,. Because interesting gain control properties arise here we will analyze this before going on with the rest of the circuit. In Figure 2b, we model P and Ql as current sources with associated drain/collector resistances. Now, 721 722 Delbro.ck and Mead rp and rQl physically arise from the familiar Early effect, a variation of depletion region thickness causing a variation in the channel length or base thickness. It is a reasonable approximation to model the drain or collector resistance due to the Early effect as r = Vel I, where Ve is the Early voltage and is typically tens of volts, and I is the value of the current source. I ~1 rQl ~1 12: Vp rp 12: ~ Vp Vp a V 1 ... Figure 2. The first stage of amplification. a: The curves show the current through the phototransistor P and feedback transistor Ql as a function of the voltage Vp . Since Ip = IQl the intersection gives the voltage Vp. b: An equivalent circuit model for these transistors in the linear region. The Early effect leads to drain/collector resistances inversely proportional to the value of the current source. Substituting this approximation for rp and rQl into the above expression for 6Vp and letting 6IE = P6Is, we obtain 6V = 6Is Is (Ve,p IIVe ,Ql) where Ve,p and Ve,Ql are the Early voltages associated with the phototransistor and the feedback transistor QlI respectively. In other words, the change in Vp is just proportional to the "contrast" 6 Is / Is. Figure 4a shows test results which support this model. A detector which encodes the intensity logarithmically (so that the output V in response to an input I is V = log I) would also give 6V = 61/1. Our in our circuit the gain control properties for transients arise from an unrelated property of the conductances of the sensor and the feedback element. Comparing the gains for DC and for transients in our circuit and using the expression for the DC output given earlier, we find that the ratio of the gains is transient gain ~ Ve,p IIVe ,Ql ~ 200 2kT/q DC gain 723 An Electronic Photoreceptor Sensitive to Small Changes assuming Ve ,pllVe ,Ql = lOV and kT/q = 25mV. Finally, let us consider the operation of the rest of the circuit. The second stage of amplification is done by the transconductance amplifier A. A produces a current which is pr~ortional to the tanh of the difference between the two inputs, I G tanh( v;k;7~b). When the output of the amplifier is taken as a voltage, the voltage gain is typically a few hundred. Following the transconductance amplifier there is a pair of diode connected transistors, Q4 and Q5 (Figure 3a), which we call the hysteretic element. This pair of transistors has an I-V characteristic which is similar to that of Figure 3. The hysteretic element conducts very little until the voltage across it becomes substantial. Thus the transconductance amplifier works in a dual voltage-output/current-output mode. Small changes in the output voltage result in little change in the feedback voltage. Larger changes in the output voltage cause current to flow through the hysteretic element, completing the feedback loop. This represents a form of memory, or hysteresis, for the past state of the output and a sensitivity to small changes in the input around the past history of the input. = I v I a b Figure 3. a: The hysteretic element. b: 1- l': characteristic. COMPARISON OF CmCUIT AND RETINAL CELLS We felt that since the circuit was motivated by biology it might be interesting to compare the operational characteristics of the circuit and of retinal cells. Since the circuit has no spatial extent it cannot model any of the spatially mediated effects (such as center4Jurround) seen in retinas. Nonetheless we had hoped to capture some of the temporal effects seen in retinal cells. In Figure 4 we compare the responses of the circuit and responses of a retinal bipolar cell to diffuse flashes of light. The circuit has response characteristics closest to those of retinal bipolar cells. The circuit's gain control properties are very similar to those of bipolar cells. Figure 5 shows that both the circuit and bipolar cells tend to control their gain so that they maintain a constant output amplitude for a given change in the log of the intensity. 724 DelbIilck and Mead The response characteristics of the circuit differ from those of bipolar cells in the following ways. First, the gain of the circuit for transients is much larger than that of the bipolar cell, as can be seen in Figure 5, with concomitantly much smaller dynamic range. The dynamic range of the bipolar cell is about 1.5 - 2 log units around the steady intensity, while for the circuit the dynamic range is only about 0.1 log unit. 2.5 Input BACKGROUND -1.0 background IV~ 0.2ms ~% \to% ch~ge 1% -3.0 background (2 decades attenuated) Noise level 20ms a Figure 4. The responses of the circuit compared with responses of a retinal bipolar cell. a: The output of the circuit in response to changes in the intensity. The background levels refer to the same scale as shown in Figure 5. The bottom curve shows the noise level; from this one can see that a detection criterion of signal/ noise ratio equals 2 is satisfied for increments of 1- 2%, in agreement with Figure 6. Note that a 2 decade attenuation hardly changes the response amplitude but the time constant increases by a factor of a hundred. b: The response of a bipolar cell (from Werblin, 1974). The numbers next to the responses are the log of the intensity of the flash substituted for the intial value of the intensity. Note the bidirectionality of the response compared to the circuit. An Electronic Photoreceptor Sensitive to Small Changes 1 ~V) (mV) '0 ~ 6 } 4 2 6V~tOr-~-+~~~~~ -2 -5 -4 -3 -2 -1 log (Intensity) a -4 log (Intensity) b Figure 5. The operating curves of the circuit (a) compared with retinal bipolar cells (b) (adapted from Werblin, 1974). The curves show the height of the peak of the response to flashes substituted for the initial intensity. The initial intensity is given by the intersection of the curves with the abscissa. Note the difference of the gain and the dynamic range. The squares show the DC responses. The slope of the DC response for the circuit is less than expected, probably because there is a leakage current through the hysteretic element. Second, the response of bipolar cells is symmetrical for increases and decreases in the intensity. This can probably be traced to the symmetrical responses of the cones from which they receive direct input. The circuit, on the other hand, only responds strongly to increases in the light intensity. The response in our circuit only becomes symmetrical for output voltage swings comparable to leT / q, probably because the limiting process is recombination in the base of the phototransistor. Third, the control of time constants is dramatically different. In Figure 4a the top set of responses is on a time scale 100 times expanded relative to the bottom scale. The circuit's time constant, in other words, is roughly inversely proportional to the light intensity. This is not the case for bipolar cells. Although we do not show it here, the time constant of the responses of bipolar cells hardly varies with light intensity over at least 4 orders of magnitude (Werblin, 1974). The circuit's action differs much more from that of photoreceptors, amacrine, or ganglion cells. Cones show a much larger sustained response relative to their transient response. Amacrine and ganglion cells spike; our circuit does not. And the circuit differs from on/off amacrine and ganglion cells in the asymmetry of its response to increases and decreases in light intensity. 725 726 Delbruck and Mead EYE vs. CHIP We compared the sensitivity to small changes in the light intensity for one of us and for the circuit in order to get an idea of the performance of the circuit relative to a subjective scale. The thresholds for detection of intensity increments are shown in Figure 6. 10 ? Human ? Photoreceptor 1.6% 9 8 log(Increment) 7 6 5 ? 6 7 8 9 10 11 log( Absorbable Quanta/ sec) Figure 6. Thresholds for detection of flicker. The subject (TD) sat in a darkened room foveating a flickering yellow (583nm) LED at a distance of 75cm. The LED subtended 22' of arc and the frequency of flicker was 5 Hz. Threshold determination was made by a series of trials, indicated by a buzzer, in which the computer either caused the LED to flicker or not to flicker. The percentage of flicker was started at some large amount for which determination was unambiguous. For each trial the subject pressed a button if he thought he saw the LED flickering. A incorrect response would cause the percentage of flicker for the next trial to be increased. A correct response would cause the percentage of flicker to be decreased with a probability of 0.44. The result after a hundred trials would be a curve of percentage flicker vs. trial number which started high and then leveled off around the 75% correct level, taken to be the threshold (Levitt, 1971). The threshold for the circuit was determined by shining the same LED onto the chip directly. The threshold was defined as the smallest amount of flicker that would result in a signal-to-noise ratio of 2 for the output. The two sets of thresholds were scaled relative to each other using a Tektronics photometer and were both scaled to read in terms of absorbable quanta, defined for the human as the number of photons hitting the cornea and for the circuit as the number of quanta that hit the area of the phototransistor. There is a bias here favoring the circuit, since the other parts An Electronic Photoreceptor Sensitive to Small Changes of the circuit have not been included in this area. Including the rest of the circuit would raise the thresholds for the circuit by a factor of about 3. The results show that both the circuit and the human approximately obey Weber's law (61/1 at threshold is a constant), and the sensitivities are nearly the same. The highest sensitivity for the circuit, measured at an incident intensity of 660#-, W / cm2 , was 1.2%. This is about half the intensity in a brightly lit office. APPLICATIONS We are trying to develop these sensors for use in neurophysiological optical dye recording. These experiments require a sensor capable of recording changes in the incident intensity of about 1 part in 103 - 4 (Grinvald, 1985). The current technique is to use integrated arrays of photodiodes each with a dedicated rack-mounted low noise amplifier. We will try to replace this arrangement with arrays of receptors of the type discussed in this paper. Currently we are 1 to 2 orders of magnitude short of the required sensitivity, but we hope to improve the performance by using hybrid bipolar/FET technology. CONCLUSION This circuit represents an example of an idea from biology directly and simply synthesised in silicon. The resulting circuit incorporated not only the intended idea, sensitivity to changes in illumination, but also gave rise to an unexpected gain control mechanism unrelated to exponential feedback. The circuit differs in several ways from its possible biological analogy but remains an interesting and potentially useful device. Aeknowledgements This work was supported by the System Development Foundation and the Office of Naval Research. Chips were fabricated through the MOSIS foundry. We thank Frank Werblin for helpful comments and Mary Ann Maher for editorial assistance. Refereneea A. Grinvald. (1985) Real time optical ma~ping of neuronal activity: from single growth cones to the intact mammalian braili. Ann. Rev. Neurolci. 8:263-305. H. Levitt. (1971) 'Iransformed up-down methods in psychoacoustics. J. Acoud. Soc. Am. 49:467-477. C. Mead. (1985) A Sensitive Electronic Photoreceptor. In 1985 Chapel Hill Conference on VLSI. 463-471. F. Werblin. (1974) Control of Retinal Sensitivity II: Lateral Interactions at the Outer Plexiform Layer. J. PhYliology. 68:62-87. 727 

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MELONET I: Neural Nets for Inventing Baroque-Style Chorale Variations Dominik Hornel dominik@ira.uka.de Institut fur Logik, Komplexitat und Deduktionssysteme Universitat Fridericiana Karlsruhe (TH) Am Fasanengarten 5 D-76128 Karlsruhe, Germany Abstract MELONET I is a multi-scale neural network system producing baroque-style melodic variations. Given a melody, the system invents a four-part chorale harmonization and a variation of any chorale voice, after being trained on music pieces of composers like J. S. Bach and J . Pachelbel. Unlike earlier approaches to the learning of melodic structure, the system is able to learn and reproduce high-order structure like harmonic, motif and phrase structure in melodic sequences. This is achieved by using mutually interacting feedforward networks operating at different time scales, in combination with Kohonen networks to classify and recognize musical structure. The results are chorale partitas in the style of J. Pachelbel. Their quality has been judged by experts to be comparable to improvisations invented by an experienced human organist. 1 INTRODUCTION The investigation of neural information structures in music is a rather new, exciting research area bringing together different disciplines such as computer science, mathematics, musicology and cognitive science. One of its aims is to find out what determines the personal style of a composer. It has been shown that neural network models - better than other AI approaches - are able to learn and reproduce styledependent features from given examples, e.g., chorale harmonizations in the style of Johann Sebastian Bach (Hild et al., 1992) . However when dealing with melodic sequences, e.g., folk-song style melodies, all of these models have considerable difficulties to learn even simple structures. The reason is that they are unable to capture high-order structure such as harmonies , motifs and phrases simultaneously occurring at multiple time scales. To overcome this problem, Mozer (Mozer, 1994) D. Hamel 888 proposes context units that learn reduced descriptions of a sequence of individual notes. A similar approach in MELONET (Feulner et Hornel, 1994) uses delayed update units that do not fire each time their input changes but rather at discrete time intervals. Although these models perform well on artificial sequences , they produce melodies that suffer from a lack of global coherence. The art of melodic variation has a long tradition in Western music . Almost every great composer has written music pieces inventing variations of a given melody, e.g., Mozart's famous variations KV 265 on the melody "Ah! Vous dirai-je, Maman", also known as "Twinkle twinkle little star". At the beginning of this tradition there is the baroque type of chorale variations. These are organ or harpsichord variations of a chorale melody composed for use in the Protestant church. A prominent representative of this kind of composition is J. Pachelbel (1653 - 1706) who wrote about 50 chorale variations or partitas on various chorale melodies. 2 TASK DESCRIPTION Given a chorale melody, the learning task is achieved in two steps: 1. A chorale harmonization of the melody is invented. 2. One of the voices of the resulting chorale is chosen and provided with melodic variations. Both subtasks are directly learned from music examples composed by J. Pachelbel and performed in an interactive composition process which results in a chorale variation of the given melody. The first task is performed by HARMONET, a neural network system which is able to harmonize melodies in the style of various composers like J. S. Bach. The second task is performed by the neural network system MELONET I, presented in the following . For simplicity we have considered melodic variations consisting of 4 sixteenth notes for each melody quarter note . This is the most common variation type used by baroque composers and presents a good starting point for even more complex variation types, since there are enough music examples for training and testing the networks, and because it allows the representation of higher-scale elements in a rather straightforward way. HARMONET is a system producing four-part chorales in various harmonization styles, given a one-part melody. It solves a musical real-world problem on a performance level appropriate for musical practice. Its power is based on a coding scheme capturing musically relevant information. and on the integration of neural networks and symbolic algorithms in a hierarchical system, combining the advantages of both. The details are not discussed in this paper. See (Hild et aI., 1992) or (Hornel et Ragg, 1996a) for a detailed account . 3 A MULTI-SCALE NEURAL NETWORK MODEL The learning goal is twofold. On the one hand, the results produced by the system should conform to musical rules. These are melodic and harmonic constraints such as the correct resolving of dissonances or the appropriate use of successive interval leaps. On the other hand, the system should be able to capture stilistic features from the learning examples, e.g., melodic shapes preferred by J. Pachelbel. The observation of musical rules and the aesthetic conformance to the learning set can be achieved by a multi-scale neural network model. The complexity of the learning task is reduced by decomposition in three subtasks (see Figure 1): MELONEI' I: Neural Netsfor Inventing Baroque-Style Chorale Variations Harmony T 889 T D, T S, T, D T k=tmod4 : MelodIc Vanallon Figure 1: Structure of the system and process of composing a new melodic variation. A melody (previously harmonized by HARMONET) is passed to the supernet which predicts the current motif class MGT from a local window given by melody notes MT to MT +2 and preceding motif class MGT-I. A similar procedure is performed at a lower time scale by the su bnet which predicts the next motif note Nt based on M CT, current harmony HT and preceding motif note Nt-I. The result is then returned to the supernet through the motif classifier to be considered when computing the next motif class MCT +1 . 1. A melody variation is considered at a higher time scale as a sequence of melodic groups, so-called motifs. Each quarter note of the given melody is varied by one motif. Before training the networks, motifs are classified according to their similarity. 2. One neural network is used to learn the abstract sequence of motif classes. Motif classes are represented in a l-of-n coding form where n is a fixed number of classes. The question it solves is: What kind of motif 'fits' a melody note depending on melodic context and the motif that has occurred before? No concrete notes are fixed by this network. It works at a higher scale and will therefore be called stlpernet in the following. 3. Another neural network learns the implementation of abstract motif classes into concrete notes depending on a given harmonic context. It produces a sequence of sixteenth notes - four notes per motif - that result in a melodic variation of the given melody. Because it works one scale below the supernet, it is called stlbnet. 4. The subnet sometimes invents a sequence of notes that does not coincide 890 D. Homel with the motif class determined by the supernet. This motif will be considered when computing the next motif class , however. and should therefore match the notes previously formed by the subnet. It is therefore reclassified by the motif classifier before the supernet determines the next motif class. The motivation of this separation into supernet and subnet arised from the following consideration : Having a neural network that learns sequences of sixteenth notes, it. would be easier for this network to predict notes given a contour of each motif. i.e. a sequence of interval directions to be produced for each quarter note. Consider a human organist who improvises a melodic variation of a given melody in real time. Because he has to take his decisions in a fraction of a second, he must at least have some rough idea in mind about what kind of melodic variation should be applied to the next melody note to obtain a meaningful continuation of the variation. Therefore, a neural network was introduced at a higher time scale , the training of which really improved the overall behavior of the system and not just shifted the learning problem to another time scale. 4 MOTIF CLASSIFICATION AND RECOGNITION In order to realize learning at different time scales as described above , we need a recognition component to find a suitable classification of motifs . This can be achieved using unsupervised learning, e.g. , agglomerative hierarchical clustering or Kohonen's topological feature maps (Kohonen, 1990). The former has the disadvantage however that an appropriate distance measure is needed which determines the similarity between small sequences of notes respectively intervals, whereas the latter allows to obtain appropriate motif classes through self-organization within a twodimensional surface. Figure 2 displays the motif representation and distribution of motif contours over a 10xlO Kohonen feature map. In MELONET I, the Kohonen algorithm is applied to all motifs contained in the training set. Afterwards a corresponding motif classification tree is recursively built from the Kohonen map. While cutting this classification tree at lower levels we can get more and more classes. One important problem remains to find an appropriate number of classes for the given learning task. This will be discussed in section 6. ~ ... -" ....... .... .. .... ... ...... ....... .. ... .... ... . , jJ 3jl ', Winner 1st interval 2nd interval 3rd interval -1 Figure 2: Motifrepresentation example (left) and motif contour distribution (right) over a 10xlO Kohonen feature map developed from one Pachelbel chorale variation (initial update area 6x6, initial adaptation height 0.95, decrease factor 0.995). Each cell corresponds to one unit in the KFM. One can see the arrangement of regions responding to motifs having different motif contours. 891 MELONEr I: Neural Nets for Inventing Baroque-Style Chorale Variations 5 REPRESENTATION In general one can distinguish two groups of motifs: Melodic motifs prefer small intervals, mainly seconds, harmonic motifs prefer leaps and harmonizing notes (chord notes) . Both motif groups heavily rely on harmonic information. In melodic motifs dissonances should be correctly resolved, in harmonic motifs notes must fit the given harmony. Small deviations may have a significant effect on the quality of musical results. Thus our idea was to integrate musical knowledge about interval and harmonic relationships into an appropriate interval representation. Each note is represented by its interval to the first motif note, the so-called reference note. This is an important element contributing to the success of MELONET I. A similar idea for Jazz improvisation was followed in (Baggi, 1992) . The interval coding shown in Table 1 considers several important relationships: neighboring intervals are realized by overlapping bits, octave invariance is represented using a special octave bit. The activation of the overlapping bit was reduced from 1 to 0.5 in order to allow a better distinction of the intervals. 3 bits are used to distinguish the direction of the interval , 7 bits represent interval size. Complementary intervals such as ascending thirds and descending sixths have similar representations because they lead to the same note and can therefore be regarded as harmonically equivalent. A simple rhythmic element was then added using a tenuto bit (not shown -in Table 1) which is set when a note is tied to its predecessor. This final 3+1+7+1=12 bit coding gave the best results in our simulations. Table 1: Complementary Interval Coding direction ninth octave seventh sixth fifth fourth third second pnme second third fourth fifth sixth seventh octave ninth \. \. \. \. \. \. \. \. -+ /' /' /' /' /' /' /' /' o0 o0 o0 1 1 1 100 100 1 0 0 1 o 0 1 o 0 010 o0 1 0 0 1 0 0 1 o0 1 o0 1 o0 1 0 0 1 o0 1 octave 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 interval size 0 1 0.5 0 0 0 0 0 1 0.5 0 0 0 0 0 1 0.5 0 0 1 0.5 0 0 0 0 0 1 0.5 0 0 0 0 0 1 0 0 0 1 0.5 0 0 0 0 0 1 0.5 0 0 0 0 0 0 0 0 0 1 0.5 0 0 0 0 0 1 0.5 0 0 0 0 0 0 0 0 0 1 0.5 0 0 0 0 0 1 0.5 0 0 0 0.5 0 0 0 0 0 1 0.5 0 0 0 0 0 1 0.5 0 0 1 0.5 0 0 0 0 0 1 0.5 0 0 0 0 0 1 0.5 0 Now we still need a representation for harmony. It can be encoded as a harmonic field which is a vector of chord notes of the diatonic scale. The tonic T in C major for example contains 3 chord notes - C, E and G - which correspond to the first, third and fifth degree of the C major scale (1010100). This representation may be further improved. We have already mentioned that each note is represented by the interval to the first motif note (reference note). We can now encode the harmonic field starting with the first motif note instead of the first degree of the scale . This is equivalent to rotating the bits of the harmonic field vector. An example is displayed in Figure 3. The harmony of the motif is the dominant D, the first motif note is B which corresponds to the seventh degree of the C major scale. Therefore the D.Homel 892 harmonic field for D (0100101) is rotated by one position to the right resulting in (1010010). Starting with the first note B. the harmonic field indicates the intervals that lead to harmonizing notes B, D and G. In the right part of Figure 3 one can see a correspondance between bits activated in the harmonic field and bits set to 1 in the three interval codings. This kind of representation helps the neural network to directly establish a relationship between intervals and given harmony. ,J J 3d I D third up sixth up pnme o0 o0 1 1 010 harmonic field 0 0 0 0 0 1 0.5 0 0 1 0 0 0 0 0 0 0.5 0 0 1 0 0 0 0.5 1 0 1 0 0 1 0 Figure 3: Example illustrating the relationship between interval coding and rotat.ed harmonic field. Each note is represented by its interval to the first note. 6 PERFORMANCE We carried out several simulations to evaluate the performance of the system. Many improvements could be found however by just listening to the improvisations produced by the neural organist. One important problem was to find an appropriate number of classes for the given learning task . The following table lists the classification rate on the learning and validation set of the supernet and the subnet using 5, 12 and 20 motif classes. The learning set was automatically built from 12 Pachelbel chorale variations corresponding to 2220 patterns for the subnet and 555 for the supernet. The validation set includes 6 Pachelbel variations corresponding to 1396 patterns for the subnet and 349 for the supernet. Supernet and subnet were then trained independently with the RPROP learning algorithm . learning set validation set 5 classes 91.17% 49.85% s'Upernet 12 classes 86.85% 40.69% 20 classes 87.57% 37.54% 5 classes 86.31% 79.15% s'Ubnet 12 classes 93.92% 83.38% 20 classes 95 .68% 86.96% The classification rate of both networks strongly depends on the number of classes, esp. on the validation set of the supernet. The smaller the number of classes, the better is the classification of the supernet because there are less alternatives to choose from. We can also notice an opposite development of the classification behavior for the subnet. The bigger the number of classes. the easier the subnet will be able to determine concrete motif notes for a given motif class. One can imagine that the optimal number of classes lies somewhere in the middle. Another idea is to form a committee of networks each of which is trained with different number of classes. We have also tested MELONET I on melodies that do not belong to the baroque era. Figure 4 shows a harmonization and variation of the melody "Twinkle twinkle little star" used by Mozart in his famous piano variations. It was produced by a network committee formed by 3*2=6 networks trained with 5, 12 and 20 classes. 7 CONCLUSION We have presented a neural network system inventing baroque-style variations on given melodies whose qualities are similar to those of an experienced human organ- MELONEI'L? Neural Nets for Inventing Baroque-Style Chorale Variations 893 !. Figure 4: Melodic variation on "Twinkle twinkle little star" ist. The complex musical task could be learned introducing a multi-scale network model with two neural networks cooperating at different time scales , together with an unsupervised learning mechanism able to classify and recognize relevant musical structure. We are about to test this multi-scale approach on learning examples of other epochs, e.g., on compositions of classical composers like Haydn and Mozart or on Jazz improvisations. First results confirm that the system is able to reproduce stylespecific elements of other kinds of melodic variation as well. Another interesting question is whether the global coherence of the musical results may be further improved adding another network working at a higher level of abstraction, e.g., at. a phrase level. In summary, we believe that this approach presents an important step towards the learning of complete melodies. References Denis L. Baggi. NeurSwing: An Intelligent Workbench for the Investigation of Swing in Jazz. In: Readings in Computer-Generated Music, IEEE Computer Society Press, pp. 79-94, 1992. Johannes Feulner, Dominik Hornel. MELONET: Neural networks that learn harmony-based melodic variations. In: Proceedings of the 1994 International Computer Music Conference. ICMA Arhus , pp. 121-124, 1994. Hermann Hild, Johannes Feulner, Wolfram Menzel. HARMONET: A Neural Net for Harmonizing Chorales in the Style of J. S. Bach. In: Advances in Neural Information Processing 4 (NIPS 4), pp. 267-274. 1992. Dominik Hornel, Thomas Ragg . Learning Musical Structure and Style by Recognition, Prediction and Evolution. In: Proceedings of the 1996 International Computer Music Conference. ICMA Hong Kong , pp. 59-62, 1996. Dominik Hornel , Thomas Ragg. A Connectionist Model for the Evolution of Styles of Harmonization. In: Proceedings of the 1996 International Conference on Music Perception and Cognition. Montreal, 1996. Teuvo Kohonen. The Self-Organizing Map. In: Proceedings of the IEEE, Vol. 78, no. 9, pp. 1464-1480 , 1990. Michael C. Mozer . Neural Network music composition by prediction . In: Connection Science 6(2,3), pp. 247-280 , 1994. 

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Detection of first and second order motion Alexander Grunewald Division of Biology California Institute of Technology Mail Code 216-76 Pasadena, CA 91125 alex@vis.caltech.edu Heiko Neumann Abteilung Neuroinformatik Vniversitat VIm 89069 VIm Germany hneumann@neuro.informatik.uni-ulm.de Abstract A model of motion detection is presented. The model contains three stages. The first stage is unoriented and is selective for contrast polarities. The next two stages work in parallel. A phase insensitive stage pools across different contrast polarities through a spatiotemporal filter and thus can detect first and second order motion. A phase sensitive stage keeps contrast polarities separate, each of which is filtered through a spatiotemporal filter, and thus only first order motion can be detected. Differential phase sensitivity can therefore account for the detection of first and second order motion. Phase insensitive detectors correspond to cortical complex cells, and phase sensitive detectors to simple cells. 1 INTRODUCTION In our environment objects are constantly in motion, and the visual system faces the task of identifying the motion of objects. This task can be subdivided into two components: motion detection and motion integration. In this study we will look at motion detection. Recent psychophysics has made a useful distinction between first and second order motion. In first order motion an absolute image feature is moving. For example, a bright bar moving on a dark background is an absolute feature because luminance is moving. In second order motion a relative image feature is moving, for example a contrast reversing bar. No longer is it possible to identify the moving object through its luminance, but only that it has different luminance with respect to the background. Humans are very sensitive to first order motion, but can they detect second order motion? Chubb & Sperling (1988) showed that subjects are in fact able to detect second order motion. These findings have since been confirmed in many psychophysical experiments, and it has become clear that the parameters that yield detection of first and second order motion are different, suggesting that separate motion detection systems exist. A. Grunewald and H. Neumann 802 1.1 Detection of first and second order motion First order motion, which is what we encounter in our daily lives, can be easily detected by finding the peak in the Fourier energy distribution. The motion energy detector developed by Adelson & Bergen (1985) does this explicitly, and it turns out that it is also equivalent to a Reichardt detector (van Santen & Sperling, 1985). However, these detectors cannot adequately detect second order motion, because second order motion stimuli often contain the maximum Fourier energy in the opposite direction (possibly at a different velocity) as the actual motion. In other words, purely linear filters, should have opposite directional tuning for first and second order motion. This is further illustrated in Figure 1. FIRST ORDER MOTION Stimulus Energy Reconstructed 80 G) E 60 =40 20 0 20 4060 80100 20 40 60 80 100 20 40 60 80 100 space SECOND ORDER MOTION Stimulus Energy 80 80 ~60 60 ;:;40 40 20 20 0 20 40 60 80 100 space 0 20 40 60 80 100 Reconstructed 20 40 60 80 100 Figure 1: Schematic of first and second order motion, their peak Fourier energy, and the reconstruction. The peak Fourier energy is along the direction of motion for first order motion, and in the opposite direction for second order motion. For this reason a linear filter cannot detect second order motion. One way to account for second order motion detection is to transform the second order motion signal into a first order signal. H second order motion is defined by contrast reversals, then detecting contrast edges and then rectifying the resulting signal of contrast will yield a first order motion signal. Thus this approach includes three steps: orientation detection, rectification and finally motion detection (Wilson et al., 1992) . 803 Detection of First and Second Order Motion 1.2 Visual physiology Cells in the retina and the lateral geniculate nucleus (LGN) have concentric (and hence unoriented) receptive fields which are organized in an opponent manner. While the center of such an ON cell is excited by a light increment, the surround is excited by a light decrement, and vice versa for OFF cells. It is only at the cortex that direction and orientation selectivity arise. Cortical simple cells are sensitive to the phase of the stimulus, while complex cells are not (Hubel & Wiesel, 1962). Most motion models take at least partial inspiration from known physiology and anatomy, by relating the kernels of the motion detectors to the physiology of cortical cells. The motion energy model in particular detects orientation and first order motion at the same time. Curiously, all motion models essentially ignore the concentric opponency of receptive fields in the LG N. This is usually justified by pointing to the linearity of simple cells with respect to stimulus parameters. However, it has been shown that simple cells in fact exhibit strong nonlinearities (Hammond & MacKay, 1983). Moreover, motion detection does require at least one stage of nonlinearity (Poggio & Reichardt, 1973). The present study develops a model of first and second order motion detection which explicitly includes an unoriented processing stage, and phase sensitive and phase insensitive motion detectors are built from these unoriented signals. The former set of detectors only responds to first order motion, while the second set of detectors responds to both types of motion. We further show the analogies that can be drawn between these detector types and simple and complex cells in cat visual cortex. 2 MODEL DESCRIPTION The model is two-dimensional, one dimension is space, which means that space has been collapsed onto a line, and the other dimension is time. The input image to the model is a space-time matrix of luminances, as shown in figure 1. At each processing stage essentially the same operations are performed. First the input signal is convolved with the appropriate kernel. At each stage there are multiple kernels, to generate the different signal types at that stage. For example, there are ON and OFF signals at the unoriented stage. Next the convolved responses are subtracted from each other. At the unoriented stage this means ON-OFF and OFF-ON. In the final step these results are half-wave rectified to only yield positive signals. Unoriented Phase insensitive Phase sensitive space Figure 2: The kernels in the model. For the unoriented (left plot) and phase sensitive (right plot) kernel plots black indicates OFF regions, white ON regions, and grey zero input. For the phase insensitive plot (middle) grey denotes ON and OFF input, and black denotes zero input. 804 A. Grunewald and H. Neumann At the unoriented stage the input pattern is convolved with a difference of Gaussians kernel. This kernel has only a spatial dimension, no temporal dimension (see figure 2). As described earlier, competition is between ON and OFF signals, followed by half-wave rectification. This ensures that at each location only one set of unoriented signals is present. A simulation of the signals at the unoriented stage is shown in figure 3. For first order motion, ON signals are at locations corresponding to the inside of the moving bar. With each shift of the bar the signals also move. Similarly, the OFF signals correspond to the outside of the bar, and also move with the bar. For second order motion the contrast polarity reverses. Thus ON signals correspond to the inside when the bar is bright, and to the outside when the bar is dark, and vice versa for OFF signals. Thus any ON or OFF signals to the leading edge of the bar will remain active after the bar moves. Stimulus 80 .:: 20 o ." ...- Unoriented ON OFF .rl' 20 40 60 80 100 space Stimulus 20 40 60 80 100 20 40 60 80 100 OFF ON 80 60 40 20 20 40 60 80 100 20 40 60 80 100 0 20 40 60 80 100 space Figure 3: Unoriented signals to first and second order motion. ON signals are at the bright side of any contrast transition, while OFF signals are at the dark side. In first order motion ON and OFF move synchronously to the moving stimulus. In second order motion ON and OFF signals persist, since the leading edge becomes the trailing edge, and at the same time the contrast reverses, which means that at a particular spatial location the contrast remains constant. At the phase insensitive stage the unoriented ON and OFF signals are added, and then the result is convolved with an energy detection filter.- The pooling of ON and OFF signals means that the contrast transitions in the image are essentially full-wave rectified. This causes phase insensitivity. These pooled signals are then convolved with a space-time oriented filter (see figure 2). Competition between opposite directions of motion ensures that only one direction is active. A consequence of the pooling of unoriented ON and OFF signals at this stage is that the resulting signals are invariant to first or second order motion. Thus phase insensitivity Detection of First and Second Order Motion 805 makes this stage able to detect both first and second order motion. These signals are shown in figure 4. In a two-dimensional extension of this model these detectors would also be orientation selective. The simplest way to obtain this would be via elongation along the preferred orientation. Phase insensitive Stimulus left 20 40 60 80 100 right 20 40 60 80 100 space Stimulus 20 40 60 80 100 left right 80 ::?? 60 40 20 20 40 60 80 100 space o 20 40 60 80 100 ? ? / 20 40 60 80 100 Figure 4: Phase insensitive signals to first and second order motion. For both stimuli there are no leftwards signals, and robust rightwards signals. At the phase sensitive stage unoriented ON and OFF signals are separately convolved with space-time oriented kernels which are offset with respect to each other (see figure 2). The separate treatment of ON and OFF signals yields phase sensitivity. At each location there are four kernels: two for the two directions of motion, and two for the two phases. Competition occurs between signals of opposite direction tuning, and opposite phase preference. To avoid activation in the opposite direction of motion slightly removed from the location of the edge spatially broadly tuned inhibition is necessary. This is provided by the phase insensitive signals, thus avoiding feedback loops among phase sensitive detectors. First order signals from the unoriented stage match the spatiotemporal filters in the preferred direction, and thus phase sensitive signals arise. However, due to their phase reversal, second order motion input, provides poor motion signals, which are quenched through phase insensitive inhibition. These signals are shown in figure 5. These simulations show that first and second order motion are detected differently. First order motion is detected by phase sensitive and phase insensitive motion detectors, while second order motion is only detected by the latter. From this we conclude that first order motion is a more potent stimulus, and that the detection of second order is more restricted, since it depends on a single type of detector. In particular, the size of the stimulus and its velocity have to be matched to the energy A. Grunewald and H. Neumann 806 Phase sensitive Stimulus 20 40 60 80 100 space Stimulus 20 40 60 80 100 DL left DL right 20 40 60 80 100 DL left 20 40 60 80 100 DL right 20 40 60 80 100 space Figure 5: Phase sensitive signals to first and second order motion. Only the darklight signals are shown. First order motion causes a consistent rightward motion signal, while second order motion does not. filters for motion signals to arise. 3 RELATION TO PHYSIOLOGY The relationship between the model and physiology is straightforward. Unoriented signals correspond to LGN responses, phase insensitive signals to complex cell responses, and phase sensitive signals to simple cell responses. Thus the model suggests that both simple and some complex cells receive direct LGN input. Moreover these complex cells inhibit simple cells. With an additional threshold in simple cells this inhibition could also be obtained via complex to simple cell excitation. We stress that we are not ruling out that many complex cells receive only simple cell input. Rather, the present research shows that if all complex cells receive only simple cell input, second order motion cannot be detected. Hence at least some complex cell responses need to be built up directly from LGN responses. Several lines of evidence from cat physiology support this suggestion. First, the mean latencies of simple and complex cells are about equal (Bullier & Henry, 1979), suggesting that at least some complex cells receive direct LGN input. Second, noise stimuli can selectively activate complex cells, without activation of simple cells (Hammond, 1991). Third, cross-correlation analyses show that complex cells do receive simple cell input (Ghose et ai., 1994). The present model predicts that some cortical complex cells should respond to Detection of First and Second Order Motion 807 second order motion. Zhou & Baker (1993) investigated this, and found that some complex cells in area 17 respond to second order motion. Moreover, they found that simple cells of a particular first order motion preference did not reverse their motion preference when stimulated with second order motion, which would occur if simple cells were just linear filters. We interpret this as further evidence that complex cells provide inhibitory input to simple cells. If complex cells are built up from LGN input, then orientation selectivity in two-dimensional space cannot be obtained based on simple cell input, but rather requires complex cells with elongated receptive fields. Thus we predict that there ought to be a correlation in complex cells between elongated receptive fields and dependence on direct LGN input. In conclusion we have shown how the phase sensitivity of motion detectors can be mapped onto the ability to detect only first order motion, or both first and second order motion. This suggests that it is not necessary to introduce a orientation detection stage before motion detection can take place, thus simplifying the model of motion detection. Furthermore we have shown that the proposed model is in accord with known physiology. Acknow ledgments This work was supported by the McDonnell-Pew program in Cognitive Neuroscience. References Adelson, E. & Bergen (1985). Spatiotemporal energy models for the perception of motion. J. Opt. Soc. Am. A, 2, 284-299. Bullier, J. & Henry, G. H. (1979). Ordinal position of neurons in cat striate cortex. J. Neurophys., 42, 1251-1263. Chubb, C. & Sperling, G. (1988). Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception. J. Opt. Soc . Am. A, 5, 1986-2007. Ghose, G. M., Freeman, R. D. & Ohzawa, I. {1994}. Local intracortical connections in the cat's visual cortex: postnatal development and plasticity. J. Neurophys., 12, 1290-1303. Hammond, P. (1991). On the response of simple and complex cells to random dot patterns. Vis. Res., 31,47-50. Hammond, P. & MacKay, D. (1983). Influence of luminance gradient reversal On simple cells in feline striate cortex. J. Physiol., 331, 69-87. Hubel, D. H. & Wiesel, T. N. (1962). Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. J. Physiol., 160, 106-154. Poggio, T. & Reichardt, W. (1973). Considerations on models of movement detection. Kybernetik, 12, 223-227. van Santen & Sperling, G. (1985). Elaborated Reichardt detectors. J. Opt. Soc. Am. A, 2, 300-321. Wilson, H. R., Ferrera, V. P. & Yo, C. (1992). A psychophysically motivated model for two-dimensional motion perception. Vis . Neurosci., 9, 79-97. Zhou, Y.X. & Baker, C. L. (1993). A processing stream in mammalian visual cortex neurons for non-Fourier responses. Science, 261, 98-101. 

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From Regularization Operators to Support Vector Kernels Alexander J. Smola Bernhard Scholkopf GMDFIRST Rudower Chaussee 5 12489 Berlin, Germany smola@first.gmd.de Max-Planck-Institut fur biologische Kybernetik Spemannstra.Be 38 72076 Ttibingen, Germany bs-@mpik-tueb.mpg.de Abstract We derive the correspondence between regularization operators used in Regularization Networks and Hilbert Schmidt Kernels appearing in Support Vector Machines. More specifica1ly, we prove that the Green's Functions associated with regularization operators are suitable Support Vector Kernels with equivalent regularization properties. As a by-product we show that a large number of Radial Basis Functions namely conditionally positive definite functions may be used as Support Vector kernels. 1 INTRODUCTION Support Vector (SV) Machines for pattern recognition, regression estimation and operator inversion exploit the idea of transforming into a high dimensional feature space where they perform a linear algorithm. Instead of evaluating this map explicitly, one uses Hilbert Schmidt Kernels k(x, y) which correspond to dot products of the mapped data in high dimensional space, i.e. (I) k(x, y) = (<I>(x) ? <I>(y)) with <I> : .!Rn --* :F denoting the map into feature space. Mostly, this map and many of its properties are unknown. Even worse, so far no general rule was available. which kernel should be used, or why mapping into a very high dimensional space often provides good results, seemingly defying the curse of dimensionality. We will show that each kernel k(x, y) corresponds to a regularization operator P, the link being that k is the Green's function of P* P (with F* denoting the adjoint operator of F). For the sake of simplicity we shall only discuss the case of regression - our considerations, however, also hold true foi the other cases mentioned above. ~ We start by briefly reviewing the concept of SV Machines (section 2) and of Regularization Networks (section 3). Section 4 contains the main result stating the equivalence of both A. J. Smola and B. Schollwpf 344 methods. In section 5, we show some applications of this finding to known SV machines. Section 6 introduces a new class of possible SV kernels, and, finally, section 7 concludes the paper with a discussion. 2 SUPPORT VECTOR MACHINES The SV algorithm for regression estimation, as described in [Vapnik, 1995] and [Vapnik et al., 1997], exploit~ the idea of computing a linear function in high dimensional feature space F (furnished with a dot product) and thereby computing a nonlinear function in the space of the input data !Rn. The functions take the form f(x) = (w ? <ll(x)) + b with ell : !Rn -+ :F and w E F. In order to infer f from a training set {(xi, Yi) I i = 1, ... , f., Xi E !Rn, Yi E IR}, one tries to minimize the empirical risk functional Remp[f] together with a complexity term l!wll 2 , thereby enforcingflatness in feature space, i.e. to minimize 1 Rreg[/] = Remp[!] l + Allwll 2 =f. :L;c(f(xi),yi) + Allwll 2 (2) i=l with c(f(xi),yi) being the cost function determining how deviations of f(xi) from the target values Yi should be penalized, and A being a regularization constant. As shown in [Vapnik, 1995] for the case of ?-insensitive cost functions, c(f(x) ) = { lf(x)- yl- ? for lf(_x)- Yl ;::: e 0 otherwtse ' (3) 'Y (2) can be minimized by solving a quadratic programming problem formulated in terms of dot products in :F. It turns out that the solution can be expressed in terms of Support Vectors, w = :Ef=I Cti<ll(xi), and therefore f(x) = l l i=l i=l L ai(<ll(xi) ? <ll(x)) + b = L aik(xi, x) + b, (4) where k(xi, x) is a kernel function computing a dot product in feature space (a concept introduced by Aizerrnan et al. [ 1964]). The coefficients ai can be found by solving a quadratic programming problem (with Kii := k(xi, Xj) and ai = f3i - {3i): l minimize !L l (f3i- f3i)(f3j- f3J)Kij- "?, (!3i- f3i)Yi- (f3i i,j=l i=l f. subject to "?, f3i - !3i = + f3i)e - (5) 0, f3i, !3i E [0, ft] ? i=l Note that (3) is not the only possible choice of cost functions resulting in a quadratic programming problem (in fact quadratic parts and infinities are admissible, too). For a detailed discussion see [Smola and Scholkopf, 1998]. Also note that any continuous symmetric function k(x, y) E L 2 ? L 2 may be used as an admissible Hilbert-Schmidt kernel if it satisfies Mercer's condition / / k(x,y)g(x)g(y)dxdy;::: 0 for all g E 3 L 2 (IRn)~ (6) REGULARIZATION NETWORKS Here again we start with minimizing the empirical risk functional Remp[!] plus a regularization term liP /11 2 defined by a regularization operator Pin the sense of Arsenin and From Regularization Operators to Support Vector Kernels 345 Tikhonov [1977]. Similar to (2), we minimize Rreg[f] = Remp 1"' ~ ?_-c(f(xi),yi) + -XIIPJII ? l ~ 2 + .\IIPJII = f 2 (7) i=1 Using an expansion off in terms of some symmetric function k(xi, Xj) (note here, that k need not fulfil Mercer's condition), f(x) = 2: aik(xi, x) + b, (8) and the cost function defined in (3 ), this leads to a quadratic programming problem similar to the one for SVs: by computing Wolfe's dual (for details of the calculations see [Smola and SchOlkopf, 1998]), and using Dij := ((Fk)(xi, .) ? (Fk)(xj, .)) (9) ((f ? g) denotes the dot product of the functions f and g in Hilbert Space, t.e. I !(x)g(x)dx), we get n- 1 K(i]- /3*), with f3i, f3i being the solution of a= l mm1m1Ze ! 2: l (f3i- f3i)(f3j- {3j)(KD- 1 K)ij- i,j=1 f_ subject to L L (f3i- f3i)Yi- (f3i + f3i)E i=l f3i - f3i = 0, f3i, f3i E [0, A] i=1 (I 0) Unfortunately this setting of the problem does not preserve sparsity in terms of the coefficients, as a potentially sparse decomposition in terms of f3i and f3i is spoiled by n- 1 K, which in general is not diagonal (the expansion (4) on the other hand does typically have many vanishing coefficients). 4 THE EQUIVALENCE OF~BOTH METHODS Comparing (5) with (10) leads to the question if and under which condition the two methods might be equivalent and therefore also under which conditions regularization networks might lead to sparse decompositions (i.e. only a few of the expansion coefficients in f would differ from zero). A sufficient condition is D = K (thus K n- 1 K = K), i.e. (11) Our goal now is twofold: ? Given a regularization operator P, find a kernel k such that a SV machine using k will not only enforce flatness in feature space, but also correspond to minimizing a regularized risk functional with P as regularization operator. ? Given a Hilbert Schmidt kernel k, find a regularization operator P such that a SV machine using this kernel can be viewed as a Regularization Network using P. These two problems can be solved by employing the concept of Green's functions as described in [Girosi et al., 1993]. These functions had been introduced in the context of solving differential equations. For our purpose, it is sufficient to know that the Green's functions Gx, (x) ofF* P satisfy (12) Here, 8xi (x) is the 8-distribution (not to be confused with the Kronecker symbol8ij) which has the property that (f ? 8x.) = f(xi). Moreover we require for all Xi the projection of Gx, (x) onto the null space of F* P to be zero. The relationship between kernels and regularization operators is formalized in the following proposition. A. I. Smola and B. Schtilkopf 346 Proposition 1 Let P be a regularization operator, and G be the Green's function of P* P. Then G is a Hilbert Schmidt-Kernel such that D = K. SV machines using G minimize risk functional (7) with Pas regularization operator. Proof: Substituting ( 12) into GxJ (xi) = (GxJ (.) ? 8x? (.)) yields (13) Gxi (xi) = ( (PGx, ){.) ? (PGxJ(.)) = Gx; {xj), hence G(xi,Xj) := Gx,{xj) is symmetric and satisfies (11). Thus the SVoptimization problem (5) is equivalent to the regularization network counterpart ( 10). Furthermore G is an admissible positive kernel, as it can be written as a dot product in Hilbert Space, namely (14) In the following we will exploit this relationship in both ways: to compute Green's functions for a given regularization operator P and to infer the regularization operator from a given kernel k. 5 TRANSLATION INVARIANT KERNELS Let us now more specifically consider regularization operators multiplications in Fourier space [Girosi et al., 1993] (Pi? Pg ) = 1 (21l')n/2 f Jn P that may be written as f{w)fJ(w) dw P(w) (15) with ](w) denoting the Fourier transform of j(x), and P(w) = P( -w) real valued, nonnegative and converging uniformly to 0 for lwl --+ oo and supp[P(w)]. Small values of P(w) correspond to a strong attenuation of the corresponding frequencies. n= For regularization operators defined in Fourier Space by (15) it can be shown by exploiting P(w) = P(-w) = P(w) that G(Xi x) = l 1 {27!' )n/2 f }JR.n eiw(x;-x) P(w)dw (16) is a corresponding Green's function satisfying translational invariance, i.e. G(xi,xj) = G (Xi - xi), and G(w) P (w). For the proof, one only has to show that G satisfies (11 ). = This provides us with an efficient tool for analyzing SV kernels and the types of capacity control they exhibit Example 1 (Bq-splines) Vapnik et al. [ 1997] propose to use Bq-splines as building blocks for kernels, i.e. (17) i=l with x E !Rn. For the sake of simplicity, we consider the case n 1. Recalling the definition (18) . Bq = ?q+ 1 1[-o.5,o.5] (? denotes the convolution and Ix the indicator function on X), we can utilize the above result and the Fourier-Plancherel identity to construct the Fourier representation of the corresponding regularization operator. Up to a multiplicative constant, it equals P(w) = k(w) = sinc(q+l)(Wi). 2 (19) From Regularization Operators to Support W?ctor Kernels 347 This shows that only B-splines of odd order are admissible, as the even ones have negative parts in the Fourier spectrum (which would result in an amplification of the corresponding frequency components). The zeros ink stem from the fact that B1 has only compact support [-(k+ 1)/2, (k+ 1)/2). By using this kernel we trade reduced computational complexity in calculatingf(we only have to take points with llxi- xi II :S cfrom some limited neighborhood determined by c into account)for a possibly worse performance of the regularization operator as it completely ~emovesfrequencies wp with k(wp) = 0. Example 2 (Dirichlet kernels) In [Vapnik et al., 1997], a class of kernels generating Fourier expansions was introduced, k(x) = sin(2N + 1)x/2. (20) sinx/2 (As in example 1 we consider x E ~ 1 to avoid tedious notation.) By construction, this kernel corresponds to P(w) = ~ L~-N 6(w- i). A regularization operator with these properties, however; may not be desirable as it only damps a finite number offrequencies and leaves all other frequencies unchanged which can lead to overjitting (Fig. 1). \ !\ 'i 'I I. I? ,\ I l 05 I \ :\ -I .I 'I -? _, -.:.,,\ 1\ .'i \ I.\ r ""t-4 -15 -10 -? \I ~ 10 15 Figure 1: Left: Interpolation with a Dirichlet Kernel of order N = 10. One can clearly observe the overfitting (dashed line: interpolation, solid line: original data points, connected by lines). Right: Interpolation of the same data with a Gaussian Kernel of width CT 2 = 1. Example 3 (Gaussian kernels) Following the exposition of Yuille and Grzywacz [1988] as described in [Girosi et al., 1993], one can see that for 11P!II 2 = I dx L 2m ~!2m com f(x)) 2 (21) m with 6 2 m = 6. m and 6 2m+l tor; we get Gaussians kernels = V' 6. m. 6. being the Laplacian and V' the Gradient operak(x) = exp ( -~~~~2 ). (22) Moreover; we can provide an equivalent representation of P in terms of its Fourier properties, i.e. P(w) = exp(- u2 \kxll 2 ) up to a multiplicative constant. Training a SV machine with Gaussian RBF kernels [Scholkopf et al., 1997] corresponds to minimizing the specific costfunction with a regularization operator of type (21 ). This also explains the good performance ofSV machines in this case, as it is by no means obvious that choosing a flat fum:;tion in high dimensional space will correspond to a simple function in low dimensional space, as showed in example 2. Gaussian kernels tend to yield good performance under g'eneral smoothness assumptions and should be considered especially if no additional knowledge of the data is available. A. J. Smola and B. Scholkopf 348 6 A NEW CLASS OF SUPPORT VECTOR KERNELS We will follow the lines of Madych and Nelson [ 1990] as pointed out by Girosi et al. [ 1993]. Our main statement is that conditionally positive definite functions (c.p.d.) generate admissible SV kernels. This is very useful as the property of being c.p.d. often is easier to verify than Mercer's condition, especially when combined with the results of Schoenberg and Micchelli on the connection between c.p.d. and completely monotonic functions [Schoenberg, 1938, Micchelli, 1986]. Moreover c.p.d. functions lead to a class of SV kernels that do not necessarily satisfy Mercer's condition. Definition 1 (Conditionally positive definite functions) A continuous function h, defined on [0, oo), is said to be conditionally positive definite ? (c.p.d.) of order m on m.n iffor any distinct points x1, ... , Xt E m.n and scalars c1, ... , Ct the quadratic form Eri=l cicih(llxi- Xj II) is nonnegative provided that E~=l Cip(xi) = 0 for all polynomials p on m.n of degree lower than m. Proposition 2 (c.p.d. functions and admissible kernels) Define II~ the space of polynomials of degree lower than m on IRn. Every c.p.d. function h of order m generates an admissible Kernel for SV expansions on the space offunctions f orthogonal to II~ by setting k(xi, Xj) := h(llxi- Xjll 2 ). Proof: In [Dyn, /991] and [Madych and Nelson, 1990] it was shown that c.p.d. functions h generate semi-norms 11-llh by (23) Provided that the projection off onto the space ofpolynomials of degree lower than m is zero. For these functions, this, however. also defines a dot product in some feature space. Hence they can be used as SV kernels. Only c.p.d. functions of order m up to 2 are of practical interest for SV methods (for details see [Smola and Scholkopf, 1998]). Consequently, we may use kernels like the ones proposed in [Girosi et al., 1993] as SV kernels: k(x,y) = e-.BIJx-yJJ2 k(x,y) = -v'llx- Yll 2 + c2 k(x,y) = k(x,y) = 1 y'Jix-yJ12+c2 llx- Yll 2 ln llx- Yll = 0); multiquadric, (m = 1) Gaussian, (m inverse multiquadric, (m thin plate splines, (m (24) (25) = 0) = 2) (26) (27) 7 DISCUSSION We have pointed out a connection between SV kernels and regularization operators. As one of the possible implications of this result, we hope that it will deepen our understanding of SV machines and of why they have been found to exhibit high generalization ability. In Sec. 5, we have given examples where only the translation into the regularization framework provided insight in why certain kernels are preferable to others. Capacity control is one of the strengths of SV machines; however, this does not mean that the structure of the learning machine, i.e. the choice of a suitable kernel for a given task, should be disregarded. On the contrary, the rather general class of admissible SV kernels should be seen as another strength, provided that we have a means of choosing the right kernel. The newly established link to regularization theory can thus be seen as a tool for constructing the structure consisting of sets of functions in which the SV machine (approximately) performs structural From Regularization Operators to Support \iector Kernels 349 risk minimization (e.g. [Vapnik, 1995]). For a treatment of SV kernels in a Reproducing Kernel Hilbert Space context see [Girosi, 1997]. Finally one should leverage the theoretical results achieved for regularization operators for a better understanding of SVs (and vice versa). By doing so this theory might serve as a bridge for connecting two (so far) separate threads of machine learning. A trivial example for such a connection would be a Bayesian interpretation of SV machines. In this case the choice of a special kernel can be regarded as a prior on the hypothesis space with P[f] ex exp{ ->.IIF111 2 ). A more subtle reasoning probably will be necessary for understanding the capacity bounds [Vapnik, 1995] from a Regularization Network point of view. Future work will include an analysis of the family of polynomial kernels, which perform very well in Pattern Classification [SchOlkopf et al., 1995]. Acknowledgements AS is supported by a grant of the DFG (# Ja 379/51 ). BS is supported by the Studienstiftung des deutschen Volkes. The authors thank Chris Burges, Federico Girosi, Leo van Hemmen, Klaus-Robert Muller and Vladimir Vapnik for helpful discussions and comments. References M.A. Aizerman, E. M. Braverman, and L. I. Rozonoer. Theoretical foundations of the potential function method in pattern recognition learning. Automation and Remote Control, 25:821-837, 1964. N. Dyn. Interpolation and approximation by radial and related functions. In C.K. Chui, L.L. Schumaker, and D.J. Ward, editors, Approximation Theory, VI, pages 211-234. Academic Press, New York, 1991. F. Girosi. An equivalence between sparse approximation and suppm1 vector machines. A.I. Memo No. 1606, MIT, 1997. F. Girosi, M. Jones, and T. Poggio. Priors, stabilizers and basis functions: From regularization to radial, tensor and additive splines. A.I. Memo No. 1430, MIT, 1993. W.R. Madych and S.A. Nelson. Multivariate interpolation and conditionally positive definite functions. II. Mathematics of Computation, 54(189):211-230, 1990. C. A. Micchelli. Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constructive Approximation, 2:11-22, 1986. I.J. Schoenberg. Metric spaces and completely monotone functions. Ann. of Math., 39: 811-841, 1938. B. Scholkopf, C. Burges, and V. Vapnik. Extracting support data for a given task. In U. M. Fayyad and R. Uthurusamy, editors, Proc. KDD I, Menlo Park, 1995. AAAI Press. B. SchOlkopf, K. Sung, C. Burges, F. Girosi, P. Niyogi, T. Poggio, and V. Vapnik. Comparing support vector machines with gaussian kernels to radial basis function classifiers. IEEE Trans. Sign. Processing, 45:2758-2765, 1997. A. J. Smola and B. SchOlkopf. On a kernel-based method for pattern recognition, regression, approximation and operator inversion. Algorithmica, 1998. see also GMD Technical Report 1997- I 064, URL: http://svm.first.gmd.de/papers.html. V. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York, 1995. V. Vapnik, S. Golowich, and A. Smola: Support vector method for function approximation, regression estimation, and signal processing. In NIPS 9, San Mateo, CA, 1997. A. Yuille and N. Gr:z;ywacz. The motion coherence theory. In Proceedings of the International Conference on Computer Vision, pages 344-354, Washington, D.C., 1988. IEEE Computer Society Press. 

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Globally Optimal On-line Learning Rules Magnus Rattray*and David Saad t Department of Computer Science & Applied Mathematics, Aston University, Birmingham B4 7ET, UK. Abstract We present a method for determining the globally optimal on-line learning rule for a soft committee machine under a statistical mechanics framework. This work complements previous results on locally optimal rules, where only the rate of change in generalization error was considered. We maximize the total reduction in generalization error over the whole learning process and show how the resulting rule can significantly outperform the locally optimal rule. 1 Introduction We consider a learning scenario in which a feed-forward neural network model (the student) emulates an unknown mapping (the teacher), given a set of training examples produced by the teacher. The performance of the student network is typically measured by its generalization error, which is the expected error on an unseen example. The aim of training is to reduce the generalization error by adapting the student network's parameters appropriately. A common form of training is on-line learning, where training patterns are presented sequentially and independently to the network at each learning step. This form of training can be beneficial in terms of both storage and computation time, especially for large systems. A frequently used on-line training method for networks with continuous nodes is that of stochastic gradient descent, since a differentiable error measure can be defined in this case. The stochasticity is a consequence of the training error being determined according to only the latest, randomly chosen, training example. This is to be contrasted with batch learning, where all the training examples would be used to determine the training error leading to a deterministic algorithm. Finding an effective algorithm for discrete networks is less straightforward as the error measure is not differentiable. ? rattraym@aston.ac.uk t saadd@aston.ac.uk Globally Optimal On-line Learning Rules 323 Often, it is possible to improve on the basic stochastic gradient descent algorithm and a number of modifications have been suggested in the literature. At late times one can use on-line estimates of second order information (the Hessian or its eigenvalues) to ensure asymptotically optimal performance (e.g., [1, 2]). A number of heuristics also exist which attempt to improve performance during the transient phase of learning (for a review, see [3]). However, these heuristics all require the careful setting of parameters which can be critical to their performance. Moreover, it would be desirable to have principled and theoretically well motivated algorithms which do not rely on heuristic arguments. Statistical mechanics allows a compact description for a number of on-line learning scenarios in the limit of large input dimension, which we have recently employed to propose a method for determining globally optimal learning rates for on-line gradient descent [4]. This method will be generalized here to determine globally optimal on-line learning rules for both discrete and continuous machines. That is, rules which provide the maximum reduction in generalization error over the whole learning process. This provides a natural extension to work on locally optimal learning rules [5, 6], where only the rate of change in generalization error is optimized. In fact, for simple systems we sometimes find that the locally optimal rule is also globally optimal. However, global optimization seems to be rather important in more complex systems which are characterized by more degrees of freedom and often require broken permutation symmetries to learn perfectly. We will outline our general formalism and consider two simple and tractable learning scenarios to demonstrate the method. It should be pointed out that the optimal rules derived here will often require knowledge of macroscopic properties related to the teacher's structure which would not be known in general. In this sense these rules do not provide practical algorithms as they stand, although some of the required macroscopic properties may be evaluated or estimated on the basis of data gathered as the learning progresses. In any case these rules provide an upper bound on the performance one could expect from a real algorithm and may be instrumental in designing practical training algorithms. 2 The statistical mechanics framework For calculating the optimal on-line learning rule we employ the statistical mechanics description of the learning process. Under this framework, which may be employed for both smooth [7,8] and discrete sy.stems (e.g. [9]), the learning process is captured by a small number of self-averaging statistics whose trajectory is deterministic in the limit of large input dimension. In this analysis the relevant statistics are overlaps between weight vectors associated with different nodes of the student and teacher networks. The equations of motion for the evolution of these overlaps can be written in closed form and can be integrated numerically to describe the dynamics. We will consider a general two-layer soft committee machine l . The desired teacher mapping is from an N-dimensional input space E RN onto a scalar ( E R, which the student models through a map a(J,e) 2:~l g(Ji ?e), where g(x) is the activation function for the hidden layer, J == {Jih<i<K is the set of input-to-hidden adaptive weights for the K hidden nodes and the hidden-to-output weights are set to 1. The activation of hidden node i under presentation of the input pattern eJ' is = J i . eJ'? denoted e = xr IThe general result presented here also applies to the discrete committee machine, but we will limit our discussion to the soft-committee machine. M. Rattray and D. Saad 324 Training examples are of the form (el', (I') where J.L = 1,2, ... , P. The components of the independently drawn input vectors el' are uncorrelated random variables with zero mean and unit variance. The corresponding output (I' is given by a deterministic teacher of a similar configuration to the student except for a possible difference in the number M of hidden units and is of the form (I' = l:!1 g(B n . el'), where B == {Bnh<n<M is the set of input-to-hidden adaptive weights. The activation of hidden node n under presentation of the input pattern el' is denoted Y~ = Bn . el'. We will use indices i, j, k, I ... to refer to units in the student network and n, m, ... for units in the teacher network. We will use the commonly used quadratic deviation E(J,e) == ~ [ a(J,e) - (]2, as the measure of disagreement between teacher and student. The most basic learning rule is to perform gradient descent on this quantity. Performance on a typical input defines the generalization error Eg(J) == (E(J,e?{(} through an average over all possible input vectors e. The general form of learning rule we will consider is, J~+l = J~ + ~FfJ(xl' ~I') el' , 'N' ,." ~ (1) where F == {Fi} depends only on the student activations and the teacher's output, and not on the teacher activations which are unobservable. Note that gradient descent on the error takes this general form, as does Hebbian learning and other training algorithms commonly used in discrete machines. The optimal F can also depend on the self-averaging statistics which describe the dynamics, since we know how they evolve in time. Some of these would not be available in a practical application, although for some simple cases the unobservable statistics can be deduced from observable quantities. This is therefore an idealization rather than a practical algorithm and provides a bound on the performance of a real algorithm. The activations are distributed according to a multivariate Gaussian with covariances: (XiXk) = Ji?Jk == Qik, (XiYn) = Ji?Bn == R in , and (YnYm) = Bn?B m == Tnm , measuring overlaps between student and teacher vectors. Angled brackets denote averages over input patterns. The covariance matrix completely describes the state of the system and in the limit of large N we can write equations of motion for each macroscopic (the Tnm are fixed and define the teacher): dRin dO! = (FiYn ) dQik do: = (Fixk + FkXi + FiFk) , (2) where angled brackets now denote the averages over activations, replacing the averages over inputs, and 0: = J.LIN plays the role of a continuous time variable. 3 The globally optimal rule Carrying out the averaging over input patterns one obtains an expression for the generalization error which depends exclusively on the overlaps R,Q and T. Using the dependence of their dynamics (Eq. 2) on F one can easily calculate the locally optimal learning rule [5] by taking the functional derivative of dEg(F)/do: to zero, looking for the rule that will maximize the reduction in generalization error at the present time step. This approach has been shown to be successful in some training scenarios but is likely to fail where the learning process is characterized by several phases of a different natures (e.g., multilayer networks). The globally optimal learning rule is found by minimizing the total change in generalization error over a fixed time window, ~fg(F)= la l ao dE dgdo: = 0: la ao l ?(F, 0:) do:. (3) Globally Optimal On-line Learning Rules 325 This is a functional of the learning rule which we minimize by a variational approach. First we can rewrite the integrand by expanding in terms of the equations of motion, each constrained by a Lagrange multiplier, ?(F ,a ) = ~ 8f g dRin ~ 8R . da in In ~ 8f g dQik +~ 8Q. ik Ik +L Vik ik da (d~ik ~..\ . (d~n _ (F.- +~ in In d a IYn ?) - (FiXk + FkXi + FiFk?) . (4) a The expression for ? still involve two multidimensional integrations over x and y, so taking variations in F, which may depend on x and ( but not on y, we find an expression for the optimal rule in terms of the Lagrange multipliers: F 1 -1 _ = -x -"2v .\y (5) where v = [Vij] and .\ = [..\in]. We define y to be the teacher's expected field given the teacher's output and the student activations, which are observable quantities: y =/ dyyp(ylx,() . (6) Now taking variations in the overlaps w.r.t. the integral in Eq. (3) we find a set of differential equations for the Lagrange multipliers1 d..\km da - _ L,,\ . 8{FiYn) _ LV .. 8{Fi Xj + FjXi + FiFj) . In 8R km .. I} 8Rkm In IJ (7) where F takes its optimal value defined in Eq. (5). The boundary conditions for the Lagrange multipliers are, and (8) which are found by minimizing the rate of change in generalization error at ai, so that the globally optimal solution reduces to the locally optimal solution at this point, reflecting the fact that changes at al have no affect at other times. IT the above expressions do not yield an explicit formula for the optimal rule then the rule can be determined iteratively by gradient descent on the functional Llfg(F). To determine all the quantities necessary for this procedure requires that we first integrate the equations for the overlaps forward and then integrate the equations for the Lagrange multipliers backwards from the boundary conditions in Eq. (8). 4 Two tractable examples In order to apply the above results we must be able to carry out the average in Eq. (6) and then in Eq. (7). These averages are also required to determine the locally optimal learning rule, so that the present method can be extended to any of the problems which have already been considered under the criteria of local optimality. Here we present two examples where the averages can be computed in closed form. The first problem we consider is a boolean perceptron learning a M. Rattray and D. Saad 326 linearly separable task where we retrieve the locally optimal rule [5]. The second problem is an over-realizable task, where a soft committee machine student learns a perceptron with a sigmoidal response. In this example the globally optimal rule significantly outperforms the locally optimal rule and exhibits a faster asymptotic decay. Boolean perceptron: For the boolean perceptron we choose the activation function g(x) = sgn(x) and both teacher and student have a single hidden node (M = K = 1). The locally optimal rule was determined by Kinouchi and Caticha [5] and they supply the expected teacher field given the teacher output ( sgn(y) and the student field x (we take the teacher length T = 1 without loss of generality), = _ R ( y= x Q (~exp(-if-?) + ---'-----,,..-----:,erfc (9) (-5?) Substituting this expression into the Lagrange multiplier dynamics in Eq. (7) shows that the ratio of .x to v is given by .x/v = -2Q / R, and Eq. (5) then returns the locally optimal value for the optimal rule: F = ( fi.. exp( _ , 2 z 2 ) Vi 2. ,erfc (-~,) (10) This rule leads to modulated Hebbian learning and the resulting dynamics are discussed in [5]. We also find that the locally optimal rule is retrieved when the teacher is corrupted by output or weight noise [9]. Soft committee machine learning a continuous perceptron: In this example the teacher is an invertible perceptron (M = 1) while the student is a soft committee machine with an arbitrary number (K) of hidden nodes. We choose the activation function g(x) = erf(x/v'2) for both the student and teacher since this allows the generalization error to be determined in closed form [7]. This is an example of an over-realizable task, since the student has greater complexity than is required to learn the teacher's mapping. The locally optimal rule for this scenario was determined recently [6]. Since the teacher is invertible, the expected teacher activation fi is trivially equal to the true activation y. This leads to a particularly simple form for the dynamics (the n suffix is dropped since there is only one teacher node), dRi dQik do: = biT - Ri do: = bibk T - Qik , (11) where we have defined bi = - Ej vi/ .xj /2 and the optimal rule is given by Fi = biy - Xi. The Lagrange multiplier dynamics in Eq. (7) then show that the relative ratios of each Lagrange multiplier remain fixed over time, so that bi is determined by its boundary value (see Eq. (8?. It is straightforward to find solutions for long times, since the bi approach limiting values for very small generalization error (there are a number of possible solutions because of symmetries in the problem but any such solution will have the same performance for long times). For example, one possible solution is to have b1 = 1 and bi = 0 for all i f:. 1, which leads to an exponential decay of weights associated with all but a single node. This shows how the optimal performance is achieved when the complexity of the student matches that of the teacher. Figure 1 shows results for a three node student learning a continuous perceptron. Clearly, the locally optimal rule performs poorly in comparison to the globally Globally Optimal On-line Learning Rules 327 100 --.__ .- 10-2 ?g 0.8 ..... '" .... 10-4 ~n '" .... .... ......... 0.6 0.4 (" . 10'" .... i ,. 0.2 ......... 0 " , '., -0.2 10'" -0.4 , .... -0.6 10- '0 " -0.8 10- '2 0 5 10 15 20 25 a -1 0 5 10 15 20 25 a Figure 1: A three node soft committee machine student learns from an continuous percept ron teacher. The figure on the left shows a log plot of the generalization error for the globally optimal (solid line) and locally optimal (dashed line) algorithms. The figure on the right shows the student-teacher overlaps for the locally optimal rule, which exhibit a symmetric plateau before specialization occurs. The overlaps where initialized randomly and uniformly with Qii E [0,0.5] and ~,Qi*j E [0,10- 6 ]. optimal rule. In this example the globally optimal r.ule arrived at was one in which two nodes became correlated with the teacher while a third became anti-correlated, showing another possible variation on the optimal rule (we determined this rule iteratively by gradient descent in order to justify our general approach, although the observations above show how one can predict the final result for long times). The locally optimal rule gets caught in a symmetric plateau, characterized by a lack of differentiation between student vectors associated with different nodes, and also displays a slower asymptotic decay. 5 Conclusion and future work We have presented a method for determining the optimal on-line learning rule for a soft committee machine under a statistical mechanics framework. This result complements previous work on locally optimal rules which sought only to optimize the rate of change in generalization error. In this work we considered the global optimization problem of minimizing the total change in generalization error over the whole learning process. We gave two simple examples for which the rule could be determined in closed form, for one of which, an over-realizable learning scenario, it was shown how the locally optimal rule performed poorly in comparison to the globally optimal rule. It is expected that more involved systems will show even greater difference in performance between local and global optimization and we are currently applying the method to more general teacher mappings. The main technical difficulty is in computing the expected teacher activation in Eq. (6) and this may require the use of approximate methods in some cases. It would be interesting to compare the training dynamics obtained by the globally optimal rules to other approaches, heuristic and principled, aimed at incorporating information about the curvature of the error surface into the parameter modification rule. In particular we would like to examine rules which are known to be optimal asymptotically (e.g. [10]). Another important issue is whether one can apply these results to facilitate the design of a practical learning algorithm. 328 M. Rattray and D. Saad Acknowledgement This work was supported by the EPSRC grant GR/L19232. References [1] G. B. Orr and T. K. Leen in Advances in Neural Information Processing Systems, vol 9, eds M. C. Mozer, M. I. Jordan and T. Petsche (MIT Press, Cambridge MA, 1997) p 606. [2] Y. LeCun, P. Y. Simard and B. Pearlmutter in Advances in Neural Information Processing Systems, vol 5, eds S. J. Hanson, J. D. Cowan and C. 1. Giles (Morgan Kaufman, San Mateo, CA, 1993) P 156. [3] C. M. Bishop, Neural networks for pattern recognition, (Oxford University Press, Oxford, 1995). [4] D. Saad and M. Rattray, Phys. Rev. Lett. 79, 2578 (1997). [5] O. Kinouchi and N. Caticha J. Phys. A 25, 6243 (1992). [6] R. Vicente and N. Caticha J. Phys. A 30, L599 (1997). [7] D. Saad and S. A. Solla, Phys. Rev. Lett. 74, 4337 (1995) and Phys. Rev. E 52 4225 {1995}. [8] M. Biehl and H. Schwarze, J. Phys. A 28, 643 (1995). [9] M. Biehl, P. Riegler and M. Stechert, Phys. Rev. E 52, R4624 (1995). [10] S. Amari in Advances in Neural Information Processing Systems, vol 9, eds M. C. Mozer, M. I. Jordan and T. Petsche (MIT Press, Cambridge MA, 1997). 

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Just One View: Invariances in Inferotemporal Cell Thning Maximilian Riesenhuber Tomaso Poggio Center for Biological and Computational Learning and Department of Brain and Cognitive Sciences Massachusetts Institute of Techno)ogy, E25-201 Cambridge, MA 02139 {max,tp }@ai.mit.edu Abstract In macaque inferotemporal cortex (IT), neurons have been found to respond selectively to complex shapes while showing broad tuning ("invariance") with respect to stimulus transformations such as translation and scale changes and a limited tuning to rotation in depth. Training monkeys with novel, paperclip-like objects, Logothetis et al. 9 could investigate whether these invariance properties are due to experience with exhaustively many transformed instances of an object or if there are mechanisms that allow the cells to show response invariance also to previously unseen instances of that object. They found object-selective cells in anterior IT which exhibited limited invariance to various transformations after training with single object views. While previous models accounted for the tuning of the cells for rotations in depth and for their selectivity to a specific object relative to a population of distractor objects,14,1 the model described here attempts to explain in a biologically plausible way the additional properties of translation and size invariance. Using the same stimuli as in the experiment, we find that model IT neurons exhibit invariance properties which closely parallel those of real neurons. Simulations show that the model is capable of unsupervised learning of view-tuned neurons. We thank Peter Dayan, Marcus Dill, Shimon Edelman, Nikos Logothetis, Jonathan Mumick and Randy O'Reilly for useful discussions and comments. 216 M. RiesenhuberandT. Poggio 1 Introduction Neurons in macaque inferotemporal cortex (IT) have been shown to respond to views of complex objects,8 such as faces or body parts, even when the retinal image undergoes size changes over several octaves, is translated by several degrees of visual angle 7 or rotated in depth by a certain amount 9 (see [13] for a review). These findings have prompted researchers to investigate the physiological mechanisms underlying these tuning properties. The original model 14 that led to the physiological experiments of Logothetis et al. 9 explains the behavioral view invariance for rotation in depth through the learning and memory of a few example views, each represented by a neuron tuned to that view. Invariant recognition for translation and scale transformations have been explained either as a result of object-specific learning4 or as a result of a normalization procedure ("shifter") that is applied to any image and hence requires only one object-view for recognition. 12 A problem with previous experiments has been that they did not illuminate the mechanism underlying invariance since they employed objects (e.g., faces) with which the monkey was quite familiar, having seen them numerous times under various transformations. Recent experiments by Logothetis et al. 9 addressed this question by training monkeys to recognize novel objects ("paperclips" and amoeba-like objects) with which the monkey had no previous visual experience. After training, responses of IT cells to transformed versions of the training stimuli and to distractors of the same type were collected. Since the views the monkeys were exposed to during training were tightly controlled, the paradigm allowed to estimate the degree of invariance that can be extracted from just one object view. In partiCUlar, Logothetis et al. 9 tested the cells' responses to rotations in depth, translation and size changes. Defining "in variance" as yielding a higher response to test views than to distractor objects, they report 9 ,10 an average rotation invariance over 30?, translation invariance over ?2?, and size invariance of up to ?1 octave around the training view. These results establish that there are cells showing some degree of invariance even after training with just one object view, thereby arguing against a completely learning-dependent mechanisms that requires visual experience with each transformed instance that is to be recognized. On the other hand, invariance is far from perfect but rather centered around the object views seen during training. 2 The Model Studies of the visual areas in the ventral stream of the macaque visual system8 show a tendency for cells higher up in the pathway (from VI over V2 and V4 to anterior and posterior IT) to respond to increasingly complex objects and to show increasing invariance to transformations such as translations, size changes or rotation in depth. 13 We tried to construct a model that explains the receptive field properties found in the experiment based on a simple feedforward model. Figure 1 shows a cartoon of the model: A retinal input pattern leads to excitation of a set of "VI" cells, in the figure abstracted as having derivative-of-Gaussian receptive field profiles. These "VI" cells are tuned to simple features and have relatively small receptive fields. While they could be cells from a variety of areas, e.g., VI or V2 (cf. Discussion), for simplicity, we label them as "VI" cells (see figure). Different cells differ in preferred feature, e.g., orientation, preferred spatial frequency (scale), and receptive field location. "VI" cells of the same type (i.e., having the same preferred stimulus, but of different preferred scale and receptive field location) feed into the same neuron in an intermediate layer. These intermediate neurons could be complex cells in VI or V2 or V4 or even posterior IT: we label them as "V4" cells, in the Just One View: Invariances in Inferotemporal Cell Tuning 217 same spirit in which we labeled the neurons feeding into them as "VI" units. Thus, a "V4" cell receives inputs from "VI" cells over a large area and different spatial scales ([8] reports an average receptive field size in V4 of 4.4? of visual angle, as opposed to about I ? in VI; for spatial frequency tuning, [3] report an average FWHM of 2.2 octaves, compared to 1.4 (foveally) to 1.8 octaves (parafoveally) in VI 5). These "V4" cells in turn feed into a layer of "IT" neurons, whose invariance properties are to be compared with the experimentally observed ones. Retina Figure 1: Cartoon of the model. See text for explanation. A crucial element of the model is the mechanism an intermediate neuron uses to pool the activities of its afferents. From the computational point of view, the intermediate neurons should be robust feature detectors, i.e., measure the presence of specific features without being confused by clutter and context in the receptive field. More detailed considerations (Riesenhuber and Poggio, in preparation) show that this cannot be achieved with a response function that just summates over all the afferents (cf. Results). Instead, intermediate neurons in our model perform a "max" operation (akin to a "Winner-Take-AII") over all their afferents, i.e., the response ofan intermediate neuron is determined by its most strongly excited afferent. This hypothesis appears to be compatible with recent data,15 that show that when two stimuli (gratings of different contrast and orientation) are brought into the recepti ve field of a V4 cell, the cell's response tends to be close to the stronger of the two individual responses (instead of e.g., the sum as in a linear model). 218 M. Riesenhuber and T. Poggio Thus, the response function 0i of an intennediate neuron i to stimulation with an image v IS (1) with Ai the set of afferents to neuron i, aU) the receptive field center of afferent j, v a(j) the (square-nonnalized) image patch centered at aU) that corresponds in size to the receptive field, ~j (also square-nonnalized) of afferent j and "." the dot product operation. Studies have shown that V 4 neurons respond to features of "intennediate" complexity such as gratings, corners and crosses. 8 In V4 the receptive fields are comparatively large (4.4 0 of visual angle on average8 ), while the preferred stimuli are usually much smaller. 3 Interestingly, cells respond independently of the location of the stimulus within the receptive field. Moreover, average V4 receptive field size is comparable to the range of translation invariance of IT cells (:S ?2?) observed in the experiment. 9 For afferent receptive fields ~j, we chose features similar to the ones found for V 4 cells in the visual system: 8 bars (modeled as second derivatives of Gaussians) in two orientations, and "corners" of four different orientations and two different degrees of obtuseness. This yielded a total of lO intennediate neurons. This set of features was chosen to give a compact and biologically plausible representation. Each intennediate cell received input from cells with the same type of preferred stimulus densely covering the visual field of 256 x 256 pixels (which thus would correspond to about 4.40 of visual angle, the average receptive field size in V48 ), with receptive field sizes of afferent cells ranging from 7 to 19 pixels in steps of 2 pixels. The features used in this paper represent the first set of features tried, optimizing feature shapes might further improve the model's performance. The response tj oftop layer neuron j with connecting weights Wj to the intennediate layer was set to be a Gaussian, centered on Wj, tj = ~exp (_IIO;~jI12) 271'0'2 (2) 0' ? where is the excitation of the intennediate layer and 0' the variance of the Gaussian, which was chosen based on the distribution of responses (for section 3.1) or learned (for section 3.2). The stimulus images were views of 21 randomly generated "paperclips" of the type used in the physiology experiment. 9 Distractors were 60 other paperclip images generated by the same method. Training size was 128 x 128 pixels. 3 3.1 Results Invariance of Representation In a first set of simulations we investigated whether the proposed model could indeed account for the observed invariance properties. Here we assumed that connection strengths from the intennediate layer cells to the top layer had already been learned by a separate process, allowing us to focus on the tolerance of the representation to the above-mentioned transformations and on the selectivity of the top layer cells. To {,?,tablish the tuning properties of view-tuned model neurons, the connections Wj between the intermediate layer and top layer unit j were set to be equal to the excitation 0training in the intermediate layer caused by the training view. Figure 2 shows the "tuning curve" for rotation in depth and Fig. 3 the response to changes in stimulus size of one such neuron . The neuron shows rotation invariance (i.e., producing a higher response than to any distractor) over about 44 0 and invariance to scale changes over the whole range tested. For translation Just One View: Invariances in InJerotemporal Cell Tuning 219 1~----------------Q.J U'J oc: O. 0.5 Q.. til ~ 60 80 100 120 angle 20 40 60 distractor Figure 2: Responses of a sample top layer neuron to different views of the training stimulus and to distractors. The left plot shows the rotation tuning curve, with the training view (900 view) shown in the middle image over the plot. The neighboring images show the views of the paperclip at the borders of the rotation tuning curve, which are located where the response to the rotated clip falls below the response to the best distractor (shown in the plot on the right). The neuron exhibits broad rotation tuning over more than 40? . (not shown), the neuron showed invariance over translations of ?96 pixels around the center in any direction, corresponding to ? 1.7? of visual angle. The average invariance ranges for the 21 tested paperclips were 35? of rotation angle, 2.9 octaves of scale invariance and ? 1.8 0 of translation invariance. Comparing this to the experimentally observed 10 30 0 ,2 octaves and ?2?, resp., shows a very good agreement of the invariance properties of model and experimental neurons. 3.2 Learning In the previous section we assumed that the connections from the intermediate layer to a view-tuned neuron in the top layer were pre-set to appropriate values. In this section, we investigate whether the system allows unsupervised learning of view-tuned neurons. Since biological plausibility of the learning algorithm was not our primary focus here, we chose a general, rather abstract learning algorithm, viz. a mixture of Gaussians model trained with the EM algorithm. Our model had four neurons in the top level, the stimuli were views of four paperclips, randomly selected from the 21 paperclips used in the previous experiments. For each clip, the stimulus set contained views from 17 different viewpoints, spanning 34 0 of viewpoint change. Also, each clip was included at 11 different scales in the stimulus set, covering a range of two octaves of scale change. = Connections Wi and variances O'i, i 1, ... ,4, were initialized to random values at the beginning of training. After a few iterations of the EM algorithm (usually less than 30), a stationary state was reached, in which each model neuron had become tuned to views of one paperclip: For each paperclip, all rotated and scaled views were mapped to (i.e., activated most strongly) the same model neuron and views of different paperclips were mapped to different neurons. Hence, when the system is presented with multiple views of different objects, receptive fields of top level neurons self-organize in such a way that different neurons become tuned to different objects. M. Riesenhuber and T. Poggio 220 1r-----------------~ 0.8 0.5 0.6 0.4 100 0.20 stimulus size 200 20 40 60 distractor Figure 3: Responses of the same top layer neuron as in Fig. 2 to scale changes of the training stimulus and to distractors. The left plot shows the size tuning curve, with the training size (128 x 128 pixels) shown in the middle image over the plot. The neighboring images show scaled versions of the paperclip. Other elements as in Fig. 2. The neuron exhibits scale invariance over more than 2 octaves. 4 Discussion Object recognition is a difficult problem because objects must be recognized irrespective of position, size, viewpoint and illumination. Computational models and engineering implementations have shown that most of the required invariances can be obtained by a relatively simple learning scheme, ba<;ed on a small set of example views. 14 ,17 Quite sensibly, the visual system can also achieve some significant degree of scale and translation invariance from just one view. Our simulations show that the maximum response function is a key component in the performance ofthe model. Without it - i.e., implementing a direct convolution of the filters with the input images and a subsequent summation - invariance to rotation in depth and translation both decrease significantly. Most dramatically, however, invariance to scale changes is abolished completely, due to the strong changes in afferent cell activity with changing stimulus size. Taking the maximum over the afferents, as in our model, always picks the best matching filter and hence produces a more stable response. We expect a maximum mechanism to be essential for recognition-in-context, a more difficult task and much more common than the recognition of isolated objects studied here and in the related psychophysical and physiological experiments. The recognition of a specific paperclip object is a difficult, subordinate level classification task. It is interesting that our model sol ves it well and with a performance closely resembling the physiological data on the same task. The model is a more biologically plausible and complete model than previous ones14, 1 but it is still at the level of a plausibility proof rather than a detailed physiological model. It suggests a maximum-like response of intermediate cells as a key mechanism for explaining the properties of view-tuned IT cells, in addition to view-based representations (already described in (1,9]). Neurons in the intermediate layer currently use a very simple set of features . While this appears to be adequate for the class of paperclip objects, more complex filters might be necessary for more complex stimulus classes like faces. Consequently, future work will aim to improve the filtering step ofthe model and to test it on more real world stimuli. One can imagine a hierarchy of cell layers, similar to the "S" and "C" layers in Fukushima's Just One View: Invariances in In!erotemporal Cell Tuning 221 Neocognitron,6 in which progressively more complex features are synthesized from simple ones. The corner detectors in our model are likely candidates for such a scheme. We are currently investigating the feasibility of such a hierarchy of feature detectors. The demonstration that unsupervised learning of view-tuned neurons is possible in this representation (which is not clear for related view-based models 14 , 1) shows that different views of one object tend to form distinct clusters in the response space of intermediate neurons. The current learning algorithm, however, is not very plausible, and more realistic learning schemes have to be explored, as, for instance, in the attention-based model of Riesenhuber and Dayan 16 which incorporated a learning mechanism using bottom-up and top-down pathways. Combining the two approaches could also demonstrate how invariance over a wide range of transformations can be learned from several example views, as in the case of familiar stimuli. We also plan to simulate detailed physiological implementations of several aspects of the model such as the maximum operation (for instance comparing nonlinear dendritic interactions l l with recurrent excitation and inhibition). As it is, the model can already be tested in additional physiological experiments, for instance involving partial occlusions. References [1] Bricolo. E, Poggio, T & Logothetis, N (1997). 3D object recognition: A model of view-tuned neurons. In Advances In Neural Information Processing 9,41-47. MIT Press. [2] Biilthoff, H & Edelman, S (1992). Psychophysical support for a two-dimensional view interpolation theory of object recognition. Proc. Nat. Acad. Sci. USA 89, 60-64. [3] Desimone, R & Schein, S (1987). Visual properties of neurons in area V4 of the macaque: Sensitivity to stimulus fonn . 1. Neurophys. 57, 835-868. [4] Foldiak, P (1991). Learning invariance from transfonnation sequences. Neural Computation 3, 194-200. [5] Foster, KH, Gaska, JP, Nagler, M & Pollen, DA (1985). Spatial and temporal selectivity of neurones in visual cortical areas VI and V2 of the macaque monkey. 1. Phy. 365,331-363 . [6] Fukushima, K (1980). Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position. Biological Cybernetics 36, 193-202. [7] Ito, M, Tamura, H, Fujita, I & Tanaka, K (1995). Size and position invariance of neuronal responses in monkey inferotemporal cortex. 1. Neurophys. 73, 218-226. [8] Kobatake, E & Tanaka, K (1995). Neuronal selectivities to complex object features in the ventral visual pathway of the macaque cerebral cortex 1. Neurophys., 71, 856-867. [9] Logothetis, NK, Pauls, J & Poggio, T (1995). Shape representation in the inferior temporal cortex of monkeys. Current Biology, 5, 552-563. [10] Nikos Logothetis, personal communication. [11] Mel, BW, Ruderman, DL & Archie, KA (1997). Translation-invariant orientation tuning in visual 'complex' cells could derive from intradendritic computations. Manuscript in preparation. [12] Olshausen, BA, Anderson, CH & Van Essen, DC (1993). A neurobiological model of visual attention and invariant pattern recognition based on dynamic routing of infonnation. 1. Neurosci. 13,4700-4719. [13] Perret, D & Oram, M (1993). Neurophysiology of shape processing. Image Vision Comput. 11, 317-333. [14] Poggio, T & Edelman, S (1990). A Network that learns to recognize 3D objects. Nature 343, 263-266. [15] Reynolds, JH & Desimone, R (1997). Attention and contrast have similar effects on competitive interactions in macaque area V4. Soc. Neurosc. Abstr. 23,302. [16] Riesenhuber, M & Dayan, P (1997). Neural models for part-whole hierarchies. In Advances In Neural Information Processing 9, 17-23. MIT Press. [17] Ullman, S (1996). High-level vision: Object recognition and visual cognition. MIT Press. 

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Classification by Pairwise Coupling * Stanford University and TREVOR HASTIE ROBERT TIBSHIRANI t University of Toronto Abstract We discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes, and then coupling the estimates together. The coupling model is similar to the Bradley-Terry method for paired comparisons. We study the nature of the class probability estimates that arise, and examine the performance of the procedure in simulated datasets. The classifiers used include linear discriminants and nearest neighbors: application to support vector machines is also briefly described. 1 Introduction We consider the discrimination problem with J{ classes and N training observations. The training observations consist of predictor measurements x = (Xl, X2, ... Xp) on p predictors and the known class memberships. Our goal is to predict the class membership of an observation with predictor vector Xo Typically J{ -class classification rules tend to be easier to learn for J{ = 2 than for f{ > 2 - only one decision boundary requires attention. Friedman (1996) suggested the following approach for the the K-class problem: solve each of the two-class problems, and then for a test observation, combine all the pairwise decisions to form a J{ -class decision. Friedman's combination rule is quite intuitive: assign to the class that wins the most pairwise comparisons. Department of Statistics, Stanford University, Stanford California 94305; trevor@playfair.stanford.edu bepartment of Preventive Medicine and Biostatistics, and Department of Statistics; tibs@utstat.toronto.edu 508 T. Hastie and R. Tibshirani Friedman points out that this rule is equivalent to the Bayes rule when the class posterior probabilities Pi (at the test point) are known : argm~[Pd = argma~[LI(pd(Pi + Pj) > Pj/(Pi + Pj?] Jti Note that Friedman's rule requires only an estimate of each pairwise decision. Many (pairwise) classifiers provide not only a rule, but estimated class probabilities as well. In this paper we argue that one can improve on Friedman's procedure by combining the pairwise class probability estimates into a joint probability estimate for all J{ classes. This leads us to consider the following problem. Given a set of events AI, A 2 , ... A.K, some experts give us pairwise probabilities rij = Prob(AilA or Aj) . Is there a set of probabilities Pi = Prob(Ai) that are compatible with the 1'ij? In an exact sense, the answer is no. Since Prob( A d A i or Aj) = Pj /(Pi + pj) and 2: Pi = 1, we are requiring that J{ -1 free parameters satisfy J{ (/{ -1) /2 constraints and , this will not have a solution in general. For example, if the 1'ij are the ijth entries in the matrix . ( 0.1 0.6 0.9 0.4) . 0.7 0.3 . then they are not compatible with any pi's. This is clear since but also r31 > .5. (1) r12 > .5 and 1'23 > .5, The model Prob(A i IAi or Aj) = Pj /(Pi + pj) forms the basis for the BradleyTerry model for paired comparisons (Bradley & Terry 1952) . In this paper we fit this model by minimizing a Kullback- Leibler distance criterion to find the best approximation foij = pd('Pi + pj) to a given set of 1'il's. We carry this out at each predictor value x, and use the estimated probabilities to predict class membership at x. In the example above, the solution is p = (0.47, 0.25, 0.28). This solution makes qualitative sense since event Al "beats" A 2 by a larger margin than the winner of any of the other pairwise matches. Figure 1 shows an example of these procedures in action . There are 600 data points in three classes, each class generated from a mixture of Gaussians. A linear discriminant model was fit to each pair of classes, giving pairwise probability estimates 1'ij at each x. The first panel shows Friedman's procedure applied to the pairwise rules. The shaded regions are areas of indecision , where each class wins one vote. The coupling procedure described in the next section was then applied , giving class probability estimates f>(x) at each x. The decision boundaries resulting from these probabilities are shown in the second panel. The procedure has done a reasonable job of resolving the confusion, in this case producing decision boundaries similar to the three-class LDA boundaries shown in panel 3. The numbers in parentheses above the plots are test-error rates based on a large test sample from the same population. Notice that despite the indeterminacy, the max-wins procedure performs no worse than the coupling procedure. and both perform better than LDA. Later we show an example where the coupling procedure does substantially better than max-wms. Classification by Pairwise Coupling 509 Pairwise LDA + Max (0.132) Pairwise LOA + Coupling (0.136) 3?Class LOA (0.213) Figure 1: A three class problem, with the data in each class generated from a mixture of Gaussians. The first panel shows the maximum-win procedure. The second panel shows the decision boundary from coupling of the pairwise linear discriminant rules based on d in (6). The third panel shows the three-class LDA boundaries. Test-error rates are shown in parentheses. This paper is organized as follows. The coupling model and algorithm are given in section 2. Pairwise threshold optimization, a key advantage of the pairwise approach, is discussed in section 3. In that section we also examine the performance of the various methods on some simulated problems, using both linear discriminant and nearest neighbour rules. The final section contains some discussion . 2 Coupling the probabilities Let the probabilities at feature vector x be p(x) = (PI (x) , ... PK (x)). In this section we drop the argument x , since the calculations are done at each x separately. \Ve assume that for each i -# j, there are nij observations in the training set and from these we have estimated conditional probabilities Tij = Prob( iii or j). Our model is J..Lij Binomial( nij , J-Lij ) Pi Pi + Pj (2) or equivalently log J-Lij = log (Pi ) - log (Pi + Pj), (3) a log-nonlinear model. We wish to find Pi'S so that the Uij'S are close to the Tij'S. There are K - 1 independent parameters but K(I{ - 1)/2 equations, so it is not possible in general to find .Pi's so that {iij = Tij for all i, j. Therefore we must settle for {iij'S that are close to the observed Tij'S. Our closeness criterion is the average (weighted) Kullback-Leibler distance between Tij and J-Lij : (4) T. Hastie and R. libshirani 510 and we find p to minimize this function . This model and criterion is formally equivalent to the Bradley-Terry model for preference data. One observes a proportion fij of nij preferences for item i, and the sampling model is binomial, as in (2) . If each of the fij were independent, then R(p) would be equivalent to the log-likelihood under this model. However our fij are not independent as they share a common training set and were obtained from a common set of classifiers. Furthermore the binomial models do not apply in this case; the fij are evaluations of functions at a point, and the randomness arises in the way these functions are constructed from the training data. We include the nij as weights in (4); this is a crude way of accounting for the different precisions in the pairwise probability estimates. The score (gradient) equations are: Lnijj1ij = Lnijfij; jti subject to L Pi = 1. (5) i= 1,2 .... K j#i We use the following iterative procedure to compute the iN's: Algorithm 1. Start with some guess for the Pi, and corresponding Pij. 2. Repeat (i = 1,2, . .. , K, 1, ... ) until convergence: renormalize the Pi, and recompute the Pij. The algorithm also appears in Bradley & Terry (1952). The updates in step 2 attempt to modify p so that the sufficient statistics match their expectation, but go only part of the way. We prove in Hastie & Tibshirani (1996) that R(p) increases at each step. Since R(p) is bounded above by zero, the procedure converges. At convergence, the score equations are satisfied, and the PijS and p are consistent. This algorithm is similar in flavour to the Iterative Proportional Scaling (IPS) procedure used in log-linear models. IPS has a long history, dating back to Deming & Stephan (1940). Bishop, Fienberg & Holland (1975) give a modern treatment and many references. The resulting classification rule is (6) Figure 2 shows another example similar to Figure 1, where we can compare the performance of the rules d and d. The hatched area in the top left panel is an indeterminate region where there is more than one class achieving max(pd. In the top right panel the coupling procedure has resolved this indeterminacy in favor of class 1 by weighting the various probabilities. See the figure caption for a description of the bottom panels. Classification by Pairwise Coupling 511 PallWlse LOA + Max (0.449) Pairwise LOA + Coupling (0 358) LOA (0.457) aDA (0.334) Figure 2: A three class problem similar to that in figure 1, with the data in each class generated from a mixture of Gaussians. The first panel shows the maximumwins procedure d). The second panel shows the decision boundary from coupling of the pairwise linear discriminant rules based on d in (6). The third panel shows the three-class LDA boundaries, and the fourth the QDA boundaries. The numbers in the captions are the error rates based on a large test set from the same population. 3 Pairwise threshold optimization As pointed out by Friedman (1996), approaching the classification problem in a pairwise fashion allows one to optimize the classifier in a way that would be computationally burdensome for a J< -class classifier . Here we discuss optimization of the classification threshold. = For each two class problem, let logit Pij(X) dij(x). Normally we would classify to class i if d ij (x) > O. Suppose we find that d ij (x) > tij is better. Then we define dij (x) = d ij (x) - tij, and hence pij (x) = logiC 1 di/x). We do this for all pairs, and then apply the coupling algorithm to the P~j (x) to obtain probabilities pi( x) . In this way we can optimize over J?J< - 1)/2 parameters separately, rather than optimize jointly over J< parameters. With nearest neigbours, there are other approaches to threshold optimization, that bias the class probability estimates in different ways. See Hastie & Tibshirani (1996) for details. An example of the benefit of threshofd optimization is given next. Example: ten Gaussian classes with unequal covariance In this simulated example taken from Friedman (1996), there are 10 Gaussian classes in 20 dimensions. The mean vectors of each class were chosen as 20 independent uniform [0,1] random variables . The covariance matrices are constructed from eigenvectors whose square roots are uniformly distributed on the 20-dimensional unit sphere (subject to being mutually orthogonal) , and eigenvalues uniform on [0.01,1.01]. There are 100 observations per class in the training set, and 200 per T. Hastie and R ..1ibshirani 512 class in the test set. The optimal decision boundaries in this problem are quadratic, and neither linear nor nearest-neighbor methods are well-suited. Friedman states that the Bayes error rate is less than 1%. Figure 3 shows the test error rates for linear discriminant analysis, J -nearest neighbor and their paired versions using threshold optimization. We see that the coupled classifiers nearly halve the error rates in each case. In addition, the coupled rule works a little better than Friedman's max rule in each task. Friedman (1996) reports a median test error rate of about. 16% for his thresholded version of pairwise nearest neighbor. Why does the pairwise t.hresholding work in this example? We looked more closely at the pairwise nearest neighbour rules rules that were constructed for this problem. The thresholding biased the pairwise distances by about 7% on average. The average number of nearest neighbours used per class was 4.47 (.122), while t.he standard Jnearest neighbour approach used 6.70 (.590) neighbours for all ten classes. For all ten classes, the 4.47 translates into 44.7 neighbours. Hence relative to t.he standard J - NN rule, the pairwise rule, in using the threshold optimization to reduce bias, is able to use about six times as many near neighbours. It) C\I ci o C\I ci T, , II 1 o : c....l..." ci I ! I 1 T I , 1.' c....l..." J-nn nn/max nn/coup Ida Ida/max Ida/coup Figure 3: Test errors for 20 simulations of ten-class Gaussian example. 4 Discussion Due to lack of space, there are a number of issues that we did not discuss here. In Hastie & Tibshirani (1996), we show the relationship between the pairwise coupling and the max-wins rule: specifically, if the classifiers return 0 or Is rather than probabilities, the two rules give the same classification. We also apply the pairwise coupling procedure to nearest neighbour and support vector machines. In the latter case, this provides a natural way of extending support vector machines, which are defined for two-class problems, to multi-class problems. Classification by Pairwise Coupling 513 The pairwise procedures, both Friedman 's max-win and our coupling, are most likely to offer improvements when additional optimization or efficiency gains are possible in the simpler 2-class scenarios. In some situations they perform exactly like the multiple class classifiers. Two examples are: a) each of the pairwise rules are based on QDA: i.e. each class modelled by a Gaussian distribution with separate covariances, and then the rijS derived from Bayes rule; b) a generalization of the above, where the density in each class is modelled in some fashion, perhaps nonparametrically via density estimates or near-neighbor methods, and then the density estimates are used in Bayes rule. Pairwise LDA followed by coupling seems to offer a nice compromise between LDA and QDA, although the decision boundaries are no longer linear. For this special case one might derive a different coupling procedure globally on the logit scale , which would guarantee linear decision boundaries. Work of this nature is currently in progress with Jerry Friedman. Acknowledgments We thank Jerry Friedman for sharing a preprint of his pairwise classification paper with us, and acknowledge helpful discussions with Jerry, Geoff Hinton, Radford Neal and David Tritchler. Trevor Hastie was partially supported by grant DMS-9504495 from the National Science Foundation, and grant ROI-CA-72028-01 from the National Institutes of Health. Rob Tibshirani was supported by the Natural Sciences and Engineering Research Council of Canada and the iRIS Centr of Excellence. References Bishop, Y., Fienberg, S. & Holland, P. (1975), Discrete multivariate analysis, MIT Press, Cambridge. Bradley, R. & Terry, M. (1952), 'The rank analysis of incomplete block designs . i. the method of paired comparisons', Biometrics pp . 324-345 . Deming, W. & Stephan, F . (1940), 'On a least squares adjustment of a sampled frequency table when the expected marginal totals are known', A nn. Math . Statist. pp. 427-444. Friedman, J . (1996), Another approach to polychotomous classification, Technical report, Stanford University. Hastie, T. & Tibshirani, R. (1996), Classification by pairwise coupling, Technical report, University of Toronto. 

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Agnostic Classification of Markovian Sequences Ran EI-Yaniv Shai Fine Naftali Tishby* Institute of Computer Science and Center for Neural Computation The Hebrew University Jerusalem 91904, Israel E-Dlail: {ranni,fshai,tishby}Ocs.huji.ac.il Category: Algorithms. Abstract Classification of finite sequences without explicit knowledge of their statistical nature is a fundamental problem with many important applications. We propose a new information theoretic approach to this problem which is based on the following ingredients: (i) sequences are similar when they are likely to be generated by the same source; (ii) cross entropies can be estimated via "universal compression"; (iii) Markovian sequences can be asymptotically-optimally merged. With these ingredients we design a method for the classification of discrete sequences whenever they can be compressed. We introduce the method and illustrate its application for hierarchical clustering of languages and for estimating similarities of protein sequences. 1 Introd uction While the relationship between compression (minimal description) and supervised learning is by now well established, no such connection is generally accepted for the unsupervised case. Unsupervised classification is still largely based on ad-hock distance measures with often no explicit statistical justification. This is particularly true for unsupervised classification of sequences of discrete symbols which is encountered in numerous important applications in machine learning and data mining, such as text categorization, biological sequence modeling, and analysis of spike trains. The emergence of "universal" (Le. asymptotically distribution independent) se?Corresponding author. R. EI-Yaniv, S. FineandN. TlShby 466 quence compression techniques suggests the existence of "universal" classification methods that make minimal assumptions about the statistical nature of the data. Such techniques are potentially more robust and appropriate for real world applications. In this paper we introduce a specific method that utilizes the connection between universal compression and unsupervised classification of sequences. Our only underlying assumption is that the sequences can be approximated (in the information theoretic sense) by some finite order Markov sources. There are three ingredients to our approach. The first is the assertion that two sequences are statistically similar if they are likely to be independently generated by the same source. This likelihood can then be estimated, given a typical sequence of the most likely joint source, using any good compression method for the sequence samples. The third ingredient is a novel and simple randomized sequence merging algorithm which provably generates a typical sequence of the most likely joint source of the sequences, under the above Markovian approximation assumption. Our similarity measure is also motivated by the known "two sample problem" [Leh59] of estimating the probability that two given samples are taken from the same distribution. In the LLd. (Bernoulli) case this problem was thoroughly investigated and the. optimal statistical test is given by the sum of the empirical cross entropies between the two samples and their most likely joint -source. We argue that this measure can be extended for arbitrary order Markov sources and use it to construct and sample the most likely joint source. The similarity measure and the statistical merging algorithm can be naturally combined into classification algorithms for sequences. Here we apply the method to hierarchical clustering of short text segments in 18 European languages and to evaluation of similarities of protein sequences. A complete analysis of the method, with further applications, will be presented elsewhere [EFT97]. 2 Measuring the statistical similarity of sequences Estimating the statistical similarity of two individual sequences is traditionally done by training a statistical model for each sequence and then measuring the likelihood of the other sequence by the model. Training a model entails an assumption about the nature of the noise in the data and this is the rational behind most "edit distance" measures, even when the noise model is not explicitly stated. Estimating the log-likelihood of a sequence-sample over a discrete alphabet E by a statistical model can be done through the Cross Entropy or Kullback-Leibler Divergence[CT91] between the sample empirical distribution p and model distribution q, defined as: DKL (pllq) = L P (0-) log P((:? q . (1) uEE The KL-divergence, however, has some serious practical drawbacks. It is nonsymmetric and unbounded unless the model distribution q is absolutely continuous with respect to p (Le. q = 0 ::::} P = 0). The KL-divergence is therefore highly sensitive to low probability events under q. Using the "empirical" (sample) distributions for both p and q can result in very unreliable estimates of the true divergences. Essentially, D K L [Pllq] measures the asymptotic coding inefficiency when coding the sample p with an optimal code for the model distribution q. The symmetric divergence, i.e. D (p, q) = DKL [Pllq] + DKL [qllp], suffers from Agnostic Classification of Markovian Sequences 467 similar sensitivity problems and lacks the clear statistical meaning. 2.1 The "two sample problem" Direct Bayesian arguments, or alternately the method of types [CK81], suggest that the probability that there exists one source distribution M for two independently drawn samples, x and y [Leh59], is proportional to ! dJ-L (M) Pr (xIM) . Pr (yIM) = ! dJ-L (M) . 2-(lzIDKdp",IIM1+lyIDKL[P1l1IM]), (2) where dJ-L(M) is a prior density of all candidate distributions, pz and Py are the empirical (sample) distributions, and Ixl and Iyl are the corresponding sample sizes. For large enough samples this integral is dominated (for any non-vanishing prior) by the maximal exponent in the integrand, or by the most likely joint source of x and y, M>.., defined as M>.. = argmin {lxIDKL (PzIIM') + IYIDKL (pyIlM')}. (3) M' where 0 ~ A = Ixl/(lxl + Iyl) ~ 1 is the sample mixture ratio. The convexity of the KL-divergence guarantees that this minimum is unique and is given by M>.. = APz + (1 - A) PY' the A - mixture of pz and py. The similarity measure between two samples, d(x, y), naturally follows as the minimal value of the above exponent. That is, Definition 1 The similarity measure, d(x, y) = V>..(Pz,Py), of two samples x and y, with empirical distributions pz and Py respectively, is defined as d(x, y) = V>..(Pz,Py) = ADKL (PzIIM>..) + (1- A) DKL (pyIIM>..) (4) where M>.. is the A-mixture of pz and Py. The function V>.. (p, q) is an extension of the Jensen-Shannon divergence (see e.g. [Lin91]) and satisfies many useful analytic properties, such as symmetry and boundedness on both sides by the L1-norm, in addition to its clear statistical meaning. See [Lin91, EFT97] for a more complete discussion of this measure. 2.2 Estimating the V>.. similarity measure The key component of our classification method is the estimation of V>.. for individual finite sequences, without an explicit model distribution. Since cross entropies, D K L, express code-length differences, they can be estimated using any efficient compression algorithm for the two sequences. The existence of "universal" compression methods, such as the Lempel-Ziv algorithm (see e.g. [CT91]) which are provably asymptotically optimal for any sequence, give us the means for asymptotically optimal estimation of V>.., provided that we can obtain a typical sequence of the most-likely joint source, M >... We apply an improvement of the method of Ziv and Merhav [ZM93] for the estimation of the two cross-entropies using the Lempel-Ziv algorithm given two sample sequences [BE97]. Notice that our estimation of V>.. is as good as the compression method used, namely, closer to optimal compression yields better estimation of the similarity measure. It remains to show how a typical sequence of the most-likely joint source can be generated. 468 3 R. El-Yaniv, S. Fine and N. Tishby Joint Sources of Markovian Sequences In this section we first explicitly generalize the notion of the joint statistical source to finite order Markov probability measures. We identify the joint source of Markovian sequences and show how to construct a typical random sample of this source. More precisely, let x and y be two sequences generated by Markov processes with distributions P and Q, respectively. We present a novel algorithm for the merging the two sequences, by generating a typical sequence of an approximation to the most likely joint source of x and y. The algorithm does not require the parameters of the true sources P and Q and the computation of the sequence is done directly from the sequence samples x and y. As before, r; denotes a finite alphabet and P and Q denote two ergodic Markov sources over r; of orders Kp and KQ, respectively. By equation 3, the :A-mixture joint source M>.. of P and Q is M>.. = argminM' :ADKdPIIM')+(I-:A)DKdQIIM') , where for sequences DKdPIIM) = limsuPn-too ~ L:zE!:n P(x) log The following theorem identifies the joint source of P and Q. :1:))' Theorem 1 The unique :A-mixture joint source M>.. of P and Q, of order K = max {K p, K Q}, is given by the following conditional distribution. For each s E r;K,aEE, (1 - :A)Q(s) :AP(s) M>..(als) = :AP(s) + (1 _ :A)Q(s) P(als) + :AP(s) + (1- :A)Q(s) Q(als) . This distribution can be naturally extended to n sources with priors :At, ... ,:An. 3.1 The "sequence merging" algorithm The above theorem can be easily translated into an algorithm. Figure 1 describes a randomized algorithm that generates from the given sequences x and y, an asymptotically typical sequence z of the most likely joint source, as defined by Theorem 1, of P and Q. Initialization: ? z [OJ = choose a symbol from x with probability ,x or y with probability 1 - ,x ? i = 0 Loop: Repeat until the approximation error is lower then a prescribe threshold ? s", := max length suffix of z appearing somewhere in x ? Sy := max length suffix of ? A(,x , S "" Sy } - z appearing somewhere in y >.Pr .. (s .. ) >.Pr .. (s.,)+(l->.) Pr ll(s\I) ? r = choose x with probability A(,x, s"" Sy} or y with probability 1-A(,x, S"" S1/} ? r (Sr) = randomly choose one of the occurrences of Sr in r ? z [i + 1) = the symbol appearing immediately after r (Sr) at r ? i=i+1 End Repeat Figure 1: The most-likely joint source algorithm Agnostic Classification of Markovian Sequences 469 Notice that the algorithm is completely unparameterized, even the sequence alphabets, which may differ from one sequence to another, are not explicitly needed. The algorithm can be efficiently implemented by pre-preparing suffix trees for the given sequences, and the merging algorithm is naturally generalizable to any number of sequences. 4 Applications There are several possible applications of our sequence merging algorithm and similarity measure. Here we focus on three possible applications: the source merging problem, estimation of sequence similarity, and bottom-up sequence-classification. These algorithms are different from most existing approaches because they rely only on the sequenced data, similar to universal compression, without explicit modeling assumptions. Further details, analysis, and applications of the method will be presented elsewhere [EFT97]. 4.1 Merging and synthesis of sequences An immediate application of the source merging algorithm is for synthesis of typical sequences of the joint source from some given data sequences, without any access to an explicit model of the source. To illustrate this point consider the sequence in Figure 2. This sequence was randomly generated, character by character, from two natural excerpts: a 47,655character string from Dickens' Tale of Two Cities, and a 59,097-character string from Twain's The King and the Pauper. Do your way to her breast, and sent a treason's sword- and not empty. "I am particularly and when the stepped of his ovn commits place. No; yes, of course, and he passed behind that by turns ascended upon him, and my bone to touch it, less to say: 'Remove thought, everyone! Guards! In miness?" The books third time. There was but pastened her unave misg his ruined head than they had knovn to keep his saw whether think" The feet our grace he called offer information? [Twickens, 1997] Figure 2: A typical excerpt of random text generated by the "joint source" of Dickens and Twain. 4.2 Pairwise similarity of proteins The joint source algorithm, combined with the new similarity measure, provide natural means for computing the similarity of sequences over any alphabet. In this section we illustrate this application l for the important case of protein sequences (sequences over the set of the 20 amino-acids). From a database of all known proteins we selected 6 different families and within each family we randomly chose 10 proteins. The families chosen are: Chaperonin, MHC1, Cytochrome, Kinase, Globin Alpha and Globin Beta. Our pairwise distances between all 60 proteins were computed using our agnostic algorithm and are depicted in the 6Ox60 matrix of Figure 3. As can be seen, the algorithm succeeds to IThe protein results presented here are part of an ongoing work with G. Yona and E . Ben-Sasson. R. El-Yaniv, S. Fine and N. TlShby 470 identify the families (the success with the Kinase and Cytochrome families is more limited). PairwIse Distances of Protein Sequences chaperonin MHC I cytochrome kinase globin a globin b Figure 3: A 60x60 symmetric matrix representing the pairwise distances, as computed by our agnostic algorithm, between 60 proteins, each consecutive 10 belong to a different family. Darker gray represent higher similarity. In another experiment we considered all the 200 proteins of the Kinase family and computed the pairwise distances of these proteins using the agnostic algorithm. For comparison we computed the pairwise similarities of these sequences using the widely used Smith-Waterman algorithm (see e.g. [HH92]).2 The resulting agnostic similarities, computed with no biological information whatsoever, are very similar to the Smith-Waterman similarities. 3 Furthermore, our agnostic measure discovered some biological similarities not detected by the Smith-Waterman method. 4.3 Agnostic classification of languages The sample of the joint source of two sequences can be considered as their "average" or "centroid", capturing a mixture of their statistics. Averaging and measuring distance between objects are sufficient for most standard clustering algorithms such as bottom-up greedy clustering, vector quantization (VQ), and clustering by deterministic annealing. Thus, our merging method and similarity measure can be directly applied for the classification of finite sequences via standard clustering algorithms. To illustrate the power of this new sequence clustering method we give the result of a rudimentary linguistic experiment using a greedy bottom-up (conglomerative) clustering of short excerpts (1500 characters) from eighteen languages. Specifically, we took sixteen random excerpts from the following Porto-Indo-European languages: Afrikaans, Catalan, Danish, Dutch, English, Flemish, French, German, Italian, Latin, Norwegian, Polish, Portuguese, Spanish, Swedish and Welsh, together with 2we applied the Smith-Waterman for computing local-alignment costs using the stateof-the-art blosum62 biological cost matrix. 3These results are not given here due to space limitations and will be discussed elsewhere. Agnostic Classification of Markovian Sequences 471 two artificial languages: Esperanto and Klingon4. The resulting hierarchical classification tree is depicted in Figure 4. This entirely unsupervised method, when applied to these short random excerpts, clearly agrees with the "standard" philologic tree of these languages, both in terms of the grouping and the levels of similarity (depth of the split) of the languages (the Polish-Welsh "similarity" is probably due to the specific transcription used). Figure 4: Agnostic bottom-up greedy clustering of eighteen languages Acknowledgments We sincerely thank Ran Bachrach and Golan Yona for helpful discussions. We also thank Sageev Oore for many useful comments. References [BE97] R. Bachrach and R. EI-Yaniv, An Improved Measure of Relative Entropy Between Individual Sequences, unpublished manuscript. [CK81] 1. Csiszar and J . Krorner. Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press, New-York 1981. [CT91] T. M. Cover and J. A. Thomas. Elements of Information Theory, John Wiley & Sons, New-York 1991. [EFT97] R. EI-Yaniv, S. Fine and N. Tishby. Classifying Markovian Sources, in preparations, 1997. [HH92] S. Henikoff and J . G. Henikoff (1992) . Amino acid substitution matrices from protein blocks. Proc. Natl. Acad. Sci. USA 89, 10915-10919. [Leh59] E. L. Lehmann. Testing Statistical Hypotheses, John Wiley & Sons, NewYork 1959. [Lin91] J. Lin, 1991. Divergence measures based on the Shannon entropy. IEEE Transactions on In/ormation Theory, 37(1):145-15l. [ZM93] J . Ziv and N. Merhav, 1993. A Measure of Relative Entropy Between Individual Sequences with Application to Universal Classification, IEEE Transactions on In/ormation Theory, 39(4). 4Klingon is a synthetic language that was invented for the Star-Trek TV series. 

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Bidirectional Retrieval from Associative Memory Friedrich T. Sommer and Gunther Palm Department of Neural Information Processing University of Ulm, 89069 Ulm, Germany {sommer,palm}~informatik.uni-ulm.de Abstract Similarity based fault tolerant retrieval in neural associative memories (N AM) has not lead to wiedespread applications. A drawback of the efficient Willshaw model for sparse patterns [Ste61, WBLH69], is that the high asymptotic information capacity is of little practical use because of high cross talk noise arising in the retrieval for finite sizes. Here a new bidirectional iterative retrieval method for the Willshaw model is presented, called crosswise bidirectional (CB) retrieval, providing enhanced performance. We discuss its asymptotic capacity limit, analyze the first step, and compare it in experiments with the Willshaw model. Applying the very efficient CB memory model either in information retrieval systems or as a functional model for reciprocal cortico-cortical pathways requires more than robustness against random noise in the input: Our experiments show also the segmentation ability of CB-retrieval with addresses containing the superposition of pattens, provided even at high memory load. 1 INTRODUCTION From a technical point of view neural associative memories (N AM) provide data storage and retrieval. Neural models naturally imply parallel implementation of storage and retrieval algorithms by the correspondence to synaptic modification and neural activation. With distributed coding of the data the recall in N AM models is fault tolerant: It is robust against noise or superposition in the addresses and against local damage in the synaptic weight matrix. As biological models N AM F. T. Sommer and G. Palm 676 have been proposed as general working schemes of networks of pyramidal cells in many places of the cortex. An important property of a NAM model is its information capacity, measuring how efficient the synaptic weights are used. In the early sixties Steinbuch realized under the name "Lernmatrix" a memory model with binary synapses which is now known as Wills haw model [Ste6I, WBLH69]. The great variety of NAM models proposed since then, many triggered by Hopfield's work [Hop82], do not reach the high asymptotic information capacity of the Willshaw model. For finite network size, the Willshaw model does not optimally retrieve the stored information, since the inner product between matrix colum and input pattern determines the activity for each output neuron independently. For autoassociative pattern completion iterative retrieval can reduce cross talk noise [GM76, GR92, PS92, SSP96]. A simple bidirectional iteration - as in bidirectional associative memory (BAM) [Kos87] - can, however, not improve heteroassociative pattern mapping. For this task we propose CB-retrieval where each retrieval step forms the resulting activity pattern in an autoassociative process that uses the connectivity matrix twice before thresholding, thereby exploiting the stored information more efficiently. 2 WILLSHAW MODEL AND CB EXTENSION Here pattern mapping tasks XV -+ yV are considered for a set of memory patterns: {(XV,yV): XV E {O,I}n,yv E {o,I}m,v = I, ... ,M}. The number of I-components in a pattern is called pattern activity. The Willshaw model works efficiently, if the memories are sparse, i.e., if the memory patterns have the same activities: Ixvi = 2:~=I xi = a,lyvl = 2::1 Yi b V v with a ? nand b ?m. During learning the set of memory patterns is transformed to the weight matrix by = Cij = min(I, L xiv}) = supxiy'j? V V For a given initial pattern XJ1. the retrieval yields the output pattern yJ1. by forming in each neuron the dendritic sum [C xJ1.]j = 2:i Ci/if and by calculating the activity value by threshold comparison yj = H([C xJ1.j j - 9) Vj, (1) with the global threshold value 9 and H(x) denoting the Heaviside function. For finite sizes and with high memory load, Le., 0? PI := Prob[Cij = 1] ? 0.5), the Willshaw model provides no tolerance with respect to errors in the address, see Fig. 1 and 2. A bidirectional iteration of standard simple retrieval (1), as proposed in BAM models [Kos87], can therefore be ruled out for further retrieval error reduction [SP97j. In the energy function of the Willshaw BAM E(x,y) =- LCijXiYj ij + 8' LXi + 8 i LYj j we now indroduce a factor accounting for the magnitudes of dendritic potentials at acti vated neurons (2) Bidirectional Retrieval from Associative Memory 677 Differentiating the energy function (2) yields the gradient descent equations yr W = + L 'LCi j CikXi Yk H( [CxU k - 8 ) (3) i "'-v---' =:Wjk X~ew H( [CT y); + = L "LPiiCljYi Xl I 8' ) (4) i --------=:wfr As new terms in (3) and (4) sums over pattern components weighted with the quantities wjk and wft occur. wjk is the overlap between the matrix columns j and k conditioned by the pattern X, which we call a conditioned link between y-units. Restriction on the conditioned link terms yields a new iterative retrieval scheme which we denote as crosswise bidirectional (eB) retrieval y(r+ I)i = 'L Cij[CT y(r-I))i - H( 8) (5) L Cij[Cx(r-I))j - 8') (6) iEx(r) H( iEy(r) For r = 0 pattern y(r:-I) has to be replaced by H([Cx(O)] - 0), for r > 2 Boolean ANDing with results from timestep r - 1 can be applied which has been shown to improve iterative retrieval in the Willshaw model for autoassociation [SSP96]. 3 MODEL EVALUATION Two possible retrieval error types can be distinguished: a "miss" error converts a I-entry in Y~ to '0' and a "add" error does the opposite. 35 30 25 20 15 10 5 0 ]. "2. ,." 2. ]. 40 simple r. add error ... . . C8-r. add error CB-r. miss error ..... 5 10 15 20 25 30 Figure 1: Mean retrieval error rates for n = 2000, M = 15000, a = b = 10 corresponding to a memory load of H = 0.3. The x-axes display the address activity: lilLl = 10 corresponds to a errorfree learning pattern, lower activities are due to miss errors, higher activities due to add errors. Left: Theory - Add errors for simple retrieval, eq. (7) (upper curve) and lower bound for the first step of CB-retrieval, eq. (9). Right: Simulations - Errors for simple and CB retrieval. The analysis of simple retrieval from the address i~ yields with optimal threshold setting 0 = k the add error rate, i.e, the expectation of spurious ones: & = (m - b)Prob [r ~ k] , (7) F. T. Sommer and G. Palm 678 with the binomial random variable Prob[r=l] = B(Lit'I,Pt}I, where B(n,p), := (7)pl(1 - p)n-l. a denotes the add error rate and k = lit'l - a the number of correct 1-s in the address. For the first step of CB-retrieval a lower bound of the add error rate a(l) can be derived by the analysis of CB-retrieval with fixed address x(O) = iIJ. and the perfect learning pattern ylJ. as starting patterns in the y-Iayer. In this case the add error rate is: (8) where the random variables rl and r2 have the distributions: Prob [rl = lib] = B(k, PI), and Prob [r2 = 1] = B(ab, (PI )2) l" Thus, a(l) ~ (m - b) k L B(k, PdsBS [ab, (PI )2, (k - s)b) , (9) 8=0 where BS [n,p, t] := L:~t B(n,p), is the binomial sum. In Fig. 1 the analytic results for the first step (7) and (9) can be compared with simulations (left versus right diagram) . In the experiments simple retrieval is performed with threshold () = k. CB-retrieval is iterated in the y-Iayer (with fixed address x) starting with three randomly chosen 1-s from the simple retrieval result yt'. The iteration is stopped, if a stable pattern at threshold e = bk is reached. The memory capacity can be calculated per pattern component under the assumption that in the memory patterns each component is independent, i.e., the probabilities for a 1 are p = a/n or q = b/m respectively, and the probabilities of an add and a miss error are simply the renormalized rates denoted by a', {3' and a', {3' for x-patterns and by,', 6' for y-patterns. The information about the stored pattern contained in noisy initial or retrieved patterns is then given by the transinformation t(p,a',{3') := i(p) -i(p,a',{3'), where i(p) is the Shannon information, and i (p, a', {3') the conditional information. The heteroassociative mapping is evaluated by the output capacity: A(a', {3') := Mm t(q, ,', 6')/mn (in units bit/synapse). It depends on the initial noise since the performance drops with growing initial errors and assumes the maximum, if no fault tolerance is provided, that is, with noiseless initial patterns, see Fig. 2. Autoassociative completion of a distorted x-pattern is evaluated by the completion capacity: C(a', {3') := Mn(t(p, a', {3')-t(p, a', {3'))/mn. A BAM maps and completes at the same time and should be therefore evaluated by the search capacity S := C + A. The asymptotic capacity of the Willshaw model is strikingly high: The completion capacity (for autoassociation) is C+ = In[2] /4, the mapping capacity (for heteroassociation with input noise) is A+ = In[2] /2 bit/syn [Pal91]' leading to a value for the search capacity of (3 In[2])/4 = 0.52 bit/syn. To estimate S for general retrieval procedures one can consider a recognition process of stored patterns in the whole space of sparse initial patterns; an initial pattern is "recognized", if it is invariant under a bidirectional retrieval cycle. The so-called recognition capacity of this process is an upper bound of the completion capacity and it had been determined as In [2J/2, see [PS92]. This is achieved again with parameters M, p, q providing A = In[2] /2 yielding In[2] bit/syn as upper bound of the asymptotic search capacity. In summary, we know about the asymptotic search capacity of the CB-model: 0.52 ::; S+ ::; 0.69 bit/syn. For experimental results, see Fig. 4. Bidirectional Retrieval from Associative Memory 4 679 EXPERIMENTAL RESULTS The CB model has been tested in simulations and compared with the Willshaw model (simple retrieval) for addresses with random noise (Fig. 2) and for addresses composed by two learning patterns (Fig. 3). In Fig. 2 the widely enlarged range of high qualtity retrieval in the CB-model is demonstrated for different system sizes. output miss errors 6 5 4 3 2 1 0 10 simple r. .. ". : CB?r. - .. : " 6 4 .' , .. "... 10 15 20 25 30 .' , .. ~ '" ....... ::' 2 5 7 6 5 4 3 2 1 0 8 0 5 10 15 20 25 30 7 6 5 4 3 2 101214161820 0 101214161820 output add errors 6 5 4 3 r--..--.--,--.----.---, 14 12 simple r. " ." CB?r.- 10 8 /" 2 6 4 2 0 ',' 1 o ~'--'--'--'--1----'-""'-'-----' 5 rcr-..---,----.~----, 10 15 20 25 30 10 10 8 8 6 6 .,.,.; , ?.-:: .... os?,? .:'1 ...... ? /", L-.C..:....I..!.-...I.-_~=:::::::..J 5 4 4 2 0 2 0 1015202530 101214161820 L..:....L-I.........."--I 101214161820 transinformation in output pattern (bit) 50 ;c;::r;:::::r:::=r::=7---"J 100 45 L ~ 100 r-r--,-,--,--,1 00 r-r--.-.-T'"""1 ~ 35 30 25 ...... . ~ 15 10 5 s imple r. ..". ?CB-r. - ". 20 o '---''---'----1---'----'---' 5 10 15 20 25 30 ~ ~ 60 60 ~ 40 20 20 o 5 10 15 20 25 30 Fig. 2: Retrieval from addresses with random noise. The x-axis labeling is as in Fig. 1. Small system with n = 100, M = 35 (left), system size as in Fig. 1, two trials (right). Output activities adjusted near Iyl = k by threshold setting. 101214161820 0 L-.I.---L.--1-.I..--J 101214161820 Fig. 3: Retrieval from addresses composed by two learning patterns. Parameters as in right column of Fig. 2, explanation of left and right column, see text. In Fig. 3 the address contains one learning pattern and I-components of a second learning pattern successively added with increasing abscissae. On the right end of each diagram both patterns are completely superimposed. Diagrams in the left column show errors and transinformation, if retrieval results are compared with the learning pattern which is for li~ I < 20 dominantly addressed. Simple retrieval errors behave similiar as for random noise in the address (Fig. 2) while the error level of CB-retrieval raises faster if more than 7 adds from the second pattern are present. Diagrams in the right column show the same quantities, if the retrieval result is compared with the closest of the two learning patterns. It can be observed i) that a learning pattern is retrieved even if the address is a complete superposition and ii) if the second pattern is almost complete in the address the retrieved pattern corresponds in some cases to the second pattern. However, in all cases CBretrieval yields one of the learning pattern pairs and it could be used to generate a good address for further retrieval of the other by deletion of the corresponding I-components in the original address. 680 F. T. Sommer and G. Palm 0.48 0.46 0.44 0.42 0.4 output c. ..... searchc. - 0.38 . 8 ..... 10 12 14 16 18 Fig. 4: Output and search capacity of CB retrieval in bit/syn with x-axis labeling as in Fig. 2 for n = m = 2000, a = b = 10 M = 20000. The difference between both curves is the contribution due to x-pattern completion, the completion capacity C. It is zero for Ix(O}1 = 10, if the initial pattern is errorfree. The search capacity of the CB model in Fig. 4 is close to the theoretical expectations from Sect. 3, increasing with input noise due to the address completion. 5 SPARSE CODING To apply the proposed N AM model, for instance, in information retrieval, a coding of the data to be accessed into sparse binary patterns is required. A useful extraction of sparse features should take account of statistical data properties and the way the user is acting on them. There is evidence from cognitive psychology that such a coding is typically quite easy to find. The feature encoding, where a person is extracting feature sets to characterize complex situations by a few present features, is one of the three basic classes of cognitive processes defined by Sternberg [Ste77]. Similarities in the data are represented by feature patterns having a large number of present features in common, that is a high overlap: o(x, x'} := L:i XiX'i' For text retrieval word fragments used in existing indexing techniques can be directly taken as sparse binary features [Geb87]. For image processing sparse coding strategies [Zet90], and neural models for sparse feature extraction by anti-Hebbian learning [F6l90] have been proposed. Sparse patterns extracted from different data channels in heterogeneous data can simply be concatenated and processed simultaneously in N AM. If parts of the original data should be held in a conventional memory, also these addresses have to be represented by distributed and sparse patterns in order to exploit the high performance of the proposed NAM. 6 CONCLUSION A new bidirectional retrieval method (CB-retrieval) has been presented for the Willshaw neural associative memory model. Our analysis of the first CB-retrieval step indicates a high potential for error reduction and increased input fault tolerance. The asymptotic capacity for bidirectional retrieval in the binary Willshaw matrix has been determined between 0.52 and 0.69 bit/syn. In experiments CB-retrieval showed significantly increased input fault tolerance with respect to the standard model leading to a practical information capacity in the order of the theoretical expectations (0.5 bit/syn). Also the segmentation ability of CB-retrieval with ambiguous addresses has been shown. Even at high memory load such input patterns can be decomposed and corresponding memory entries returned individually. The model improvement does not require sophisticated individual threshold setting [GW95], strategies proposed for BAM like more complex learning procedures, or "dummy augmentation" in the pattern coding [WCM90, LCL95]. The demonstrated performance of the CB-model encourages applications as massively parallel search strategies in Information Retrieval. The sparse coding requirement has been briefly discussed regarding technical strategies and psychological plausibility. Biologically plausible variants of CB-retrieval contribute to more Bidirectional Retrieval from Associative Memory 681 refined cell assembly theories, see [SWP98]. Acknowledgement: One of the authors (F.T.S.) was supported by grant S0352/3-1 of the Deutsche Forschungsgemeinschaft. References [F6190] P. F6ldiak. Forming sparse representations by local anti-hebbian learning. Biol. Cybern., 64:165-170, 1990. [Geb87] F. Gebhardt. Text signatures by superimposed coding of letter triplets and quadruplets. Information Systems, 12(2):151-156, 1987. [GM76] A.R. Gardner-Medwin. The recall of events through the learning of associations between their parts. Proceedings of the Royal Society of London B, 194:375-402, 1976. [GR92] W.G. Gibson and J. Robinson. Statistical analysis of the dynamics of a sparse associative memory. Neural Networks, 5:645-662, 1992. [GW95] B. Graham and D. Willshaw. Improving recall from an associative memory. Biological Cybernetics, 72:337-346, 1995. [Hop82] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, USA, 79, 1982. [Kos87] B. Kosko. Adaptive bidirectional associative memories. 26(23):4947-4971, 1987. [LCL95] C.-S. Leung, L.-W. Chan, and E. Lai. Stability, capacity and statistical dynamics of second-order bidirectional associative memory. IEEE Trans. Syst, Man Cybern., 25(10):1414-1424, 1995. [Pal91] G. Palm. Memory Capacities of Local Rules for Synaptic Modification. Concepts in Neuroscience, 2:97-128, 1991. [PS92] G. Palm and F. T. Sommer. Information capacity in recurrent McCulloch-Pitts networks with sparsely coded memory states. Network, 3:1-10, 1992. [SP97] F. T. Sommer and G. Palm. Improved bidirectional retrieval of sparse patterns stored by Hebbian learning. Submitted to Neural Networks, 1997. [SSP96] F. Schwenker, F. T. Sommer, and G. Palm. Iterative retrieval of sparsely coded associative memory patterns. Neural Networks, 9(3) :445 - 455, 1996. [Ste61] K. Steinbuch. Die Lernmatrix. Kybernetik, 1:36-45, 1961. [Ste77] R. J. Sternberg. Intelligence, information processing and analogical reasoning. Hillsdale, NJ, 1977. [SWP98] F. T. Sommer, T. Wennekers, and G. Palm. Bidirectional completion of Cell Assemblies in the cortex. In Computational Neuroscience: Trends in Research. Plenum Press, 1998. Applied Optics, [WBLH69] D. J. Willshaw, O. P. Buneman, and H. C. Longuet-Higgins. Nonholographic associative memory. Nature, 222:960-962, 1969. [WCM90] Y. F. Wang, J. B. Cruz, and J. H. Mulligan. Two coding stragegies for bidirectional associative memory. IEEE Trans. Neural Networks, 1(1):81-92, 1990. [Zet90] C. Zetsche. Sparse coding: the link between low level vision and associative memory. In R. Eckmiller, G. Hartmann, and G. Hauske, editors, Parallel Processing in Neural Systems a.nd Computers. Elsevier Science Publishers B. V. (North Holland), 1990. 

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Using Expectation to Guide Processing: A Study of Three Real-World Applications Shumeet 8aluja Justsystem Pittsburgh Research Center & School of Computer Science, Carnegie Mellon University baluja@cs.cmu.edu Abstract In many real world tasks, only a small fraction of the available inputs are important at any particular time. This paper presents a method for ascertaining the relevance of inputs by exploiting temporal coherence and predictability. The method proposed in this paper dynamically allocates relevance to inputs by using expectations of their future values. As a model of the task is learned, the model is simultaneously extended to create task-specific predictions of the future values of inputs. Inputs which are either not relevant, and therefore not accounted for in the model, or those which contain noise, will not be predicted accurately. These inputs can be de-emphasized, and, in turn, a new, improved, model of the task created. The techniques presented in this paper have yielded significant improvements for the vision-based autonomous control of a land vehicle, vision-based hand tracking in cluttered scenes, and the detection of faults in the etching of semiconductor wafers. 1 Introduction In many real-world tasks, the extraneous information in the input can be easily confused with the important features, making the specific task much more difficult. One of the methods in which humans function in the presence of many distracting features is to selectively attend to only portions of the input signal. A means by which humans select where to focus their attention is through the use of expectations. Once the important features in the current input are found, an expectation can be formed of what the important features in the next inputs will be, as well as where they will be. The importance of features must be determined in the context of a specific task; different tasks can require the processing of different subsets of the features in the same input. There are two distinct uses of expectations. Consider Carnegie Mellon's Navlab autonomous navigation system. The road-following module [Pomerleau, 1993] is separate from the obstacle avoidance modules [Thorpe, 1991]. One role of expectation, in which unexpected features are de-emphasized, is appropriate for the road-following module in which the features to be tracked, such as lane-markings, appear in predictable locations. This use of expectation removes distractions from the input scene. The second role of expectation, to emphasize unexpected features, is appropriate for the obstacle avoidance modules. This use of expectation emphasizes unanticipated features of the input scene. 2 Architectures for Attention In many studies of attention, saliency maps (maps which indicate input relevance) have been constructed in a bottom-up manner. For example, in [Koch & Ullman, 1985], a s. Baluja 860 saliency map, which is not task-specific, is created by emphasizing inputs which are different from their neighbors. An alternate approach, presented in [Clark & Ferrier, 1992], places mUltiple different, weighted, task-specific feature detectors around the input image. The regions of the image which contain high weighted sums of the detected features are the portion of the scene which are focused upon. Top-down knowledge of which features are used and the weightings of the features is needed to make the procedure task-specific. In contrast, the goal of this study is to learn which task-specific features are relevant without requiring top-down knowledge. In this study, we use a method based on Input Reconstruction Reliability Estimation (IRRE) [Pomerleau, 1993] to detennine which portions of the input are important for the task. IRRE uses the hidden units of a neural network (NN) to perfonn the desired task and to reconstruct the inputs. In its original use, IRRE estimated how confident a network's outputs were by measuring the similarity between the reconstructed and current inputs. Figure 1(Left) provides a schematic ofIRRE. Note that the weights between the input and hidden layers are trained to reduce both task and reconstruction error. Because the weights between the input and hidden layers are trained to reduce both task and reconstruction error, a potential drawback of IRRE is the use of the hidden layer to encode all of the features in the image, rather than only the ones required for solving the particular task [Pomerleau, 1993]. This can be addressed by noting the following: if a strictly layered (connections are only between adjacent layers) feed-forward neural network can solve a given task, the activations of the hidden layer contain, in some fonn, the important infonnation for this task from the input layer. One method of detennining what is contained in the hidden layer is to attempt to reconstruct the original input image, based solely upon the representation developed in the hidden layer. Like IRRE, the input image is reconstructed from the activations of the units in the hidden layer. Unlike IRRE, the hidden units are not trained to reduce reconstruction error, they are only trained to solve the panicular task. The network's allocation of its limited representation capacity at the hidden layer is an indicator of what it deems relevant to the task. Information which is not relevant to the task will not be encoded in the hidden units. Since the reconstruction of the inputs is based solely on the hidden units' activations, and the irrelevant portions of the input are not encoded in the hidden units' activations, the inputs which are irrelevant to the task cannot be reconstructed. See Figure I(Right). By measuring which inputs can be reconstructed accurately, we can ascertain which inputs the hidden units have encoded to solve the task. A synthetic task which demonstrates this idea is described here. Imagine being given a lOxlO input retina such as shown in Figure 2a&b. The task is to categorize many such examples into one of four classes. Because of the random noise in the examples, the simple underlying process, of a cross being present in one of four locations (see Figure 2c), is not easily discernible, although it is the feature on which the classifications are to be based. Given enough examples, the NN will be able to solve this task. However, even after the model of the task is learned, it is difficult to ascertain to which inputs the network is attending. To detennine this, we can freeze the weights in the trained network and connect a input-reconstruction layer to the hidden units, as shown in Figure 1(Right). After training these connections, by measuring where the reconstruction matches the actual input, we can detennine which inputs the network has encoded in its hidden units, and is therefore attending. See Figure 2d. weights trained to reduce task error only weights trained to reduce reconstruction error only. error. Figure 1: weights trained to reduce task error only. (Left) IRRE. (Right) Modified IRRE. "- weights trained to reduce reconstruction error only. Using Expectation to Guide Processing B: 861 C: D: 2 3 4 + ir+ Figure 2: (A & B): Samples of training data (cross appears in position 4 & 1 respectively). Note the large amounts of noise. (C): The underlying process puts a cross in one of these four locations. (D): The black crosses are where the reconstruction matched the inputs; these correspond exactly to the underlying process. IRRE and this modified IRRE are related to auto-encoding networks [Cottrell, 1990] and principal components analysis (PeA). The difference between auto-encoding networks and those employed in this study is that the hidden layers of the networks used here were trained to perfonn well on the specific task, not to reproduce the inputs accurately. 2.1 Creating Expectations A notion of time is necessary in order to focus attention in future frames. Instead of reconstructing the current input, the network is trained to predict the next input; this corresponds to changing the subscript in the reconstruction layer of the network shown in Figure 1(Right) from t to t+ 1. The prediction is trained in a supervised manner, by using the next set of inputs in the time sequence as the target outputs. The next inputs may contain noise or extraneous features. However, since the hidden units only encode infonnation to solve the task, the network will be unable to construct the noise or extraneous features in its prediction. To this point, a method to create a task-specific expectation of what the next inputs will be has been described. As described in Section 1, there are two fundamentally different ways in which to interpret the difference between the expected next inputs and the actual next inputs. The first interpretation is that the difference between the expected and the actual inputs is a point of interest because it is a region which was not expected. This has applications in anomaly detection; it will be explored in Section 3.2. In the second interpretation, the difference between the expected and actual inputs is considered noise. Processing should be de-emphasized from the regions in which the difference is large. This makes the assumption that there is enough infonnation in the previous inputs to specify what and where the important portions of the next image will be. As shown in the road-following and hand-tracking task, this method can remove spurious features and noise. 3 Real-World Applications 1bree real-world tasks are discussed in this section. The first, vision-based road following, shows how the task-specific expectations developed in the previous section can be used to eliminate distractions from the input. The second, detection of anomalies in the plasmaetch step of wafer fabrication, shows how expectations can be used to emphasize the unexpected'features in the input. The third, visual hand-tracking, demonstrates how to incorporate a priori domain knowledge about expectations into the NN. 3.1 Application 1: Vision-Based Autonomous Road Following In the domain of autonomous road following, the goal is to control a robot vehicle by analyzing the image of the road ahead. The direction of travel should be chosen based on the location of important features like lane markings and r~ad edges. On highways and dirt roads, simple techniques, such as feed-forward NNs, have worked well for mapping road images to steering commands [Pomerleau, 1993]. However, on city streets, where there are distractions like old lane-.narkings, pedestrians, and heavy traffic, these methods fail. The purpose of using attention in this domain is to eliminate features of the road which the NN may mistake as lane markings. Approximately 1200 images were gathered from a 862 S. Baluja E F Figure 3: (Top) : Four samples of training images . Left most shows the position of the lane-marking which was hand-marked. (Right): In each triplet: Left: raw input imagtt. Middle: the network's prediction of the inputs at time t; this prediction was made by a network with input ofimaget_I ' Right: a pixel-by-pixel filtered image (see text). This image is used as the input to the NN. G camera mounted on the left side of the CMU-Navlab 5 test vehicle, pointed downwards and slightly ahead of the vehicle. The car was driven through city and residential neighborhoods around Pittsburgh, PA. The images were gathered at 4-5 hz. The images were subsampled to 30x32 pixels. In each of these images, the horizontal position of the lane marking in the 20th row of the input image was manually identified. The task is to produce a Gaussian of activation in the outputs centered on the horizontal position of the lane marking in the 20th row of the image, given the entire input image. Sample images and target outputs are shown in Figure 3. In this task, the ANN can be confused by road edges (Figure 3a), by extraneous lane markings (Figure 3b), and reflections on the car itself (since the camera was positioned on the side of the car), as shown in Figure 3c. The network architecture shown in Figure 4 was used; this is the same architecture as in Figure l(right) with the feedback shown. The feedback is used during both training and simulation. In each time-step, a steering direction and a prediction of the next inputs is produced. For each time-step, the magnitude of the difference' between the input's expected value (computed in the previous time-step) and its actual value is computed. Each input pixel can be moved towards its background value l in proportion to this difference-value. The larger the difference value, the more weight is given to the background value. If the difference value is small, the actual inputs are used. This has the effect of deemphasizing the unexpected inputs. The results of using this method were very promising. The lane tracker removed distracting features from the images. In Figure 3G, a distracting lane-marking is removed: the lane marker on the right was correctly tracked in images before the distractor lane-marker appeared. In Figure 3F, a passing car is de-emphasized: the network does not have a model to predict the movement of passing cars, since these are not relevant for the lane-marker detection task. In Figure 3E, the side of the road appears brighter than expected; therefore it is de-emphasized. Note that the expectation-images (shown in the middle of each triplet weights trained to reduce task error only. Delayed I time step I ...~~.... __ weights --- trained to reduce prediction error only. m~tii;:mrt;,._--1 Weight bkgd. and actual inputs according to difference ima e. Figure 4: t - - - t Difference between inputs t & predicted in uts Architecture used to track the lane marking in cluttered scenes. Signal Transfer (Connections are not trainable) I. A simple estimate of the background value for each pixel is its average activation across the training set. For the road-following domain, it is possible to use a background activation of 0.0 (when the entire image is scaled to activations of +1.0 to -1.0) since the road often appears as intermediate grays. Using Expectation to Guide Processing 863 in Figure 3) show that the expected lane-marker and road edge locations are not precisely defined. This is due to the training method, which attempts to model the many possible transitions from one time step to the next to account for inter- and intra-driver variability with a limited training set [Baluja, 1996]. In summary, by eliminating the distractions in the input images, the lane-tracker with the attention mechanisms improved performance by 20% over the standard lane-tracker, measured on the difference between the estimated and hand-marked position of the lanemarker in each image. This improvement was seen on multiple runs, with random initial weights in the NN and different random translations chosen for the training images. 3.2 Application 2: Fault Detection in the Plasma-Etch Wafer Fabrication Plasma etch is one of the many steps in the fabrication of semiconductor wafers. In this study, the detection of four faults was attempted. Descriptions of the faults can be found in [Baluja, 1996][Maxion, 1996]. For the experiments conducted here, only a single sensor was used, which measured the intensity of light emitted from the plasma at the 520nm wavelength. Each etch was sampled once a second, providing approximately 140 samples per wafer wavefonn. The data-collection phase of this experiment began on October 25, 1994, and continued until April 4, 1995. The detection of faults is a difficult problem because the contamination of the etch chamber and the degradation parts keeps the sensor's outputs, even for fault-free wafers, changing over time. Accounting for machine state should help the detection process. Expectation is used as follows: Given the waveform signature of waferT_I' an expectation of waferT can be fonned. The input to the prediction-NN is the wavefonn signature of waferT_I; the output is the prediction of the signature of waferT. The target output for each example is the signature of the next wafer in sequence (the full 140 parameters). Detection of the four faults is done with a separate network which used as input: the expectation of the wafer's wavefonn, the actual wafer's wavefonn, and the point-by-point difference of the two. In this task, the input is not filtered as in the driving domain described previously; the values of the point-by-point difference vector are used as extra inputs. The perfonnance of many methods and architectures were compared on this task, details can be found in [Baluja, 1996]. The results using the expectation based methods was a 98.7% detection rate, 100% classification rate on the detected faults (detennining which of the four types of faults the detected fault was), and a 2.3% false detection rate. For comparison, a simple perceptron had an 80% detection rate, and a 40% false-detection rate. A fully-connected network which did not consider the state of the machine achieved a 100% detection rate, but a 53% false detection rate. A network which considered state by using the last-previous no-fault wafer for comparison with the current wafer (instead of an expectation for the current wafer) achieved an 87.9% detection rate, and a 1.5% falsedetection rate. A variety of neural and non-neural methods which examined the differences between the expected and current wafer, as well those which examined the differences between the last no-fault wafer and the current wafer, perfonned poorly. In summary, methods which did not use expectations were unable to obtain the false-positives and detection rates of the expectation-based methods. 3.3 Application 3: Hand-Tracking in Cluttered Scenes In the tasks described so far, the transition rules were learned by the NN. However, if the transition rules had been known a priori, processing could have been directed to only the relevant regions by explicitly manipulating the expectations. The ability to incorporate a priori rules is important in many vision-based tasks. Often the constraints about the environment in which the tracking is done can be used to limit the portions of the input scene which need to be processed. For example, consider visually tracking a person's hand. Given a fast camera sampling rate, the person's hand in the current frame will be close to 864 S. Baluja B. A. Figure 5: Typical input images used for the hand-tracking experiments. The target is to track the subject's right hand. Without expectation, in (A) both hands were found in X outputs, and the wrong hand was found in the Y outputs. In (8) Subject's right hand and face found in the X outputs. where it appeared in the previous frame. Although a network can learn this constraint by developing expectations of future inputs (as with the NN architecture shown in Figure 4), training the expectations can be avoided by incorporating this rule directly. In this task, the input layer is a 48*48 image. There are two output layers of 48 units; the desired outputs are two gaussians centered on the (X,Y) position of the hand to be tracked. See Figure 5. Rather than creating a saliency map based upon the difference between the actual and predicted inputs, as was done with autonomous road following, the saliency map was explicitly created with the available domain knowledge. Given the sampling rate of the camera and the size of the hand in the image, the salient region for the next timestep was a circular region centered on the estimated location of the hand in the previous image. The activations of the inputs outside of the salient region were shifted towards the background image. The activations inside the salient region were not modified. After applying the saliency map to the inputs, the filtered inputs were fed into the NN. This system was tested in very difficult situations; the testing set contained images of a person moving both of his hands and body throughout the sequence (see Figure 5). Therefore, both hands and body are clearly visible in the difference images used as input into the network. All training was done on much simpler training sets in which only a single hand was moving. To gauge the perfonnance of an expectation-based system, it was compared to a system which used the following post-processing heuristics to account for temporal coherence. First, before a gaussian was fit to either of the output layers, the activation of the outputs was inversely scaled with the distance away from the location of the hand in the previous time step. This reduces the probability of detecting a hand in a location very different than the previous detection. This helps when both hands are detected, as shown in Figure 5. The second heuristic was that any predictions which differ from the previous prediction by more than half of the dimension of the output layer were ignored, and the previous prediction used instead. See Table I for the results. In summary, by using the expectation based methods, perfonnance improved from 66% to 90% when tracking the left hand, and 52% to 91 % when tracking the right hand. Table I: Performance: Number of frames in which each hand was located (283 total images). Method No Heuristics, No Expect. Heuristics Expectation Expectation + Heuristics Target: Find Left Hand Target: Find Right Hand Which Hand Was Found Which Hand Was Found % Correct L 52% 66% 91% 90% 146 187 258 256 R 44 22 3 3 None % Correct L 93 74 22 24 16% 52% 90% 91% 143 68 3 2 R 47 147 255 257 None 93 68 25 24 [Nowlan & Platt, 1995] presented a convolutional-NN based hand-tracker which used separate NNs for intensity and differences images with a rule-based integration of the multiple network outputs. The integration of this expectation-based system should improve the performance of the difference-image NN. Using Expectation to Guide Processing 865 4 Conclusions A very closely related procedure to the one described in this paper is the use of Kalman Filters to predict the locations of objects of interest in the input retina. For example, Dickmanns uses the prediction of the future state to help guide attention by controlling the direction of a camera to acquire accurate position of landmarks [Dickmanns, 1992]. Strong models of the vehicle motion, the appearance of objects of interest (such as the road, road-signs, and other vehicles), and the motion of these objects are encoded in the system. The largest difference in their system and the one presented here is the amount of a priori knowledge that is used. Many approaches which use Kalman Filters require a large amount of problem specific information for creating the models. In the approach presented in this paper, the main object is to automatically learn this information from examples. First, the system must learn what the important features are, since no top-down information is assumed. Second, the system must automatically develop the control strategy from the detected features. Third, the system must also learn a model for the movements of all of the relevant features. In deciding whether the approaches described in this paper are suitable to a new problem, two criteria must be considered. First, if expectation is to be used to remove distractions from the inputs, then given the current inputs, the activations of the relevant inputs in the next time step must be predictable while the irrelevant inputs are either unrelated to the task or are unpredictable. In many visual object tracking problems, the relevant inputs are often predictable while the distractions are not. In the cases in which the distractions are predictable, if they are unrelated to the main task, these methods can work. When using expectation to emphasize unexpected or potentially anomalous features, the activations of the relevant inputs should be unpredictable while the irrelevant ones are predictable. This is often the case for anomaly/fault detection tasks. Second, when expectations are used as a filter, it is necessary to explicitly define the role of the expected features. In particular, it is necessary to define whether the expected features should be considered relevant or irrelevant, and therefore, whether they should be emphasized or de-emphasized, respectively. We have demonstrated the value of using task-specific expectations to guide processing in three real-world tasks. In complex, dynamic, environments, such as driving, expectations are used to quickly and accurately discriminate between the relevant and irrelevant features. For the detection of faults in the plasma-etch step of semiconductor fabrication, expectations are used to account for the underlying drift of the process. Finally, for visionbased hand-tracking, we have shown that a priori knowledge about expectations can be easily integrated with a hand-detection model to focus attention on small portions of the scene, so that distractions in the periphery can be ignored. Acknowledgments The author would like to thank Dean Pomerleau, Takeo Kanade, Tom Mitchell and Tomaso Poggio for their help in shaping this work. References Baluja, S. 1996, Expectation-Based Selective Attention. Ph.D. Thesis, School of Computer Science, CMU. Clark, J. & Ferrier, N (1992), Attentive Visual Servoing, in: Active Vision. Blake & Yuille, (MIT Press) 137-154. Cottrell, G.W., 1990, Extracting Features from Faces using Compression Network, Connectionist Models, Morgan Kaufmann 328-337. Dickmanns, 1992, Expectation-based Dynamic Scene Understanding, in: Active Vision. A. Blake & A.Yuille, MIT Press . Koch, C. & Ullman, S. (1985) "Shifts in Selective Visual Attention: Towards the Underlying Neural Circuitry", in: Human Neurobiology 4 (1985) 219-227. Maxion, R. (1995) The Semiconductor Wafer Plasma-Etch Data Set. Nowlan, S. & Platt, J., 1995, "A Convolutional Neural Network Hand Tracker". NIPS 7. MIT Press . 901-908. Pomerleau, D.A., 1993. Neural Network Perception for Mobile Robot Guidance, Kluwer Academic. Thorpe, C., 1991, Outdoor Visual Navigation for Autonomous Robots, in: Robotics and Autonomous Systems 7. 

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Shared Context Probabilistic Transducers Yoshua Bengio* Dept. IRO , Universite de Montreal, Montreal (QC) , Canada, H3C 3J7 bengioyOiro.umontreal.ca Samy Bengio t Microcell Labs, 1250 , Rene Levesque Ouest, Montreal (QC) , Canada, H3B 4W8 samy.bengioOmicrocell.ca Jean-Fran~ois Isabelle t Microcell Labs, 1250, Rene Levesque Ouest , Montreal (QC), Canada, H3B 4W8 jean-francois.isabelleCmicrocell.ca Yoram Singer AT&T Laboratories, Murray Hill , NJ 07733, USA, singerOresearch.att.com Abstract Recently, a model for supervised learning of probabilistic transducers represented by suffix trees was introduced. However, this algorithm tends to build very large trees, requiring very large amounts of computer memory. In this paper, we propose anew, more compact, transducer model in which one shares the parameters of distributions associated to contexts yielding similar conditional output distributions . We illustrate the advantages of the proposed algorithm with comparative experiments on inducing a noun phrase recogmzer . 1 Introduction Learning algorithms for sequential data modeling are important in many applications such as natural language processing and time-series analysis, in which one has to learn a model from one or more sequences of training data. Many of these algorithms can be cast as weighted transducers (Pereira, Riley and Sproat, 1994), which associate input sequences to output sequences, with weights for each input/output * Yoshua Bengio is also with AT&T Laboratories, Holmdel, NJ 07733, USA. t This work was performed while Samy Bengio was at INRS-Telecommunication, Iledes-Soeurs, Quebec, Canada, H3E IH6 t This work was performed while Jean-Franc;ois Isabelle was at INRS-Telecommunication, Ile-des-Soeurs, Quebec, Canada, H3E IH6 y. Bengio, S. Bengio, J-F. Isabelle and Y. Singer 410 sequence pair. When these weights are interpreted as probabilities, such models are called probabilistic transducers. In particular, a probabilistic transducer can represent the conditional probability distribution of output sequences given an input sequence. For example, algorithms for combining several transducers were found useful in natural language and speech processing (Riley and Pereira, 1994). Very often, weighted transducers use an intermediate variable that represents "context", such as the state variable of Hidden Markov Models (Baker, 1975; Jelinek, 1976). A particular type of weighted transducer, called Input/Output Hidden Markov Model, is one in which the input-to-context distribution and context-to-output distribution are represented by flexible parameterized models (such as neural networks) (Bengio and Frasconi, 1996) . In this paper, we will study probabilistic transducers with a deterministic input-to-state mapping (i.e., a function from the past input subsequence to the current value of the context variable). One such transducer is the one which assigns a value of the context variable to every value of the past input subsequence already seen in the data. This input-to-state mapping can be efficiently represented by a tree. Such transducers are called suffix tree transducers (Singer, 1996). A problem with suffix tree transducers is that they tend to yield very large trees (whose size may grow as O(n 2 ) for a sequence of data of length n). For example, in the application studied in this paper, one obtains trees requiring over a gigabyte of memory. Heuristics may be used to limit the growth of the tree (e.g., by limiting the maximum depth of the context, i.e., of the tree, and by limiting the maximum number of contexts, i.e., nodes of the tree). In this paper, instead, we propose a new model for a probabilistic transducer with deterministic input-to-state function in which this function is compactly represented, by sharing parameters of contexts which are associated to similar output distributions. Another way to look at the proposed algorithm is that it searches for a clustering of the nodes of a suffix tree transducer. The data structure that represents the contexts is not anymore a tree but a single-root acyclic directed graph. 2 Background: Suffix Tree Probabilistic Transducers The learning algorithm for suffix tree probabilistic transducers (Singer, 1996) constructs the model P(Yilxi) from discrete input sequences xi {Xl,X2, ... ,Xn } to output sequences yi = {Y1, Y2, ... , Yn}, where Xt are elements of a finite alphabet Ein. This distribution is represented by a tree in which each internal node may have a child for every element of Ein, therefore associating a label E Ein to each arc. A node at depth d is labeled with the sequence ut of labels on arcs from root to node, corresponding to a particular input context, e.g., at some position n in the sequence a context of length d is the value ut of the preceding subsequence x~_d l' Each node at depth d is therefore associated with a model of the output distribution in this context, P(Yn IX~-d+1 = ut) (independent of n). = To obtain a local output probability for Yn (i.e., given xi), one follows the longest possible path from the root to a node a depth d according to the labels x n , xn -1, ... Xn-d+1. The local output probability at this node is used to model Yn' Since p(yflxf) can always be written n~=1 P(Yn Ixi)' the overall input/output conditional distribution can be decomposed, according to this model, as follows: T p(yflxf) = II P(YnIX~_d(x~)+l)' (1) n=1 where d(xi) is the depth of the node of the tree associated with the longest suffix uf = x~_d+1 of xi. Figure 1 gives a simple example of a suffix tree transducer. 411 Shared Context Probabilistic Transducers Pfaj"O.S Pfb)~OJ P(c)", 0 2 p(a)",O.6 p(b) _O.) p(c:)= O I 11 Figure 1: ExampJe of suffix tree transducer (Singer, 1996). The input alphabet, E in = {O, I} and the output aJphabet, EotJt = {a, b, c}. For instance, P(aIOOllO) = P(alllO) = 0.5 . 3 Proposed Model and Learning Algorithm In the model proposed here, the input/output conditional distribution p(yT IxI) is represented by a single-root acyclic directed graph. Each node of this graph is associated with a set of contexts C node = {(jt?}, corresponding to all the paths i (of various lengths di ) from the root of the tree to this node. All these contexts are associated with the same local output distribution P(Yn Ix? has a suffix in Cnode). Like in suffix tree transducers, each internal node may have a child for every element of E in . The arc is labeled with the corresponding element of ~in . Also like in suffix tree transducers, to obtain P(Ynlx~), one follows the path from the root to the deepest node called deepest(x?) according to the labels Xn, Xn-l, etc .. . The local output distribution at this node is used to predict Yn or its probability. The overall conditional distribution is therefore given by T P(yilxf) = II P(Ynldeepest(x~)) (2) n=l where the set of contexts Cdeepe3t(x~) associated to the deepest node deepest(xl) contains a suffix of x? The model can be used both to compute the conditional probability of a given input/output sequence pair, or to guess an output sequence given an input sequence. Note that the input variable can contain delayed values of the output variable (as in Variable Length Markov Models). 3.1 Proposed Learning Algorithm We present here a constructive learning algorithm for building the graph of the model and specify which data points are used to update each local output model (associated to nodes of the graph). The algorithm is on-line and operates according to two regimes: (1) adding new nodes and simply updating the local output distributions at existing nodes, and (2) merging parts of the graph which represent similar distributions. If there are multiple sequences in the training data they are concatenated in order to obtain a single input/output sequence pair. (1) After every observation (xn, Yn), the algorithm updates the output distributions y. Bengio, S. Bengio, J-F. Isabelle and Y. Singer 412 of the nodes for which Cnode(x~) contains a suffix of Xl, possibly adding new nodes (with labels x~_d.) until xl E Cnode for some node. (2) Every Tmerge observations, the algorithm attempts to merge sub-graphs which are found similar enough, by comparing the N (N - 1) /2 pairs of sub-graphs rooted at the N nodes that have seen at least minn observations. Merging two subgraphs is equivalent to forcing them to share parameters (as well as reducing the size of the representation of the distribution). A merge is performed between the graphs rooted at nodes a and b if Ll(a, b) < mina and the merge succeeds. The details of the similarity measure and merging algorithm are given in the next subsections. 3.2 Similarity Measure Between Rooted Subgraphs In order to compare (asymmetrically) output distributions P(yla) and P(ylb) at two nodes a and b, one can use the Kullback-Liebler divergence: _ ~ P(ylb) (3) Ii. L(a, b) L...J P(ylb) log P(yla) = yEE out However, we want to compare the whole acyclic graphs rooted at these 2 nodes. In order to do so, let us define the following. Let s be a string of input labels, and b a node. Define desc(b, s) as the most remote descendant of b obtained by following from b the arcs whose labels correspond to the sequence s. Let descendents(a) be the set of strings obtained by following the arcs starting from node a until reaching the leaves which have a as an ancestor. Let P(sla) be the probability offollowing the arcs according to string s, starting from node a. This distribution can be estimated by counting the relative number of descendents through each of the children of each node. To compare the graphs rooted at two nodes a and b, we extend the KL divergence by weighing each of the descendents of a, as follows: L W K L(a, b) = P(sla)K L(desc(a, s), desc(b, s)) (4) de3cendent8 (a) Finally, to obtain a symmetric measure, we define Ll(a,b) = WKL(a,b) + WKL(b,a) (5) that is used in the merge phase of the constructive learning algorithm to decide whether the subgraphs rooted at a and b should be merged. 3 3.3 E Merging Two Rooted Subgraphs If Ll (a, b) < mina (a predefined threshold) we want to merge the two subgraphs rooted at a and b and create a new subgraph rooted at c. The local output distribution at c is obtained from the local output distributions at a and b as follows: P(Yn Ic) = P(Ynla)P(ala or b) where we define + P(Ynlb)P(bla or b) ad(a) P(ala or b) = ad(a) + ad(b) , (6) (7) where d(a) is the length ofthe longest path from the root to node a, and a represents a prior parameter (between 0 and 1) on the depth of the acyclic graphs . This prior parameter can be used to induce a prior distribution over possible rooted acyclic graphs structures which favors smaller graphs and shorter contexts (see the mixture of probabilistic transducers of (Singer, 1996)). The merging algorithm can then be summarized as follows: Shared Context Probabilistic Transducers 413 ? The parents of a and b become parents for c. ? Some verifications are made to prevent merges which would yield to cycles in the graph. The nodes a and b are not merged if they are parents of one another. ? We make each child of a a child of c. For each child u of b (following an arc labeled l), look for the corresponding child v of c (also following the arc labeled l) . If there is no such child, and u is not a parent of c, make u a new child of c. Else, if u and v are not parents of each other, recursively merge them. ? Delete nodes a and b, as well as all the links from and to these nodes. This algorithm is symmetric with respect to a and b except when a merge cannot be done because a and b are parents of one another. In this case, an asymmetric decision must be taken: we chose to keep only a and reject b. Figure 2 gives a simple example of merge. Figure 2: This figure shows how two nodes are merged. The result is no longer a tree, but a directed graph. Some verifications are done to avoid cycles in the graph. Each node can have multiple labels, corresponding to the multiple possible paths from the root to the node. 4 Comparative Experiments We compared experimentally our model to the one proposed in (Singer, 1996) on mixtures of suffix tree transducers, using the same task. Given a text where each word is assigned an appropriate part-of-speech value (verb, noun, adjective, etc), the task is to identify the noun phrases in the text. The UPENN tree-bank corpus database was used in these experiments. The input vocabulary size, IEinl 41, is the number of possible part-of-speech tags, and the output vocabulary size is IEouti = 2. The model was trained over 250000 marked tags, constraining the tree to be of maximal depth 15. The model was then tested (freezing the model structure and its parameters) over 37000 other tags. Using the mixture of suffix tree transducers (Singer, 1996) and thresholding the output probability at 0.5 to take output decisions, yielded an accuracy rate of 97.6% on the test set, but required over 1 gigabyte of computer memory. = To make interesting comparisons with the shared context transducers, we chose the following experimental scheme. Not only did we fix the maximal depth of the directed graph to 15, but we also fixed the maximal number of allocated nodes, i.e., simulating fixed memory resources. When this number was reached, we froze the structure but continued to update the parameters of the model until the end of the training database was reached. For the shared context version, whenever a merge freed some nodes, we let the graph grow again to its maximal node size. At the end of this process, we evaluated the model on the test set. Y. Bengio, S. Bengio, J-F. Isabelle and Y. Singer 414 We tested this method for various values of the maximum number of nodes in the graph. For each experiment, we tried different values of the other parameters (the similarity threshold min~ for merging, the minimum number of observations miIln at a node before it can be considered for a merge, and the delay Tmerge between two merging phases), and we picked the one which performed the best on the training set. Results are reported in figure 3. maximal number of nodes 20 50 100 500 1000 2000 5000 with merge (%) 0.762 0.827 0.861 0.924 0.949 0.949 0.952 without merge (%) 0.584 0.624 0.727 0.867 0.917 0.935 0.948 005 09 085 01 075 07 085 01 055 Figure 3: This figure shows the generalization accuracy rate of a transducer with merges (shared contexts graph) against one without merges (suffix tree), with different maximum number of nodes. The maximum number of nodes are in a logarithmic scale, and the accuracy rates are expressed in relative frequency of correct classification. As can be seen from the results, the accuracy rate over the test set is better for transducers with shared contexts than without. More precisely, the gain is greater when the maximum number of nodes is smaller. When we fix the maximum number of nodes to a very small value (20), a shared context transducer performs 1.3 times better (in classification error) than a non-shared one. This gain becomes smaller and smaller as the maximum size increases. Beyond a certain maximum size, there is almost no gain, and one could probably observe a loss for some large sizes. We also need to keep in mind that the larger the transducer is, the slower the program to create the shared context transducer is, compared to the non-shared one. Finally, it is interesting to note that using only 5000 nodes, we were able to obtain 95.2% accuracy, which is only 2.4% less than those obtained with no constraint on the number of nodes. 5 Conclusion In this paper, we have presented the following: ? A new probabilistic model for probabilistic transducers with deterministic input-to-state function, represented by a rooted acyclic directed graph with nodes associated to a set of contexts and children associated to the different input symbols. This is a generalization of the suffix tree transducer. ? A constructive learning algorithm for this model, based on construction and merging phases. The merging is obtained by clustering parts of the graph which represent a similar conditional distribution. ? Experimental results on a natural-language task showing that when the size of the graph is constrained, this algorithm performs better than the purely constructive (no merge) suffix tree algorithm. Shared Context Probabilistic Transducers 4}5 References Baker, J. (1975). Stochastic modeling for automatic speech understanding. In Reddy, D., editor, Speech Recognition, pages 521-542. Academic Press, New York. Bengio, S. and Bengio, Y. (1996). An EM algorithm for asynchronous input/output hidden markov models. In Proceedings of the International Conference on Neural Information Processing, Honk Kong . Bengio, Y. and Frasconi, P. (1996). Input/Output HMMs for sequence processing. IEEE Transactions on Neural Networks , 7(5):1231-1249. Jelinek, F. (1976). Continuous speech recognition by statistical methods. Proceedings of the IEEE, 64:532-556 . Pereira, F. , Riley, M., and Sproat, R. (1994). Weighted rational transductions and their application to human language processing. In ARPA Natural Language Processing Workshop. Riley, M. and Pereira, F. (1994). Weighted-finite-automata tools with applications to speech and language processing. Technical Report Technical Memorandum 11222-931130-28TM, AT&T Bell Laboratories. Singer, Y . (1996). Adaptive mixtures of probabilistic transducers. In Mozer, M., Touretzky, D., and Perrone, M., editors, Advances in Neural Information Processing Systems 8. MIT Press, Cambridge, MA. 

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402 MODELING THE OLFACTORY BULB - COUPLED NONLINEAR OSCILLATORS Zhaoping Lit J. J. Hopfield? t Division of Physics, Mathematics and Astronomy ?Division of Biology, and Division of Chemistry and Chemical Engineering t? California Institute of Technology, Pasadena, CA 91125, USA ? AT&T Bell Laboratories ABSTRACT The olfactory bulb of mammals aids in the discrimination of odors. A mathematical model based on the bulbar anatomy and electrophysiology is described. Simulations produce a 35-60 Hz modulated activity coherent across the bulb, mimicing the observed field potentials. The decision states (for the odor information) here can be thought of as stable cycles, rather than point stable states typical of simpler neuro-computing models. Analysis and simulations show that a group of coupled non-linear oscillators are responsible for the oscillatory activities determined by the odor input, and that the bulb, with appropriate inputs from higher centers, can enhance or suppress the sensitivity to partiCUlar odors. The model provides a framework in which to understand the transform between odor input and the bulbar output to olfactory cortex. 1. INTRODUCTION The olfactory system has a simple cortical intrinsic structure (Shepherd 1979), and thus is an ideal candidate to yield insight on the principles of sensory information processing. It includes the receptor cells, the olfactory bulb, and the olfactory cortex receiving inputs from the bulb (Figure [1]). Both the bulb and the cortex exhibit similar 35-90 Hz rhythmic population activity modulated by breathing. Efforts have been made to model the bulbar information processing function (Freeman 1979b, 1979c; Freeman and Schneider 1982; Freeman and Skarda 1985; Baird 1986j Skarda and Freeman 1987), which is still unclear (Scott 1986). The bulbar position in the olfactory pathway, and the linkage of the oscillatory activity with the sniff cycles suggest that the bulb and the oscillation play important roles in the olfactory information processing. We will examine how the bulbar oscillation pattern, which can be thought of as the decision state about odor information, originates and how it depends on the input odor. We then show that with appropriate inputs from the higher centers, the bulb can suppress or enhance the its sensitivity to particular odors. Much more details of our work are described in other two papers (Li and Hopfield 1988a, 1988b). 403 Modeling the Olfactory Bulb-Coupled Nonlinear Oscillators The olfactory bulb has mainly the excitatory mitral and the inhibitory granule cells located on different parallel lamina. Odor receptors effectively synapse on the mitral cells which interact locally with the granule cells and carry the bulbar outputs (Fig 1, Shepherd 1979). A rabbit has about 50,000 mitral, and 10,000,000 granule cells (Shepherd 1979). With short odor pulses, the receptor firing rate increases in time, and terminates quickly after the odor pulse terminates (Getchell and Shepherd 1978). Most inputs from higher brain centers are directed to the granule cells, and little is know about them. The surface EEG wave (generated by granule activities, Freeman 1978j Freeman and Schneider 1982), depending on odor stimulations and animal motivation, shows a high amplitude oscillation arising during the inhalation and stopping early in the exhalation. The oscillation is an intrinsic property of the bulb itself, and is influenced by central inputs (Freeman 1979aj Freeman and Skarda 1985). It has a peak frequency (which is the same across the bulb) in the range of 35-90 Hz, and rides on a slow background wave phase locked with the respiratory wave. "'-J 2. MODEL ORGANIZATION we only include ( N excitatory) mitral and ( M For simplicity, inhibitory ) Iodor" + Ibaekgrotmd,i, granule cells in the model. The Receptor input I is Ii for 1 ... , N, a superposition of an odor signal Iodor and a background input Ibaekgrov,nd. Iodor > 0 increases in time during inhalation, and return exponentially during exhalation toward the ambient. The central input to the granule cells is vector Ie with components Ie,j for 1 < j < M. For now, it is assumed that Ie = 0.1 and Ibaekgrov,nd = 0.243 do not change during a sniff (Li and Hopfield 1988a). = Each cell is one unit with its internal state level described by a single variable, and its output a continuous function of the internal state level. The internal states and outputs are respectively X = {Xl' X2, ... , XN} and G z ( X) = {gz(xd, gz(x2),?.? ,gz(XN)} (Y = {YI' Y2,??? ,YM} and Gy{Y) = {gy(Yd, gy(Y2), ... ,gy(YM)}) for the mitral (granule) cells, where gz > 0 and gy > 0 are the neurons' non-linear sigmoid output functions essential for the bulbar oscillation dynamics (Freeman and Skarda 1985) to be studied. Receptor inputs other brain ... ... ... ... ... ... ... ... (t)(t)(t)(t)(t)(t)(t)(t) It' t ~ , +)IfInt~~:~lion 22222222 It. It. It. It. It. It. Central inputs It. ... Fig.l. Left: olfactory system; Right: bulbar structure Cells marked "+" are mitral cells, "_" are granule cells The geometry of bulbar structure is simplified to a one dimensional ring. Each cell is specified by an index, e.g. ith mitral cell, and jth granule cell for all i, i 404 Li and Hopfield indicating cell locations on the ring (Fig 1). N X M matrix Ho and M X N matrix Wo are used respectively to describe the synaptic strengths (postsynaptic input: presynaptic output) from granule cells to mitral cells and vice versa. The bulb model system has equations of motion: X= Y -HoGy(Y) - = WoGz(X) - O:zX + I, (2.1) O:yY + Ie. where O:z = 1/'rz , O:y = 1/'ry, and 'rz = 'ry = 7 msec are the time constants of the mitral and granule cells respectively (Freeman and Skarda 1985; Shepherd 1988). In simulation, weak random noise is added to I .and Ie to simulate the fluctuations in the system. 3. SIMULATION RESULT Computer simulation was done with 10 mitral and granule cells, and show that the model can capture the major effects of the real bulb. The rise and fall of oscillations with input and the baseline shift wave phase locked with sniff cycles are obvious (Fig.2). The simulated EEG (calculated using the approximation by Freeman (1980)) and the measured EEG are shown for comparison. During a sniff, all the cells oscillate coherently with the same frequency as physiologically observed. 7\ .'""1 B GrwIule Output ~J1 .0 ~EEGW'" lOOms H EEG Wave Respiratory Wave looms ~ Fig.2. A: Simulation result; B: measured result from Freeman and Schneider 1982. The model also shows the capability of a pattern classifier. During a sniff, some input patterns induce oscillation, while others do not, and different inputs induce different oscillation patterns. We showed (Li and Hopfield 1988a) that the bulb amplifies the differences between the different inputs to give different output patterns, while the responses to same odor inputs with different noise samples differ negligibly. 4. MATHEMATICAL ANALYSIS A (damped) oscillator with frequency w can be described by the equations X = -wy - o:x iI = wx - o:y or (4.1) Modeling the Olfactory Bulb-Coupled Nonlinear Oscillators The solution orbit in (x, y) space is a circle if a = 0 (non-damped oscillator), and spirals into the origin otherwise (damped oscillator). IT a mitral cell and a granule cell are connected to each other, with inputs i(t) and ie(t) respectively, then x = -h . gy(y) - azx + i(t), y = w . gz(x) - ayy + ie(t). (4.2) This is the scalar version of equation (2.1) with each upper case letter representing a vector or matrix replaced by a lower case letter representing a scalar. It is assumed that i(t) has a much slower time course than X or y (frequency of sniffs ~ characteristic neural oscillation frequency). Use the adiabatic approximation, and define the equilibrium point (xo, Yo) as Xo ~ 0 Yo ~ 0 Define x' = x - Xo, = -h . gy(yo) - azxo + i, = w . gz(x o) - ayyo + ie' y' - y - Yo' Then x' = -h(gy(y) il = w(gz(x) (cf. equation (4.1)). IT a z =f gy(yo)) - azx', gz(x o)) - ayy'. = ay = 0, then the solution orbit zo+z' R (4.3) yo+Y' w(gz(s) - gz(xo))ds + Zo f h(gy(s) - gy(Yo))ds = constant Yo is a closed curve in the original (x, y) space surrounding the point (xo, Yo), i.e., (x, y) oscillates around the point (xo, Yo). When the dissipation is included, dR/ dt < 0, the orbit in (x, y) space will spiral into the point (xo, Yo). Thus a connected pair of mitral and granule cells behaves as a damped non-linear oscillator, whose oscillation center (xo, Yo) is determined by the external inputs i and ie' For small oscillation amplitudes, it can be approximated by a sinusoidal oscillator via linearization around the (xo, Yo): x = -h . g~(yo)Y - azx iI = w . g~(xo)x - a 1l y (4.4) where (x, y) is the deviation from (xo, Yo . The solution is X where a = (az + a y)/2 and w = hwg~(xo)g~(yo) = Toe-at sin(wt+<p) + (a z - a y)2/4. IT az = a y, which is about right in the bulb, w = Jhwg~(xo)g~(yo). For the bulb, a ~ 0.3w. The oscillation frequency depends on the synaptic strengths hand w, and is modulated by the receptor and central input via (xo, Yo). 405 406 Li and Hopfield N such mitral-granule pairs with cell interconnections between the pairs represent a group of N coupled non-linear damped oscillators. This is exactly the situation in the olfactory bulb. The locality of synaptic connections in the bulb implies that the oscillator coupling is also local. (That there are many more granule cells than mitral cells only means that there is more than one granule cell in each oscillator.) Corresponding to equation (4.2) and (4.4), we have equation (2.1) and , . X = -HoGy(Yo)Y - o.zX y = WoG~(Xo)X - ayY = -HY - o.zX, = (4.5) WX - o.yY. where (X, Y) are now deviations from (Xo, Yo) and G~(Xo) and G~(Yo) are diagonal matrices with elements: [G~(Xo)lii = g~(Xi,o) g~(Yilo) > > 0, [G~(Yo)lii = 0, for all i,j. Eliminating Y, (4.6) = where A HW the equation Xi = HoG~(Yo)WoG~(Xo). The ith oscillator (mitral cell) follows + (o.z + o.y)Xi + (Aii + o.zo.y)Xi + L AijXj = 0 (4.7) jt.i (cf.equation (4.1)), the the last term describes the coupling between oscillators. Non-linear effect occurs when the amplitude is large, and make the oscillation wave form non-sinusoidal. If X k is one of the eigenvectors of A with eigenvalue Ak, equation (4.6) has kth oscillation mode Components of Xk indicate oscillators' relative amplitudes and phases (for each k = 1,2, ... , N independent mode). For simplicity, we set 0.2: = 0.1/ = 0., then X ex: Xke-at?i../X,.t. Each mode has frequency Re~k' where Re means the real part of a complex number. If Re( -0. ? i~k) > 0 is satisfied for some k, then the amplitude of the kth mode will increase with time, i.e. growing oscillation. Starting from an initial condition of arbitrary small amplitudes in linear analysis, the mode with the fastest growing amplitude will dominate the output, and the whole bulb will oscillate in the same frequency as observed physiologically (Freeman 1978; Freeman and Schneider 1982) as well as in the simulation. With the non-linear effect, the strongest mode will suppress the others, and the final activity output will be a single "mode" in a non-linear regime. Modeling the Olfactory Bulb-Coupled Nonlinear Oscillators Because of the coupling between the (damped) oscillators, the equilibrium point (Xo, Yo) of a group of oscillators is no longer always stable with the possibility of growing oscillation modes. Ak must be complex in order to have kth mode grow. For this, a necessary (but not sufficient) condition is that matrix A is nonsymmetric. Those inputs that make matrix A less symmetric will more likely induce the oscillatory output and thus presumpably be noticed by the following olfactory cortex (see Li and Hopfield 1988a for details). The consequences (also observed physiologically) of our model are (Freeman 1975,1978; Freeman and Schneider 1982; Li and Hopfield 1988a): 1): local mitral cells' oscillation phase leads that of the local granule cells by a quarter cycle; 2): oscillations across the bulb have the same dominant frequency whose range possible should be narrow; 3): there should be a non-zero phase gradient field across the bulb; 4): the oscillation activity will rise during the inhale and fall at exhale, and rides on a slow background baseline shift wave phase locked with the sniff cycles. This model of the olfactory bulb can be generalized to other masses of interacting excitatory and inhibitory cells such as those in olfactory cortex, neocortex and hippocampus (Shepherd 1979) etc. where there may be connections between the excitatory cells as well as the inhibitory cells (Li and Hopfield 1988a). Suppose that Bo and Co are excitatory-to-excitatory and inhibitory-to-inhibitory connection matrices respectively, then equation (4.6) becomes x + (az - B + a y + C)k + (A + (a z - B)(a y + C))X = 0 (4.9) 5. COMPUTATIONS IN THE OLFACTORY BULB Receptor input I influences (Xo, Yo) as follows dXo ~ (a 2 + HW)-l(adI + di) dYo ~ (a 2 + W H)-l(W dI - aH-1di) (5.1) This is how the odor input determines the bulbar output. Increasing Iodor not only raises the mean activity level (Xo, Yo) (and thus the gain (G~(Xo), G~(Yo))), but also slowly changes the oscillation modes by structurally changing the oscillation equation (4.6) through matrix A = HoG~(Yo)WoG~(Xo). If (Xo, Yo) is raised to such an extent that Re( -a?iv'Ak) > 0 is satisfied for some mode k, the equilibrium point (Xo, Yo) becomes unstable and this mode emerges with oscillatory bursts. Different oscillation modes that emerge are indicative of the different odor inputs controlling the system parameters (Xo, Yo), and can be thought of as the decision states reached for odor information, i.e., the oscillation pattern classifies odors. When (Xo, Yo) is very low (e.g. before inhale), all modes are damped, and only small amplitUde oscillations occur, driven by noise and the weak time variation of the odor input. The absence of oscillation can be interpreted by higher processing 407 408 Li and Hopfield centers as the absence of an odor (Skarda and Freeman 1987). Detailed analysis shows how the bulb selectively responds (or not to respond) to certain input patterns (Li and Hopfield 1988a) by choosing the synaptic connections appropriately. This means the bulb can have non-uniform sensitivities to different odor receptor inputs and achieve better odor discriminations. 6. PERFORMANCE OPTIMIZATION IN THE BULB We discussed (Li and Hopfield 1988a) how the olfactory bulb makes the least information contamination between sniffs and changes the motivation level for odor discrimination. We further postulate with our model that the bulb, with appropriate inputs from the higher centers, can enhance or suppress the sensitivity to particular odors (details in Li and Hopfield 1988b). When the central input Ie is not fixed, it can control the bulbar output by shifting (Xo, Yo), just as the odor input I can, equation (5.1) becomes: dXo ~ (a 2 + HW)-l(adI + di - HdIe + aW-ldic) dYo ~ (a 2 + W H)-l(W dI - aH-ldi + adIe + die) (6.1) = Suppose that Ie Ie,ba.ekground + Ie,eontrol where Ie,eontrol is the control signal which changes during a sniff. Olfactory adaptation is achieved by having an Ie,eontrol = Iga.neel which cancels the effect of Iodor on Xo - cancelling. This keeps the mitral cells baseline output Gz(Xo) and gain G~(Xo) low, and thus makes the oscillation output impossible as if no odor exists. We can then expect that reversing the sign of Iga.neel will cause the bulb to have an enhanced, instead of reduced (adapted), response to Iodor - anti-cancelling, and achieve the olfactory enhancement. We can derive further phenomena such as recognizing an odor component in an odor mixture, cross-adaptation and cross-enhancement (Li and Hopfield 1988b). Computer simulations confirmed the expected results. 7. DISCUSSION Our model of the olfactory bulb is a simplification of the known anatomy and physiology. The net of the mitral and granule cells simulates a group of coupled non-linear oscillators which are the sources of the rhythmic activities in the bulb. The coupling makes the oscillation coherent across the bulb surface for each sniff. The model suggests, in agreement with Freeman and coworkers, that stability change bifurcation is used for the bulbar oscillator system to decide primitively on the relevance of the receptor input information. Different non-damping oscillation modes emerged are used to distinguish the different odor input information which is the driving source for the bifurcations, and are approximately thought of as the (unitary) decision states of the system for the odor information. With the extra information represented in the oscillation phases of the cells, the bulb emphasizes the differences between different input patterns (section 4). Both the analysis and simulation show that the bulb is selectively sensitive to different receptor input patterns. This selectivity as well as the motivation level of the animal could also be Modeling the Olfactory Bulb-Coupled Nonlinear Oscillators modulated from higher centers. This model also successfully applies to bulbar ability to use input from higher centers to suppress or enhance sensitivity to particular target or to mask odors. This model does not exclude the possibility that the information be coded in the non-oscillatory slow wave Xo which is also determined by the odor input. The chief behaviors do not depend on the number of cells in the model. The model can be generalized to olfactory cortex, hippocampus and neocortex etc. where there are more varieties of synaptic organizations. Acknowledgements This research was supported by ONR contract NOO014-87-K-0377. We would also like to acknowledge discussions with J.A. Bower. References Baird B. Nonlinear dynamics of pattern formation and pattern recognition in rabbit olfactory bulb. Physica 22D, 150-175 (1986) Freeman W.J. Mass action in the nervous system. New York: Academic Press 1975 Freeman W.J. Spatial properties of an EEG event in the olfactory bulb and cortex. Electroencephalogr. Clin. Neurophysiol. 44, 586-605 (1978) Freeman W.J. Nonlinear Gain mediating cortical stimulus-response relations. BioI. Cybernetics 33, 237-247 (1979a) Freeman W.J. Nonlinear dynamics of paleocortex manifested in the olfactory EEG. BioI. Cybernetics 35, 21-37 (1979b) Freeman W.J. EEG analysis gives model of neuronal template-matching mechanism for sensory search with olfactory bulb. BioI. Cybernetics 35, 221-234 (1979c) Freeman W.J. Use of spatial deconvolution to compensate for distortion of EEG by volume conduction. IEEE Trans. Biomed. Engineering 21,421-429 (1980) Freeman W.J., Schneider W.S. Changes in spatial patterns of rabbit olfactory EEG with conditioning to odors. Psychophysiology 19, 44-56 (1982) Freeman W.J., Skarda C.A. Spatial EEG patterns, non-linear dynamics and perception: the Neo-Sherringtonian view. Brain Res. Rev. 10, 147-175 (1985) Getchell T.V., Shepherd G.M. Responses of olfactory receptor cells to step pulses of odour at different concentrations in the salamender. J. Physiol. 282, 521-540 (1978) Lancet D. Vertebrate olfactory reception Ann. Rev. Neurosci. 9, 329-355 (1986) Li Z., Hopfield J.J. Modeling the olfactory bulb. Submitted to Biological Cybernetics (1988a) Li Z., Hopfield J.J. A model of olfactory adaptation and enhancement in the olfactory bulb. In preparation. (1988b) Scott J. W. The olfactory bulb and central pathways. Experientia 42,223-232 (1986) Shepherd G.M. The synaptic organization of the brain. New York: Oxford University Press 1979 Shepherd G.M. Private communications. (1988) Skarda C.A., Freeman W.J. How brains make chaos in order to make sense of the world. Behavioral and Brain Sciences 10, 161-195 (1987) 409 

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Estimating Dependency Structure as a Hidden Variable Marina Meill and Michael I. Jordan {mmp, jordan}@ai.mit.edu Center for Biological & Computational Learning Massachusetts Institute of Technology 45 Carleton St. E25-201 Cambridge, MA 02142 Abstract This paper introduces a probability model, the mixture of trees that can account for sparse, dynamically changing dependence relationships. We present a family of efficient algorithms that use EM and the Minimum Spanning Tree algorithm to find the ML and MAP mixture of trees for a variety of priors, including the Dirichlet and the MDL priors. 1 INTRODUCTION A fundamental feature of a good model is the ability to uncover and exploit independencies in the data it is presented with. For many commonly used models, such as neural nets and belief networks, the dependency structure encoded in the model is fixed, in the sense that it is not allowed to vary depending on actual values of the variables or with the current case. However, dependency structures that are conditional on values of variables abound in the world around us. Consider for example bitmaps of handwritten digits. They obviously contain many dependencies between pixels; however, the pattern of these dependencies will vary across digits. Imagine a medical database recording the body weight and other data for each patient. The body weight could be a function of age and height for a healthy person, but it would depend on other conditions if the patient suffered from a disease or was an athlete. Models that are able to represent data conditioned dependencies are decision trees and mixture models, including the soft counterpart of the decision tree, the mixture of experts. Decision trees however can only represent certain patterns of dependecy, and in particular are designed to represent a set of conditional probability tables and not a joint probability distribution. Mixtures are more flexible and the rest of this paper will be focusing on one special case called the mixtures of trees. We will consider domains where the observed variables are related by pairwise dependencies only and these dependencies are sparse enough to contain no cycles. Therefore they can Estimating Dependency Structure as a Hidden Variable 585 be represented graphically as a tree. The structure of the dependencies may vary from one instance to the next. We index the set of possible dependecy structures by a discrete structure variable z (that can be observed or hidden) thereby obtaining a mixture. In the framework of graphical probability models, tree distributions enjoy many properties that make them attractive as modelling tools: they have a flexible topology, are intuitively appealing, sampling and computing likelihoods are linear time, simple efficient algorithms for marginalizing and conditioning (O(1V1 2 ) or less) exist. Fitting the best tree to a given distribution can be done exactly and efficiently (Chow and Liu, 1968). Trees can capture simple pairwise interactions between variables but they can prove insufficient for more complex distributions. Mixtures of trees enjoy most of the computational advantages of trees and, in addition, they are universal approximators over the space of all distributions. Therefore, they are fit for domains where the dependency patterns become tree like when a possibly hidden variable is instantiated. Mixture models have been extensively used in the statistics and neural network literature. Of relevance to the present work are the mixtures of Gaussians, whose distribution space, in the case of continuous variables overlaps with the space of mixtures of trees. Work on fitting a tree to a distribution in a Maximum-Likelihood (ML) framework has been pioneered by (Chow and Liu, 1968) and was extended to poly trees by (Pearl, 1988) and to mixtures of trees with observed structure variable by (Geiger, 1992; Friedman and Goldszmidt, 1996). Mixtures of factorial distributions were studied by (Kontkanen et al., 1996) whereas (Thiesson et aI., 1997) discusses mixtures of general belief nets. Multinets (Geiger, 1996) which are essentially mixtures of Bayes nets include mixtures of trees as a special case. It is however worth studying mixtures of trees separately for their special computational advantages. This work presents efficient algorithms for learning mixture of trees models with unknown or hidden structure variable. The following section introduces the model; section 3 develops the basic algorithm for its estimation from data in the ML framework. Section 4 discusses the introduction of priors over mixtures of trees models and presents several realistic factorized priors for which the MAP estimate can be computed by a modified versions of the basic algorithm. The properties of the model are verified by simulation in section 5 and section 6 concludes the paper. 2 THE MIXTURE OF TREES MODEL In this section we will introduce the mixture of trees model and the notation that will be used throughout the paper. Let V denote the set of variables of interest. According to the graphical model paradigm, each variable is viewed as a vertex of a graph. Let Tv denote the number of values of variable v E V, XV a particular value of V, XA an assignment to the variables in the subset A of V. To simplify notation Xv will be denoted by x. We use trees as graphical representations for families of probability distributions over V that satisfy a common set of independence relationships encoded in the tree topology. In this representation, an edge of the tree shows a direct dependence, or, more precisely, the absence of an edge between two variables signifies that they are independent, conditioned on all the other variables in V. We shall call a graph that has no cycles a tree I and shall denote by E the set of its (undirected) edges. A probability distribution T that is conformal with the tree (V, E) is a distribution that can be factorized as: T (X) = IT(u,v)EE Tuv (xu, xv) IT vEV T,v (x v )degv-l (1) Here deg v denotes the degree of v, e.g. the number of edges incident to node v E V. The l In the graph theory literature, our definition corresponds to a forest. The connected components of a forest are called trees. M. MeillJ and M. I. Jordan 586 factors Tuv and Tv are the marginal distributions under T: Tuv(xu,xv) = 2: T(xu , xv,XV-{u ,v}), 2: T(xv,xv-{ v}) ' Tv(xv) = (2) Xv-tv} XV-{u.v} The distribution itself will be called a tree when no confusion is possible. Note that a tree distribution has for each edge (u, v) E E a factor depending on xu, Xv onlyl If the tree is connected, e.g. it spans all the nodes in V , it is often called a spanning tree. An equivalent representation for T in terms of conditional probabilities is T(x) = II Tvlpa(v)(xvlxpa(v?) (3) vEV The form (3) can be obtained from (1) by choosing an arbitrary root in each connected component and recursively substituting Tv t';V) by Tvlpa(v) starting from the root. pa(v) represents the parent of v in the thus directed tree or the empty set if v is the root of a connected component. The directed tree representation has the advantage of having independent parameters. The total number of free parameters in either representation is E(u,v)EET rurv - EVEv(degv - l)rv . Now we define a mixture of trees to be a distribution of the form m Q(X) = 2: AkTk(x); Ak 2: 0, k = 1, . .. , m; (4) k=1 From the graphical models perspecti ve, a mixture of trees can be viewed as a containing an unobserved choice variable z, taking value k E {I, ... m} with probability Ak. Conditioned on the value of z the distribution of the visible variables X is a tree. The m trees may have different structures and different parameters. Note that because of the structure variable, a mixture of trees is not properly a belief network, but most of the results here owe to the belief network perspective. 3 THE BASIC ALGORITHM: ML FITIING OF MIXTURES OF TREES This section will show how a mixture of trees can be fit to an observed dataset in the Maximum Likelihood paradigm via the EM algorithm (Dempster et al., 1977). The observations are denoted by {xl , x 2 , ... , xN}; the corresponding values of the structure variable are { zi,i=I, ... N}. Following a usual EM procedure for mixtures, the Expectation (E) step consists in estimating the posterior probability of each tree to generate datapoint xi Pr[zi = klxl ,.. .,N, model] = 'Yk(i) = AkTk(x:). Lkl AklTk (x') (5) Then the expected complete log-likelihood to be maximized by the M step of the algorithm is m E[Ic Ixl ,...N , model] N L rk[log Ak + L k=1 pk(x i ) 10gTk(xi )] (6) i=1 N rk = 2: 'Yk(X i ), (7) i=1 The maximizing values for the parameters A are Akew = rk/ N. To obtain the new distributions Tk, we have to maximize for each k the expression that is the negative of the 587 Estimating Dependency Structure as a Hidden Variable Figure 1: The Basic Algorithm: ML Fitting of a Mixture of Trees Input:Dataset {xl, ... xN} Initial model m, Tk, ,\k, k = I, . .. m Procedure MST( weights) that fits a maximum weight spanning tree over V Iterate until convergence Estep: compute'Y~, pk(X') fork = I, . .. m, i= 1, . . . Nby(5),(7) Mstep: Ml. '\k +- rk/N, k = I, ... m M2. compute marginals P:, p!v, U, v E V, k = I , ... m MJ. compute mutual information I!v u, v E V , k I , ... m M4. call MST( { I!v }) to generate ETk for k = I, ... m M5. T!v +- p!v, ; T: +- P: for (u, v) E ETk, k = I, ... m = crossentropy between pk and Tk. N L pk(x i ) 10gTk(xi) (8) i=l This problem can be solved exactly as shown in (Chow and Liu, 1968). Here we will give a brief description of the procedure. First, one has to compute the mutual information between each pair of variables in V under the target distribution P JUti '"" ( ) PUtl(x u , Xtl) = Jvu = L.J P uv Xu, Xv log Pu(xu)PtI(x v )' u, v E V, u f=v . (9) X",X v Second, the optimal tree structure is found by a Maximum Spanning Tree (MST) algorithm using JUti as the weight for edge (u, v), \lu, v E V.Once the tree is found, its marginals Tutl (or Tul v ), (u, v) E ET are exactly equal to the corresponding marginals P Uti of the target distribution P. They are already computed as an intermediate step in the computation of the mutual informations JUti (9). In our case, the target distribution for Tk is represented by the posterior sample distribution pk. Note that although each tree fit to pk is optimal, for the encompassing problem of fitting a mixture of trees to a sample distribution only a local optimum is guaranteed to be reached. The algorithm is summarized in figure 1. This procedure is based on one important assumption that should be made explicit now. It is the Parameter independence assumption: The distribution T:1pa( tI) for any k, v and value of pa( v) is a multinomial with rv - 1 free parameters that are independent of any other parameters of the mixture. It is possible to constrain the m trees to share the same structure, thus constructing a truly Bayesian network. To achieve this, it is sufficient to replace the weights in step M4 by Lk J~tI and run the MST algorithm only once to obtain the common structure ET. The tree stuctures obtained by the basic algorithm are connected. The following section will give reasons and ways to obtain disconnected tree structures. 4 MAP MIXTURES OF TREES In this section we extend the basic algorithm to the problem of finding the Maximum a Posteriori (MAP) probability mixture of trees for a given dataset. In other words, we will consider a nonuniform prior P[mode/] and will be searching for the mixture of trees that maximizes log P[model\x1 ,.. .N] = 10gP[xl, ... N\model] + log P[model] + constant. (10) Factorized priors The present maximization problem differs from the ML problem solved in the previous section only by the addition of the term log P[model]. We can as well M. Meilii and M. l. Jordan 588 approach it from the EM point of view, by iteratively maximizing E [logP[modelJxl ,...N, ZI , ...NJ] = E[lc{xl ,...N, zl ,... NJmodel)] + 10gP[model] (11) It is easy to see that the added term does not have any influence on the E step,which will proceed exactly as before. However, in the M step, we must be able to successfully maximize the r.h.s. of (11). Therefore, we look for priors of the form m P[model] = P[AI, .. .m] II P[Tkl (12) k=1 This class of priors is in agreement with the parameter independence assumption and includes the conjugate prior for the multinomial distribution which is the Dirichlet prior. A Dirichlet prior over a tree can be represented as a table of fictitious marginal probabilities P~~ for each pair u , v of variables plus an equivalent sample size Nt that gives the strength of the prior (Heckerman et al., 1995). However, for Dirichlet priors, the maximization over tree structures (corresponding to step M4) can only be performed iteratively (Meilli et al., 1997). MDL (Minimum Description Length) priors are less informative priors. They attempt to balance the number of parameters that are estimated with the amount of data available, usually by introducing a penalty on model complexity. For the experiments in section 5 we used edge pruning. More smoothing methods are presented in (Meilli et al., 1997). To penalize the number of parameters in each component we introduce a prior that penalizes each edge that is added to a tree, thus encouraging the algorithm to produce disconnected trees. The edge pruning prior is P [T] <X exp [-(3 L:utJ EET L\utJ] . We choose a uniform penalty L\utJ = 1. Another possible choice is L\utJ = (r u - 1)( rtJ - 1) which is the number of parameters introduced by the presence of edge (u , v) w.r.t. a factorized distribution. Using this prior is equivalent to maximizing the following expression in step M4 of the Basic Algorithm (the index k being dropped for simplicity) argmax L: ET utJEET max[O, r1utJ - (3~tJ] = argmax ET L: WutJ (13) utJEET 5 EMPIRICAL RESULTS We have tested our model and algorithms for their ability to retrieve the dependency structure in the data, as classifiers and as density estimators. For the first objective, we sampled 30,000 datapoints from a mixture of 5 trees over 30 variables with rtJ = 4 for all vertices. All the other parameters of the generating model and the initial points for the algorithm were picked at random. The results on retrieving the original trees were excellent: out of 10 trials, the algorithm failed to retrieve correctly only 1 tree in 1 trial. This bad result can be accounted for by sampling noise. The tree that wasn't recovered had a A of only 0.02. The difference between the log likelihood of the samples of the generating tree and the approximating tree was 0.41 bits per example. For classification, we investigated the performance of mixtures of trees on a the Australian Credit dataset from the UCI repository2. The data set has 690 instances of 14-dimensional attribute vectors. Nine attributes are discrete ( 2 - 14 values) and 5 are continuous. The class variable has 6 values. The continuous variables were discretized in 3 - 5 uniform bins each. We tested mixtures with different values for m and for the edge pruning parameter (3. For comparison we tried also mixtures of factorial distributions of different sizes. One tenth of the data, picked randomly at each trial, was used for testing and the rest for training. In the training phase, we learned a MT model of the joint distribution of all the 15 variables. 2 http://www.ics . uci.edu/-mlearn/MLRepository.html Estimating Dependency Structure as a Hidden Variable 589 Figure 2: Performance of different algorithms on the Australian Credit dataset. - is mixture of trees with j3 = 10, - - is mixture of trees with beta = 11m, -.- is mixture of factorial distributions. 88 87 86 C1) 584 u 81 , ,, ~'i .- ~83 82 , -,, (385 I ~ '? t7 .., I ... I I' I ' 800 5 10 15 m 20 25 30 Table 1: a) Mixture of trees compression rates [log lte&t! N te &t1. b) Compression rates (bits/digit) for the single digit (Digit) and double digit (Pairs) datasets. MST is mixtures of trees, MF is a mixture of factorial distributions, BR is base rate model, H-WS is Helmholtz Machine trained with the wake-sleep algorithm (Frey et aI., 1996), H-MF is Helmholtz Machine trained with the Mean Field approximation, FV is a fully visible bayes net. (*=best) (a) (b) Algorithm Digits Pairs gzip 44.3 89.2 m Digits Pairs BR 59.2 118.4 16 *34.72 79.25 MF 37.5 92.7 32 34.48 *78.99 H-MF 80.7 39.5 64 79.70 34.84 H-WS 39.1 80.4 128 34.88 81 .26 FV 35.9 *72.9 *34.7 79.0 MT In the testing phase, the output of our classifier was chosen to be the class value with the largest posterior probability given the inputs. Figure 2 shows that the results obtained for mixtures of trees are superior to those obtained for mixtures of factorial distributions.For comparison, correct classification rates obtained and cited in (Kontkanen et aI., 1996) on training/test sets of the same size are: 87.2next best model (a decision tree called CaI50). We also tested the basic algorithm as a density estimator by running it on a subset of binary vector representations of handwritten digits and measuring the compression rate. One dataset contained images of single digits in 64 dimensions, the second contained 128 dimensional vectors representing randomly paired digit images. The training, validation and test set contained 6000, 2000, and 5000 exemplars respectively. The data sets, the training conditions and the algorithms we compared with are described in (Frey et aI., 1996). We tried mixtures of 16, 32, 64 and 128 trees, fitted by the basic algorithm. The results (shown in table1 averaged over 3 runs) are very encouraging: the mixture of trees is the absolute winner for compressing the simple digits and comes in second as a model for pairs of digits. This suggests that our model (just like the mixture of factorized distributions) is able to perform good compression of the digit data but is unable to discover the independency in the double digit set. 590 M. Meilli and M. I. Jordan 6 CONCLUSIONS This paper has shown a method of modeling and exploiting sparse dependency structure that is conditioned on values of the data. By using trees, our method avoids the exponential computation demands that plague both inference and structure finding in wider classes of belief nets. The algorithms presented here are linear in m and N and quadratic in IV I. Each M step is performing exact maximization over the space of all the tree structures and parameters. The possibility of pruning the edges of the components of a mixture of trees can playa role in classification, as a means of automatically selecting the variables that are relevant for the task. The importance of using the right priors in constructing models for real-world problems can hardly be understated. In this context, the present paper has presented a broad class of priors that are efficiently handled in the framework of our algorithm and it has shown that this class includes important priors like the MDL prior and the Dirichlet prior. Acknowledgements Thanks to Quaid Morris for running the digits and structure finding experiments and to Brendan Frey for providing the digits datasets. References Chow, C. K. and Liu, C. N. (1968). Approximating discrete probability distributions with dependence trees. "IEEE Transactions on Information Theory ", IT-14(3 ):462-467. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B, 39: 1-38. Frey, B. J., Hinton, G. E., and Dayan, P. (1996). Does the wake-sleep algorithm produce good density estimators? In Touretsky, D., Mozer, M., and Hasselmo, M., editors, Neural Information Processing Systems, number 8, pages 661-667. MIT Press. Friedman, N. and Goldszmidt, M. (1996). Building classifiers using Bayesian networks. In Proceedings of the National Conference on Artificial Intelligence (AAAI 96), pages 1277-1284, Menlo Park, CA. AAAI Press. Geiger, D. (1992). An entropy-based learning algorithm of bayesian conditional trees. In Proceedings of the 8th Conferenceon Uncertainty in AI, pages 92-97. Morgan Kaufmann Publishers. Geiger, D. (1996). Knowledge representation and inference in similarity networks and bayesian multinets. "Artificial Intelligence", 82:45-74. Heckerman, D., Geiger, D., and Chickering, D. M. (1995). Learning Bayesian networks: the combination of knowledge and statistical data. Machine Learining, 20(3): 197-243. Kontkanen, P., Myllymaki, P., and Tirri, H. (1996). Constructing bayesian finite mixture models by the EM algorithm. Technical Report C-1996-9, Univeristy of Helsinky. Department of Computer Science. Meilli, M., Jordan, M. I., and Morris, Q. D. (1997). Estimating dependency structure as a hidden variable. Technical Report AIM-1611 ,CBCL-151, Massachusetts Institute of Technology, Artificial Intelligence Laboratory. Pearl, 1. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufman Publishers, San Mateo, CA. Thiesson, B., Meek, C., Chickering, D. M., and Heckerman, D. (1997). Learning mixtures of Bayes networks . Technical Report MSR-POR-97-30, Microsoft Research. 

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Hippocampal Model of Rat Spatial Abilities Using Temporal Difference Learning David J Foster* Centre for Neuroscience Edinburgh University Richard GM Morris Centre for Neuroscience Edinburgh University Peter Dayan E25-210, MIT Cambridge, MA 02139 Abstract We provide a model of the standard watermaze task, and of a more challenging task involving novel platform locations, in which rats exhibit one-trial learning after a few days of training. The model uses hippocampal place cells to support reinforcement learning, and also, in an integrated manner, to build and use allocentric coordinates. 1 INTRODUCTION Whilst it has long been known both that the hippocampus of the rat is needed for normal performance on spatial tasks l3 , 11 and that certain cells in the hippocampus exhibit place-related firing,12 it has not been clear how place cells are actually used for navigation. One of the principal conceptual problems has been understanding how the hippocampus could specify or learn paths to goals when spatially tuned cells in the hippocampus respond only on the basis of the rat's current location. This work uses recent ideas from reinforcement learning to solve this problem in the context of two rodent spatial learning results. Reference memory in the watermaze l l (RMW) has been a key task demonstrating the importance of the hippocampus for spatial learning. On each trial, the rat is placed in a circular pool of cloudy water, the only escape from which is a platform which is hidden (below the water surface) but which remains in a constant position. A random choice of starting pOSition is used for each trial. Rats take asymptotically short paths after approximately 10 trials (see figure 1 a). Delayed match-to-place (DMP) learning is a refined version in which the platform'S location is changed on each day. Figure 1b shows escape latencies for rats given four trials per day for nine days, with the platform in a novel position on each day. On early days, acquisition ?Crichton Street, Edinburgh EH8 9LE, United Kingdom. Funded by Edin. Univ. Holdsworth Scholarship, the McDonnell-Pew foundation and NSF grant IBN-9634339. Email: djf@cfn.ed.ac.uk D. J Foster; R. G. M. Mo"is and P. Dayan 146 100 100 a 90 90 b 80 _ 70 '" i oo '" 50 -' ~40 ~30 20 10 13 17 21 2S Figure 1: a) Latencies for rats on the reference memory in the watermaze (RMW) task (N=8). b) Latencies for rats on the Delayed Match-to-Place (DMP) task (N=62). is gradual but on later days, rats show one-trial learning, that is, near asymptotic performance on the second trial to a novel platform position. The RMW task has been extensively modelled. 6,4,5,20 By contrast, the DMP task is new and computationally more challenging. It is solved here by integrating a standard actor-critic reinforcement learning system2 ,7 which guarantees that the rat will be competent to perform well in arbitrary mazes, with a system that learns spatial coordinates in the maze. Temporal difference learning1 7 (TO) is used for actor, critic and coordinate learning. TO learning is attractive because of its generality for arbitrary Markov decision problems and the fact that reward systems in vertebrates appear to instantiate it. 14 2 THEMODEL The model comprises two distinct networks (figure 2): the actor-critic network and a coordinate learning network. The contribution of the hippocampus, for both networks, is to provide a state-space representation in the form of place cell basis functions. Note that only the activities of place cells are required, by contrast with decoding schemes which require detailed information about each place cell. 4 COORDINATE SYSTEM ACTOR-CRITIC SYSTEM r------------Remembered 1 Goal coordinates 1 1 1 VECTOR COMPUTA nONI ~ Coordinate Representation 1 1______ -------1 Figure 2: Model diagram showing the interaction between actor-critic and coordinate system components. Hippocampal Model of Rat Spatial Abilities Using TD Learning 147 2.1 Actor-Critic Learning Place cells are modelled as being tuned to location. At position p, place cell i has an output given by h(p) = exp{ -lip - sdI2/2(12}, where Si is the place field centre, and (1 = 0.1 for all place fields. The critic learns a value function V(p) wih(p) which comes to represent the distance of p from the goal, using = L:i the TO rule 6.w~ ex: 8t h(pt), where (1) is the TD error, pt is position at time t, and the reward r(pt, pt+I) is 1 for any move onto the platform, and 0 otherwise. In a slight alteration of the original rule, the value V (p) is set to zero when p is at the goal, thus ensuring that the total future rewards for moving onto the goal will be exactly 1. Such a modification improves stability in the case of TD learning with overlapping basis functions. The discount factor, I' was set to 0.99. Simultaneously the rat refines a policy, which is represented by eight action cells. Each action cell (aj in figure 2) receives a parameterised input at any position p: aj (p) = L:i qjdi (p). An action is chosen stochastically with probabilities given by P(aj) = exp{2aj}/ L:k exp{2ak}. Action weights are reinforced according to: 2 (2) where 9j((Jt) is a gaussian function of the difference between the head direction (Jt at time t and the preferred direction of the jth action cell. Figure 3 shows the development of a policy over a few trials. V(p)l Triall V(p) 1 0.5 0.5 0. 0.5 01 0.5 TrialS Triall3 V(P)l 0.5 I 0 0.5 0: 0.5 0.5 -0.5 -------0.5 .---'--- 0 0.5 Figure 3: The RMW task: the value function gradually disseminates information about reward proximity to all regions of the environment. Policies and paths are also shown. There is no analytical guarantee for the convergence of TD learning with policy adaptation. However our simulations show that the algorithm always converges for the RMW task. In a simulated arena of diameter 1m and with swimming speeds of 20cm/s, the simulation matched the performance of the real rats very closely (see figure S). This demonstrates that TD-based reinforcement learning is adequately fast to account for the learning performance of real animals. D. 1. Foster, R. G. M Morris and P. Dayan 148 2.2 Coordinate Learning Although the learning of a value function and policy is appropriate for finding a fixed platform, the actor-critic model does not allow the transfer of knowledge from the task defined by one goal position to that defined by any other; thus it could not generate the sort of one-trial learning that is shown by rats on the DMP task (see figure 1b). This requires acquisition of some goal-independent know ledge about s~ace. A natural mechanism for this is the path integration or self-motion system. 0,10 However, path integration presents two problems. First, since the rat is put into the maze in a different position for each trial, how can it learn consistent coordinates across the whole maze? Second, how can a general, powerful, but slow, behavioral learning mechanism such as TO be integrated with a specific, limited, but fast learning mechanism involving spatial coordinates? Since TO critic learning is based on enforcing consistency in estimates of future reward, we can also use it to learn spatially consistent coordinates on the basis of samples of self-motion. It is assumed that the rat has an allocentric frame of reference. 1s The model learns parameterised estimates of the x and y coordinates of all positions p: x(p) = Li w[ fi(P) and y(p) = Li wY h(p), Importantly, while place cells were again critical in supporting spatial representation, they do not embody a map of space. The coordinate functions, like the value function previously, have to be learned. As the simulated rat moves around, the coordinate weights {w[} are adjusted according to: t Llwi ()( (Llxt + X(pt+l ) - X(pt)) LA t- k h(pk) (3) k=1 where Llxt is the self-motion estimate in the x direction. A similar update is applied to {wn. In this case, the full TO(A) algorithm was used (with A = 0.9); however TD(O) could also have been used, taking slightly longer. Figure 4a shows the x and y coordinates at early and late phases of learning. It is apparent that they rapidly become quite accurate - this is an extremely easy task in an open field maze. An important issue in the learning of coordinates is drift, since the coordinate system receives no direct information about the location of the origin. It turns out that the three controlling factors over the implicit origin are: the boundary of the arena, the prior setting of the coordinate weights (in this case all were zero) and the position and prior value of any absorbing area (in this case the platform). If the coordinate system as a whole were to drift once coordinates have been established, this would invalidate coordinates that have been remembered by the rat over long periods. However, since the expected value of the prediction error at time steps should be zero for any self-consistent coordinate mapping, such a mapping should remain stable. This is demonstrated for a single run: figure 4b shows the mean value of coordinates x evolving over trials, with little drift after the first few trials. We modeled the coordinate system as influencing the choice of swimming direction in the manner of an abstract action. I5 The (internally specified) coordinates of the most recent goal position are stored in short term memory and used, along with the current coordinates, to calculate a vector heading. This vector heading is thrown into the stochastic competition with the other possible actions, governed by a single weight which changes in a similar manner to the other action weights (as in equation 2, see also fig 4d), depending on the TO error, and on the angular proximity of the current head direction to the coordinate direction. Thus, whether the the coordinate-based direction is likely to be used depends upon its past performance. One simplification in the model is the treatment of extinction. In the DMP task, Hippocampal Model ofRat Spatial Abilities Using 1D Learning 149 ,. " TJUAL d III .1 TRIAL 26 16 i: ~Ol ~o !" ~o Figure 4: The evolution of the coordinate system for a typical simulation run: a.) coordinate outputs at early and late phases of learning, b.) the extent of drift in the coordinates, as shown by the mean coordinate value for a single run, c.) a measure A2 ~ ~ {X (Pr.)-X -X(pr.)}2 ? o f coord mate error for the same run (7E = r r. (Np-l)N ' where k r r r indexes measurement points (max N p ) and r indexes runs (max N r ), Xr(Pk) is the model estimate of X at position Pk, X(Pk) is the ideal estimate for a coordinate system centred on zero, and Xr is the mean value over all the model coordinates, d.) the increase during training of the probability of choosing the abstract action. This demonstrates the integration of the coordinates into the control system. real rats extinguish to a platform that has moved fairly quickly whereas the actorcritic model extinguishes far more slowly. To get around this, when a simulated rat reaches a goal that has just been moved, the value and action weights are reinitialised, but the coordinate weights wf and wf, and the weights for the abstract action, are not. 3 RESULTS The main results of this paper are the replication by simulation of rat performance on the RMW and DMP tasks. Figures la and b show the course of learning for the rats; figures Sa and b for the model. For the DMP task, one-shot acquisition is apparent by the end of training. 4 DISCUSSION We have built a model for one-trial spatial learning in the watermaze which uses a single TD learning algorithm in two separate systems. One system is based on a reinforcement learning that can solve general Markovian decision problems, and the other is based on coordinate learning and is specialised for an open-field water maze. Place cells in the hippocampus offer an excellent substrate for learning the actor, the critic and the coordinates. The model is explicit about the relationship between the general and specific learning systems, and the learning behavior shows that they integrate seamlessly. As currently constituted, the coordinate system would fail if there were a barrier in the maze. We plan to extend the model to allow the coordinate system to specify abstract targets other than the most recent platform position - this could allow it fast navigation around a larger class of environments. It is also important to improve the model of learning 'set' behavior - the information about the nature of D. 1. Foster; R. G. M. Mo"is and P. Dayan 150 14 a b 12 10 . . ?. z j:10\ ~ ~ ~ 12 ~ S> .. '"~ ............................................ 0~D.~yl~~y~2~D~.y~3~D~.y~47~~yS~~~.~~~y~7~~~y~.7D.~y9~ Figure 5: a.) Performance of the actor-critic model on the RMW task, and b.) performance of the full model on the DMP task. The data for comparison is shown in figures la and b. the DMP task that the rats acquire over the course of the first few days of training. Interestingly, learning set is incomplete - on the first trial of each day, the rats still aim for the platform position on the previous day, even though this is never correct. 16 The significant differences in the path lengths on the first trial of each day (evidence in figure Ib and figure 5b) come from the relative placements of the platforms. However, the model did not use the same positions as the empirical data, and, in any case, the model of exploration behavior is rather simplistic. The model demonstrates that reinforcement learning methods are perfectly fast enough to match empirical learning curves. This is fortunate, since, unlike most models specifically designed for open-field navigation,6,4,5,2o RL methods can provably cope with substantially more complicated tasks with arbitrary barriers, etc, since they solve the temporal credit assignment problem in its full generality. The model also addresses the problem that coordinates in different parts of the same environment need to be mutually consistent, even if the animal only experiences some parts on separate trials. An important property of the model is that there is no requirement for the animal to have any explicit knowledge of the relationship between different place cells or place field position, size or shape. Such a requirement is imposed in various models. 9 ,4,6,2o Experiments that are suggested by this model (as well as by certain others) concern the relationship between hippocampally dependent and independent spatial learning. First, once the coordinate system has been acquired, we predict that merely placing the rat at a new location would be enough to let it find the platform in one shot, though it might be necessary to reinforce the placement e.g. by first placing the rat in a bucket of cold water. Second, we know that the establishment of place fields in an environment happens substantiallr faster than establishment of one-shot or even ordinary learning to a platform. 2 We predict that blocking plasticity in the hippocampus following the establishment of place cells (possibly achieved without a platform) would not block learning of a platform. In fact, new experiments show that after extensive pre-training, rats can perform one-trial learning in the same environment to new platform positions on the DMP task without hippocampal synaptic plasticity. 16 This is in contrast to the effects of hippocampal lesion, which completely disrupts performance. According to the model, coordinates will have been learned during pre-training. The full prediction remains untested: that once place fields have been established, coordinates could be learned in the absence of hippocampal synaptic plasticity. A third prediction follows from evidence that rats with restricted hippocampal lesions can learn the fixed platform Hippocampal Model of Rat Spatial Abilities Using TD Learning 151 task, but much more slowly, based on a gradual "shaping" procedure. 22 In our model, they may also be able to learn coordinates. However, a lengthy training procedure could be required, and testing might be complicated if expressing the knowledge required the use of hippocampus dependent short-term memory for the last platform location. I6 One way of expressing the contribution of the hippocampus in the model is to say that its function is to provide a behavioural state space for the solution of complex tasks. Hence the contribution of the hippocampus to navigation is to provide place cells whose firing properties remain consistent in a given environment. It follows that in different behavioural situations, hippocampal cells should provide a representation based on something other than locations - and, indeed, there is evidence for this. 8 With regard to the role of the hippocampus in spatial tasks, the model demonstrates that the hippocampus may be fundamentally necessary without embodying a map. References [1] Barto, AG & Sutton, RS (1981) BioI. Cyber., 43:1-8. [2] Barto, AG, Sutton, RS & Anderson, CW (1983) IEEE Trans. on Systems, Man [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] and Cybernetics 13:834-846. Barto, AG, Sutton, RS & Watkins, CJCH (1989) Tech Report 89-95, CAIS, Univ. Mass., Amherst, MA. Blum, KI & Abbott, LF (1996) Neural Computation, 8:85-93. Brown, MA & Sharp, PE (1995) Hippocampus 5:171-188. Burgess, N, Reece, M & O'Keefe, J (1994) Neural Networks, 7:1065-1081. Dayan, P (1991) NIPS 3, RP Lippmann et aI, eds., 464-470. Eichenbaum, HB (1996) CurroOpin. Neurobiol., 6:187-195. Gerstner, W & Abbott, LF (1996) J. Computational Neurosci. 4:79-94. McNaughton, BL et a1 (1996) J. Exp. BioI., 199:173-185. Morris, RGM et al (1982) Nature, 297:681-683. O'Keefe, J & Dostrovsky, J (1971) Brain Res., 34(171). Olton, OS & Samuelson, RJ (1976) J. Exp. Psych: A.B.P., 2:97-116. Rudy, JW & Sutherland, RW (1995) Hippocampus, 5:375-389. SchUltz, W, Dayan, P & Montague, PR (1997) Science, 275, 1593-1599. Singh, SP Reinforcement learning with a hierarchy of abstract models. Steele, RJ & Morris, RGM in preparation. Sutton, RS (1988) Machine Learning, 3:9-44. Taube, JS (1995) J. Neurosci. 15(1):70-86. Tsitsiklis, IN & Van Roy, B (1996) Tech Report LIDS-P-2322, M.LT. Wan, HS, Touretzky, OS & Redish, AD (1993) Proc. 1993 Connectionist Models Summer School, Lawrence Erlbaum, 11-19. Watkins, CJCH (1989) PhD Thesis, Cambridge. Whishaw, IQ & Jarrard, LF (1996) Hippocampus Wilson, MA & McNaughton, BL (1993) Science 261:1055-1058. 

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Hippocampal Model of Rat Spatial Abilities Using Temporal Difference Learning David J Foster* Centre for Neuroscience Edinburgh University Richard GM Morris Centre for Neuroscience Edinburgh University Peter Dayan E25-210, MIT Cambridge, MA 02139 Abstract We provide a model of the standard watermaze task, and of a more challenging task involving novel platform locations, in which rats exhibit one-trial learning after a few days of training. The model uses hippocampal place cells to support reinforcement learning, and also, in an integrated manner, to build and use allocentric coordinates. 1 INTRODUCTION Whilst it has long been known both that the hippocampus of the rat is needed for normal performance on spatial tasks l3 , 11 and that certain cells in the hippocampus exhibit place-related firing,12 it has not been clear how place cells are actually used for navigation. One of the principal conceptual problems has been understanding how the hippocampus could specify or learn paths to goals when spatially tuned cells in the hippocampus respond only on the basis of the rat's current location. This work uses recent ideas from reinforcement learning to solve this problem in the context of two rodent spatial learning results. Reference memory in the watermaze l l (RMW) has been a key task demonstrating the importance of the hippocampus for spatial learning. On each trial, the rat is placed in a circular pool of cloudy water, the only escape from which is a platform which is hidden (below the water surface) but which remains in a constant position. A random choice of starting pOSition is used for each trial. Rats take asymptotically short paths after approximately 10 trials (see figure 1 a). Delayed match-to-place (DMP) learning is a refined version in which the platform'S location is changed on each day. Figure 1b shows escape latencies for rats given four trials per day for nine days, with the platform in a novel position on each day. On early days, acquisition ?Crichton Street, Edinburgh EH8 9LE, United Kingdom. Funded by Edin. Univ. Holdsworth Scholarship, the McDonnell-Pew foundation and NSF grant IBN-9634339. Email: djf@cfn.ed.ac.uk D. J Foster; R. G. M. Mo"is and P. Dayan 146 100 100 a 90 90 b 80 _ 70 '" i oo '" 50 -' ~40 ~30 20 10 13 17 21 2S Figure 1: a) Latencies for rats on the reference memory in the watermaze (RMW) task (N=8). b) Latencies for rats on the Delayed Match-to-Place (DMP) task (N=62). is gradual but on later days, rats show one-trial learning, that is, near asymptotic performance on the second trial to a novel platform position. The RMW task has been extensively modelled. 6,4,5,20 By contrast, the DMP task is new and computationally more challenging. It is solved here by integrating a standard actor-critic reinforcement learning system2 ,7 which guarantees that the rat will be competent to perform well in arbitrary mazes, with a system that learns spatial coordinates in the maze. Temporal difference learning1 7 (TO) is used for actor, critic and coordinate learning. TO learning is attractive because of its generality for arbitrary Markov decision problems and the fact that reward systems in vertebrates appear to instantiate it. 14 2 THEMODEL The model comprises two distinct networks (figure 2): the actor-critic network and a coordinate learning network. The contribution of the hippocampus, for both networks, is to provide a state-space representation in the form of place cell basis functions. Note that only the activities of place cells are required, by contrast with decoding schemes which require detailed information about each place cell. 4 COORDINATE SYSTEM ACTOR-CRITIC SYSTEM r------------Remembered 1 Goal coordinates 1 1 1 VECTOR COMPUTA nONI ~ Coordinate Representation 1 1______ -------1 Figure 2: Model diagram showing the interaction between actor-critic and coordinate system components. Hippocampal Model of Rat Spatial Abilities Using TD Learning 147 2.1 Actor-Critic Learning Place cells are modelled as being tuned to location. At position p, place cell i has an output given by h(p) = exp{ -lip - sdI2/2(12}, where Si is the place field centre, and (1 = 0.1 for all place fields. The critic learns a value function V(p) wih(p) which comes to represent the distance of p from the goal, using = L:i the TO rule 6.w~ ex: 8t h(pt), where (1) is the TD error, pt is position at time t, and the reward r(pt, pt+I) is 1 for any move onto the platform, and 0 otherwise. In a slight alteration of the original rule, the value V (p) is set to zero when p is at the goal, thus ensuring that the total future rewards for moving onto the goal will be exactly 1. Such a modification improves stability in the case of TD learning with overlapping basis functions. The discount factor, I' was set to 0.99. Simultaneously the rat refines a policy, which is represented by eight action cells. Each action cell (aj in figure 2) receives a parameterised input at any position p: aj (p) = L:i qjdi (p). An action is chosen stochastically with probabilities given by P(aj) = exp{2aj}/ L:k exp{2ak}. Action weights are reinforced according to: 2 (2) where 9j((Jt) is a gaussian function of the difference between the head direction (Jt at time t and the preferred direction of the jth action cell. Figure 3 shows the development of a policy over a few trials. V(p)l Triall V(p) 1 0.5 0.5 0. 0.5 01 0.5 TrialS Triall3 V(P)l 0.5 I 0 0.5 0: 0.5 0.5 -0.5 -------0.5 .---'--- 0 0.5 Figure 3: The RMW task: the value function gradually disseminates information about reward proximity to all regions of the environment. Policies and paths are also shown. There is no analytical guarantee for the convergence of TD learning with policy adaptation. However our simulations show that the algorithm always converges for the RMW task. In a simulated arena of diameter 1m and with swimming speeds of 20cm/s, the simulation matched the performance of the real rats very closely (see figure S). This demonstrates that TD-based reinforcement learning is adequately fast to account for the learning performance of real animals. D. 1. Foster, R. G. M Morris and P. Dayan 148 2.2 Coordinate Learning Although the learning of a value function and policy is appropriate for finding a fixed platform, the actor-critic model does not allow the transfer of knowledge from the task defined by one goal position to that defined by any other; thus it could not generate the sort of one-trial learning that is shown by rats on the DMP task (see figure 1b). This requires acquisition of some goal-independent know ledge about s~ace. A natural mechanism for this is the path integration or self-motion system. 0,10 However, path integration presents two problems. First, since the rat is put into the maze in a different position for each trial, how can it learn consistent coordinates across the whole maze? Second, how can a general, powerful, but slow, behavioral learning mechanism such as TO be integrated with a specific, limited, but fast learning mechanism involving spatial coordinates? Since TO critic learning is based on enforcing consistency in estimates of future reward, we can also use it to learn spatially consistent coordinates on the basis of samples of self-motion. It is assumed that the rat has an allocentric frame of reference. 1s The model learns parameterised estimates of the x and y coordinates of all positions p: x(p) = Li w[ fi(P) and y(p) = Li wY h(p), Importantly, while place cells were again critical in supporting spatial representation, they do not embody a map of space. The coordinate functions, like the value function previously, have to be learned. As the simulated rat moves around, the coordinate weights {w[} are adjusted according to: t Llwi ()( (Llxt + X(pt+l ) - X(pt)) LA t- k h(pk) (3) k=1 where Llxt is the self-motion estimate in the x direction. A similar update is applied to {wn. In this case, the full TO(A) algorithm was used (with A = 0.9); however TD(O) could also have been used, taking slightly longer. Figure 4a shows the x and y coordinates at early and late phases of learning. It is apparent that they rapidly become quite accurate - this is an extremely easy task in an open field maze. An important issue in the learning of coordinates is drift, since the coordinate system receives no direct information about the location of the origin. It turns out that the three controlling factors over the implicit origin are: the boundary of the arena, the prior setting of the coordinate weights (in this case all were zero) and the position and prior value of any absorbing area (in this case the platform). If the coordinate system as a whole were to drift once coordinates have been established, this would invalidate coordinates that have been remembered by the rat over long periods. However, since the expected value of the prediction error at time steps should be zero for any self-consistent coordinate mapping, such a mapping should remain stable. This is demonstrated for a single run: figure 4b shows the mean value of coordinates x evolving over trials, with little drift after the first few trials. We modeled the coordinate system as influencing the choice of swimming direction in the manner of an abstract action. I5 The (internally specified) coordinates of the most recent goal position are stored in short term memory and used, along with the current coordinates, to calculate a vector heading. This vector heading is thrown into the stochastic competition with the other possible actions, governed by a single weight which changes in a similar manner to the other action weights (as in equation 2, see also fig 4d), depending on the TO error, and on the angular proximity of the current head direction to the coordinate direction. Thus, whether the the coordinate-based direction is likely to be used depends upon its past performance. One simplification in the model is the treatment of extinction. In the DMP task, Hippocampal Model ofRat Spatial Abilities Using 1D Learning 149 ,. " TJUAL d III .1 TRIAL 26 16 i: ~Ol ~o !" ~o Figure 4: The evolution of the coordinate system for a typical simulation run: a.) coordinate outputs at early and late phases of learning, b.) the extent of drift in the coordinates, as shown by the mean coordinate value for a single run, c.) a measure A2 ~ ~ {X (Pr.)-X -X(pr.)}2 ? o f coord mate error for the same run (7E = r r. (Np-l)N ' where k r r r indexes measurement points (max N p ) and r indexes runs (max N r ), Xr(Pk) is the model estimate of X at position Pk, X(Pk) is the ideal estimate for a coordinate system centred on zero, and Xr is the mean value over all the model coordinates, d.) the increase during training of the probability of choosing the abstract action. This demonstrates the integration of the coordinates into the control system. real rats extinguish to a platform that has moved fairly quickly whereas the actorcritic model extinguishes far more slowly. To get around this, when a simulated rat reaches a goal that has just been moved, the value and action weights are reinitialised, but the coordinate weights wf and wf, and the weights for the abstract action, are not. 3 RESULTS The main results of this paper are the replication by simulation of rat performance on the RMW and DMP tasks. Figures la and b show the course of learning for the rats; figures Sa and b for the model. For the DMP task, one-shot acquisition is apparent by the end of training. 4 DISCUSSION We have built a model for one-trial spatial learning in the watermaze which uses a single TD learning algorithm in two separate systems. One system is based on a reinforcement learning that can solve general Markovian decision problems, and the other is based on coordinate learning and is specialised for an open-field water maze. Place cells in the hippocampus offer an excellent substrate for learning the actor, the critic and the coordinates. The model is explicit about the relationship between the general and specific learning systems, and the learning behavior shows that they integrate seamlessly. As currently constituted, the coordinate system would fail if there were a barrier in the maze. We plan to extend the model to allow the coordinate system to specify abstract targets other than the most recent platform position - this could allow it fast navigation around a larger class of environments. It is also important to improve the model of learning 'set' behavior - the information about the nature of D. 1. Foster; R. G. M. Mo"is and P. Dayan 150 14 a b 12 10 . . ?. z j:10\ ~ ~ ~ 12 ~ S> .. '"~ ............................................ 0~D.~yl~~y~2~D~.y~3~D~.y~47~~yS~~~.~~~y~7~~~y~.7D.~y9~ Figure 5: a.) Performance of the actor-critic model on the RMW task, and b.) performance of the full model on the DMP task. The data for comparison is shown in figures la and b. the DMP task that the rats acquire over the course of the first few days of training. Interestingly, learning set is incomplete - on the first trial of each day, the rats still aim for the platform position on the previous day, even though this is never correct. 16 The significant differences in the path lengths on the first trial of each day (evidence in figure Ib and figure 5b) come from the relative placements of the platforms. However, the model did not use the same positions as the empirical data, and, in any case, the model of exploration behavior is rather simplistic. The model demonstrates that reinforcement learning methods are perfectly fast enough to match empirical learning curves. This is fortunate, since, unlike most models specifically designed for open-field navigation,6,4,5,2o RL methods can provably cope with substantially more complicated tasks with arbitrary barriers, etc, since they solve the temporal credit assignment problem in its full generality. The model also addresses the problem that coordinates in different parts of the same environment need to be mutually consistent, even if the animal only experiences some parts on separate trials. An important property of the model is that there is no requirement for the animal to have any explicit knowledge of the relationship between different place cells or place field position, size or shape. Such a requirement is imposed in various models. 9 ,4,6,2o Experiments that are suggested by this model (as well as by certain others) concern the relationship between hippocampally dependent and independent spatial learning. First, once the coordinate system has been acquired, we predict that merely placing the rat at a new location would be enough to let it find the platform in one shot, though it might be necessary to reinforce the placement e.g. by first placing the rat in a bucket of cold water. Second, we know that the establishment of place fields in an environment happens substantiallr faster than establishment of one-shot or even ordinary learning to a platform. 2 We predict that blocking plasticity in the hippocampus following the establishment of place cells (possibly achieved without a platform) would not block learning of a platform. In fact, new experiments show that after extensive pre-training, rats can perform one-trial learning in the same environment to new platform positions on the DMP task without hippocampal synaptic plasticity. 16 This is in contrast to the effects of hippocampal lesion, which completely disrupts performance. According to the model, coordinates will have been learned during pre-training. The full prediction remains untested: that once place fields have been established, coordinates could be learned in the absence of hippocampal synaptic plasticity. A third prediction follows from evidence that rats with restricted hippocampal lesions can learn the fixed platform Hippocampal Model of Rat Spatial Abilities Using TD Learning 151 task, but much more slowly, based on a gradual "shaping" procedure. 22 In our model, they may also be able to learn coordinates. However, a lengthy training procedure could be required, and testing might be complicated if expressing the knowledge required the use of hippocampus dependent short-term memory for the last platform location. I6 One way of expressing the contribution of the hippocampus in the model is to say that its function is to provide a behavioural state space for the solution of complex tasks. Hence the contribution of the hippocampus to navigation is to provide place cells whose firing properties remain consistent in a given environment. It follows that in different behavioural situations, hippocampal cells should provide a representation based on something other than locations - and, indeed, there is evidence for this. 8 With regard to the role of the hippocampus in spatial tasks, the model demonstrates that the hippocampus may be fundamentally necessary without embodying a map. References [1] Barto, AG & Sutton, RS (1981) BioI. Cyber., 43:1-8. [2] Barto, AG, Sutton, RS & Anderson, CW (1983) IEEE Trans. on Systems, Man [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] and Cybernetics 13:834-846. Barto, AG, Sutton, RS & Watkins, CJCH (1989) Tech Report 89-95, CAIS, Univ. Mass., Amherst, MA. Blum, KI & Abbott, LF (1996) Neural Computation, 8:85-93. Brown, MA & Sharp, PE (1995) Hippocampus 5:171-188. Burgess, N, Reece, M & O'Keefe, J (1994) Neural Networks, 7:1065-1081. Dayan, P (1991) NIPS 3, RP Lippmann et aI, eds., 464-470. Eichenbaum, HB (1996) CurroOpin. Neurobiol., 6:187-195. Gerstner, W & Abbott, LF (1996) J. Computational Neurosci. 4:79-94. McNaughton, BL et a1 (1996) J. Exp. BioI., 199:173-185. Morris, RGM et al (1982) Nature, 297:681-683. O'Keefe, J & Dostrovsky, J (1971) Brain Res., 34(171). Olton, OS & Samuelson, RJ (1976) J. Exp. Psych: A.B.P., 2:97-116. Rudy, JW & Sutherland, RW (1995) Hippocampus, 5:375-389. SchUltz, W, Dayan, P & Montague, PR (1997) Science, 275, 1593-1599. Singh, SP Reinforcement learning with a hierarchy of abstract models. Steele, RJ & Morris, RGM in preparation. Sutton, RS (1988) Machine Learning, 3:9-44. Taube, JS (1995) J. Neurosci. 15(1):70-86. Tsitsiklis, IN & Van Roy, B (1996) Tech Report LIDS-P-2322, M.LT. Wan, HS, Touretzky, OS & Redish, AD (1993) Proc. 1993 Connectionist Models Summer School, Lawrence Erlbaum, 11-19. Watkins, CJCH (1989) PhD Thesis, Cambridge. Whishaw, IQ & Jarrard, LF (1996) Hippocampus Wilson, MA & McNaughton, BL (1993) Science 261:1055-1058. Reinforcement Learning for Call Admission Control and Routing in Integrated Service Networks Oliver Mihatsch Siemens AG Corporate Technology, ZT IK 4 0-81730 Munich, Germany email:oliver.mihatsch@ mchp.siemens.de Peter Marbach" LIDS MIT Cambridge, MA, 02139 email: marbach@mi t . edu Miriam Schulte Zentrum Mathematik Technische UniversWit Miinchen D-80290 Munich Germany John N. Tsitsiklis LIDS MIT Cambridge, MA, 02139 email: jnt@mit. edu Abstract In integrated service communication networks, an important problem is to exercise call admission control and routing so as to optimally use the network resources. This problem is naturally formulated as a dynamic programming problem, which, however, is too complex to be solved exactly. We use methods of reinforcement learning (RL), together with a decomposition approach, to find call admission control and routing policies. The performance of our policy for a network with approximately 10 45 different feature configurations is compared with a commonly used heuristic policy. 1 Introduction The call admission control and routing problem arises in the context where a telecommunication provider wants to sell its network resources to customers in order to maximize long term revenue. Customers are divided into different classes, called service types. Each service type is characterized by its bandwidth demand, its average call holding time and the immediate reward the network provider obtains, whenever a call of that service type is ? Author to whom correspondence should be addressed. Reinforcement Learning for Call Admission Control and Routing 923 accepted. The control actions for maximizing the long term revenue are to accept or reject new calls (Call Admission Control) and, if a call is accepted, to route the call appropriately through the network (Routing). The problem is naturally formulated as a dynamic programming problem, which, however, is too complex to be solved exactly. We use the methodology of reinforcement learning (RL) to approximate the value function of dynamic programming. Furthermore, we pursue a decomposition approach, where the network is viewed as consisting of link processes, each having its own value function. This has the advantage, that it allows a decentralized implementation of the training methods of RL and a decentralized implementation of the call admission control and routing policies. Our method learns call admission control and routing policies which outperform the commonly used heuristic "Open-Shortest-Path-First" (OSPF) policy. In some earlier related work, we applied RL to the call admission problem for a single communication link in an integrated service environment. We found that in this case, RL methods performed as well, but no better than, well-designed heuristics. Compared with the single link problem, the addition of routing decisions makes the network problem more complex and good heuristics are not easy to derive. 2 Call Admission Control and Routing We are given a telecommunication network consisting of a set of nodes N = {I, ... , N} and a set of Iinks .c = {I, ... , L}, where link I has a a total capacity of B(l) units of bandwidth. We support a set M = {I, "', M} of different service types, where a service type m is characterized by its bandwidth demand b(m), its average call holding time I/v(m) (here we assume that the call holding times are exponentially distributed) and the immediate reward c( m) we obtain, whenever we accept a call of that service type. A link can carry simultaneously any combination of calls, as long as the bandwidth used by these calls does not exceed the total bandwidth of the link (Capacity Constraint). When a new call of service type m requests a connection between a node i and a node j, we can either reject or accept that request (Call Admission Control). If we accept the call, we choose a route out of a list of predefined routes (Routing). The call then uses b(m) units of bandwidth on each link along that route for the duration of the call. We can, therefore, only choose a route, which does not violate the capacity constraints of its links, if the call is accepted. Furthermore, if we accept the call, we obtain an immediate reward c( m). The objective is to exercise call admission control and routing in such a way that the long term revenue obtained by accepting calls is maximized. We can formulate the call admission control and routing problem using dynamic programming (e. g. Bertsekas, 1995). Events w which incur state transitions, are arrivals of new calls and call terminations. The state Xt at time t consists of a list for each route, indicating how many calls of each service type are currently using that route. The decision/control Ut applied at the time t of an arrival of a new call is to decide, whether to reject or accept the call, and, if the call is accepted, how to route it through the network. The objective is to learn a policy that assigns decisions to each state so as to where E{?} is the expectation operator, tk is the time when the kth event happens, g( Xtk' Wk, Ut,,) is the immediate reward associated with the kth event, and f3 is a discount factor that makes immediate rewards more valuable than future ones. 924 P Marbach, O. Mihatsch, M. Schulte and 1. N. Tsitsiklis 3 Reinforcement Learning Solution RL methods solve optimal control (or dynamic programming) problems by learning good approximations to the optimal value function r, given by the solution to the Bellman optimality equation which takes the following form for the caB admission control and routing problem J*(x) = Er {e- th } Ew { max [g(x,w, u) ueU(x) + J*(X I )]} where U ( x) is the set of control actions available in the current state x, T is the time when the first event w occurs and x' is the successor state. Note that x' is a deterministic function of the current state x, the control u and the event w. RL uses a compact representation j (', 0) to learn and store an estimate of J" (.). On each event, i(., 0) is both used to make decisions and to update the parameter vector e. In the caB admission control and routing problem, one has only to choose a control action when a new call requests a connection. In such a case, J(,,0) is used to choose a control action according to the formula u=arg max [g(x,w,u) ueU(x) + J(X', e)] (1) This can be expressed in words as follows. Decision Making: When a new call requests a connection, use J (', e) to evaluate, for each permissible route, the successor state x' we transit to, when we choose that route, and pick a route which maximizes that value. If the sum of the immediate reward and the value associated with this route is higher than the value of the current state, route the call over that route; otherwise reject the call. Usually, RL uses a global feature extractor f(x) to form an approximate compact representation of the state of the system, which forms the input to a function approximator i(., e). Sutton's temporal difference (TO()'? algorithms (Sutton, 1988) can then be used to train i(., 0) to learn an estimate of J*. Using ID(O), the update at the kth event takes the following form where dk e-/J(t/c-t/c-d (g(Xt/c, Wk, Ut/c) + J(!(Xt/c), ek-I)) -J(I(Xt/C_l)' Ok-I) and where 'Yk is a small step size parameter and Utk is the control action chosen according to the decision making rule described above. Here we pursue an approach where we view the network as being composed of link processes. Furthermore, we decompose immediate rewards g( Xtk' Wk, Ut/c) associated with the kth event, into link rewards g(l) (Xt/c, Wk, Ut/c) such that L g(Xtk' Wk. Ut/c) = L gil) (Xtl:' Wk, UtI:) 1=1 We then define, for each link I, a value function J(I) (I(l) (x), e( I?), which is interpreted as an estimate of the discounted long term revenue associated with that link. Here, f(l) defines a local feature, which forms the input to the value function associated with link I. To obtain 925 Reinforcement Learning for Call Admission Control and Routing an approximation of J* (x), the functions ](1) (J(l) (x), 0(1)) are combined as follows L L ](1) (J(I) (x), (J(l)). 1=1 At each event, we update the parameter vector (J(l) of link 1, only if the event is associated with the link. Events associated with a link 1 are arrivals of new calls which are potentially routed over link 1 and termination of calls which were routed over the link I. The update rule of the parameter vector 0(1) is very similar to the TD(O) algorithm described above (J(l) k - (J(l) k-l + "V(I)d(I)V ()](l) (/(1) (x ) (J(l) ) Ik k 9 I tk_1 , k-l (2) where e - i3(t~') -t~/~ I) (g(l) (x t(l), Wkl ) , Ut(l)) k _](1) (J(l) (x t (/) ), + ](l) (J(l) (xt(l?), (Jkl~ 1)) k (3) k (Jkl~l) k-I and where Ill) is a small step size parameter and tr) is the time when the kth event will associated with link 1 occurs. Whenever a new call of a service of type m is routed over a route r which contains the link i, the immediate reward g(l) associated with the link i is equal to c( m) / #r, where #r is the number of links along the route r. For all other events, the immediate reward associated with link 1 is equal to O. The advantage of this decomposition approach is that it allows decentralized training and decentralized decision making. Furthermore, we observed that this decomposition approach leads to much shorter training times for obtaining an approximation for J* than the approach without decomposition. All these features become very important if one considers applying methods of RL to large integrated service networks supporting a fair number of different service types. We use exploration to obtain the states at which we update the parameter vector O. At each state, with probability p == 0.5, we apply a random action, instead of the action recommended by the current value function, to generate the next state in our training trajectory. However, the action Ut(I), that is used in the update rule (3), is still the one chosen ack cording to the rule given in (1). Exploration during the training significantly improved the performance of the policy. Table I: Service Types. SERVICE TYPE m BANDWIDTH DEMAND b( m) AVERAGE HOLDING TIME l/v(m) IMMEDIATE REWARD c( m) 1 2 3 1 10 3 10 2 5 2 50 1 4 Experimental Results In this section, we present experimental results obtained for the case of an integrated service network consisting of 4 nodes and 12 unidirectional links. There are two different classes of links with a total capacity of 60 and 120 units of bandwidth, respectively (indicated by thick and thin arrows in Figure 1). We assume a set M == {I, 2, 3} of three different service types. The corresponding bandwidth demands, average holding times and immediate 926 P. Marbach, O. Mihatsch, M Schulte and 1. N. Tsitsiklis Figure 1: Telecommunication Network Consisting of 4 Nodes and 12 Unidirectional Links. PERFORMANCE DURING LEAANING Figure 2: Average Reward per TIme Unit During the Whole Training Phase of 10 7 Steps (Solid) and During Shorter Time Windows of 10 5 Steps (Dashed). rewards are given in Table 1. Call arrivals are modeled as independent Poisson processes, with a separate mean for each pair of source and destination nodes and each service type. Furthermore, for each source and destination node pair, the list of possible routes consists of three entries: the direct path and the two alternative 2-hop-routes. We compare the policy obtained through RL with the commonly used heuristic OSPF (Open Shortest Path First). For every pair of source and destination nodes, OSPF orders the list of predefined routes. When a new call arrives, it is routed along the first route in the corresponding list, that does not violate the capacity constraint; if no such a route exists, the call is rejected. We use the average reward per unit time as performance measure to compare the two policies. For the RL approach, we use a quadratic approximator, which is linear with respect to the parameters ()(I), as a compact representation of ](1). Other approximation architectures were tried, but we found that the quadratic gave the best results with respect to both the speed of convergence and the final performance. As inputs to the compact representation Reinforcement Learning for Call Admission Control and Routing 927 AVERAGE REWARD potential reward reward obtained by RL reward obtained by OSPF o 50 150 100 reward per time 200 250 un~ COMPARISON OF REJECTION RATES o 5 10 15 20 25 30 35 40 45 50 percentage of calls rejected Figure 3: Comparison of the Average Rewards and Rejection Rates of the RL and OSPF Policies. ROUTING (OSPF) direct link alternative route no. 1 alternative route no. 2 o 10 20 30 ~ .. ----~~-----.0:-= 50 ~ ro ~ 90 100 80 90 100 percentage of calls routed on direct and alternative paths ROUTING (RL) direct link alternative route no. 1 alternative route no. 2 o 10 20 30 40 50 60 70 percentage of calls routed on direct and alternative paths Figure 4: Comparison of the Routing Behaviour of the RL and OSPF Policies. ](/), we use a set of local features, which we chose to be the number of ongoing calls of each service type on link l. For the 4-node network, there are approximately 1.6. 1045 different feature configurations. Note that the total number of possible states is even higher. The results of the case studies are given in in Figure 2 (Training Phase), Figure 3 (Performance) and Figure 4 (Routing Behaviour). We give here a summary of the results. Training Phase: Figure 2 shows the average reward of the RL policy as a function of the training steps. Although the average reward increases during the training, it does not exceed 141, the average reward of the heuristic OSPF. This is due to the high amount of exploration in the training phase. Performance Comparison: The policy obtained through RL gives an average reward of 212, which as about 50% higher than the one of 141 achieved by OSPF. Furthermore, the RL policy reduces the number of rejected calls for all service types. The most significant reduction is achieved for calls of service type 3, the service type, which has the highest P. Marbach, O. Mihatsch, M Schulte and I. N. Tsitsiklis 928 immediate reward. Figure 3 also shows that the average reward of the RL policy is close to the potential average reward of 242, which is the average reward we would obtain if all calls were accepted. This leaves us to believe that the RL policy is close to optimal. Figure 4 compares the routing behaviour of the RL control policy and OSPF. While OSPF routes about 15% - 20% of all calls along one of the alternative 2-hop-routes, the RL policy almost uses alternative routes for calls of type 3 (about 25%) and routes calls of the other two service types almost exclusively over the direct route. This indicates, that the RL policy uses a routing scheme, which avoids 2-hop-routes for calls of service type 1 and 2, and which allows us to use network resources more efficiently. 5 Conclusion The call admission control and routing problem for integrated service networks is naturally formulated as a dynamic programming problem, albeit one with a very large state space. Traditional dynamic programming methods are computationally infeasible for such large scale problems. We use reinforcement learning, based on Sutton's (1988) T D(O), combined with a decomposition approach, which views the network as consisting of link processes. This decomposition has the advantage that it allows decentralized decision making and decentralized training, which reduces significantly the time of the training phase. We presented a solution for an example network with about 10 45 different feature configurations. Our RL policy clearly outperforms the commonly used heuristic OSPF. Besides the game of backgammon (Tesauro, 1992), the elevator scheduling (Crites & Barto, 1996), the jop-shop scheduling (Zhang & Dietterich, 1996) and the dynamic channel allocation (Singh & Bertsekas, 1997), this is another successful application of RL to a large-scale dynamic programming problem for which a good heuristic is hard to find. References Bertsekas, D. P; (1995) Dynamic Programming and Optimal Control. Athena Scientific, Belmont, MA. Crites, R. H., Barto, A. G. (1996) Improving elevator performance using reinforcement learning. In D. S. Touretzky, M. C. Mozer and M. E. Hasselmo (eds.), Advances in Neural Information Processing Systems 8, pp. 1017-1023. Cambridge, MA: MIT Press. Singh, S., Bertsekas, D. P. (1997) Reinforcement learning for dynamic channel allocation in cellular telephone systems. To appear in Advances in Neural Information Processing Systems 9, Cambridge, MA: MIT Press. Sutton, R. S. (1988) Learning to predict by the method of temporal differences. Machine Learning, 3:9-44. Tesauro, G. J. (1992) Practical issues in temporal difference learning. Machine Learning, 8(3/4):257-277. Zhang, W., Dietterich, T. G. (1996) High performance job-shop scheduling with a timedelay TD(>.) network. In D. S. Touretzky, M. C. Mozer and M. E. Hasselmo (eds.), Advances in Neural Information Processing Systems 8, pp. 1024-1030. Cambridge. MA: MIT Press. 

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The Asymptotic Convergence-Rate of Q-Iearning es. Szepesvari* Research Group on Artificial Intelligence, "Jozsef Attila" University, Szeged, Aradi vrt. tere 1, Hungary, H-6720 szepes@math.u-szeged.hu Abstract In this paper we show that for discounted MDPs with discount factor, > 1/2 the asymptotic rate of convergence of Q-Iearning is O(1/t R (1-1') if R(1 - ,) < 1/2 and O( Jlog log tit) otherwise provided that the state-action pairs are sampled from a fixed probability distribution. Here R = Pmin/Pmax is the ratio of the minimum and maximum state-action occupation frequencies. The results extend to convergent on-line learning provided that Pmin > 0, where Pmin and Pmax now become the minimum and maximum state-action occupation frequencies corresponding to the stationary distribution. 1 INTRODUCTION Q-Iearning is a popular reinforcement learning (RL) algorithm whose convergence is well demonstrated in the literature (Jaakkola et al., 1994; Tsitsiklis, 1994; Littman and Szepesvari, 1996; Szepesvari and Littman, 1996). Our aim in this paper is to provide an upper bound for the convergence rate of (lookup-table based) Q-Iearning algorithms. Although, this upper bound is not strict, computer experiments (to be presented elsewhere) and the form of the lemma underlying the proof indicate that the obtained upper bound can be made strict by a slightly more complicated definition for R. Our results extend to learning on aggregated states (see (Singh et al., 1995? and other related algorithms which admit a certain form of asynchronous stochastic approximation (see (Szepesv<iri and Littman, 1996?. Present address: Associative Computing, Inc., Budapest, Konkoly Thege M. u. 29-33, HUNGARY-1121 The Asymptotic Convergence-Rate of Q-leaming 2 1065 Q-LEARNING Watkins introduced the following algorithm to estimate the value of state-action pairs in discounted Markovian Decision Processes (MDPs) (Watkins, 1990): Here x E X and a E A are states and actions, respectively, X and A are finite. It is assumed that some random sampling mechanism (e.g. simulation or interaction with a real Markovian environment) generates random samples of form (Xt, at, Yt, rt), where the probability of Yt given (xt,at) is fixed and is denoted by P(xt,at,Yt), E[rt I Xt, at] = R(x, a) is the immediate average reward which is received when executing action a from state x, Yt and rt are assumed to be independent given the history of the learning-process, and also it is assumed that Var[rt I Xt, at] < C for some C > O. The values 0 ~ at(x,a) ~ 1 are called the learning rate associated with the state-action pair (x, a) at time t. This value is assumed to be zero if (x,a) =J (xt,at), i.e. only the value of the actual state and action is reestimated in each step. If 00 L at(x, a) = 00 (2) 00 (3) t=l and 00 L a;(x, a) < t=l then Q-Iearning is guaranteed to converge to the only fixed point Q* of the operator T : lR X x A ~ lR XxA defined by (TQ)(x,a) = R(x, a) +, L P(x,a,y)mFQ(y,b) yEX (convergence proofs can be found in (Jaakkola et al., 1994; TSitsiklis, 1994; Littman and Szepesv.hi, 1996; Szepesvari and Littman, 1996)). Once Q* is identified the learning agent can act optimally in the underlying MDP simply by choosing the action which maximizes Q* (x, a) when the agent is in state x (Ross, 1970; Puterman, 1994). 3 THE MAIN RESULT Condition (2) on the learning rate at(x, a) requires only that every state-action pair is visited infinitely often, which is a rather mild condition. In this article we take the stronger assumption that {(Xt, at) h is a sequence of independent random variables with common underlying probability distribution. Although this assumption is not essential it simplifies the presentation of the proofs greatly. A relaxation will be discussed later. We further assume that the learning rates take the special form at ( x, a ) = { l , if (x,a) = (xt,a); ~ Ol,x,a, otherwise, 0, where St (x, a) is the number of times the state-action pair was visited by the process (xs, as) before time step t plus one, i.e. St(x, a) = 1 + #{ (xs, as) = (x, a), 1 ~ s ~ C. Szepesvari 1066 t }. This assumption could be relaxed too as it will be discussed later. For technical reasons we further assume that the absolute value of the random reinforcement signals Tt admit a common upper bound. Our main result is the following: 3.1 Under the above conditions the following relations hold asymptotically and with probability one: THEOREM IQt(x, a) - Q*(x, a)1 ~ tR(~-'Y) (4) and * JIOg log t IQt(x,a) - Q (x,a)1 ~ B t' (5) for some suitable constant B > O. Here R = Pmin/Pmax, where Pmin = min(z,a) p(x, a) and Pmax = max(z,a) p(x, a), where p(x, a) is the sampling probability of (x, a). Note that if'Y 2: 1 - Pmax/2pmin then (4) is the slower, while if'Y < 1 - Pmax/2Pmin then (5) is the slower. The proof will be presented in several steps. Step 1. Just like in (Littman and Szepesvari, 1996) (see also the extended version (Szepesvciri and Littman, 1996)) the main idea is to compare Qt with the simpler process Note that the only (but rather essential) difference between the definition of Qt and that of Qt is the appearance of Q* in the defining equation of Qt. Firstly, notice that as a consequence of this change the process Qt clearly converges to Q* and this convergence may be investigated along each component (x, a) separately using standard stochastic-approximation techniques (see e.g. (Was an , 1969; Poljak and Tsypkin, 1973)). Using simple devices one can show that the difference process At(x, a) = IQt(x, a)- at(x, a)1 satisfies the following inequality: A t + 1 (x, a) ~ (1 - Ot(x, a))At(x, a) + 'Y0t(x, a)(IIAtll + lIat - Q*II). (7) Here 11?11 stands for the maximum norm. That is the task of showing the convergence rate of Qt to Q* is reduced to that of showing the convergence rate of At to zero. Step 2. We simplify the notation by introducing the abstract process whose update equation is (8) where i E 1,2, ... , n can be identified with the state-action pairs, Xt with At, with Qt - Q*, etc. We analyze this process in two steps. First we consider processes when the "perturbation-term" f.t is missing. For such processes we have the following lemma: f.t 3.2 Assume that 771,1]2, ... ,'TIt, . .. are independent random variables with a common underlying distribution P{TJt = i) = Pi > O. Then the process Xt defined LEMMA The Asymptotic Convergence-Rate of Q-leaming 1067 by (9) satisfies IIxtil = wi~h OCR(~--Y?) probability one (w.p.1), where R = mini Pi/ maxi Pi. Proof. (Outline) Let To Tk+l = 0 and = min{ t ~ Tk IVi = 1 . . . n, 3s = s(i) : 1]8 = i}, i.e. Tk+1 is the smallest time after time Tk such that during the time interval [Tk + 1, Tk+d all the components of XtO are "updated" in Equation (9) at least once. Then (10) where Sk = maxi Sk(i) . This inequality holds because if tk(i) is the last time in [Tk + 1, T k+1] when the ith component is updated then XT"+l+1(i) Xtk(i)+l(i) = (1-1/St/o(i?Xt,,(i)(i) + ,/St,,(i) II X t,,(i) 011 < (l-l/S t,,(i?lI x t/o(i)OIl +,/St,,(i)lIxt,,(i)OIl = = < (1 -1St,,(i) -,) IIXt,,(i) 011 (1- 1;k') IIXT,,+1011, where it was exploited that time yields Ilxtll is decreasing. Now, iterating (10) backwards in X7Hl(-)::: IIxolin (1- 1~ 'Y). Now, consider the following approximations: Tk ~ Ck, where C ~ 1/Pmin (C can be computed explicitly from {Pi}), Sk ~ PmaxTk+1 ~ Pmax/Pmin(k + 1) ~ (k + 1)/ Ro, where Ro = 1/CPmax' Then, using Large Deviation's Theory, IT k-l ( j=O 1_,) ~II k-l ( 1-Sj 1- j=O holds w.p.1. Now, by defining s = Tk Ro(1 _ . J ,?) ~(1) Ro(l--Y) +1 + 1 so that (11) k siC ~ k we get which holds due to the monotonicity of Xt and l/k Ro (l--y) and because R Pmin/Pmax ~ Ro. 0 Step 3. Assume that, > 1/2. Fortunately, we know by an extension of the Law of the Iterated Logarithm to stochastic approximation processes that the convergence C. Szepesvari 1068 rate of IIOt -Q*II is 0 (y'loglogt/t) (the uniform boundedness ofthe random reinforcement signals must be exploited in this step) (Major, 1973). Thus it is sufficient to provide a convergence rate estimate for the perturbed process, Xt, defined by (8), when f.t = Cy'loglogt/t for some C > O. We state that the convergence rate of f.t is faster than that of Xt. Define the process ZHI (i) = { (1 - ~~l)) Zt(i), Zt (i), if 7Jt = i; if 7Jt f. i. (12) This process clearly lower bounds the perturbed process, Xt. Obviously, the convergence rate of Zt is O(l/tl-'Y) which is slower than the convergence rate of f.t provided that, > 1/2, proving that f.t must be faster than Xt. Thus, asymptotically f.t ~ (1/, - l)xt, and so Ilxtll is decreasing for large enough t. Then, by an argument similar to that of used in the derivation of (10), we get XTIo+1+1(i) ~ (1- 1~k') II XTk +1 II ~ ~ f.Tk, (13) where Sk = mini Sk(i). By some approximation arguments similar to that of Step 2, together with the bound (l/n71) s71- 3 / 2Jloglogs ~ s-1/2Jloglogs, 1 > 7J > 0, which follows from the mean-value theorem for integrals and the law of integration by parts, we get that Xt ~ O(l/t R (l-'Y?). The case when , ~ 1/2 can be treated similarly. 2:: Step 5. Putting the pieces together and applying them for At = Ot - Qt yields Theorem 3.1. 4 DISCUSSION AND CONCLUSIONS The most restrictive of our conditions is the assumption concerning the sampling of (Xt, at). However, note that under a fixed learning policy the process (Xt, at) is a (non-stationary) Markovian process and if the learning policy converges in the sense that limt-+oo peat 1Ft) = peat I Xt) (here Ft stands for the history of the learning process) then the process (Xt, at) becomes eventually stationary Markovian and the sampling distribution could be replaced by the stationary distribution of the underlying stationary Markovian process. If actions become asymptotically optimal during the course of learning then the support of this stationary process will exclude the state-action pairs whose action is sub-optimal, i.e. the conditions of Theorem 3.1 will no longer be satisfied. Notice that the proof of convergence of such processes still follows very similar lines to that of the proof presented here (see the forthcoming paper (Singh et al., 1997)), so we expect that the same convergence rates hold and can be proved using nearly identical techniques in this case as well. A further step would be to find explicit expressions for the constant B of Theorem 3.1. Clearly, B depends heavily on the sampling of (Xt, at), as well as the transition probabilities and rewards of the underlying MDP. Also the choice of harmonic learning rates is arbitrary. If a general sequence at were employed then the artificial "time" Tt (x, a) = 1/IT}=o (1 - at (x, a)) should be used (note that for the harmonic sequence Tt(x, a) ~ t). Note that although the developed bounds are asymptotic in their present forms, the proper usage of Large Deviation's Theory would enable us to develope non-asymptotic bounds. The Asymptotic Convergence-Rate ofQ-learning 1069 Other possible ways to extend the results of this paper may include Q-Iearning when learning on aggregated states (Singh et al., 1995), Q-Iearning for alternating/simultaneous Markov games (Littman, 1994; Szepesvari and Littman, 1996) and any other algorithms whose corresponding difference process At satisfies an inequality similar to (7). Yet another application of the convergence-rate estimate might be the convergence proof of some average reward reinforcement learning algorithms. The idea of those algorithms follows from a kind of Tauberian theorem, Le. that discounted sums converge to the average value if the discount rate converges to one (see e.g. Lemma 1 of (Mahadevan, 1994; Mahadevan, 1996) or for a value-iteration scheme relying on this idea (Hordjik and Tijms, 1975)). Using the methods developed here the proof of convergence of the corresponding Q-learning algorithms seems quite possible. We would like to note here that related results were obtained by Bertsekas et al. et. al (see e.g. (Bertsekas and Tsitsiklis, 1996)). Finally, note that as an application of this result we immediately get that the convergence rate of the model-based RL algorithm, where the transition probabilities and rewards are estimated by their respective averages, is clearly better than that of for Q-Iearning. Indeed, simple calculations show that the law of iterated logarithm holds for the learning process underlying model-based RL. Moreover, the exact expression for the convergence rate depends explicitly on how much computational effort we spend on obtaining the next estimate of the optimal value function, the more effort we spend the faster is the convergence. This .bound thus provides a direct way to control the tradeoff between the computational effort and the convergence rate. Acknowledgements This research was supported by aTKA Grant No. F20132 and by a grant provided by the Hungarian Educational Ministry under contract no. FKFP 1354/1997. I would like to thank Andras Kramli and Michael L. Littman for numerous helpful and thought-provoking discussions. References Bertsekas, D. and Tsitsiklis, J. (1996). Scientific, Belmont, MA. Neuro-Dynamic Programming. Athena Hordjik, A. ~nd Tijms, H. (1975). A modified form of the iterative method of dynamic programming. Annals of Statistics, 3:203-208. Jaakkola, T., Jordan, M., and Singh, S. (1994). On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6(6):11851201. Littman, M. (1994). Markov games as a framework for multi-agent reinforcement learning. In Proc. of the Eleventh International Conference on Machine Learning, pages 157-163, San Francisco, CA. Morgan Kauffman. Littman, M. and Szepesvciri, C. (1996). A Generalized Reinforcement Learning Model: Convergence and applications. In Int. Con/. on Machine Learning. http://iserv.ikLkfki.hu/ asl-publs.html. 1070 C. Szepesvari Mahadevan, S. (1994). To discount or not to discount in reinforcement learning: A case study comparing R learning and Q learning. In Proceedings of the Eleventh International Conference on Machine Learning, pages 164-172, San Francisco, CA. Morgan Kaufmann. Mahadevan, S. (1996). Average reward reinforcement learning: Foundations, algorithms, and empirical results. Machine Learning, 22(1,2,3):124-158. Major, P. (1973). A law of the iterated logarithm for the Robbins-Monro method. Studia Scientiarum Mathematicarum Hungarica, 8:95-102. Poljak, B. and Tsypkin, Y. (1973). Pseudogradient adaption and training algorithms. Automation and Remote Control, 12:83-94. Puterman, M. L. (1994). Markov Decision Processes - Discrete Stochastic Dynamic Programming. John Wiley & Sons, Inc., New York, NY. Ross, S. (1970). Applied Probability Models with Optimization Applications. Holden Day, San Francisco, California. Singh, S., Jaakkola, T., and Jordan, M. (1995). Reinforcement learning with soft state aggregation. In Proceedings of Neural Information Processing Systems. Singh, S., Jaakkola, T., Littman, M., and Csaba Szepesva ri (1997). On the convergence of single-step on-policy reinforcement-learning al gorithms. Machine Learning. in preparation. Szepesvari, C. and Littman, M. (1996). Generalized Markov Decision Processes: Dynamic programming and reinforcement learning algorithms. Machine Learning. in preparation, available as TR CS96-10, Brown Univ. Tsitsiklis, J. (1994). Asynchronous stochastic approximation and q-learning. Machine Learning, 8(3-4):257-277. Wasan, T. (1969). Stochastic Approximation. Cambridge University Press, London. Watkins, C. (1990). Learning /rom Delayed Rewards. PhD thesis, King's College, Cambridge. QLEARNING. 

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Reinforcement Learning with Hierarchies of Machines * Ronald Parr and Stuart Russell Computer Science Division, UC Berkeley, CA 94720 {parr,russell}@cs.berkeley.edu Abstract We present a new approach to reinforcement learning in which the policies considered by the learning process are constrained by hierarchies of partially specified machines. This allows for the use of prior knowledge to reduce the search space and provides a framework in which knowledge can be transferred across problems and in which component solutions can be recombined to solve larger and more complicated problems. Our approach can be seen as providing a link between reinforcement learning and "behavior-based" or "teleo-reactive" approaches to control. We present provably convergent algorithms for problem-solving and learning with hierarchical machines and demonstrate their effectiveness on a problem with several thousand states. 1 Introduction Optimal decision making in virtually all spheres of human activity is rendered intractable by the complexity of the task environment. Generally speaking, the only way around intractability has been to provide a hierarchical organization for complex activities. Although it can yield suboptimal policies, top-down hierarchical control often reduces the complexity of decision making from exponential to linear in the size of the problem. For example, hierarchical task network (HTN) planners can generate solutions containing tens of thousands of steps [5], whereas "fiat" planners can manage only tens of steps. HTN planners are successful because they use a plan library that describes the decomposition of high-level activities into lower-level activities. This paper describes an approach to learning and decision making in uncertain environments (Markov decision processes) that uses a roughly analogous form of prior knowledge. We use hierarchical abstract machines (HAMs), which impose constraints on the policies considered by our learning algorithms. HAMs consist of nondeterministic finite state machines whose transitions may invoke lower-level machines. Nondeterminism is represented by choice states where the optimal action is yet to be decided or learned. The language allows a variety of prior constraints to be expressed, ranging from no constraint all the way to a fully specified solution. One *This research was supported in part by ARO under the MURI program "Integrated Approach to Intelligent Systems," grant number DAAH04-96-1-0341. R. Parr and S. Russell 1044 0_1. (a) (b) (c) Figure 1: (a) An MOP with ~ 3600 states. The initial state is in the top left. (b) Closeup showing a typical obstacle. (c) Nondetenninistic finite-state controller for negotiating obstacles. useful intennediate point is the specification of just the general organization of behavior into a layered hierarchy, leaving it up to the learning algorithm to discover exactly which lower-level activities should be invoked by higher levels at each point. The paper begins with a brief review of Markov decision processes (MOPs) and a description of hierarchical abstract machines. We then present, in abbreviated fonn, the following results: 1) Given any HAM and any MOP, there exists a new MOP such that the optimal policy in the new MOP is optimal in the original MOP among those policies that satisfy the constraints specified by the HAM. This means that even with complex machine specifications we can still apply standard decision-making and learning methods. 2) An algorithm exists that detennines this optimal policy, given an MOP and a HAM. 3) On an illustrative problem with 3600 states, this algorithm yields dramatic perfonnance improvements over standard algorithms applied to the original MOP. 4) A reinforcement learning algorithm exists that converges to the optimal policy, subject to the HAM constraints, with no need to construct explicitly a new MOP. 5) On the sample problem, this algorithm learns dramatically faster than standard RL algorithms. We conclude with a discussion of related approaches and ongoing work. 2 Markov Decision Processes We assume the reader is familiar with the basic concepts of MOPs. To review, an MOP is a 4-tuple, (5, A, T, R) where 5 is a set of states, A is a set of actions, T is a transition model mapping 5 x A x 5 into probabilities in [0, I J, and R is a reward function mapping 5 x A x 5 into real-valued rewards. Algorithms for solving MOPs can return a policy 7r that maps from 5 to A, a real-valued value function V on states, or a real-valued Q-function on state-action pairs. In this paper, we focus on infinite-horizon MOPs with a discount factor /3. The aim is to find an optimal policy 7r* (or, equivalently, V* or Q*) that maximizes the expected discounted total reward of the agent. Throughout the paper, we will use as an example the MOP shown in Figure l(a). Here A contains four primitive actions (up, down, left, right). The transition model, T, specifies that each action succeeds 80% of time, while 20% of the time the agent moves in an unintended perpendicular direction. The agent begins in a start state in the upper left corner. A reward of 5.0 is given for reaching the goal state and the discount factor /3 is 0.999. 3 Hierarchical abstract machines A HAM is a program which, when executed by an agent in an environment, constrains the actions that the agent can take in each state. For example, a very simple machine might dictate, "repeatedly choose right or down," which would eliminate from consideration all policies that go up or left. HAMs extend this simple idea of constraining policies by providing a hierarchical means of expressing constraints at varying levels of detail and Reinforcement Learning with Hierarchies of Machines 1045 specificity. Machines for HAMs are defined by a set of states, a transition function, and a start function that detennines the initial state of the machine. Machine states are of four types: Action states execute an action in the environment. Call states execute another machine as a subroutine. Choice states nondetenninistically select a next machine state. Stop states halt execution of the machine and return control to the previous call state. The transition function detennines the next machine state after an action Or call state as a stochastic function of the current machine state and some features of the resulting environment state. Machines will typically use a partial description of the environment to detennine the next state. Although machines can function in partially observable domains, for the purposes of this paper we make the standard assumption that the agent has access to a complete description as well. A HAM is defined by an initial machine in which execution begins and the closure of all machines reachable from the initial machine. Figure I(c) shows a simplified version of one element of the HAM we used for the MDP in Figure I. This element is used for traversing a hallway while negotiating obstacles of the kind shown in Figure 1(b). It runs until the end of the hallway or an intersection is reached. When it encounters an obstacle, a choice point is created to choose between two possible next machine states. One calls the backoff machine to back away from the obstacle and then move forward until the next one. The other calls the follow-wall machine to try to get around the obstacle. The follow-wall machine is very simple and will be tricked by obstacles that are concave in the direction of intended movement; the backoff machine, on the other hand, can move around any obstacle in this world but could waste time backing away from some obstacles unnecessarily and should be used sparingly. Our complete "navigation HAM" involves a three-level hierarchy, somewhat reminiscent of a Brooks-style architecture but with hard-wired decisions replaced by choice states. The top level of the hierarchy is basically just a choice state for choosing a hallway navigation direction from the four coordinate directions. This machine has control initially and regains control at intersections or corners. The second level of the hierarchy contains four machines for moving along hallways, one for each direction. Each machine at this level has a choice state with four basic strategies for handling obstacles. Two back away from obstacles and two attempt to follow walls to get around obstacles. The third level of the hierarchy implements these strategies using the primitive actions. The transition function for this HAM assumes that an agent executing the HAM has access to a short-range, low-directed sonar that detects obstacles in any of the four axis-parallel adjacent squares and a long-range, high-directed sonar that detects larger objects such as the intersections and the ends of hallways. The HAM uses these partial state descriptions to identify feasible choices. For example, the machine to traverse a hallway northwards would not be called from the start state because the high-directed sonar would detect a wall to the north. Our navigation HAM represents an abstract plan to move about the environment by repeatedly selecting a direction and pursuing this direction until an intersection is reached. Each machine for navigating in the chosen direction represents an abstract plan for moving in a particular direction while avoiding obstacles. The next section defines how a HAM interacts with a specific MDP and how to find an optimal policy that respects the HAM constraints. 4 Defining and solving the HAM-induced MDP A policy for a model, M, that is HAM-consistent with HAM H is a scheme for making choices whenever an agent executing H in M, enters a choice state. To find the optimal HAM-consistent policy we apply H to M to yield an induced MDP, HoM. A somewhat simplified description of the construction of HoM is as follows: 1) The set of states in HoM is the cross-product of the states of H with the states of M. 2) For each state in HoM where the machine component is an action state, the model and machine transition R. Parr and S. Russell 1046 WllItoIIlHAM - - WillloutHAM WittlHAM ....... WtthHAM .... -- r-20 ~ 1 I~ : 10 o ____~________________~~. o SOO 1(0) lSOO 2:C)X) l500 lOOO 3SOO .4QOO ..soD ~ RuntiJne(.ca::ondI' (a) (b) Figure 2: Experimental results showing policy value (at the initial state) as a function of runtime on the domain shown in Figure 1. (a) Policy iteration with and without the HAM. (b) Q-learning with and without the HAM (averaged over 10 runs). functions are combined. 3) For each state where the machine component is a choice state, actions that change only the machine component of the state are introduced. 4) The reward is taken from M for primitive actions, otherwise it is zero. With this construction, we have the following (proof omitted): Lemma 1 For any Markov decision process M and any! HAM H, the induced process HoM is a Markov decision process. Lemma 2 If 7r is an optimal policy for HoM , then the primitive actions specified by constitute the optimal policy for M that is HAM-consistent with H. 7r Of course, HoM may be quite large. Fortunately, there are two things that will make the problem much easier in most cases. The first is that not all pairs of HAM states and environment states will be possible, i.e., reachable from an initial state. The second is that the actual complexity of the induced MOP is determined by the number of choice points, i.e., states of HoM in which the HAM component is a choice state. This leads to the following: Theorem 1 For any MOP, M, and HAM, H, let C be the set of choice points in HoM . There exists a decision process, reduce(H 0 M), with states C such that the optimal policy for reduce(H 0 M) corresponds to the optimal policy for M that is HAM-consistent with H. Proof sketch We begin by applying Lemma 1 and then observing that in states of HoM where the HAM component is not a choice state, only one action is permitted. These states can be removed to produce an equivalent Semi-Markov decision process (SMOP). (SMOPs are a generalization of Markov decision processes that permit different discount rates for different transitions.) The optimal policy for this SMOP will be the same as the optimal policy for HoM and by Lemma 2, this will be the optimal policy for M that is HAM-consistent with H. 0 This theorem formally establishes the mechanism by which the constraints embodied in a HAM can be used to simplify an MDP. As an example of the power of this theorem, and to demonstrate that this transformation can be done efficiently, we applied our navigation HAM to the problem described in the previous section. Figure 2(a) shows the results of applying policy iteration to the original model and to the transformed model. Even when we add in the cost of transformation (which, with our rather underoptimized code, takes ITo preserve the Markov property, we require that if a machine has more than one possible caller in the hierarchy, that each appearance is treated as a distinct machine. This is equivalent to requiring that the call graph for the HAM is a tree. It follows from this that circular calling sequences are also forbidden. Reinforcement Learning with Hierarchies of Machines 1047 866 seconds), the HAM method produces a good policy in less than a quarter of the time required to find the optimal policy in the original model. The actual solution time is 185 seconds versus 4544 seconds. An important property of the HAM approach is that model transformation produces an MDP that is an accurate model of the application of the HAM to the original MDP. Unlike typical approximation methods for MDPs, the HAM method can give strict performance guarantees. The solution to the transformed model Teduce(H 0 M) is the optimal solution from within a well-defined class of policies and the value assigned to this solution is the true expected value of applying the concrete HAM policy to the original MDP. 5 Reinforcement learning with HAMs HAMs can be of even greater advantage in a reinforcement learning context, where the effort required to obtain a solution typically scales very badly with the size of the problem. HAM contraints can focus exploration of the state space, reducing the "blind search" phase that reinforcement learning agents must endure while learning about a new environment. Learning will also be fasterfor the same reason policy iteration is faster in the HAM-induced model; the agent is effectively operating in a reduced state space. We now introduce a variation of Q-learning called HAMQ-1earning that learns directly in the reduced state space without performing the model transformation described in the previous section. This is significant because the the environment model is not usually known a priori in reinforcement learning contexts. A HAMQ-learning agent keeps track of the following quantities: t, the current environment state; n, the current machine state; Se and me, the environment state and machine state at the previous choice point; a, the choice made at the previous choice point; and Te and 13e, the total accumulated reward and discount since the previous choice point. It also maintains an extended Q-table, Q([s, m], a), which is indexed by an environment-state/machine-state pair and by an action taken at a choice point. For every environment transition from state s to state t with observed reward T and discount ~ Te + 13eT and 13e ~ 13l3e. For each transition to a choice point, the agent does 13, the HAMQ-Iearning agent updates: Te Q([se, me], a) and then Te ~ 0, 13e ~ ~ Q([se, mc], a) + a[Te + 13e V([t, n]) - Q([Se, mc], a)], 1. Theorem 2 For any finite-state MDP, M, and any HAM, H, HAMQ-Iearning will converge to the optimal choice for every choice point in Teduce(H 0 M) with probability l. Proof sketch We note that the expected reinforcement signal in HAMQ-Iearning is the same as the expected reinforcement signal that would be received if the agent were acting directly in the transformed model of Theorem 1 above. Thus, Theorem 1 of [11] can be applied to prove the convergence of the HAMQ-learning agent, provided that we enforce suitable constraints on the exploration strategy and the update parameter decay rate. 0 We ran some experiments to measure the performance of HAMQ-learning on our sample problem. Exploration was achieved by selecting actions according to the Boltzman distribution with a temperature parameter for each state. We also used an inverse decay for the update parameter a. Figure 2(b) compares the learning curves for Q-Iearning and HAMQlearning. HAMQ-Iearning appears to learn much faster: Q-Iearning required 9,000,000 iterations to reach the level achieved by HAMQ-learning after 270,000 iterations. Even after 20,000,000 iterations, Q-Iearning did not do as well as HAMQ-learning.2 2Speedup techniques such as eligibility traces could be applied to get better Q-Ieaming results; such methods apply equally well to HAMQ-Iearning. 1048 6 R. Parr and S. Russell Related work State aggregation (see, e.g., [18] and [7]) clusters "similar" states together and assigns them the same value, effectively reducing the state space. This is orthogonal to our approach and could be combined with HAMs. However, aggregation should be used with caution as it treats distinct states as a single state and can violate the Markov property leading to the loss of performance guarantees and oscillation or divergence in reinforcement learning. Moreover, state aggregation may be hard to apply effectively in many cases. Dean and Lin [8] and Bertsekas and Tsitsiklis [2], showed that some MDPs are loosely coupled and hence amenable to divide-and-conquer algorithms. A machine-like language was used in [13] to partition an MDP into decoupled subproblems. In problems that are amenable to decoupling, this could approaches could be used in combinated with HAMs. Dayan and Hinton [6] have proposedJeudal RL which specifies an explicit subgoal structure, with fixed values for each sub goal achieved, in order to achieve a hierarchical decomposition of the state space. Dietterich extends and generalizes this approach in [9]. Singh has investigated a number of approaches to subgoal based decomposition in reinforcement learning (e.g. [17] and [16]). Subgoals seem natural in some domains, but they may require a significant amount of outside knowledge about the domain and establishing the relationship between the value of subgoals with respect to the overall problem can be difficult. Bradtke and Duff [3] proposed an RL algorithm for SMDPs. Sutton [19] proposes temporal abstractions, which concatenate sequences of state transitions together to permit reasoning about temporally extended events, and which can thereby form a behavioral hierarchy as in [14] and [15]. Lin's somewhat informal scheme [12] also allows agents to treat entire policies as single actions. These approaches can be emcompassed within our framework by encoding the events or behaviors as machines. The design of hierarchically organized, "layered" controllers was popularized by Brooks [4]. His designs use a somewhat different means of passing control, but our analysis and theorems apply equally well to his machine description language. The "teleo-reactive" agent designs of Benson and Nilsson [I] are even closer to our HAM language. Both of these approaches assume that the agent is completely specified, albeit self-modifiable. The idea of partial behavior descriptions can be traced at least to Hsu's partial programs [10], which were used with a deterministic logical planner. 7 Conclusions and future work We have presented HAMs as a principled means of constraining the set of policies that are considered for a Markov decision process and we have demonstrated the efficacy of this approach in a simple example for both policy iteration and reinforcement learning. Our results show very significant speedup for decision-making and learning-but of course, this reflects the provision of knowledge in the form of the HAM. The HAM language provides a very general method of transferring knowledge to an agent and we only have scratched the surface of what can be done with this approach. We believe that if desired, subgoal information can be incorporated into the HAM structure, unifying subgoal-based approaches with the HAM approach. Moreover, the HAM structure provides a natural decomposition of the HAM-induced model, making it amenable to the divide-and-conquer approaches of [8] and [2]. There are opportunities for generalization across all levels of the HAM paradigm. Value function approximation can be used for the HAM induced model and inductive learning methods can be used to produce HAMs or to generalize their effects upon different regions of the state space. Gradient-following methods also can be used to adjust the transition probabilities of a stochastic HAM. HAMs also lend themselves naturally to partially observable domains. They can be applied directly when the choice points induced by the HAM are states where no confusion about Reinforcement Learning with Hierarchies of Machines 1049 the true state of the environment is possible. The application of HAMs to more general partially observable domains is more complicated and is a topic of ongoing research. We also believe that the HAM approach can be extended to cover the average-reward optimality criterion. We expect that successful pursuit of these lines of research will provide a formal basis for understanding and unifying several seemingly disparate approaches to control, including behavior-based methods. It should also enable the use of the MDP framework in real-world applications of much greater complexity than hitherto attacked, much as HTN planning has extended the reach of classical planning methods. References [1] S. Benson and N. Nilsson. Reacting, planning and learning in an autonomous agent. In K. Furukawa, D. Michie, and S. Muggleton, editors, Machine Intelligence 14. Oxford University Press, Oxford, 1995. [2] D. C. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation: Numerical Metlwds. Prentice-Hall, Englewood Cliffs, New Jersey, 1989. [3] S. J. Bradtke and M. O. Duff. Reinforcement learning methods for continuous-time Markov decision problems. In Advances in Neurallnfonnation Processing Systems 7: Proc. of the 1994 Conference, Denver, Colorado, December 1995. MIT Press. [4] R. A. Brooks. A robust layered control system for a mobile robot. IEEE Journal of Robotics and Automation, 2, 1986. [5] K. W. Currie and A. Tate. O-Plan: the Open Planning Architecture. Artificial Intelligence, 52(1), November 1991. [6] P. Dayan and G. E. Hinton. Feudal reinforcement learning. In Stephen Jose Hanson, Jack D. Cowan, and C. Lee Giles, editors, Neural Information Processing Systems 5, San Mateo, California, 1993. Morgan Kaufman. [7] T. Dean, R. Givan, and S. Leach. Model reduction techniques for computing approximately optimal solutions for markov decision processes. In Proc. of the Thirteenth Conference on Uncertainty in Artificial Intelligence , Providence, Rhode Island, August 1997. Morgan Kaufmann. [8] T. Dean and S.-H. Lin. Decomposition techniques for planning in stochastic domains. In Proc. of the Fourteenth Int. Joint Conference on Artificial Intelligence, Montreal, Canada, August 1995. Morgan Kaufmann. [9] Thomas G. Dietterich. Hierarchical reinforcement learning with the MAXQ value function decomposition. Technical report, Department of Computer Science, Oregon State University, Corvallis, Oregon, 1997. [10] Y.-J. Hsu. Synthesizing efficient agents from partial programs. In Metlwdologiesfor Intelligent Systems: 6th Int. Symposium, ISMIS '91, Proc., Charlotte, North Carolina, October 1991. Springer-Verlag. [11] T. Jaakkola, M.l. Jordan, and S.P. Singh. On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6(6), 1994. [12] L.-J. Lin. Reinforcement Learning for Robots Using Neural Networks. PhD thesis, Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania, 1993. [13] Shieu-Hong Lin. Exploiting Structure for Planning and Control. PhD thesis, Computer Science Department, Brown University, Providence, Rhode Island, 1997. [14] A. McGovern, R. S. Sutton, and A. H. Fagg. Roles of macro-actions in accelerating reinforcement learning. In 1997 Grace Hopper Celebration of Women in Computing, 1997. [15] D. Precup and R. S. Sutton. Multi-time models fortemporally abstract planning. In This Volume . [16] S. P. Singh. Scaling reinforcement learning algorithms by learning variable temporal resolution models. In Proceedings of the Ninth International Conference on Machine Learning, Aberdeen, July 1992. Morgan Kaufmann. [17] S. P. Singh. Transfer of learning by composing solutions of elemental sequential tasks. Machine Learning, 8(3), May 1992. [18] S. P. Singh, T. Jaakola, and M. I. Jordan. Reinforcement learning with soft state aggregation. In G. Tesauro, D. S. Touretzky, and T. K. Leen, editors, Neural Information Processing Systems 7, Cambridge, Massachusetts, 1995. MIT Press. [19] R. S. Sutton. Temporal abstraction in reinforcement learning. In Proc. of the Twelfth Int. Conference on Machine Learning, Tahoe City, CA, July 1995. Morgan Kaufmann. 

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The Observer-Observation Dilemma in Neuro-Forecasting Hans Georg Zimmermann Ralph Neuneier SiemensAG Corporate Technology D-81730 Munchen, Germany Georg.Zimmermann@mchp.siemens.de Siemens AG Corporate Technology D-81730 Munchen, Germany Ralph.Neuneier@mchp.siemens.de Abstract We explain how the training data can be separated into clean information and unexplainable noise. Analogous to the data, the neural network is separated into a time invariant structure used for forecasting, and a noisy part. We propose a unified theory connecting the optimization algorithms for cleaning and learning together with algorithms that control the data noise and the parameter noise. The combined algorithm allows a data-driven local control of the liability of the network parameters and therefore an improvement in generalization. The approach is proven to be very useful at the task of forecasting the German bond market. 1 Introduction: The Observer-Observation Dilemma Human beings believe that they are able to solve a psychological version of the ObserverObservation Dilemma. On the one hand, they use their observations to constitute an understanding of the laws of the world, on the other hand, they use this understanding to evaluate the correctness of the incoming pieces of information. Of course, as everybody knows, human beings are not free from making mistakes in this psychological dilemma. We encounter a similar situation when we try to build a mathematical model using data. Learning relationships from the data is only one part of the model building process. Overrating this part often leads to the phenomenon of overfitting in many applications (especially in economic forecasting). In practice, evaluation of the data is often done by external knowledge, i. e. by optimizing the model under constraints of smoothness and regularization [7]. If we assume, that our model summerizes the best knowledge of the system to be identified, why should we not use the model itself to evaluate the correctness of the data? One approach to do this is called Clearning [11]. In this paper, we present a unified approach of the interaction between the data and a neural network (see also [8]). It includes a new symmetric view on the optimization algorithms, here learning and cleaning, and their control by parameter and data noise. The Observer-Observation Dilemma in Neuro-Forecasting 993 2 Learning 2.1 Learning reviewed We are especially interested in using the output of a neural network y( x, w), given the input pattern, x, and the weight vector, w, as a forecast of financial time series. In the context of neural networks learning nonnally means the minimization of an error function E by changing the weight vector w in order to achieve good generalization performance. Typical error functions can be written as a sum of individual terms over all T training patterns, E = ~ 'L,;=1 E t . For example, the maximum-likelihood principle leads to E t = 1/2 (y(x, w) - yt)2 , (1) yt with as the given target pattern. If the error function is a nonlinear function of the parameters, learning has to be done iteratively by a search through the weight space, changing the weights from step T to T + 1 according to: (2) There are several algorithms for choosing the weight increment ~W(T) , the most easiest being gradient descent. After each presentation of an input pattern, the gradient gt := VEt Iw of the error function with respect to the weights is computed. In the batch version of gradient descent the increments are based on all training patterns 1 ~W(T) = -"1g = -"1 T T L gt, (3) t=l whereas the pattern-by-pattern version changes the weights after each presentation of a pattern Xt (often randomly chosen from the training set): (4) The learning rate "1 is typically held constant or follows an annealing procedure during training to assure convergence. Our experiments have shown that small batches are most useful, especially in combination with Vario-Eta, a stochastic approximation of a QuasiNewton method [3]: ~W(T) = - N "1 J+'L,(gt - g)2 . ~L gt, (5) N t=l with and N ~ 20. Learning pattern-by-pattern or with small batches can be viewed as a stochastic search process because we can write the weight increments as: (6) These increments consist of the terms 9 with a drift to a local minimum and of noise terms 'L,~ 1 gt - g) disturbing this drift. (k 2.2 Parameter Noise as an Implicit Penalty Function Consider the Taylor expansion of E (w) around some point w in the weight space E(w + ~w) = E(w) 1 + VE ~w + 2~W'H ~w (7) H. G. Zimmermann and R. Neuneier 994 with H as the Hessian of the error function. Assume a given sequence of T disturbance vectors ~ Wt, whose elements ~ Wt ( i) are identically, independently distributed (i .i.d.) with zero mean and variance (row-)vector var(~wi) to approximate the expectation (E( w) by 1", (E(w) ~ T L...J E(w 1", . + ~Wt) = E(w) + :2 L...J var(~w(z))Hii' t (8) i with Hii as the diagonal elements of H. In eq. 8, noise on the weights acts implicitly as a penalty term to the error function given by the second derivatives H ii . The noise variances var( ~ w( i)) operate as penalty parameters. As a result of this flat minima solutions which may be important for achieving good generalization performance are favored [5]. Learning pattern-by-pattem introduces such noise in the training procedure i.e., ~ Wt = -1] ? gt? Close to convergence, we can assume that gt is i.i.d. with zero mean and variance vector var(gi) so that the expected value can be approximated by (E(w) ~ E(w) 82E Lvar(gd8Wi2' 2 I. TJ2 +- (9) This type of learning introduces to a local penalty parameter var( ~ w (i) ), characterizing the stability of the weights w = [Wdi=l, ... ,k. The noise effects due to Vario-Eta learning ~wt(i) 1]2 (E(w) ~ E(w) + 2 = -R .gti leads to E :L 88w~' 2 i (10) I By canceling the term var(gi) in eq. 9, Vario-Eta achieves a simplified uniform penalty parameter, which depends only on the learning rate 1]. Whereas pattern-by-pattern learning is a slow algorithm with a locally adjusted penalty control, Vario-Eta is fast only at the cost of a simplified uniform penalty term. We summarize this section by giving some advice on how to learn to flat minima solutions: ? Train the network to a minimal training error solution with Vario-Eta, which is a stochastic approximation of a Newton method and therefore very fast. ? Add a final phase of pattem-by-pattern learning with uniform learning rate to fine tune the local curvature structure by the local penalty parameters (eq. 9). ? Use a learning rate 1] as high as possible to keep the penalty effective. The training error may vary a bit, but the inclusion of the implicit penalty is more important. 3 3.1 Cleaning Cleaning reviewed When training neural networks, one typically assumes that the data is noise-free and one forces the network to fit the data exactly. Even the control procedures to minimize overfitting effects (i.e., pruning) consider the inputs as exact values. However, this assumption is often violated, especially in the field of financial analysis, and we are taught by the phenomenon of overfitting not to follow the data exactly. Clearning, as a combination of cleaning and learning, has been introduced in the paper of [11]. The motivation was to minimize overfitting effects by considering the input data as corrupted by noise whose distribution has also to be learned. The Cleaning error function for the pattern t is given by the sum of two terms yx E t' ="21[(Yt - Ytd)2 + (Xt - x td)2] = E ty + E tx (11) The Observer-Observatio?n Dilemma in Neuro-Forecasting 995 with xf, yt as the observed data point. In the pattem-by-pattem learning, the network output y( x t, w) determines the adaptation as usual, (12) We have also to memorize correction vectors present the cleaned input Xt to the network, Xt = &t for all input data of the training set to xf + &t (13) = 0 can be described as The update rule for the corrections, initialized with ~x~O) ~X~T+I) (1 - 1])~X!T) - 1](Yt - yt) ~ (14) All the necessary quantIties, i. e. (Yt - yt) &Y~;w) are computed by typical backpropagation algorithms, anyway. We experienced, that the algorithms work well, if the same learning rate 1] is used for both, the weight and cleaning updates. For regression, cleaning forces the acceptance of a small error in x. which can in turn decrease the error in Y dramatically, especially in the case of outliers. Successful applications of Cleaning are reported in [11] and [9]. Although the network may learn an optimal model for the cleaned input data, there is no easy way to work with cleaned data on the test set. As a consequence, the model is evaluated on a test set with a different noise characteristic compared to the training set. We will later propose a combination of learning with noise and cleaning to work around this serious disadvantage. 3.2 Data Noise reviewed Artificial noise on the input data is often used during training because it creates an infinite number of training examples and expands the data to empty parts of the input space. As a result, the tendency of learning by heart may be limited because smoother regression functions are produced. Now, we are considering again the Taylor expansion, this time applied to E (x) around some point x in the input space. The expected value (E (x)) is approximated by (E(x)} ~ ~L E(x + &t} = E(x) +~L t var(&(j))Hjj, (15) j with Hjj as the diagonal elements of the Hessian Hxx of the error function with respect to the inputs x. Again, in eq. 15, noise on the inputs acts implicitly as a penalty term to the error function with the noise variances var( & (j)) operating as penalty parameters. Noise on the input improve generalization behavior by favoring smooth models [1]. The noise levels can be set to a constant value, e. g. given by a priori knowledge, or adaptive as described now. We will concentrate on a uniform or normal noise distribution. Then, the adaptive noise level ~j is estimated for each input j individually. Suppressing pattern indices, we define the noise levels ~j or ~J as the average residual errors: uniform residual error:..t). Gaussian residual error: ..t)2. = _1 '~ " T ___ 1 T t '~ " t IOEY I, (16) (OEY) 2 (17) OXj OXj Actual implementations use stochastic approximation, e. g. for the uniform residual error ?~T+1) J = (1 _ T ~ )~~T) + ~ IoEY I. ) T OXj (18) H. G. Zimmennann and R. Neuneier 996 The different residual error levels can be interpreted as follows: A small level ~j may indicate an unimportant input j or a perfect fit of the network concerning this input j. In both cases, a small noise level is appropriate. On the other hand, a high value of ~j for an input j indicates an important but imperfectly fitted input. In this case high noise levels are advisable. High values of ~j lead to a stiffer regression model and may therefore increase the generalization perfonnance of the network. 3.3 Cleaning with Noise Typically, training with noisy inputs takes a data point and adds a random variable drawn from a fixed or adaptive distribution. This new data point Xt is used as an input to the network. If we assume, that the data is corrupted by outliers and other influences, it is preferable to add the noise tenn to the cleaned input. For the case of Gaussian noise the resulting new input is: Xt = + ~Xt + ~?, (19) xf with ? drawn from the nonnal distribution. The cleaning of the data leads to a corrected mean of the data and therefore to a more symmetric noise distribution, which also covers the observed data x t ? We propose a variant which allows more complicated noise distributions: (20) with k as a random number drawn from the indices of the correction vectors (~xtlt=l , ... ,T. In this way we use a possibly asymmetric and/or dependent noise distribution, which still covers the observed data Xt by definition of the algorithm. xf One might wonder, why to disturb the cleaned input + ~Xt with an additional noisy tenn ~x k. The reason for this is, that we want to benefit from representing the whole input distribution to the network instead of only using one particular realization. 4 A Unifying Approach 4.1 The Separation of Structure and Noise In the previous sections we explained how the data can be separated into clean infonnation and unexplainable noise. Analogous, the neural network is described as a time invariant structure (otherwise no forecasting is possible) and a noisy part. - t cleaned data + time invariant data noise data neural network-ttime invariant parameters+parameter noise We propose to use cleaning and adaptive noise to separate the data and to use learning and stochastic search to separate the structure of the neural network. ~ cleaning(neural network) + adaptive noise (neural network) data neural network~learning (data) + stochastic search(data) The algorithms analyzing the data depend directly o"n the network whereas the methods searching for structure are directly related to the data. It should be clear that the model building process should combine both aspects in an alternate or simultaneous manner. The interaction of algorithms concerning data analysis and network structure enables the realization of the the concept of the Observer-Observation Dilemma. The Observer-Observation Dilemma in Neuro-Forecasting 997 The aim of the unified approach can be described, exemplary assuming here a Gaussian noise model, as the minimization of the error due to both, the structure and the data: T 2~ I: [(Yt - y1)2 + {Xt - x1)2] -+ ~~fJ (21) t=l Combining the algorithms and approximating the cumulative gradient 9 by g, we receive data (22) structure (1 - 0: WIT) - )gH + O:(Yt - 7]g(T) '-....-' learning yt) P- -7](9t - g0-)) ---......-.noise The cleaning of the data by the network computes an individual correction term for each training pattern. The adaptive noise procedure according to eq. 20 generates a potentially asymmetric and dependent noise distribution which also covers the observed data. The implied curvature penalty, whose strength depends on the individual liability of the input variables, can improve the generalization performance of the neural network. The learning of the structure searches for time invariant parameters characterized by -j; L 9t = O. The parameter noise supports this exploration as a stochastic search to find better "global" minima. Additionally, the generalization performance may be further improved by the implied curvature penalty depending on the local liability of the parameters. Note that, although the description of the weight updates collapses to the simple form of eq. 4, we preferred the formula above to emphasize the analogy between the mechanism which handles the data and the structure. In searching for an optimal combination of data and parameters, the noise of both parts is not a disastrous failure to build a perfect model but it is an important element to control the interaction of data and structure. 4.2 Pruning The neural network topology represents only a hypothesis of the true underlying class of functions. Due to possible misspecification, we may have defects of the parameter noise distribution. Pruning algorithms are not only a way to limit the memory of the network, but they also appear useful to correct the noise distribution in different ways. Stochastic-Pruning [2] is basically a t-test on the weights w. Weights with low testw values constitute candidates for pruning to cancel weights with low liability measured by the size of the weight divided by the standard deviation of its fluctuations. By this, we get a stabilization of the learning against resampling of the training data. A further weight pruning method is EBD, Early-Brain-Damage [10], which is based on the often cited OBD pruning method of [6]. In contrast to OBD, EBD allows its application before the training has reached a local minimum. One of the advantages of EBD over OBD is the possibility to perform the testing while being slidely away from a local minimum. In our training procedure we propose to use noise even in the final part of learning and therefore we are only nearby a local minimum. Furthermore, EBD is also able to revive already pruned weights. Similar to Stochastic Pruning, EBD favors weights with a low rate of fluctuations. If a weight is pushed around by a high noise, the implicit curvature penalty would favor a flat minimum around this weight which leads to its elimination by EBD. 998 H. G. Zimmermann and R. Neuneier 5 Experiments In a research project sponsored by the European Community we are applying the proposed approach to estimate the returns of 3 financial markets for each of the G7 countries subsequently using these estimations in an asset allocation scheme to create a Markowitz-optimal portfolio [4]. This paper reports the 6 month forecasts of the German bond rate, which is one of the more difficult tasks due to the reunification of Germany and GDR. The inputs consist of 39 variables achieved by preprocessing 16 relevant financial time series. The training set covers the time from April, 1974 to December, 1991, the test set runs from J anuary, 1992 to May, 1996. The network arcitecture consists of one hidden layer (20 neurons, tanh transfer function) and one linear output. First, we trained the neural network until convergence with pattern-by-pattern learning using a small batch size of 20 patterns (classical approach). Then, we trained the network using the unified approach as described in section 4.1 using pattern-by-pattern learning. We compare the resulting predictions of the networks on the basis of four performance measures (see table). First, the hit rate counts how often the sign of the return of the bond has been correctly predicted. As to the other measures, the step from the forecast model to a trading system is here kept very simple. If the output is positive, we buy shares of the bond, otherwise we sell them. The potential realized is the ratio of the return to the maximum possible return over the test (training) set. The annualized return is the average yearly profit of the trading systems. Our approach turns out to be superior: we almost doubled the annualized return from 4.5% to 8.5% on the test set. The figure compares the accumulated return of the two approaches on the test set. The unified approach not only shows a higher profitability, but also has by far a less maximal draw down. 35 i -- -- ~ -- -.., - I approach II our 30\ classical - , - -. 81%(96%) 66%(93%) realized potential 75%(100%) 44%(96%) annualized return 8.5% (11.2%) 4.5%(10.1%) !!!20 ~'5'JI j 'O' ., ' . 1i I " '- - -. . I ~~~/.// .3: 25 r hit rate - - " " ., ::. .... . ..... ... . .{--'0? - 20 -" 30 ... 40 - sO I - -So 11... References [I] Christopher M. Bishop. Neural Networks for Pattern Recognition. Clarendon Press, 1994. [2] W. Finnoff, F. Hergert, and H. G. Zimmennann. Improving generalization perfonnance by nonconvergentmodel selection methods. In proc. of ICANN-92, 1992. [3] W. Finnoff, F. Hergert, and H. G. Zimmennann. Neuronale Lemverfahren mit variabler Schrittweite. 1993. Tech. report, Siemens AG. [4] P. Herve, P. Nairn, and H. G. Zimmennann. Advanced Adaptive Architectures for Asset Allocation: A Trial Application. In Forecasting Financial Markets, 1996. [5] S. Hochreiter and J. Schmid huber. Flat minima. Neural Computation, 9(1): 1-42, 1997. [6] Y. Ie Cun, J. S. Denker, and S. A. Solla. Optimal brain damage. NIPS*89, 1990. [7] J. E. Moody and T. S. Rognvaldsson . Smoothing regularizers for projective basis function networks. NIPS 9, 1997. [8] R. Neuneier and H. G. Zimmennann. How to Train Neural Networks. In Tricks of the Trade: How to make algorithms really to work. Springer Verlag, Berlin, 1998. [9] B. Tang, W. Hsieh, and F. Tangang. Cleaming neural networks with continuity constraints for prediction of noisy time series. ICONIP '96, 1996. [10] V. Tresp, R. Neuneier, and H. G. Zimmennann. Early brain damage. NIPS 9, 1997. [II] A. S. Weigend, H. G. Zimmennann, and R. Neuneier. Cleaming. Neural Networks in Financial Engineering, (NNCM95), 1995. 

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Automated Aircraft Recovery via Reinforcement Learning: Initial Experiments Jeffrey F. Monaco Barron Associates, Inc. Jordan Building 1160 Pepsi Place, Suite 300 Charlottesville VA 22901 monaco@bainet.com David G. Ward Barron Associates, Inc. Jordan Building 1160 Pepsi Place, Suite 300 Charlottesville VA 22901 ward@bainet.com Andrew G. Barto Department of Computer Science University of Massachusetts Amherst MA 01003 barto@cs.umass.edu Abstract Initial experiments described here were directed toward using reinforcement learning (RL) to develop an automated recovery system (ARS) for high-agility aircraft. An ARS is an outer-loop flight-control system designed to bring an aircraft from a range of out-of-control states to straightand-level flight in minimum time while satisfying physical and physiological constraints. Here we report on results for a simple version of the problem involving only single-axis (pitch) simulated recoveries. Through simulated control experience using a medium-fidelity aircraft simulation, the RL system approximates an optimal policy for pitch-stick inputs to produce minimum-time transitions to straight-and-Ievel flight in unconstrained cases while avoiding ground-strike. The RL system was also able to adhere to a pilot-station acceleration constraint while executing simulated recoveries. Automated Aircraft Recovery via Reinforcement Learning 1023 1 INTRODUCTION An emerging use of reinforcement learning (RL) is to approximate optimal policies for large-scale control problems through extensive simulated control experience. Described here are initial experiments directed toward the development of an automated recovery system (ARS) for high-agility aircraft. An ARS is an outer-loop flight control system designed to bring the aircraft from a range of initial states to straight, level, and non-inverted flight in minimum time while satisfying constraints such as maintaining altitude and accelerations within acceptable limits. Here we describe the problem and present initial results involving only single-axis (pitch) recoveries. Through extensive simulated control experience using a medium-fidelity simulation of an F-16, the RL system approximated an optimal policy for longitudinal-stick inputs to produce near-minimum-time transitions to straight and level flight in unconstrained cases, as well as while meeting a pilot-station acceleration constraint. 2 AIRCRAFT MODEL The aircraft was modeled as a dynamical system with state vector x = {q, 0, p, r, {3, Vi}, where q = body-axes pitch rate, 0 = angle of attack, p = body-axes roll rate, r = body-axes yaw rate, {3 = angle of sideslip, Vi = total airspeed, and control vector fl = {fl se , flae, fla/' Orud} of effector and pseudo-effector displacements. The controls are defined as: flse = symmetric elevon, oae = asymmetric elevon, oal = asymmetric flap, and Orud = rudder. (A pseudo-effector is a mathematically convenient combination of real effectors that, e.g., contributes to motion in a limited number of axes.) The following additional descriptive variables were used in the RL problem formulation: h = altitude, h = vertical component of velocity, e = pitch attitude, N z = pilot-station normal acceleration. For the initial pitch-axis experiment described here, five discrete actions were available to the learning agent in each state; these were longitudinal-stick commands selected from {-6, -3,0, +3, +6} lbf. The command chosen by the learning agent was converted into a desired normal-acceleration command through the standard F-16 longitudinal-stick command gradient with software breakout. This gradient maps pounds-of-force inputs into desired acceleration responses. We then produce an approximate relationship between normal acceleration and body-axes pitch rate to yield a pitch-rate flying-qualities model. Given this model, an inner-loop linear-quadratic (LQ) tracking control algorithm determined the actuator commands to result in optimal model-following of the desired pitch-rate response. The aircraft model consisted of complete translational and rotational dynamics, including nonlinear terms owing to inertial cross-coupling and orientation-dependent gravitational effects. These were obtained from a modified linear F-16 model with dynamics of the form j; = Ax + Bfl + b + N where A and B were the F-16 aero-inertial parameters (stability derivatives) and effector sensitivities (control derivatives). These stability and control derivatives and the bias vector, b, were obtained from linearizations of a high-fidelity nonlinear, six-degree-of-freedom model. Nonlinearities owing to inertial cross-coupling and orientation-dependent gravitational effects were accounted for through the term N, which depended nonlinearly on the state. Nonlinear actuator dynamics were modeled via the incorporation ofF-16 effector-rate and effector-position limits. See Ward et al. (1996) for additional details. 3 PROBLEM FORMULATION The RL problem was to approximate a minimum-time control policy capable of bringing the aircraft from a range of initial states to straight, level, and non-inverted flight, while satisfying given constraints, e.g., maintaining the normal acceleration at the pilot station within 1. F. Monaco, D. G. Ward and A G. Barto 1024 an acceptable range. For the single-axis (pitch-axis) flight control problem considered here, recovered flight was defined by: q = q = it = h = i't = o. (1) Successful recovery was achieved when all conditions in Eq. 1 were satisfied simultaneously within pre-specified tolerances. Because we wished to distinguish between recovery supplied by the LQ tracker and that learned by the RL system, special attention was given to formulating a meaningful test to avoid falsely attributing successes to the RL system. For example, if initial conditions were specified as off-trim perturbations in body-axes pitch rate, pitch acceleration, and true airspeed, the RL system may not have been required because the LQ controller would provide all the necessary recovery, i.e., zero longitudinal-stick input would result in a commanded body-axes pitch rate of zero deg./ sec. Because this controller is designed to be highly responsive, its tracking and integral-error penalties usually ensure that the aircraft responses attain the desired state in a relatively short time. The problem was therefore formulated to demand recovery from aircraft orientations where the RL system was primarily responsible for recovery, and the goal state was not readily achieved via the stabilizing action of the LQ control law. A pitch-axis recovery problem of interest is one in which initial pitch attitude, e, is selected to equal e trim +U(80 ,n' 8 0Tn a:l:)' where e trim == atrim (by definition), U is a uniformly distributed random number, and eO Tnin and eoTnaz define the boundaries of the training region, and other variables are set so that when the aircraft is parallel to the earth (8 0 = 0), it is "pancaking" toward the ground (with positive trim angle of attack). Other initial conditions correspond to purely-translational climb or descent of the aircraft. For initial conditions where eo < atrim, the flight vehicle will descend, and in the absence of any corrective longitudinal-stick force, strike the ground or water. Because it imposes no constraints on altitude or pitch-angle variations, the stabilizing response of the LQ controller is inadequate for providing the necessary recovery. Tn 4 REINFORCEMENT LEARNING ALGORITHM Several candidate RL algorithms were evaluated for the ARS. Initial efforts focused primarily on (1) Q-Learning, (2) alternative means for approximating the action-value function (Q function), and (3) use of discrete versus continuous-action controls. During subsequent investigations, an extension of Q-Learning called Residual Advantage Learning (Baird, 1995; Harmon & Baird, 1996) was implemented and successfully applied to the pitch-axis ARS problem. As with action-values in Q-Learning, the advantage function, A(x, u), may be represented by a function approximation system of the form A(x,u) = ?(x,ufO, (2) where ?( x, u) is a vector of relevant features and 0 are the corresponding weights. Here, the advantage function is linear in the weights, 0, and these weights are the modifiable, learned parameters. For advantage functions of the form in Eq. 2, the update rule is: Ok+l Ok - a ((r ? ( + "Y~t A(y, b*)) K~t + (1 - K~t) A(x, a*) - ~"Y~t?(y, b*) K~t + ~ (1 - K~t) ?(x, a*) - A(x, a)) ?(x, a)) , where a* = argminaA(x, a) and b* = argminbA(y, b), !l.t is the system rate (0.02 sec. in the ARS), "Y~t is the discount factor, and K is an fixed scale factor. In the above notation, Automated Aircraft Recovery via Reinforcement Learning 1025 y is the resultant state, i.e., the execution of action a results in a transition from state x to its successor y. The Residual Advantage Learning update collapses to the Q-Learning update for the case ~ = 0, K = The parameter ~ is a scalar that controls the trade-off between residualgradient descent when ~ = 1, and a faster, direct algorithm when ~ = O. Harmon & Baird (1996) address the choice of ~, suggesting the following computation of ~ at each time step: L. ;F,. 'J!'= l:o WdWrg +J-L l:o(Wd - wrg)wrg where Wd and Wrg are traces (one for each (J of the function approximation system) associated with the direct and residual gradient algorithms, respectively, and J-L is a small, positive constant that dictates how rapidly the system forgets. The traces are updated during each cycle as follows Wd f- (1-J-L)Wd-J-L[(r+'Y~tA(y,b*)) K~t+(1- K~t)A(X,a*)] ? [- :(JA(x, a*)] wrg f- (1-J-L)Wrg-J-L[(r+'Y~tA(y,b*?K~t+(1- K~t)A(x,a*)-A(X,a)] ? ['Y~t ;(JA(y,b*) K~t + (1- K~t) ;(JA(x,a*) - ;(JA(X, a)] . Advantage Learning updates of the weights, including the calculation of an adaptive ~ as discussed above, were implemented and interfaced with the aircraft simulation. The Advantage Learning algorithm consistently outperformed its Q-Learning counterpart. For this reason, most of our efforts have focused on the application of Advantage Learning to the solution of the ARS. The feature vector 4>(x, u) consisted of normalized (dimensionless) states and controls, and functions ofthese variables. Use ofthese nondimensionalized variables (obtained via the Buckingham 7r-theorem; e.g., Langharr, 1951) was found to enhance greatly the stability and robustness of the learning process. Furthermore, the RL system appeared to be less sensitive to changes in parameters such as the learning rate when these techniques were employed. 5 TRAINING Training the RL system for arbitrary orientations was accomplished by choosing random initial conditions on e as outlined above. With the exception of h, all other initial conditions corresponded to trim values for a Mach 0.6, 5 kIt. flight condition. Rewards were -1 per-time-step until the goal state was reached. In preliminary experiments, the training region was restricted to ? 0.174 rad.(l0 deg.) from the trim pitch angle. For this range of initial conditions, the system was able to learn an appropriate policy given only a handful of features (approximately 30). The policy was significantly mature after 24 hours oflearning on an HP-730 workstation and appeared to be able to achieve the goal for arbitrary initial conditions in the aforementioned domain. We then expanded the training region and considered initial e values within ? 0.785 rad. (45 deg .) of trim. The policy previously learned for the more restricted training domain performed well here too, and learning to recover for these more drastic off-trim conditions was trivial. No boundary restrictions were imposed on the system, but a report of whether the aircraft would have struck the ground was maintained. It was noted 1. R Monaco, D. G. Ward and A. G. Barto 1026 that recovery from all possible initial conditions could not be achieved without hitting the ground. Episodes in which the ground would have been encountered were a result of inadequate control authority and not an inadequate RL policy. For example, when the initial pitch angle was at its maximum negative value, maximum-allowable positive stick (6 lbf.) was not sufficient to pull up the aircraft nose in time. To remedy this in subsequent experiments, the number of admissible actions was increased to include larger-magnitude commands: {-12, -9, -6, -3,0, +3, +6, +9, +12} lbf. Early attempts at solving the pitch-axis recovery problem with the expanded initial conditions in conjunction with this augmented action set proved challenging. The policy that worked well in the two previous experiments was no longer able to attain the goal state; it was only able to come close and oscillate indefinitely about the goal region. The agent learned to pitch up and down appropriately, e.g., when h was negative it applied a corrective positive action, and vice versa. However, because of system and actuator dynamics modeled in the simulation, the transient response caused the aircraft to pass through the goal state. Once beyond the goal region, the agent applied an opposite action, causing it to approach the goal state again, repeating the process indefinitely (until the system was reset and a new trial was started). Thus, the availability of large-amplitude commands and the presence of actuator dynamics made it difficult for the agent to fonnulate a consistent policy that afforded all goal state criteria being satisfied simultaneously. One might remedy the problem by removing the actuator dynamics; however, we did not wish to compromise simulation fidelity, and chose to use an expanded feature set to improve RL perfonnance. Using a larger collection offeatures with approximately 180 inputs, the RL agent was able to formulate a consistent recovery policy. The learning process required approximately 72 hours on an HP-730 workstation. (On this platform, the combined aircraft simulation and RL software execution rate was approximately twice that of real-time.) At this point performance was evaluated. The simulation was run in evaluation mode, i.e., learning rate was set to zero and random exploration was disabled. Performance is summarized below. 6 6.1 RESULTS UNCONSTRAINED PITCH-AXIS RECOVERY Fig. 1 shows the transition times from off-trim orientations to the goal state as a function of initial pitch (inclination) angle. Recovery times were approximately 11-12 sec. for the worst-case scenarios. i.e .? 180 1= 45 deg. off-trim. and decrease (almost) monotonically for points closer to the unperturbed initial conditions. The occasional "blips" in the figure suggest that additional learning would have improved the global RL performance slightly. For 180 1 = 45 deg. off-trim, maximum altitude loss and gain were each approximately 1667 ft. (0.33 x 5000 f t. ). These excursions may seem substantial. but when one looks atthe time histories for these maneuvers, it is apparent that the RL-derived policy was perfonning well. The policy effectively minimizes any altitude variation; the magnitude of these changes are principally governed by available control authority and the severity of the flight condition from which the policy must recover. Fig. 2 shows time histories of relevant variables for one of the limiting cases. The first column shows body-axes pitch rate (Qb) and commanded body-axes pitch rate (Qbmodel) in (deg./sec.), pilot station nonnal acceleration (Nz) in (g), angle of attack (Alpha) in (deg.). and pitch attitude (Theta) in (deg.), respectively. The second column shows the longitudinal stick action executed by the RL system (lbf.), the left and right elevator deflections (deg.). total airspeed (ft./ sec.), and altitude (ft.). The majority ofthe 1600+ ft. altitude loss occurs between zero and five sec.; during this time, the RL system is applying maximum (allowable) positive stick. Thus, this altitude excursion is principally attributed to limited control authority as well as significant off-trim initial orientations. Automated Aircraft Recovery via Reinforcement Learning 1027 20 18 16 14 Recovery Time (sec.) 12 10 8 6 4 2 O~~rn~~~~~~~~~~~~~~~~ ?50 ?40 ?30 ?20 ?10 0 10 30 20 40 50 Figure 1: Simulated Aircraft Recovery Times for Unconstrained Pitch-Axis ARS - _. 00.- 00. L ? - ~ ? 2 ? " , 12 j 11? ? - -IU jV ~ ? 2 ? ? ? 2 ? ? -:t: ~ , " 12 1. 12 ? 2 ? ? ? " cC." -_ ,'z ....... ........... he o -- 2 ? I 10 , -.- ~, ",,-- , " t== ? 2 ? -"0 Figure 2: TIme Histories During Unconstrained Pitch-Axis Recovery for 8 45 deg . 0 = 8 trim - I. R Monaco, D. G. Ward and A G. Barto 1028 6.2 CONSTRAINED PITCH-AXIS RECOVERY The requirement to execute aircraft recoveries while adhering to pilot-safety constraints was a deciding factor in using RL to demonstrate the automated recovery system concept. The need to recover an aircraft while minimizing injury and, where possible, discomfort to the flight crew, requires that the controller incorporate constraints that can be difficult or impossible to express in forms suitable for linear and nonlinear programming methods. In subsequent ARS investigations, allowable pilot-station normal acceleration was restricted to the range -1.5 9 ~ N z ~ 3.5 g. These values were selected because the unconstrained ARS was observed to exceed these limits. Several additional features (for a total of 189) were chosen, and the learning process was continued. Initial weights for the original 180 inputs corresponded to those from the previously learned policy; the new features were chosen to have zero weights initially. Here, the RL system learned to avoid the normal acceleration limits and consistently reach the goal state for initial pitch angles in the region [-45 + 8 trim , 35 + 8 trim ] deg. Additional learning should result in improved recovery policies in this bounded acceleration domain for all initial conditions. Nonetheless, the results showed how an RL system can learn to satisfy these kinds of constraints. 7 CONCLUSION In addition to the results reported here, we conducted extensive analysis of the degree to which the learned policy successfully generalized to a range of initial conditions not experienced in training. In all cases, aircraft responses to novel recovery scenarios were stable and qualitatively similar to those previously executed in the training region. We are also conducting experiments with a multi-axes ARS, in which longitudinal-stick and lateral-stick sequences must be coordinated to recover the aircraft. Initial results are promising, but substantially longer training times are required. In summary, we believe that the results presented here demonstrate the feasibility of using RL algorithms to develop robust recovery strategies for high-agility aircraft, although substantial further research is needed. Acknowledgments This work was supported by the Naval Air Warfare Center Aircraft Division (NAWCAD), Flight Controls/Aeromechanics Division under Contract N62269-96-C-0080. The authors thank Marc Steinberg, the Program Manager and Chief Technical Monitor. The authors also express appreciation to Rich Sutton and Mance Harmon for their valuable help, and to Lockheed Martin Tactical Aircraft Systems for authorization to use their ATLAS software, from which F-16 parameters were extracted. References Baird, L. C. (1995) Residual algorithms: reinforcement learning with function approximation. In A. Prieditis and S. Russell (eds.), Machine Learning: Proceedings of the Twelfth International Conference, pp. 30-37. San Francisco, CA: Morgan Kaufmann. Harmon, M. E. & Baird, L. C. (1996) Multi-agent residual advantage learning with general function approximation. Wright Laboratory Technical Report, WPAFB, OH. Langharr, H. L. (1951) Dimensional Analysis and Theory of Models. New York: Wiley and Sons. Ward, D. G., Monaco, J. E, Barron, R. L., Bird, R.A., Virnig, J.C., & Landers, T.E (1996) Self-designing controller. Final Tech. Rep. for Directorate of Mathematics and Computer Sciences, AFOSR, Contract F49620-94-C-0087. Barron Associates, Inc. 

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Local Dimensionality Reduction Stefan Schaal 1,2,4 sschaal@usc.edu http://www-slab.usc.edulsschaal Sethu Vijayakumar 3, I Christopher G. Atkeson 4 sethu@cs.titech.ac.jp http://ogawawww.cs.titech.ac.jp/-sethu cga@cc.gatech.edu http://www.cc.gatech.edul fac/Chris.Atkeson IERATO Kawato Dynamic Brain Project (IST), 2-2 Hikaridai, Seika-cho, Soraku-gun, 619-02 Kyoto 2Dept. of Comp. Science & Neuroscience, Univ. of South. California HNB-I 03, Los Angeles CA 90089-2520 3Department of Computer Science, Tokyo Institute of Technology, Meguro-ku, Tokyo-I 52 4College of Computing, Georgia Institute of Technology, 801 Atlantic Drive, Atlanta, GA 30332-0280 Abstract If globally high dimensional data has locally only low dimensional distributions, it is advantageous to perform a local dimensionality reduction before further processing the data. In this paper we examine several techniques for local dimensionality reduction in the context of locally weighted linear regression. As possible candidates, we derive local versions of factor analysis regression, principle component regression, principle component regression on joint distributions, and partial least squares regression. After outlining the statistical bases of these methods, we perform Monte Carlo simulations to evaluate their robustness with respect to violations of their statistical assumptions. One surprising outcome is that locally weighted partial least squares regression offers the best average results, thus outperforming even factor analysis, the theoretically most appealing of our candidate techniques. 1 INTRODUCTION Regression tasks involve mapping a n-dimensional continuous input vector x E ~n onto a m-dimensional output vector y E ~m ? They form a ubiquitous class of problems found in fields including process control, sensorimotor control, coordinate transformations, and various stages of information processing in biological nervous systems. This paper will focus on spatially localized learning techniques, for example, kernel regression with Gaussian weighting functions. Local learning offer advantages for real-time incremental learning problems due to fast convergence, considerable robustness towards problems of negative interference, and large tolerance in model selection (Atkeson, Moore, & Schaal, 1997; Schaal & Atkeson, in press). Local learning is usually based on interpolating data from a local neighborhood around the query point. For high dimensional learning problems, however, it suffers from a bias/variance dilemma, caused by the nonintuitive fact that " ... [in high dimensions] if neighborhoods are local, then they are almost surely empty, whereas if a neighborhood is not empty, then it is not local." (Scott, 1992, p.198). Global learning methods, such as sigmoidal feedforward networks, do not face this 634 S. School, S. Vijayakumar and C. G. Atkeson problem as they do not employ neighborhood relations, although they require strong prior knowledge about the problem at hand in order to be successful. Assuming that local learning in high dimensions is a hopeless, however, is not necessarily warranted: being globally high dimensional does not imply that data remains high dimensional if viewed locally. For example, in the control of robot anns and biological anns we have shown that for estimating the inverse dynamics of an ann, a globally 21dimensional space reduces on average to 4-6 dimensions locally (Vijayakumar & Schaal, 1997). A local learning system that can robustly exploit such locally low dimensional distributions should be able to avoid the curse of dimensionality. In pursuit of the question of what, in the context of local regression, is the "right" method to perfonn local dimensionality reduction, this paper will derive and compare several candidate techniques under i) perfectly fulfilled statistical prerequisites (e.g., Gaussian noise, Gaussian input distributions, perfectly linear data), and ii) less perfect conditions (e.g., non-Gaussian distributions, slightly quadratic data, incorrect guess of the dimensionality of the true data distribution). We will focus on nonlinear function approximation with locally weighted linear regression (L WR), as it allows us to adapt a variety of global linear dimensionality reduction techniques, and as L WR has found widespread application in several local learning systems (Atkeson, Moore, & Schaal, 1997; Jordan & Jacobs, 1994; Xu, Jordan, & Hinton, 1996). In particular, we will derive and investigate locally weighted principal component regression (L WPCR), locally weighted joint data principal component analysis (L WPCA), locally weighted factor analysis (L WF A), and locally weighted partial least squares (LWPLS). Section 2 will briefly outline these methods and their theoretical foundations, while Section 3 will empirically evaluate the robustness of these methods using synthetic data sets that increasingly violate some of the statistical assumptions of the techniques. 2 METHODS OF DIMENSIONALITY REDUCTION We assume that our regression data originate from a generating process with two sets of observables, the "inputs" i and the "outputs" y. The characteristics of the process ensure a functional relation y = f(i). Both i and yare obtained through some measurement device that adds independent mean zero noise of different magnitude in each observable, such that x == i + Ex and y = y + Ey ? For the sake of simplicity, we will only focus on one-dimensional output data (m=l) and functions / that are either linear or slightly quadratic, as these cases are the most common in nonlinear function approximation with locally linear models. Locality of the regression is ensured by weighting the error of each data point with a weight from a Gaussian kernel: Wi = exp(-O.5(Xi - Xqf D(Xi - Xq)) (1) Xtt denotes the query point, and D a positive semi-definite distance metric which determmes the size and shape of the neighborhood contributing to the regression (Atkeson et aI., 1997). The parameters Xq and D can be determined in the framework of nonparametric statistics (Schaal & Atkeson, in press) or parametric maximum likelihood estimations (Xu et aI, 1995}- for the present study they are determined manually since their origin is secondary to the results of this paper. Without loss of generality, all our data sets will set !,q to the zero vector, compute the weights, and then translate the input data such that the locally weighted mean, i = L WI Xi / L Wi , is zero. The output data is equally translated to be mean zero. Mean zero data is necessary for most of techniques considered below. The (translated) input data is summarized in the rows of the matrix X, the corresponding (translated) outputs are the elements of the vector y, and the corresponding weights are in the diagonal matrix W. In some cases, we need the joint input and output data, denoted as Z=[X y). Local Dimensionality Reduction 635 2.1 FACTORANALYSIS(LWFA) Factor analysis (Everitt, 1984) is a technique of dimensionality reduction which is the most appropriate given the generating process of our regression data. It assumes the observed data z was produced. by a mean zero independently distributed k -dimensional vector of factors v, transformed by the matrix U, and contaminated by mean zero independent noise f: with diagonal covariance matrix Q: z=Uv+f:, where z=[xT,yt and f:=[f:~,t:yr (2) If both v and f: are normally distributed, the parameters Q and U can be obtained iteratively by the Expectation-Maximization algorithm (EM) (Rubin & Thayer, 1982). For a linear regression problem, one assumes that z was generated with U=[I, f3 Y and v = i, where f3 denotes the vector of regression coefficients of the linear model y = f31 x, and I the identity matrix. After calculating Q and U by EM in joint data space as formulated in (2), an estimate of f3 can be derived from the conditional probability p(y I x). As all distributions are assumed to be normal, the expected value ofy is the mean of this conditional distribution. The locally weighted version (L WF A) of f3 can be obtained together with an estimate of the factors v from the joint weighted covariance matrix 'I' of z and v: E{[:] +[} ~ ~ ~,,~,;'x, Q+UU T = UT [ where ~ ~ [ZT, VT~~Jft: w; ~ (3) U] ['I'II(=n x n) 'I'12(=nX(m+k?)] I = '?21(= (m + k) x n) '1'22(= (m + k) x (m + k?) where E { .} denotes the expectation operator and B a matrix of coefficients involved in estimating the factors v. Note that unless the noise f: is zero, the estimated f3 is different from the true f3 as it tries to average out the noise in the data. 2.2 JOINT-SPACE PRINCIPAL COMPONENT ANALYSIS (LWPCA) An alternative way of determining the parameters f3 in a reduced space employs locally weighted principal component analysis (LWPCA) in the joint data space. By defining the . largest k+ 1 principal components of the weighted covariance matrix ofZ as U: U= [eigenvectors(I Wi (Zi - ZXZi - Z)T II Wi)] (4) max(l :k+1l and noting that the eigenvectors in U are unit length, the matrix inversion theorem (Hom & Johnson, 1994) provides a means to derive an efficient estimate of f3 T T( T f3=U x(Uy -Uy UyUy -I )-1 UyU yt T\ [Ux(=nXk)] where U= Uy(=mxk) (5) In our one dimensional output case, U y is just a (1 x k) -dimensional row vector and the evaluation of (5) does not require a matrix inversion anymore but rather a division. If one assumes normal distributions in all variables as in LWF A, LWPCA is the special case of L WF A where the noise covariance Q is spherical, i.e., the same magnitude of noise in all observables. Under these circumstances, the subspaces spanned by U in both methods will be the same. However, the regression coefficients of LWPCA will be different from those of LWF A unless the noise level is zero, as LWFA optimizes the coefficients according to the noise in the data (Equation (3? . Thus, for normal distributions and a correct guess of k, LWPCA is always expected to perform worse than LWFA. S. Schaal, S. Vijayakumar and C. G. Atkeson 636 2.3 PARTIAL LEAST SQUARES (LWPLS, LWPLS_I) Partial least squares (Wold, 1975; Frank & Friedman, 1993) recursively computes orthogonal projections of the input data and performs single variable regressions along these projections on the residuals of the previous iteration step. A locally weighted version of partial least squares (LWPLS) proceeds as shown in Equation (6) below. As all single variable regressions are ordinary uniFor Training: For Lookup: variate least-squares minim izations, L WPLS Initialize: Initialize: makes the same statistical assumption as ordinary Do = X, eo = y do = x, y= linear regressions, i.e., that only output variables have additive noise, but input variables are noiseFor i = 1 to k: For i = 1 to k: less. The choice of the projections u, however, ins. = dT.u. troduces an element in LWPLS that remains statistically still debated (Frank & Friedman, 1993), although, interestingly, there exists a strong similarity with the way projections are chosen in Cascade Correlation (Fahlman & Lebiere, 1990). A peculiarity of LWPLS is that it also regresses the inputs of the previous step against the projected inputs s in order to ensure the orthogonality of all the projections u. Since LWPLS chooses projections in a (6) very powerful way, it can accomplish optimal function fits with only one single projections (i.e., k= 1) for certain input distributions. We will address this issue in our empirical evaluations by comparing k-step LWPLS with I-step LWPLS, abbreviated LWPLS_I. ? I 1- I 2.4 PRINCIPAL COMPONENT REGRESSION (L WPCR) Although not optimal, a computationally efficient techniques of dimensionality reduction for linear regression is principal component regression (LWPCR) (Massy, 1965). The inputs are projected onto the largest k principal components of the weighted covariance matrix of the input data by the matrix U: U = [eig envectors(2: Wi (Xi - xX xt /2: Xi - (7) Wi )] max(l:k) The regression coefficients f3 are thus calculated as: f3 = (UTXTwxUtUTXTWy (8) Equation (8) is inexpensive to evaluate since after projecting X with U, UTXTWXU becomes a diagonal matrix that is easy to invert. LWPCR assumes that the inputs have additive spherical noise, which includes the zero noise case. As during dimensionality reduction LWPCR does not take into account the output data, it is endangered by clipping input dimensions with low variance which nevertheless have important contribution to the regression output. However, from a statistical point of view, it is less likely that low variance inputs have significant contribution in a linear regression, as the confidence bands of the regression coefficients increase inversely proportionally with the variance of the associated input. If the input data has non-spherical noise, L WPCR is prone to focus the regression on irrelevant projections. 3 MONTE CARLO EVALUATIONS In order to evaluate the candidate methods, data sets with 5 inputs and 1 output were randomly generated. Each data set consisted of 2,000 training points and 10,000 test points, distributed either uniformly or nonuniformly in the unit hypercube. The outputs were Local Dimensionality Reduction 637 generated by either a linear or quadratic function. Afterwards, the 5-dimensional input space was projected into a to-dimensional space by a randomly chosen distance preserving linear transformation. Finally, Gaussian noise of various magnitudes was added to both the 10-dimensional inputs and one dimensional output. For the test sets, the additive noise in the outputs was omitted. Each regression technique was localized by a Gaussian kernel (Equation (1)) with a to-dimensional distance metric D=IO*I (D was manually chosen to ensure that the Gaussian kernel had sufficiently many data points and no "data holes" in the fringe areas of the kernel) . The precise experimental conditions followed closely those suggested by Frank and Friedman (1993): {g.I for: 131.. = [I, I, I, I, If , ? 2 kinds of linear functions y = ? 2 kinds of quadratic functions y = f3J.I + f3::.aAxt ,xi i) ? 1311. i) = [I, I, I, I, Wand f3q.ad = 0.1 [I, I, I, I, If, and ii) ii) I3Ii. = [1,2,3,4, sf ,xi ,X;,X;]T for: 131.. = [1,2,3,4, sf and f3q uad = 0.1 [I, 4, 9, 16, 2sf 3 kinds of noise conditions, each with 2 sub-conditions: local signal/noise ratio Isnr=20, i) only output noise: a) low noise: and b) high noise: Isnr=2, ii) equal noise in inputs and outputs: a) low noise Ex ?? = Sy = N(O,O.Ot2), n e[I,2, ... ,10], and b) high noise Ex ?? =sy=N(0,0.1 2),ne[I,2, ... ,10], iii) unequal noise in inputs and outputs: a) low noise : Ex .? = N(0,(0.0In)2), n e[I,2, . .. ,1O] and Isnr=20, and ? b) high noise: Ex .? = N(0,(0.0In)2), n e[I,2, ... ,1O] and Isnr=2, 2 kinds of input distributions: i) uniform in unit hyper cube, ii) uniform in unit hyper cube excluding data points which activate a Gaussian weighting function (I) at c = [O.S,O,o,o,of with D=IO*I more than w=0.2 (this forms a "hyper kidney" shaped distribution) Every algorithm was run * 30 times on each of the 48 combinations of the conditions. Additionally, the complete test was repeated for three further conditions varying the dimensionality--called factors in accordance with LWFA-that the algorithms assumed to be the true dimensionality of the to-dimensional data from k=4 to 6, i.e., too few, correct, and too many factors. The average results are summarized in Figure I. Figure I a,b,c show the summary results of the three factor conditions. Besides averaging over the 30 trials per condition, each mean of these charts also averages over the two input distribution conditions and the linear and quadratic function condition, as these four cases are frequently observed violations of the statistical assumptions in nonlinear function approximation with locally linear models. In Figure I b the number of factors equals the underlying dimensionality of the problem, and all algorithms are essentially performing equally well. For perfectly Gaussian distributions in all random variables (not shown separately), LWFA's assumptions are perfectly fulfilled and it achieves the best results, however, almost indistinguishable closely followed by LWPLS. For the ''unequal noise condition", the two PCA based techniques, LWPCA and LWPCR, perform the worst since--as expected-they choose suboptimal projections. However, when violating the statistical assumptions, L WF A loses parts of its advantages, such that the summary results become fairly balanced in Figure lb. The quality of function fitting changes significantly when violating the correct number of factors, as illustrated in Figure I a,c. For too few factors (Figure la), LWPCR performs worst because it randomly omits one of the principle components in the input data, without respect to how important it is for the regression. The second worse is LWF A: according to its assumptions it believes that the signal it cannot model must be noise, leading to a degraded estimate of the data's subspace and, consequently, degraded regression results. LWPLS has a clear lead in this test, closely followed by LWPCA and LWPLS_I. * Except for LWFA, all methods can evaluate a data set in non-iterative calculations. LWFA was trained with EM for maximally 1000 iterations or until the log-likelihood increased less than I.e-lOin one iteration. S. Schaal, S. Vljayakumar and C. G. Atkeson 638 For too many factors than necessary (Figure Ie), it is now LWPCA which degrades. This effect is due to its extracting one very noise contaminated projection which strongly influences the recovery of the regression parameters in Equation (4). All other algorithms perform almost equally well, with LWF A and LWPLS taking a small lead. Equal NoIse In ell In puIS end OutpUIS OnlyOutpul Noise Unequel NoIse In ell Inputs end OutpulS 0.1 c o ~ ::::;; 0.01 c II> C> ~ 0.001 ~ 0.0001 fl- I. E>O ~I. ? >>(I ~ J. &>O ~J , E ? O ~ J .E>O fl- I.&? O ~I . & >O ~ I . & > >O ~I.& >O ~ I .?>>o p,. 1. s>O tJ-J .?>>O e) RegressIon Results with 4 Factors ? LWFA ? LWPCA ? LWPCR 0 LWPLS ? LWPLS_1 0.1 c:: o W ~c:: 0.01 II> C) ~ 0.001 ~ 0.0001 ~ ~ 0.1 W 0.01 8 ~c:: g, ~ 0.001 ~ 0.0001 c) RegressIon Results with 6 Feclors jj il 0.1 f- a ~ ::::;; 0.01 c II> C) ~ 0.001 ~ 0.0001 d) Summery Results Figure I: Average summary results of Monte Carlo experiments. Each chart is primarily divided into the three major noise conditions, cf. headers in chart (a). In each noise condition, there are four further subdivision: i) coefficients of linear or quadratic model are equal with low added noise; ii) like i) with high added noise; iii) coefficients oflinear or quadratic model are different with low noise added; iv) like iii) with high added noise. Refer to text and descriptions of Monte Carlo studies for further explanations. Local Dimensionality Reduction 639 4 SUMMARY AND CONCLUSIONS Figure 1d summarizes all the Monte Carlo experiments in a final average plot. Except for LWPLS, every other technique showed at least one clear weakness in one of our "robustness" tests. It was particularly an incorrect number of factors which made these weaknesses apparent. For high-dimensional regression problems, the local dimensionality, i.e., the number of factors, is not a clearly defined number but rather a varying quantity, depending on the way the generating process operates. Usually, this process does not need to generate locally low dimensional distributions, however, it often "chooses" to do so, for instance, as human ann movements follow stereotypic patterns despite they could generate arbitrary ones. Thus, local dimensionality reduction needs to find autonomously the appropriate number of local factor. Locally weighted partial least squares turned out to be a surprisingly robust technique for this purpose, even outperforming the statistically appealing probabilistic factor analysis. As in principal component analysis, LWPLS's number of factors can easily be controlled just based on a variance-cutoff threshold in input space (Frank & Friedman, 1993), while factor analysis usually requires expensive cross-validation techniques. Simple, variance-based control over the number of factors can actually improve the results of LWPCA and LWPCR in practice, since, as shown in Figure I a, LWPCR is more robust towards overestimating the number of factors, while L WPCA is more robust towards an underestimation. If one is interested in dynamically growing the number of factors while obtaining already good regression results with too few factors, L WPCA and, especially, L WPLS seem to be appropriate-it should be noted how well one factor LWPLS (LWPLS_l) already performed in Figure I! In conclusion, since locally weighted partial least squares was equally robust as local weighted factor analysis towards additive noise in. both input and output data, and, moreover, superior when mis-guessing the number of factors, it seems to be a most favorable technique for local dimensionality reduction for high dimensional regressions. Acknowledgments The authors are grateful to Geoffrey Hinton for reminding them of partial least squares. This work was supported by the ATR Human Information Processing Research Laboratories. S. Schaal's support includes the German Research Association, the Alexander von Humboldt Foundation, and the German Scholarship Foundation. S. Vijayakumar was supported by the Japanese Ministry of Education, Science, and Culture (Monbusho). C. G. Atkeson acknowledges the Air Force Office of Scientific Research grant F49-6209410362 and a National Science Foundation Presidential Young Investigators Award. References Atkeson, C. G., Moore, A. W., & Schaal, S, (1997a). "Locally weighted learning." ArtifiCial Intelligence Review, 11, 1-5, pp.II-73. Atkeson, C. G., Moore, A. W., & Schaal, S, (1997c). "Locally weighted learning for control." ArtifiCial Intelligence Review, 11, 1-5, pp.75-113. Belsley, D. A., Kuh, E., & Welsch, R. E, (1980). Regression diagnostics: Identifying influential data and sources of collinearity. New York: Wiley. Everitt, B. S, (1984). An introduction to latent variable models. London: Chapman and Hall. Fahlman, S. E. ,Lebiere, C, (1990). "The cascadecorrelation learning architecture." In: Touretzky, D. S. (Ed.), Advances in Neural Information Processing Systems II, pp.524-532. Morgan Kaufmann. Frank, I. E., & Friedman, 1. H, (1993). "A statistical view of some chemometric regression tools." Technometrics, 35, 2, pp.l09-135. Geman, S., Bienenstock, E., & Doursat, R. (1992). "Neural networks and the bias/variance dilemma." Neural Computation, 4, pp.I-58. Hom, R. A., & Johnson, C. R, (1994). Matrix analySis. Press Syndicate of the University of Cambridge. Jordan, M.I., & Jacobs, R, (1994). "Hierarchical mix- tures of experts and the EM algorithm." Neural Computation, 6, 2, pp.181-214. Massy, W. F, (1965). "Principle component regression in exploratory statistical research." Journal ofthe American Statistical Association, 60, pp.234-246. Rubin, D. B., & Thayer, D. T, (l982). "EM algorithms for ML factor analysis." Psychometrika, 47, I, 69-76. Schaal, S., & Atkeson, C. G, (in press). "Constructive incremental learning from only local information." Neural Computation. Scott, D. W, (1992). Multivariate Density Estimation. New York: Wiley. Vijayakumar, S., & Schaal, S, (1997). "Local dimensionality reduction for locally weighted learning." In: International Conference on Computational Intelligence in Robotics and Automation, pp.220-225, Monteray, CA, July 10-11, 1997. Wold, H. (1975). "Soft modeling by latent variables: the nonlinear iterative partial least squares approach." In: Gani, J. (Ed.), Perspectives in Probability and Statistics, Papers in Honour ofM S. Bartlett. Aca<j. Press. Xu, L., Jordan, M.l., & Hinton, G. E, (1995). "An alternative model for mixture of experts." In: Tesauro, G., Touretzky, D. S., & Leen, T. K. (Eds.), Advances in Neural Information Processing Systems 7, pp.633-640. Cambridge, MA: MIT Press. 

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On Efficient Heuristic Ranking of Hypotheses Steve Chien, Andre Stechert, and Darren Mutz Jet Propulsion Laboratory, California Institute of Technology 4800 Oak Grove Drive, MIS 525-3660, Pasadena, CA 91109-8099 steve.chien@jpl.nasa.gov, Voice: (818) 306-6144 FAX: (818) 306-6912 Content Areas: Applications (Stochastic Optimization),Model Selection Algorithms Abstract This paper considers the problem of learning the ranking of a set of alternatives based upon incomplete information (e.g., a limited number of observations). We describe two algorithms for hypothesis ranking and their application for probably approximately correct (PAC) and expected loss (EL) learning criteria. Empirical results are provided to demonstrate the effectiveness of these ranking procedures on both synthetic datasets and real-world data from a spacecraft design optimization problem. 1 INTRODUCTION In many learning applications, the cost of information can be quite high, imposing a requirement that the learning algorithms glean as much usable information as possible with a minimum of data. For example: ? In speedup learning, the expense of processing each training example can be significant [Tadepalli921. ? In decision tree learning, ihe cost of using all available training examples when evaluating potential attributes for partitioning can be computationally ex.pensive [Musick93]. ? In evaruating medical treatment policies, additional training examples imly suboptimal treatment of human subjects. ? n data-poor applications, training data may be very scarce and learning as well as possible from limited data may be key. E This paper provides a statistical decision-theoretic framework for the ranking of parametric distributions. This framework will provide the answers to a wide range of questions about algorithms such as: how much information is enough? At what point do we have adequate information to rank the alternatives with some requested confidence? 445 On Efficient Heuristic Ranking of Hypotheses The remainder of this paper is structured as follows. First, we describe the hypothesis ranking problem more formally, including definitions for the probably approximately correct (PAC) and expected loss (EL) decision criteria. We then define two algorithms for establishing these criteria for the hypothesis ranking problem - a recursive hypothesis selection algorithm and an adjacency based algorithm. Next, we describe empirical tests demonstrating the effectiveness of these algorithms as well as documenting their improved performance over a standard algorithm from the statistical ranking literature. Finally, we describe related work and future extensions to the algorithms. 2 HYPOTHESIS RANKING PROBLEMS Hypothesis ranking problems, an extension of hypothesis selection problems, are an abstract class of learning problems where an algorithm is given a set of hypotheses to rank according to expected utility over some unknown distribution, where the expected utility must be estimated from training data. In many of these applications, a system chooses a single alternative and never revisits the decision. However, some systems require the ability to investigate several options (either serially or in parallel), such as in beam search or iterative broadening, where the ranking formulation is most appropriate. Also, as is the case with evolutionary approaches, a system may need to populate future alternative hypotheses on the basis of the ranking of the current population[Goldberg89] . In any hypothesis evaluation problem, always achieving a correct ranking is impossible in practice, because the actual underlying probability distributions are unavailable and there is always a (perhaps vanishingly) small chance that the algorithms will be unlucky because only a finite number of samples can be taken. Consequently, rather than always requiring an algorithm to output a correct ranking, we impose probabilistic criteria on the rankings to be produced. While several families of such requirements exist, in this paper we examine two, the probably approximately correct (PAC) requirement from the computational learning theory community [Valiant84] and the expected loss (EL) requirement frequently used in decision theory and gaming problems [Russe1l92] . The expected utility of a hypothesis can be estimated by observing its values over a finite set of training examples. However, to satisfy the PAC and EL requirements, an algorithm must also be able to reason about the potential difference between the estimated and true utilities of each hypotheses. Let Ui be the true expected utility of hypothesis i and let Ui be the estimated expected utility of hypothesis i. Without loss of generality, let us presume that the proposed ranking of hypotheses is U1 > U2 >, ... , > Uk-I> Uk. The PAC requirement states that for some userspecified ?. with probability 1 - 8: k-l /\ [(Ui + f) > MAX(Ui+I, ... ,UIe)] (1) ;=1 Correspondingly, let the loss L of selecting a hypothesis HI to be the best from a set of k hypotheses HI, ... , Hk be as follows . L(HI' {HI, ... ,HIe}) = MAX(O, MAX(U2 , ... ,UIe) - UI) (2) and let the loss RL of a ranking H 1, ... , H k be as follows. Ie-I RL(Hl, ... , Hie) = L L(Hi, {Hi+l, .. ., Hie}) (3) i=1 A hypothesis ranking algorithm which obeys the expected loss requirement must produce rankings that on average have less than the requested expected loss bound. S. Chien, A. Stechert and D. Mutz 446 Consider ranking the hypotheses with expected utilities: U1 = 1.0, U2 = 0.95, U3 = 0.86. The ranking U2 > U1 > U3 is a valid PAC ranking for { = 0.06 but not for { = 0.01 and has an observed loss of 0.05 + 0 = 0.05. However, while the confidence in a pairwise comparison between two hypotheses is well understood, it is less clear how to ensure that desired confidence is met in the set of comparisons required for a selection or the more complex set of comparisons required for a ranking. Equation 4 defines the confidence that Ui + { > Uj, when the distribution underlying the utilities is normally distributed with unknown and unequal variances. (4) where ? represents the cumulative standard normal distribution function, and n, Ui-j, and Si-j are the size, sample mean, and sample standard deviation of the blocked differential distribution, respectively 1 . Likewise, computation of the expected loss for asserting an ordering between a pair of hypotheses is well understood, but the estimation of expected loss for an entire ranking is less clear. Equation 5 defines the expected loss for drawing the conclusion Ui > Uj, again under the assumption ~fnormality (see [Chien95] for further details). EL(Ui > Ujl ~ 'e U'-i -O . 6 n ( ) :l Si_j = .-] ';21rn -j fJ +~ .,j2; oc _ \~irn e- O ? 6 ? :l dz (5) '-J In the next two subsections, we describe two interpretations for estimating the likelihood that an overall ranking satisfies the PAC or EL requirements by estimating and combining pairwise PAC errors or EL estimates. Each of these interpretations lends itself directly to an algorithmic implementation as described below. 2.1 RANKING AS RECURSIVE SELECTION One way to determine a ranking HI, ... , Hk is to view ranking as recursive selection from the set of remaining candidate hypotheses. In this view, the overall ranking error, as specified by the desired confidence in PAC algorithms and the loss threshhold in EL algorithms, is first distributed among k - 1 selection errors which are then further subdivided into pairwise comparison errors. Data is then sampled until the estimates of the pairwise comparison error (as dictated by equation 4 or 5) satisfy the bounds set by the algorithm. Thus, another degree of freedom in the design of recursive ranking algorithms is the method by which the overall ranking error is ultimately distributed among individual pairwise comparisons between hypotheses. Two factors influence the way in which we compute error distribution. First, our model of error combination determines how the error allocated for individual comparisons or selections combines into overall ranking error and thus how many candidates are available as targets for the distribution. Using Bonferroni's inequality, one combine errors additively, but a more conservative approach might be to assert that because the predicted "best" hypothesis may change during sampling in the worst case the conclusion might depend on all possible pairwise comparisons and thus the error should be distributed among all (~) pairs of hypotheses 2 ). INote that in our approach we block examples to further reduce sampling complexity. Blocking forms estimates by using the difference in utility between competing hypotheses on each observed example. Blocking can significantly reduce the variance in the data when the hypotheses are not independent. It is trivial to modify the formulas to address the cases in which it is not possible to block data (see [Moore94, Chien95] for further details). 2For a discussion of this issue, see pp. 18-20 of [Gratch93]. 447 On Efficient Heuristic Ranking of Hypotheses Second, our policy with respect to allocation of error among the candidate comparisons or selections determines how samples will be distributed. For example, in some contexts, the consequences of early selections far outweigh those of later selections. For these scenarios, we have implemented ranking algorithms that divide overall ranking error unequally in favor of earlier selections3 . Also, it is possible to divide selection error into pairwise error unequally based on estimates of hypothesis parameters in order to reduce sampling cost (for example, [Gratch94] allocates error rationally) . Within the scope of this paper, we only consider algorithms that: (1) combine pairwise error into selection error additively, (2) combine selection error into overall ranking error additively and (3) allocate error equally at each level. One disadvantage of recursive selection is that once a hypothesis has been selected, it is removed from the pool of candidate hypotheses. This causes problems in rare instances when, while sampling to increase the confidence of some later selection, the estimate for a hypothesis' mean changes enough that some previously selected hypothesis no longer dominates it. In this case, the algorithm is restarted taking into account the data sampled so far. These assumptions result in the following formulations (where d(U11>? {U2' ... , Uk}) is used to denote the error due to the action of selecting hypothesis 1 under Equation 1 from the set {HI, ... , Hk} and d(UII>{U2, ... , Uk}) denotes the error due to selection loss in situations where Equation 2 applies): t5 rec (UI > U2 > ... > Uk) = t5 rec (U2 > U3 > ... > Uk) +t5(UI t>. {U2 , ??? ,Uk}) (6) where drec(Uk) = 0 (the base case for the recursion) and the selection error is as defined in [Chien95]: k t5(Ul t>. {U2 , ??? ,Uk}) = L 15 1 ,. (7) .=2 using Equation 4 to compute pairwise confidence. Algorithmically, we implement this by: 1. sampling a default number of times to seed the estimates for each hypothesis mean and variance, 2. allocating the error to selection and pairwise comparisons as indicated above, 3. sampling until the desired confidences for successive selections is met, and 4. restarting the algorithm if any of the hypotheses means changed significantly enough to change the overall ranking. An analogous recursive selection algorithm based on expected loss is defined as follows. EL rec (U2 > U3 > ... > Uk) (8) +EL(U1 t> {U2 , ??? ,Uk}) where ELrec(Uk) = 0 and the selection EL is as defined in [Chien95]: k EL(U1 I> {U2, ... , Uk}) =L i=2 3Space constraints preclude their description here. EL(Ut, Ud (9) 448 S. Chien, A. StecheT1 and D. Mutz 2.2 RANKING BY COMPARISON OF ADJACENT ELEMENTS Another interpretation of ranking confidence (or loss) is that only adjacent elements in the ranking need be compared. In this case, the overall ranking error is divided directly into k -1 pairwise comparison errors. This leads to the following confidence equation for the PAC criteria: k-l (10) dac(i(Ul > U2 > ... > Uk) = Ldi,i+1 i=l And the following equation for the EL criteria.k_l ELac(i(Ul > U2 > ... > Uk) = '2: EL (Ui,Ui+d (11) i=l Because ranking by comparison of adjacent hypotheses does not establish the dominance between non-adjacent hypotheses (where the hypotheses are ordered byobserved mean utility), it has the advantage of requiring fewer comparisons than recursive selection (and thus may require fewer samples than recursive selection). However, for the same reason, adjacency algorithms may be less likely to correctly bound probability of correct selection (or average loss) than the recursive selection algorithms. In the case of the PAC algorithms, this is because f-dominance is not necessarily transitive. In the case of the EL algorithms, it is because expected loss is not additive when considering two hypothesis relations sharing a common hypothesis. For instance, the size of the blocked differential distribution may be different for each of the pairs of hypotheses being compared. 2.3 OTHER RELEVANT APPROACHES Most standard statistical ranking/selection approaches make strong assumptions about the form of the problem (e.g., the variances associated with underlying utility distribution of the hypotheses might be assumed known and equal). Among these, Turnbull and Weiss [Turnbull84] is most comparable to our PAC-based approach4. Turnbull and Weiss treat hypotheses as normal random variables with unknown mean and unknown and unequal variance. However, they make the additional stipulation that hypotheses are independent. So, while it is still reasonable to use this approach when the candidate hypotheses are not independent, excessive statistical error or unnecessarily large training set sizes may result. 3 EMPIRICAL PERFORMANCE EVALUATION We now turn to empirical evaluation of the hypothesis ranking techniques on realworld datasets. This evaluation serves three purposes. First, it demonstrates that the techniques perform as predicted (in terms of bounding the probability of incorrect selection or expected loss). Second, it validates the performance of the techniques as compared to standard algorithms from the statistical literature. Third, the evaluation demonstrates the robustness of the new approaches to real-world hypothesis ranking problems. An experimental trial consists of solving a hypothesis ranking problem with a given technique and a given set of problem and control parameters. We measure performance by (1) how well the algorithms satisfy their respective criteria; and (2) the number of samples taken. Since the performance of these statistical algorithms on any single trial provides little information about their overall behavior, each trial is repeated multiple times and the results are averaged across 100 trials. Because 4 PAC-based approaches have been investigated extensively in the statistical ranking and selection literature under the topic of confidence interval based algorithms (see [Haseeb85] for a review of the recent literature). On Efficient Heuristic Ranking of Hypotheses 449 Table 1: Estimated expected total number of observations to rank DS-2 spacecraft . sown h 'm parenthesis. designs. Achieved pro b a bTt I I yof correc t ran k'mg IS k 10 10 10 'Y !!.. 0.75 0 .90 0.95 2 2 2 TURNtlULL 534 {0 .96 667 (0 .98 793 (0.99 PAC rec 144 1.00 160 1.00 177 1.00 PACod ' 92 0.98 98 1.00 103 0.99 Table 2: Estimated expected total number of observations and expected loss of an incorrect ranking of DS-2 penetrator designs. Parameters k ~ 10 0.10 0 .05 10 0 .02 10 EL rec Samples 152 200 378 EL a d ' Loss 0.005 0 .003 0 .003 l:)amples 77 90 139 l..oss 0 .014 0 .006 0 .003 the PAC and expected loss criteria are not directly comparable, the approaches are analyzed separately. Experimental results from synthetic datasets are reported in [Chien97]. The evaluation of our approach on artificially generated data is used to show that: (1) the techniques correctly bound probability of incorrect ranking and expected loss as predicted when the underlying assumptions are valid even when the underlying utility distributions are inherently hard to rank , and (2) that the PAC techniques compare favorably to the algorithm of Thrnbull and Weiss in a wide variety of problem configurations. The test of real-world applicability is based on data drawn from an actual NASA spacecraft design optimization application. This data provides a strong test of the applicability of the techniques in that all of the statistical techniques make some form of normality assumption - yet the data in this application is highly non-normal. Tables 1 and 2 show the results of ranking 10 penetrator designs using the PACbased, Thrnbull, and expected loss algorithms In this problem the utility function is the depth of penetration of the penetrator, with those cases in which the penetrator does not penetrate being assigned zero utility. As shown in Table 1, both PAC algorithms significantly outperformed the Thrnbull algorithm, which is to be expected because the hypotheses are somewhat correlated (via impact orientations and soil densities). Table 2 shows that the EL rec expected loss algorithm effectively bounded actual loss but the ELad,i algorithm was inconsistent. 4 DISCUSSION AND CONCLUSIONS There are a number of areas of related work. First, there has been considerable analysis of hypothesis selection problems. Selection problems have been formalized using a Bayesian framework [Moore94, Rivest88] that does not require an initial sample, but uses a rigorous encoding of prior knowledge. Howard [Howard70] also details a Bayesian framework for analyzing learning cost for selection problems. If one uses a hypothesis selection framework for ranking, allocation of pairwise errors can be performed rationally [Gratch94]. Reinforcement learning work [Kaelbling93] with immediate feedback can also be viewed as a hypothesis selection problem. In su~mary, this paper has described the hypothesis ranking problem, an extension to the hypothesis selection problem. We defined the application of two decision criteria, probably approximately correct and expected loss, to this problem. We then defined two families of algorithms, recursive selection and adjacency, for solution of hypothesis ranking problems. Finally, we demonstrated the effectiveness of these algorithms on both synthetic and real-world datasets, documenting improved performance over existing statistical approaches. 450 S. Chien, A. Stechert and D. Mutz References [Bechhofer54] R.E. Bechhofer, "A Single-sample Multiple Decision Procedure for Ranking Means of Normal Populations with Known Variances," Annals of Math. Statistics (25) 1, 1954 pp. 16-39. [Chien95] S. A. Chien, J . M. Gratch and M. C. Burl, "On the Efficient Allocation of Resources for Hypothesis Evaluation: A Statistical Approach," IEEE Trans. Pattern Analysis and Machine Intelligence 17 (7), July 1995, pp. 652-665. [Chien97] S. Chien, A. Stechert, and D. Mutz, "Efficiently Ranking Hypotheses in Machine Learning," JPL-D-14661, June 1997. Available online at http://wwwaig.jpl.nasa.gov/public/www/pas-bibliography.html [Goldberg89] D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Add. Wes., 1989. [Govind81] Z. Govindarajulu, "The Sequential Statistical Analysis," American Sciences Press, Columbus, OH, 1981. [Gratch92] J . Gratch and G. Dejong, "COMPOSER: A Probabilistic Solution to the Utility Problem in Speed-up Learning," Proc. AAAI92, San Jose, CA, July 1992, pp. 235240. [Gratch93] J. Gratch, "COMPOSER: A Decision-theoretic Approach to Adaptive Problem Solving," Tech. Rep. UIUCDCS-R-93-1806, Dept. Compo Sci., Univ. Illinois, May 1993. [Gratch94] J. Gratch, S. Chien, and G. Dejong, "Improving Learning Performance Through Rational Resource Allocation," Proc. AAAI94, Seattle, WA, August 1994, pp. 576-582. [Greiner92] R. Greiner and I. Jurisica, "A Statistical Approach to Solving the EBL Utility Problem," Proc. AAAI92, San Jose, CA, July 1992, pp. 241-248. [Haseeb85] R. M. Haseeb, Modern Statistical Selection, Columbus, OH: Am. Sciences Press, 1985. [Hogg78] R. V. Hogg and A. T. Craig, Introduction to Mathematical Statistics, Macmillan Inc., London, 1978. [Howard70] R. A. Howard, Decision Analysis: Perspectives on Inference, Decision, and Experimentation," Proceedings of the IEEE 58, 5 (1970), pp. 823-834. [Kaelbling93] L. P. Kaelbling, Learning in Embedded Systems, MIT Press, Cambridge, MA,1993. [Minton88] S. Minton, Learning Search Control Knowledge: An Explanation-Based Approach, Kluwer Academic Publishers, Norwell, MA, 1988. [Moore94] A. W. Moore and M. S. Lee, "Efficient Algorithms for Minimizing Cross Validation Error," Proc. ML94, New Brunswick, MA, July 1994. [Musick93] R. Musick, J. Catlett and S. Russell, "Decision Theoretic Subsampling for Induction on Large Databases," Proc. ML93, Amhert, MA, June 1993, pp. 212-219. [Rivest88] R. L. Rivest and R. Sloan, A New Model for Inductive Inference," Proc. 2nd Conference on Theoretical Aspects of Reasoning about Knowledge, 1988. [Russell92] S. Russell and E . Wefald, Do the Right Thing: Studies in Limited Rationality, MIT Press, MA. [Tadepalli92] P. Tadepalli, "A theory of unsupervised speedup learning," Proc. AAAI92" pp. 229-234. [Turnbull84] Turnbull and Weiss, "A class of sequential procedures for k-sample problems concerning normal means with unknown unequal variances," in Design of Experiments: ranking and selection, T. J. Santner and A. C. Tamhane (eds. ), Marcel Dekker, 1984. [Valiant84] L. G. Valiant, "A Theory of the Learnable," Communications of the ACM 27, (1984), pp. 1134-1142. 

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Local Dimensionality Reduction Stefan Schaal 1,2,4 sschaal@usc.edu http://www-slab.usc.edulsschaal Sethu Vijayakumar 3, I Christopher G. Atkeson 4 sethu@cs.titech.ac.jp http://ogawawww.cs.titech.ac.jp/-sethu cga@cc.gatech.edu http://www.cc.gatech.edul fac/Chris.Atkeson IERATO Kawato Dynamic Brain Project (IST), 2-2 Hikaridai, Seika-cho, Soraku-gun, 619-02 Kyoto 2Dept. of Comp. Science & Neuroscience, Univ. of South. California HNB-I 03, Los Angeles CA 90089-2520 3Department of Computer Science, Tokyo Institute of Technology, Meguro-ku, Tokyo-I 52 4College of Computing, Georgia Institute of Technology, 801 Atlantic Drive, Atlanta, GA 30332-0280 Abstract If globally high dimensional data has locally only low dimensional distributions, it is advantageous to perform a local dimensionality reduction before further processing the data. In this paper we examine several techniques for local dimensionality reduction in the context of locally weighted linear regression. As possible candidates, we derive local versions of factor analysis regression, principle component regression, principle component regression on joint distributions, and partial least squares regression. After outlining the statistical bases of these methods, we perform Monte Carlo simulations to evaluate their robustness with respect to violations of their statistical assumptions. One surprising outcome is that locally weighted partial least squares regression offers the best average results, thus outperforming even factor analysis, the theoretically most appealing of our candidate techniques. 1 INTRODUCTION Regression tasks involve mapping a n-dimensional continuous input vector x E ~n onto a m-dimensional output vector y E ~m ? They form a ubiquitous class of problems found in fields including process control, sensorimotor control, coordinate transformations, and various stages of information processing in biological nervous systems. This paper will focus on spatially localized learning techniques, for example, kernel regression with Gaussian weighting functions. Local learning offer advantages for real-time incremental learning problems due to fast convergence, considerable robustness towards problems of negative interference, and large tolerance in model selection (Atkeson, Moore, & Schaal, 1997; Schaal & Atkeson, in press). Local learning is usually based on interpolating data from a local neighborhood around the query point. For high dimensional learning problems, however, it suffers from a bias/variance dilemma, caused by the nonintuitive fact that " ... [in high dimensions] if neighborhoods are local, then they are almost surely empty, whereas if a neighborhood is not empty, then it is not local." (Scott, 1992, p.198). Global learning methods, such as sigmoidal feedforward networks, do not face this 634 S. School, S. Vijayakumar and C. G. Atkeson problem as they do not employ neighborhood relations, although they require strong prior knowledge about the problem at hand in order to be successful. Assuming that local learning in high dimensions is a hopeless, however, is not necessarily warranted: being globally high dimensional does not imply that data remains high dimensional if viewed locally. For example, in the control of robot anns and biological anns we have shown that for estimating the inverse dynamics of an ann, a globally 21dimensional space reduces on average to 4-6 dimensions locally (Vijayakumar & Schaal, 1997). A local learning system that can robustly exploit such locally low dimensional distributions should be able to avoid the curse of dimensionality. In pursuit of the question of what, in the context of local regression, is the "right" method to perfonn local dimensionality reduction, this paper will derive and compare several candidate techniques under i) perfectly fulfilled statistical prerequisites (e.g., Gaussian noise, Gaussian input distributions, perfectly linear data), and ii) less perfect conditions (e.g., non-Gaussian distributions, slightly quadratic data, incorrect guess of the dimensionality of the true data distribution). We will focus on nonlinear function approximation with locally weighted linear regression (L WR), as it allows us to adapt a variety of global linear dimensionality reduction techniques, and as L WR has found widespread application in several local learning systems (Atkeson, Moore, & Schaal, 1997; Jordan & Jacobs, 1994; Xu, Jordan, & Hinton, 1996). In particular, we will derive and investigate locally weighted principal component regression (L WPCR), locally weighted joint data principal component analysis (L WPCA), locally weighted factor analysis (L WF A), and locally weighted partial least squares (LWPLS). Section 2 will briefly outline these methods and their theoretical foundations, while Section 3 will empirically evaluate the robustness of these methods using synthetic data sets that increasingly violate some of the statistical assumptions of the techniques. 2 METHODS OF DIMENSIONALITY REDUCTION We assume that our regression data originate from a generating process with two sets of observables, the "inputs" i and the "outputs" y. The characteristics of the process ensure a functional relation y = f(i). Both i and yare obtained through some measurement device that adds independent mean zero noise of different magnitude in each observable, such that x == i + Ex and y = y + Ey ? For the sake of simplicity, we will only focus on one-dimensional output data (m=l) and functions / that are either linear or slightly quadratic, as these cases are the most common in nonlinear function approximation with locally linear models. Locality of the regression is ensured by weighting the error of each data point with a weight from a Gaussian kernel: Wi = exp(-O.5(Xi - Xqf D(Xi - Xq)) (1) Xtt denotes the query point, and D a positive semi-definite distance metric which determmes the size and shape of the neighborhood contributing to the regression (Atkeson et aI., 1997). The parameters Xq and D can be determined in the framework of nonparametric statistics (Schaal & Atkeson, in press) or parametric maximum likelihood estimations (Xu et aI, 1995}- for the present study they are determined manually since their origin is secondary to the results of this paper. Without loss of generality, all our data sets will set !,q to the zero vector, compute the weights, and then translate the input data such that the locally weighted mean, i = L WI Xi / L Wi , is zero. The output data is equally translated to be mean zero. Mean zero data is necessary for most of techniques considered below. The (translated) input data is summarized in the rows of the matrix X, the corresponding (translated) outputs are the elements of the vector y, and the corresponding weights are in the diagonal matrix W. In some cases, we need the joint input and output data, denoted as Z=[X y). Local Dimensionality Reduction 635 2.1 FACTORANALYSIS(LWFA) Factor analysis (Everitt, 1984) is a technique of dimensionality reduction which is the most appropriate given the generating process of our regression data. It assumes the observed data z was produced. by a mean zero independently distributed k -dimensional vector of factors v, transformed by the matrix U, and contaminated by mean zero independent noise f: with diagonal covariance matrix Q: z=Uv+f:, where z=[xT,yt and f:=[f:~,t:yr (2) If both v and f: are normally distributed, the parameters Q and U can be obtained iteratively by the Expectation-Maximization algorithm (EM) (Rubin & Thayer, 1982). For a linear regression problem, one assumes that z was generated with U=[I, f3 Y and v = i, where f3 denotes the vector of regression coefficients of the linear model y = f31 x, and I the identity matrix. After calculating Q and U by EM in joint data space as formulated in (2), an estimate of f3 can be derived from the conditional probability p(y I x). As all distributions are assumed to be normal, the expected value ofy is the mean of this conditional distribution. The locally weighted version (L WF A) of f3 can be obtained together with an estimate of the factors v from the joint weighted covariance matrix 'I' of z and v: E{[:] +[} ~ ~ ~,,~,;'x, Q+UU T = UT [ where ~ ~ [ZT, VT~~Jft: w; ~ (3) U] ['I'II(=n x n) 'I'12(=nX(m+k?)] I = '?21(= (m + k) x n) '1'22(= (m + k) x (m + k?) where E { .} denotes the expectation operator and B a matrix of coefficients involved in estimating the factors v. Note that unless the noise f: is zero, the estimated f3 is different from the true f3 as it tries to average out the noise in the data. 2.2 JOINT-SPACE PRINCIPAL COMPONENT ANALYSIS (LWPCA) An alternative way of determining the parameters f3 in a reduced space employs locally weighted principal component analysis (LWPCA) in the joint data space. By defining the . largest k+ 1 principal components of the weighted covariance matrix ofZ as U: U= [eigenvectors(I Wi (Zi - ZXZi - Z)T II Wi)] (4) max(l :k+1l and noting that the eigenvectors in U are unit length, the matrix inversion theorem (Hom & Johnson, 1994) provides a means to derive an efficient estimate of f3 T T( T f3=U x(Uy -Uy UyUy -I )-1 UyU yt T\ [Ux(=nXk)] where U= Uy(=mxk) (5) In our one dimensional output case, U y is just a (1 x k) -dimensional row vector and the evaluation of (5) does not require a matrix inversion anymore but rather a division. If one assumes normal distributions in all variables as in LWF A, LWPCA is the special case of L WF A where the noise covariance Q is spherical, i.e., the same magnitude of noise in all observables. Under these circumstances, the subspaces spanned by U in both methods will be the same. However, the regression coefficients of LWPCA will be different from those of LWF A unless the noise level is zero, as LWFA optimizes the coefficients according to the noise in the data (Equation (3? . Thus, for normal distributions and a correct guess of k, LWPCA is always expected to perform worse than LWFA. S. Schaal, S. Vijayakumar and C. G. Atkeson 636 2.3 PARTIAL LEAST SQUARES (LWPLS, LWPLS_I) Partial least squares (Wold, 1975; Frank & Friedman, 1993) recursively computes orthogonal projections of the input data and performs single variable regressions along these projections on the residuals of the previous iteration step. A locally weighted version of partial least squares (LWPLS) proceeds as shown in Equation (6) below. As all single variable regressions are ordinary uniFor Training: For Lookup: variate least-squares minim izations, L WPLS Initialize: Initialize: makes the same statistical assumption as ordinary Do = X, eo = y do = x, y= linear regressions, i.e., that only output variables have additive noise, but input variables are noiseFor i = 1 to k: For i = 1 to k: less. The choice of the projections u, however, ins. = dT.u. troduces an element in LWPLS that remains statistically still debated (Frank & Friedman, 1993), although, interestingly, there exists a strong similarity with the way projections are chosen in Cascade Correlation (Fahlman & Lebiere, 1990). A peculiarity of LWPLS is that it also regresses the inputs of the previous step against the projected inputs s in order to ensure the orthogonality of all the projections u. Since LWPLS chooses projections in a (6) very powerful way, it can accomplish optimal function fits with only one single projections (i.e., k= 1) for certain input distributions. We will address this issue in our empirical evaluations by comparing k-step LWPLS with I-step LWPLS, abbreviated LWPLS_I. ? I 1- I 2.4 PRINCIPAL COMPONENT REGRESSION (L WPCR) Although not optimal, a computationally efficient techniques of dimensionality reduction for linear regression is principal component regression (LWPCR) (Massy, 1965). The inputs are projected onto the largest k principal components of the weighted covariance matrix of the input data by the matrix U: U = [eig envectors(2: Wi (Xi - xX xt /2: Xi - (7) Wi )] max(l:k) The regression coefficients f3 are thus calculated as: f3 = (UTXTwxUtUTXTWy (8) Equation (8) is inexpensive to evaluate since after projecting X with U, UTXTWXU becomes a diagonal matrix that is easy to invert. LWPCR assumes that the inputs have additive spherical noise, which includes the zero noise case. As during dimensionality reduction LWPCR does not take into account the output data, it is endangered by clipping input dimensions with low variance which nevertheless have important contribution to the regression output. However, from a statistical point of view, it is less likely that low variance inputs have significant contribution in a linear regression, as the confidence bands of the regression coefficients increase inversely proportionally with the variance of the associated input. If the input data has non-spherical noise, L WPCR is prone to focus the regression on irrelevant projections. 3 MONTE CARLO EVALUATIONS In order to evaluate the candidate methods, data sets with 5 inputs and 1 output were randomly generated. Each data set consisted of 2,000 training points and 10,000 test points, distributed either uniformly or nonuniformly in the unit hypercube. The outputs were Local Dimensionality Reduction 637 generated by either a linear or quadratic function. Afterwards, the 5-dimensional input space was projected into a to-dimensional space by a randomly chosen distance preserving linear transformation. Finally, Gaussian noise of various magnitudes was added to both the 10-dimensional inputs and one dimensional output. For the test sets, the additive noise in the outputs was omitted. Each regression technique was localized by a Gaussian kernel (Equation (1)) with a to-dimensional distance metric D=IO*I (D was manually chosen to ensure that the Gaussian kernel had sufficiently many data points and no "data holes" in the fringe areas of the kernel) . The precise experimental conditions followed closely those suggested by Frank and Friedman (1993): {g.I for: 131.. = [I, I, I, I, If , ? 2 kinds of linear functions y = ? 2 kinds of quadratic functions y = f3J.I + f3::.aAxt ,xi i) ? 1311. i) = [I, I, I, I, Wand f3q.ad = 0.1 [I, I, I, I, If, and ii) ii) I3Ii. = [1,2,3,4, sf ,xi ,X;,X;]T for: 131.. = [1,2,3,4, sf and f3q uad = 0.1 [I, 4, 9, 16, 2sf 3 kinds of noise conditions, each with 2 sub-conditions: local signal/noise ratio Isnr=20, i) only output noise: a) low noise: and b) high noise: Isnr=2, ii) equal noise in inputs and outputs: a) low noise Ex ?? = Sy = N(O,O.Ot2), n e[I,2, ... ,10], and b) high noise Ex ?? =sy=N(0,0.1 2),ne[I,2, ... ,10], iii) unequal noise in inputs and outputs: a) low noise : Ex .? = N(0,(0.0In)2), n e[I,2, . .. ,1O] and Isnr=20, and ? b) high noise: Ex .? = N(0,(0.0In)2), n e[I,2, ... ,1O] and Isnr=2, 2 kinds of input distributions: i) uniform in unit hyper cube, ii) uniform in unit hyper cube excluding data points which activate a Gaussian weighting function (I) at c = [O.S,O,o,o,of with D=IO*I more than w=0.2 (this forms a "hyper kidney" shaped distribution) Every algorithm was run * 30 times on each of the 48 combinations of the conditions. Additionally, the complete test was repeated for three further conditions varying the dimensionality--called factors in accordance with LWFA-that the algorithms assumed to be the true dimensionality of the to-dimensional data from k=4 to 6, i.e., too few, correct, and too many factors. The average results are summarized in Figure I. Figure I a,b,c show the summary results of the three factor conditions. Besides averaging over the 30 trials per condition, each mean of these charts also averages over the two input distribution conditions and the linear and quadratic function condition, as these four cases are frequently observed violations of the statistical assumptions in nonlinear function approximation with locally linear models. In Figure I b the number of factors equals the underlying dimensionality of the problem, and all algorithms are essentially performing equally well. For perfectly Gaussian distributions in all random variables (not shown separately), LWFA's assumptions are perfectly fulfilled and it achieves the best results, however, almost indistinguishable closely followed by LWPLS. For the ''unequal noise condition", the two PCA based techniques, LWPCA and LWPCR, perform the worst since--as expected-they choose suboptimal projections. However, when violating the statistical assumptions, L WF A loses parts of its advantages, such that the summary results become fairly balanced in Figure lb. The quality of function fitting changes significantly when violating the correct number of factors, as illustrated in Figure I a,c. For too few factors (Figure la), LWPCR performs worst because it randomly omits one of the principle components in the input data, without respect to how important it is for the regression. The second worse is LWF A: according to its assumptions it believes that the signal it cannot model must be noise, leading to a degraded estimate of the data's subspace and, consequently, degraded regression results. LWPLS has a clear lead in this test, closely followed by LWPCA and LWPLS_I. * Except for LWFA, all methods can evaluate a data set in non-iterative calculations. LWFA was trained with EM for maximally 1000 iterations or until the log-likelihood increased less than I.e-lOin one iteration. S. Schaal, S. Vljayakumar and C. G. Atkeson 638 For too many factors than necessary (Figure Ie), it is now LWPCA which degrades. This effect is due to its extracting one very noise contaminated projection which strongly influences the recovery of the regression parameters in Equation (4). All other algorithms perform almost equally well, with LWF A and LWPLS taking a small lead. Equal NoIse In ell In puIS end OutpUIS OnlyOutpul Noise Unequel NoIse In ell Inputs end OutpulS 0.1 c o ~ ::::;; 0.01 c II> C> ~ 0.001 ~ 0.0001 fl- I. E>O ~I. ? >>(I ~ J. &>O ~J , E ? O ~ J .E>O fl- I.&? O ~I . & >O ~ I . & > >O ~I.& >O ~ I .?>>o p,. 1. s>O tJ-J .?>>O e) RegressIon Results with 4 Factors ? LWFA ? LWPCA ? LWPCR 0 LWPLS ? LWPLS_1 0.1 c:: o W ~c:: 0.01 II> C) ~ 0.001 ~ 0.0001 ~ ~ 0.1 W 0.01 8 ~c:: g, ~ 0.001 ~ 0.0001 c) RegressIon Results with 6 Feclors jj il 0.1 f- a ~ ::::;; 0.01 c II> C) ~ 0.001 ~ 0.0001 d) Summery Results Figure I: Average summary results of Monte Carlo experiments. Each chart is primarily divided into the three major noise conditions, cf. headers in chart (a). In each noise condition, there are four further subdivision: i) coefficients of linear or quadratic model are equal with low added noise; ii) like i) with high added noise; iii) coefficients oflinear or quadratic model are different with low noise added; iv) like iii) with high added noise. Refer to text and descriptions of Monte Carlo studies for further explanations. Local Dimensionality Reduction 639 4 SUMMARY AND CONCLUSIONS Figure 1d summarizes all the Monte Carlo experiments in a final average plot. Except for LWPLS, every other technique showed at least one clear weakness in one of our "robustness" tests. It was particularly an incorrect number of factors which made these weaknesses apparent. For high-dimensional regression problems, the local dimensionality, i.e., the number of factors, is not a clearly defined number but rather a varying quantity, depending on the way the generating process operates. Usually, this process does not need to generate locally low dimensional distributions, however, it often "chooses" to do so, for instance, as human ann movements follow stereotypic patterns despite they could generate arbitrary ones. Thus, local dimensionality reduction needs to find autonomously the appropriate number of local factor. Locally weighted partial least squares turned out to be a surprisingly robust technique for this purpose, even outperforming the statistically appealing probabilistic factor analysis. As in principal component analysis, LWPLS's number of factors can easily be controlled just based on a variance-cutoff threshold in input space (Frank & Friedman, 1993), while factor analysis usually requires expensive cross-validation techniques. Simple, variance-based control over the number of factors can actually improve the results of LWPCA and LWPCR in practice, since, as shown in Figure I a, LWPCR is more robust towards overestimating the number of factors, while L WPCA is more robust towards an underestimation. If one is interested in dynamically growing the number of factors while obtaining already good regression results with too few factors, L WPCA and, especially, L WPLS seem to be appropriate-it should be noted how well one factor LWPLS (LWPLS_l) already performed in Figure I! In conclusion, since locally weighted partial least squares was equally robust as local weighted factor analysis towards additive noise in. both input and output data, and, moreover, superior when mis-guessing the number of factors, it seems to be a most favorable technique for local dimensionality reduction for high dimensional regressions. Acknowledgments The authors are grateful to Geoffrey Hinton for reminding them of partial least squares. This work was supported by the ATR Human Information Processing Research Laboratories. S. Schaal's support includes the German Research Association, the Alexander von Humboldt Foundation, and the German Scholarship Foundation. S. Vijayakumar was supported by the Japanese Ministry of Education, Science, and Culture (Monbusho). C. G. Atkeson acknowledges the Air Force Office of Scientific Research grant F49-6209410362 and a National Science Foundation Presidential Young Investigators Award. References Atkeson, C. G., Moore, A. W., & Schaal, S, (1997a). "Locally weighted learning." ArtifiCial Intelligence Review, 11, 1-5, pp.II-73. Atkeson, C. G., Moore, A. W., & Schaal, S, (1997c). "Locally weighted learning for control." ArtifiCial Intelligence Review, 11, 1-5, pp.75-113. Belsley, D. A., Kuh, E., & Welsch, R. E, (1980). Regression diagnostics: Identifying influential data and sources of collinearity. New York: Wiley. Everitt, B. S, (1984). An introduction to latent variable models. London: Chapman and Hall. Fahlman, S. E. ,Lebiere, C, (1990). "The cascadecorrelation learning architecture." In: Touretzky, D. S. (Ed.), Advances in Neural Information Processing Systems II, pp.524-532. Morgan Kaufmann. Frank, I. E., & Friedman, 1. H, (1993). "A statistical view of some chemometric regression tools." Technometrics, 35, 2, pp.l09-135. Geman, S., Bienenstock, E., & Doursat, R. (1992). "Neural networks and the bias/variance dilemma." Neural Computation, 4, pp.I-58. Hom, R. A., & Johnson, C. R, (1994). Matrix analySis. Press Syndicate of the University of Cambridge. Jordan, M.I., & Jacobs, R, (1994). "Hierarchical mix- tures of experts and the EM algorithm." Neural Computation, 6, 2, pp.181-214. Massy, W. F, (1965). "Principle component regression in exploratory statistical research." Journal ofthe American Statistical Association, 60, pp.234-246. Rubin, D. B., & Thayer, D. T, (l982). "EM algorithms for ML factor analysis." Psychometrika, 47, I, 69-76. Schaal, S., & Atkeson, C. G, (in press). "Constructive incremental learning from only local information." Neural Computation. Scott, D. W, (1992). Multivariate Density Estimation. New York: Wiley. Vijayakumar, S., & Schaal, S, (1997). "Local dimensionality reduction for locally weighted learning." In: International Conference on Computational Intelligence in Robotics and Automation, pp.220-225, Monteray, CA, July 10-11, 1997. Wold, H. (1975). "Soft modeling by latent variables: the nonlinear iterative partial least squares approach." In: Gani, J. (Ed.), Perspectives in Probability and Statistics, Papers in Honour ofM S. Bartlett. Aca<j. Press. Xu, L., Jordan, M.l., & Hinton, G. E, (1995). "An alternative model for mixture of experts." In: Tesauro, G., Touretzky, D. S., & Leen, T. K. (Eds.), Advances in Neural Information Processing Systems 7, pp.633-640. Cambridge, MA: MIT Press. Serial Order in Reading Aloud: Connectionist Models and Neighborhood Structure Jeanne C. Milostan Computer Science & Engineering 0114 University of California San Diego La Jolla, CA 92093-0114 Garrison W. Cottrell Computer Science & Engineering 0114 University of California San Diego La Jolla, CA 92093-0114 Abstract Dual-Route and Connectionist Single-Route models ofreading have been at odds over claims as to the correct explanation of the reading process. Recent Dual-Route models predict that subjects should show an increased naming latency for irregular words when the irregularity is earlier in the word (e.g. chef is slower than glow) - a prediction that has been confirmed in human experiments. Since this would appear to be an effect of the left-to-right reading process, Coltheart & Rastle (1994) claim that Single-Route parallel connectionist models cannot account for it. A refutation of this claim is presented here, consisting of network models which do show the interaction, along with orthographic neighborhood statistics that explain the effect. 1 Introduction A major component of the task of learning to read is the development of a mapping from orthography to phonology. In a complete model of reading, message understanding must playa role, but many psycholinguistic phenomena can be explained in the context of this simple mapping task. A difficulty in learning this mapping is that in a language such as English, the mapping is quasiregular (Plaut et al., 1996); there are a wide range of exceptions to the general rules. As with nearly all psychological phenomena, more frequent stimuli are processed faster, leading to shorter naming latencies. The regularity of mapping interacts with this variable, a robust finding that is well-explained by connectionist accounts (Seidenberg and M.cClelland, 1989; Taraban and McClelland, 1987). In this paper we consider a recent effect that seems difficult to account for in terms of the standard parallel network models. Coltheart & Rastle (1994) have shown 1. C. Milostan and G. W. Cottrell 60 Position of 2 Irregular 3 Phoneme 1 4 5 Irregular Regular Control Difference 554 502 52 542 516 26 530 518 12 529 523 6 537 525 12 Irregular Regular Control Difference Avg. Difl'. 545 500 45 48.5 524 503 21 23.5 528 503 25 18.5 526 515 11 8.5 528 524 4 8 Filler Nonword Exception Table 1: Naming Latency vs. Irregularity Position that the amount of delay experienced in naming an exception word is related to the phonemic position of the irregularity in pronunciation. Specifically, the earlier the exception occurs in the word, the longer the latency to the onset of pronouncing the word. Table 1, adapted from (Coltheart and Rastle, 1994) shows the response latencies to two-syllable words by normal subjects. There is a clear left-to-right ranking of the latencies compared to controls in the last row of the Table. Coltheart et al. claim this delay ranking cannot be achieved by standard connectionist models. This paper shows this claim to be false, and shows that the origin of the effect lies in a statistical regularity of English, related to the number of "friends" and "enemies" of the pronunciation within the word's neighborhood 1. 2 Background Computational modeling of the reading task has been approached from a number of different perspectives. Advocates of a dual-route model of oral reading claim that two separate routes, one lexical (a lexicon, often hypothesized to be an associative network) and one rule-based, are required to account for certain phenomena in reaction times and nonword pronunciation seen in human subjects (Coltheart et al., 1993). Connectionist modelers claim that the same phenomena can be captured in a single-route model which learns simply by exposure to a representative dataset (Seidenberg and McClelland, 1989). In the Dual-Route Cascade model (DRC) (Coltheart et al., 1993), the lexical route is implemented as an Interactive Activation (McClelland and Rumelhart, 1981) system, while the non-lexical route is implemented by a set of grapheme-phoneme correspondence (GPC) rules learned from a dataset. Input at the letter identification layer is activated in a left-to-right sequential fashion to simulate the reading direction of English, and fed simultaneously to the two pathways in the model. Activation from both the GPC route and the lexicon route then begins to interact at the output (phoneme) level, starting with the phonemes at the beginning of the word. If the GPC and the lexicon agree on pronunciation, the correct phonemes will be activated quickly. For words with irregular pronunciation, the lexicon and GPC routes will activate different phonemes: the GPC route will try to activate the regular pronunciation while the lexical route will activate the irregular (correct) 1 Friends are words with the same pronunciations for the ambiguous letter-ta-sound correspondence; enemies are words with different pronunciations. Serial Ortier in Reading Aloud 61 pronunciation. Inhibitory links between alternate phoneme pronunciations will slow down the rise in activation, causing words with inconsistencies to be pronounced more slowly than regular words. This slowing will not occur, however, when an irregularity appears late in a word. This is because in the model the lexical node spreads activation to all of a word's phonemes as soon as it becomes active. If an irregularity is late in a word, the correct pronunciation will begin to be activated before the GPC route is able to vote against it. Hence late irregularities will not be as affected by conflicting information. This result is validated by simulations with the one-syllable DRC model (Coltheart and Rastle, 1994). Several connectionist systems have been developed to model the orthography to phonology process (Seidenberg and McClelland, 1989; Plaut et al., 1996). These connectionist models provide evidence that the task, with accompanying phenomena, can be learned through a single mechanism. In particular, Plaut et al. (henceforth PMSP) develop a recurrent network which duplicates the naming latencies appropriate to their data set, consisting of approximately 3000 one-syllable English words (monosyllabic words with frequency greater than 1 in the Kucera & Francis corpus (Kucera and Francis, 1967?. Naming latencies are computed based on time-t~settle for the recurrent network, and based on MSE for a feed-forward model used in some simulations. In addition to duplicating frequency and regularity interactions displayed in previous human studies, this model also performs appr~ priately in providing pronunciation of pronounceable nonwords. This provides an improvement over, and a validation of, previous work with a strictly feed-forward network (Seidenberg and McClelland, 1989). However, to date, no one has shown that Coltheart's naming latency by irregularity of position interaction can be accounted for by such a model. Indeed, it is difficult to see how such a model could account for such a phenomenon, as its explanation (at least in the DRC model) seems to require the serial, left-t~right nature of processing in the model, whereas networks such as PMSP present the word orthography all at once. In the following, we fill this gap in the literature, and explain why a parallel, feed-forward model can account for this result. Experiments & Results 3 3.1 The Data Pronunciations for approximately 100,000 English words were obtained through an electronic dictionary developed by CMU 2 . The provided format was not amenable to an automated method for distinguishing the number of syllables in the word. To obtain syllable counts, English tw~syllable words were gathered from the Medical Research Council (MRC) Psycholinguistic Database (Coltheart and Rastle, 1994), which is conveniently annotated with syllable counts and frequency (only those with Kucera-Francis written frequency of one or greater were selected). Intersecting the two databases resulted in 5,924 tw~syllable words. There is some noise in the data; ZONED and AERIAL, for example, are in this database of purported tw~syllable words. Due to the size of the database and time limitations, we did not prune the data of these errors, nor did we eliminate proper nouns or foreign words. Singlesyllable words with the same frequency criterion were also selected for comparison with previous work. 3,284 unique single-syllable words were obtained, in contrast to 2,998 words used by PMSP. Similar noisy data as in the tw~syllable set exists in this database. Each word was represented using the orthography and phonology representation scheme outlined by PMSP. 2 Available via ftp://ftp.cs.cmu.edu/project/fgdata/dict/ 1. C. Milostan and G. W Cottrell 62 1.0 S 0.8 I 0.6 ""it. J... 0.4 r!I 02 >::' Ii Figure 1: I-syllable network latency differences & neighborhood statistics 3.2 Methods For the single syllable words, we used an identical network to the feed-forward network used by PMSP, i.e., a 105-100-61 network, and for the two syllable words, we simply used the same architecture with the each layer size doubled. We trained each network for 300 epochs, using batch training with a cross entropy objective function, an initial learning rate of 0.001, momentum of 0.9 after the first 10 epochs, weight decay of 0.0001, and delta-bar-delta learning rate adjustment. Training exemplars were weighted by the log of frequency as found in the Kucera-Francis corpus. After this training, the single syllable feed-forward networks averaged 98.6% correct outputs, using the same evaluation technique outlined in PMSP. Two syllable networks were trained for 1700 epochs using online training, a learning rate of 0.05, momentum of 0.9 after the first 10 epochs, and raw frequency weighting. The two syllable network achieved 85% correct. Naming latency was equated with network output MSE; for successful results, the error difference between the irregular words and associated control words should decrease with irregularity position. 3.3 Results Single Syllable Words First, Coltheart's challenge that a single-route model cannot produce the latency effects was explored. The single-syllable network described above was tested on the collection of single-syllable words identified as irregular by (Taraban and McClelland, 1987). In (Coltheart and Rastle, 1994), control words are selected based on equal number of letters, same beginning phoneme, and Kucera-Francis frequency between 1 and 20 (controls were not frequency matched). For single syllable words used here, the control condition was modified to allow frequency from 1 to 70, which is the range of the "low frequency" exception words in the Taraban & McClelland set. Controls were chosen by drawing randomly from the words meeting the control criteria. Each test and control word input vector was presented to the network, and the MSE at the output layer (compared to the expected correct target) was calculated. From these values, the differences in MSE for target and matched control words were calculated and are shown in Figure 1. Note that words with an irregularity in the first phoneme position have the largest difference from their control words, with this (exception - regular control) difference decreasing as phoneme position increases. Contrary to the claims of the Dual-Route model, this network does show the desired rank-ordering of MSE/latency. 63 Serial Order in Reading Aloud 1.0 02 I' d I 1;l 0.1 0.0 ::I! o 4 I'boaeme 1........ larit)' PooIIioa O.O+--~--r----.---~--' 6 o 2 " Pbonane lnegularlty PosItioIl 6 Figure 2: 2-syllable network latency differences & neighborhood statistics Two Syllable Words Testing of the two-syllable network is identical to that of the one-syllable network. The difference in MSE for each test word and its corresponding control is calculated, averaging across all test pairs in the position set. Both test words and their controls are those found in (Coltheart and Rastle, 1994). The 2-syllable network appears to produce approximately the correct linear trend in the naming MSE/latency (Figure 2), although the results displayed are not monotonically decreasing with position. Note, however, that the results presented by Coltheart, when taken separately, also fail to exhibit this trend (Table 1). For correct analysis, several "subject" networks should be trained, with formal linear trend analysis then performed with the resulting data. These further simulations are currently being undertaken. 4 Why the network works: Neighborhood effects A possible explanation for these results relies on the fact that connectionist networks tend to extract statistical regularities in the data, and are affected by regularity by frequency interactions. In this case, we decided to explore the hypothesis that the results could be explained by a neighborhood effect: Perhaps the number of "friends" and "enemies" in the neighborhood (in a sense to be defined below) of the exception word varies in English in a position-dependent way. If there are more enemies (different pronunciations) than friends (identical pronunciations) when the exception occurs at the beginning of a word than at the end, then one would expect a network to reflect this statistical regularity in its output errors. In particular, one would expect higher errors (and therefore longer latencies in naming) if the word has a higher proportion of enemies in the neighborhood. To test this hypothesis, we created some data search engines to collect word neighborhoods based on various criteria. There is no consensus on the exact definition of the "neighborhood" of a word. There are some common measures, however, so we explored several of these. Taraban & McClelland (1987) neighborhoods (T&M) are defined as words containing the same vowel grouping and final consonant cluster. These neighborhoods therefore tend to consist of words that rhyme (MUST, DUST, TRUST). There is independent evidence that these word-body neighbors are psychologically relevant for word naming tasks (i.e., pronunciation) (Treiman and Chafetz, 1987). The neighborhood measure given by Coltheart (Coltheart and Rastle, 1994), N, counts same-length words which differ by only one letter, taking string position into account. Finally, edit-distance-1 (ED1) neighborhoods are those words which can be generated from the target word by making one change J. C. Milostan and G. W Cottrell 64 (Peereman, 1995): either a letter substitution, insertion or deletion. This differs from the Coltheart N definition in that "TRUST" is in the EDI neighborhood (but not the N neighborhood) of "RUST" , and provides a neighborhood measure which considers both pronunciation and spelling similarity. However, the N and the ED-l measure have not been shown to be psychologically real in terms of affecting naming latency (Treiman and Chafetz, 1987). We therefore extended T&M neighborhoods to multi-syllable words. Each vowel group is considered within the context of its rime, with each syllable considered separately. Consonant neighborhoods consist of orthographic clusters which correspond to the same location in the word. This results in 4 consonant cluster locations: first syllable onset, first syllable coda, second syllable onset, and second syllable coda. Consonant cluster neighborhoods include the preceeding vowel for coda consonants, and the following vowel for onset consonants. The notion of exception words is also not universally agreed upon. Precisely which words are exceptions is a function of the working definition of pronunciation and regularity for the experiment at hand. Given a definition of neighborhood, then, exception words can be defined as those words which do not agree with the phonological mapping favored by the majority of items in that particular neighborhood. Alternatively, in cases assuming a set of rules for grapheme-phoneme correspondence, exception words are those which violate the rules which define thp majority of pronunciations. For this investigation, single syllable exception words are those defined as exception by the T&M neighborhood definition. For instance, PINT would be considered an exception word compared to its neighbors MINT, TINT, HINT, etc. Coltheart, on the other hand, defines exception words to be those for which his G PC rules produce incorrect pronunciation. Since we are concerned with addressing Coltheart's claims, these 2-syllable exception words will also be used here. 4.1 Results Single syllable words For each phoneme position, we compare each word with irregularity at that position with its neighbors, counting the number of enemies (words with alternate pronunciation at the supposed irregularity) and friends (words with pronunciation in agreement) that it has. The T &M neighborhood numbers (words containing the same vowel grouping and final consonant cluster) used in Figure 1 are found in (Taraban and McClelland, 1987). For each word , we calculate its (enemy) / (friend+enemy) ratio; these ratios are then averaged over all the words in the position set. The results using neighborhoods as defined in Taraban & McClelland clearly show the desired rank ordering of effect. First-position-irregularity words have more "enemies" and fewer "friends" than third-position-irregularity words, with the second-position words falling in the middle as desired. We suggest that this statistical regularity in the data is what the above networks capture. However convincing these results may be, they do not fully address Coltheart's data, which is for two syllable words of five phonemes or phoneme clusters, with irregularities at each of five possible positions. Also, due to the size of the T&M data set, there are only 2 members in the position I set, and the single-syllable data only goes up to phoneme position 3. The neighborhoods for the two-syllable data set were thus examined. Two syllable results Recall that the two-syllable test words are those used in the (Coltheart and Rastle, 1994) subject study, for which naming latency differences are shown in Table 1. CoItheart's I-letter-different neighborhood definition Serial Order in Reading Aloud 65 is not very informative in this case, since by this criterion most of the target words provided in (Coltheart and Rastle, 1994) are loners (i.e., have no neighbors at all). However, using a neighborhood based on T&M-2 recreates the desired ranking (Figure 2) as indicated by the ratio of hindering pronunciations to the total of the helping and hindering pronunciations. As with the single syllable words, each test word is compared with its neighbor words and the (enemy)/(friend+enemy) ratio is calculated. Averaging over the words in each position set, we again see that words with early irregularities are at a support disadvantage compared to words with late irregularities. 5 Summary Dual-Route models claim the irregularity position effect can only be accounted for by two-route models with left-to-right activation of phonemes, and interaction between GPC rules and the lexicon. The work presented in this paper refutes this claim by presenting results from feed-forward connectionist networks which show the same rank ordering of latency. Further, an analysis of orthographic neighborhoods shows why the networks can do this: the effect is based on a statistical interaction between friend/enemy support and position. Words with irregular orthographicphonemic correspondence at word beginning have less support from their neighbors than words with later irregularities; it is this difference which explains the latency results. The resulting statistical regularity is then easily captured by connectionist networks exposed to representative data sets. References Coltheart, M., Curitis, B., Atkins, P., and Haller, M. (1993). Models of reading aloud: Dual-route and parallel-distributed-processing approaches. Psychological Review, 100(4):589-608. Coltheart, M. and Rastle, K. (1994). Serial processing in reading aloud: Evidence for dual route models of reading. Journal of Experimental Psychology: Human Perception and Performance, 20(6):1197-1211. Kucera, H. and Francis, W. (1967). Computational Analysis of Present-Day American English. Brown University Press, Providence, RI. McClelland, J. and Rumelhart, D. (1981). An interactive activation model of context effects in letter perception: Part 1. an account of basic findings. Psychological Review, 88:375-407. Peereman, R. (1995). Naming regular and exception words: Further examination of the effect of phonological dissension among lexical neighbours. European Journal of Cognitive Psychology, 7(3):307-330. Plaut, D., McClelland, J., Seidenberg, M., and Patterson, K. (1996). Understanding normal and impaired word reading: Computational principles in quasi-regular domains. Psychological Review, 103(1):56-115. Seidenberg, M. and McClelland, J. (1989). A distributed, developmental model of word recognition and naming. Psychological Review, 96:523-568. Taraban, R. and McClelland, J. (1987). Conspiracy effects in word pronunciation. Journal of Memory and Language, 26:608-631. Treiman, R. and Chafetz, J. (1987). Are there onset- and rime-like units in printed words? In Coltheart, M., editor, Attention and Performance XII: The Psychology of Reading. Erlbaum, Hillsdale, NJ. 

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11 AN OPTIMALITY PRINCIPLE FOR UNSUPERVISED LEARNING Terence D. Sanger MIT AI Laboratory, NE43-743 Cambridge, MA 02139 (tds@wheaties.ai.mit.edu) ABSTRACT We propose an optimality principle for training an unsupervised feedforward neural network based upon maximal ability to reconstruct the input data from the network outputs. We describe an algorithm which can be used to train either linear or nonlinear networks with certain types of nonlinearity. Examples of applications to the problems of image coding, feature detection, and analysis of randomdot stereograms are presented. 1. INTRODUCTION There are many algorithms for unsupervised training of neural networks, each of which has a particular optimality criterion as its goal. (For a partial review, see (Hinton, 1987, Lippmann, 1987).) We have presented a new algorithm for training single-layer linear networks which has been shown to have optimality properties associated with the Karhunen-Loeve expansion (Sanger, 1988b). We now show that a similar algorithm can be applied to certain types of nonlinear feedforward networks, and we give some examples of its behavior. The optimality principle which we will use to describe the algorithm is based on the idea of maximizing information which was first proposed as a desirable property of neural networks by Linsker (1986, 1988). Unfortunately, measuring the information in network outputs can be difficult without precise knowledge of the distribution on the input data, so we seek another measure which is related to information but which is easier to compute. If instead of maximizing information, we try to maximize our ability to reconstruct the input (with minimum mean-squared error) given the output of the network, we are able to obtain some useful results. Note that this is not equivalent to maximizing information except in some special cases. However, it contains the intuitive notion that the input data is being represented by the network in such a way that very little of it has been "lost". 12 Sanger 2. LINEAR CASE We now summarize some of the results in (Sanger, 1988b). A single-layer linear feedforward network is described by an M xN matrix C of weights such that if x is a vector of N inputs and y is a vector of M outputs with M < N, we have y Cx. As mentioned above, we choose an optimality principle defined so that we can best reconstruct the inputs to the network given the outputs. We want to minimize the mean squared error E[(x - x)2] where x is the actual input which is zero-mean with correlation matrix Q E[xxT], and x is a linear estimation of this input given the output y. The linear least squares estimate (LLSE) is given by = = and we will assume that x is computed in this way for any matrix C of weights which we choose. The mean-squared error for the LLSE is given by and it is well known that this is minimized if the rows of C are a linear combination of the first M eigenvectors of the correlation matrix Q. One optimal choice of C is the Singular Value Decomposition (SVD) of Q, for which the output correlation matrix E[yyT] CQCT will be the diagonal matrix of eigenvalues of Q. In this case, the outputs are uncorrelated and the sum of their variances (traceE[yyT]) is maximal for any set of M un correlated outputs. We can thus think of the eigenvectors as being obtained by any process which maximizes the output variance while maintaining the outputs uncorrelated. = We now define the optimal single-layer linear network as that network whose weights represent the first M eigenvectors of the input correlation matrix Q. The optimal network thus minimizes the mean-squared approximation error E[(x - x)2] given the shape constraint that M < N. 2.1 LINEAR ALGORITHM We have previously proposed a weight-update rule called the "Generalized Hebbian Algorithm" , and proven that this algorithm causes the rows of the weight matrix C to converge to the eigenvectors of the input correlation matrix Q (Sanger, 1988a,b). The algorithm is given by: C(t + 1) = C(t) + I (y(t)xT(t) - LT[y(t)yT(t)]C(t?) (1) where I is a rate constant which decreases as l/t, x(t) is an input sample vector, yet) = C(t)x(t), and LTD is an operator which makes its matrix argument lower triangular by setting all entries above the diagonal to zero. This algorithm can be implemented using only a local synaptic learning rule (Sanger, 1988b). Since the Generalized Hebbian Algorithm computes the eigenvectors of the input correlation matrix Q, it is related to the Singular Value Decomposition (SVD), An Optimality Principle for Unsupervised Learning c. Figure 1: (aJ original image. (bJ image coded at .36 bits per pixel. (cJ masks learned by the network which were used for vector quantized coding of 8x8 blocks of the image. Principal Components Analysis (PCA), and the Karhunen-Loeve Transform (KLT). (For a review of several related algorithms for performing the KLT, see (Oja, 1983).) 2.2 IMAGE CODING We present one example of the behavior of a single-layer linear network. (This example appears in (Sanger, 1988b).) Figure 1a shows an original 256x256x8bit image which was used for training a network. 8x8 blocks of the image were chosen by scanning over the image, and these were used as training inputs to a network with 64 inputs and 8 outputs. After training, the set of weights for each output (figure lc) represents a vector quantizing mask. Each 8x8 block of the input image is then coded using the outputs of the network. Each output is quantized with a number of bits related to the log of the variance, and the original figure is approximated from the quantized outputs. The reconstruction of figure 1b uses a total of 23 bits per 8x8 region, which gives a data rate of 0.36 bits per pixel. The fact that the image could be represented using such a low bit rate indicates that the masks that were found represent significant features which are useful for recognition. This image coding technique is equivalent to block-coded KLT methods common in the literature. 13 14 Sanger 3. NONLINEAR CASE In general, training a nonlinear unsupervised network to approximate nonlinear functions is very difficult. Because of the large (infinite-dimensional) space of possible functions, it is important to have detailed knowledge of the class of functions which are useful in order to design an efficient network algorithm. (Several people pointed out to me that the talk implied such knowledge is not necessary, but unfortunately such an implication is false.) The network structure we consider is a linear layer represented by a matrix C (which is perhaps an interior layer of a larger network) followed by node nonlinearities (1'(Yi) where Yi is the ith linear output, followed by another linear layer (perhaps followed by more layers). We assume that the nonlinearities (1'0 are fixed, and that the only parameters susceptible to training are the linear weights C. If z is the M-vector of outputs after the nonlinearity, then we can write each component Zi = (1'(Yi) (1'(CiX) where Ci is the ith row of the matrix C. Note that the level contours of each function Zi are determined entirely by the vector Ci, and that the effect of (1'0 is limited to modifying the output value. Intuitively, we thus expect that if Yi encodes a useful parameter of the input x, then Zi will encode the same parameter, although scaled by the nonlinearity (1'0. = This can be formalized, and if we choose our optimality principle to again be minimum mean-squared linear approximation of the original input x given the output z, the best solution remains when the rows of C are a linear combination of the first M eigenvectors of the input correlation matrix Q (Bourlard and Kamp, 1988) . In two of the simulations, the nonlinearity (1'0 which we use is a rectification nonlinearity, given by Yi if Yi 20 (1'(Yd { 0 if Yi <0 = Note that at most one of {(1'(Yi), (1'( -Yi)} is nonzero at any time, so these two values are uncorrelated. Therefore, if we maximize the variance of y (before the nonlinearity) while maintaining the elements of Z (after the nonlinearity) uncorrelated, we need 2M outputs in order to represent the data available from an M-vector y. Note that 2M may be greater than the number of inputs N, so that the "hidden layer" Z can have more elements than the input. 3.1 NONLINEAR ALGORITHM The nonlinear Generalized Hebbian Algorithm has exactly the same form as for the linear case, except that we substitute the use of the output values after the nonlinearity for the linear values. The algorithm is thus given by: C(t + 1) = C(t) + 'Y (z(t)xT(t) - LT[z(t)zT(t)]C(t)) where the elements of z are given by Zi(t) = (1'(Yi(t)), with y(t) (2) = C(t)x(t). Although we have not proven that this algorithm converges, a heuristic analysis of its behavior (for a rectification nonlinearity and Gaussian input distribution) An Optimality Principle for Unsupervised Learning shows that stable points may exist for which each row of C is proportional to an eigenvector of Q, and pairs of rows are either the negative of each other or orthogonal. In practice, the rows of C are ordered by decreasing output variance, and occur in pairs for which one member is the negative of the other. This choice of C will maximize the sum of the output variances for uncorrelated outputs, so long as the input is Gaussian. It also allows optimal linear estimation of the input given the output, so long as both polarities of each of the eigenvectors are present. 3.2 NONLINEAR EXAMPLES 3.2.1 Encoder Problem We compare the performance of two nonlinear networks which have learned to perform an identity mapping (the "encoder" problem). One is trained by backpropagation, (Rumelhart et a/., 1986) and the other has two hidden layers trained using the unsupervised Hebbian algorithm, while the output layer is trained using a supervised LMS algorithm (Widrow and Hoff, 1960). The network has 5 inputs, two hidden layers of 3 units each, and 5 outputs. There is a sigmoid nonlinearity at each hidden layer, but the thresholds are all kept at zero. The task is to minimize the mean-squared difference between the inputs and the outputs. The input is a zero-mean correlated Gaussian random 5-vector, and both algorithms are presented with the same sequence of inputs. The unsupervised-trained network converged to a steady state after 1600 examples, and the backpropagation network converged after 2400 (convergence determined by no further decrease in average error). The RMS error at steady state was 0.42 for both algorithms (this figure should be compared to the sum of the variances of the inputs, which was 5.0). Therefore, for this particular task, there is no significant difference in performance between backpropagation and the Generalized Hebbian Algorithm. This is an encouraging result, since if we can use an unsupervised algorithm to solve other problems, the training time will scale at most linearly with the number of layers. 3.2.2 Nonlinear Receptive Fields Several investigators have shown that Hebbian algorithms can discover useful image features related to the receptive fields of cells in primate visual cortex (see for example (Bienenstock et a/., 1982, Linsker, 1986, Barrow, 1987?. One of the more recent methods uses an algorithm very similar to the one proposed here to find the principal component of the input (Linsker, 1986). We performed an experiment to find out what types of nonlinear receptive fields could be learned by the Generalized Hebbian Algorithm if provided with similar input to that used by Linsker. We used a single-layer nonlinear network with 4096 inputs arranged in a 64x64 grid, and 16 outputs with a rectification nonlinearity. The input data consisted of images of low-pass filtered white Gaussian noise multiplied by a Gaussian window. After 5000 samples, the 16 outputs learned the masks shown in figure 2. These masks possess qualitative similarity to the receptive fields of cells found in the visual cortex of cat and monkey (see for example (Andrews and Pollen, 1979?. They are equivalent to the masks learned by a purely linear network (Sanger, 1988b), except that both positive and negative polarities of most mask shapes are present here. 15 16 Sanger Figure 2: Nonlinear receptive fields ordered from left-to-right and top-to-bottom. 3.2.3 Stereo We now show how the nonlinear Generalized Hebbian Algorithm can be used to train a two-layer network to detect disparity edges. The network has 128 inputs, 8 types of unit in the hidden layer with a rectification nonlinearity, and 4 types of output unit. A 128x128 pixel random-dot stereo pair was generated in which the left half had a disparity of two pixels, and the right half had zero disparity. The image was convolved with a vertically-oriented elliptical Gaussian mask to remove high-frequency vertical components. Corresponding 8x8 blocks of the left and right images (64 pixels from each image) were multiplied by a Gaussian window function and presented as input to the network, which was allowed to learn the first layer according to the unsupervised algorithm. After 4000 iterations, the first layer had converged to a set of 8 pairs of masks. These masks were convolved with the images (the left mask was convolved with the left image, and the right mask with the right image, and the two results were summed and rectified) to produce a pattern of activity at the hidden layer. (Although there were only 8 types of hidden unit, we now allow one of each type to be centered at every input image location to obtain a pattern of total activity.) Figure 3 shows this activity, and we can see that the last four masks are disparity-sensitive since they respond preferentially to either the 2 pixel disparity or the zero disparity region of the image. An Optimality Principle for Unsupervised Learning Figure 3: Hidden layer response for a two-layer nonlinear network trained on stereo images. The left half of the input random dot image has a 2 pixel disparity, and the right half has zero disparity. Figure 4: Output layer response for a two-layer nonlinear network trained on stereo Images. Since we were interested in disparity, we trained the second layer using only the last four hidden unit types. The second layer had 1024 (=4x16x16) inputs organized as a 16x16 receptive field in each of the four hidden unit "planes". The outputs did not have any nonlinearity. Training was performed by scanning over the hidden unit activity pattern (successive examples overlapped by 8 pixels) and 6000 iterations were used to produce the second-layer weights. The masks that were learned were then convolved with the hidden unit activity pattern to produce an output unit activity pattern, shown in figure 4. The third output is clearly sensitive to a change in disparity (a depth edge). If we generate several different random-dot stereograms and average the output results, 17 18 Sanger Figure 5: Output layer response averaged over ten stereograms with a central 2 pixel disparity square and zero disparity surround. we see that the other outputs are also sensitive (on average) to disparity changes, but not as much as the third. Figure 5 shows the averaged response to 10 stereograms with a central 2 pixel disparity square against a zero disparity background. Note that the ability to detect disparity edges requires the rectification nonlinearity at the hidden layer, since no linear function has this property. 4. CONCLUSION We have shown that the unsupervised Generalized Hebbian Algorithm can produce useful networks. The algorithm has been proven to converge only for single-layer linear networks. However, when applied to nonlinear networks with certain types of nonlinearity, it appears to converge to good results. In certain cases, it operates by maintaining the outputs uncorrelated while maximizing their variance. We have not investigated its behavior on nonlinearities other than rectification or sigmoids, so we can make no predictions about its general utility. Nevertheless, the few examples presented for the nonlinear case are encouraging, and suggest that further investigation of this algorithm will yield interesting results. Acknowledgements I would like to express my gratitude to the many people at the NIPS conference and elsewhere whose comments, criticisms, and suggestions have increased my understanding of these results. In particular, thanks are due to Ralph Linsker for pointing out to me an important error in the presentation and for his comments on the manuscript, as well as to John Denker, Steve Nowlan, Rich Sutton, Tom Breuel, and my advisor Tomaso Poggio. This report describes research done at the MIT Artificial Intelligence Laboratory, and sponsored by a grant from the Office of Naval Research (ONR), Cognitive and Neural Sciences Division; 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 An Optimality Principle for Unsupervised Learning Advanced Research Projects Agency of the Department of Defense under Army contract DACA76-85-C-0010, and in part by ONR contract N00014-85-K-0124. The author was supported during part of this research by a National Science Foundation Graduate fellowship, and later by a Medical Scientist Training Program grant. References Andrews B. W., Pollen D. A., 1979, Relationship between spatial frequency selectivity and receptive field profile of simple cells, J. Physiol., 287:163-176. Barrow H. G., 1987, Learning receptive fields, In Pmc. IEEE 1st Ann. Conference on Neural Networks, volume 4, pages 115-121, San Diego, CA. Bienenstock E. L., Cooper L. N., Munro P. W., 1982, Theory for the development of neuron selectivity: Orientation specificity and binocular interaction in visual cortex, J. Neuroscience, 2(1):32-48. 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Online learning from finite training sets in nonlinear networks David Barber t Peter Sollich* Department of Physics University of Edinburgh Edinburgh ERg 3JZ, U.K. Department of Applied Mathematics Aston University Birmingham B4 7ET, U.K. P.Sollich~ed.ac.uk D.Barber~aston . ac.uk Abstract Online learning is one of the most common forms of neural network training. We present an analysis of online learning from finite training sets for non-linear networks (namely, soft-committee machines), advancing the theory to more realistic learning scenarios. Dynamical equations are derived for an appropriate set of order parameters; these are exact in the limiting case of either linear networks or infinite training sets. Preliminary comparisons with simulations suggest that the theory captures some effects of finite training sets, but may not yet account correctly for the presence of local minima. 1 INTRODUCTION The analysis of online gradient descent learning, as one of the most common forms of supervised learning, has recently stimulated a great deal of interest [1, 5, 7, 3]. In online learning, the weights of a network ('student') are updated immediately after presentation of each training example (input-output pair) in order to reduce the error that the network makes on that example. One of the primary goals of online learning analysis is to track the resulting evolution of the generalization error - the error that the student network makes on a novel test example, after a given number of example presentations. In order to specify the learning problem, the training outputs are assumed to be generated by a teacher network of known architecture. Previous studies of online learning have often imposed somewhat restrictive and ? Royal Society Dorothy Hodgkin Research Fellow tSupported by EPSRC grant GR/J75425: Novel Developments in Learning Theory for Neural Networks P. SolIich and D. Barber 358 unrealistic assumptions about the learning framework. These restrictions are, either that the size of the training set is infinite, or that the learning rate is small[l, 5, 4]. Finite training sets present a significant analytical difficulty as successive weight updates are correlated, giving rise to highly non-trivial generalization dynamics. For linear networks, the difficulties encountered with finite training sets and noninfinitesimal learning rates can be overcome by extending the standard set of descriptive ('order') parameters to include the effects of weight update correlations[7]. In the present work, we extend our analysis to nonlinear networks. The particular model we choose to study is the soft-committee machine which is capable of representing a rich variety of input-output mappings. Its online learning dynamics has been studied comprehensively for infinite training sets[l, 5]. In order to carry out our analysis, we adapt tools originally developed in the statistical mechanics literature which have found application, for example, in the study of Hopfield network dynamics[2]. 2 MODEL AND OUTLINE OF CALCULATION For an N-dimensional input vector x, the output of the soft committee machine is given by (I) where the nonlinear activation function g(hl ) = erf(hz/V2) acts on the activations hi = wtxl.JFi (the factor 1/.JFi is for convenience only). This is a neural network with L hidden units, input to hidden weight vectors WI, 1 = I..L, and all hidden to output weights set to 1. In online learning the student weights are adapted on a sequence of presented examples to better approximate the teacher mapping. The training examples are drawn, with replacement, from a finite set, {(X/",yl-') ,j.t I..p}. This set remains fixed piN. during training. Its size relative to the input dimension is denoted by a We take the input vectors xl-' as samples from an N dimensional Gaussian distribution with zero mean and unit variance. The training outputs y'" are assumed to be generated by a teacher soft committee machine with hidden weight vectors w~, m = I..M, with additive Gaussian noise corrupting its activations and output. = = The discrepancy between the teacher and student on a particular training example (x, y), drawn from the training set, is given by the squared difference of their corresponding outputs, E= H~9(hl) -yr = H~9(hl) - ~g(km +em) -eor where the student and teacher activations are, respectively h, em, and m = I..M and output respectively. = {J;wtx km = {J;(w:n?x, (2) eo are noise variables corrupting the teacher activations and Given a training example (x, y), the student weights are updated by a gradient descent step with learning rate "I, w; - W, = -"I\1wIE = - JNx8h E l (3) 359 On-line Learning from Finite Training Sets in Nonlinear Networks The generalization error is defined to be the average error that the student makes on a test example selected at random (and uncorrelated with the training set), which we write as ?g = (E). Although one could, in principle, model the student weight dynamics directly, this will typically involve too many parameters, and we seek a more compact representation for the evolution of the generalization error. It is straightforward to show that the generalization error depends, not on a detailed description of all the network weights, but only on the overlap parameters Qll' = ~ W WI' and Rim = ~ W w':n [1, 5, 7]. In the case of infinite 0, it is possible to obtain a closed set of equations governing the overlap parameters Q, R [5]. For finite training sets, however, this is no longer possible, due to the correlations between successive weight updates[7]. r r In order to overcome this difficulty, we use a technique developed originally to study statistical physics systems [2] . Initially, consider the dynamics of a general vector of order parameters, denoted by 0, which are functions of the network weights w. If the weight updates are described by a transition probability T(w -+ w'), then an approximate update equation for 0 is 0' - 0 = IfdW' (O(w') - O(w)) T(w -+ \ W')) (4) P(w)oc6(O(w)-O) Intuitively, the integral in the above equation expresses the average change l of 0 caused by a weight update w -+ w', starting from (given) initial weights w. Since our aim is to develop a closed set of equations for the order parameter dynamics, we need to remove the dependency on the initial weights w. The only information we have regarding w is contained in the chosen order parameters 0, and we therefore average the result over the 'subshell' of all w which correspond to these values of the order parameters. This is expressed as the 8-function constraint in equation(4). It is clear that if the integral in (4) depends on w only through O(w), then the average is unnecessary and the resulting dynamical equations are exact. This is in fact the case for 0 -+ 00 and 0 = {Q, R}, the standard order parameters mentioned above[5]. If this cannot be achieved, one should choose a set of order parameters to obtain approximate equations which are as close as possible to the exact solution. The motivation for our choice of order parameters is based on the linear perceptron case where, in addition to the standard parameters Q and R, the overlaps projected onto eigenspaces of the training input correlation matrix A = ~ E:=l xl' (xl') T are required 2 . We therefore split the eigenvalues of A into r equal blocks ('Y = 1 ... r) containing N' = N Ir eigenvalues each, ordering the eigenvalues such that they increase with 'Y. We then define projectors p'Y onto the corresponding eigenspaces and take as order parameters: 'Y R1m _ - 1 Tp'Y .. N'w, wm UI.'Y - ~ Nt W,Tp'Yb II (5) where the b B are linear combinations of the noise variables and training inputs, (6) 1 Here we assume that the system size N is large enough that the mean values of the parameters alone describe the dynamics sufficiently well (i. e., self-averaging holds). 2The order parameters actually used in our calculation for the linear perceptron[7] are Laplace transforms of these projected order parameters. P. Sollich and D. Barber 360 As r -+ 00, these order parameters become functionals of a continuous variable3 . The updates for the order parameters (5) due to the weight updates (3) can be found by taking the scalar products of (3) with either projected student or teacher weights, as appropriate. This then introduces the following activation 'components', k'Y m = VNi ff(w* )Tp'"Yx m = so that the student and teacher activations are h, = ~ E'"Y hi and km ~ E'"Y k~, respectively. For the linear perceptron, the chosen order parameters form a complete set - the dynamical equations close, without need for the average in (4). For the nonlinear case, we now sketch the calculation of the order parameter update equations (4). Taken together, the integral over Wi (a sum of p discrete terms in our case, one for each training example) and the subshell average in (4), define an average over the activations (2), their components (7), and the noise variables ~m, ~o. These variables turn out to be Gaussian distributed with zero mean, and therefore only their covariances need to be worked out. One finds that these are in fact given by the naive training set averages. For example, = (8) where we have used p'"Y A = a'"YP'"Y with a'"Y 'the' eigenvalue of A in the ,-th eigenspace; this is well defined for r -+ 00 (see [6] for details of the eigenvalue spectrum). The correlations of the activations and noise variables explicitly appearing in the error in (3) are calculated similarly to give, (h,h,,) = ~ L:; Q~, '"Y (h,km) = ~L :; Rim (9) '"Y (h,~s) = ~ L ~U,~ '"Y where the final equation defines the noise variances. The T~m' are projected overlaps between teacher weight vectors, T~m' = ~ (w~)Tp'"Yw:n,. We will assume that the teacher weights and training inputs are uncorrelated, so that T~m' is independent of ,. The required covariances of the 'component' activations are a'"YR'"Y (kinh,) (c] h,) (hi h" ) a 'm - a'"YU'"Y - a'"YQ'"Y a a ls II' (k~km') = a'"YT'"Y (c]k m, ) - 0 - a'"YR'"Y (hJkm,) a a mm' 'm (k~~s) - 0 (C]~8' ) - a'"Y 2 -(7s588 , = .!.U'"Y (hJ~s) a a 's (10) 3Note that the limit r -+ 00 is taken after the thermodynamic limit, i.e., r ~ N. This ensures that the number of order parameters is always negligible compared to N (otherwise self-averaging would break down). On-line Learning from Finite Training Sets in Nonlinear Networks 0.03 r f I I : - - - - -........- - - - - - - , 0.025 I (a) 0.25 (b) o 00 OOOOOOOOC 0.2 000000000000000000000000 0000000 00 0.02 L.. ...o~ooo ~ I 0.15 0.01 361 '------~-----~ o t 50 \ 0000 'NNNoaa oa aaaoaaaaaaaaaaaaaaaac ,,------------ o 100 50 t 100 Figure 1: fg vs t for student and teacher with one hidden unit (L = M = 1); a = 2, 3, 4 from above, learning rate "I = 1. Noise of equal variance was added to both activations and output (a) O'~ = 0'5 = 0.01, (b) O'~ = 0'5= 0.1. Simulations for N = 100 are shown by circles; standard errors are of the order of the symbol size. The bottom dashed lines show the infinite training set result for comparison. r = 10 was used for calculating the theoretical predictions; the curved marked "+" in (b), with r = 20 (and a = 2), shows that this is large enough to be effectively in the r -+ 00 limit. Using equation (3) and the definitions (7), we can now write down the dynamical equations, replacing the number of updates n by the continuous variable t = n/ N in the limit N -+ 00: -"I (k-:nOh,E) OtRim OtU?s -"I (c~oh,E) OtQIz, -"I (h7 Oh" E) - "I (h~ Oh, E) + "12 a-y (Oh,Eoh" E) (11) a where the averages are over zero mean Gaussian variables, with covariances (9,10). Using the explicit form of the error E, we have oh,E = g'(h,) [L9(hl') - Lg(km I' + em) - eo] (12) m which, together with the equations (11) completes the description of the dynamics. The Gaussian averages in (11) can be straightforwardly evaluated in a manner similar to the infinite training set case[5], and we omit the rather cumbersome explicit form of the resulting equations. We note that, in contrast to the infinite training set case, the student activations hI and the noise variables C and are now correlated through equation (10). Intuitively, this is reasonable as the weights become correlated, during training, with the examples in the training set. In calculating the generalization error, on the other hand, such correlations are absent, and one has the same result as for infinite training sets. The dynamical equations (11), together with (9,10) constitute our main result. They are exact for the limits of either a linear network (R, Q, T -+ 0, so that g(x) ex: x) or a -+ 00, and can be integrated numerically in a straightforward way. In principle, the limit r -+ 00 should be taken but, as shown below, relatively small values of r can be taken in practice. s 3 es RESULTS AND DISCUSSION We now discuss the main consequences of our result (11), comparing the resulting predictions for the generalization dynamics, fg(t), to the infinite training set theory P. Sollich and D. Barber 362 k (a) 0.4 ..----------~--~----, 0.25 02 . 100000000000000000000000 1 ______________ 0.3 0.15 , 0.2 0.1 \ 0.05 ... ,'-- ~ --- --- 0.1 O~--~------~----~~~ o (b) 10 20 30 40 t 50 ~ooooooooooooooooooo o W 100 1W t 200 OL---~----------~----~ Figure 2: ?g VS t for two hidden units (L = M = 2). Left: a = 0.5, with a = 00 shown by dashed line for comparison; no noise. Right: a = 4, no noise (bottom) and noise on teacher activations and outputs of variance 0.1 (top). Simulations for N = 100 are shown by small circles; standard errors are less than the symbol size. Learning rate fJ = 2 throughout. and to simulations. Throughout, the teacher overlap matrix is set to (orthogonal teacher weight vectors of length V'ii). Tij = c5ij In figure(l), we study the accuracy of our method as a function of the training set size for a nonlinear network with one hidden unit at two different noise levels. The learning rate was set to fJ = 1 for both (a) and (b). For small activation and output noise (0'2 = 0.01), figure(la) , there is good agreement with the simulations for a down to a = 3, below which the theory begins to underestimate the generalization error, compared to simulations. Our finite a theory, however, is still considerably more accurate than the infinite a predictions. For larger noise (0'2 = 0.1, figure(lb?, our theory provides a reasonable quantitative estimate of the generalization dynamics for a > 3. Below this value there is significant disagreement, although the qualitative behaviour of the dynamics is predicted quite well, including the overfitting phenomenon beyond t ~ 10. The infinite a theory in this case is qualitatively incorrect. In the two hidden unit case, figure(2), our theory captures the initial evolution of ?g(t) very well, but diverges significantly from the simulations at larger t; nevertheless, it provides a considerable improvement on the infinite a theory. One reason for the discrepancy at large t is that the theory predicts that different student hidden units will always specialize to individual teacher hidden units for t --+ 00, whatever the value of a. This leads to a decay of ?g from a plateau value at intermediate times t. In the simulations, on the other hand, this specialization (or symmetry breaking) appears to be inhibited or at least delayed until very large t. This can happen even for zero noise and a 2:: L, where the training data should should contain enough information to force student and teacher weights to be equal asymptotically. The reason for this is not clear to us, and deserves further study. Our initial investigations, however, suggest that symmetry breaking may be strongly delayed due to the presence of saddle points in the training error surface with very 'shallow' unstable directions. When our theory fails, which of its assumptions are violated? It is conceivable that multiple local minima in the training error surface could cause self-averaging to break down; however, we have found no evidence for this, see figure(3a). On the other hand, the simulation results in figure(3b) clearly show that the implicit assumption of Gaussian student activations - as discussed before eq. (8) - can be violated. On-line Learning from Finite Training Sets in Nonlinear Networks (a) 363 (b) / Variance over training histories 10"'" ' - - - - - - - - - - - - - - - ' 102 N Figure 3: (a) Variance of fg(t = 20) vs input dimension N for student and teacher with two hidden units (L = M = 2), a = 0.5, 'fJ = 2, and zero noise. The bottom curve shows the variance due to different random choices of training examples from a fixed training set ('training history'); the top curve also includes the variance due to different training sets. Both are compatible with the liN decay expected if selfaveraging holds (dotted line). (b) Distribution (over training set) of the activation hI of the first hidden unit of the student. Histogram from simulations for N = 1000, all other parameter values as in (a). In summary, the main theoretical contribution of this paper is the extension of online learning analysis for finite training sets to nonlinear networks. Our approximate theory does not require the use of replicas and yields ordinary first order differential equations for the time evolution of a set of order parameters. Its central implicit assumption (and its Achilles' heel) is that the student activations are Gaussian distributed. In comparison with simulations, we have found that it is more accurate than the infinite training set analysis at predicting the generalization dynamics for finite training sets, both qualitatively and also quantitatively for small learning times t. Future work will have to show whether the theory can be extended to cope with non-Gaussian student activations without incurring the technical difficulties of dynamical replica theory [2], and whether this will help to capture the effects of local minima and, more generally, 'rough' training error surfaces. Acknowledgments: We would like to thank Ansgar West for helpful discussions. References [1] M. Biehl and H. Schwarze. Journal of Physics A, 28:643-656, 1995. [2] A. C. C. Coolen, S. N. Laughton, and D. Sherrington. In NIPS 8, pp. 253-259, MIT Press, 1996; S.N. Laughton, A.C.C. Coolen, and D. Sherrington. Journal of Physics A, 29:763-786, 1996. [3] See for example: The dynamics of online learning. Workshop at NIPS'95. [4] T. Heskes and B. Kappen. Physical Review A, 44:2718-2762, 1994. [5] D. Saad and S. A. Solla Physical Review E, 52:4225, 1995. [6] P. Sollich. Journal of Physics A, 27:7771-7784, 1994. [7] P. Sollich and D. Barber. In NIPS 9, pp.274-280, MIT Press, 1997; Europhysics Letters, 38:477-482, 1997. 

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Effects of Spike Timing Underlying Binocular Integration and Rivalry in a Neural Model of Early Visual Cortex Erik D. Lumer Wellcome department of Cognitive Neurology Institute of Neurology, University College of London 12 Queen Square, London, WC1N 3BG, UK Abstract In normal vision, the inputs from the two eyes are integrated into a single percept. When dissimilar images are presented to the two eyes, however, perceptual integration gives way to alternation between monocular inputs, a phenomenon called binocular rivalry. Although recent evidence indicates that binocular rivalry involves a modulation of neuronal responses in extrastriate cortex, the basic mechanisms responsible for differential processing of con:6.icting and congruent stimuli remain unclear. Using a neural network that models the mammalian early visual system, I demonstrate here that the desynchronized firing of cortical-like neurons that first receive inputs from the two eyes results in rivalrous activity patterns at later stages in the visual pathway. By contrast, synchronization of firing among these cells prevents such competition. The temporal coordination of cortical activity and its effects on neural competition emerge naturally from the network connectivity and from its dynamics. These results suggest that input-related differences in relative spike timing at an early stage of visual processing may give rise to the phenomena both of perceptual integration and rivalry in binocular vision. 1 Introduction The neural determinants of visual perception can be probed by subjecting the visual system to ambiguous viewing conditions - stimulus configurations that admit more E. D.Lumer 188 than one perceptual interpretation. For example, when a left-tilted grating is shown to the left eye and a right-tilted grating to the right eye, the two stimuli are momentarily perceived together as a plaid pattern, but soon only one line grating becomes visible, while the other is suppressed. This phenomenon, known as binocular rivalry, has long been thought to involve competition between monocular neurons within the primary visual cortex (VI), leading to the suppression of information from one eye (Lehky, 1988; Blake, 1989). It has recently been shown, however, that neurons whose activity covaries with perception during rivalry are found mainly in higher cortical areas and respond to inputs from both eyes, thus suggesting that rivalry arises instead through competition between alternative stimulus interpretations in extrastriate cortex (Leopold and Logothetis, 1996). Because eye-specific information appears to be lost at this stage, it remains unclear how the stimulus conditions (i.e. conflicting monocular stimuli) yielding binocular rivalry are distinguished from the conditions (i.e. matched monocular inputs) that produce stable single vision. I propose here that the degree of similarity between the images presented to the two eyes is registered by the temporal coordination of neuronal activity in VI, and that changes in relative spike timing within this area can instigate the differential responses in higher cortical areas to conflicting or congruent visual stimuli. Stimulus and eye-specific synchronous activity has been described previously both in the lateral geniculate nucleus (LGN) and in the striate cortex (Gray et al., 1989; Sillito et al., 1994; Neuenschwander and Singer, 1996). It has been suggested that such synchrony may serve to bind together spatially distributed neural events into coherent representations (Milner, 1974; von der Malsburg, 1981; Singer, 1993). In addition, reduced synchronization of striate cortical responses in strabismic cats has been correlated with their perceptual inability to combine signals from the two eyes or to incorporate signals from an amblyopic eye (Konig et al., 1993; Roelfsema et al., 1994). However, the specific influences of interocular input-similarity on spike coordination in the striate cortex, and of spike coordination on competition in other cortical areas, remain unclear. To examine these influences, a simplified neural model of an early visual pathway is simulated. In what follows, I first describe the anatomical and physiological constraints incorporated in the model, and then show that a temporal patterning of neuronal activity in its primary cortical area emerges naturally. By manipulating the relative spike timing of neuronal discharges in this area, I demonstrate its role in inducing differential responses in higher visual areas to conflicting or congruent visual stimulation. Finally, I discuss possible implications of these results for understanding the neural basis of normal and ambiguous perception in vivo. 2 Model overview The model has four stages based on the organization of the mammalian visual pathway (Gilbert, 1993). These stages represent: (i) sectors of an ipsilateral ('left eye') and a contralateral ('right eye') lamina of the LGN, which relay visual inputs to the cortex; (ii) two corresponding monocular regions in layer 4 of VI with different ocular dominance; (iii) a primary cortical sector in which the monocular inputs are first combined (called Vp in the model); and (iv) a secondary visual area of cortex in which higher-order features are extracted (Vs in the model; Fig. 1). Each stage consists of 'standard' integrate-and-fire neurons that are incorporated in synaptic networks. At the cortical stages, these units are grouped in local recurrent circuits that are similar to those used in previous modeling studies (Douglas et al., 1995; Somers et al., 1995). Synaptic interactions in these circuits are both excitatory and inhibitory between cells with similar orientation selectivity, but are restricted to in- Spike Timing Effects in Binocular Integration and Rivalry 189 (L.L ___ , _.. ___ , ( )__ ... ___ -.. , :9 : : Exp' . ~1 ; ; ElI2p' Iaver 4 : Ex1 t-.~...J . :?~...J : - - ~ In~~~; :.~~~ ; : Inhl..1j :,~~ J (L) Figure 1: Architecture of the model. Excitatory and inhibitory connections are represented by lines with arrowheads and round terminals , respectively, Each lamina in the LGN consists of 100 excitatory units (Ex) and 100 inhibitory units (Inh) , coupled via local inhibition. Cortical units are grouped into local recurrent circuits (stippled boxes) , each comprising 200 Ex units and 100 Inh units. In each monocular patch of layer 4, one cell group (Exl and Inh1) responds to left-tilted lines (orientation 1), whereas a second group (Ex2 and Inh2) is selective for right-tilted lines (orientation 2) . The same orientation selectivities are remapped onto Vp and Vs, although cells in these areas respond to inputs from both eyes. In addition, convergent inputs from Vp to Vs establish a third selectivity in Vs, namely for line crossings (Ex+ and Inh+) . hibition only between cell groups with orthogonal orientation preference (Kisvarday and Eysel, 1993) . Two orthogonal orientations (orientation 1 and 2) are mapped in each monocular sector of layer 4, and in Vp. To account for the emergence of more complex response properties at higher levels in the visual system (Van Essen and Gallant, 1994), forward connectivity patterns from Vp to Vs are organized to support three feature selectivities in Vs , one for orientation 1, one for orientation 2, and one for the conjunction of these two orientations, i.e. for line crossings. These forward projections are reciprocated by weaker backward projections from Vs to Vp. As a general rule , connections are established at random within and between interconnected populations of cells, with connection probabilities between pairs of cells ranging from 1 to 10 %, consistent with experimental estimates (Thomson et al., 1988; Mason et al., 1991) . Visual stimulation is achieved by applying a stochastic synaptic excitation independently to activated cells in the LGN . A quantitative description of the model parameters will be reported elsewhere. 190 3 E. D. Lumer Results In a first series of simulations, the responses of the model to conflicting and congruent visual stimuli are compared. When the left input consists of left-tilted lines (orientation 1) and the right input of right-tilted lines (orientation 2), rivalrous response suppression occurs in the secondary visual area. At any moment, only one of the three feature-selective cell groups in Vs can maintain elevated firing rates (Fig. 2a). By contrast, when congruent plaid patterns are used to stimulate the two monocular channels, these cell groups are forced in a regime in which they all sustain elevated firing rates (Fig. 2b). This concurrent activation of cells selective for orthogonal orientations and for line crossings can be interpreted as a distributed representation of the plaid pattern in Vs 1. A quantitative assessment of the degree of competition in Vs is shown in Figure 2c. The rivalry index of two groups of neurons is defined as the mean absolute value of the difference between their instantaneous group-averaged firing rates divided by the highest instantaneous firing rate among the two cell groups. This index varies between 0 for nonrivalrous groups of neurons and 1 for groups of neurons with mutually exclusive patterns of activity. Groups of cells with different selectivity in Vs have a significantly higher rivalry index when stimulated by conflicting rather than by congruent visual inputs (p < 0.0001) (Fig. 2c). Note that, in the example shown in Figure 2a, the differential responses to conflicting inputs develop from about 200 ms after stimulus onset and are maintained over the remainder of the stimulation epoch. In other simulations, alternation between dominant and suppressed responses was also observed over the same epoch as a result of fluctuations in the overall network dynamics. A detailed analysis of the dynamics of perceptual alternation during rivalry, however, is beyond the scope of this report. Although Vp exhibits a similar distribution of firing rates during rivalrous and nonrivalrous stimulation, synchronization between the two cell groups in Vp is more pronounced in the nonrivalrous than in the rivalrous case (Fig. 2d, upper plots). Subtraction of the shift predictor demonstrates that the units are not phase-locked to the stimuli. The changes in spike coordination among Vp units reflects the temporal patterning of their layer 4 inputs. During rivalry, Vp cells with different orientation selectivity are driven by layer 4 units that belong to separate monocular pathways, and hence, are uncorrelated (Fig. 2d, lower left). By contrast, cells in Vp receive convergent inputs from both eyes during nonrivalrous stimulation. Because of the synchronization of discharges among cells responsive to the same eye within layer 4 (Fig. 2d, lower right), the paired activities from the two monocular channels are also synchronized, a.nd provide synchronous inputs to cells with different orienta.tion selectivity in Vp. To establish unequivocally that changes in spike coordination within Vp are sufficient to trigger differential responses in Vs to conflicting and congruent stimuli, the model can be modified as follows. A single group of cells in layer 4 is used to drive with equal strength both orientation-selective populations of neurons in Vp. The outputs from layer 4, however, is relayed to these two target populations with average transmission delays that differ by either 10 ms or by 0 ms. In the first case, competition prevails among cells in the secondary visual area. This contrasts with the nonrivalrous activity in this area when similar transmission delays are used at an earlier stage (data not shown). This test confirms that changes in relative spike lTo discount possible effects of binocular snmmmation, synaptic strengths from layer 4 to Vp are reduced during congruent stimulation so as to produce a feedforward activation of Vp comparable to that elicited by conflicting monocular inputs. Spike Timing Effects in Binocular Integration and Rivalry conflicting visual inputs a 191 congruent visual inputs b 40~ .S , 30 _6 N 3 2 -::~ 4 N l: J: 40 2 3 C)2 -Vs+ C)2 2 .s ' I- 3 o 1 2 <;'--t " -" - : - " 3 4 L41 -R42 -ee- L42 R41 2 3 .. Time (ms) Time (ms) c -Yp1 -Yp2 -3..-------:4 ir. :[4 :u~ at-.. . t.~ " oC"_ ~' 'b'---~ 1 -~2- -Ys1 -Ys2 .. 4 C 'C d congruent conflicting fHdNkd 1m gAd~50 >< CI) "c ~ l a: 1 I- 11.I"'\Jo/"1IfV!1'Y I- u:: 'b ~eo~vp 1U G) 1& . CJ 0.1 o Vs1:Vs2 max(Vs}:Vs+ -80 0 80 -80 0 80 Time lag (ms) Figure 2: A, Instantaneous firing rates in response to conflicting inputs for cell groups in layer 4, in Vp, and in Vs (stimulus onset at t = 250ms). Discharge rates of layer 4 cells driven by different 'eyes' are similar (lower plot). By contrast, Vs exhi bi ts com peti ti ve firing pat terns soon after stimulus onset (upper plot). Feed back influence from Vs to Vp results in comparatively weaker competition in Vp (middle plot). B, Responses to congruent inputs. All cell groups in layer 4 are activated by visual inputs. Nonrivalrous firing patterns ensue in Vp and Vs. C, Rivalry indices during conflicting and congruent stimulation, are calculated for the two orientationselective cell groups and for the dominant and cross-selective cell group in Vs. D, Interocular responses are uncorrelated in layer 4 (lower left), whereas intraocular activities are synchronous at this stage (lower right). Enhanced synchronization of discharges ensues between cell groups in Vp during congruent stimulation (upper right), relative to the degree of coherence during conflicting stimulation (upper left). E. D.Lumer 192 timing are sufficient to switch the outcome of neural network interactions involving strong mutual inhibition from competitive to cooperative. 4 Conclusion In the present study, a simplified model of a visual pathway was used to gain insight into the neural mechanisms operating during binocular vision. Simulations of neuronal responses to visual inputs revealed a stimulus-related patterning of relative spike timing at an early stage of cortical processing. This patterning reflected the degree of similarity between the images presented to the two 'eyes', and, in turn, it altered the outcome of competitive interactions at later stages along the visual pathway. These effects can help explaining how the same cortical networks can exhibit both rivalrous and nonrivalrous activity, depending on the temporal coordination of their synaptic inputs. These results bear on the interpretation of recent empirical findings about the neuronal correlates of rivalrous perception. In experiments with awake monkeys, Logothetis and colleagues (Sheinberg et al., 1995; Leopold and Logothetis, 1996) have shown that neurons whose firing rate correlates with perception during rivalry are distributed at several levels along the primate visual pathways, including Vl/V2, V4, and IT. Importantly, the fraction of modulated responses is lower in VI than in extrastriate areas, and it increases with the level in the visual hierarchy. Simulations of the present model exhibit a behavior that is consistent with these observations. However, these simulations also predict that both rivalrous and nonrivalrous perception may have a clear neurophysiological correlate in VI, i.e. at the earliest stage of visual cortical processing. Accordingly, congruent stimulation of both eyes will synchronize the firing of binocular cells with overlapping receptive fields in Vl. By contrast, conflicting inputs to the two eyes will cause a desynchronization between their corresponding neural events in Vl. Because this temporal registration of stimulus dissimilarity instigates competition among binocular cells in higher visual areas and not between monocular pathways, the ensuing pattern of response suppression and dominance is independent of the eyes through which the stimuli are presented. Thus, the model can in principle account for the psychophysical finding that a single phase of perceptual dominance during rivalry can span multiple interocular exchanges of the rival stimuli (Logothetis et al., 1996). The present results also reveal a novel property of canonical cortical-like circuits interacting through mutual inhibition, i.e. the degree of competition among such circuits exhibits a remarkable sensitivity to the relative timing of neuronal action potentials. This suggests that the temporal patterning of cortical activity may be a fundamental mechanism for selecting among stimuli competing for the control of attention and motor action. Acknowledgements This work was supported in part by an IRSIA visiting fellowship at the Center for Nonlinear Phenomena and Complex Systems, Universite Libre de Bruxelles. I thank Professor Gregoire Nicolis for his hospitality during my stay in Brussels; and David Leopold and Daniele Piomelli for helpful discussions and comments on an earlier version of the manuscript. References Blake R (1989) A neural theory of binocular vision. Psychol Rev 96:145-167. Spike Timing Effects in Binocular Integration andRivalry 193 Douglas RJ, Koch C, Mahowald M, Martin K, Suarez H (1995) Recurrent excitation in neocortical circuits. Science 269:981-985. Gilbert C (1993) Circuitry, architecture, and functional dynamics of visual cortex. Cereb Cortex 3:373-386. Gray CM, Konig P, Engel AK, Singer, W (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338 :334-337. Kisvarday ZF, Eysel UT (1993) Functional and structural topography of horizontal inhibitory connections in cat visual cortex. Europ J Neurosci 5:1558-1572. Konig P, Engel AK, Lowel S, Singer, W (1993) Squint affects synchronization of oscillatory responses in cat visual cortex. Eur J Neurosci 5:501-508. Lehky SR (1988) An astable multivibrator model of binocular rivalry. Perception 17: 215- 228. Leopold DA, Logothetis NK (1996) Activity changes in early visual cortex reflect monkeys percepts during binocular rivalry. Nature 379:549-553. Logothetis NK, Leopold DA, Sheinberg DL (1996) What is rivalling during rivalry? Nature 380:621-624. Neuenschwander S, Singer W (1996) Long-range synchronization of oscillatory light responses in the cat retina and lateral geniculate nucleus. Nature 379:728-733. Milner PM (1974) A model of visual shape recognition. Psychol Rev 81:521-535. Roelfsema PR, Konig P, Engel AK, Sireteanu R , Singer W (1994) Reduced synchronization in the visual cortex of cats with strabismic amblyopia. Eur J Neurosci 6:1645-1655. Sheinberg DL, Leopold DA, Logothetis NK (1995) Effects of binocular rivalry on face cell activity in monkey temporal cortex. Soc Neurosci Abstr 21:15.12. Sillito AM, Jones HE, Gerstein GL, West DC (1994) Feature-linked synchronization of thalamic relay cell firing induced by feedback from the visual cortex. Nature 369:479-482. Singer W (1993) Synchronization of cortical activity and its putative role in information processing. Annu Rev Physiol 55:349-374. Somers D, Nelson S, Sur M (1995) An emergent model of orientation selectivity in cat visual cortical simple cells. J Neurosci 15:5448-5465. Van Essen DC, Gallant JL (1994) Neural mechanisms ofform and motion processing in the primate visual system . Neuron 13:1-10. von der Malsburg C (1981) The correlation theory of the brain. Internal Report 81-2, Max Planck Institute for Biophysical Chemistry, Gottingen. 

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Approximating Posterior Distributions in Belief Networks using Mixtures Christopher M. Bishop Neil Lawrence Neural Computing Research Group Dept. Computer Science & Applied Mathematics Aston University Binningham, B4 7ET, U.K. Tommi Jaakkola Michael I. Jordan Center for Biological and Computational Learning Massachusetts Institute of Technology 79 Amherst Street, ElO-243 Cambridge, MA 02139, U.S.A. Abstract Exact inference in densely connected Bayesian networks is computationally intractable, and so there is considerable interest in developing effective approximation schemes. One approach which has been adopted is to bound the log likelihood using a mean-field approximating distribution. While this leads to a tractable algorithm, the mean field distribution is assumed to be factorial and hence unimodal. In this paper we demonstrate the feasibility of using a richer class of approximating distributions based on mixtures of mean field distributions. We derive an efficient algorithm for updating the mixture parameters and apply it to the problem of learning in sigmoid belief networks. Our results demonstrate a systematic improvement over simple mean field theory as the number of mixture components is increased. 1 Introduction Bayesian belief networks can be regarded as a fully probabilistic interpretation of feedforward neural networks. Maximum likelihood learning for Bayesian networks requires the evaluation of the likelihood function P(VIO) where V denotes the set of instantiated (visible) variables, and 0 represents the set of parameters (weights and biases) in the network. Evaluation of P(VIO) requires summing over exponentially many configurations of Approximating Posterior Distributions in Belief Networks Using Mixtures 417 the hidden variables H, and is computationally intractable except for networks with very sparse connectivity, such as trees. One approach is to consider a rigorous lower bound on the log likelihood, which is chosen to be computationally tractable, and to optimize the model parameters so as to maximize this bound instead. If we introduce a distribution Q(H), which we regard as an approximation to the true posterior distribution, then it is easily seen that the log likelihood is bounded below by F[Q] = L Q(H) In P(V~H). Q( {H} (1) ) The difference between the true log likelihood and the bound given by (1) is equal to the Kullback-Leibler divergence between the true posterior distribution P(HIV) and the approximation Q(H). Thus the correct log likelihood is reached when Q(H) exactly equals the true posterior. The aim of this approach is therefore to choose an approximating distribution which leads to computationally tractable algorithms and yet which is also flexible so as to permit a good representation of the true posterior. In practice it is convenient to consider parametrized distributions, and then to adapt the parameters to maximize the bound. This gives the best approximating distribution within the particular parametric family. 1.1 Mean Field Theory Considerable simplification results if the model distribution is chosen to be factorial over the individual variables, so that Q(H) = Q(hd, which gives meanfieid theory. Saul et al. (1996) have applied mean field theory to the problem of learning in sigmoid belief networks (Neal. 1992). These are Bayesian belief networks with binary variables in which the probability of a particular variable Si being on is given by ni P(S, = Ilpa(S,? =" ( ~ ],;S; + b) (2) where u(z) == (1 + e-Z)-l is the logistic sigmoid function, pa(Si) denote the parents of Si in the network. and Jij and bi represent the adaptive parameters (weights and biases) in the model. Here we briefly review the framework of Saul et ai. (1996) since this forms the basis for the illustration of mixture modelling discussed in Section 3. The mean field distribution is chosen to be a product of Bernoulli distributions of the form Q(H) = II p,~i (1 _ p,;)l-h i (3) in which we have introduced mean-field parameters J.Li. Although this leads to considerable simplification of the lower bound, the expectation over the log of the sigmoid function. arising from the use of the conditional distribution (2) in the lower bound (I), remains intractable. This can be resolved by using variational methods (Jaakkola, 1997) to find a lower bound on F(Q), which is therefore itself a lower bound on the true log likelihood. In particular, Saul et al. (1996) make use of the following inequality (In[l + e Zi ]) ::; ei(Zi) + In(e-~iZi + e(1-~;)Zi) (4) where Zi is the argument of the sigmoid function in (2), and ( ) denotes the expectation with respect to the mean field distribution. Again, the quality of the bound can be improved by adjusting the variational parameter ei. Finally, the derivatives of the lower bound with respect to the J ij and bi can be evaluated for use in learning. In summary. the algorithm involves presenting training patterns to the network. and for each pattern adapting the P,i and to give the best approximation to the true posterior within the class of separable distributions of the form (3). The gradients of the log likelihood bound with respect to the model parameters Jij and bi can then be evaluated for this pattern and used to adapt the parameters by taking a step in the gradient direction. ei 418 2 C. M. Bishop, N. LAwrence, T. Jaakkola and M I. Jordan Mixtures Although mean field theory leads to a tractable algorithm, the assumption of a completely factorized distribution is a very strong one. In particular, such representations can only effectively model posterior distributions which are uni-modal. Since we expect multi-modal distributions to be common, we therefore seek a richer class of approximating distributions which nevertheless remain computationally tractable. One approach (Saul and Jordan, 1996) is to identify a tractable substructure within the model (for example a chain) and then to use mean field techniques to approximate the remaining interactions. This can be effective where the additional interactions are weak or are few in number, but will again prove to be restrictive for more general, densely connected networks. We therefore consider an alternative approach I based on mixture representations of the form M Qmix(H) = L (5) amQ(Hlm) m=l in which each of the components Q(Hlm) is itself given by a mean-field distribution, for example of the form (3) in the case of sigmoid belief networks. Substituting (5) into the lower bound (1) we obtain F[Qmix] = L amF[Q(Hlm)] + f(m, H) (6) m where f(m, H) is the mutual information between the component label m and the set of hidden variables H, and is given by f(m,H) = L L m {H} Q(Hlm) amQ(Hlm) In Q . (H)' (7) mix The first tenn in (6) is simply a convex combination of standard mean-field bounds and hence is no greater than the largest of these and so gives no useful improvement over a single mean-field distribution. It is the second term, i.e. the mutual infonnation, which characterises the gain in using mixtures. Since f(m, H) ~ 0, the mutual information increases the value of the bound and hence improves the approximation to the true posterior. 2.1 Smoothing Distributions As it stands, the mutual infonnation itself involves a summation over the configurations of hidden variables, and so is computationally intractable. In order to be able to treat it efficiently we first introduce a set of 'smoothing' distributions R(Hlm), and rewrite the mutual infonnation (7) in the form f(m, H) - LLamQ(Hlm)lnR(Hlm) - LamInam {H} m - L m m L amQ(Hlm) In {R(H1m) Qmix(H) } . am Q(Hlm) (8) {H} It is easily verified that (8) is equivalent to (7) for arbitrary R(Hlm). We next make use of the following inequality (9) - In x ~ - ~x + In ~ + 1 lHere we outline the key steps. A more detailed discussion can be found in Jaakkola and Jordan (1997). Approximating Posterior Distributions in Belief Networks Using Mixtures 419 to replace the logarithm in the third term in (8) with a linear function (conditionally on the component label m). This yields a lower bound on the mutual information given by J(m,H) ~ J),(m,H) where h(m,H) I:I:amQ(Hlm)lnR(Hlm)- I: am In am m {H} m - I: Am I: R(Hlm)Qmix(H) m {H} + I: am InAm + 1. (10) m With J),(m, H) substituted for J(m, H) in (6) we again obtain a rigorous lower bound on the true log likelihood given by F),[Qmix(H)] = I: amF[Q(Hlm)] + h(m, H). (11) m The summations over hidden configurations {H} in (10) can be performed analytically if we assume that the smoothing distributions R(Hlm) factorize. In particular, we have to consider the following two summations over hidden variable configurations II I: R(hilm)Q(hilk) ~ 7rR,Q(m, k) I: R(Hlm)Q(Hlk) {H} i I: I: Q(hilm) InR(hilm) ~f H(QIIRlm). I: Q(Hlm) InR(Hlm) {H} (12) h. (13) h. We note that the left hand sides of (12) and (13) represent sums over exponentially many hidden configurations, while on the right hand sides these have been re-expressed in terms of expressions requiring only polynomial time to evaluate by making use of the factorization of R(Hlm). It should be stressed that the introduction of a factorized form for the smoothing distributions still yields an improvement over standard mean field theory. To see this, we note that if R(Hlm) = const. for all {H, m} then J(m, H) = 0, and so optimization over R(Hlm) can only improve the bound. 2.2 Optimizing the Mixture Distribution In order to obtain the tightest bound within the class of approximating distributions, we can maximize the bound with respect to the component mean-field distributions Q(Hlm), the mixing coefficients am, the smoothing distributions R(Hlm) and the variational parameters Am' and we consider each of these in turn. We will assume that the choice of a single mean field distribution leads to a tractable lower bound, so that the equations 8F[Q] 8Q(h j ) = const (14) can be solved efficiently2. Since h(m, H) in (10) is linear in the marginals Q(hjlm), it follows that its derivative with respect to Q(hj 1m) is independent of Q(hjlm), although it will be a function of the other marginals, and so the optimization of (11) with respect to individual marginals again takes the form (14) and by assumption is therefore soluble. Next we consider the optimization with respect to the mixing coefficients am. Since all of the terms in (11) are linear in am, except for the entropy term, we can write F),[Qmix(H)] = I:am(-Em) - I:amlnam + 1 m (15) m 2In standard mean field theory the constant would be zero, but for many models of interest the slightly more general equations given by (14) will again be soluble. C. M. Bishop, N. Lawrence, T. Jaakkola and M. L Jordan 420 where we have used (10) and defined F[Q(Hlm)] +L Q(Hlm) InR(Hlm) {H} + LAk LR(Hlk)Q(Hlm) +lnA m . k (16) {H} Maximizing (15) with respect to am, subject to the constraints 0 ~ am ~ 1 and Lm am = 1, we see that the mixing coefficients which maximize the lower bound are given by the Boltzmann distribution exp(-Em) am = Lk exp(-Ek )' (17) We next maximize the bound (11) with respect to the smoothing marginals R(h j 1m). Some manipulation leads to the solution R(hilm) = amQA~ilm) [~>'~'Q(m'k)Q(hi'k)l-1 (18) in which 7r~,Q(m, k) denotes the expression defined in (12) but with the j term omitted from the product. The optimization of the JLmj takes the form of a re-estimation formula given by an extension of the result obtained for mean-field theory by Saul et al. (1996). For simplicity we omit the details here. Finally, we optimize the bound with respect to the Am, to give 1 1 m am ~= - L 7rR,Q(m, k). (19) k Since the various parameters are coupled, and we have optimized them individually keeping the remainder constant, it will be necessary to maximize the lower bound iteratively until some convergence criterion is satisfied. Having done this for a particular instantiation of the visible nodes, we can then determine the gradients of the bound with respect to the parameters governing the original belief network, and use these gradients for learning. 3 Application to Sigmoid Belief Networks We illustrate the mixtures formalism by considering its application to sigmoid belief networks of the form (2). The components of the mixture distribution are given by factorized Bernoulli distributions of the form (3) with parameters JLmi. Again we have to introduce variational parameters ~mi for each component using (4). The parameters {JLmi, ~mi} are optimized along with {am, R(hjlm), Am} for each pattern in the training set. We first investigate the extent to which the use of a mixture distribution yields an improvement in the lower bound on the log likelihood compared with standard mean field theory. To do this, we follow Saul et al. (1996) and consider layered networks having 2 units in the first layer, 4 units in the second layer and 6 units in the third layer, with full connectivity between layers. In all cases the six final-layer units are considered to be visible and have their states clamped at zero. We generate 5000 networks with parameters {Jij, bi } chosen randomly with uniform distribution over (-1, 1). The number of hidden variable configurations is 26 = 64 and is sufficiently small that the true log likelihood can be computed directly by summation over the hidden states. We can therefore compare the value of 421 Approximating Posterior Distributions in Belief Networks Using Mixtures the lower bound F with the true log likelihood L, using the nonnalized error (L - F)/ L. Figure 1 shows histograms of the relative log likelihood error for various numbers of mixture components, together with the mean values taken from the histograms. These show a systematic improvement in the quality of the approximation as the number of mixture components is increased. 5 components, mean: 0 .011394 4 components, mean: 0.012024 3~r-----~----~----~---. 3000r-----~----~----~--_. 0 .02 0.04 0.06 0.08 3 components, mean : 0.01288 0.02 0.04 0.06 0.08 2 components, mean: 0.013979 3000r-----~----~----~--_. 2000 0.04 0.02 0.04 0.06 0.08 1 component, mean: 0.015731 g0.014 o III o c as ~ 0.012 0.06 0.01 0.08 0.08 0.01 Gn..:-p----~----~----~----..., ~r-----~----~----~--~ 0.04 0.06 o '------~----~----~-----' 1 2 3 4 5 no. of components Figure 1: Plots of histograms of the normalized error between the true log likelihood and the lower bound. for various numbers of mixture components. Also shown is the mean values taken from the histograms. plotted against the number of components. Next we consider the impact of using mixture distributions on learning. To explore this we use a small-scale problem introduced by Hinton et al. (1995) involving binary images of size 4 x 4 in which each image contains either horizontal or vertical bars with equal probability, with each of the four possible locations for a bar occupied with probability 0.5. We trained networks having architecture 1-8-16 using distributions having between 1 and 5 components. Randomly generated patterns were presented to the network for a total of 500 presentations, and the J-tmi and ~mi were initialised from a unifonn distribution over (0,1). Again the networks are sufficiently small that the exact log likelihood for the trained models can be evaluated directly. A Hinton diagram of the hidden-to-output weights for the eight units in a network trained with 5 mixture components is shown in Figure 2. Figure 3 shows a plot of the true log likelihood versus the number M of components in the mixture for a set of experiments in which, for each value of M, the model was trained 10 times starting from different random parameter initializations. These results indicate that, as the number of mixture components is increased, the learning algorithm is able to find a set of network parameters having a larger likelihood, and hence that the improved flexibility of the approximating distribution is indeed translated into an improved training algorithm. We are currently applying the mixture fonnalism to the large-scale problem of hand-written digit classification. 422 C. M. Bishop, N. Lawrence, T. Jaakkola and M. I. Jordan .? 0. t? ? ? ? ???? ? ?? o ? ? o j? ? ? - 0 0.0 jO 0 . 0 10 0 . 0 ??? 10 ??. 0 ?? o ? 0 -0 Figure 2: Hinton diagrams of the hidden-to-output weights for each of the 8 hidden units in a network trained on the 'bars' problem using a mixture distribution having 5 components. -5 -6 o -7 8 0 '80 :? -8 Q) ~ CI .S! -9 Q) o o ~ 8o o o o 6 o B 8 8 o o o o o 0 0 0 o E -10 -11 -12 0 234 5 6 no. of components Figure 3: True log likelihood (divided by the number of patterns) versus the number M of mixture components for the 'bars' problem indicating a systematic improvement in performance as M is increased. References Hinton, G. E., P. Dayan, B. 1. Frey, and R. M. Neal (1995). The wake-sleep algorithm for unsupervised neural networks. Science 268, 1158-1161. Jaakkola, T. (1997). Variational Methods for Inference and Estimation in Graphical Models. Ph.D. thesis, MIT. Jaakkola, T. and M. I. Jordan (1997). Approximating posteriors via mixture models. To appear in Proceedings NATO ASI Learning in Graphical Models, Ed. M. I. Jordan. Kluwer. Neal, R. (1992) . Connectionist learning of belief networks. Artificial Intelligence 56, 71-113. Saul, L. K., T. Jaakkola, and M. I. Jordan (1996). Mean field theory for sigmoid belief networks. Journal of Artificial Intelligence Research 4,61-76. Saul, L. K. and M. I. Jordan (1996). Exploiting tractable substructures in intractable networks. In D. S. Touretzky, M . C. Mozer, and M. E. Hasselmo (Eds.), Advances in Neural Information Processing Systems, Volume 8, pp. 486-492. MIT Press. 

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Comparison of Human and Machine Word Recognition M. Schenkel Dept of Electrical Eng. University of Sydney Sydney, NSW 2006, Australia schenkel@sedal.usyd.edu.au C. Latimer Dept of Psychology University of Sydney Sydney, NSW 2006, AustTalia M. Jabri Dept of Electrical Eng. University of Sydney Sydney, NSW 2006, Australia marwan@sedal. usyd.edu.au Abstract We present a study which is concerned with word recognition rates for heavily degraded documents. We compare human with machine reading capabilities in a series of experiments, which explores the interaction of word/non-word recognition, word frequency and legality of non-words with degradation level. We also study the influence of character segmentation, and compare human performance with that of our artificial neural network model for reading. We found that the proposed computer model uses word context as efficiently as humans, but performs slightly worse on the pure character recognition task. 1 Introduction Optical Character Recognition (OCR) of machine-print document images ?has matured considerably during the last decade. Recognition rates as high as 99.5% have been reported on good quality documents. However, for lower image resolutions (200 Dpl and below), noisy images, images with blur or skew, the recognition rate declines considerably. In bad quality documents, character segmentation is as big a problem as the actual character recognition. fu many cases, characters tend either to merge with neighbouring characters (dark documents) or to break into several pieces (light documents) or both. We have developed a reading system based on a combination of neural networks and hidden Markov models (HMM), specifically for low resolution and degraded documents. To assess the limits of the system and to see where possible improvements are still to be Comparison of Human and Machine Word Recognition 95 expected, an obvious comparison is between its performance and that of the best reading system known, the human reader. It has been argued, that humans use an extremely wide range of context information, such as current topics, syntax and semantic analysis in addition to simple lexical knowledge during reading. Such higher level context is very hard to model and we decided to run a. first comparison on a word recognition task, excluding any context beyond word knowledge. The main questions asked for this study are: how does human performance compare with our system when it comes to pure character recognition (no context at all) of bad quality documents? How do they compare when word context can be used? Does character segmentation information help in reading? 2 Data Preparation We created as stimuli 36 data sets, each containing 144 character strings, 72 words and 72 non-words, all lower case. The data sets were generated from 6 original sets, each COI!taining 144 unique wordsjnon-words. For each original set we used three ways to divide the words into the different degradation levels such that each word appears once in each degradation level. We also had two ways to pick segmented/non-segmented so that each word is presented once segmented and once non-segmented. This counterbalancing creates the 36 sets out of the six original ones. The order of presentation within a test set was randomized with respect to degradation, segmentation and lexical status. All character strings were printed in 'times roman 10 pt' font. Degradation was achieved by photocopying and faxing the printed docJiment before scanning it at 200Dpl. Care was taken to randomize the print position of the words such that as few systematic degradation differences as possible were introduced. Words were picked from a dictionary of the 44,000 most frequent words in the 'Sydney Morning Herald'. The length of the words was restricted to be between 5 and 9 characters. They were divided in a 3x3x2 mixed factorial model containing 3 word-frequency groups, 3 stimulus degradation levels and visually segmented/non-segmented words. The three word-frequency groups were: 1 to 10 occurences/million (o/m) as low frequency, 11 to 40 ojm as medium frequency and 41 or more ojm as high frequency. Each participant was presented with four examples per stimulus class (e.g. four high frequency words in medium degradation level, not segmented). The non-words conformed to a 2x3x2 model containing legal/illegal non-words, 3 stimulus degradation levels and visually segmented/non-segmented strings. The illegal non-words (e.g. 'ptvca') were generated by randomly selecting a word length between 5 and 9 characters (using the same word length frequencies as the dictionary has) and then randomly picking characters (using the same character frequencies as the dictionary has) and keeping the unpronouncable sequences. The legal non-words (e.g. 'slunk') were generated by using trigrams (using the dictionary to compute the trigram probabilities) and keeping pronouncable sequences. Six examples per non-word stimulus class were used in each test set. (e.g. six illegal non-words in high degradaton level, segmented). 3 Human Reading There were 36 participants in the study. Participants were students and staff of the University of Sydney, recruited by advertisement and paid for their service. They were all native English speakers, aged between 19 and 52 with no reported uncorrected visual deficits. The participants viewed the images, one at a time, on a computer monitor and were asked to type in the character string they thought would best fit the image. They had been 96 M. Schenkel, C. Latimer and M. Jabri instructed that half of the character strings were English words and half non-words, and they were informed about the degradation levels and the segmentation hints. Participants were asked to be as fast and as accurate as possible. After an initial training session of 30 randomly picked character strings not from an independent training set, the participants had a short break and were then presented with the test set, one string at a time. After a Carriage Return was typed, time was recorded and the next word was displayed. Training and testing took about one hour. The words were about 1-1.5cm large on the screen and viewed at a distance of 60cm, which corresponds to a viewing angle of 1?. 4 Machine Reading For the machine reading tests, we used our integrated segmentation/recognition system, using a sliding window technique with a combination of a neural network and an HMM [6). In the following we describe the basic workings without going into too much detail on the specific algorithms. For more detailed description see (6]. A sliding window approach to word recognition performs no segmentation on the input data of the recognizer. It consists basically of sweeping a window over the input word in small steps. At each step the window is taken to be a tentative character and corresponding character class scores are produced. Segmentation and recognition decisions are then made on the basis of the sequence of character scores produced, possibly taking contextual information into account. In the preprocessing stage we normalize the word to a fixed height. The result is a grey-normalized pixel map of the word. This pixel map is the input to a neural network which estimates a posteriori probabilities of occurrence for each character given the input in the sliding window whose length corresponds approximately to two characters. We use a space displacement neural network (SDNN) which is a multi-layer feed-forward network with local connections and shared weights, the layers of which perform successively higherlevel feature extraction. SDNN's are derived from Time Delay Neural Networks which have been successfully used in speech recognition (2] and handwriting recognition (4, 1]. Thanks to its convolutional structure the computational complexity of the sliding window approach is kept tractable. Only about one eighth of the network connections are reevaluated for each new input window. The outputs of the SDNN are processed by an HMM. In our case the HMM implements character duration models. It tries to align the best scores of the SDNN with the corresponding expected character durations. The Viterbi algorithm is used for this alignment, determining simultaneously the segmentation and the recognition of the word. Finding this state sequence is equivalent to finding the most probable path through the graph which represents the HMM. Normally additive costs are used instead of multiplicative probabilities. The HMM then selects the word causing the smallest costs. Our best architecture contains 4 convolutional layers with a total of 50,000 parameters (6]. The training set consisted of a subset of 180,000 characters from the SEDAL database, a low resuloution degraded document database which was collected earlier and is independent of any data used in this experiment. 4.1 The Dictionary. Model A natural way of including a dictionary in this process, is to restrict the solution space of the HMM to words given by the dictionary. Unfortunately this means calculating the cost for each word in the dictionary, which becomes prohibitively slow with increasing dictionary size (we use a combination of available dictionaries, with a total size of 98,000 words). We thus chose a two step process for the dictionary search: in a first step a list of the most probable words is generated, using a fast-matcher technique. In the second step the HMM costs are calculated for the words in the proposed list. Comparison of Human and Machine Word Recognition 97 To generate the word list, we take the character string as found by the HMM without the dictionary and calculate the edit-distance between that string and all the words in the dictionary. The edit-distance measues how many edit operations (insertion, deletion and substitution) are necessary to convert a given input string into a target word [3, 5]. We now select all dictionary words that have the smallest edit-distance to the string recognized without using the dictionary. The composed word list contains on average 10 words, and its length varies considerably depending on the quality of the initial string. For all words in the word list the HMM cost is now calculated and the word with the smallest cost is the proposed dictionary word. As the calculation of the edit-distance is much faster than the calculation of the HMM costs, the recognition speed is increased substantially. In a last step the difference in cost between the proposed dictionary word and the initial string is calculated. If this difference is smaller than a threshold, the system will return the dictionary word, otherwise the original string is returned. This allows for the recognition of non-dictionary words. The value for the threshold determines the amount of reliance on the dictionary. A high value will correct most words but will also force non-words to be recognized as words. A low value, on the other hand, leaves the non-words unchanged but doesn't help for words either. Thus the value of the threshold influences the difference between word and non-word recognition. We chose the value such that the over-all error rate is optimized. 4.2 The Case of Segmented data When character segmentation is given, we know how many characters we have and where to look for them. There is no need for an HMM and we just sum up the character probabilities over the x-coordinate in the region corresponding to a segment. This leaves a vector of 26 scores {the whole alphabet) for each character in the input string. With no dictionary constraints, we simply pick the label corresponding to the highest probability for each character. The dictionary is used in the same way, replacing the HMM scores by calculating the word scores directly from the corresponding character probabilities. Results 5 Recognition Performance Machine Reading Human Reading 0.6 0.6 -- ... X Non-Segmental 0 S.pncnled -- - ~0.5 ~::;0.4 1:: ,..- -------lC' r?3 IX'! :::::=---Non-Words ao.2 0.1 0.1 Words o~------------~2------------~ Degradation Figure 1: Human Reading Performance. Words 0?~------------~2------------~3~ Degradation Figure 2: Machine Reading Performance. M. Schenkel, C. Latimer and M. Jabri 98 Figure 1 depicts the recognition results for human readers. All results are per character error rates counted by the edit-distance. All results reported as significant pass an F -test with p < .01. As expected there was a significant interaction between error rate and degradation and clearly non-words have higher error rates than words. Also character segmentation has also an influence on the error rate. Segmentation seems to help slightly more for higher degradations. Figure 2 shows performance of the machine algorithm. Again greater degradation leads to higher error rates and non-words have higher error rates than words. Segmentation hints lead to significantly better recognition for all degradation levels; in fact there is no interaction between degradation and segmentation for the machine algorithm. In general the machine benefited more from segmentation than humans. One would expect a smaller gain from lexical knowledge for higher en;or rates (i.e. higher degradation) as in the limit of complete degradation all error rates will be 100%. Both humans and machine show this 'closing of the gap . Segmented Recognition 0.6 Non-Segmented Recognition 0.6 -Human -Human - - - ? Mac;binc 0.1 Words o~------------~2------------~3~ Degradation Figure 3: Segmented Data. o~------------~2------------~3~ Degradation Figure 4: Non-Segmented Data. More interesting is the direct comparison between the error rates for humans and machine as shown in figure 3 and figure 4. The difference for non-words reflects the difference in ability to recognize the geometrical shape of characters without context. For degradation levels 1 and 2, the machine has the same reading abilities as humans for segmented data and looses only about 7% in the non-'segmented case. For degradation level 3; the machine clearly performs worse than human readers. The difference between word and non-word error rates reflects the ability of the participant to use lexical knowledge. Note that the task contains word/non-word discrimination as well as recognition. It is striking how similar the behaviour for humans and machine is for degradation levels 1 and 2. Timing Results Figure 5 shows the word entry times for humans. As the main goal was to compare recognition rates, we did not emphasize entry speed when instructing the participants. However, we recorded the word entry time for each word (which includes inspection time and typing). When analysing the timimg data the only interest was in relative difference between word groups. Times were therefore? converted for each participant into a z-score (zero mean with a standard deviation of one) and statistics were made over the z-scores of all participants. Non-words generally took longer to recognize than words and segmented data took longer Comparison of Human and Machine Woni Recognition o.s,..-------,---------.--, ?---- ---- -------====---------------x -? Non-Segmented 0.5,..-------,.--------~ Humau Reading Times ___ . -------0 99 -Human ;..., 0.3 ---- ~ -------- ------=~~~= .!:!. 0 I ~ -- . b Jj -{).I "E ~ -{)3 -{).S'-;-------2:-------~3:-' -{).S'-:-1------2:-------~3--l Degradation Desradation Figure 5: Human Reading Times. Figure 6: Times. Non-Segmented Reading to recognize than non-segmented for humans which we believe stems from participants not being used to reading segmented data. When asked, participants reported difficulties in using the segmentation lines. Interestingly this segmentation effect is significant only for words but not for non-words. As predicted there is also an interaction between time and degradation. Greater degradations take longer to recognize. Again, the degradation effect for time is only significant for words but barely for non-words. Our machine reading algorithm behaves differently in segmented and non-segmented mode with respect to time consumption. In segmented mode, the time for evaluating the word list in our system is very short compared to the base recognition time, as there is no HMM involved. Accordingly we found no or very little effects on timing for our system for segmented data. All the timing information for the machine refer to the non-segmented case (see Figure 6). Frequency and Legality Table 5 shows word frequencies, legality of non-words and entry-time. Our experiment confirmed the well known frequency and legality effect for humans in recognition rate as well as time and respectively for frequency. The only exception is that there is no difference in error rate for middle and low frequency words. The machine shows (understandably) no frequency effect in error rate or time, as all lexical words had the same prior probability. Interestingly even when using the correct prior probabilities we could not produce a strong word frequency? effect for the machine. Also no legality effect was observed for the error rate. One way to incorporate legality effects would be the use of Markov chains such as n-grams. Note however, how the recognition time for non-words is higher than for words and the legality effect for the recognition time. Recognition times for our system in non-segmented mode depend mainly on the time it takes to evaluate the word list. Non-words generally produce longer word lists than words, because there are no good quality matches for a non-word in the dictionary (on average a word list length of 8.6 words was found for words and of 14.5 for non-words). Also illegal non-words produce longer word lists than legal ones, again because the match quality for illegal non-words is worse than for legal ones (average length for illegal non-words 15.9 and for legal non-words 13.2). The z-scores for the word list length parallel nicely the recognition time scores. In segmented mode, the time for evaluating the word list is very short compared to the M. Schenkel, C. Latimer and M. Jabri 100 base recognition time, as there is no HMM involved. Accordingly we found no or very little effects on timing for our system in the segmented case. Error l%J Words 41+ Words 11-40 Words 1-10 Legal Non-W. lllegal Non-W. Humans Error z-Time 0.22 -0.37 -0.13 0.27 0.26 -0.06 0.36 0.07 0.46 0.31 Machine Error z-Time 0.36 -0.14 0.34 -0.19 -0.22 0.33 0.47 0.09 0.49 0.28 Table 1: Human and Machine Error rates for the different word and non-word classes. The z-times for the machine are for the non-segmented data only. 6 Discussion .The ability to recognize the geometrical shape of characters without the possibility to use any sort of context information is reflected in the error rate of illegal non-words. The difference between the error rate for illegal non-words and the one for words reflects the ability to use lexical knowledge. To our surprise the behavior of humans and machine is very similar for both tasks, indicating a near to optimal machine recognition system. Clearly this does not mean our system is a good model for human reading. Many effects such as semantic and repetition priming are not reproduced and call for a system which is able to build semantic classes and memorize the stimuli presented. Nevertheless, we believe that our experiment validates empirically the verification model we implemented, using real world data. Acknowledgments This research is supported by a grant from the Australian Research Council (grant No A49530190). References [1] I. Guyon, P. Albrecht, Y. Le Cun, J. Denker, and W. Hubbard. Design of a neural network character recognizer for a touch terminal. Pattern Recognition, 24(2):105-119, 1991. [2] K. J. Lang and G. E. Hinton. A Time Delay Neural Network architecture for speech recognition. Technical Report CMU-cs-88-152, Carnegie-Mellon University, Pittsburgh PA, 1988. [3] V.I. Levenshtein. Binary codes capable of correcting deletions, insertions and reversals. Soviet Physics-Doklady, 10(8):707-710, 1966. [4] 0. Matan, C. J. C. Burges, Y. Le Cun, and J. Denker. Multi-digit recognition using a Space Dispacement Neural Network. In J. E. Moody, editor, Advances in Neural Information Processing Systems 4, pages 488-495, Denver, 1992. Morgan Kaufmann. f5] T. Okuda, E. Tanaka, and K. Tamotsu. A method for the correction of garbled words based on the Levenshtein metric. IEEE Transactions on Computers, c-25(2):172-177, 1976. [6] M. Schenkel and M. Jabri. Degraded printed document recognition using convolutional neural networks and hidden markov models. In Proceedings of the A CNN, Melbourne, 1997. 

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Combining Classifiers Using Correspondence Analysis Christopher J. Merz Dept. of Information and Computer Science University of California, Irvine, CA 92697-3425 U.S.A. cmerz@ics.uci.edu Category: Algorithms and Architectures. Abstract Several effective methods for improving the performance of a single learning algorithm have been developed recently. The general approach is to create a set of learned models by repeatedly applying the algorithm to different versions of the training data, and then combine the learned models' predictions according to a prescribed voting scheme. Little work has been done in combining the predictions of a collection of models generated by many learning algorithms having different representation and/or search strategies. This paper describes a method which uses the strategies of stacking and correspondence analysis to model the relationship between the learning examples and the way in which they are classified by a collection of learned models. A nearest neighbor method is then applied within the resulting representation to classify previously unseen examples. The new algorithm consistently performs as well or better than other combining techniques on a suite of data sets. 1 Introduction Combining the predictions of a set of learned models! to improve classification and regression estimates has been an area of much research in machine learning and neural networks [Wolpert, 1992, Merz and Pazzani, 1997, Perrone, 1994, Breiman, 1996, Meir, 1995]. The challenge of this problem is to decide which models to rely on for prediction and how much weight to give each. The goal of combining learned models is to obtain a more accurate prediction than can be obtained from any single source alone. 1 A learned model may be anything from a decision/regression tree to a neural network. C. l Men 592 Recently, several effective methods have been developed for improving the performance of a single learning algorithm by combining multiple learned models generated using the algorithm. Some examples include bagging [Breiman, 1996], boosting [Freund, 1995], and error correcting output codes [Kong and Dietterich, 1995]. The general approach is to use a particular learning algorithm and a model generation technique to create a set of learned models and then combine their predictions according to a prescribed voting scheme. The models are typically generated by varying the training data using resampling techniques such as bootstrapping [Efron and Tibshirani, 1993J or data partitioning [Meir, 1995] . Though these methods are effective, they are limited to a single learning algorithm by either their model generation technique or their method of combining. Little work has been done in combining the predictions of a collection of models generated by many learning algorithms each having different representation and/or search strategies. Existing approaches typically place more emphasis on the model generation phase rather than the combining phase [Opitz and Shavlik, 1996]. As a result, the combining method is rather limited. The focus of this work is to present a more elaborate combining scheme, called SCANN, capable of handling any set of learned models , and evaluate it on some real-world data sets. A more detailed analytical and empirical study of the SCANN algorithm is presented in [Merz, 1997] . This paper describes a combining method applicable to model sets that are homogeneous or heterogeneous in their representation and/or search techniques. Section 2 describes the problem and explains some of the caveats of solving it. The SCANN algorithm (Section 3), uses the strategies of stacking [Wolpert, 1992J and correspondence analysis (Greenacre, 1984] to model the relationship between the learning examples and the way in which they are classified by a collection of learned models . A nearest neighbor method is then applied to the resulting representation to classify previously unseen examples. In an empirical evaluation on a suite of data sets (Section 4), the naive approach of taking the plurality vote (PV) frequently exceeds the performance of the constituent learners. SCANN, in turn, matches or exceeds the performance of PV and several other stacking-based approaches. The analysis reveals that SCANN is not sensitive to having many poor constituent learned models , and it is not prone to overfit by reacting to insignificant fluctuations in the predictions of the learned models. 2 Problem Definition and Motivation The problem of generating a set of learned models is defined as follows. Suppose two sets of data are given: a learning set C = {(Xi, Yi), i = 1, .. . ,I} and a test set T = {(Xt, yd, t = 1, .. . , T}. Xi is a vector of input values which are either nominal or numeric values, and Yi E {Cl , ... , Cc} where C is the number of classes. Now suppose C is used to build a set of N functions, :F {fn (x)}, each element of which approximates f(x) , the underlying function. = The goal here is to combine the predictions of the members of :F so as to find the best approximation of f(x). Previous work [Perrone, 1994] has indicated that the ideal conditions for combining occur when the errors of the learned models are uncorrelated. The approaches taken thus far attempt to generate learned models which make uncorrelated errors by using the same algorithm and presenting different samples of the training data [Breiman, 1996, Meir, 1995], or by adjusting the search heuristic slightly [Opitz and Shavlik, 1996, Ali and Pazzani, 1996J. No single learning algorithm has the right bias for a broad selection of problems. Combining Classifiers Using Correspondence Analysis 593 Therefore, another way to achieve diversity in the errors of the learned models generated is to use completely different learning algorithms which vary in their method of search and/or representation. The intuition is that the learned models generated would be more likely to make errors in different ways. Though it is not a requirement of the combining method described in the next section, the group of learning algorithms used to generate :F will be heterogeneous in their search and/or representation methods (i.e., neural networks, decision lists, Bayesian classifiers, decision trees with and without pruning, etc .). In spite of efforts to diversify the errors committed, it is still likely that some of the errors will be correlated because the learning algorithms have the same goal of approximating f, and they may use similar search strategies and representations. A robust combining method must take this into consideration. Approach 3 The approach taken consists of three major components: Stacking, Correspondence Analysis, and Nearest Neighbor (SCANN) . Sections 3.1-3.3 give a detailed description of each component, and section 3.4 explains how they are integrated to form the SCANN algorithm. 3.1 Stacking Once a diverse set of models has been generated, the issue of how to combine them arises. Wolpert (Wolpert, 1992] provided a general framework for doing so called stacked genemiization or stacking. The goal of stacking is to combine the members of:F based on information learned about their particular biases with respect to ?2 . The basic premise of stacking is that this problem can be cast as another induction problem where the input space is the (approximated) outputs of the learned models, and the output space is the same as before, i.e., The approximated outputs of each learned model, represented as jn(Xi), are generated using the following in-sample/out-of-sample approach: 1. Divide the ?0 data up into V partitions. 2. For each partition, v, ? Train each algorithm on all but partition v to get {j;V}. ? Test each learned model in {j;V} on partition v. ? Pair the predictions on each example in partition v (i.e., the new input space) with the corresponding output, and append the new examples to ?1 3. Return ?1 3.2 Correspondence Analysis Correspondence Analysis (CA) (Greenacre, 1984] is a method for geometrically exploring the relationship between the rows and columns of a matrix whose entries are categorical. The goal here is to explore the relationship between the training 2Henceforth ? will be referred to as ?0 for clarity. c. 1. Men 594 Stage 1 Symbol N n r 2 3 c P Dc Dr A A F G Table 1? Correspondence Analysis calculations. Description Definition Records votes of learned models . (I x J) indicator matrix Grand total of table N. i=1 jJ=1 nij Row masses. ri = ni+/n Column masses . cj=n+j/n Correspondence matrix. (1/n)N (J x J) diagonal matrix Masses c on diagonal. Masses r on diagonal. (I X I) diagonal matrix Dr -1/2(p _ rcT)Dc -1/2 Standardized residuals. urv'! SVD of A. Dr -1/2ur Principal coordinates of rows. Dc -1/2vr Principal coordinates of columns. 2:1 2: examples and how they are classified by the learned models. To do this, the prediction matrix, M, is explored where min = in (xd (1 ::; i ::; I, and 1 ::; n ::; N). It is also important to see how the predictions for the training examples relate to their true class labels, so the class labels are appended to form M' , an (I x J) matrix (where J = N + 1). For proper application of correspondence analysis, M' must be converted to an (I x (J . C)) indicator matrix, N, where ni,(joJ+e) is a one exactly when mij = ee, and zero otherwise. The calculations of CA may be broken down into three stages (see Table 1). Stage one consists of some preprocessing calculations performed on N which lead to the standardized residual matrix, A . In the second stage, a singular value decomposition (SVD) is performed on A to redefine it in terms ofthree matrices: U(lXK), r(KxK) ' and V(KXJ), where K = min(I - 1, J - 1) . These matrices are used in the third stage to determine F(lXK) and G(JxK) , the coordinates of the rows and columns of N, respectively, in the new space . It should be noted that not all K dimensions are necessary. Section 3.4, describes how the final number of dimensions, K *, is determined . Intuitively, in the new geometric representation , two rows, f p* and fq*, will lie close to one another when examples p and q receive similar predictions from the collection of learned models. Likewise, rows gr* and gu will lie close to to one another when the learned models corresponding to r and s make similar predictions for the set of examples. Finally, each column, r, has a learned model, j', and a class label, c', with which it is associated; f p * will lie closer to gr* when model j' predicts class c'. 3.3 Nearest Neighbor The nearest neighbor algorithm is used to classify points in a weighted Euclidean space. In this scenario, each possible class will be assigned coordinates in the space derived by correspondence analysis. Unclassified examples will be mapped into the new space (as described below) , and the class label corresponding to the closest class point is assigned to the example . Since the actual class assignments for each example reside in the last C columns of N, their coordinates in the new space can be found by looking in the last Crows of G. For convenience, these class points will be called Class!, . .. , Classc . To classify an unseen example, XTest, the predictions of the learned models on XTest must be converted to a row profile, rT , oflength J . C, where r& oJ+e) is 1/ J exactly Combining Classifiers Using Correspondence Analysis Data set abalone bal breast credit dementia glass heart ionosphere .. 1flS krk liver lymphography musk retardation sonar vote wave wdbc Table 2: Experimental results. PV SCANN S-BP S-BAYES vs PV vs PV vs PV ratio ratio ratio error 80.35 .490 .499 .487 13.81 .900 .859 .992 4.31 .886 .881 .920 13.99 .999 1.012 1.001 32.78 .989 .932 1.037 1.158 1.215 31.44 1.008 18.17 .964 .998 .972 3.05 .691 1.289 1.299 4.44 1.017 1.467 1.033 1.030 1.080 1.149 6.67 1.024 1.035 1.077 29.33 1.162 1.100 17.78 1.017 .812 .889 .835 13.51 32.64 .970 .960 .990 23.02 1.079 .990 1.007 .903 .908 .893 5.24 21.94 1.008 1.109 1.008 4.27 1.000 1.103 1.007 = ee, 595 Best Ind. vs PV ratio .535 111' .911 BP .938BP 1.054 BP 1.048c4 . 5 1.155 0C1 .962BP 2.175 c4 .5 1.150 oc1 1. 159 NN 1.138 cN2 .983Pebl8 1. 113Peb13 .936Baye3 1.048 BP .927 c4 .5 1.200Pebl8 1. 164NN when mij and zero otherwise. However, since the example is unclassified, XTe3t is of length (J - 1) and can only be used to fill the first (( J - 1) . entries in iT. For this reason , C different versions are generated, i.e., iT, . .. , i c , where each one "hypothesizes" that XTe3t belongs to one of the C classes (by putting 1/ J in t~e appropr~ate col~~) .. Loc~ting thes=l.rofiles in the scale~ sp~ce is a matter of s1mple matflx multIphcatIOn, 1.e., f'[ = re Gr- 1. The f'[ wh1ch lies closest to a class point, say Classc') is considered the "correct" hypothesized class, and XTe3t is assigned the class label c' . 3.4 C) The SCANN Algorithm Now that the three main parts of the approach have been described, a summary of the SCANN algorithm can be given as a function of Co and the constituent learning algorithms, A. The first step is to use Co and A to generate the stacking data, C 1 , capturing the approximated predictions of each learned model. Next, C1 is used to form the indicator matrix, N. A correspondence analysis is performed on N to derive the scaled space, A = urvT. The number of dimensions retained from this new representation, K *, is the value which optimizes classification on C 1 . The resulting scaled space is used to derive the row/column coordinates F and G, thus geometrically capturing the relationships between the examples, the way in which they are classified, and their position relative to the true class labels. Finally, the nearest neighbor strategy exploits the new representation by predicting which class is most likely according to the predictions made on a novel example. 596 4 C. J Merz Experimental Results The constituent learning algorithms, A, spanned a variety of search and/or representation techniques: Backpropagation (BP) [Rumelhart et al., 1986], CN2 [Clark and Niblett, 1989], C4.5 [Quinlan, 1993], OC1 [Salzberg; and Beigel, 1993], PEBLS [Cost, 1993], nearest neighbor (NN), and naive Bayes. Depending on the data set, anywhere from five to eight instantiations of algorithms were applied. The combining strategies evaluated were PV, SCANN, and two other learners trained on ?1: S-BP, and S-Bayes. The data sets used were taken from the UCI Machine Learning Database Repository [Merz and Murphy, 1996], except for the unreleased medical data sets: retardation and dementia. Thirty runs per data set were conducted using a training/test partition of 70/30 percent. The results are reported in Table 2. The first column gives the mean error rate over the 30 runs of the baseline combiner, PV. The next three columns ("SCANN vs PV", "S-BP vs PV", and "S-Bayes vs PV") report the ratio of the other combining strategies to the error rate of PV. The column labeled "Best Ind . vs PV" reports the ratio with respect to the model with the best average error rate. The superscript of each entry in this column denotes the winning algorithm. A value less than 1 in the "a vs b" columns represents an improvement by method a over method b. Ratios reported in boldface indicate the difference between method a and method b is significant at a level better than 1 percent using a two-tailed sign test. It is clear that, over the 18 data sets, SCANN holds a statistically significant advantage on 7 sets improving upon PV's classification error by 3-50 percent. Unlike the other combiners, SCANN posts no statistically significant losses to PV (i.e., there were 4 losses each for S-BP and S-Bayes). With the exception of the retardation data set, SCANN consistently performs as well or better than the best individual learned model. In the direct comparison of SCANN with the S-BP and S-Bayes, SCANN posts 5 and 4 significant wins, respectively, and no losses. The most dramatic improvement of the combiners over PV came in the abalone data set. A closer look at the results revealed that 7 of the 8 learned models were very poor classifiers with error rates around 80 percent, and the errors of the poor models were highly correlated. This empirically demonstrates PV's known sensitivity to learned models with highly correlated errors. On the other hand, PV performs well on the glass and wave data sets where the errors of the learned models are measured to be fairly uncorrelated. Here, SCANN performs similarly to PV, but S-BP and S-Bayes appear to be overfitting by making erroneous predictions based on insignificant variations on the predictions of the learned models. 5 Conclusion A novel method has been introduced for combining the predictions of heterogeneous or homogeneous classifiers. It draws upon the methods of stacking, correspondence analysis and nearest neighbor. In an empirical analysis, the method proves to be insensitive to poor learned models and matches the performance of plurality voting as the errors of the learned models become less correlated. References [Ali and Pazzani, 1996) Ali, K. and Pazzani, M. (1996). Error reduction through learning multiple descriptions. Machine Learning, 24:173. Combining Classifiers Using Correspondence Analysis [Breiman, 1996] Breiman, L. (1996). 24(2):123-40. Bagging predictors. 597 Machine Learning, [Clark and Niblett, 1989] Clark, P. and Niblett, T. (1989). The CN2 induction algorithm. Machine Learning, 3(4):261-283. [Cost, 1993] Cost, S.; Salzberg, S. (1993). A weighted nearest neighbor algorithm for learning with symbolic features. Machine Learning, 10(1):57-78. [Efron and Tibshirani, 1993] Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman and Hall, London and New York. [Freund, 1995] Freund, Y. (1995). Boosting a weak learning algorithm by majority. Information and Computation, 121(2):256-285. Also appeared in COLT90. [Greenacre, 1984] Greenacre, M. J. (1984). Theory and Application of Correspondence Analysis. Academic Press, London. [Kong and Dietterich, 1995] Kong, E. B. and Dietterich, T. G. (1995). Errorcorrecting output coding corrects bias and variance. In Proceedings of the 12th International Conference on Machine Learning, pages 313-321. Morgan Kaufmann. [Meir, 1995] Meir, R. (1995). Bias, variance and the combination ofleast squares estimators. In Tesauro, G., Touretzky, D., and Leen, T., editors, Advances in Neural Information Processing Systems, volume 7, pages 295-302. The MIT Press. [Merz, 1997] Merz, C. (1997). Using correspondence analysis to combine classifiers. Submitted to Machine Learning. [Merz and Murphy, 1996] Merz, C. and Murphy, P. (1996). UCI repository of machine learning databases. [Merz and Pazzani, 1997] Merz, C. J. and Pazzani, M. J. (1997). Combining neural network regression estimates with regularized linear weights. In Mozer, M., Jordan, M., and Petsche, T., editors, Advances in Neural Information Processing Systems, volume 9. The MIT Press. [Opitz and Shavlik, 1996] Opitz, D. W. and Shavlik, J. W. (1996). Generating accurate and diverse members of a neural-network ensemble. In Touretzky, D. S., Mozer, M. C., and Hasselmo, M. E., editors, Advances in Neural Information Processing Systems, volume 8, pages 535-541. The MIT Press. [Perrone, 1994] Perrone, M. P. (1994). Putting it all together: Methods for combining neural networks. In Cowan, J. D., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems, volume 6, pages 1188-1189. Morgan Kaufmann Publishers, Inc. [Quinlan, 1993] Quinlan, R. (1993). G..4-5 Programs for Machine Learning. Morgan Kaufmann, San Mateo, CA. [Rumelhart et al., 1986] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning internal representations by error propagation. In Rumelhart, D. E., McClelland, J. 1., and the PDP research group., editors, Parallel distributed processing: Explorations in the microstructure of cognition, Volume 1: Foundations. MIT Press. [Salzberg; and Beigel, 1993] Salzberg;, S. M. S. K. S. and Beigel, R. (1993). OC1: Randomized induction of oblique decision trees. In Proceedings of AAAI-93. AAAI Pres. [Wolpert, 1992] Wolpert, D. H. (1992). Stacked generalization. Neural Networks, 5:241-259. 

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A 1,OOO-Neuron System with One Million 7-bit Physical Interconnections Yuzo Hirai Institute of Information Sciences and Electronics University of Tsukuba 1-1-1 Ten-nodai, Tsukuba, Ibaraki 305, Japan e-mail: hirai@is.tsukuba.ac.jp Abstract An asynchronous PDM (Pulse-Density-Modulating) digital neural network system has been developed in our laboratory. It consists of one thousand neurons that are physically interconnected via one million 7-bit synapses. It can solve one thousand simultaneous nonlinear first-order differential equations in a fully parallel and continuous fashion. The performance of this system was measured by a winner-take-all network with one thousand neurons. Although the magnitude of the input and network parameters were identical for each competing neuron, one of them won in 6 milliseconds. This processing speed amounts to 360 billion connections per second. A broad range of neural networks including spatiotemporal filtering, feedforward, and feedback networks can be run by loading appropriate network parameters from a host system. 1 INTRODUCTION The hardware implementation of neural networks is crucial in order to realize the real-time operation of neural functions such as spatiotemporal filtering, learning and constraint processings. Since the mid eighties, many VLSI chips and systems have been reported in the literature, e.g. [1] [2]. Most of the chips and the systems including analog and digital implementations, however, have focused on feedforward neural networks. Little attention has been paid to the dynamical aspect of feedback neural networks, which is especially important in order to realize constraint processings, e.g. [3]. Although there were a small number of exceptions that used analog circuits [4] [5], their network sizes were limited as compared to those of their feedforward counterparts because of wiring problems that are inevitable in regard to full and physical interconnections. To relax this problem, a pulse-stream system has been used in analog [6] and digital implementations [7]. Y. Hirai 706 The author developed a fully interconnected 54-neuron system that uses an asynchronous PDM (Pulse-Density-Modulating) digital circuit system [8]. The present paper describes a thousand-neuron system in which all of the neurons are physically interconnected via one million 7-bit synapses in order to create a fully parallel feedback system. The outline of this project was described in [10]. In addition to the enlargement of system size, synapse circuits were improved and time constant of each neuron was made variable. The PDM system was used because it can accomplish faithful analog data transmission between neurons and can relax wiring problems. An asynchronous digital circuit was used because it can solve scaling problems, and we could also use it to connect more than one thousand VLSI chips, as described below. 2 NEURON MODEL AND THE CIRCUITS 2.1 SINGLE NEURON MODEL The behavior of each neuron in the system can be described by the following nonlinear first-order differential equation: dyi(t) Iti--;];t Yi{ t) <pta] N = = = -viet) + L WijYj{t) + li{t), j=l <p[yi{t)], and if a > 0 {~ (1) (2) (3) otherwise, where Iti is a time constant of the i-th neuron, y;(t) is an internal potential of the i-th neuron at time t, Wij is a synaptic weight from the j-th to the i-th neurons, and li(t) is an external input to the i-th neuron. <pta] is an analog threshold output function which becomes saturated at a given maximum val~e. The system solves Eq.{l) in the following integral form: yi{t) = (t {-VieT) 10 + t j=l WijYj(T) + h(T)} d~Itl + yi(O), (4) where y;(O) is an initial value. An analog output of a neuron is expressed by a pulse stream whose frequency is proportional to the positive, instantaneous internal potential. 2.2 2.2.1 SINGLE NEURON CmCUIT Synapse circuits The circuit diagrams for a single neuron are shown in Fig. 1. As shown in Fig.l(a), it consists of synapse circuits, excitatory and inhibitory dendrite OR circuits, and a cel~ body circuit. Each synapse circuit transforms the instantaneous frequency of the input pulses to a frequency that is proportional to the synaptic weight. This transformation is carried out by a 6-bit rate multiplier, as shown in Fig.l{b). The behavior of a rate multiplier is illustrated in Fig.l(c) using a 3-bit case for brevity. A rate multiplier is a counter and its state transits to the next state when an input pulse occurs. Each binary bit of a given weight specifies at which states the output pulses are generated. When the LSB is on, an output pulse is generated at the fourth state. When the second bit is on, output pulses are generated at the second A I,OOO-Neuron System with One Million 7-bit Physical Interconnections Neuron circuit Dendrite extension terminals ~ U 0_0 J - J JUL : Synaptic weight (7-blt) ~~ U Rate multiplier (&-bit) wt 1 + ~ j ~ Synapse ~ j I circuit I - a: a: nIl!.. '.:;g::. -~ InJ)Ut pulses JUL ~.? tp@ Synapse circuit -'L r,:,. ~ @ (Circuit sign C CI@l?rJUL I. II Cell body circuit f<:IrI \51 I @. JlJlJL Output pulses (a) 'w 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1111111 (c) sign Cell body circuit - 1 1 1 101 110 111 (b) """ states 01234567 100 J ~ weight 000 010 011 ia _.ISy~pse rI e ~ Lt . : : \ @ , ,.=. ~ 3-blt rate multiplier 001 (Osw<1) i .. 707 r-=-~+'_:_--, UDr-----., Sampling ~ Up-down f-- Rate circuit Down counter ~ ~ multiplier 2f Output ;. gate and f~~~_(1~2-b-+t=)__I--(_12~-br....It)_ =~ Main clock =4fma (2OMHz) -:-='112 I~ r=----,~-..~ Down ? L-----:~---II X (-1) r Up' Rate multiplier (&-bit) I If f.J for2f H slgn=O (d) Figure 1: Circuit diagram of a single neuron. (a) Circuit diagram of a single neuron and (b) that of a synapse circuit. (c) To illustrate the function of a rate multiplier, the multiplication table for a 3-bit case is shown. (d) Circuit diagram of a cell body circuit. See details in text. and at the sixth states. When the MSB is on, they are generated at all of the odd states. Therefore, the magnitude of synaptic weight that can be represented by a rate multiplier is less than one. In our circuit, this limitation was overcome by increasing the frequency of a neuron output by a factor of two, as described below. 2.2.2 Dendrite circuits Output pulses from a synapse circuit are fed either to an excitatory dendrite OR circuit or to an inhibitory one, according to the synaptic weight sign. In each dendrite OR circuit, the synaptic output pulses are summed by OR gates, as is shown along the right side of Fig.1(a). Therefore, if these output pulses are synchronized, they are counted as one pulse and linear summation cannot take place. In our circuit, each neuron is driven by an individual clock oscillator. Therefore, they will tend to become desynchronized. The summation characteristic was analysed in [9], and it was shown to have a saturation characteristic that is similar to the positive part of a hyperbolic tangent function. 2.2.3 Cell body circuit A cell body circuit performs the integration given by Eq.( 4) as follows. As shown in Fig.1(d), integration was performed by a 12-bit up-down counter. Input pulses from an excitatory dendrite OR circuit are fed into the up-input of the counter and those from an inhibitory one are fed into the down-input after conflicts between Y. Hirai 708 excitatory and inhibitory pulses have been resolved by a sampling circuit. A 12-bit rate multiplier produces internal pulses whose frequency is 21, where 1 is proportional to the absolute value of the counter. The rate multiplier is driven by a main clock whose frequency is 4/max, Imax being the maximum output frequency. When the counter value is positive, an output pulse train whose frequency is either 1 or 2/, according to the scale factor is transmitted from a cell body circuit. The negative feedback term that appeared in the integrand of Eq.( 4) can be realized by feeding the internal pulses into the down-input of the counter when the counter value is positive and feeding them into the up-input when it is negative. The 6bit rate multiplier inserted in this feedback path changes the time constant of a neuron. Let f3i be the rate value of the rate multiplier, where 0 ~ f3i < 26 . The Eq.(4) becomes: yi(t) (5) 26 211 Therefore, the time constant changes to l!i?, where Pi was given by -,seconds. /3 max It should be noted that, since the magnitUde of the total input was increased by a factor of ~, the strength of the input should be decreased by the inverse of that factor in order to maintain an appropriate output level. If it is not adjusted, we can increase the input strength. Therefore, the system has both input and output scaling functions. The time constant varies from about 416psec for f3 = 63 to 26.2msec for f3 = 1. When f3 = 0, the negative feedback path is interrupted and the circuit operates as a simple integrator, and every feedforward network can be run in this mode of operation. 3 3.1 THE 1,OOO-NEURON SYSTEM VLSI CHIP A single type of VLSI chip was fabricated using a 0.7 pm CMOS gate array with 250,000 gates. A single chip contains 18 neurons and 51 synapses for each neuron. Therefore, each chip has a total of 918 synapses. About 85% of the gates in a gate array could be used, which was an extremely efficient value. A chip was mounted on a flat package with 256 pins. Among them, 216 pins were used for signals and the others were used for twenty pairs of V cc( =3.3V) and GND. 3.2 THE SYSTEM As illustrated in Fig.2(a), this system consists of 56 x 20 = 1,120 chips. 56 chips are used for both cell bodies and synapses, and the others are used to extend dendrite circuits and increase the number of synapses. In order to extend the dendrites, the dendrite signals in a chip can be directly transmitted to the dendrite extention terminals of another chip by bypassing the cell body circuits. There are 51 x 20 1,020 synapses per neuron. Among them, 1,008 synapses are used for fully hard-wired interconnections and the other 12 synapses are used to receive external signals. There are a total of 1,028,160 synapses in this system. It is controlled by a personal computer. The synaptic weights, the contents of the up-down counters = A 1,OOO-Neuron System with One Million 7-bit Physical Interconnections Host computer 709 High-speed bus 12 external inputs m Neural chips o Neural chips for neurons for synapses (a) (b) Figure 2: Structure of the system. (a) System configuration. The down arrows emitted from the open squares designate signal lines that are extending dendrites. The others designate neuron outputs. (b) Exterior of the system. It is controlled by a personal computer. and the control registers can be read and written by the host system. It takes about 6 seconds to set all the network parameters from the host system. The exterior of this system is shown in Fig.2(b). Inside the cabinet, there are four shelves. In each shelf, fourteen circuit boards were mounted and on each board 20 chips were mounted. One chip was used for 18 neurons and the other chips were used to extend the dendrites. Each neuron is driven by an individual 20MHz clock oscillator. 4 SYSTEM PERFORMANCE In order to measure the performance of this system, one neuron was used as a signal generator. By setting all the synaptic weights and the internal feedback gain of a signal neuron to zero, and by setting the content of the up-down counter to a given value, it can produce an output with a constant frequency that is proportional to the counter value. The input strength of the other neurons can be adjusted by changing the counter value of a signal neuron or the synaptic weights from it. The step reponses of a neuron to different inputs are shown in Fig.3(a). As seen in the figure, the responses exactly followed Eq.(l) and the time constant Was about 400psec. Figure 3(b) shows responses with different time constants. The inputs were identical for all cases. Figure 3(c) shows the response of a temporal filter that was obtained by the difference between a fast and a slow neuron. By combining two low-pass filters that had different cutoff frequencies, a band-pass filter was created. A variety of spatiotem- Y. Hirai 710 r: rc-------:: _ - - - - - -- '&. ::J 768 500 512 256 OL--~--~--~--~--~--~ o s;.-~ /~.~------- 1792 ... ,I ~ 1500 ...?. " 32. / S 16 i.? 1 1000 /.'/:/. 1 / I' 8 /~- _____ /' ----- "-- _.---. -- ___ ---- - --- ~ ~ ~~-:::-::~~-----'--o 10000 20000 30000 40000 50000 10000 20000 30000 40000 50000 line (x 100 Il8IlO58conds) time (x 100 nanoeeconds) (b) (a) 1~~-~~-~--~---~--. 2000 Bela_53 ---::=--------:j 800 1&600 i 1500 S ! 1000 1 400 i ~P--"""IiiiiiiiOi;~ 1000 %?200 500 -400 OIL....--~---'---~......::::.==--..-J o 10000 20000 30000 40000 50000 :L_'---_'---_~~~ lime (x 100 nanoeeconds) (c) o 20000 40000 60000 time (x 100 nanoseconds) 80000 (d) Figure 3: Responses obtained by the system. (a) Step responses to different input levels. Parameters are the values that are set in the up-down counter of a signal neuron. (b) Step responses for different time constants. Parameters are the values of f3i in Eq.5. Inputs were identical in all cases. (c) Response of a temporal filter that was ol>taind by the difference between a fast and a slow neuron. (d) Response of a winner-take-all network among 1,007 neurons. The responses of a winner neuron and 24 of the 1,006 defeated neurons are shown. poral filters can be implemented in this way. Figure 3(d) shows the responses of a winner-take-all network among 1,007 neurons. The time courses of the responses of a winner neuron and 24 of the 1,006 defeated neurons are shown in the figure. The strength of all of the inhibitory synaptic weights between neurons was set to 2 x (- ::), where 2 is an output scale factor. The synaptic weights from a signal neuron to the 1,007 competing ones were identical and were ~;. Although the network parameters and the inputs to all competing neurons were identical, one of them won in 6 msec. Since the system operates asynchronously and the spatial summation of the synaptic output pulses is probabilistic, one of the competing neurons can win in a stochastic manner. In order to derive the processing speed in terms of connections per second, the same winner-take-all network was solved by the Euler method on a latest workstation. Since it took about 76.2 seconds and 2,736 iterations to converge, the processing speed of the workstation was about 36 million connections per second l007xI0<17x2736) . ( ::::::! 76 .2 ! ? S'mce t h'1S system .IS 10000' , times f aster t han t h e wor kstatlOn, A I,OOO-Neuron System with One MillioK7-bit Physical Interconnections 711 the processing speed amounts to 360 billion connections per second. Various kinds of neural networks including spatiotemporal filtering, feedforward and feedback neural networks can be run in this single system by loading appropriate network parameters from the host system. The second version of this system, which can be used via the Internet, will be completed by the end of March, 1998. Acknowledgements The author is grateful to Mr. Y. Kuwabara and Mr. T. Ochiai of Hitachi Microcomputer System Ltd. for their collaboration in developing this system and to Dr. M. Yasunaga and Mr. M. Takahashi for their help in testing it. The author is also grateful to Mr. H. Toda for his collaboration in measuring response data. This work was supported by "Proposal-Based Advanced Industrial Technology R&D Program" from NEDO in Japan. References [1] C. Mead: Analog VLSI and Neural Systems. Addison-Wesley Publishing Company, Massachusetts, 1989 [2] K.W.Przytula and V.K.Prasanna, Eds.: Parallel Digital Implementations of Neural Networks. Prentice Hall, New Jersey, 1993 [3] J.J. Hopfield: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. U.S.A., 81, pp.3088-3092, 1984 [4] P. Mueller, J. van der Spiegel, V. Agami, D. Blackman, P. Chance, C. Donham, R. Etienne, J. Kim. M. Massa and S. Samarasekera: Design and performance of a prototype analog neural computer. Proc. the 2nd International Conf. on Microelectronics for Neural Networks, pp.347-357, 1991 [5] G. Cauwenberghs: A learning analog neural network chip with continuous-time recurrent dynamics. In J. D. Cowan, G. Tesauro and J. Alspector, Eds., Advances in Neural Information Processing Systems 6, Morgan Kaufmann Publishers, San Mateo, CA, pp.858-865, 1994 [6] S. Churcher, D. J. Baxter, A. Hamilton, A. F. Murry, and H. M. Reekie: Generic analog neural computation - The EPSILON chip. In S. J. Hanson, J. D. Cowan and C. L. Giles, Eds., Advances in Neural Information Processing Systems 6, Morgan Kaufmann Publishers, San Mateo, CA, pp.773-780, 1993 [7] H. Eguchi, T. Furuta, H. Horiguchi, S. Oteki and T. Kitaguchi: Neural network LSI chip with on-chip learning. Proceedings of IJCNN'91 Seattle, Vol.I/453-456, 1991 [8] Y. Hirai, et al.: A digital neuro-chip with unlimited connectability for large scale neural networks. Proc. International Joint Conf. on Neural Networks'89 Washington D.C., Vo1.11/163-169, 1989 [9] Y.Hirai, VLSI Neural Network Systems (Gordon and Breach Science Publishers, Birkshire, 1992) [10] Y. Hirai and M. Yasunaga: A PDM digital neural network system with 1,000 neurons fully interconnected via 1,000,000 6-bit synapses. Proc. International Conference on Neural Information Processings'96, Vo1.ll/1251, 1996 

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Multiple Threshold Neural Logic Jehoshua Bruck Vasken Bohossian ~mail: California Institute of Technology Mail Code 136-93 Pasadena, CA 91125 {vincent, bruck}~paradise.caltech.edu Abstract We introduce a new Boolean computing element related to the Linear Threshold element, which is the Boolean version of the neuron. Instead of the sign function, it computes an arbitrary (with polynornialy many transitions) Boolean function of the weighted sum of its inputs. We call the new computing element an LT M element, which stands for Linear Threshold with Multiple transitions. The paper consists of the following main contributions related to our study of LTM circuits: (i) the creation of efficient designs of LTM circuits for the addition of a multiple number of integers and the product of two integers. In particular, we show how to compute the addition of m integers with a single layer of LT M elements. (ii) a proof that the area of the VLSI layout is reduced from O(n 2 ) in LT circuits to O(n) in LTM circuits, for n inputs symmetric Boolean functions, and (iii) the characterization of the computing power of LTM relative to LT circuits. 1 Introduction Human brains are by far superior to computers in solving hard problems like combinatorial optimization and image and speech recognition, although their basic building blocks are several orders of magnitude slower. This observation has boosted interest in the field of artificial neural networks [Hopfield 82], [Rumelhart 82]. The latter are built by interconnecting artificial neurons whose behavior is inspired by that of biological neurons. In this paper we consider the Boolean version of an artificial neuron, namely, a Linear Threshold (LT) element, which computes a neural-like MUltiple Threshold Neural Logic 0 WI -Wo Wn 253 1 0 LT gate 0 t3 1 t2 WI t3 1 t2 t1 Wn t1 0 1 SYM gate 0 1 LTM gate Figure 1: Schematic representation of LT, SYM and LTM computing elements. Boolean function of n binary inputs [Muroga 71]. An LT element outputs the sign of a weighted sum of its Boolean inputs. The main issues in the study of networks (circuits) consisting of LT elements, called LT circuits, include the estimation of their computational capabilities and limitations and the comparison of their properties with those of traditional Boolean logic circuits based on AND, OR and NOT gates (called AON circuits). For example, there is a strong evidence that LT circuits are more efficient than AON circuits in implementing a number of important functions including the addition, product and division of integers [Siu 94], [Siu 93]. Motivated by our recent work on the VLSI implementation of LT elements [Bohossian 95b], we introduce in this paper a more powerful computing element, a multiple threshold neuron, which we call LTM, which stands for Linear Threshold with Multiple transitions, see [Haring 66] and [Olafsson 88]. Instead of the sign function in the LT element it computes an arbitrary (with polynomialy many transitions) Boolean function of the weighted sum of its inputs. The main issues in the study of LTM circuits (circuits consisting of LTM elements) include the estimation of their computational capabilities and limitations and the comparison of their properties to those of AON circuits. A natural approach in this study is first to understand the relation between LT circuits and LT M circuits. Our main contributions in this paper are: ? We demonstrate the power of LTM by deriving efficient designs of LTM circuits for the addition of m integers and the product of two integers. ? We show that LT M circuits are more amenable in implementation than LT circuits. In particular, the area of the VLSI layout is reduced from O(n 2 ) in LT circuits to O(n) in LTM circuits, for n input symmetric Boolean functions. ? We characterize the computing power of LT M relative to LT circuits. Next we describe the formal definitions of LT and LT M elements. 1.1 Definitions and Examples Definition 1 (Linear Threshold Gate - LT) A linear threshold gate computes a Boolean function of its binary inputs : n f(X) = sgn(wo + L WiXi) i=l V. Bohossian and J. Bruck 254 where the Wi are integers and sgn(.) outputs 1 if its argument is greater or equal to 0, and 0 otherwise. Figure 1 shows an-input LT element; if L~ WiXi ~ -Wo the element outputs 1, otherwise it outputs o. A single LT gate is unable to compute parity. The latter belongs to the general class of symmetric functions - SY M. Definition 2 (Symmetric Functions - SY M) A Boolean function f is symmetric if its value depends only on the number of ones in the input denoted by IX I. Figure 1 shows an example of a symmetric function; it has three transitions, it outputs 1 for IXI < tl and for t2 ~ IXI < t3, and 0 otherwise. AND, OR and parity are examples of symmetric functions. A single LT element can implement only a limited subset of symmetric functions. We define LT M as a generalization of SY M. That is, we allow the weights to be arbitrary as in the case of LT, rather than fixed to 1 (see Figure 1 ). Definition 3 (Linear Threshold Gate with Multiple Transitions - LT M) A function f is in LTM if there exists a set of weights Wi E Z, 1 ~ i ~ n and a function h : Z ---+ {O, 1} such that n f(X) = h(L wixd for all X E {O,l}n i=l The only constraint on h is that it undergoes polynomialy many transitions as its input scans [- L~=l IWi I, L~=l IWi I]? Notice that without the constraint on the number of transitions, an LTM gate is capable of computing any Boolean function. Indeed, given an arbitrary function f, let Wi = 2i - 1 and h(L~ 2i - 1 xd = f(xt, .? ?, x n ). Example 1 (XOR E LTM) XOR(X) outputs 1 if lXI, the number of l's in X, is odd. Otherwise it outputs O. To implement it choose Wi = 1 and h(k) = ~(1 - (_l)k) for 0 ~ k ~ n. Note that h(k) needs not be defined for k < 0 and k > n, and has polynomialy many transitions. Another useful function that LTM can compute is ADD (X, Y), the sum of two n-bit integers X and Y. Example 2 (ADD E LT M) To implement addition we set fl (X, Y) = h, (L~=l 2i (Xi + yd) where h, (k) = 1 for k E [2 ' ,2 x 2' - 1] U [3 X 2 ' , +00). Defined thus, fl computes the m-th bit of X + Y. 1.2 Organization The paper is organized as follows . In Section 2, we study a number of applications as well as the VLSI implementations of LTM circuits. In particular, we show how to compute the addition of m integers with a single layer of LT!vI elements. In Section 3, we prove J..he characterization results of LT M - inclusion relations, in particular LTM ~ LT2. In addition, we indicate which inclusions are proper and exhibit functions to demonstrate the separations. 255 Multiple Threshold Neural Logic 2 LT M Constructions The theoretical results about LTM can be applied to the VLSI implementation of Boolean functions. The idea of a gate with multiple thresholds came to us as we were looking for an efficient VLSI implementation of symmetric Boolean functions. Even though a single LT gate is not powerful enough to implement any symmetric function, a 2-layer LT circuit is. FUrthermore, it is well known that such a circuit performs much better than the traditional logic circuit based on AND, OR and NOT gates. The latter has exponential size (or unbounded depth) [Wegener 91]. Proposition 4 (LT2 versus LT M for symmetric function implementation) The LT2 layout of a symmetric function requires area of O(n 2), while using LT M one needs only area of O( n). PROOF: Implementing a generalized symmetric function in LT2 requires up to n LT gates in the first layer. Those have the same weights Wi except for the threshold Woo Instead of laying out n times the same linear sum E~ WiXi we do it once and compare the result to n different thresholds. The resulting circuit corresponds to a single LTM gate. 0 The LT2 layout is redundant, it has n copies of each weight, requiring area of at least O(n 2). On the other hand, LTM performs a single weighted sum, its area requirement is O(n). A single LT M gate can compute the addition of m n-bit integers M ADD. The only constraint is that m be polynomial in n. Theorem 5 (MADD E LTM) A single layer of LT M gates can compute the sum of m n-bit integers, provided that m is at most polynomial in n. PROOF: MAD D returns an integer of at most n + log m bits. We need one LT M gate per bit. The least significant bit is computed by a simple m-bit XOR. For all other bits we use h(X(l), .. ,x(m?) = hl(E~=12i Ej=l x~j?) to compute the l-th bit ofthe mm. 0 Corollary 6 (PRODUCT E PTM) A single layer of PTM (which is defined below) gates, can compute the product of m n-bit integers, provided that m is at most polynomial in n. PROOF: By analogy with PTb defined in [Bruck 90], in PT Ml (or simply PT M) we allow a polynomial rather than a linear sum: f(X) = h(WIXl + ... +wnxn +W(1,2)XIX2+ ... ) However we restrict the sum to have polynomialy many terms (else, any Boolean function could be realized with a single gate). The product of two n-bit integers X and Y can be written as PRODUCT(X, Y) = E~=l XiY. We use the construction of MADD in order to implement PRODUCT. PRODUCT(X, Y) = MADD(x 1Y,x2 Y , ... ,xnY). fleX, Y) = hi (LJj=l LJi=12i XjYi) b outputs the l-th bit of the product. 0 "n "I V. Bohossian and J Bruck 256 Figure 2: Relationship between Classes 3 Classification of LTM .- --- We me a hat to indicate small (polynomialy growing) weights, e.g. LT, LT M [Bohossian 95a], [Siu 91], and a subscript to indicate the depth (number of layers) of the circuit of more than a single layer. All the circuits we consider in this paper are of polynomial size (number of elements) in n (number of inputs). For example, the class fr2 consists of those B0...2!ean functions that can be implemented by a depth-2 polynomial size circuit of LT elements. Figure 2 depicts the membership relations between five classes of Boolean functions, including, LT, ilr, LTM, LTM and ilr2, along with the functions used to establish the separations. In this section we will prove the relations illustrated by Figure 2 . Theorem 7 (Classification of LTM ) The inclusions and separations shown in Figure 12 hold. That is, .- 1. LT ~ LT ; LTM 12. LT ~ LTM ; LTM --- .- 9. LTM; LT2 4? XOR E CTM but XOR tJ. LT 5. CaMP E LT but CaMP tJ. LTM 6. ADD E LTM but ADD tJ. LTULTM 7. IPk E fr2 but 1Pk tJ. LTM PROOF: We show only the outline of the proof. The complete version can be found in [Bohossian 96]. Claims 1 and 2 follow from the definition. The first part of Claim 4 was shown in Example 1 and the second is well known. In Claim 5, CaMP stands for the Comparison functio!lt the proof mes the pigeonhole principle and is related to the proof of CaMP tJ. LT which can be found in [Siu 91]. In Claim 6 to show that ADD tJ. LTM we use the same idea as for CaMP. Claim 3 is proved using a result from [Goldman 93]: a single LT gate with arbitrary weights can be realized by an LT2 circuit. Claim 7 introduces the function IPk(X, Y) = 1 iff L:~ XiYi ~ k, 257 Multiple Threshold Neural Logic o otherwise. .- If IPk E LTM, using the result from [Goldman 93], we can construct a LT2 circuit that computes IP2 (Inner Product mod 2) which is known to be false 0 [Hajnal 94]. What remains to be shown in order to complete the classification picture is LT n LTM. We conjecture that this is true. 4 fr = Conclusions Our original goal was to use theoretical results in order to efficiently layout a generalized symmetric function. During that process we came to the conclusion that the LT2 implementation is partially redundant, which lead to the definition of LTM, a new, more powerful computing element. We characterized the power of LTM relative to LT. We showed how it can be used to reduce the area of VLSI layouts from O(n 2 ) to O(n) and derive efficient designs for multiple addition and product. Interesting directions for future investigation are (i) to prove the conjecture: fr = LT n LTM, (ii) to apply spectral techniques ([Bruck 90)) to the analysis of LT M, in particular show how PT M fits into the classification picture (Figure 2 ). Another direction for future research consists in introducing the ideas described above in the domain of VLSI. We have fabricated a programmable generalized symmetric function on a 2J,L, analog chip using the model described above. Floating gate technology is used to program the weights. We store a weight on a single transistor by injecting and tunneling electrons on the floating gate [Hasler 95]. Acknowledgments This work was supported in part by the NSF Young Investigator Award CCR9457811 and by the Sloan Research Fellowship. References [Bohossian 95a] V. Bohossian and J. Bruck. On Neural Networks with Minimal Weights. In Advances in Neural Information Processing Systems 8, MIT Press, Cambridge, MA, 1996, pp.246-252. [Bohossian 95b] V. Bohossian, P. Hasler and J. Bruck. Programmable Neural Logic. Proceedings of the second annual IEEE International Conference on Innovative Systems in Silicon, pp. 13-21, October 1997. [Bohossian 96] V. Bohossian and J. Bruck. ral Logic. Technical Report, ETR010, http://paradise.caltech.edu/ETR.html) Multiple Threshold N euJune 1996. (available at [Bruck 90] J. Bruck. Harmonic Analysis of Polynomial Threshold Functions. SIAM J. Disc. Math, Vol. 3(No. 2)pp. 168- 177, May 1990. [Goldman 93] M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In Proc. 25th ACM STOC, pages pp. 551-560, 1993. 258 V. Bohossian and J Bruck [Hajnal 94] A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, G. Turan. Threshold Circuits of Bounded Depth. Journal of Computer and System Sciences, Vol. 46(No. 2):pp. 129-154, April 1993. [Haring 66] D.R. Haring. Multi-Threshold Threshold Elements. IEEE Transactions on Electronic Computers, Vol. EC-15, No.1, February 1966. [Hasler 95] P. Hasler, C. Diorio, B.A. Minch and C.A. Mead. Single Transistor Learning Synapses. Advances in Neural Information Processing Systems 7, MIT Press, Cambridge, MA, 1995, pp.817-824. [Hastad 94] J. Hastad. On the size of weights for threshold gates. SIAM. J. Disc. Math., 7:484-492, 1994. [Hofmeister 96] T. Hofmeister. A Note on the Simulation of Exponential Threshold Weights. 1996 CONCOON conference. [Hopfield 82] J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. of the USA National Academy of Sciences, 79:2554-2558, 1982. [Muroga 71] M. Muroga. Threshold Logic and its Applications. Wiley-Interscience, 1971. [Olafsson 88] S. Olafsson and Y.S. Abu-Mostafa. The Capacity of Multilevel Threshold Functions. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.10, No.2, March 1988. [Rumelhart 82] D. Rumelhart and J. McClelland. Parallel distributed processing: Explorations in the microstructure of cognition. MIT Press, 1982. [Siu 91] K. Siu and J. Bruck. On the power of threshold circuits with small weights. SIAM J. Disc. Math., Vol. 4(No. 3):pp. 423-435, August 1991. [Siu 93] K. Siu, J. Bruck, T. Kailath, and T. Hofmeister. Depth Efficient Neural Networks for Division and Related Problems. IEEE Trans. on Information Theory, Vol. 39(No. 3), May 1993. [Siu 94] K. Siu and V.P. Roychowdhury. On Optimal Depth Threshold Circuits for Multiplication and Related Problems. SIAM J. Disc. Math., Vol. 7(No. 2):pp. 284-292, May 94. [Szegedy 89] M. Szegedy. Algebraic Methods in Lower Bounds for Computational Models with Limited Communication. PhD Thesis, Dep. Computer Science, Chicago Univ., December 1989. [Wegener 91] 1. Wegener. The complexity of the parity function in unbounded fanin unbounded depth circuits. In Theoretical Computer Science, Vol. 85, pp. 155-170, 1991. 

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Function Approximat.ion with the Sweeping Hinge Algorithm Don R. Hush, Fernando Lozano Dept. of Elec. and Compo Engg. University of New Mexico Albuquerque, NM 87131 Bill Horne MakeWaves, Inc. 832 Valley Road Watchung, NJ 07060 Abstract We present a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the method of fitting the residual. The task of fitting individual nodes is accomplished using a new algorithm that searchs for the best fit by solving a sequence of Quadratic Programming problems. This approach offers significant advantages over derivative-based search algorithms (e.g. backpropagation and its extensions). Unique characteristics of this algorithm include: finite step convergence, a simple stopping criterion, a deterministic methodology for seeking "good" local minima, good scaling properties and a robust numerical implementation. 1 Introduction The learning algorithm developed in this paper is quite different from the traditional family of derivative-based descent methods used to train Multilayer Perceptrons (MLPs) for function approximation. First, a constructive approach is used, which builds the network one node at a time. Second, and more importantly, we use piecewise linear sigmoidal nodes instead of the more popular (continuously differentiable) logistic nodes. These two differences change the nature of the learning problem entirely. It becomes a combinatorial problem in the sense that the number of feasible solutions that must be considered in the search is finite. We show that this number is exponential in the input dimension, and that the problem of finding the global optimum admits no polynomial-time solution. We then proceed to develop a heuristic algorithm that produces good approximations with reasonable efficiency. This algorithm has a simple stopping criterion, and very few user specified parameters. In addition, it produces solutions that are comparable to (and sometimes better than) those produced by local descent methods, and it does so D. R. Hush, R Lozano and B. Horne 536 using a deterministic methodology, so that the results are independent of initial conditions. 2 Background and Motivation We wish to approximate an unknown continuous function f(x) over a compact set with a one-hidden layer network described by n f~(x) = ao + L (1) aiU(x, Wi) i=l where n is the number of hidden layer nodes (basis functions), x E ~d is the input vector, and {u(x, w)} are sigmoidal functions parameterized by a weight vector w. A set of example data, S = {Xi, Yi}, with a total of N samples is available for training and test. The models in (1) have been shown to be universal approximators. More importantly, (Barron, 1993) has shown that for a special class of continuous functions, r c, the generalization error satisfies ~ E[lIf - fn,NII2] IIf - fnll 2 + E[lIfn - fn,NII2] = 0 (*) +0 ( nd ~g N ) where 11?11 is the appropriate two-norm, f n is the the best n-node approximation to f, and fn,N is the approximation that best fits the samples in S. In this equation IIf - fnll 2 and E[lIfn - fn,NII2] correspond to the approximation and estimation error respectively. Of particular interest is the O(l/n) bound on approximation error, which for fixed basis functions is of the form O(1/n 2 / d ) (Barron, 1993). Barron's result tells us that the (tunable) sigmoidal bases are able to avoid the curse of dimensionality (for functions in rc). Further, it has been shown that the O(l/n) bound can be achieved constructively (Jones, 1992), that is by designing the basis functions (nodes) one at a time. The proof of this result is itself constructive, and thus provides a framework for the development of an algorithm which can (in principle) achieve this bound. One manifestation of this algorithm is shown in Figure 1. We call this the iterative approximation algorithm (I1A) because it builds the approximation by iterating on the residual (Le. the unexplained portion of the function) at each step. This is the same algorithmic strategy used to form bases in numerous other settings, e.g. Grahm-Schmidt, Conjugate Gradient, and Projection Pursuit. The difficult part of the I1A algorithm is in the determination of the best fitting basis function Un in step 2. This is the focus of the remainder of this paper. 3 Algorithmic Development We begin by defining the hinging sigmoid (HS) node on which our algorithms are based. An HS node performs the function w+, Uh(X, w) = { - Tw, X, w_, -T> w, X _ w+ - Tw_ ~ w, X ~ w+ T< w, x _ w_ (2) where w T = [WI w+ w_] and x is an augmented input vector with a 1 in the first component. An example of the surface formed by an HS node on a two-dimensional input is shown in Figure 2. It is comprised of three hyperplanes joined pairwise Function Approximation with the Sweeping Hinge Algorithm 537 Initialization: fo(x) = 0 for n 1 to nma:c do 1. Compute Residual: 2. Fit Residual: = en(x) = f(x) - fn-l (x) un(x) = argminO"EE lIen(x) - u(x)11 3. Update Estimate: fn(x) = o:fn-l (x) + f3u n (x) where 0: and f3 are chosen to minimize IIf(x) - fn(x)1I endloop Figure 1: Iterative Approximation Algorithm (rIA). Figure 2: A Sigmoid Hinge function in two dimensions . continuously at two hinge locations. The upper and middle hyperplanes are joined at "Hinge I" and the lower and middle hyperplanes are joined at "Hinge 2". These hinges induce linear partitions on the input space that divide the space into three regions, and the samples in 5 into three subsets, 5+ = {(Xi,Yi): Wr-Xi ~ w+} 5, = {(Xi,Yi): w_ ~ WTXi ~ w+} 5_ = {(Xi,Yi): WTXi ~ w_} (3) These subsets, and the corresponding regions of the input space, are referred to as the PLUS, LINEAR and MINUS subsets/regions respectively. We refer to this type of partition as a sigmoidal partition. A sigmoidal partition of 5 will be denoted P = {5+, 5" 5_}, and the set of all such partitions will be denoted II = {Pd. Input samples that fall on the boundary between two regions can be assigned to the set on either side. These points are referred to as hinge samples and playa crucial role in subsequent development. Note that once a weight vector w is specified, the partition P is completely determined, but the reverse is not necessarily true. That is, there are generally an infinite number of weight vectors that induce the same partition. We begin our quest for a learning algorithm with the development of an expression for the empirical risk. The empirical risk (squared error over the sample set) is defined (4) D. R Hush, F. Lozano and B. Horne 538 This expression can be expanded into three terms, one for each set in the partition, Ep(w) = ~ :E(Yi - W_)2 ~ + ~ :E(Yi - W+)2 + ~ 2)Yi - ~ WT Xi)2 ~ After further expansion and rearrangement of terms we obtain 1 Ep(w) = 2wTRw - w T r where + s; "L:s, XiX; r, = "L: s, XiYi s; = ! "L:s Y; st = "L:s+ Yi Sy = "L:s_ Yi R, = R= ( R, ~ r=un (5) (6) (7) (8) and N+ , N, and N_ are the number of samples in S+ , S, and S_ respectively. The subscript P is used to emphasize that this criterion is dependent on the partition (i.e. P is required to form Rand r). In fact, the nature of the partition plays a critical role in determining the properties of the solution. When R is positive definite (i.e. full rank), P is referred to as a stable partition, and when R has reduced rank P is referred to as an unstable partition. A stable partition requires that R, > O. For purposes of algorithm development we will assume that R, > 0 when ISti > Nmin, where Nmin is a suitably chosen value greater than or equal to d + 1. With this, a necessary condition for a stable partition is that there be at least one sample in S+ and S_ and N, ~ Nmin. When seeking a minimizing solution for Ep(w) we restrict ourselves to stable partitions because of the potential nonuniqueness associated with solutions to unstable partitions. Determining a weight vector that simultaneously minimizes E p (w) and preserves the current partition can be posed as a constrained optimization problem. This problem takes on the form - w Tr min !wTRw 2 subject to Aw ~ 0 (9) where the inequality constraints are designed to maintain the current partition defined by (3). This is a Quadratic Programming problem with inequality constraints, and because R > 0 it has a unique global minimum. The general Quadratic Programming problem is N P-hard and also hard to approximate (Bellare and Rogaway, 1993). However, the convex case which we restrict ourselves to here (i.e. R > 0) admits a polynomial time solution. In this paper we use the active set algorithm (Luenberger, 1984) to solve (9). With the proper implementation, this algorithm runs in O(k(~ + Nd)) time, where k is typically on the order of d or less. The solution to the quadratic programming problem in (9) is only as good as the current partition allows. The more challenging aspect of minimizing Ep(w) is in the search for a good partition. Unfortunately there is no ordering or arrangement of partitions that is convex in Ep(w), so the search for the optimal partition will be a computationally challenging problem. An exhaustive search is usually out of the question because of the prohibitively large number of partitions, as given by the following lemma. Lemma 1: Let S contain a total of N samples in Rd that lie in general position. Then the number of sigmoidal partitions defined in (3) is 8(Nd+l). Function Approximation with the Sweeping Hinge Algorithm 539 Proof: A detailed proof is beyond the scope of this paper, but an intuitive proof follows. It is well-known that the number of linear dichotomies of N points in d dimensions is 8(N d ) (Edelsbrunner, 1987). Each sigmoidal partition is comprised of two linear dichotomies, one formed by Hinge 1 and the other by Hinge 2, and these dichotomies are constrained to be simple translations of one another. Thus, to enumerate all sigmoidal partitions we allow one of the hinges, say Hinge 1, can take on 8(Nd) different positions. For each of these the other hinge can occupy only'" N unique positions. The total is therefore 8 (Nd+l ). The search algorithm developed here employs a Quadratic Programming (QP) algorithm at each new partition to determine the optimal weight vector for that partition (Le. the optimal orientation for the separating hyperplanes). Transitions are made from one partition to the next by allowing hinge samples to flip from one side of the hinge boundary to the next. The search is terminated when a minimum value of Ep(w) is found (Le. it can no longer be reduced by flipping hinge samples). Such an algorithm is shown in Figure 3. We call this the HingeDescent algorithm because it allows the hinges to "walk across" the data in a manner that descends the Ep(w) criterion. Note that provisions are made within the algorithm to avoid unstable partitions. Note also that it is easy to modify this algorithm to descend only one hinge at a time, simply by omitting one of the blocks of code that flips samples across the corresponding hinge boundary. {This routine is invoked with a stable feasible solution W = {w, R, r, A, S+, SI, S_ }.} procedure HingeDescent (W) { Allow hinges to walk across the data until a minimizing partition is found. } E_1wTRw-wTr - 2 do Emin = E {Flip Hinge 1 Samples.} for each ?Xi, Yi) on Hinge 1) do if ?Xi, Yi) E S+ and N+ > 1) then Move (Xi,Yi) from S+ to S" and update R, r, and A elseif ?Xi, Yi) E S, and N, > N min ) then Move (Xi, Yi) from S, to S+, and update R, r, and A endif endloop {Flip Hinge 2 Samples.} for each ?Xi, Yi) on Hinge 2) do if ?Xi,Yi) E S- and N_ > 1) then Move (Xi,Yi) from S- to S" and update R, r, and A elseif ?Xi, Yi) E S, and N, > Nmin) then Move (Xi,Yi) from S, to S-, and update R, r, and A endif endloop {Compute optimal solution for new partition.} W = QPSolve(W}; E= ~wTRw-wTr while (E < Emin) j return(W)j end; {HingeDescent} Figure 3: The HingeDescent Algorithm. Lemma 2: When started at a stable partition, the HingeDescent algorithm will D. R Hush, R Lozano and B. Horne 540 converge to a stable partition of Ep(w) in a finite number of steps. Proof: First note that when R> 0, a QP solution can always be found in a finite number of steps. The proof of this result is beyond the scope of this paper, but can easily be found in the literature (Luenberger, 1984). Now, by design, HingeDescent always moves from one stable partition to the next, maintaining the R > 0 property at each step so that all QP solutions can be produced in a finite number of steps. In addition, Ep(w) is reduced at each step (except the last one) so no partitions are revisited, and since there are a finite number of partitions (see Lemma 1) this algorithm must terminate in a finite number of steps. QED. Assume that QPSol ve runs in O(k( cP +N d)) time as previously stated. Then the run time of HingeDescent is given by O(Np((k+Nh)cP+kNd)), where Nh is the number of samples flipped at each step and Np is the total number of partitions explored. Typical values for k and Nh are on the order of d, simplifying this expression to O(Np(d 3 + NcP)). Np can vary widely, but is often substantially less than N. HingeDescent seeks a local minimum over II, and may produce a poor solution, depending on the starting partition. One way to remedy this is to start from several different initial partitions, and then retain the best solution overall. We take a different approach here, that always starts with the same initial condition, visits several local minima along the way, and always ends up with the same final solution each time. The SweepingHinge algorithm works as follows. It starts by placing one of the hinges, say Hinge 1, at the outer boundary of the data. It then sweeps this hinge across the data, M samples at a time (e.g. M = 1), allowing the other hinge (Hinge 2) to descend to an optimal position at each step. The initial hinge locations are determined as follows. A linear fit is formed to the entire data set and the hinges are positioned at opposite ends of the data so that the PLUS and MINUS regions meet the LINEAR region at the two data samples on either end. After the initial linear fit, the hinges are allowed to descend to a local minimum using HingeDescent. Then Hinge 1 is swept across the data M samples at a time. Mechanically this is achieved by moving M additional samples from S, to S+ at each step. Hinge 2 is allowed to descend to an optimal position at each of these steps using the Hinge2Descent algorithm. This algorithm is identical to HingeDescent except that the code that flips samples across Hinge 1 is omitted. The best overall solution from the sweep is retained and "fine-tuned" with one final pass through the HingeDescent algorithm to produce the final solution. The run time of SweepingHinge is no worse than N j M times that of HingeDescent. Given this, an upper bound on the (typical) run time for this algorithm (with M = 1) is O(NNp(d 3 + NcP)). Consequently, SweepingHinge scales reasonably well in both Nand d, considering the nature of the problem it is designed to solve. 4 Empirical Results The following experiment was adapted from (Breiman, 1993). The function lex) = e- lIx ll 2 is sampled at 100d points {xd such that IIxll ~ 3 and IIxll is uniform on [0,3]. The dimension d is varied from 4 to 10 (in steps of 2) and models of size 1 to 20 nodes are trained using the I1AjSweepingHinge algorithm. The number of samples traversed at each step of the sweep in SweepingHinge was set to M = 10. Nmin was set equal to 3d throughout. A refitting pass was employed after each new node was added in the I1A. The refitting algorithm used HingeDescent to "fine-tune" each node each node before adding the next node. The average sum of squared Function Approximation with the Sweeping Hinge Algorithm 3000 2500 2000 541 d=4 d=6 d=8 d=10 d=4 d=6 d=8 d=10 1500 1000 500 6 8 10 12 14 16 Number or Nodes 18 20 Figure 4: Upper (lower) curves are for training (test) data. error, e-2 , was computed for both the training data and an independent set of test data of size 200d. Plots of 1/e-2 versus the number of nodes are shown in Figure 4. The curves for the training data are clearly bounded below by a linear function of n (as suggested by inverting the O(l/n) result of Barron's). More importantly however, they show no significant dependence on the dimension d. The curves for the test data show the effect of the estimation error as they start to "bend over" around n = 10 nodes. Again however, they show no dependence on dimension. Acknowledgements This work was inspired by the theoretical results of (Barron, 1993) for Sigmoidal networks as well as the "Hinging Hyperplanes" work of (Breiman, 1993) , and the "Ramps" work of (Friedman and Breiman, 1994). This work was supported in part by ONR grant number N00014-95-1-1315. References Barron, A.R. (1993) Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory 39(3):930-945. Bellare, M. & Rogaway, P. (1993) The complexity of approximating a nonlinear program. In P.M. Pardalos (ed.), Complexity in numerical optimization, pp. 16-32, World Scientific Pub. Co. Breiman, L. (1993) Hinging hyperplanes for regression, classification and function approximation. IEEE Transactions on Information Theory 39(3):999-1013. Breiman, L. & Friedman, J.H. (1994) Function approximation using RAMPS. Snowbird Workshop on Machines that Learn. Edelsbrunner, H. (1987) In EATCS Monographs on Theoretical Computer Science V. 10, Algorithms in Combinatorial Geometry. Springer-Verlag. Jones, L.K. (1992) A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training. The Annals of Statistics, 20:608-613. Luenberger, D.G. (1984) Introduction to Linear and Nonlinear Programming. Addison-Wesley. 

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EM Algorithms for PCA and SPCA Sam Roweis? Abstract I present an expectation-maximization (EM) algorithm for principal component analysis (PCA). The algorithm allows a few eigenvectors and eigenvalues to be extracted from large collections of high dimensional data. It is computationally very efficient in space and time. It also naturally accommodates missing infonnation. I also introduce a new variant of PC A called sensible principal component analysis (SPCA) which defines a proper density model in the data space. Learning for SPCA is also done with an EM algorithm. I report results on synthetic and real data showing that these EM algorithms correctly and efficiently find the leading eigenvectors of the covariance of datasets in a few iterations using up to hundreds of thousands of datapoints in thousands of dimensions. 1 Why EM for peA? Principal component analysis (PCA) is a widely used dimensionality reduction technique in data analysis. Its popularity comes from three important properties. First, it is the optimal (in tenns of mean squared error) linear scheme for compressing a set of high dimensional vectors into a set of lower dimensional vectors and then reconstructing. Second, the model parameters can be computed directly from the data - for example by diagonalizing the sample covariance. Third, compression and decompression are easy operations to perfonn given the model parameters - they require only matrix multiplications. Despite these attractive features however, PCA models have several shortcomings. One is that naive methods for finding the principal component directions have trouble with high dimensional data or large numbers of datapoints. Consider attempting to diagonalize the sample covariance matrix of n vectors in a space of p dimensions when n and p are several hundred or several thousand. Difficulties can arise both in the fonn of computational complexity and also data scarcity. I Even computing the sample covariance itself is very costly, requiring 0 (np2) operations. In general it is best to avoid altogether computing the sample ? rowei s@cns . cal tech. edu; Computation & Neural Systems, California Institute of Tech. IOn the data scarcity front, we often do not have enough data in high dimensions for the sample covariance to be of full rank and so we must be careful to employ techniques which do not require full rank matrices. On the complexity front, direct diagonalization of a symmetric matrix thousands of rows in size can be extremely costly since this operation is O(P3) for p x p inputs. Fortunately, several techniques exist for efficient matrix diagonalization when only the first few leading eigenvectors and eigerivalues are required (for example the power method [10] which is only O(p2?. EM Algorithms for PCA and SPCA 627 covariance explicitly. Methods such as the snap-shot algorithm [7] do this by assuming that the eigenvectors being searched for are linear combinations of the datapoints; their complexity is O(n 3 ). In this note, I present a version of the expectation-maximization (EM) algorithm [1] for learning the principal components of a dataset. The algorithm does not require computing the sample covariance and has a complexity limited by 0 (knp) operations where k is the number of leading eigenvectors to be learned. Another shortcoming of standard approaches to PCA is that it is not obvious how to deal properly with missing data. Most of the methods discussed above cannot accommodate missing values and so incomplete points must either be discarded or completed using a variety of ad-hoc interpolation methods. On the other hand, the EM algorithm for PCA enjoys all the benefits [4] of other EM algorithms in tenns of estimating the maximum likelihood values for missing infonnation directly at each iteration. Finally, the PCA model itself suffers from a critical flaw which is independent of the technique used to compute its parameters: it does not define a proper probability model in the space of inputs. This is because the density is not nonnalized within the principal subspace. In other words, if we perfonn PCA on some data and then ask how well new data are fit by the model, the only criterion used is the squared distance of the new data from their projections into the principal subspace. A datapoint far away from the training data but nonetheless near the principal subspace will be assigned a high "pseudo-likelihood" or low error. Similarly, it is not possible to generate "fantasy" data from a PCA model. In this note I introduce a new model called sensible principal component analysis (SPCA), an obvious modification of PC A, which does define a proper covariance structure in the data space. Its parameters can also be learned with an EM algorithm, given below. In summary, the methods developed in this paper provide three advantages. They allow simple and efficient computation of a few eigenvectors and eigenvalues when working with many datapoints in high dimensions. They permit this computation even in the presence of missing data. On a real vision problem with missing infonnation, I have computed the 10 leading eigenvectors and eigenvalues of 217 points in 212 dimensions in a few hours using MATLAB on a modest workstation. Finally, through a small variation, these methods allow the computation not only of the principal subspace but of a complete Gaussian probabilistic model which allows one to generate data and compute true likelihoods. 2 Whence EM for peA? Principal component analysis can be viewed as a limiting case of a particular class of linearGaussian models. The goal of such models is to capture the covariance structure of an observed p-dimensional variable y using fewer than the p{p+ 1) /2 free parameters required in a full covariance matrix. Linear-Gaussian models do this by assuming that y was produced as a linear transfonnation of some k-dimensionallatent variable x plus additive Gaussian noise. Denoting the transfonnation by the p x k matrix C, and the ~dimensional) noise by v (with covariance matrix R) the generative model can be written as y = Cx+v x-N{O,I) v-N(O,R) (la) The latent or cause variables x are assumed to be independent and identically distributed according to a unit variance spherical Gaussian. Since v are also independent and nonnal distributed (and assumed independent of x), the model reduces to a single Gaussian model 2 All vectors are column vectors. To denote the transpose of a vector or matrix I use the notation x T . The determinant of a matrix is denoted by IAI and matrix inversion by A-1. The zero matrix is 0 and the identity matrix is I. The symbol", means "distributed according to". A multivariate normal (Gaussian) distribution with mean JL and covariance matrix 1:: is written as N (JL, 1::). The same Gaussian evaluated at the point x is denoted N (JL, 1::) Ix- 628 S. Roweis for y which we can write explicitly: + R) (lb) In order to save parameters over the direct covariance representation in p-space, it is necessary to choose k < p and also to restrict the covariance structure of the Gaussian noise v by constraining the matrix R.3 For example, if the shape of the noise distribution is restricted to be axis aligned (its covariance matrix is diagonal) the model is known asfactor analysis. y ",N (O,CC T 2.1 Inference and learning There are two central problems of interest when working with the linear-Gaussian models described above. The first problem is that of state inference or compression which asks: given fixed model parameters C and R, what can be said about the unknown hidden states x given some observations y? Since the datapoints are independent, we are interested in the posterior probability P (xly) over a single hidden state given the corresponding single observation. This can be easily computed by linear matrix projection and the resulting density is itself Gaussian: P( I ) = P(Ylx)P(x) = N(Cx,R)lyN(O,I)lx N(O,CCT+R)ly (2a) P (xly) = N ((3y,I - (3C) Ix , (3 = CT(CC T + R)-l (2b) from which we obtain not only the expected value (3y of the unknown state but also an estimate of the uncertainty in this value in the form of the covariance 1- (3C. Computing y from x (reconstruction) is also straightforward: P (ylx) = N (Cx, R) Iy. Finally, xy P(y) computing the likelihood of any datapoint y is merely an evaluation under (1 b). The second problem is that of learning, or parameter fitting which consists of identifying the matrices C and R that make the model assign the highest likelihood to the observed data. There are a family of EM algorithms to do this for the various cases of restrictions to R but all follow a similar structure: they use the inference formula (2b) above in the e-step to estimate the unknown state and then choose C and the restricted R in the m-step so as to maximize the expected joint likelihood of the estimated x and the observed y. 2.2 Zero noise limit Principal component analysis is a limiting case of the linear-Gaussian model as the covariance of the noise v becomes infinitesimally small and equal in all directions. Mathematically, PCA is obtained by taking the limit R = limf~O d. This has the effect of making the likelihood of a point y dominated solely by the squared distance between it and its reconstruction Cx. The directions of the columns of C which minimize this error are known as the principal components. Inference now reduces t04 simple least squares projection: P (xIY) =N ((3y,I - (3C) Ix , (3 = lim C T (CC T + d)-l f~O (3a) P (xly) = N ((CTC)-lC T y, 0) Ix = 6(x - (CTC)-lC T y) (3b) Since the noise has become infinitesimal, the posterior over states collapses to a single point and the covariance becomes zero. 3This restriction on R is not merely to save on parameters: the covariance of the observation noise must be restricted in some way for the model to capture any interesting or informative projections in the state x. If R were not restricted, the learning algorithm could simply choose C = 0 and then set R to be the covariance of the data thus trivially achieving the maximum likelihood model by explaining all of the structure in the data as noise. (Remember that since the model has reduced to a single Gaussian distribution for y we can do no better than having the covariance of our model equal the sample covariance of our data.) 4Recall that if C is p x k with p > k and is rank k then left multiplication by C T (CC T )-l (which appears not to be well defined because (CC T ) is not invertible) is exactly eqUivalent to left multiplication by (C T C) -1 CT. The intuition is that even though CC T truly is not invertible, the directions along which it is not invertible are exactly those which C T is about to project out. EM Algorithms for PCA and SPCA 3 629 An EM algorithm for peA The key observation of this note is that even though the principal components can be computed explicitly, there is still an EM algorithm for learning them. It can be easily derived as the zero noise limit of the standard algorithms (see for example [3, 2] and section 4 below) by replacing the usual e-step with the projection above. The algorithm is: ? e-step: ? m-step: x = (CTC)-lCTy cnew = YXT(XXT)-l where Y is a p x n matrix of all the observed data and X is a k x n matrix of the unknown states. The columns of C will span the space of the first k principal components. (To compute the corresponding eigenvectors and eigenvalues explicitly, the data can be projected into this k-dimensional subspace and an ordered orthogonal basis for the covariance in the subspace can be constructed.) Notice that the algorithm can be performed online using only a single datapoint at a time and so its storage requirements are only O(kp) + O(k2). The workings of the algorithm are illustrated graphically in figure 1 below. Gaussian Input Data ' .. Non-Gaussian Input Data -. ,.'" ., ' " '. l.?.? ~ , I ', ~ 0 . ~ 0 -I ',:', -I '. ';' . ~ , ": -2 : -2 .'.' " ( ,'I" ,,' -~3L--_'7'2-'-'--_~ I -~o----c~--:------: xl , I . . ~3~---~2-----~I--~O~--~--~--- xl Figure 1: Examples of iterations of the algorithm. The left panel shows the learning of the first principal component of data drawn from a Gaussian distribution, while the right panel shows learning on data from a non-Gaussian distribution. The dashed lines indicate the direction of the leading eigenvector of the sample covariance. The dashed ellipse is the one standard deviation contour of the sample covariance. The progress of the algorithm is indicated by the solid lines whose directions indicate the guess of the eigenvector and whose lengths indicate the guess of the eigenvalue at each iteration. The iterations are numbered; number 0 is the initial condition. Notice that the difficult learning on the right does not get stuck in a local minimum, although it does take more than 20 iterations to converge which is unusual for Gaussian data (see figure 2). The intuition behind the algorithm is as follows: guess an orientation for the principal subspace. Fix the guessed subspace and project the data y into it to give the values of the hidden states x. Nowfix the values ofthe hidden states and choose the subspace orientation which minimizes the squared reconstruction errors of the datapoints. For the simple twodimensional example above, I can give a physical analogy. Imagine that we have a rod pinned at the origin which is free to rotate. Pick an orientation for the rod. Holding the rod still, project every datapoint onto the rod, and attach each projected point to its original point with a spring. Now release the rod. Repeat. The direction of the rod represents our guess of the principal component of the dataset. The energy stored in the springs is the reconstruction error we are trying to minimize. 3.1 Convergence and Complexity The EM learning algorithm for peA amounts to an iterative procedure for finding the subspace spanned by the k leading eigenvectors without explicit computation of the sample S. Roweis 630 covariance. It is attractive for small k because its complexity is limited by 0 (knp) per iteration and so depends only linearly on both the dimensionality of the data and the number of points. Methods that explicitly compute the sample covariance matrix have complexities limited by 0 (np2), while methods like the snap-shot method that form linear combinations of the data must compute and diagonalize a matrix of all possible inner products between points and thus are limited by O(n 2 p) complexity. The complexity scaling of the algorithm compared to these methods is shown in figure 2 below. For each dimensionality, a random covariance matrix E was generated5 and then lOp points were drawn from N (0, E). The number of floating point operations required to find the first principal component was recorded using MATLAB'S flops function. As expected, the EM algorithm scales more favourably in cases where k is small and both p and n are large. If k ~ p ~ n (we want all the eigenvectors) then all methods are O(p3). The standard convergence proofs for EM [I] apply to this algorithm as well, so we can be sure that it will always reach a local maximum of likelihood. Furthennore, Tipping and Bishop have shown [8, 9] that the only stable local extremum is the global maximum at which the true principal subspace is found; so it converges to the correct result. Another possible concern is that the number of iterations required for convergence may scale with p or n. To investigate this question, I have explicitly computed the leading eigenvector for synthetic data sets (as above, with n = lOp) of varying dimension and recorded the number of iterations of the EM algorithm required for the inner product of the eigendirection with the current guess of the algorithm to be 0.999 or greater. Up to 450 dimensions (4500 datapoints), the number of iterations remains roughly constant with a mean of 3.6. The ratios of the first k eigenvalues seem to be the critical parameters controlling the number of iterations until convergence (For example, in figure I b this ratio was 1.0001.) Convergence Behaviour ~~metbod Sompli Covariance + Dill? Smtple Covariance only Figure 2: Time complexity and convergence behaviour of the algorithm. In all cases, the number of datapoints n is 10 times the dimensionality p. For the left panel, the number of floating point operations to find the leading eigenvector and eigenvalue were recorded. The EM algorithm was always run for exactly 20 iterations. The cost shown for diagonalization of the sample covariance uses the MATLAB functions cov and eigs. The snap-shot method is show to indicate scaling only; one would not normally use it when n > p . In the right hand panel, convergence was investigated by explicitly computing the leading eigenvector and then running the EM algorithm until the dot product of its guess and the true eigendirection was 0.999 or more. The error bars show ? one standard deviation across many runs. The dashed line shows the number of iterations used to produce the EM algorithm curve ('+') in the left panel. 5First, an axis-aligned covariance is created with the p eigenvalues drawn at random from a uniform distribution in some positive range. Then (p - 1) points are drawn from a p-dimensional zero mean spherical Gaussian and the axes are aligned in space using these points. EM Algorithms for PCA and SPCA 631 3.2 Missing data In the complete data setting, the values of the projections or hidden states x are viewed as the "missing information" for EM. During the e-step we compute these values by projecting the observed data into the current subspace. This minimizes the model error given the observed data and the model parameters. However, if some of the input points are missing certain coordinate values, we can easily estimate those values in the same fashion. Instead of estimating only x as the value which minimizes the squared distance between the point and its reconstruction we can generalize the e-step to: ? generalized e-step: For each (possibly incomplete) point y find the unique pair of points x? and y. (such that x? lies in the current principal subspace and y. lies in the subspace defined by the known information about y) which minimize the norm IICx? - y?lI. Set the corresponding column of X to x* and the corresponding column ofY to y ?. If y is complete, then y* = y and x* is found exactly as before. If not, then x* and y* are the solution to a least squares problem and can be found by, for example, QR factorization of a particular constraint matrix. Using this generalized e-step I have found the leading principal components for datasets in which every point is missing some coordinates. 4 Sensible Principal Component Analysis If we require R to be a multiple ?I of the identity matrix (in other words the covariance ellipsoid of v is spherical) but do not take the limit as E --t 0 then we have a model which I shall call sensible principal component analysis or SPCA. The columns of C are still known as the principal components (it can be shown that they are the same as in regular PC A) and we will call the scalar value E on the diagonal of R the global noise level. Note that SPCA uses 1 + pk - k(k - 1)/2 free parameters to model the covariance. Once again, inference is done with equation (2b). Notice however, that even though the principal components found by SPCA are the same as those for PCA, the mean of the posterior is not in general the same as the point given by the PCA projection (3b). Learning for SPCA also uses an EM algorithm (given below). Because it has afinite noise level E, SPCA defines a proper generative model and probability distribution in the data space: (4) which makes it possible to generate data from or to evaluate the actual likelihood of new test data under an SPCA model. Furthermore, this likelihood will be much lower for data far from the training set even if they are near the principal subspace, unlike the reconstruction error reported by a PCA model. The EM algorithm for learning an SPCA model is: ? e-step: {3 ? m-step: = CT(CCT + d)-l cnew = Y J..L~:E-l J..Lx Enew = (3Y = trace[XX T :Ex - = nI - n{3C + J..LxJ..L~ CJ..Lx yT]/n2 Two subtle points about complexity6 are important to notice; they show that learning for SPCA also enjoys a complexity limited by 0 (knp) and not worse. 6 First, since d is diagonal, the inversion in the e-step can be performed efficiently using the matrix inversion lemma: {CC T + d)-l = (I/f - C(I + CTC/f)-ICT /f 2 ). Second, since we are only taking the trace of the matrix in the m-step, we do not need to compute the fu\1 sample covariance XXT but instead can compute only the variance along each coordinate. 632 S. Roweis 5 Relationships to previous methods The EM algorithm for PCA, derived above using probabilistic arguments, is closely related to two well know sets of algorithms. The first are power iteration methods for solving matrix eigenvalue problems. Roughly speaking, these methods iteratively update their eigenvector estimates through repeated mUltiplication by the matrix to be diagonalized. In the case of PCA, explicitly forming the sample covariance and multiplying by it to perform such power iterations would be disastrous. However since the sample covariance is in fact a sum of outer products of individual vectors, we can multiply by it efficiently without ever computing it. In fact, the EM algorithm is exactly equivalent to performing power iterations for finding C using this trick. Iterative methods for partial least squares (e.g. the NIPALS algorithm) are doing the same trick for regression. Taking the singular value decomposition (SVD) of the data matrix directly is a related way to find the principal subspace. If Lanczos or Arnoldi methods are used to compute this SVD, the resulting iterations are similar to those of the EM algorithm. Space prohibits detailed discussion of these sophisticated methods, but two excellent general references are [5, 6]. The second class of methods are the competitive learning methods for finding the principal subspace such as Sanger's and Oja's rules. These methods enjoy the same storage and time complexities as the EM algorithm; however their update steps reduce but do not minimize the cost and so they typically need more iterations and require a learning rate parameter to be set by hand. Acknowledgements I would like to thank John Hopfield and my fellow graduate students for constant and excellent feedback on these ideas. In particular I am grateful to Erik Winfree for significant contributions to the missing data portion of this work, to Dawei Dong who provided image data to try as a real problem, as well as to Carlos Brody, Sanjoy Mahajan, and Maneesh Sahani. The work of Zoubin Ghahrarnani and Geoff Hinton was an important motivation for this study. Chris Bishop and Mike Tipping are pursuing independent but yet unpublished work on a virtually identical model. The comments of three anonymous reviewers and many visitors to my poster improved this manuscript greatly. References [I] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society series B, 39: 1-38, 1977. [2] B. S. Everitt. An Introducction to Latent Variable Models. Chapman and Hill, London, 1984. [3] Zoubin Ghahramani and Geoffrey Hinton. The EM algorithm for mixtures of factor analyzers. Technical Report CRG-TR-96-1 , Dept. of Computer Science, University of Toronto, Feb. 1997. [4] Zoubin Ghahramani and Michael I. Jordan. Supervised learning from incomplete data via an EM approach. In Jack D. Cowan, Gerald Tesauro, and Joshua Alspector, editors, Advances in Neural Information Processing Systems , volume 6, pages 120-127. Morgan Kaufmann, 1994. [5] Gene H. Golub and Charles F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore, MD, USA, second edition, 1989. [6] R. B. Lehoucq, D. C. Sorensen, and C. Yang. Arpack users' guide: Solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. Technical Report from http://www.caam.rice.edu/software/ARPACK/, Computational and Applied Mathematics, Rice University, October 1997. [7] L. Sirovich. Turbulence and the dynamics of coherent structures. Quarterly Applied Mathematics, 45(3):561-590, 1987. [8] Michael Tipping and Christopher Bishop. Mixtures of probabilistic principal component analyzers. Technical Report NCRG/97/003 , Neural Computing Research Group, Aston University, June 1997. [9] Michael Tipping and Christopher Bishop. Probabilistic principal component analysis. Technical Report NCRG/97/010, Neural Computing Research Group, Aston University, September 1997. [10] J. H. Wilkinson. The AlgebraiC Eigenvalue Problem. Claredon Press, Oxford, England, 1965. 

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Task and Spatial Frequency Effects on Face Specialization 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 Abstract There is strong evidence that face processing is localized in the brain. The double dissociation between prosopagnosia, a face recognition deficit occurring after brain damage, and visual object agnosia, difficulty recognizing otber kinds of complex objects, indicates tbat face and nonface object recognition may be served by partially independent mechanisms in the brain. Is neural specialization innate or learned? We suggest that this specialization could be tbe result of a competitive learning mechanism that, during development, devotes neural resources to the tasks they are best at performing. Furtber, we suggest that the specialization arises as an interaction between task requirements and developmental constraints. In this paper, we present a feed-forward computational model of visual processing, in which two modules compete to classify input stimuli. When one module receives low spatial frequency information and the other receives high spatial frequency information, and the task is to identify the faces while simply classifying the objects, the low frequency network shows a strong specialization for faces. No otber combination of tasks and inputs shows this strong specialization. We take these results as support for the idea that an innately-specified face processing module is unnecessary. 1 Background Studies of the preserved and impaired abilities in brain damaged patients provide important clues on how the brain is organized. Cases of prosop agnosia, a face recognition deficit often sparing recognition of non-face objects, and visual object agnosia, an object recognition deficit that can occur witbout appreciable impairment of face recognition, provide evidence that face recognition is served by a "special" mechanism. (For a recent review of this 18 M. N. Dailey and G. W. Cottrell evidence, see Moscovitch, Winocur, and Behrmann (1997?. In this study, we begin to provide a computational account of the double dissociation. Evidence indicates that face recognition is based primarily on holistic, configural information, whereas non-face object recognition relies more heavily on local features and analysis of the parts of an object (Farah, 1991; Tanaka and Sengco, 1997). For instance, the distance between the tip of the nose and an eye in a face is an important factor in face recognition, but such subtle measurements are rarely as critical for distinguishing, say, two buildings. There is also evidence that con figural information is highly relevant when a human becomes an "expert" at identifying individuals within other visually homogeneous object classes (Gauthier and Tarr, 1997). What role might configural information play in the development of a specialization for face recognition? de Schonen and Mancini (1995) have proposed that several factors, including different rates of maturation in different areas of cortex, an infant's tendency to track the faces in its environment, and the gradual increase in visual acuity as an infant develops, all combine to force an early specialization for face recognition. If this scenario is correct, the infant begins to form configural face representations very soon after birth, based primarily on the low spatial frequency information present in face stimuli. Indeed, Costen, Parker, and Craw (1996) showed that although both high-pass and low-pass image filtering decrease face recognition accuracy, high-pass filtering degrades identification accuracy more quickly than low-pass filtering. Furthermore, Schyns and Oliva (1997) have shown that when asked to recognize the identity of the "face" in a briefly-presented hybrid image containing a low-pass filtered image of one individual's face and a high-pass filtered image of another individual's face, subjects consistently use the lOW-frequency compone.nt of the image for the task. This work indicates that low spatial frequency information may be more important for face identification than high spatial frequency information. Jacobs and Kosslyn (1994) showed how differential availability of large and small receptive field sizes in a mixture of experts network (Jacobs, Jordan, Nowlan, and Hinton, 1991) can lead to experts that specialize for "what" and "where" tasks. In previous work, we proposed that a neural mechanism allocating resources according to their ability to perform a given task could explain the apparent specialization for face recognition evidenced by prosopagnosia (Dailey, Cottrell, and Padgett, 1997). We showed that a model based on the mixture of experts architecture, in which a gating network implements competitive learning between two simple homogeneous modules, could develop a specialization such that damage to one module disproportionately impaired face recognition compared to nonface object recognition. In the current study, we consider how the availability of spatial frequency information affects face recognition specialization given this hypothesis of neural resource allocation by competitive learning. We find that when high and low frequency information is "split" between the two modules in our system, and the task is to identify the faces while simply classifying the objects, the low-frequency module consistently specializes for face recognition. After describing the study, we discuss its results and their implications. 2 Experimental Methods We presented a modular feed-forward neural network preprocessed images of 12 different faces, 12 different books, 12 different cups, and 12 different soda cans. We gave the network two types of tasks: 1. Learning to recognize the superordinate classes of all four object types (hereafter referred to as classification). 2. Learning to distinguish the individual members of one class (hereafter referred to Task and Spatial Frequency Effects on Face Specialization 19 as identification) while simply classifying objects of the other three types. For each task, we investigated the effects of high and low spatial frequency information on identification and classification in a visual processing system with two competing modules. We observed how splitting the range of spatial frequency information between the two modules affected the specializations developed by the network. 2.1 Image Data We acquired face images from the Cottrell and Metcalfe facial expression database (1991 ) and captured multiple images of several books, cups, and soda cans with a CCD camera and video frame grabber. For the face images, we chose five grayscale images of each of 12 individuals. The images were photographed under controlled lighting and pose conditions; the subjects portrayed a different facial expression in each image. For each of the non-face object classes, we captured five different grayscale images of each of 12 books, 12 cups, and 12 cans. These images were also captured under controlled lighting conditions, with small variations in position and orientation between photos. The entire image set contained 240 images, each of which we cropped and scaled to a size of 64x64 pixels. 2.2 Image Preprocessing To convert the raw grayscale images to a biologically plausible representation more suitable for network learning and generalization, and to experiment with the effect of high and low spatial frequency information available in a stimulus, we extracted Gabor jet features from the images at multiple spatial frequency scales then performed a separate principal components analysis on the data from each filter scale separately to reduce input pattern dimensionality. 2.2.1 Gabor jet features The basic two-dimensional Gabor wavelet resembles a sinusoid grating restricted by a twodimensional Gaussian, and may be tuned to a particular orientation and sinusoidal frequency scale. The wavelet can be used to model simple cell receptive fields in cat primary visual cortex (Jones and Palmer, 1987). Buhmann, Lades, and von der Malsburg (1990) describe the Gabor "jet," a vector consisting of filter responses at multiple orientations and scales. We convolved each of the 240 images in the input data set with two-dimensional Gabor ? elg ? h t onentatlons . . (0 '8' 11' 11' 311' 11' 511' 311' 711') filters at fi ve scaIes m 4' 8' 2' T' 4' 8 an d sub sampled an 8x8 grid of the responses to each filter. The process resulted in 2560 complex numbers describing each image. 2.2.2 Principal components analysis To reduce the dimensionality of the Gabor jet representation while maintaining a segregation of the responses from each filter scale, we performed a separate PCA on each spatial frequency component of the pattern vector described above. For each of the 5 filter scales in the jet, we extracted the subvectors corresponding to that scale from each pattern in the training set, computed the eigenvectors of their covariance matrix, projected the sub vectors from each of the patterns onto these eigenvectors, and retained the eight most significant coefficients. Reassembling the pattern set resulted in 240 40-dimensional vectors. M. N. Dailey and G. W. Cottrell 20 Module 1 Inputs Figure 1: Modular network architecture. The gating network units mix the outputs of the hidden layers multiplicatively. 2.3 The Model The model is a simple modular feed-forward network inspired by the mixture of experts architecture (Jordan and Jacobs, 1995); however, it contains hidden layers and is trained by backpropagation of error rather than maximum likelihood estimation or expectation maximization. The connections to the output units come from two separate inputlhidden layer pairs; these connections are gated multiplicatively by a simple linear network with softmax outputs. Figure 1 illustrates the model's architecture. During training, the network's weights are adjusted by backpropagation of error. The connections from the softmax units in the gating network to the connections between the hidden layers and output layer can be thought of as multiplicative connections with a constant weight of 1. The resulting learning rules gate the amount of error feedback received by a module according to the gating network's current estimate of its ability to process the current training pattern. Thus the model implements a form of competitive learning in which the gating network learns which module is better able to process a given pattern and rewards the "winner" with more error feedback. 2.4 Training Procedure Preprocessing the images resulted in 240 40-dimensional vectors; four examples of each face and object composed a I92-element training set, and one example of each face and object composed a 48-element test set. We held out one example of each individual in the training set for use in determining when to stop network training. We set the learning rate for all network weights to 0.1 and their momentum to 0.5. Both of the hidden layers contained 15 units in all experiments. For the identification tasks, we determined that a mean squared error (MSE) threshold of 0.02 provided adequate classification performance on the hold out set without overtraining and allowed the gate network to settle to stable values. For the four-way classification task, we found that an MSE threshold of 0.002 was necessary to give the gate network time to stabilize and did not result in overtraining. On all runs reported in the results section, we simply trained the network until it reached the relevant MSE threshold. For each of the tasks reported in the results section (four-way classification, book identification, and face identification), we performed two experiments. In the first, as a control, both modules and the gating network were trained and tested with the fu1l40-dimensional pattern vector. In the second, the gating network received the full 40-dimensional vector, Task and Spatial Frequency Effects on Face Specialization 21 but module 1 received a vector in which the elements corresponding to the largest two Gabor filter scales were set to 0, and the elements corresponding to the middle filter scale were reduced by 0.5. Module 2, on the other hand, received a vector in which the elements corresponding to the smallest two filter scales were set to 0 and the elements corresponding to the middle filter were reduced by 0.5. Thus module 1 received mostly high-frequency information, whereas module 2 received mostly low-frequency information, with deemphasized overlap in the middle range. For each of these six experiments, we trained the network using 20 different initial random weight sets and recorded the softmax outputs learned by the gating network on each training pattern. 3 Results Figure 2 displays the resulting degree of specialization of each module on each stimulus class. Each chart plots the average weight the gating network assigns to each module for the training patterns from each stimulus class, averaged over 20 training runs with different initial random weights. The error bars denote standard error. For each of the three reported tasks (four-way claSSification, book identification, and face identification), one chart shows division of labor between the two modules in the control situation, in which both modules receive the same patterns, and the other chart shows division of labor between the two modules when one module receives low-frequency information and the other receives highfrequency information. When required to identify faces on the basis of high- or lOW-frequency information, compared with the four-way-classification and same-pattern controls, the lOW-frequency module wins the competition for face patterns extremely consistently (lower right graph). Book identification specialization, however, shows considerably less sensitivity to spatial frequency. We have also performed the equivalent experiments with a cup discrimination and a can discrimination task. Both of these tasks show a low-frequency sensitivity lower than that for face identification but higher than that for book identification. Due to space limitations, these results are not presented here. The specialized face identification networks also provide good models of prosopagnosia and visual object agnosia: when the face-specialized module's output is "damaged" by removing connections from its hidden layer to the output layer, the overall network's generalization performance on face identification drops dramatically, while its generalization performance on object recognition drops much more slowly. When the non-face-specialized (high frequency) module'S outputs are damaged, the opposite effect occurs: the overall network's performance on each of the object recognition tasks drops, whereas its performance on face identification remains high. 4 Discussion The results in Figure 2 show a strong preference for low-frequency information in the face identification task, empirically demonstrating that, given a choice, a competitive mechanism will choose a module receiving low-frequency, large receptive field information for this task. This result concurs with the psychological evidence for configural face representations based upon low spatial frequency information, and suggests how the developing brain could be biased toward a specialization for face recognition by the infant's initially low visual acuity. On the basis of these results, we predict that human subjects performing face and object 22 M. N Dailey and G. W. Cottrell Classification (split frequencies) Classification (control) i '0; ~ i : 1.0 \.0 0.8 0.8 0.6 1m Module 1 a Module 2 0.4 ~ i"11 r ~ CModule 1 (highfreq) ? Module 2 (low freq) 0.6 0.4 ~ 0.2 0.2 0.0 0.0 Faces Books Cups Cans Faces Books Cups Cans SdmulusType Stimulus Type Bookid task (control) Book id task (split frequencies) \.0 1.0 0.8 0.8 j j -; f 0.6 r:::::J Module 1 . . Module 2 0.4 -; 0.6 r 0.4 : 0.2 0.2 0.0 0.0 Faces Books Cups Cans Faces Books Cups Cans Sdmul.. Type Stimulus Type Face id task (control) r : (highfreq) . . Module 2 (low freq) ~ ~ i-; c:I Module 1 Face id task (split frequencies) 1.0 1.0 0.8 0.8 i 0.6 C!I Module 1 . . Module 2 0.4 ~ -; r Cl Module 1 0.6 (high freq) IilII Module 2 0.4 (low freq) ~ 0.2 0.2 0.0 0.0 Faces Books Cup. Cans Stimulus Type Faces Books Cups Cans Stimulus Type Figure 2: Average weight assigned to each module broken down by stimulus class. For each task, in the control experiment, each module receives the same pattern; the split-frequency charts summarize the specialization resulting when module 1 receives high-frequency Gabor filter information and module 2 receives low-frequency Gabor filter information. Task and Spatial Frequency Effects on Face Specialization 23 identification tasks will show more degradation of performance in high-pass filtered images of faces than in high-pass filtered images of other objects. To our knowledge, this has not been empirically tested, although Costen et al. (1996) have investigated the effect of highpass and low-pass filtering on face images in isolation, and Parker, Lishman, and Hughes (1996) have investigated the effect of high-pass and low-pass filtering of face and object images used as 100 ms cues for a same/different task. Their results indicate that relevant high-pass filtered images cue object processing better than low-pass filtered images, but the two types of filtering cue face processing equally well. Similarly, Schyns & Oliva's (1997) results described earlier suggest that the human face identification network preferentially responds to low spatial frequency inputs. Our results suggest that simple data-driven competitive learning combined with constraints and biases known or thought to exist during visual system development can account for some of the effects observed in normal and brain-damaged humans. The study lends support to the claim that there is no need for an innately-specified face processing module face recognition is only "special" insofar as faces form a remarkably homogeneous category of stimuli for which Within-category discrimination is ecologically beneficial. References Buhmann, J., Lades, M., 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. Costen, N., Parker, D., and Craw, I. (1996). Effects of high-pass and low-pass spatial filtering on face identification. Perception & Psychophysics, 38(4):602-612. Cottrell, G. and Metcalfe, J. (1991). Empath: Face, gender and emotion recognition using holons. In Lippman, R., Moody, J., and Touretzky, D., editors, Advances in Neural Information ProceSSing Systems 3, pages 564-571. Dailey, M., Cottrell, G., and Padgett, C. (1997). A mixture of experts model exhibiting pro sop agnosia. In Proceedings of the Nineteenth Annual Conference of the Cognitive Science Society, pp. 155-160. Stanford, CA, Mahwah: Lawrence Erlbaum. de Schonen, S. and Mancini, J. (1995). About functional brain specialization: The development of face recognition. TR 95.1, MRC Cognitive Development Unit, London, UK. Farah, M. (1991). Patterns of co-occurrence among the associative agnosias: Implications for visual object representation. Cognitive Neuropsychology, 8:1-19. Gauthier, I. and Tarr, M. (1997). Becoming a "greeble" expert: Exploring mechanisms for face recognition. Vision Research. In press. Jacobs, R. and Kosslyn, S. (1994). Encoding shape and spatial relations - The role of receptive field size in coordinating complementary representations. Cognitive Science, 18(3):361-386. Jacobs, R, Jordan, M., Nowlan, S., and Hinton, G. (1991). Adaptive mixtures of local experts. Neural Computation, 3:79-87. Jones, J. and Palmer, L. (1987). An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex. 1 Neurophys., 58(6):1233-1258. Moscovitch, M., Winocur, G., and Behrmann, M. (1997). What is special about face recognition? Nineteen experiments on a person with visual object agnosia and dyslexia but normal face recognition. Journal of Cognitive Neuroscience, 9(5):555-604. Parker, D., Lishman, J., and Hughes, J. (1996). Role of coarse and fine spatial information in face and object processing. Journal of Experimental Psychology: Human Perception and Performance, 22(6):1445-1466. Schyns, P. and Oliva, A. (1997). Dr. Angry and Mr. Smile: The multiple faces of perceptual categorizations. Submitted for publication. Tanaka, J. and Sengco, 1. (1997). Features and their configuration in face recognition. Memory and Cognition. In press. 

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301 ENCODING GEOMETRIC INVARIANCES IN HIGHER-ORDER NEURAL NETWORKS C.L. Giles Air Force Office of Scientific Research, Bolling AFB, DC 20332 R.D. Griffin Naval Research Laboratory, Washington, DC 20375-5000 T. Maxwell Sachs-Freeman Associates, Landover, MD 20785 ABSTRACT We describe a method of constructing higher-order neural networks that respond invariantly under geometric transformations on the input space. By requiring each unit to satisfy a set of constraints on the interconnection weights, a particular structure is imposed on the network. A network built using such an architecture maintains its invariant performance independent of the values the weights assume, of the learning rules used, and of the form of the nonlinearities in the network. The invariance exhibited by a firstorder network is usually of a trivial sort, e.g., responding only to the average input in the case of translation invariance, whereas higher-order networks can perform useful functions and still exhibit the invariance. We derive the weight constraints for translation, rotation, scale, and several combinations of these transformations, and report results of simulation studies. INTRODUCTION A persistent difficulty for pattern recognition systems is the requirement that patterns or objects be recognized independent of irrelevant parameters or distortions such as orientation (position, rotation, aspect), scale or size, background or context, doppler shift, time of occurrence, or signal duration. The remarkable performance of humans and other animals on this problem in the visual and auditory realms is often taken for granted, until one tries to build a machine with similar performance. Thoufh many methods have been developed for dealing with these problems, we have classified them into two categories: 1) preprocessing or transformation (inherent) approaches, and 2) case-specific or "brute force" (learned) approaches. Common transformation techniques include: Fourier, Hough, and related transforms; moments; and Fourier descriptors of the input signal. In these approaches the signal is usually transformed so that the subsequent processing ignores arbitrary parameters such as scale, translation, etc. In addition, these techniques are usually computationally expensive and are sensitive to noise in the input signal. The "brute force" approach is exemplified by training a device, such as a perceptron, to classify a pattern independent of it's position by presenting the @ American Institute of Physics 1988 302 training pattern at all possible positions. MADALINE machines 2 have been shown to perform well using such techniques. Often, this type of invariance is pattern specific, does not easily generalize to other patterns, and depends on the type of learning algorithm employed. Furthermore, a great deal of time and energy is spent on learning the invariance, rather than on learning the signal. We describe a method that has the advantage of inherent invariance but uses a higher-order neural network approach that must learn only the desired signal. Higher-order units have been shown to have unique computational strengths and are quite amenable to the encoding of a priori know1edge. 3 - 7 MATHEMATICAL DEVELOPMENT Our approach is similar to the group invariance approach,8,10 although we make no appeal to group theory to obtain our results. We begin by selecting a transformation on the input space, then require the output of the unit to be invariant to the transformation. The resulting equations yield constraints on the interconnection weights, and thus imply a particular form or structure for the network architecture. For the i-th unit Yi of order M defined on a discrete input space, let the output be given by Yi[YiM(X),P(x)] - f( WiO + ~ Wi 1 (X1) P(x1) + ~~ Wi 2 (X1,X2) P(x1) P(x2) + ... +~ ... ~ Wi M(X1,? ?XM) P(x1)? ?P(XM) ), (1) where p(x) is the input pattern or signal function (sometimes called a pixel) evaluated at position vector x, wim(xl, ... Xm) is the weight of order m connecting the outputs of units at Xl, x2, .. Xm to the ith unit, i.e., it correlates m values, f(u) is some threshold or sigmoid output function, and the summations extend over the input space. YiM(X) represents the entire set of weights associated with the i-th unit. These units are equivalent to the sigma-pi units a defined by Rumelhart, Hinton, and Williams. 7 Systems built from these units suffer from a combinatorial explosion of terms, hence are more complicated to build and train. To reduce the severity of this problem, one can limit the range of the interconnection weights or the number of orders, or impose various other constraints. We find that, in addition to the advantages of inherent invariance, imposing an invariance constraint on Eq. (1) reduces the number of allowed aThe sigma-pi neural networks are multi-layer networks with higher-order terms in any layer. As such, most of the neural networks described here can be considered as a special case of the sigma-pi units. However, the sigma-pi units as originally formulated did not have invariant weight terms, though it is quite simple to incorporate such invariances in these units. 303 weights, thus simplifying the architecture and shortening the training time. We now define what we mean by invariance. The output of a unit is invariant with respect to the transformation T on the input pattern if 9 (2) An example of the class of invariant response defined by Eq. (2) would be invariant detection of an object in the receptive field of a panning or zooming camera. An example of a different class would be invariant detection of an object that is moving within the field of a fixed camera. One can think of this latter case as consisting of a fixed field of "noise" plus a moving field that contains only the object of interest. If the detection system does not respond to the fixed field, then this latter case is included in Eq. (2). To illustrate our method we derive the weight constraints for one-dimensional translation invariance. We will first switch to a continuous formulation, however, for reasons of simplicity and generality, and because it is easier to grasp the physical significance of the results, although any numerical simulation requires a discrete formulation and has significant implications for the implementation of our results. Instead of an index i, we now keep track of our units with the continuous variable u. With these changes Eq. (2) now becomes y[u;wM(x),p(X)] = f( wO + JrdXl Wl(U;Xl) P(xl) + ... + f?? Jr dXl? .dXM wM(U;Xl,? ?XM) P(Xl)? .P(XM) ), (3) The limits on the integrals are defined by the problem and are crucial in what follows. Let T be a translation of the input pattern by -xO, so that (4) T[p(x)] - p(x+XO) where xo is the translation of the input pattern. Ty[u;wM(x) ,p(x)] - y[u;YM(x),p(x+XO?) = Then, from eq (2), y[u;wM(x),p(x)] (5) Since p(x) is arbitrary we must impose term-by-term equality in the argument of the threshold function; i.e., f dXl Wl(U;Xl) P(xl) = f dxl Wl(U;Xl) P(xl+XO), (Sa) Jr fdxl dX2 W2 (U;Xl,X2) P(xl) P(x2) = Jr f dXl dX2 W2 (U;Xl,X2) P(xl+XO) P(x2+XO), etc. (Sb) 304 Making the substitutions xl. xl-XO, x2 ",x2-XO, etc, we find that f dXl Wl(U;Xl) f f dxl P(xl) - f dxl WI(U;Xl-XO) P(XI) , (6a) dX2 W2 (U;XI,X2) P(xI) P(x2) - f f dXI dX2 W2 (U;XI-XO,X2-XO) P(xI) P(x2), (6b) etc. Note that the limits of the integrals on the right hand side must be adjusted to satisfy the change-of-variables. If the limits on the integrals are infinite or if one imposes some sort of periodic boundary condition, the limits of the integrals on both sides of the equation can be set equal. We will assume in the remainder of this paper that these conditions can be met; normally this means the limits of the integrals extend to infinity. (In an implementation, it is usually impractical or even impossible to satisfy these requirements, but our simulation results indicate that these networks perform satisfactorily even though the regions of integration are not identical. This question must be addressed for each class of transformation; it is an integral part of the implementation design.) Since the functions p(x) are arbitrary and the regions of integration are the same, the weight functions must be equal. This imposes a constraint on the functional form of the weight functions or, in the discrete implementation, limits the allowed connections and thus the number of weights. In the case of translation invariance, the constraint on the functional form of the weight functions requires that w1(U;XI) - wl(u;X].-XO), w2(U;XI,X2) - w2(U;XI-XO,X2-XO), (7a) (7b) etc. These equations imply that the first order weight is independent of input position, and depends only on the output position u. The second order weight is a function only of vector differences,IO i.e., w1(u;Xj) - J..(u), (8a) w2(U;X].,X2) - w2(u:X]. -Xl)? (8b) For a discrete implementation with N input units (pixels) fully connected to an output unit, this requirement reduces the number of second-order weights from order N2 to order N, i.e., only weights for differences of indexes are needed rather than all unique pair combinations. Of course, this advantage is multiplied as the number of fully-connected output units increases. FURTHER EXAMPLES We have applied these techniques to several other transformations of interest. For the case of transformation of scale 305 define the scale operator S such that Sp(x) - aIlp(ax) (9) where a is the scale factor, and x is a vector of dimension n. The factor an is used for normalization purposes, so that a given figure always contains the same "energy" regardless of its scale. Application of the same procedure to this transformation leads to the following constraints on the weights: wl(u;Xjfa) -= wl(u;~, w2(u;X1Ia,xv'a) .. w2(u;'X.l.'~)' w3(u;xlla,x2/a ,x3/a) ... w3(U;X].,X2,X3), etc. (lOa) (lOb) (lOc) Consider a two-dimensional problem viewed in polar coordinates (r,t). A set of solutions to these constraints is J.(u;q,tI) - w1(u;Q), w2(u;rl,r2;tl,t2) - w2(u;rllr2;tl,t2). w3 (u;rl,r2,r3;tl,t2,t3) - w3 (u;(rl-r2)/r3;tl,t2,t3). (lla) (llb) (llc) Note that with increasing order comes increasing freedom in the selection of the functional form of the weights. Any solution that satisfies the constraint may be used. This gives the designer additional freedom to limit the connection complexity, or to encode special behavior into the net architecture. An example of this is given later when we discuss combining translation and scale invariance in the same network. Now consider a change of scale for a two-dimensional system in rectangular coordinates, and consider only the second-order weights. A set of solutions to the weight constraint is: W2 (U;Xl,Yl;X2,Y2) - W2 (U;Xl/Yl;X2/Y2), - W2 (U;Xl/X2;Yl/Y2), W2 (U;Xl,Yl;X2,Y2) - w2 (U;(Xl-X2)/(Yl-Y2)), etc. W2 (U;Xl,Yl;X2,Y2) (12a) (l2b) (12c) We have done a simulation using the form of Eq. (12b). The simulation was done using a small input space (8x8) and one output unit. A simple least-mean-square (back-propagation) algorithm was used for training the network. When taught to distinguish the letters T and C at one scale, it distinguished them at changes of scale of up to 4X with about 15 percent maximum degradation in the output strength. These results are quite encouraging because no special effort was required to make the system work, and no corrections or modifications were made to account for the boundary condition requirements as discussed near Eq. (6). This and other simulations are discussed further later. As a third example of a geometric transformation, consider the case of rotation about the origin for a two-dimensional space in polar coordinates. One can readily show that the weight constraints 306 are satisfied if wl(u;rl,tl) ~ wl(u;rl), w2(u;rl,r2;tl,t2) - w2(u;rl,r2;tl-t2), etc. (13a) (l3b) These results are reminiscent of the results for translation invariance. This is not uncommon: seemingly different problems often have similar constraint requirements if the proper change of variable is made. This can be used to advantage when implementing such networks but we will not discuss it further here. An interesting case arises when one considers combinations of invariances, e.g., scale and translation. This raises the question of the effect of the order of the transformations, i.e., is scale followed by translation equivalent to translation followed by scale? The obvious answer is no, yet for certain cases the order is unimportant. Consider first the case of change-of-scale by a, followed by a translation XC; the constraints on the weights up to second order are: Wl(U;Xl) - wl(u; (xl-xo)/a), w2 (u; Xl ,x2) 0= w2(u; (xl-xo)/a, (x2-xo)/a) , (14a) (l4b) and for translation followed by scale the constraints are: wl(u;Xl) - wl(u; (xl/a)-xo). and (lSa) w2(U;Xl,X2) = w2(u;(xl/a)-xo,(x2Ia )-XO) . (lSb) Consider only the second-order weights for the two-dimensional case. Choose rectangular coordinate variables (x,y) so that the translation is given by (xO,YO). Then W2 (U;Xl,Yl;X2,Y2) = w2 (u;(xl/a)-xO,(Yl/a)-YO;(x2/a)-xO'(Y2/a)-yO)' (l6a) W2 (U;Xl,Yl;X2,Y2) w2 (U;(Xl- x o)/a, (Yl-yo)/a; (x2- xo)/a, (Y2-Yo)/a). (16b) or If we take as our solution w2(U;Xl,Yl;X2,Y2) = w2(U;(X1-X2)/(Yl-Y2?, (17) then w2 is invariant to scale and translation, and the order is unimportant. With higher-order weights one can be even more adventurous. As a final example consider the case of a change of scale by a factor a and rotation about the origin by an amount to for a twodimensional system in polar coordinates. (Note that the order of transformation makes no difference.) The weight constraints up to second order are: (18a) 307 (18b) The first-order constraint requires that wI be independent of the input variables, but for the second-order term one can obtain a more useful solution: (19) This implies that with second-order weights, one can construct a unit that is insensitive to changes in scale and rotation of the input space. How useful it is depends upon the application. SIMULATION RESULTS We have constructed several higher-order neural networks that demonstrated invariant response to transformations of scale and of translation of the input patterns. The systems were small, consisting of less than 100 input units, were constructed from second-and first-order units, and contained only one, two, or three layers. We used a back-propagation algorithm modified for the higher-order (sigma-pi) units. The simulation studies are still in the early stages, so the performance of the networks has not been thoroughly investigated. It seems safe to say, however, that there is much to be gained by a thorough study of these systems. For example, we have demonstrated that a small system of second-order units trained to distinguish the letters T and C at one scale can continue to distinguish them over changes in scale of factors of at least four without retraining and with satisfactory performance. Similar performance has been obtained for the case of translation invariance. Even at this stage, some interesting facets of this approach are becoming clear: 1) Even with the constraints imposed by the invariance, it is usually necessary to limit the range of connections in order to restrict the complexity of the network. This is often cited as a problem with higher-order networks, but we take the view that one can learn a great deal more about the nature of a problem by examining it at this level rather than by simply training a network that has a general-purpose architecture. 2) The higher-order networks seem to solve problems in an elegant and simple manner. However, unless one is careful in the design of the network, it performs worse than a simpler conventional network when there is noise in the input field. 3) Learning is often "quicker" than in a conventional approach, although this is highly dependent on the specific problem and implementation design. It seems that a tradeoff can be made: either faster learning but less noise robustness, or slower learning with more robust performance. DISCUSSION We have shown a simple way to encode geometric invariances into neural networks (instead of training them), though to be useful the networks must be constructed of higher-order units. The invariant encoding is achieved by restricting the allowable network 308 architectures and is independent of learning rules and the form of the sigmoid or threshold functions. The invariance encoding is normally for an entire layer, although it can be on an individual unit basis. It is easy to build one or more invariant layers into a multi-layer net, and different layers can satisfy different invariance requirements. This is useful for operating on internal features or representations in an invariant manner. For learning in such a net, a multi-layered learning rule such as generalized backpropagation 7 must be used. In our simulations we have used a generalized back-propagation learning rule to train a two-layer system consisting of a second-order, translation-invariant input layer and a first-order output layer. Note that we have not shown that one can not encode invariances into layered first-order networks, but the analysis in this paper implies that such invariance would be dependent on the form of the sigmoid function. When invariances are encoded into higher-order neural networks, the number of interconnections required is usually reduced by orders of powers of N where N is the size of the input. For example, a fully connected, first-order, single-layer net with a single output unit would have order N interconnections; a similar second-order net, order N2 . If this second-order net (or layer) is made shift invariant, the order is reduced to N. The number of multiplies and adds is still of order N2 . We have limited our discussion in this paper to geometric invariances, but there seems to be no reason why temporal or other invariances could not be encoded in a similar manner. REFERENCES 1. D.H. Ballard and C.M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, NJ, 1982). 2. B. Widrow, IEEE First Int1. Conf. on Neural Networks, 87TH019l7, Vol. 1, p. 143, San Diego, CA, June 1987. 3. J.A. Feldman, Biological Cybernetics 46, 27 (1982). 4. C.L. Giles and T. Maxwell, App1. Optics 26, 4972 (1987). 5. G.E. Hinton, Proc. 7th IntI. Joint Conf. on Artificial Intelligence, ed. A. Drina, 683 (1981). 6. Y.C. Lee, G. Doolen, H.H. Chen, G.Z. Sun, T. Maxwell, H.Y. Lee, C.L. Giles, Physica 22D, 276 (1986). 7. D.E. Rume1hart, G.E. Hinton, and R.J. Williams, Parallel Distributed Processing, Vol. 1, Ch. 8, D.E. Rume1hart and J.L. McClelland, eds., (MIT Press, Cambridge, 1986). 309 8. T . Maxwell, C.L. Giles, Y.C. Lee, and H.H. Chen, Proc. IEEE IntI. Conf. on Systems, Man, and Cybernetics, 86CH2364-8, p. 627, Atlanta, GA, October 1986. 9. W. Pitts and W.S. McCulloch, Bull. Math. Biophys. 9, 127 (1947). 10. M. Minsky and S, Papert, Perceptrons (MIT Press, Cambridge, Mass., 1969). 

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20 ASSOCIATIVE LEARNING VIA INHIBITORY SEARCH David H. Ackley Bell Communications Research Cognitive Science Research Group ABSTRACT ALVIS is a reinforcement-based connectionist architecture that learns associative maps in continuous multidimensional environments. The discovered locations of positive and negative reinforcements are recorded in "do be" and "don't be" subnetworks, respectively. The outputs of the subnetworks relevant to the current goal are combined and compared with the current location to produce an error vector. This vector is backpropagated through a motor-perceptual mapping network. to produce an action vector that leads the system towards do-be locations and away from don 't-be locations. AL VIS is demonstrated with a simulated robot posed a target-seeking task. INTRODUCTION The "backpropagation algorithm" or generalized delta rule (Rumelhart, Hinton, & Williams, 1986) is sometimes criticized on the grounds that it is a "supervised" learning algorithm, which requires a "teacher" to provide correct outputs, and apparently leaves open the question of how the teacher learned the right answers. However, work. by Rumelhart (personal communication, 1987) and Miyata (1988) has shown how the environment that a system is embedded in can serve as the "teacher." If, as in this paper, a backpropagation network is posed the task of mapping from a vector 9 of robot arm joint angles to the resulting vector X of arm coordinates in space (the "forward kinematics problem"), then input-output training data can be obtained by supplying sets of joint angles to the arm and observing the resulting configurations. Although this "environment as teacher" strategy shows how a "teacher" can come to possess useful information without an infinite regress learning it, it is not a complete solution. There are problems for which the "laws of physics" of an environment do not suffice to determine the solution. Suppose, for example, that a robot is posed the problem of learning to reach for different positions in space depending on which of a set of signals is currently presented, and that the only feedback available from the environment is success or failure information about the current arm configuration. Associative Learning via Inhibitory Search d Figure 1. A "trunk" robot. What is needed in such a case is a mechanism to search through the space of possible arm configurations, recording the successful configurations associated with the various inputs. ALVIS - Associative Learning Via Inhibitory Search - provides one such mechanism. The next section applies backpropagation to the 9 --+ X mapping and shows how the resulting network can sometimes be used to solve X --+ 9 problems. The third section, "Self-supervision and inhibitory search," integrates that network into the overall ALVIS algorithm. The final section contains some discussion and conclusions. An expanded version of this paper may be found in Ackley (1988). FORWARD AND INVERSE KINEMATICS The in verse kinematics problem in controlling an arm is the problem of determining what joint angles are needed to produce a specific position and orientation of a hand. In the general case it is a difficult problem. An itch on your back suggests the kinds of questions that arise. Which hand should you use? Should you go up from around your waist, or down from over your shoulders? Can you be sure you know what will work without actually trying it? From a computational standpoint, forward kinematics - deciding where your limbs will end up given a set of joint angles - is an easier problem. Figure 1 depicts the planar "robot" that was used in this work. I call it the "trunk" robot. (The work discussed in Ackley (1988) also used a two-handed "pincer" robot .) Of course, the trunk is a far cry from a real robot, and the only significant constraint is that the possible joint angles are limited, but this suffices to pose non-trivial kinematics problems. The trunk has five joints, and each joint angle is limited to a range of 4 0 to 176 0 with respect to the previous limb. I simulated a backpropagation network with five real-valued input units (the joint angles), sixty hidden units in a single layer, and twelve linear output units (the Cartesian joint positions). Joint angles were expressed in radians, so the range of input unit values was from about 0.07 to about 3.07. The configuration vector X was represented by twelve output units corresponding to six pairs of (x, y) coordi- 21 22 Ackley Figure 2. A "logarithmic strobe" display of the trunk's asymptotic convergence on a specified position and orientation. The arm position is displayed after iterations 1,2,4,8, ... ,256. nates, one pair for each joint a through f. With the trunk robot (though not with the "pincer") f:e and fy have constant values, since they end up being part of the "anchor." The state of an output unit equals the sum of its inputs, and the error propagated out of an output unit equals the error propagated into it. Errors were defined by the difference between the predicted configuration and the actual configuration, and after extensive training on the trunk robot forward kinematics problem, the network achieved high accuracy over most of the joint ranges. In typical backpropagation applications, once the desired mapping has been learned, the backward "error channels" in the network are no longer used. However, suppose some other error computation, different from that used to train the weights, was then incorporated. Those errors can be propagated from the outputs of the trained network all the way back to the inputs. The goal is no longer to change weights in the network - since they already represent a useful mapping - but to use the trained network to translate output-space errors, however defined, into errors at the inputs to the network. Figure 2 illustrates one use of this process, showing how a trained forward kinematics network can be used to perform a cheap kind of inverse kinematics. The figure shows superimposed outputs of the trained network under the influence of a task-specific error computationj in this case the trunk is trying to reduce the distances between the front and back of a "target arrow" and the front and back of its first arm section. The target arrow is defined by a head (h:e, hy) and a tail (t:e, ty). The errors for output units a:e and ay are defined by e(a:e} = h:e - a:e and e(ay ) hy-ay , the errors for output units b:e and by are defined by e(b:e) t:e-b:e and e(by ) = ty - by, and the errors at all other outputs are set to zero. = = The algorithm used to generate this behavior has the following steps: 1. Compute errors for one or more output units based on the current positions of the joints and the desired positioning and orienting information. If "close Associative Learning via Inhibitory Search o Figure 3. The trunk kinking itself. enough" to the target, exit, otherwise, store these errors on the selected output units, and set the other error terms to zero. 2. Backpropagate the errors all the way through the network to the input units. This produces an error term e(9i) for each joint angle 9i. Produce new joint angles: 9~ = 9i + ke(9i) where k is a scaling constant. Clip the joint angles against their minimum and maximum values. 3. Forward propagate through the network based on the new joint angles, to produce new current positions for the joints. Go to step 1. Whereas the training phase has forward propagation of activations (states) followed by back propagation of errors, this usage reverses the order. Backpropagation of errors is followed by changes in inputs followed by forward propagation of activations. This is a general gradient descent technique usable when a backpropagation network can learn to map from a control space 9 to an error or evaluation space X. Figure 3 illustrates how gradient descent's familiar limitation can manifest itself: The target is reachable but the robot fails to reach it. The initial configuration was such that while approaching the target, the trunk kinked itself too short to reach. If the robot had "thought" to open 93 instead of closing it, it could have succeeded. In that sense, the problem arises because the error computation only specified errors for the tip of the trunk, and not for the rest of the arm. If, instead, there were indications where all of the joints were to be placed, failures due to local minima could be greatly reduced. SELF-SUPERVISION AND INHIBITORY SEARCH The feedback control network of the previous section locally minimizes joint position errors - however they are generated - by translating them into joint angle space and moving downhill. AL VIS uses the feedback control network for arm control; this section shows how ALVIS learns to generate appropriate joint position space errors given only a reinforcement signal. There are two key points. The first is this: Once an action producing a positive reinforcement has somehow been found, 23 24 Ackley the problem reduces to associative mapping between the input and the discovered correct output. In ALVIS, "do-be units" are used to record such successes. The second point is this: When negative reinforcement occurs, the current configuration can be associated with the input in a behavior-reversed fashion - as a place to avoid in the future. In ALVIS, "don't-be units" are used to record such failures. The overall idea, then, is to perform inhibitory search by remembering failures as they occur and avoiding them in the future, and to perform associative learning by remembering successful configurations as the search process uncovers them and recreating them in the future. In effect, ALVIS constructs input-dependent "attractors" at arm configurations associated with success and "repellors" at configurations associated with failure. Figure 4 summarizes the algorithm. A few points to note are these: ? The do-be and don't-be units use the spherical non-linear function explored by Burr & Hanson (1987). The response of a spherical unit is maximal and equal to one when the input vector and the weight vector are identical. The response of the unit decreases monotonically with the Euclidean distance between the two vectors, and the radius r governs the rate of decay. ? The don't-be units of each subnetwork (i.e., relevant to one goal) are in a competitive network (see, e.g., Feldman 1982). The don't-be unit with the largest activation value (which is a function of both the distance from the current position and the radius) is the only don't-be unit that has effects on the rest of the system. In the simulations reported here, I used m 4 don't-be units per goal. = ? In addition to the parameters associated with spherical units, each do-be and don't-be unit has a strength parameter a that specifies how much influence the unit has over the behavior of the arm. Do-be strength (at) grows logarithmically with positive reinforcement and shrinks linearly with negative; don't-be strength (a;i) grows logarithmically with negative reinforcement and shrinks linearly with positive. Figure 5 illustrates a situation from early in a run of the system. From left to right, the three displays show the state of the relevant do-be unit, the relevant don't-be subnetwork, and the current configuration of the arm. Since this particular goal has never been achieved before, the do-be map provides no useful information its weight vector contains small random values (as it happens, the origin is below and right of the display) and its strength is zero. The display of the don't-be map shows the positions of all four relevant don't-be units, with the currently selected don't-be (unit number 3) drawn somewhat darker. The don't-be units are spread around configuration space, creating "hills" that push away the arm if it comes too close. As the arm moves about without reaching the target, different don't-be units win the competition and take control. Negative reinforcement accrues, and the winning don't-be consequently moves toward the various current configurations and gets stronger, until the arm is pushed elsewhere. Figure 6 illustrates the behavior of the system after more extensive learning. The Associative Learning via Inhibitory Search Figure 4. AL VIS O. (Initialize) Given: a space X of h dimensions, a backpropagation network trained on 8 -+ X, a set G of goals and a mapping from G to regions of space. Create an AL VIS network with n = IGI goal units 91, ... ,9n, n dobe/don't-be subnetworks consisting of one do-be unit and m don't-be units dii, ... , dfu., and h current position units ZI,'" Zh. Create modifiable connections Wzi from z's to d's, Wiz from d's to z's, and a modifiable strength s for each d. Set all do-be strengths st and all don't-be strengths sti to zero. Set all weights Wzi and Wiz to small random values. Set 8 to a random legal vector and produce a current configuration X. 1. (New stimulus) Choose t at random from 1 ... n. dt 2. (Do's/Don'ts) Compute activations for do-be's and don't-be's using the spherical function: let dt: d= 1/ (1 + ~JL./:=1 (Zi - wz il 2 ) . In subnetwork t, be the unit with the largest activation. dt 3. (Errors) Let wt denote the weights from and w~ denote the weights from dt:, and similarly for strengths st and st:. Compute errors for each component of X: e(zi) = si(wt - Zi) + Si:.(Zi - w~). 4. (Move) Backpropagate to produce e(8). Generate angle changes: A8i = min(q, max( -q, k,.e(8 i ))), with parameters q and k,.. Generate new angles respecting the maximum vt and minimum vi possible joint angles: 8~ = min( vt, max(vi ,8i + A8i )). Forward propagate to produce a new configuration X'. 5. (Positive reinforcement) Determine whether X' satisfies goal t. If it does not, go to step 6. Otherwise, +, - I d wzi +, -- zi'I 5 . 1 ror 1 - 1, ??? , hIt , e wiz - zi an 5.2 Let st' = min (5, si + p+ / (1 + si)), for positive reinf p+ > O. 5.3 Let ri = k,/ max(.I, st'), with parameter k,. 5.4 For i = 1, ... , m, let Sti' = max(O, sti-p+), and rti = k,f max(.I, sti'l. 5.5 Go to step 1. 6. (Negative reinforcement) Perform the following: = w~ + 77(zi - w~) and = w;i + + 6.1 For i = 1, ... , h, let 77(Z; - w~) with parameter 77, where is a uniform random variable between ?O.O1. 6.2 Let = min (5, s;: + p- / (1 + s;:)), for negative reinf p- > O. 6.3 Let r t = k,/ max ( .1, S;:'). 6.4 Let st' = max(O, si - p-), and ri = k,/ max(O.I, Si'l. 6.5 Go to step 2. t;\ ? - wi: S;:' e w;/ e 25 26 Ackley ~~~Be~:~0~.~00~O~OO~0___________ I_D_on_'_t_~~:_3~1~.~29~4~70~3________ I_B~e____________________ 1 Figure 5. A display of the internal state of the trunk robot in the process of learning to associate a set of twelve arbitrary stimuli with specified positions in space. The current signal (though the system has not discovered this yet) means "touch 5". Do Be: 4.998667 Don't Be: 3 0.007976 ~~~~~~----------I~~~~~~--------- Be Figure 6. A display of the internal state of the same trunk robot later in the learning process. The previous goal was "touch *" and the current goal is "touch 3." \ do-be map now contains an accurate image of a successful configuration for the "touch 3" goal, and its strength is high. The strength of the selected don't-be unit is low. The current configuration map in Figure 6 shows each iteration of the algorithm between the time it achieved its previous goal and the time it achieved the current goal. Finally, Figure 7 displays the average time-per-goal as a function of the number of goals achieved. For 75 repetitions, the trunk network was initialized and run until 500 goals had been achieved, and the resulting time-per-goal data was averaged to produce the graph. The average time-per-goal declines rapidly as goals are presented, then seems to rise slightly, and then stabilizes around an average value of about 300. To have some kind of standard of comparison, albeit unsophisticated, if the joint angles are simply changed by uniform random values between q and -q (see Figure 4) on each iteration, the average time-per-goal is observed to be about 490. Associative Learning via Inhibitory Search :nOD 24DO 2100 liDO 15DD 1200 900 600 300 Figure 7. A graph of the average time taken per goal as a function of the number of goals achieved. The horizontal line shows the average performance of random joint changes. DISCUSSION ALVIS is a preliminary, exploratory system. Of course, the ALVIS environment is but a pale shadow of the real world, but even granting the limited scope of the problem formulation, several aspects of AL VIS are incompletely satisfying and in need of improvement. To my eye, the biggest drawback of the current implementation is the local goal representation - which essentially requires that the set of goals be enumerable at network definition time. Related problems include the inability to share information between one goal and another and the inability to pursue more than one goal simultaneously. To determine behavior, the constraints of goals must be integrated with the possibilities of the current situation. In AL VIS this is done as a strictly two-step process: the goal selects a subnetwork, and the current situation selects units within the subnetwork. Work such as Jordan (1986) and Miyata (1988) shows how goal information and context information can be integrated by supplying both as inputs to a single network. ALVIS is a pure feedback control model, and can suffer from the traditional problem of that approach: when the errors are small, the resulting joint angle changes are small, and the arm converges only slowly. If the gain at the joints is increased to speed con vergence, overshoot and oscillation become more likely. However, in ALVIS oscillations gradually die out, as don't-be units shift positions under the negative reinforcement, and sometimes such temporary oscillations actually help with the search, causing the tip of the arm to explore a variety of different points. 27 28 Ackley The aspect of AL VIS behavior that I find most irritating reveals something about the approach in general. In some cases - usually on more "peripheral" targets - AL VIS learns to hit the very edge of the target region. While approaching such targets, ALVIS experiences negative reinforcement, and the don't-be units, consequently, gain a little strength. The resulting interference occasionally causes a very long search for a goal that had previously been rapidly achieved. AL VIS has a representation only for its own body; a better system would also be able to represent other objects in the world, and useful relations on the expanded set. The "mostly motor" emphasis evident in the present system needs to be balanced by more sophistication on the perceptual side. Though limited in scope, ALVIS demonstrates three ideas I think worth highlightmg: ? The reuse of the error channel of a backpropagation network, after training, for translating arbitrary output-space gradients into input-space gradients. ? The recording of previous actual outputs to be used as future desired outputs. ? The use of "repellors" (don 't-be units) as well as attractors in defining errors, and the resulting process of search-by-inhibition generated by negative reinforcement. Characterizing the behavior of a machine in terms of attractor dynamics is a familiar notion, but "repellor dynamics" seems to be largely unknown territory. Indeed, in ALVIS there is an ephemeral quality to the don't-be units: When all answers have been discovered, all strength accrues to the do-be's, good performances become routine, and AL VIS behavior is essentially attractor-based. In watching such a "grown-up" AL VIS, it is easy to forget how it was in the beginning, when the world was big and answers were scarce, and ALVIS was doing well just to discover a new mistake. References Ackley, D.H. (1988). Associative learning via inhibitory search. Teclmical memorandum TM ARH-012509. Morristown. NJ: Bell Communications Research. Burr. D.J .? & Hanson. S.J. (1987). Knowledge representation in connectionist networks. Technical memorandum TM-ARH-008733. Morristown. NJ: Bell Communications Research. Feldman. J.A. (1982). Dynamic connections in neural networks. Biological Cybernetics, 36, 193202. Jordan. M.1. (1986). Serial order: A parallel. distributed processing approach. Technical report ICS-86M. La Jolla, CA: University of California. Institute for Cognitive Science. Miyata, Y. (1988). The learning and planning of actions. Unpublished doctoral dissertation in psychology, University of California San Diego. Rumelhart, D.E .? Hinton. G.E .? & Williams. R.J. (1986). propagating errors. Nature, 3Z3. 533-536. Learning representations by back- Rumelhart. D.E. (personal communication. 1987). Also cited as personal communication in Miyata (1988). 

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Synchronized Auditory and Cognitive 40 Hz Attentional Streams, and the Impact of Rhythmic Expectation on Auditory Scene Analysis Bill Baird Dept Mathematics, U.C.Berkeley, Berkeley, Ca. 94720. baird@math.berkeley.edu Abstract We have developed a neural network architecture that implements a theory of attention, learning, and trans-cortical communication based on adaptive synchronization of 5-15 Hz and 30-80 Hz oscillations between cortical areas. Here we present a specific higher order cortical model of attentional networks, rhythmic expectancy, and the interaction of hi~her? order and primar?, cortical levels of processing. It accounts for the' mismatch negativity' of the auditory ERP and the results of psychological experiments of Jones showing that auditory stream segregation depends on the rhythmic structure of inputs. The timing mechanisms of the model allow us to explain how relative timing information such as the relative order of events between streams is lost when streams are formed. The model suggests how the theories of auditory perception and attention of Jones and Bregman may be reconciled. 1 Introduction Amplitude patterns of synchronized "gamma band" (30 to 80 Hz) oscillation have been observed in ilie ensemble activity (local field potentials) of vertebrate olfactory, visual, auditory, motor, and somatosensory cortex, and in the retlna, thalamus, hippocampus, reticular formation, and EMG. Such activity has not only been found in primates, cats, rabbits and rats, but also insects, slugs, fish, amphibians, reptiles, and birds. This suggests that gamma oscillation may be as fundamental to neural processing at the network level as action potentials are at the cellular level. We have shown how oscillatory associative memories may be coupled to recognize and generate sequential behavior, and how a set of novel mechanisms utilizing these complex dynamics can be configured to solve attentional and perceptual processing problems. For pointers to full treatment wi th mathematics and complete references see [BaIrG et al., 1994]. An important element of intra-cortical communication in the brain, and between modules in this architecture, is the ability of a module to detect and respond to the proper input signal from a particular module, when inputs from other modules which are irrelevant to tlie present computation are contributing crosstalk noise. We have demonstrated that selective control of synchronization, which we hypothesize to be a model of "attention", can be used to solve this coding problem and control program flow in an architecture with dynamic attractors [Baird et al., 1994]. Using dynamical systems theory, the architecture is constructed from recurrently interconnected oscillatory associative memory modules that model higher order sensory and motor areas of cortex. The modules learn connection weights between themselves which cause the system to evolve under a 5-20 Hz clocked sensory-motor processing cycle by a sequence 4 B. Baird of transitions of synchronized 30-80 Hz oscillatory attractors within the modules. The architecture employ's selective"attentional" control of the synchronization of the 30-80 Hz gamma band OSCIllations between modules to direct the flow of computation to recognize and generate sequences. The 30-80 Hz attractor amplitude patterns code the information content of a cortIcal area, whereas phase and frequency are used to "softwire" the network, since only the synchronized areas communicate by exchanging amplitude information. The system works like a broadcast network where the unavoidable crosstalk to all areas from previous learned connections is overcome by frequency coding to allow the moment to moment operation of attentional communication only between selected task-relevant areas. The behavior of the time traces in different modules of the architecture models the temporary appearance and switching of the synchronization of 5-20 and 30-80 Hz oscillations between cortical areas that is observed during sensorimotor tasks in monkeys and numans. The architecture models the 5-20 Hz evoked potentials seen in the EEG as the control signals which determine the sensory-motor processing cycle. The 5-20 Hz clocks which drive these control signals in the archItecture model thalamic pacemakers which are thought to control the excitabili ty of neocortical tissue through similar nonspecific biasing currents that cause the cogni tive and sensory evoked potentials of the EEG. The 5-20 Hz cycles "quantize time" and form the basis of derived somato-motor rhythms with periods up to seconds that entrain to each other in motor coordination and to external rhythms in speech perception [Jones et al., 1981]. 1.1 Attentional Streams of Synchronized 40 Hz Activity There is extensive evidence for the claim of the model that the 30-80 Hz gamma band activity in the brain accomplishes attentional processing, since 40 Hz appears in cortex when and where attention is required. For example, it is found in somatosensory, motor and premotor cortex of monkeys when they must pick a rasin out of a small box, but not when a habitual lever press delivers the reward. In human attention experiments, 30-80 Hz activity goes up in the contralateral auditory areas when subjects are mstructed to pay attention to one ear and not the other. Gamma activity declines in the dominant hemisphere along with errors in a learnable target and distractors task, but not when the distractors and target vary at random on each trial. Anesthesiologists use the absence of 40 Hz activity as a reliable indicator of unconsciousness. Recent work has shown that cats with convergent and divergent strabismus who fail on tasks where perceptual binding is required also do not exhibit cortical synchrony. This is evidence that gamma synchronization IS perceptually functional and not epiphenomenal. The architecture illustrates the notion that synchronization of gamma band activity not only"binds" the features of inputs in primary sensory cortex into "objects", but further binds the activity of an attended object to oscillatory activity in associational and higher-order sensory and motor cortical areas to create an evolving attentional network of intercommunicating cortical areas that directs behavior. The binding of sequences of attractor transitions between modules of the architecture by synchronization of their activity models the physiological mechanism for the formation of perceptual and cognitive "streams" investigated by Bregman [Bregman, 1990], Jones [Jones et aI., 1981], and others. In audition, according to Bregman's work, successive events of a sound source are bound together into a distinct sequence or "stream" and segy:egated from other sequences so that one pays attention to onl?, one sound source at a time (the cocktail party problem). Higher order cortical or "cognitive' streams are in evidence when subjects are unable to recall the relative order of the telling of events between two stories told in alternating segments. MEG tomographic observations show large scale rostral to caudal motor-sensory sweeps of coherent thalamo-cortical40Hz activity accross the entire brain, the phase of which is reset by sensory input in waking, but not in dream states [Llinas and Ribary, 1993]. This suggests an inner higher order "attentional stream" is constantly cycling between motor (rostral) and sensory (caudal) areas in the absence of input. It may be interrupted by input "pop out" from primary areas or it may reach down as a "searchlight" to synchromze with particular ensembles of primary activity to be attended. 2 Jones Theory of Dynamic Attention Jones [Jones et al., 1981] has developed a psychological theory of attention,perception, and motor timing based on the hypotheSIS that these processes are organized by neural rhythms in the range of 10 to .5 Hz - the range within which subjects perceive pen odic events as a rhythm. These rhythms provide a multi scale representation of time and selectively synchronize with the prominant periodiCities of an input to provide a -t emporal expectation mechanism for attention to target particular points in time. 40 Hz Attentional Streams, Rhythmic Expectation, and Auditory Scene Analysis 5 For example, some work suggests that the accented parts of speech create a rhythm to which listeners entrain. Attention can then be focused on these expected locations as recognition anchor points for inference of less prominant parts of the speech stream. This is the temporal analog of the body centered spatial coordinate frame and multiscale covert attention window system in vision. Here the body centered temporal coordinates of the internal time base orient by entrainment to the external rhythm, and the window of covert temporal attention can then select a level of the multiscale temporal coordinates. In this view, just as two cortical areas must synchronize to communicate, so must two nervous systems. Work using frame by frame film analysis of human verbal interaction, shows evidence of "interactional synchrony" of gesture and body movement changes and EEG of both speaker and listener With the onsets of phonemes in speech at the level of a 10 Hz "microrhythm" - the base clock rate of our models. Normal infants synchronize their spontaneous body ftailings at this 10 Hz level to the mothers voice accents, while autistic and s~hitzophrenic children fail to show interactional synchrony. Autistics are unable to tap in orne t9 a metronome. Neural expectation rhythms that support Jones' theory have been found in the auditory EEG. In experiments where the arrival time of a target stimulus is regular enough to be learned by an experimental subject, it has been shown that the 10 Hz activity in advance of the stimulus becomes phase locked to that expected arrival time. This fits our model of rhythmic expectation where the 10Hz rhythm IS a fast base clock that is shifted in phase and frequency to produce a match in timmg between the stimulus arrival and the output of longer period cycles derived from this base clock. 2.1 Mismatch Negativity The "mismatch negativity" (MNN) [Naatanen, 1992] of the auditory evoked potential appears to be an important physiological indicator of the action of a neural expectancy system like that proposed by Jones. It has been localized to areas within primary auditory cortex by MEG studies [Naatanen, 1992] and it appears as an increased negativity of the ERP in the region of the N200 peak whenever a psycbologically discriminable deviation of a repetitive auditory stimulus occurs. Mismatch IS caused by deviations in onset or offset time, rise time, frequency, loudness, timbre, phonetic structure, or spatial location of a tone in the sequence. The mismatch is abolished by blockers of the action ofNMDA channels [Naatanen, 1992] which are important for the synaptic changes underlying the kind of Hebbian learning which is used in the model. MNN is not a direct function of echoic memory because it takes several repetitions for the expectancy to begin to develop, and it decays 10 2 - 4 seconds. It appears only for repetition periods greater that 50-100 msec and less than 2-4 seconds. Thus the time scale of its operation is 10 the appropriate range for Jones' expectancy system. Stream formation also takes several cycles of stimulus repetition to builo up over 2-4 seconds and decays away within 2-4 seconds in the absence of stimulation. Those auditory stimulus features which cause streaming are also features which cause mismatch. This supports the hypothesis in the model that these phenomena are functionally related. Finally, MNN can occur independent of attention - while a subject is reading or doing a visual discrimination task. ThIS implies that the auditory system at least must have its own timing system that can generate timmg and expectancies independent of other behavior. We can talk or do internal verbal thinking while doing other tasks. A further component of this negativity appears in prefrontal cortex and is thought by Nataanen to initiate attentional switchmg toward the deVIant event causing perceptual "pop out" [Naatanen, 1992]. Stream formation is known to affect rhythm perception. The galloping rhythm of high H and low L tones - HLH-HLH-HLH, for example becomes two separate Isochronous rhythmic streams of H-H-H-Hand L-L-L-L when the H and L tones are spread far enough apart [Bregman, 1990]. Evidence for the effect of in'putrhythms on stream formation, however, is more sparse, and we focus here on the simulatIOn of a particular set of experiments by Jones [J ones et al., 1981] and Bregman [Bregman, 1990] where this effect has been demonstrated. 2.2 Jones-Bregman Experiment Jones [Jones et al., 1981] replicated and altered a classic streaming experiment of Bregman and Rudnicky [Bregman, 1990], and found that their result depended on a specific choice of the rhythm of presentation. The experiment required human subjects to determine of the order of presentation of a pair of high target tones AB or BA of slIghtly different frequencies. Also presented before and after the target tones were a series of identical much lower frequency tones called the capture tones CCC and two identical tones of intermediate fre- B. Baird 6 quency before and after the target tones called the flanking tones F - CCCFABFCCC. Bregman and Rudnicky found that target order determination performance was best when tfie capture tones were near to the flanking tones in frequency, and deteriorated as the captor tones were moved away. Their explanation was that the flanking tones were captured by the background capture tone stream when close in frequency, leav10g the target tones to stand out by themselves in the attended stream. When the captor tones were absent or far away in frequency, the flanking tones were included in the attended stream and obscured the target tones. Jones noted that the flanking tones and the capture stream were presented at a stimulus onset rate of one per 240 ms and the targets appeared at 80 ms intervals. In her experiments, when the captor and flanking tones were given a rhythm in common with the targets, no effect of the distance of captor and flanking tones appeared. This suggested that rfiythmic distinction of targets and dlstractors was necessary 10 addition to the frequency diStinction to allow selective attention to segregate out the target stream. Because performance in the single rhythm case was worse than that for the control condition without captors, it appeared that no stream segregation of targets and captors and flanking tones was occurring until the rhythmic difference was added. From this evidence we malie the assumption in the model that the distance of a stimulus in time from a rhythmic expectancy acts like the distance between stimuli in pitch, loudness, timbre, or spatial location as/actor for theformation of separate streams. 3 Architecture and Simulation To implement Jones's theory in the model and account for her data, subsets of the oscillatory modules are dedicated to form a rhythmic temporal coordinate frame or time base by dividing down a thalamic 10 Hz base clock rate in steps from 10 to .5 Hz. Each derived clock is created by an associative memory module that has been specialized to act stereotypically as a counter or shift register by repeatedly cycling through all its attractors at the rate of one for each time step of its clock. Its overall cycle time is therefore determined by the number of attractors. Each cycle is guaranteed to be identical, as required for clocklike function, because of the strong attractors that correct the perturbing effect of noise. Only one step of the cycle can send output back to primary cortex - the one with the largest weight from receiving the most matco to incoming stimuli. Each clock derived in this manner from a thalamic base clock will therefore phase reset itself to get the best match to incoming rhythms. The match can be further refined by frequency and phase entrainment of the base clock itself. Three such counters are sufficient to model the rhythms in Jones' experiment as shown in the architecture of figure 1. The three counters divide the 12.5 Hz clock down to 6.25 and 4.16 Hz. The first contains one attractor at the base clock rate which has adapted to entrain to the 80 msec period of target stimulation (12.5 Hz). The second cycles at 12.5/2 = 6.25 Hz, alternating between two attractors, and the third steps through three attractors, to cycle at 12.5/3 = 4.16 Hz, which is the slow rhythm of the captor tones. The modules of the time base send theirinternal30-80 Hz activity to {>rimary auditory cortex in 100msec bursts at these different rhythmic rates through fast adapting connections (which would use NMDA channels in the brain) that continually attempt to match incoming stimulus patterns using an incremental Hebbian learning rule. The weights decay .to zero over 2-4 sec to simulate the data on the rise and fall of the mismatch negativity. These weights effectively compute a low frequency discrete Fourier transform over a sliding window of several seconds, and the basic periodic structure of rhythmic patterns is quickly matched. This serves to establish a quantized temporal grid of expectations against which expressive timing deviations in speech and music can be experienced. Following Jones [Jones et al., 1981], we hypothesize that this happens automatically as a constant adaptation to environmental rhythms, as suggested by the mismatch negativity. Retained in these weights of the timebase is a special k10d of short term memory of the activity which includes temporal information since the timebase will partially regenerate the prevIous activity in primary cortex at the expected recurrence time. This top-down input causes enchanced sensitivity in target units by increasing their gain. Those patterns which meet these established rhythmic expectancy signals in time are thereby boosted in amplitude and pulled into synchrony with the 30-80 Hz attentional searchlight stream to become part of the attentional network sending input to higher areas. In accordance with Jones' theory, voluntary top-down attention can probe input at different hierarchical levels of periodicity by selectively synchronizing a particular cortical column in the time base set to the 40 Hz frequency of the inner attention stream. Then the searchlight into primary cortex is synchro- 40 Hz Attentional Streams, Rhythmic Expectation, and Auditory Scene Analysis 7 Dynamic Attention Architecture Higher Order AuditoryCortex Synchronization Timebase 10 Hz Clock w=, :J'~:' : '0" ~ ~" Pitch .' :' ~ ~ Cycle ..... ~: : :" : :. '---...,. Attentionallnput Stream Rhythmic Searchlight -.-.- . -.-.-.-.-.-.- . -.-.-.-.-.~ High _._._._._._._._._.~ Target Tones " -'1 B ~. ? ? - . II. @H I _ . - ' ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? -~ Fast Weights :-. 1 '1 _ . ~F I '_._._._._ . _._._._._._._._._._.~ High Flanking Tones Low Captor Tones ? - f ? - _.-._. C Input ~ I I 1':'."", .:::::::(~~ ~::-:-: I._._._._._._._._ ..._. __ ? I .............. \ I __.:-._. I - '. C ._._._._._._ . _.r ' C ._._._._~ ' ?- "1 Primary Auditory Cortex ~----------------------------------------------------~Time Figure 1: Horizontally arrayed units at the top model higher order auditory and motor cortical columns which are sequentially clocked by the (thalamic) base clock on the right to alternate attractor transitions between upper hidden (motor) and lower context (sensory) layers to act as an Elman net. Three cortical regions are shown - sequence representation memory, attentional synchronization control, and a rhythmic timebase of three counters. The hidden and context layers consist of binary "units" composed of two oscillatory attractors. Activity levels oscillate up and down through the plane of the paper. Dotted hnes show frequency shifting outputs from the synchromzation (attention) control modules. The lower vertical set of units IS a sample of primary auditory cortex frequency channels at the values used in the Jones-Bregman expenment. The dashed lines show the rhythmic pattern of the target, flanking, and captor tones moving in time from left to right to Impact on auditory cortex. nizing and reading in activity occuring at the peaks of that particular time base rhythm. 3.1 Cochlear and Primary Cortex Model At present, we have modeled only the minimal aspects of primary auditory conex sufficient to qualitatively simulate the Jones-Bregman experiment, but the principles at work allow expansion to larger scale models with more stimulus features. We simulate four sites in auditory conex corresponding to the four frequencies of stimuli used in the experiment, as shown in figure 1. There are two close high frequency target tones, one high flanking frequency location, and the low frequency location of the captor stream. These cortical locations are modeled as oscillators with the same equations used for associative memory modules [Baird et al., 1994], with full linear cross coupling weights. This lateral connectivity is sufficient to promote synchrony among simultaneously activated oscillators, but insufficient to activate them strongly in the absence of externallOput. This makes full synchrony of activated units the default condition in the model conex, as in Brown's model [Brown and Cooke, 1996]. so that the background activation is coherent, and can be read into higher order cortical levels which synchronize with it. The system assumes that all input is due to the same environmental source in the absence of evidence for segregation [Bregman, 1990]. Brown and Cooke [Brown and Cooke, 1996] model the cochlear and brain stem nuclear output as a set of overlapping bandpass ("gammatone") filters consistent with auditory nerve responses and psychophysical "critical bands". A tone can excite several filter outputs at once. We apprOXImate this effect of the gammatone filters as a lateral fan out of input activations with weights that spread the activation in the same way as the overlapping gammatone 8 B. Baird filters do. Experiments show that the intrinsic resonant or "natural" frequencies or "eigenfr~uencies" of cortical tissue within the 30-80 Hz gamma band vary within individuals on different trials of a task, and that neurotransmitters can quickly alter these resonant frequencies of neural clocks. Following the evidence that the oscillation frequency of binding in vision goes up with the speed of motion of an object, we assume that unattended activity in auditory cortex synchromzes at a default background fr~uency of 35 Hz, while the higher order attentional stream is at a higher frequency of 40 Hz. Just as fast motion in vision can cause stimulus driven capture of attention, we hypothesize that expectancy mismatch in audition causes the deviant activity to be boosted above the default background frequency to facilitate synchronization with the attentional stream at 40 Hz. This models the mechanism of involuntary stimulus driven attentional "pop out". Multiple streams of primary cortex activity synchronized at different eigenfrequencies can be selectively attended by unifonnly sweeping the eigenfrequencies of all primary ensembles through the passband of the 40 Hz higner order attentional stream to "tune in" each in turn as a radio reciever does. Following, but modifing the approach of Brown and Cooke [Brown and Cooke, 1996], the core of our primary cortex stream fonning model is a fast learning rule that reduces the lateral coupling and (in our model) spreads apart the intrinsic cortIcal frequencies of sound frequency cflannels that do not exhibit the same amplitude of activity at the same time. This coupling and eigenfrequency difference recovers between onsets. In the absence of lateral synchronizing connectlons or coherent top down driving, synchrony between cortical streams is rapidly lost because of their distant resonant frequencIes. Activity not satisfying the Gestalt prinCIple of "common fate" [Bregman, 1990] is thus decorrelated. The trade off of the effect of temporal and sound frequency proximity on stream segregation follows because close stimulus frequencies excite each other's channel filters. Each produces a similar output in the other, and their activitites are not decorrelated by coupling reduction and resonant frequency shifts. On the other hand, to the extent that they are distant enough in sound frequency, each tone onset weakens the weights and shifts the eigenfrequencies of the other channels that are not simultaneously active. This effect is greater, the faster the presentation rate, because the weight recovery rate is overcome. This recovery rate can then be adjusted to yield stream segregation at the rates reported by van Noorden [Bregman, 1990] for given sound frequency separations. 3.2 Sequential Grouping by Coupling and Resonant Frequency Labels In the absence of rhythmic structure in the input, the temporary weights and resonant frequency "labels" serve as a short tenn "stream memory" to brid~e time (up to 4 seconds) so that the next nearby input is "captured" or "sequentially bound into the same ensemble of synchronized activity. This pattern of synchrony in primary cortex has been made into a temporary attractor by the temporary weight and eigenfrequency changes from the previous stimulation. This explains the single tone capture expenments where a series of ioentical tones captures later nearby tones. For two points in tIme to be sequentially grouped by this mechanIsm, there is no need for activity to continue between onsets as in Browns model [Brown and Cooke, 1996], or to be held in multiple spatial locations as Wang [Wang, 1995] does. Since the gamma band response to a single auditory input onset lasts only 100 - 150 ms, there is no 40 Hz activity available in prim~ cortex (at most stimulus rates) for succesive inputs to synchronize with for sequential bmding by these mechanisms. Furthermore, the decorrelation rule, when added to the mechanism of timing expectancies, explains the loss of relative timing (order) between streams, since the lateral connections that normally broadcast actual and expected onsets accross auditory cortex, are cut between two streams by the decorrelating weight reduction. Expected and actual onset events in different streams can no longer be directly (locally) compared. Experimental evidence for the broadcast of expectancies comes from the fast generalization to other frequencies of a learned expectancy for the onset time of a tone of a particular frequency (Schreiner lab personal commumcation). When rhythmic structure is present, the expectancy system becomes engaged, and this becomes an additional feature dimension along which stimuli can be segregated. Distance from expected timing as well as sound quality is now an added factor causing stream formation by decoupling and eigenfrequency ShIft. Feedback of expected input can also partially"fill in" missing input for a cycle or two so that the expectancy protects the binding of features of a stimulus and stabilizes a perceptual stream accross seconds of time. 40 Hz AItentional Streams, Rhythmic Expectation, and Auditory Scene Analysis 3.3 9 Simulation or the Jones-Bregman Experiment Figure 2 shows the architecture used to simulate the Jones-Bregman experiment. The case shown is where the flanking tones are in the same stream as the targets because the captor stream is at the lower sound frequency channel. At the particular pomt in time shown here, the first flanking tone has just fimshed, and the first target tone has arrived. Both channels are therfore active, and synchronized with the attentional stream into the higher order sequence recognizer. Our mechanistic explanation of the Bregman result is that the early standard target tones arriving at the 80 msec rate first prime ttie dynamic attention system by setting the 80 msec clock to oscillate at 40 Hz and depressing the oscillation frequency of other auditory cortex background activi~y. Then the slow captor tones at the 240 msec period establish a background stream at 30 Hz with a rhythmic expectancy that is later violated by the appearance of the fast target tones. These now fall outside the correlation attractor basin of the background stream because the mismatch increases their cortical oscillation frequency. They are explicitly brought into the 40 Hz foreground frequency by the mismatch pop out mechanism. This allows the attentional stream into the Elman sequence recognition units to synchronize and read in activity due to the target tones for order determination. It is assisted by the timebase searchlight at the 80 msec period which synchronizes and enhances activity arriving at that rhythm. In the absence of a rhythmic distmction for the target tones, their sound frequency difference alone is insufficient to separate them from the background stream, and the targets cannot be reliably discriminated. In this simulation, the connections to the first two Elman associative memory units are hand wired to the A and B primary cortex oscillators to act as a latching, order determining switch. If sy-nchronized to the memory unit at the attentional stream frequency, the A target tone OSCillator will drive the first memory unit into the 1 attractor whicfi then inhibits the second unit from being driven to 1 by the B target tone. The second unit has similar wiring from the B tone oscillator, so that the particular higher order (intermediate term) memory unit which is left in the 1 state after a tnal indicates to the rest of the brain which tone came first. The flanking and high captor tone oscillator is connected equally to both memory units, so that a random attractor transition occurs before the targets amve, when it is interfering at the 40 Hz attentional frequency, and poor order determination results. If the flanking tone oscillator is in a separate stream along with the captor tones at the background eigenfrequency of 35 Hz, it is outside the recieving passband of the memory units and cannot cause a spurious attractor transition. This architecture demonstrates mechanisms that integrate the theories of Jones and Bregman about auditory perception. Stream formation is a preattentive process that works well on non-rhythmic inputs as Bregman asserts, but an equally primary and preattentive rhythmic expectancy process is also at work as Jones asserts and the mismatch negativity indicates. This becomes a factor in stream formation when rhythmic structure is present in stimuli as demonstrated by Jones. References [Baird et al., 1994] Baird, B., Troyer, T., and Eeckman, F. H. (1994). Gramatical inference by attentional control of synchronization in an oscillating elman network. In Hanson, S., Cowan, J., and Giles, C., editors, Advances in Neural InjormationProcessing Systems 6, pages 67-75. Morgan Kaufman. [Bregman, 1990] Bregman, A. S. (1990). Auditory Scene Analysis. MIT Press, Cambridge. [Brown and Cooke, 1996] Brown, G. and Cooke, M. (1996). A neural oscillator model of auditory stream segregation. In JJCAI Workshop on Computational Auditory Scene Analysis. to appear. [Jones et al., 1981] Jones, M., Kidd, G., and Wetzel, R. (1981). Evidence for rhythmic attention. Journal 0/ Experimental Psychology: Human Perception and Performance, 7: 1059-1073. [Llinas and Ribary, 1993] Llinas, R. and Ribary, U. (1993). Coherent 40-hz oscillation characterizes dream state in humans. Proc. Natl. Acad. Sci. USA,90:2078-2081. [Naatanen, 1992] Naatanen, R. (1992). Attention and Brain Function. Erlbaum, New Jersey. [Wang, 1~95] Wang, D. (1995). ~n oscillatory correlation theory of te~poral pattern segmentatIOn. In Covey, E., Hawkms, H., McMullen, T., and Port, R., edaors, Neural Representations o/Temporal Patterns. Plenum. to appear. 

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Coding of Naturalistic Stimuli by Auditory Midbrain Neurons H. Attias* and C.E. Schreiner t Sloan Center for Theoretical Neurobiology and W.M. Keck Foundation Center for Integrative Neuroscience University of California at San Francisco San Francisco, CA 94143-0444 Abstract It is known that humans can make finer discriminations between familiar sounds (e.g. syllables) than between unfamiliar ones (e.g. different noise segments). Here we show that a corresponding enhancement is present in early auditory processing stages. Based on previous work which demonstrated that natural sounds had robust statistical properties that could be quantified, we hypothesize that the auditory system exploits those properties to construct efficient neural codes. To test this hypothesis, we measure the information rate carried by auditory spike trains on narrow-band stimuli whose amplitude modulation has naturalistic characteristics, and compare it to the information rate on stimuli with non-naturalistic modulation. We find that naturalistic inputs significantly enhance the rate of transmitted information, indicating that auditiory neural responses are matched to characteristics of natural auditory scenes. 1 Natural Scene Statistics and the Neural Code A primary goal of hearing research is to understand how complex sounds that occur in natural scenes are processed by the auditory system. However, natural sounds are difficult to describe quantitatively and the complexity of auditory responses they evoke makes it hard to gain insight into their processing. Hence, most studies of auditory physiology are restricted to pure tones and noise stimuli, resulting in a limited understanding of auditory encoding. In this paper we pursue a novel approach to the study of natural sound encoding in auditory spike trains. Our ? Corresponding author. E-mail: hagai@phy.ucsf.edu. t E-mail: chris@phy.ucsf.edu . H. Attias and C. E. Schreiner 104 ~ 11111111 I II I II I I 111111 ~ I 111111 III 11111111 III II III UII Figure 1: Left: amplitude modulation stimulus drawn from a naturalistic stimulus set, and the evoked spike train of an inferior colliculus neuron. Right: amplitude modulation from a non-naturalistic set and the evoked spike train of the same neuron. method consists of measuring statistical characteristics of natural auditory scenes, and incorporating them into simple stimuli in a systematic manner, thus creating 'naturalistic' stimuli which enable us to study the encoding of natural sounds in a controlled fashion. The first stage of this program has been described in (Attias and Schreiner 1997); the second is reported below. Fig. 1 shows two segments of long stimuli and the corresponding spike trains of the same neuron, elicited by pure tones that were amplitude-modulated by these stimuli. While both stimuli appear to be random and to have the same mean and both spike trains have the same firing rate, one may observe that high and low amplitudes are more likely to occur in the stimulus on the left; indeed, these stimuli are drawn from two stimulus sets with different statistical properties. Our present study of auditory coding focuses on assessing the efficiency of this neural code: for a given stimulus set, how well can the animal reconstruct the input sound and discriminate between similar sound segments, based on the evoked spike train, and how those abilities are affected by changing the stimulus statistics. We quantify the discrimination capability of auditory neurons in the inferior colliculus of the cat using concepts from information theory (Bialek et al. 1991; Rieke et al. 1997). This leads to the issue of optimal coding (Atick 1992). Theoretically, given an auditory scene with particular statistical properties, it is possible to design an encoding scheme that would exploit those properties, resulting in a neural code that is optimal for that scene but is consequently less efficient for other scenes. Here we investigate the hypothesis that the auditory system uses a code that is adapted to natural auditory scenes. This question is addressed by comparing the discrimination capability of auditory neurons between sound segments drawn from a naturalistic stimulus set, to the one for a non-naturalistic set. 2 Statistics of Natural Sounds As a first step in investigating the relation between neural responses and auditory inputs, we studied and quantified temporal statistics of natural auditory scenes {Attias and Schreiner 1997}. It is well-known that different locations on the basal membrane respond selectively to different frequency components of the incoming sound x{t) (e.g., Pickles 1988), hence the frequency v corresponds to a spatial coordinate, in analogy with retinal location in vision. We therefore analyzed a large database of sounds, including speech, music, animal vocalizations, and background sounds, using various filter banks comprising 0 -10kHz. In each frequency band v, the amplitude a{t) ~ 0 and phase r/>{t) ofthe band-limited signal xv(t) = a{t) cos{vt+r/>{t)) were extracted, and the amplitude probability distribution p(a) and auto-correlation Coding of Naturalistic Stimuli by Auditory Midbrain Neurons Piano music 105 Symphonic: music: 0.6 0.5 Cat vQcalizatlona Bird 80ngs Ba.okground sound. 0 .5 0 .5 0 .4 0 .2 o. ~ o~~--------~~ -4 Figure 2: Log-amplitude distribution in several sound ensembles. Different curves for a given ensemble correspond to different frequency bands. The low amplitude peak in the cat plot reflect abundance of silent segments. The theoretical curve p(a) (1) is plotted for comparison (dashed line). function c(r) = (a(t)a(t + r)) were computed, as well as those of the instantaneous frequency d?(t)/dt. Those statistics were found to be nearly identical in all bands and across all examined sounds. In particular, the distribution of the log-amplitude a = log a, normalized to have zero mean and unit variance, could be well-fitted to the form p(a) = ,8 exp (,8a + Q - e.Bii+t:t) (1) (with normalization constants Q = -.578 and ,8 = 1.29), which should, however, be corrected at large amplitude (> 5a). Several examples are displayed in Fig. 1. The log-amplitude distribution (1) corresponds mathematically to the amplitude distribution of musical instruments and vocalizations, found to be p(a) = e- a (known as the Laplace distribution in speech signal processing), as well as that of background sounds, where p(a) <X ae- a2 (which can be shown to be the band amplitude distribution for a Gaussian signal). The power spectra of a(t) (Fourier transform of c(r)) were found to have a modified 1/ f form. Together with the results for ?(t), those findings show that natural sounds are distinguished from arbitrary ones by robust characteristics. In the present paper we explore to what extent the auditory system exploits them in constructing efficient neural codes. Another important point made by (Attias and Schreiner 1997), as well as by (Ruderman and Bialek 1994) regarding visual signals, is that natural inputs are very often not Gaussian (e.g. (1)), unlike the signals used by conventional system-identification methods often applied to the nervous system. In this paper we use non-Gaussian stimuli to study auditory coding. 3 3.1 Measuring the Rate of Information Transfer Experiment Based on our results for temporal statistics of natural auditory scenes, we can construct 'naturalistic' stimuli by starting with a simple signal and systematically incorporate successively more complicated characteristics of natural sounds into it. H. Attias and C. E. Schreiner 106 We cQ.ose to use narrow-band stimuli consisting of amplitude-modulated carriers a(t) cos(vt) at sound frequencies v = 2 - 9kHz with no phase modulation. Focusing on one-point amplitude statistics, we constructed a white naturalistic amplitude by choosing a(t) from an exponential distribution with a cutoff, p(O ::; a ::; ae) ex: e- a , p(a > ae) = 0 at each time point t independently, using a cutoff modulation frequency of fe = 100Hz (i.e., 1a(J ::; fe) 1= const., 1a(J > fe) 1= 0, where a(J) is the Fourier transform of a{t)). We also used a non-naturalistic stimulus set where a(t) was chosen from a uniform distribution p(O ::; a ~ be) = 1lbe, p(a > be) = 0, with be adjusted so that both stimulus sets had the same mean. A short segment from each set is shown in Fig. 1, and the two distributions are plotted in Figs. 3,4 (right) . Stimuli of 15 - 20min duration were played to ketamine-anesthetized cats. To minimize adaptation effects we alternated between the two sets using 10sec long segments. Single-unit recordings were made from the inferior colliculus (IC), a subthalamic auditory processing stage (e.g., Pickles 1988). Each IC unit responds best to a narrow range of sound frequencies, the center of which is called its 'best frequency' (BF). Neighboring units have similar BF's, in accord with the topographic frequency organization of the auditory system. For each unit, stimuli with carrier frequency v at most 500Hz away from the unit's BF were used. Firing rates in response to those stimuli were between 60 - 100Hz. The stimulus and the electrode signal were recorded simultaeneously at a sampling rate of 24kHz. After detecting and sotring the spikes and extracting the stimulus amplitude, both amplitude and spike train were down-sampled to 3kHz. 3.2 Analysis In order to assess the ability to discriminate between different inputs based on the observed spike train, we computed the mutual information Ir,s between the spike train response r(t) = Li o(t - ti), where ti are the spike times, and the stimulus amplitude s(t). I consists of two terms, Ir,s = Hs - Hslr' where Hs is the stimulus entropy (the log-number of different stimuli) and Hslr is the entropy of the stimulus conditioned on the response (the log-number of different stimuli that could elicit a given response, and thus could not be discriminated based on that response, averaged over all responses). Our approach generally follows the ideas of (Bialek et al. 1991; Rieke et al. 1997). To simplify the calculation, we first modified the stimuli s(t) to get s'(t) = f(s(t?, where the function f(s) was chosen so that s' was Gaussian. Hence for exponential stimuli f(s) = y'(2)erfi(1-2e- S ) and for uniform stimuli f(s) = y'(2)erfi(2slb e-1), where erfi is the inverse error function. This Gaussianization has two advantages: first, the expression for the mutual information Ir,s' (= Ir,s) is now simpler, being given by the frequency-dependent signal-to-noise ratio SNR(J) (see below), since Hs' depends only on the power spectrum of s'(t); second and more importantly, the noise distribution was observed to become closer to Gaussian following this transformation. ftc dfH[s'(J) 1 f(J)], the calculation of To compute Hs'lr we bound it from above by which requires the conditional distribution p[s'(J) 1 f(J)] (note that these variables are complex, hence this is the joint ditribution of the real and imaginary parts). The latter is approximated by a Gaussian with mean s~(J) and variance Nr(f). This variance is, in fact, the power spectrum of the noise, Nr(J) = (I nr(J) 12 ), which we define by nr(t) = s'(t) - s~(t). Computing the mutual information for those Gaussian distributions is straightforward and provides a lower bound on the Coding of Naturalistic Stimuli by Auditory Midbrain Neurons 107 1 I, 0.8 I, ", I I I I , , , 1.6 60 100 f Figure 3: Left: signal-to-noise ratio SNR(f) vs. modulation frequency I for naturalistic stimuli. Right: normalized noise distribution (solid line), amplitude distribution of stimuli (dashed line) and of Gaussianized stimuli (dashed-dotted line). true Ir,s, Ir,s = Ir,s' ~ Ie J dllog 2 SNR(f) . (2) o The signal-to-noise ratio is given by SNR(f) = S'(J)j(Nr(f))r, where S'(f) = (I s'(J) 12} is the spectrum of the Gaussianized stimulus and the averaging Or is performed over all responses. The main object here is s~(J), which is an estimate of the stimulus from the elicited spike train, and would optimally be given by the conditional mean J ds's'p(s' 1 f) at each I (Kay 1993). For Gaussian p( S' ,f) this estimator, which is generally non-linear, becomes linear in f(f) and is given by h(J)f(J), where h(J) (s'(J)f*(J)}j(f(J)f*(f?) is the Wiener filter. However, since our distributions were only approximately Gaussians we used the conditional mean, obtained by the kernel estimate = s~(f) = (3) where k is a Gaussian kernel, R(J) is the spectrum of the spike train, and i indexes the data points obtained by computing FFT using a sliding window. The scaling by y'Si,.,fii reflects the assumption that the distributions at all I differ only by their variance, which enables us to use the data points at all frequencies to estimate s~ at a given I. Our estimate produced a slightly higher SNR(f) than the Wiener estimate used by (Bialek et al. 1991;Rieke et al. 1997) and others. 4 Information on Naturalistic Stimuli The SNR(f) for exponential stimuli is shown in Fig. 3 (left) for one of our units. Ie neurons have a preferred modulation frequency 1m (e.g., Pickles 1988), which is about 40Hz for this unit; notice that generally SNR(J) ~ 1, with equality when the stimulus and response are completely independent. Thus, stimulus components at frequencies higher than 60Hz effectively cannot be estimated from the spike train. The stimulus amplitude distribution is shown in Fig. 3 (right, dashed line), together with the noise distribution (normalized to have unit variance; solid line) which is nearly Gaussian. H Attias and C. E. Schreiner J08 4,-----------------------, .. .. .. 1 3.5 0.8 3 ~2 . 6 . '. ~o.e en 0.4 2 0.2 1.5 100 0 -5 . Figure 4: Left: signal-to-noise ratio SNR(f) vs. modulation frequency f for nonnaturalistic stimuli (solid line) compared with naturalistic stimuli (dotted line). Right: normalized noise distribution (solid line), amplitude distribution of stimuli (dashed line) compared with that of naturalistic stimuli (dotted line), and of Gaussianized stimuli (dashed-dotted line). Using (2) we obtain an information rate of Ir,s ~ 114bit/sec. For the spike rate of 82spike/sec measured in this unit, this translates into 1.4bit/spike. Averaging across units, we have 1.3 ? 0.2bit/spike for naturalistic stimuli. Although this information rate was computed using the conditional mean estimator (3), it is interesting to examine the Wiener filter h{t) which provides the optimal linear estimator of the stimulus, as discussed in the previous section. This filter is displayed in Fig. 5 (solid line) and has a temporal width of several tens of milliseconds. 5 Information on Non-Naturalistic Stimuli The SNR(f) for uniform stimuli is shown in Fig. 4 (left, solid line) for the same unit as in Fig. 3, and is significantly lower than the corresponding SNR(f) for exponential stimuli plotted for comparison (dashed line). For the mutual information rate we obtain Ir,B ~ 77bit/sec, which amounts to 0.94bit/spike. Averaging across units, we have 0.8 ? 0.2bit/spike for non-naturalistic stimuli. The stimulus amplitude distribution is shown in Fig. 4 (right, dashed line), together with the exponential distribution (dotted line) plotted for comparison, as well as the noise distribution (normalized to have unit variance). The noise in this case is less Gaussian than for exponential stimuli, suggesting that our calculated bound on Ir,s may be lower for uniform stimuli. Fig. 5 shows the stimulus reconstruction filter (dashed line). It has a similar time course as the filter for exponential stimuli, but the decay is significantly slower and its temporal width is more than 100msec. 6 Conclusion We measured the rate at which auditory neurons carry information on simple stimuli with naturalistic amplitude modulation, and found that it was higher than for stimuli with non-naturalistic modulation. A result along the same lines for the frog was obtained by (Rieke et al. 1995) using Gaussian signals whose spectrum was shaped according to the frog call spectrum. Similarly, work in vision (Laughlin 1981; Field 1987; Atick and Redlich 1990; Ruderman and Bialek 1994; Dong and Atick 1995) suggests that visual receptive field properties are consistent with optimal coding predictions based on characteristics of natural images. Future work will explore coding of stimuli with more complex natural statistical characteristics and Coding a/Naturalistic Stimuli by Auditory Midbrain Neurons 109 0 .6 , - - - - - - - - - - - - - - - - - - - - - - - - . 0 .4 0 .3 ~ 0 .2 0 . '" __ -.r..1 -'-=----::;::-.::: v Vrr-~------==---__l o 1.-___ -0.'" -0 .2 L---_-=o=-...,=-------::0::-------::0,--...,=------' t Figure 5: Impulse response of Wiener reconstruction filter for naturalistic stimuli (solid line) and non-naturalistic stimuli (dashed line). will extend to higher processing stages. Acknowledgements We thank W. Bialek, K. Miller, S. Nagarajan, and F. Theunissen for useful discussions and B. Bonham, M. Escabi, M. Kvale, L. Miller, and H. Read for experimental support. Supported by The Office of Naval Research (NOOOI4-94-1-0547), NIDCD (ROI-02260), and the Sloan Foundation. References J.J. Atick and N. Redlich (1990). Towards a theory of early visual processing. Neural Comput. 2,308-320. J .J. Atick (1992). Could information theory provide an ecological theory of sensory processing. Network 3, 213-251. H. Attias and C.E. Schreiner (1997). Temporal low-order statistics of natural sounds. In Advances in Neural Information Processing Systems 9, MIT Press. W. Bialek, F. Rieke, R. de Ruyter van Steveninck, and D. Warland (1991). Reading the neural code. Science 252, 1854-1857. D.W. Dong and J.J. Atick (1995). Temporal decorrelation: a theory of lagged and non-lagged responses in the lateral geniculate nucleus. Network 6, 159-178. D.J. Field (1987). Relations between the statistics of natural images and the response properties of cortical cells. J. Opt. Soc. Am. 4, 2379-2394. S.M. Kay (1993). Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice-Hall, New Jersey. S.B. Laughlin (1981). A simple coding procedure enhances a neuron's information capacity. Z. Naturforsch. 36c, 910-912. J.O. Pickles (1988). An introduction to the physiology of hearing (2nd Ed.). San Diego, CA: Academic Press. F. Rieke, D. Bodnar, and W. Bialek (1995). Naturalistic stimuli increase the rate and efficiency of information transmission by primary auditory neurons. Proc. R. Soc . Lond. B, 262, 259-265. F. Rieke, D. Warland, R. de Ruyter van Steveninck, and W. Bialek (1997). Spikes: Exploring the Neural Code. MIT Press, Cambridge, MA. D.L. Ruderman and W. Bialek (1994). Statistics of natural images: scaling in the woods. Phys. Rev. Lett. 73,814-817. 

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The Rectified Gaussian Distribution N. D. Socci, D. D. Lee and H. S. Seung Bell Laboratories, Lucent Technologies Murray Hill, NJ 07974 {ndslddleelseung}~bell-labs.com Abstract A simple but powerful modification of the standard Gaussian distribution is studied. The variables of the rectified Gaussian are constrained to be nonnegative, enabling the use of nonconvex energy functions. Two multimodal examples, the competitive and cooperative distributions, illustrate the representational power of the rectified Gaussian. Since the cooperative distribution can represent the translations of a pattern, it demonstrates the potential of the rectified Gaussian for modeling pattern manifolds. 1 INTRODUCTION The rectified Gaussian distribution is a modification of the standard Gaussian in which the variables are constrained to be nonnegative. This simple modification brings increased representational power, as illustrated by two multimodal examples of the rectified Gaussian, the competitive and the cooperative distributions. The modes of the competitive distribution are well-separated by regions of low probability. The modes of the cooperative distribution are closely spaced along a nonlinear continuous manifold. Neither distribution can be accurately approximated by a single standard Gaussian. In short, the rectified Gaussian is able to represent both discrete and continuous variability in a way that a standard Gaussian cannot. This increased representational power comes at the price of increased complexity. While finding the mode of a standard Gaussian involves solution of linear equations, finding the modes of a rectified Gaussian is a quadratic programming problem. Sampling from a standard Gaussian can be done by generating one dimensional normal deviates, followed by a linear transformation. Sampling from a rectified Gaussian requires Monte Carlo methods. Mode-finding and sampling algorithms are basic tools that are important in probabilistic modeling. Like the Boltzmann machine[l], the rectified Gaussian is an undirected graphical model. The rectified Gaussian is a better representation for probabilistic modeling 351 The Rectified Gaussian Distribution (a) (c) Figure 1: Three types of quadratic energy functions. (a) Bowl (b) Trough (c) Saddle of continuous-valued data. It is unclear whether learning will be more tractable for the rectified Gaussian than it is for the Boltzmann machine. A different version of the rectified Gaussian was recently introduced by Hinton and Ghahramani[2, 3]. Their version is for a single variable, and has a singularity at the origin designed to produce sparse activity in directed graphical models. Our version lacks this singularity, and is only interesting in the case of more than one variable, for it relies on undirected interactions between variables to produce the multimodal behavior that is of interest here. The present work is inspired by biological neural network models that use continuous dynamical attractors[4]. In particular, the energy function of the cooperative distribution was previously studied in models of the visual cortex[5], motor cortex[6], and head direction system[7]. 2 ENERGY FUNCTIONS: BOWL, TROUGH, AND SADDLE The standard Gaussian distribution P(x) is defined as P(x) E(x) = Z - l e -{3E(;r:) , 1 _xT Ax - bTx . 2 (1) (2) The symmetric matrix A and vector b define the quadratic energy function E(x). The parameter (3 = lIT is an inverse temperature. Lowering the temperature concentrates the distribution at the minimum of the energy function. The prefactor Z normalizes the integral of P(x) to unity. Depending on the matrix A, the quadratic energy function E(x) can have different types of curvature. The energy function shown in Figure l(a) is convex. The minimum of the energy corresponds to the peak of the distribution. Such a distribution is often used in pattern recognition applications, when patterns are well-modeled as a single prototype corrupted by random noise. The energy function shown in Figure 1(b) is flattened in one direction. Patterns generated by such a distribution come with roughly equal1ikelihood from anywhere along the trough. So the direction of the trough corresponds to the invariances of the pattern. Principal component analysis can be thought of as a procedure for learning distributions of this form. The energy function shown in Figure 1(c) is saddle-shaped. It cannot be used in a Gaussian distribution, because the energy decreases without limit down the N. D. Socci, D. D. Lee and H. S. Seung 352 sides of the saddle, leading to a non-normalizable distribution. However, certain saddle-shaped energy functions can be used in the rectified Gaussian distribution, which is defined over vectors x whose components are all nonnegative. The class of energy functions that can be used are those where the matrix A has the property x T Ax > 0 for all x > 0, a condition known as copositivity. Note that this set of matrices is larger than the set of positive definite matrices that can be used with a standard Gaussian. The nonnegativity constraints block the directions in which the energy diverges to negative infinity. Some concrete examples will be discussed shortly. The energy functions for these examples will have multiple minima, and the corresponding distribution will be multimodal, which is not possible with a standard Gaussian. 3 MODE-FINDING Before defining some example distributions, we must introduce some tools for analyzing them. The modes of a rectified Gaussian are the minima of the energy function (2), subject to nonnegativity constraints. At low temperatures, the modes of the distribution characterize much of its behavior. Finding the modes of a rectified Gaussian is a problem in quadratic programming. Algorithms for quadratic programming are particularly simple for the case of nonnegativity constraints. Perhaps the simplest algorithm is the projected gradient method, a discrete time dynamics consisting of a gradient step followed by a rectification (3) The rectification [x]+ = max(x, 0) keeps x within the nonnegative orthant (x ~ 0). If the step size 7J is chosen correctly, this algorithm can provably be shown to converge to a stationary point of the energy function[8]. In practice, this stationary point is generally a local minimum. Neural networks can also solve quadratic programming problems. We define the synaptic weight matrix W = I - A, and a continuous time dynamics x+x = [b+ Wx]+ (4) For any initial condition in the nonnegative orthant, the dynamics remains in the nonnegative orthant, and the quadratic function (2) is a Lyapunov function of the dynamics. Both of these methods converge to a stationary point of the energy. The gradient of the energy is given by 9 = Ax - b. According to the Kiihn-Tucker conditions, a stationary point must satisfy the conditions that for all i, either gi = 0 and Xi > 0, or gi > 0 and Xi = O. The intuitive explanation is that in the interior of the constraint region, the gradient must vanish, while at the boundary, the gradient must point toward the interior. For a stationary point to be a local minimum, the Kiihn-Tucker conditions must be augmented by the condition that the Hessian of the nonzero variables be positive definite. Both methods are guaranteed to find a global minimum only in the case where A is positive definite, so that the energy function (2) is convex. This is because a convex energy function has a unique minimum. Convex quadratic programming is solvable in polynomial time. In contrast, for a nonconvex energy function (indefinite A), it is not generally possible to find the global minimum in polynomial time, because of the possible presence of local minima. In many practical situations, however, it is not too difficult to find a reasonable solution. The Rectified Gaussian Distribution 353 (a) (b) Figure 2: The competitive distribution for two variables. (a) A non-convex energy function with two constrained minima on the x and y axes. Shown are contours of constant energy, and arrows that represent the negative gradient of the energy. (b) The rectified Gaussian distribution has two peaks. The rectified Gaussian happens to be most interesting in the nonconvex case, precisely because of the possibility of multiple minima. The consequence of multiple minima is a multimodal distribution, which cannot be well-approximated by a standard Gaussian. We now consider two examples of a multimodal rectified Gaussian. 4 COMPETITIVE DISTRIBUTION The competitive distribution is defined by Aij bi = -dij 1; +2 (5) (6) We first consider the simple case N = 2. Then the energy function given by X2 +y2 E(x,y)=2 +(x+y)2_(x+y) (7) has two constrained minima at (1,0) and (0,1) and is shown in figure 2(a). It does not lead to a normalizable distribution unless the nonnegativity constraints are imposed. The two constrained minima of this nonconvex energy function correspond to two peaks in the distribution (fig 2(b)). While such a bimodal distribution could be approximated by a mixture of two standard Gaussians, a single Gaussian distribution cannot approximate such a distribution. In particular, the reduced probability density between the two peaks would not be representable at all with a single Gaussian. The competitive distribution gets its name because its energy function is similar to the ones that govern winner-take-all networks[9]. When N becomes large, the N global minima of the energy function are singleton vectors (fig 3), with one component equal to unity, and the rest zero. This is due to a competitive interaction between the components. The mean of the zero temperature distribution is given by (8) The eigenvalues of the covariance (XiXj) - (Xi)(Xj) 1 = N dij - 1 N2 (9) 354 N. D. Socci, D. D. Lee and H. S. Seung - .:(a) .: (b) : (c) - ., r- ? ?n 0 , 2 J a I ? ., ? ? to '2 S ??? ., ?? 'I r- III n n I ? . . , . . . , . u. . Figure 3: The competitive distribution for N = 10 variables. (a) One mode (zero temperature state) of the distribution. The strong competition between the variables results in only one variable on. There are N modes of this form, each with a different winner variable. (b) A sample at finite temperature (13 ~ 110) using Monte Carlo sampling. There is still a clear winner variable. (c) Sample from a standard Gaussian with matched mean and covariance. Even if we cut off the negative values this sample still bears little resemblance to the states shown in (a) and (b), since there is no clear winner variable. all equal to 1/N, except for a single zero mode. The zero mode is 1, the vector of all ones, and the other eigenvectors span the N - 1 dimensional space perpendicular to 1. Figure 3 shows two samples: one (b) drawn at finite temperature from the competitive distribution, and the other (c) drawn from a standard Gaussian distribution with the same mean and covariance. Even if the sample from the standard Gaussian is cut so negative values are set to zero the sample does not look at all like the original distribution. Most importantly a standard Gaussian will never be able to capture the strongly competitive character of this distribution. 5 COOPERATIVE DISTRIBUTION To define the cooperative distribution on N variables, an angle fh = 27ri/N is associated with each variable Xi, so that the variables can be regarded as sitting on a ring. The energy function is defined by 1 4 Aij 6ij + N - N COS(Oi - OJ) (10) bi = 1; (11) The coupling Aij between Xi and Xj depends only on the separation Oi - 03. between them on the ring. The minima, or ground states, of the energy function can be found numerically by the methods described earlier. An analytic calculation of the ground states in the large N limit is also possible[5]. As shown in Figure 4(a), each ground state is a lump of activity centered at some angle on the ring. This delocalized pattern of activity is different from the singleton modes of the competitive distribution, and arises from the cooperative interactions between neurons on the ring. Because the distribution is invariant to rotations of the ring (cyclic permutations of the variables xd, there are N ground states, each with the lump at a different angle. The mean and the covariance of the cooperative distribution are given by (Xi) (XiXj) - (Xi}(Xj) = const C(Oi - OJ) (12) (13) = A given sample of x, shown in Figure 4(a), does not look anything like the mean, which is completely uniform. Samples generated from a Gaussian distribution with 355 The Rectified Gaussian Distribution , (b) '(a) (c) r Figure 4: The cooperative distribution for N = 25 variables. (a) Zero temperature state. A cooperative interaction between the variables leads to a delocalized pattern of activity that can sit at different locations on the ring. (b) A finite temperature (/3 = 50) sample. (c) A sample from a standard Gaussian with matched mean and covariance. the same mean and covariance look completely different from the ground states of the cooperative distribution (fig 4(c)). These deviations from standard Gaussian behavior reflect fundamental differences in the underlying energy function. Here the energy function has N discrete minima arranged along a ring. In the limit of large N the barriers between these minima become quite small. A reasonable approximation is to regard the energy function as having a continuous line of minima with a ring geometry[5] . In other words, the energy surface looks like a curved trough, similar to the bottom of a wine bottle. The mean is the centroid of the ring and is not close to any minimum. The cooperative distribution is able to model the set of all translations of the lump pattern of activity. This suggests that the rectified Gaussian may be useful in invariant object recognition, in cases where a continuous manifold of instantiations of an object must be modeled. One such case is visual object recognition, where the images of an object from different viewpoints form a continuous manifold. 6 SAMPLING Figures 3 and 4 depict samples drawn from the competitive and cooperative distribution. These samples were generated using the Metropolis Monte Carlo algorithm. Since full descriptions of this algorithm can be found elsewhere, we give only a brief description of the particular features used here . The basic procedure is to generate a new configuration of the system and calculate the change in energy (given by eq. 2). If the energy decreases, one accepts the new configuration unconditionally. If it increases then the new configuration is accepted with probability e-{3AE. In our sampling algorithm one variable is updated at a time (analogous to single spin flips). The acceptance ratio is much higher this way than if we update all the spins simultaneously. However, for some distributions the energy function may have approximately marginal directions; directions in which there is little or no barrier. The cooperative distribution has this property. We can expect critical slowing down due to this and consequently some sort of collective update (analogous to multi-spin updates or cluster updates) might make sampling more efficient. However, the type of update will depend on the specifics of the energy function and is not easy to determine. 356 7 N D. Socci, D. D. Lee and H. S. Seung DISCUSSION The competitive and cooperative distributions are examples of rectified Gaussians for which no good approximation by a standard Gaussian is possible. However, both distributions can be approximated by mixtures of standard Gaussians. The competitive distribution can be approximated by a mixture of N Gaussians, one for each singleton state. The cooperative distribution can also be approximated by a mixture of N Gaussians, one for each location of the lump on the ring. A more economical approximation would reduce the number of Gaussians in the mixture, but .make each one anisotropic[IO]. Whether the rectified Gaussian is superior to these mixture models is an empirical question that should be investigated empirically with specific real-world probabilistic modeling tasks. Our intuition is that the rectified Gaussian will turn out to be a good representation for nonlinear pattern manifolds, and the aim of this paper has been to make this intuition concrete. To make the rectified Gaussian useful in practical applications, it is critical to find tractable learning algorithms. It is not yet clear whether learning will be more tractable for the rectified Gaussian than it was for the Boltzmann machine. Perhaps the continuous variables of the rectified Gaussian may be easier to work with than the binary variables of the Boltzmann machine. Acknowledgments We would like to thank P. Mitra, L. Saul, B. Shraiman and H. Sompolinsky for helpful discussions. Work on this project was supported by Bell Laboratories, Lucent Technologies. References [1] D. H. Ackley, G. E. Hinton, and T. J. Sejnowski. A learning algorithm for Boltzmann machines. Cognitive Science, 9:147-169, 1985. [2] G. E. Hinton and Z. Ghahramani. Generative models for discovering sparse distributed representations. Phil. Trans. Roy. Soc., B352:1177-90, 1997. [3] Z. Ghahramani and G. E. Hinton. Hierarchical non-linear factor analysis and topographic maps. Adv. Neural Info. Proc. Syst., 11, 1998. [4] H. S. Seung. How the brain keeps the eyes still. Proc. Natl. Acad. Sci. USA, 93:13339-13344, 1996. [5] R. Ben-Yishai, R. L. Bar-Or, and H. Sompolinsky. Theory of orientation tuning in visual cortex. Proc. Nat. Acad. Sci. USA, 92:3844-3848, 1995. [6] A. P. Georgopoulos, M. Taira, and A. Lukashin. Cognitive neurophysiology of the motor cortex. Science, 260:47-52, 1993. [7] K. Zhang. Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. J. Neurosci., 16:2112-2126, 1996. [8] D. P. Bertsekas. Nonlinear programming. Athena Scientific, Belmont, MA, 1995. [9] S. Amari and M. A. Arbib. Competition and cooperation in neural nets. In J. Metzler, editor, Systems Neuroscience, pages 119-165. Academic Press, New York, 1977. [10] G. E. Hinton, P. Dayan, and M. Revow. Modeling the manifolds of images of handwritten digits. IEEE Trans. Neural Networks, 8:65-74, 1997. 

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Modeling Complex Cells in an A wake Macaque During Natural Image Viewing William E. Vinje vinjeCsocrates.berkeley.edu Department of Molecular and Cellular Biology, Neurobiology Division University of California, Berkeley Berkeley, CA, 94720 Jack L. Gallant gallantCsocrates.berkeley.edu Department of Psychology University of California, Berkeley Berkeley, CA, 94720 Abstract We model the responses of cells in visual area VI during natural vision. Our model consists of a classical energy mechanism whose output is divided by nonclassical gain control and texture contrast mechanisms. We apply this model to review movies, a stimulus sequence that replicates the stimulation a cell receives during free viewing of natural images. Data were collected from three cells using five different review movies, and the model was fit separately to the data from each movie. For the energy mechanism alone we find modest but significant correlations (rE = 0.41, 0.43, 0.59, 0.35) between model and data. These correlations are improved somewhat when we allow for suppressive surround effects (rE+G = 0.42, 0.56, 0.60, 0.37). In one case the inclusion of a delayed suppressive surround dramatically improves the fit to the data by modifying the time course of the model's response. 1 INTRODUCTION Complex cells in the primary visual cortex (area VI in primates) are tuned to localized visual patterns of a given spatial frequency, orientation, color, and drift direction (De Valois & De Valois, 1990). These cells have been modeled as linear spatio-temporal filters whose output is rectified by a static nonlinearity (Adelson & Bergen, 1985); more recent models have also included a divisive contrast gain control mechanism (Heeger, 1992; Wilson & Humanski, 1993; Geisler & Albrecht, 1997). We apply a modified form of these models to a stimulus that simulates natural vision. Our model uses relatively few parameters yet incorporates the cells' temporal response properties and suppressive influences from beyond the classical receptive field (C RF). Modeling Complex Cells during Natural Image Viewing 2 237 METHODS Data Collection: Data were collected from One awake behaving Macaque monkey, using single unit recording techniques described elsewhere (Connor et al., 1997).1 First, the cell's receptive field size and location were estimated manually, and tuning curves were objectively characterized using two-dimensional sinusoidal gratings. Next a static color image of a natural scene was presented to the animal and his eye position was recorded continuously as he freely scanned the image for 9 seconds (Gallant et al., 1998). 2 Image patches centered on the position of the cell's C RF (and 2-4 times the CRF diameter) were then extracted using an automated procedure. The sequence of image patches formed a continuous 9 second review movie that simulated all of the stimulation that had occurred in and around the C RF during free viewing. 3 Although the original image was static, the review movies contain the temporal dynamics of the saccadic eye movements made by the animal during free viewing. Finally, the review movies were played in and around the C RF while the animal performed a fixation task. During free viewing each eye position is unique, so each image patch is likely to enter the C RF only once. The review movies were therefore replayed several times and the cell's average response with respect to the movie timestream was computed from the peri-stimulus time histogram (PSTH). These review movies also form the model's stimulus input, while its output is relative spike probability versus time (the model cell's PSTH). Before applying the model each review movie was preprocessed by converting to gray scale (since the model does not consider color tuning), setting the average luminance level to zero (on a frame by frame basis) and prefiltering with the human contrast sensitivity function to more accurately reflect the information reaching cells in VI. Divisive Normalization Model: The model consists of a classical receptive field energy mechanism, ECRF, whose output is divided by two nonclassical suppressive mechanisms, a gain control field, G, and a texture contrast field, T. ECRF(t) PSTHmodel(t) ex 1 + Q G(t - d) + f3T(t - d) (1) We include a delay parameter for suppressive effects, consistent with the hypothesis that these effects may be mediated by local cortical interactions (Heeger, 1992; Wilson & Humanski, 1993). Any latency difference between the central energy mechanism and the suppressive surround will be reflected as a positive delay offset (15 > 0 in Equation 1). Classical Receptive Field Energy Mechanism: The energy mechanism, ECRF, is composed of four phase-dependent subunits, Uti>. Each subunit computes an inner product in space and a convolution in time between the model cell's space-time classical receptive field, CRFtI>(x, y, r), and the image, I(x, y, t). U<P(t) = JJJCRFtI>(x, y, r) . I(x, y, t - r) dx dydr (2) 1 Recorcling was performed under a university-approved protocol and conformed to all relevant NIH and USDA guidelines. 2 Images were taken from a Corel Corporation photo-CD library at 1280xl024 resolution. 3Eye position data were collected at 1 KHz, whereas the monitor clisplay rate was 72.5 Hz (14 ms per frame). Therefore each review movie frame was composed of the average stimulation occurring during the corresponcling 13.8 ms of free viewing. W. E. Vinje and 1. L Gallant 238 The model presented here incorporates the simplifying assumption of a space-time separable receptive field structure, CRF4>(x, y, r) = CRF4>(x, y) CRF(r). u4>(t) = L: CRF(r) (L: L: CRF4>(x, y) . I(x, y, t T X r)) (3) Y Time is discretized into frames and space is discretized into pixels that match the review movie input. CRF4>(x, y) is modeled as a sinusoidal grating that is spatially weighted by a Gaussian envelope (i.e. a Gabor function). In this paper CRF(r) is approximated as a delta function following a constant latency. This minimizes model parameters and highlights the model's responses to the stimulus present at each fixation. The latency, orientation and spatial frequency of the grating, and the size of the C RF envelope, are all determined empirically by maximizing the fit between model and data. 4 A static non-linearity ensures that the model PSTH does not become negative. We have e~amined both half-wave rectification, fj4>(t) = max[U4>(t), O], and halfsquaring, U4>(t) = (max[U4>(t) , 0])2; here we present the results from half-wave rectification. Half-squaring produces small changes in the model PSTH but does not improve the fit to the data. The energy mechanism is made phase invariant by averaging over the rectified phasedependent subunits: (4) Gain Control Field: Cells in V 1 incorporate a contrast gain control mechanism that compensates for changes in local luminance. The gain control field, G, models this effect as the total image power in a region encompassing the C RF and surround. G(t-<5) = L:CRF(r) T (L:L:VP(kx,ky,r) ) k% (5) ky P(kx, ky, r) = F FT[PG(x, y, r)] F FT*[JlG(x, y, r)] (6) JlG(x, y, r) = vG(x, y) I(x, y, (t - <5) - r) (7) P(kx, ky, r) is the spatial Fourier power of JlG(x, y, r) and VG is a two dimensional Gaussian weighting function whose width sets the size of the gain control field. Heeger's (1992) divisive gain control term sums over many discrete energy mechanisms that tile space in and around the area of the C RF. Equation 5 approximates Heeger's approach in the limiting case of dense tiling. Texture Contrast Field: Cells in area VI can be affected by the image surrounding the region of the CRF (Knierim & Van Essen, 1992) . The responses of many VI cells are highest when the optimal stimulus is presented alone within the CRF, and lowest when that stimulus is surrounded with a texture of similar orientation and frequency. The texture contrast field, T, models this effect as the image power 4 As a fit statistic we use the linear correlation coefficient (Pearson's r) between model and data. Fitting is done with a gradient ascent algorithm. Our choice of correlation as a statistic eliminates the need to explicitly consider model normalization as a variable, and is very sensitive to latency mismatches between model and data. However, linear correlation is more prone to noise contamination than is X2 ? Modeling Complex Cells during Natural Image Viewing 239 in the spatial region surrounding the C RF that matches the C RF's orientation and spatial frequency. T(t-J) = 1 4 1: 1: CRF (r) "?1: Jp4>(kx,ky,r) 90,180,270[ ( 4>=0 k", T lICRF(X, (8) ky P4>(kx, ky, r) = F FT[p~(x, y, r)] F FT?[p~(x, y, r)] J.t~(x, y, r) = ~*(x, y) (1 - )] y)) I(x, y, (t - J) - r) (9) (10) ~* is a Gabor function whose orientation and spatial frequency match those of the best' fit C RF4> (x, y). The envelope of ~* defines the size of the texture contrast field. lICRF is a two dimensional Gaussian weighting function whose width matches the C RF envelope, and which suppresses the image center. Thus the texture contrast term picks up oriented power from an annular region of the image surrounding the C RF envelope. T is made phase invariant by averaging over phase. 3 RESULTS Thus far our model has been evaluated on a small data set collected as part of a different study (Gallant et ai., 1998). Two cells, 87A and 98C, were examined with one review movie each, while cell 97 A was examined with three review movies. Using this data set we compare the model's response in two interesting situations: cell 97 A, which had high orientation-selectivity, versus cell 87 A, which had poor orientation-selectivity; and cell 98C, which was directionally-selective, versus cell 97 A, which was not directionally-selective. CRF Energy Mechanism: We separately fit the energy mechanism parameters to each of the three different cells. For cell 97A the three review movies were fit independently to test for consistency of the best fit parameters. Table 1 shows the correlation between model and data using only the C RF energy mechanism (a = f3 = 0 in Equation 1). The significance of the correlations was assessed via a permutation test. The correlation values for cells 97 A and 98C, though modest, are significant (p < 0.01). For these cells the 95% confidence intervals on the best fit parameter values are consistent with estimates from the flashed grating tests. The best fit parameter values for cell 97 A are also consistent across the three independently fit review movies. The model best accounts for the data from cell 97 A. This cell was highly selective for vertical gratings and was not directionally-selective. Figure 1 compares the PSTH obtained from cell 97 A with movie B to the model PSTH. The model generally responds to the same features that drive the real cell, though the match is imperfect. Much of the discrepancy between the model and data arises from our approximation of CRF(t) as a delta function. The model's response is roughly constant during 97A 87A 97A 97A 98C Cell B A C A A Movie Yes Yes Yes Yes Oriented No Yes No No No Directional No 0.35 NA 0.41 0.43 0.59 rE Table 1: Correlations between model and data PSTHs. Oriented cells showed orientation-selectivity in the flashed grating test while Directional cells showed directional-selectivity during manual characterization. rE is the correlation between ECRF and the data. No fit was obtained for cell 87 A. W. E. Vinje and J L. Gallant 240 1~--~~--~----~----~----T-----~--~----~----~ .-cO.8 .-. ~ ~ ? 0.6 ~ ~ 'a ~ 0.4 ~ .~ .-. ~ ~ 0.2 1 2 3 4 5 Time (seconds) 6 7 8 Figure 1: CRF energy mechanism versus data (Cell 97A, Movie B) . White indicates that the model response is greater than the data, while black indicates the data is greater than the model and gray indicates regions of overlap. A perfect match between model and data would result in the entire area under the curve being gray. Our approximation of CRF(t) leads to a relatively constant model PSTH during each fixation. In contrast the real cell generally gives a phasic response as each saccade brings a new stimulus into the CRF. In general the same movie features drive both model and cell. each fixation, which causes the model PSTH to appear stepped. In contrast the data PSTH shows a strong phasic response at the beginning of each fixation when a new stimulus patch enters the cell's CRF . The model is less successful at accounting for the responses of the directionallyselective cell, 98C. This is probably because the model's space-time separable receptive field misses motion energy cues that drive the cell. The model completely failed to fit the data from cell 87 A . This cell was not orientation-selective, so the fitting procedure was unable to find an appropriate orientation for the CRF?(x, y) Gabor function. 5 CRF Energy Mechanism with Suppressive Surround: Table 2 lists the improvements in correlation obtained by adding the gain control term (a > 0, fJ = 0 in Equation 1). For cell 97 A (all three movies) the best correlations are obtained when the surround effects are delayed by 56 ms relative to the center. The best correlation for cell 98C is obtained when the surround is not delayed. In three out of four cases the correlation values are barely improved when the surround effects are included, suggesting that the cells were not strongly surroundinhibited by these review movies. However, the improvement is quite striking in the SFor cell 87 A the correlation values in the orientation and spatial frequency parameter subspace contained three roughly equivalent maxima. Contamination by multiple cells was unlikely due to this cell's excellent isolation. 9 Modeling Complex Cells during Natural Image Viewing Cell Movie 97A A 0.42 +0.01 rE+G ~r 97A B 0.56 +0.13 241 97A C 0.60 +0.01 98C A 0.37 +0.02 Table 2: Correlation improvements due to surround gain control mechanism. rE+G gives the correlation value between the best fit model and the data. ~r gives the improvement over rEo Including G in Equation 1 leads to a dramatic correlation increase for cell 97 A, movie B, but not for the other review movies. case of cell 97 A, movie B. Figure 2 compares the data with a model using both Ecr f and G in Equation 1. Here the delayed surround suppresses the sustained responses seen in Figure 1 and results in a more phasic model PSTH that closely matches the data. We consider G and T fields both independently and in combination. For each we independently fit for Q, {3, &, and the size of the suppressive fields. However, the oriented Fourier power correlates with the total Fourier power for our sample of natural images, so that G and T are highly correlated. Combined fitting of G and T terms leads to competition and dominance by G (i.e. (3 -r 0). In this paper we only report the effects of the gain control mechanism; the texture contrast mechanism results in similar (though slightly degraded) results. 1~--~----~----~----~----~----~---.----~----~ cO.8 ..... ...... ..... ~ ~ ct 0.6 ] ..... 0- ~ 0.4 .....> ~ ...... (I) 0::: 0.2 o o 1 2 3 456 Time (seconds) 7 8 Figure 2: C RF energy mechanism with delayed surround gain control versus data (Cell 97A, Movie B). Color scheme as in Figure 1. The inclusion of the delayed G term results in a more phasic model response which greatly improves the match between model and data. 9 W. E. Vinje and 1. L. Gallant 242 4 DISCUSSION This preliminary study suggests that models of the form outlined here show great promise for describing the responses of area V1 cells during natural vision. For comparison consider the correlation values obtained from an earlier neural network model that attempted to reproduce V1 cells' responses to a variety of spatial patterns (Lehky et al. 1992) . They report a median correlation value of 0.65 for complex stimuli, whereas the average correlation score from Table 2 is 0.49. This is remarkable considering that our model has only 7 free parameters, a very hmited data set for fitting, doesn't yet consider color tuning or directional-selectivity and considers response across time. Future implementations of the model will use a more sophisticated energy mechanism that allows for nonseparable space time receptive field structure and more realistic temporal response dynamics. We will also incorporate more detail into the surround mechanisms, such as asymmetric surround structure and a broadband texture contrast term. By abstracting physiological observation into approximate functional forms our model balances explanatory power against parametric complexity. A cascaded series of these models may form the foundation for future modeling of cells in extra-striate areas V2 and V4. Natural image stimuli may provide an appropriate stimulus set for development and validation of these extrastriate models. Acknowledgements We thank Joseph Rogers for assistance in this study, Maneesh Sahani for the extremely useful suggestion of fitting the CRF parameters, Charles Connor for help with data collection and David Van Essen for support of data collection. References Adelson, E. H. & Bergen, J. R. (1985) Spatiotemporal energy models for the perception of motion. Journal of the Optical Society of America, A, 2, 284-299. Connor, C. C., Preddie, D. C., Gallant, J . L. & Van Essen, D. C. (1997) Spatial attention effects in macaque area V4 . Journal of Neuroscience, 77, 3201-3214. De Valois, R. L. & De Valois, K. K. (1990) Spatial Vision. New York: Oxford University Press. Gallant, J. L., Connor, C. E., & Van Essen, D. C. (1998) Neural Activity in Areas V1 , V2 and V4 During Free Viewing of Natural Scenes Compared to Controlled Viewing. NeuroReport, 9 . Geisler, W. S., Albrecht, D. G. (1997) Visual cortex neurons in monkeys and cats: Detection, discrimination, and identification. Visual Neuroscience, 14, 897-919. Heeger, D. J. (1992) Normalization of cell responses in cat striate cortex. Visual Neuroscience, 9, 181-198. Knierim, J . J . & Van Essen, D. C. (1992) Neuronal responses to static texture patterns in area V1 of the alert macaque monkey. Journal of Neurophysiology, 67, 961-980. Lehky, S. R., Sejnowski, T . J . & Desimone, R. (1992) Predicting Responses of Nonlinear Neurons in Monkey Striate Cortex to Complex Patterns. Journal of Neuroscience, 12, 3568-3581. Wilson, H. R. & Humanski, R. (1993) Spatial frequency adaptation and contrast gain control. Vision Research, 33, 1133-1149. 

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Reinforcement Learning for Continuous Stochastic Control Problems Remi Munos CEMAGREF, LISC, Pare de Tourvoie, BP 121, 92185 Antony Cedex, FRANCE. Rerni.Munos@cemagref.fr Paul Bourgine Ecole Polyteclmique, CREA, 91128 Palaiseau Cedex, FRANCE. Bourgine@poly.polytechnique.fr Abstract This paper is concerned with the problem of Reinforcement Learning (RL) for continuous state space and time stocha.stic control problems. We state the Harnilton-Jacobi-Bellman equation satisfied by the value function and use a Finite-Difference method for designing a convergent approximation scheme. Then we propose a RL algorithm based on this scheme and prove its convergence to the optimal solution. 1 Introduction to RL in the continuous, stochastic case The objective of RL is to find -thanks to a reinforcement signal- an optimal strategy for solving a dynamical control problem. Here we sudy the continuous time, continuous state-space stochastic case, which covers a wide variety of control problems including target, viability, optimization problems (see [FS93], [KP95])}or which a formalism is the following. The evolution of the current state x(t) E 0 (the statespace, with 0 open subset of IRd ), depends on the control u(t) E U (compact subset) by a stochastic differential equation, called the state dynamics: (1) dx = f(x(t), u(t))dt + a(x(t), u(t))dw where f is the local drift and a .dw (with w a brownian motion of dimension rand (j a d x r-matrix) the stochastic part (which appears for several reasons such as lake of precision, noisy influence, random fluctuations) of the diffusion process. For initial state x and control u(t), (1) leads to an infinity of possible traj~tories x(t). For some trajectory x(t) (see figure I)., let T be its exit time from 0 (with the convention that if x(t) always stays in 0, then T = 00). Then, we define the functional J of initial state x and control u(.) as the expectation for all trajectories of the discounted cumulative reinforcement : J(x; u(.)) = Ex,u( .) {loT '/r(x(t), u(t))dt +,,{ R(X(T))} R. Munos and P. Bourgine 1030 where rex, u) is the running reinforcement and R(x) the boundary reinforcement. 'Y is the discount factor (0 :S 'Y < 1). In the following, we assume that J, a are of class C2 , rand Rare Lipschitzian (with constants Lr and LR) and the boundary 80 is C2 . ? ? all ? II ? ? xirJ ? ? ? ? ? ? ? ? Figure 1: The state space, the discretized ~6 (the square dots) and its frontier 8~6 (the round ones). A trajectory Xk(t) goes through the neighbourhood of state ~. RL uses the method of Dynamic Program~ing (DP) which generates an optimal (feed-back) control u*(x) by estimating the value function (VF), defined as the maximal value of the functional J as a function of initial state x : (2) Vex) = sup J(x; u(.). u( .) In the RL approach, the state dynamics is unknown from the system ; the only available information for learning the optimal control is the reinforcement obtained at the current state. Here we propose a model-based algorithm, i.e. that learns on-line a model of the dynamics and approximates the value function by successive iterations. Section 2 states the Hamilton-Jacobi-Bellman equation and use a Finite-Difference (FD) method derived from Kushner [Kus90] for generating a convergent approximation scheme. In section 3, we propose a RL algorithm based on this scheme and prove its convergence to the VF in appendix A. 2 A Finite Difference scheme Here, we state a second-order nonlinear differential equation (obtained from the DP principle, see [FS93J) satisfied by the value function, called the Hamilton-JacobiBellman equation. Let the d x d matrix a = a.a' (with' the transpose of the matrix). We consider the uniformly pambolic case, Le. we assume that there exists c > 0 such that V$ E 0, Vu E U, Vy E IR d ,2:t,j=l aij(x, U)YiYj 2: c1lY112. Then V is C2 (see [Kry80J). Let Vx be the gradient of V and VXiXj its second-order partial derivatives. Theorem 1 (Hamilton-Jacohi-Bellman) The following HJB equation holds : Vex) In 'Y + sup [rex, u) uEU + Vx(x).J(x, u) + ! 2:~j=l aij V XiXj (x)] = 0 for x E 0 Besides, V satisfies the following boundary condition: Vex) = R(x) for x E 80. 1031 Reinforcement Learningfor Continuous Stochastic Control Problems Remark 1 The challenge of learning the VF is motivated by the fact that from V, we can deduce the following optimal feed-back control policy: u*(x) E arg sup [r(x, u) uEU + Vx(x).f(x, u) + ! L:7,j=l aij VXiXj (x)] In the following, we assume that 0 is bounded. Let eI, ... , ed be a basis for JRd. Let the positive and negative parts of a function 4> be : 4>+ = ma.x(4),O) and 4>- = ma.x( -4>,0). For any discretization step 8, let us consider the lattices: 8Zd = {8. L:~=1 jiei} where j}, ... ,jd are any integers, and ~6 = 8Z d n O. Let 8~6, the frontier of ~6 denote the set of points {~ E 8Zd \ 0 such that at least one adjacent point ~ ? 8ei E ~6} (see figure 1). Let U 6 cUbe a finite control set that approximates U in the sense: 8 ~ 8' => U6' C U6 and U6U 6 = U. Besides, we assume that: Vi = l..d, (3) By replacing the gradient Vx(~) by the forward and backward first-order finitedifference quotients: ~;, V(~) = l [V(~ ? 8ei) - V(~)l and VXiXj (~) by the secondorder finite-difference quotients: -b [V(~ + 8ei) + V(,' ~XiXi V(~) ~;iXj V(~) - 8ei) - 2V(O] = 2P[V(~ + 8ei ? 8ej) + V(~ - 8ei =F 8ej) -V(~ + 8ei) - V(~ - 8ei) - V(~ + 8ej) - V(~ - 8ej) + 2V(~)] in the HJB equation, we obtain the following : for ~ E :?6, V6(~)In,+SUPUEUh {r(~,u) + L:~=1 [f:(~,u)'~~iV6(~) - fi-(~,U)'~;iV6(~) + aii (~.u) ~ X,X,'" V(C) + " . . 2 . . wJ'l=~ (at; (~,'U) ~ +x,x.1'" .V(C) _ a~ (~,'U) ~ x,x V(C))] } = 0 2 2 -. . J '" Knowing that (~t In,) is an approximation of (,l:l.t -1) as ~t tends to 0, we deduce: V6(~) with T(~, SUPuEUh [,"'(~'U)L(EEbP(~,U,()V6?()+T(~,u)r(~,u)] (4) (5) u) which appears as a DP equation for some finite Markovian Decision Process (see [Ber87]) whose state space is ~6 and probabilities of transition: p(~,u,~ p(~, ? 8ei) u, ~ + 8ei ? 8ej) p(~,u,~ - 8ei ? 8ej) p(~,u,() "'~~r) [28Ift(~, u)1 + aii(~' u) - Lj=l=i laij(~, u)l] , "'~~r)a~(~,u)fori=f:j, (6) "'~~r)a~(~,u) for i =f: j, o otherwise. Thanks to a contraction property due to the discount factor" there exists a unique solution (the fixed-point) V to equation (4) for ~ E :?6 with the boundary condition V6(~) = R(~) for ~ E 8:?6. The following theorem (see [Kus90] or [FS93]) insures that V 6 is a convergent approximation scheme. R. Munos and P. Bourgine 1032 Theorem 2 (Convergence of the FD scheme) V D converges to V as 8 1 0 : lim VD(~) /)10 ~-x = Vex) un~formly on 0 Remark 2 Condition (3) insures that the p(~, u, () are positive. If this condition does not hold, several possibilities to overcome this are described in [Kus90j. 3 The reinforcement learning algorithm Here we assume that f is bounded from below. As the state dynami,:s (J and a) is unknown from the system, we approximate it by building a model f and from samples of trajectories Xk(t) : we consider series of successive states Xk = Xk(tk) and Yk = Xk(tk + Tk) such that: a - "It E [tk, tk + Tk], x(t) E N(~) neighbourhood of ~ whose diameter is inferior to kN.8 for some positive constant kN, - the control u is constant for t E [tk, tk - Tk + Tk], satisfies for some positive kl and k2, (7) Then incrementally update the model : .1 ",n n ~k=l n (Yk - Xk -;;; Lk=l 1 an(~,u) Yk - Xk Tk Tk.fn(~, u)) and compute the approximated time T( x, u) transition p(~, u, () by replacing f and a by (Yk - Xk Tk Tk?fn(~, u))' (8) ~d f the approximated probabilities of and a in (5) and (6). We obtain the following updating rule of the V D-value of state ~ : V~+l (~) = sUPuEU/) [,~/:(x,u) L( p(~, u, ()V~(() + T(x, u)r(~, u)] (9) which can be used as an off-line (synchronous, Gauss-Seidel, asynchronous) or ontime (for example by updating V~(~) as soon as a trajectory exits from the neighbourood of ~) DP algorithm (see [BBS95]). Besides, when a trajectory hits the boundary [JO at some exit point Xk(T) then update the closest state ~ E [JED with: (10) Theorem 3 (Convergence of the algorithm) Suppose that the model as well as the V D-value of every state ~ E :ED and control u E U D are regularly updated (respectively with (8) and (9)) and that every state ~ E [JED are updated with (10) at least once. Then "Ie> 0, :3~ such that "18 ~ ~, :3N, "In 2: N, sUP~EE/) IV~(~) - V(~)I ~ e with probability 1 1033 Reinforcement Learningfor Continuous Stochastic Control Problems 4 Conclusion This paper presents a model-based RL algorithm for continuous stochastic control problems. A model of the dynamics is approximated by the mean and the covariance of successive states. Then, a RL updating rule based on a convergent FD scheme is deduced and in the hypothesis of an adequate exploration, the convergence to the optimal solution is proved as the discretization step 8 tends to 0 and the number of iteration tends to infinity. This result is to be compared to the model-free RL algorithm for the deterministic case in [Mun97]. An interesting possible future work should be to consider model-free algorithms in the stochastic case for which a Q-Iearning rule (see [Wat89]) could be relevant. A Appendix: proof of the convergence Let M f ' M a, M fr. and Ma .? be the upper bounds of j, a, f x and 0' x and m f the lower bound of f. Let EO = SUP?EI:h !V0 (';) - V(';)I and E! = SUP?EI:b \V~(';) - VO(.;)\. Estimation error of the model fn and an and the probabilities Pn A.I Suppose that the trajectory Xk(t) occured for some occurence Wk(t) of the brownian motion: Xk(t) = Xk + f!k f(Xk(t),u)dt + f!" a(xk(t),U)dwk. Then we consider a trajectory Zk (t) starting from .; at tk and following the same brownian motion: Zk(t) ='; + fttk. f(Zk(t), u)dt + fttk a(zk(t), U)dWk' Let Zk = Zk(tk + Tk). Then (Yk - Xk) - (Zk -.;) = ftk [f(Xk(t), u) - f(Zk(t), u)] dt + t Tk ft [a(xk(t), u) - a(zk(t), u)J dWk. Thus, from the C1 property of f and a, :.+ II(Yk - Xk) - (Zk - ';)11 ~ (Mf'" + M aJ.kN.Tk. 8. (11) The diffusion processes has the following property ~ee for example the ItO- Taylor majoration in [KP95j) : Ex [ZkJ = ';+Tk.f(';, U)+O(Tk) which, from (7), is equivalent to: Ex [z~:g] = j(';,u) + 0(8). Thus from the law of large numbers and (11): li~-!~p Ilfn(';, u) - f(';, u)11 - li;;:s~p II~ L~=l [Yk;kX & - (Mf:r: ?.] I + 0(8) + M aJ?kN?8 + 0(8) = 0(8) w.p. 1 (12) Besides, diffusion processes have the following property (again see [KP95J): Ex [(Zk -.;) (Zk - .;)'] = a(';, U)Tk + f(';, u).f(';, U)'.T~ + 0(T2) which, from (7), is equivalent to: Ex [(Zk-?-Tkf(S'U)~(kZk-S-Tkf(S'U?/] = a(';, u) + 0(82). Let rk = = Yk - Xk - Tkfn(';, u) which satisfy (from (11) and (12? : Ilrk - ikll = (Mf:r: + M aJ.Tk.kN.8 + Tk.o(8) (13) Zk -.; - Tkf(';, u) and ik From the definition of Ci;;(';,u), we have: Ci;;(';,u) - a(';,u) Ex [r~':k] + 0(8 2 ) li~~~p = ~L~=l '\:1.' - and from the law of large numbers, (12) and (13), we have: 11~(';,u) - a(';,u)11 = Ilik -rkllli:!s!p~ li~-!~p II~ L~=l rJ./Y fl (II~II + II~II) - r~':k I + 0(8 +0(82 ) = 0(8 2 ) 2) R. Munos and P. Bourgine 1034 "In(';, u) - I(';, u)" ~ 1Ill;;(';, u) - a(';, u)1I kf?8 w.p. 1 ~ ka .82 w.p. 1 (14) Besides, from (5) and (14), we have: 2 2 c ) _ -Tn (cr.",U )1 _ < d.(k[.6 kT'UJ:2 1T(r.",U (d.m,+d.k,,6 .6)2 ) UJ:2 < _ (15) and from a property of exponential function, I,T(~.u) _ ,7' . .1?) I= kT.In ~ .8 (? 2. (16) We can deduce from (14) that: . 1( ) -( limsupp';,u,( -Pn';,u,( )1 ~ n-+oo (2.6.Mt+d.Ma)(2.kt+d.k,,)62 k J: 6mr(2.k,+d.ka)62 S; puw.p.l Estimation of IV~+l(';) - V 6(.;) 1 A.2 Mter having updated V~(';) with rule (9), IV~+l(';) - V 6(.;) From (4), (9) and (8), I. A (17) let A denote the difference < ,T(?.U) L: [P(';, u, () - p(.;, u, ()] V 6(() + (,T(?.1?) - ,7'(~'1??) L p(.;, u, () V 6 (() ( ( +,7' (?.u) . L:p(.;, u, () [V6(() - V~(()] + L:p(.;, u, ().T(';, u) [r(';, u) - F(';, u)] ( ( + L:( p(.;, u, () [T(';, u) - T(';, u)] r(';, u) for all u E U6 As V is differentiable we have : Vee) = V(';) + VX ' ( ( - . ; ) + 0(1I( - ';11). Let us define a linear function V such that: Vex) = V(';) + VX ' (x - ';). Then we have: [P(';, u, () - p(.;, u, ()] V 6(() = [P(';, u, () - p(.;, u, ()] . [V6(() - V(()] + [P(';,u,()-p(';,u,()]V((), thus: L:([p(';,u,()-p(';,u,()]V6(() = kp .E6.8 + [V(() +0(8)] = [V(7J)-VUD] + kp .E6.8 + 0(8) with: 7J = L:( p(';, u, () (( -.;) and 1j = L:( p(.;, u, () (( - .;). Besides, from the convergence of the scheme (theorem 2), we have E6.8 = L([P(';,U,()-p(.;,u,()] [V(7J) - V(1j)] + 0(8) 0(8). From the linearity of V, IL( [P(';, u, () - p(.;, u, ()] V6 (() I IV(() - I II( - ZII?Mv", V(Z) ~ = S; 2kp 82 . Thus = 0(8) and from (15), (16) and the Lipschitz prop- erty of r, A= 1'l'(?'U), L:( p(.;, u, () [V6(() - V~ (()] 1+ 0(8). As ,..,.7'(?.u) < 1 - 7'(?.U) In 1 < 1 _ T(?.u)-k.,.6 2 In 1 < 1 _ ( I - 2 'Y - 2 we have: A with k = 2d(M[~d.M,,). = (1 - k.8)E~ 'Y - + 0(8) 6 2d(M[+d.M,,) _ !ix..82) In 1 2 'Y ' (18) 1035 Reinforcement Learning for Continuous Stochastic Control Problems A.3 A sufficient condition for sUP?EE~ IV~(~) - V6(~)1 :S C2 Let us suppose that for all ~ E ~6, the following conditions hold for some a > 0 E~ > C2 =} IV~+I(O - V6(~)1 :S E~ - a (19) E~ :S c2=}IV~+I(~)_V6(~)I:Sc2 (20) From the hypothesis that all states ~ E ~6 are regularly updated, there exists an integer m such that at stage n + m all the ~ E ~6 have been updated at least once since stage n. Besides, since all ~ E 8C 6 are updated at least once with rule (10), V~ E 8C6, IV~(~) - V6(~)1 = IR(Xk(T)) - R(~)I :S 2.LR.8 :S C2 for any 8 :S ~3 = 2~lR' Thus, from (19) and (20) we have: E! > E! :S C2 =} E!+m :S E! - a C2 =} E!+m :S Thus there exists N such that: Vn ~ N, E~ :S A.4 C2 C2. Convergence of the algorithm Let us prove theorem 3. For any c > 0, let us consider Cl > 0 and C2 > 0 such that = c. Assume E~ > ?2, then from (18), A = E! - k.8'?2+0(8) :S E~ -k.8.~ for 8 :S ~3. Thus (19) holds for a = k.8.~. Suppose now that E~ :S ?2. From (18), A :S (1 - k.8)?2 + 0(8) :S ?2 for 8 :S ~3 and condition (20) is true. Cl +C2 Thus for 8 :S min {~1, ~2, ~3}, the sufficient conditions (19) and (20) are satisfied. So there exists N, for all n ~ N, E~ :S ?2. Besides, from the convergence of the scheme (theorem 2), there exists ~o st. V8:S ~o, sUP?EE~ 1V6(~) - V(~)I :S ?1? Thus for 8 :S min{~o, ~1, ~2, ~3}, "3N, Vn ~ sup IV~(~) - V(~)I :S sup IV~(~) - V6(~)1 ?EE6 ?EEh N, + sup ?EE6 1V6(~) - V(~)I :S ?1 + c2 = ?. References [BBS95j Andrew G. Barto, Steven J. Bradtke, and Satinder P. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence, (72):81138, 1995. [Ber87j Dimitri P. Bertsekas. Dynamic Programming: Deterministic and Stochastic Models. Prentice Hall, 1987. [FS93j Wendell H. Fleming and H. Mete Soner. Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics. Springer-Verlag, 1993. [KP95j Peter E. Kloeden and Eckhard Platen. Numerical Solutions of Stochastic Differential Equations. Springer-Verlag, 1995. [Kry80j N.V. Krylov. Controlled Diffusion Processes. Springer-Verlag, New York, 1980. [Kus90j Harold J. Kushner. Numerical methods for stochastic control problems in continuous time. SIAM J. Control and Optimization, 28:999-1048, 1990. [Mun97j Remi Munos. A convergent reinforcement learning algorithm in the continuous case based on a finite difference method. International Joint Conference on Art~ficial Intelligence, 1997. [Wat89j Christopher J.C.H. Watkins. Learning from delayed reward. PhD thesis, Cambridge University, 1989. 

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Hybrid NNIHMM-Based Speech Recognition with a Discriminant Neural Feature Extraction Daniel Willett, Gerhard RigoU Department of Comfuter Science Faculty of Electrica Engineering Gerhard-Mercator-University Duisburg, Germany {willett,rigoll}@tb9-ti.uni-duisburg.de Abstract In this paper, we present a novel hybrid architecture for continuous speech recognition systems. It consists of a continuous HMM system extended by an arbitrary neural network that is used as a preprocessor that takes several frames of the feature vector as input to produce more discriminative feature vectors with respect to the underlying HMM system. This hybrid system is an extension of a state-of-the-art continuous HMM system, and in fact, it is the first hybrid system that really is capable of outperforming these standard systems with respect to the recognition accuracy. Experimental results show an relative error reduction of about 10% that we achieved on a remarkably good recognition system based on continuous HMMs for the Resource Management 1OOO-word continuous speech recognition task. 1 INTRODUCTION Standard state-of-the-art speech recognition systems utilize Hidden Markov Models (HMMs) to model the acoustic behavior of basic speech units like phones or words. Most commonly the probabilistic distribution functions are modeled as mIXtures of Gaussian distributions. These mixture distributions can be regarded as output nodes of a Radial-BasisFunction (RBF) network that is embedded in the HMM system [1]. Contrary to neural training procedures the parameters of the HMM system, including the RBF network, are usually estimated to maximize the training observations' likelihood. In order to combine the time-warping abilities of HMMs and the more discriminative power of neural networks, several hybrid approaches arose during the past five years, that combine HMM systems and neural networks. The best known approach is the one proposed by Bourlard [2]. It replaces the HMMs' RBF -net with a Multi-Layer-Perceptron (MLP) which is trained to output each HMM state's posterior probability. At last year's NIPS our group presented a novel hybrid speech recognition approach that combines a discrete HMM speech recognition system and a neural quantizer (3). By maximizing the mutual information between the VQ-Iabels and the assigned phoneme-classes, this apl'roach outperforms standard discrete recognition systems. We showed that this approach IS capable of building up very accurate systems with an extremely fast likelihood computation, that only consists of a quantization and a table lookup. This resulted in a hybrid system with recognition performance equivalent to the best D. Willett and G. Rigoll 764 feature extraction x (t-P) x(t) x(t+F) Neural Network HMM-System x'(t) (linear transfonnation, MLP or recurrent MLP) (RBF-network) p(x(t)IW[) p(x(t)l~ Figure I: Architecture of the hybrid NN/HMM system continuous systems, but with a much faster decoding. Nevertheless, it has turned out that this hybrid approach is not really capable of substantially outperforming very good continuous systems with respect to the recognition accuracy. This observation is similar to experiences with Bourlard's MLP approach. For the decoding procedure, this architecture offers a very efficient pruning technique (phone deactivation pruning [4)) that is much more efficient than pruning on likelihoods, but until today this approach did not outperform standard continuous HMM systems in recognition performance. 2 HYBRID CONTINUOUS HMMlMLP APPROACH Therefore, we followed a different approach, namely the extension of a state-of-the-art continuous system that achieves extremely good recognition rates with a neural net that is trained with MMI-methods related to those in [5]. The major difference in this approach is the fact that the acoustic processor is not replaced by a neural network, but that the Gaussian probability density component is retained and combined with a neural component in an appropriate manner. A similar approach was presented in [6] to improve a speech recognitIOn system for the TIMIT database. We propose to regard the additional neural component as being part of the feature extraction, and to reuse it in recognition systems of higher complexity where discriminative training is extremely expensive. 2.1 ARCHITECTURE The basic architecture of this hybrid system is illustrated in Figure 1. The neural net functions as a feature transformation that takes several additional past and future feature vectors into account to produce an improved more discriminant feature vector that is fed into the HMM system. This architecture allows (at least) three ways of interpretation; 1. as a hybrid system that combines neural nets and continuous HMMs, 2. as an LDA-like transformation that incorporates the HMM parameters into the calculation of the transformation matrix and 3. as feature extraction method, that allows the extraction offeatures according to the underlying HMM system. The considered types of neural networks are linear transformations, MLPs and recurrent MLPs. A detailed 3escription of the possible topologies is given in Section 3. With this architecture, additional past and future feature vectors can be taken into account in the probability estimation process without increasing the dimensionality of the Gaussian mixture components. Instead of increasing the HMM system's number of parameters the neural net is trained to produce more discriminant feature vectors with respect to the trained HMM system. Of course, adding some kind of neural net increases the number of parameters too, but the increase is much more moderate than it would be when increasing each Gaussian's dimensionality. 765 Speech Recognition with a Discriminant Neural Feature Extraction 2.2 TRAINING OBJECTIVE The original purpose of this approach was the intention to transfer the hybrid approach presented in [3], based on MMI neural network, to (semi-) continuous systems. ThIS way, we hoped to be able to achieve the same remarkable improvements that we obtained on discrete systems now on continuous systems, which are the much better and more flexible baseline systems. The most natural way to do this would be the re-estimation of the codebook of Gaussian mean vectors of a semi-continuous system using the neural MMI training algorithm presented in [31. Unfortunately though, this won't work, as this codebook of a semi-continuous system does not determine a separation of the feature space, but is used as means of Gaussian densities. The MMI-principle can be retained, however, by leaving the original HMM system unmodified and instead extending it with a neural component, trained according to a frame-based MMI approach, related to the one in [3] . The MMI criterion is usually formulated in the following way: = argmaxi).(X, W) = argmax(H>.(X) - H>.(XIW)) = argmax P>.(~l~) >. >. >. P>. (1) This means that following the MMI criterion the system's free parameters ,\ have to be estimated to maximize the quotient of the observation's likelihood p>.(XIW) for the known transcription Wand its overall likelihood P>. (X). With X = (x(l), x(2), .. .x(T)) denot(w(l), w(2) , .. .w(T)) denoting the HMM statesing the training observations and W assigned to the observation vectors in a Viterbi-alignment - the frame-based MMI criterion becomes )..MMI = T )..MMI ~ arg~ax L i).(x(i), w(i)) i=1 ITT IT p>.(x(i)lw(i)) p>.(x(i)) :::::::arg~ax . =arg~ax . p>.(x(i)lw(i)) T %=1 1:1 s. 2: P>.(X(Z)lwk)p(Wk) (2) k=1 where S is the total number ofHMM states, (W1 , '" ws) denotes the HMM states and p( Wk) denotes each states' prior-probability that is estimated on the alignment of the training data or by an analysis of the lan~age model. Eq. 2 can be used to re-estImate the Gaussians of a continuous HMM system directly. In [7] we reported the slight improvements in recognition accuracy that we achieved with this parameter estimation. However, it turned out, that only the incorporation of additional features in the probability calculation pipeline can provide more discriminative emission probabilities and a major advance in recognition accuracy. Thus, we experienced it to be more convenient to train an additional neural net in order to maximize Eq. 2. Besides, this approach offers the possibility of improving a recognition system by applying a trained feature extraction network taken from a different system. Section 5 will report our positive experiences with this procedure. At first, for matter of simplicity, we will consider a linear network that takes P past feature vectors and F future feature vectors as additional input. With the linear net denoted as a (P + F + 1) x N matrix NET, each component x' (t)[c] of the network output x'(t) computes to P+F N x'(t)[c] = L L x(t - P + i)[jJ . N ET[i * N + j][c] Vc E {L.N} (3) i=O j=1 so that the derivative with respect to a component of NET easily computes to 8x'(t)[cJ _ 6 -x(t - P 8NET[i*N+j][c] - C,c + i)['J (4) J In a continuous HMM system with diagonal covariance matrices the pdf of each HMM state w is modeled by a mixture of Gaussian components like N C p>.(xlw) = Ld j=1 1 ~ (m;[I)-x[l]) 1 1U wj y'(2rr)nIUjl -"2 L, e 1=1 2 ajll) (5) 766 D. Willett and G. Rigoll A pdfs derivative with respect to a component x'[e] of the net's output becomes N 1 ~ (mj[ll-",/[I))2 C 8p,\.(x'lw) _ ~ d . (:e[e] - mj[e)) 1 e- 2 ~ 8x'[e) - ~ WJ oAe) y'(21l')nIO'jl aj[!} (6) With x(t) in Eq. 2 now replaced by the net output x'(t) the partial derivative ofEq. 2 with respect to a probabilistic distribution function p( x' (i) IWk) computes to 8h(x'(i), w(i)) _ <5W(i) ,Wk 8p,\.(x'(i)lwk) - p,\.(x(i)lwk) s (7) 2: p,\.(x(i)lwI)p(wd 1=1 Thus, using the chain rule the derivative of the net's parameters with respect to the framebased MMI criterion can be computed as displayed in Eq. 8 8h(X, W) = t(t(8h(:e(i)IW(i))) 8p,\.(x'(i)l wk) ox' (i)[c] ) 8N ET[lHe) i=l k=l op,\.(x'(i)lwk) ox' (i)[e] oN ET[lHe) (8) and a gradient descent procedure can be used to determine the optimal parameter estimates. 2.3 ADVANTAGES OF THE PROPOSED APPROACH When using a linear network, the proposed approach strongly resembles the well known Linear Discriminant Analysis (LDA) [8] in architecture and training objective. The main difference is the way the transformation is set up. In the proposed approach the transformation is computed by taking directly the HMM parameters into account whereas the LDA only tries to separate the features according to some class assignment. With the incorporation of a trained continuous HMM system the net's parameters are estimated to produce feature vectors that not only have a good separability in general, but also have a distribution that can be modeled with mixtures ofGaussians very well. Our experiments given at the end ofthis paper prove this advantage. Furthermore, contrary to LDA, that produces feature vectors that don't have much in common with the original vectors, the proposed approach only slightly modifies the input vectors. Thus, a well trained continuous system can be extended by the MMI-net approach, in order to improve its recognition performance without the need for completely rebuilding it. In addition to that, the approach offers a fairly easy extension to nonlinear networks (MLP) and recurrent networks (recurrent MLP). This will be outlined in the following Section. And, maybe as the major advantage, the approach allows keeping up the division of the input features into streams of features that are strongly uncorrelated and which are modeled with separate pdfs. The case of multiple streams is discussed in detail in Section 4. Besides) the MMI approach offers the possibility of a unified training of the HMM system and the reature extraction network or an iterative procedure of training each part alternately. 3 NETWORK TOPOLOGIES Section 2 explained how to train a linear transformation with respect to the frame-based MMI criterion. However, to exploit all the advantages of the proposed hybrid approach the network should be able to perform a nonlinear mapping, in order to produce features whose distribution is (closer to) a mixture of Gaussians although the original distribution is not. 3.1 MLP When using a fully connected MLP as displayed in Figure 2 with one hidden layer of H nodes, that perform the nonlinear function /, the activation of one of the output nodes x'(t)[e] becomes H x'(tHe) = t; P+F N L2[hJ[e)? f( BIAS. + ~ ~ x(t - P + i)li)? L1[i * N + jJ[h)) (9) 767 Speech Recognition with a Discriminant Neural Feature Extraction original features (multiple frames) : :~--:EA6 ::: I: I I I !: :':: ::: o I I x(t-l) x(t) x(t+1) 1 ~ ~ ~ ~ x'(t) I ' "''"~' ~... \NlIl,~l/\f)(JI\L.-i.M/I/ \,,~~ Ll '~~:\ . . "'~~~'~~~~~11111~ ~ p(x(t)lw ~ I I network ~~~~ ::: ::: ::: I RBF- I : : ~ _______________ 1- '--_-_-.:_-_-_-_-_-_-_-_-_-_-_-_-_ p(x(t)lwi p(x(t)lwj ?~~ transformed features - __ context-dependent Hidden Markov Model Figure 2: Hybrid system with a nonlinear feature transfonnation which is easily differentiable with respect to the nonlinear network's parameters. In our experiments we chose f to be defined as the hyperbolic tangents f(x) := tanh(x) = (2(1+ e -X) -1 - 1) so that the partial derivative with respect to i.e. a weight L 1[i . N + J] [h] of the first layer computes to ax' (t)[e] aL1[i . N + 3][h] x(t - p + i)[J] . L2[h][c] .COSh(BIAS. + %~ (10) x(t - P + i)lil? L1[i * N + j][h1r' and the gradient can be set up according to Eq. 8. 3.2 RECURRENT MLP With the incorporation of several additional past feature vectors as explained in Section 2, more discriminant feature vectors can be generated. However, this method is not capable of modeling longer term relations, as it can be achieved by extending the network with some recurrent connections. For the sake of simplicity, in our experiments we simply extended the MLP as indicated with the dashed lines in Figure 2 by propagating the output x (t) back to the input of the network (with a delay of one discrete time step). This type of recurrent neural net is often referred to as a 'Jordan' -network. Certainly, the extension of the network with additional hidden nodes in order to model the recurrence more independently would be possible as well. 4 MULTI STREAM SYSTEMS In HMM-based recognition systems the extracted features are often divided into streams that are modeled independently. This is useful the less correlated the divided features are. In this case the overall likelihood of an observation computes to M p>.(xlw) = IT p$>.(xlw)w, (11) where each of the stream pdfs p$>.(xlw) only uses a subset of the features in x. The stream weights W$ are usually set to unity. D. Willett and G. Rigoll 768 Table 1: Word error rates achieved in the experiments A multi stream system can be improved by a neural extraction for each stream and an independent training of these neural networks. However, it has to be considered that the subdivided features usually are not totally independent and by considering multiple input frames as illustrated in Figure 1 this dependence often increases. It is a common practice, for instance, to model the features' first and second order delta coefficients in independent streams. So, for sure the streams lose independence when considering multiple frames, as these coefficients are calculated using the additional frames. Nevertheless, we found it to give best results to maintain this subdivision into streams, but to consider the stronger correlation by training each stream's net dependent on the other nets' outputs. A training criterion follows straight from Eq. 11 inserted in Eq. 2. \ _ /\MMI - argmax ). rrT p).(x(i)lw(i)) _ ( (.)) i=l P). X t - argmax .). rrT 11M (P3).(X(i)IW(i)))W' i=13=1 ( (.)) Ps). X z The derivative of this equation with respect to the pdf P.;). (xlw) ofa specific stream pends on the other streams' pdfs. With the Ws set to unity it is 8h(x'(;): w(i)) = 8ps).(x (Z)IWk) (rr ps).(X(i)I~(i))) ( ~. 3r 3 P3).(X(Z)) 6w (i):Wk ps).(X(Z)IWk) (12) s de- _ s p(Wk) ) 'I\' ((')1) ( ) W P.;). x Z WI P WI 1=1 (13) Neglecting the correlation among the streams the training of each stream's net can be done independently. However, the more the incorporation of additional features increases the streams' correlation, the more important it gets to train the nets in a unified training procedure according to Eq. 13. 5 EXPERIMENTS AND RESULTS We applied the proposed approach to improve a context-independent (monophones) and a context-dependent (triphones) continuous speech reco~tion system for the 1000-wordResource Management (RM) task. The systems used lmear HMMs of three emitting states each. The tying of Gaussian mixture components was perfonned with an adaptive procedure according to [9]. The HMM states of the word-internal triphone system were clustered in a tree-based phonetic clustering procedure. Decoding was perfonned with a Viterbidecoder and the standard wordpair-grammar of perplexity 60. Training of the MLP was perfonned with the RPROP algorithm. For training the weights of the recurrent connections we chose real-time recurrent learning. The average error rates were computed using the test-sets Feb89, Oct89, Feb91 and Sep92. The table above shows the recognition results with single stream systems in its first section. These systems simply use a 12-value Cepstrum feature vector without the incorporation of delta coefficients. The systems with an input transfonnation use one additional past and one additional future feature vector as input. The proposed approach achieves the same perfonnance as the LDA, but it is not capable of outperfonning It. The second section of the table lists the recognition results with four stream systems that use the first and second order delta coefficients in additional streams plus log energy and this values' delta coefficients in a forth stream. The MLP system trained according to Eq. Speech Recognition with a Discriminant Neural Feature Extraction 769 II slightly outperforms the other approaches. The incorporation of recurrent network connections does not improve the system's performance. The third section of the table lists the recognition results with four stream systems with a context-dependent acoustic modeling (triphones). The applied LDA and the MMI-networks were taken from the monophone four stream system. On the one hand, this was done to avoid the computational complexity that the MMI training objective causes on contextdependent systems. On the other hand, this demonstrates that the feature vectors produced by the trained networks have a good discrimination for continuous systems in general. Again, the MLP system outperforms the other? approaches and achieves a very remarkable word error rate. It should be pointed out here, that the structure of the continuous system as reported in (9) is already highly optimized and it is almost impossible to further reduce the error rate by means of any acoustic modeling method. This is reflected in the fact that even a standard LDA cannot improve this system. Only the new neural approach leads to a 10% reduction in error rate which is a large improvement considering the fact that the error rate of the baseline system is among the best ever reported for the RM database, 6 CONCLUSION The paper has presented a novel approach to discriminant feature extraction. A MLP network has successfully been used to compute a feature transformation that outputs extremely suitable features for continuous HMM systems. The experimental results have proven that the proposed approach is an appropriate method for including several feature frames in the probability estimation process without increasing the dimensionality of the Gaussian mixture components in the HMM system. Furthermore did the results on the triphone speech recognition system prove that the approach provides discriminant features, not only for the system that the mapping is computed on, but for HMM systems with a continuous modeling in general: The application of recurrent networks did not improve the recognition accuracy. The longer range relations seem to be very weak and they seem to be covered well by using the neighboring feature vectors and first and second order delta coefficients. The proposed unified training procedure for multiple nets in multi-stream systems allows keeping up the subdivision of features of weak correlations, and gave us best profits in recognition accuracy. References [1) H. Ney, "Speech Recognition in a Neural Network Framework: Discriminative Training of Gaussian Models and Mixture Densities as Radial Basis Functions", Proc. IEEEICASSp, 1991, pp. 573-576. [2) H Bouriard, N. Morgan, "Connectionist Speech Recognition - A Hybrid Approach", Kluwer Academic Press, 1994. [3) G. Rigoll, C. Neukirchen, "A new approach to hybrid HMMIANN speech recognition using mutual information neural networks", Advances in Neural Information Processing Systems (NIPS-96), Denver, Dec. 1996, pp. 772-778. [4] M. M. Hochberg, G. D. Cook, S. J. Renals, A. J. Robinson, A. S. Schechtman, "The 1994 ABBOT Hybrid Connectionist-HMM Large-Vocabulary Recognition System", Proc. ARPA Spoken Language Systems Technology Workshop, 1995. [5] G. Rigoll, "Maximum Mutual Information Neural Networks for Hybrid ConnectionistHMM Speech Recognition", IEEE-Trans. Speech Audio Processing, Vol. 2, No.1, Jan. 1994,pp.175-184. [6] Y. Bengio et aI., "Global Optimization of a Neural Network - Hidden Markov Model Hybrid" IEEE-Transcations on NN, Vol. 3, No. 2, 1992, pp. 252-259. [7] D. Willett, C. Neukirchen, R. Rottland, "Dictionary-Based Discriminative HMM Parameter Estimation for Continuous Speech Recognition Systems", Proc. IEEE-ICASSp, 1997,pp.1515-1518. [8] X. Aubert, R. Haeb-Umbach, H. Ney, "Continuous mixture densities and linear discriminant analysis for improved context-dependent acoustic models", Proc. IEEE-ICASSp, 1993, pp. II 648-651. [9) D. Willett, G. Rigoll, "A New Approach to Generalized Mixture Tying for Continuous HMM-Based Speech Recognition",Proc. EUROSPEECH, Rhodes, 1997. 

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Bayesian Robustification for Audio Visual Fusion Javier Movellan * movellanOcogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92092-0515 Paul Mineiro pmineiroOcogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92092-0515 Abstract We discuss the problem of catastrophic fusion in multimodal recognition systems. This problem arises in systems that need to fuse different channels in non-stationary environments. Practice shows that when recognition modules within each modality are tested in contexts inconsistent with their assumptions, their influence on the fused product tends to increase, with catastrophic results. We explore a principled solution to this problem based upon Bayesian ideas of competitive models and inference robustification: each sensory channel is provided with simple white-noise context models, and the perceptual hypothesis and context are jointly estimated. Consequently, context deviations are interpreted as changes in white noise contamination strength, automatically adjusting the influence of the module. The approach is tested on a fixed lexicon automatic audiovisual speech recognition problem with very good results. 1 Introduction In this paper we address the problem of catastrophic fusion in automatic multimodal recognition systems. We explore a principled solution based on the Bayesian ideas of competitive models and inference robustification (Clark & Yuille, 1990; Box, 1980; O'Hagan, 1994). For concreteness, consider an audiovisual car telephony task which we will simulate in later sections. The task is to recognize spoken phone numbers based on input from a camera and a microphone. We want the recognition system to work on environments with non-stationary statistical properties: at times the video signal (V) may be relatively clean and the audio signal (A) may be contaminated by sources like the radio, the engine, and friction with the road. At other times the A signal may be more reliable than the V signal, e.g., the radio is off, but the talker's mouth is partially occluded. Ideally we want the audio-visual system to combine the A and V sources optimally given the conditions at hand, e.g., give more weight to whichever channel is more reliable at that time. At a minimum we expect that for a wide variety of contexts, the performance after fusion should not be worse than the independent unimodal systems (Bernstein & Benoit, 1996). When component modules can Significantly outperform the overall system after fusion, catastrophic fusion is said to have occurred. ? To whom correspondence should be addressed. Bayesian Robustification for Audio Visual Fusion 743 Fixed vocabulary audiovisual speech recognition (AVSR) systems typically consist of two independent modules, one dedicated to A signals and one to V signals (Bregler, Hild, Manke & Waibel, 1993; Wolff, Prasad, Stork & Hennecke, 1994; Adjondani & Benoit, 1996; Movellan & Chadderdon, 1996). From a Bayesian perspective this modularity reflects an assumption of conditional independence of A and V signals (i.e., the likelihood function factorizes) P(XaXvIWi.Aa.Av) <X P(XaIWi.Aa)P(xvlwi.Av), (1) where Xa and Xv are the audio and video data, Wi is a perceptual interpretation of the data (e.g., the word "one") and {.Aa,.A v } are the audio and video models according to which these probabilities are calculated, e.g., a hidden Markov model , a neural network, or an exemplar model. Training is also typically modularized: the A module is trained to maximize the likelihood of a sample of A signals while the V module is trained on the corresponding sample of V signals. At test time new data are presented to the system and each module typically outputs the log probability of its input given each perceptual alternative. Assuming conditional independence, Bayes' rule calls for an affine combination of modules 'Ii; argmax {logp(wilxaxv.Aa.Av)} Wi Wi where 'Ii; is the interpretation chosen by the system, and p(Wi) is the prior probability of each alternative. This fusion rule is optimal in the sense that it minimizes the expected error: no other fusion rule produces smaller error rates, provided the models {.Aa,.A v } and the assumption of conditional independence are correct. Unfortunately a naive application of Bayes' rule to AVSR produces catastrophic fusion. The A and V modules make assumptions about the signals they receive, either explicitly, e.g., a well defined statistical model, or implicitly, e.g., a blackbox trained with a particular data sample. In our notation these assumptions are reflected by the fact that the log-likelihoods are conditional on models: {.A a , .A v }. The fact that modules make assumptions implies that they will operate correctly only within a restricted context, i.e, the collection of situations that meet the assumptions. In practice one typically finds that Bayes' rule assigns more weight to modules operating outside their valid context, the opposite of what is desired. 2 Competitive Models and Bayesian Robustification Clark and Yuille (1990) and Yuille and Bulthoff (1996) analyzed information integration in sensory systems from a Bayesian perspective. Modularity is justified in their view by the need to make assumptions that disambiguate the data available to the perceptual system (Clark & Yuille, 1990, p. 5). However, this produces modules which are valid only within certain contexts. The solution proposed by Clark and Yuille (1990) is the creation of an ensemble of models each of which specializes on a restricted context and automatically checks whether the context is correct. The hope is that by working with such an ensemble of models, robustness under a variety of contexts can be achieved (Clark & Yuille, 1990, p. 13). Box (1980) investigated the problem of robust statistical inference from a Bayesian perspective. He proposed extending inference models with additional "nuisance" parameters a, a process he called Bayesian robustification. The idea is to replace an implicit assumption about the specific value of a with a prior distribution over a, representing uncertainty about that parameter. The approach here combines the ideas of competitive models and robustification. Each of the channels in the multimodal recognition system is provided with extra 744 1. Movellan and P. Mineiro parameters that represent non-stationary properties of the environment, what we call a context model. By doing so we effectively work with an infinite ensemble of models each of which compete on-line to explain the data. As we will see later even unsophisticated context models provide superior performance when the environment is non-stationary. We redefine the estimation problem as simultaneously choosing the most probable A and V context parameters and the most probable perceptual interpretation w = argmax {max P(WwaO'vIXaXvAaAv)} Wi (3) Uo,U v where 0' a and 0' v are the context parameters for the audio and visual channels and Wi are the different perceptual interpretations. One way to think of this joint decision approach is that we let all context models compete and we let only the most probable context models have an influence on the fused percept. Hereafter we refer to this approach as competitive fusion. Assuming conditional independence of the audio and video data and uninformative priors for (0' a, 0'v), we have w= ar~ax {logp(Wi) + [~~IOgP(XalwwaAa)] + [~::XIOgP(xvIWiO'vAv)]}. (4) Thus conditional independence allows a modular implementation of competitive fusion, Le., the A and V channels do not need to talk to each other until the time to make a joint decision, as follows. 1. For each Wi obtain conditional estimates of the context parameters for the audio and video signals: o-~IWi ~ argmax { 10gp(xalwWaAa) } , (5) (To. and o-~IWi ~ argmax{ 10gp(xvlwWvAv) }. (6) (Tv 2. Find the best Wi using the conditional context estimates. w= argmax {IOgp(Wi) + logp(xalwio-alwi Aa) + logp(xv IWio-vlwi Av)} Wi 3 (7) Application to AVSR Competitive fusion can be easily applied to Hidden Markov Models (HMM), an architecture closely related to stochastic neural networks and arguably the most successful for AVSR. Typical hidden Markov models used in AVSR are defined by ? Markovian state dynamics: p(qt+ll2.t ) time t and 2.t = (ql," . qt), = p(qt+llqt), where qt is the state at ? Conditionally independent sensor models linking observations to states ! (Xt Iqt), typically a mixture of multivariate Gaussian densities !(xtlqt) = L p(mi Iqt)(27l') -N/2 1~I-l/2 exp(d(xt, qt, Pi, ~)), (8) Bayesian Robustification for Audio VIsual Fusion 745 where N is the dimensionality of the data, mi is the mixture label, p(milqt) is the mixture distribution for state qt, Pi is the centroid for mixture mi, E is a covariance matrix, and d is the Mahalanobis norm (9) The approach explored here consists on modeling contextual changes as variations on the variance parameters. This corresponds to modeling non-stationary properties of the environments as variations in white noise power within each channel. Competitive fusion calls for on-line maximization of the variance parameters at the same time we optimize with respect to the response alternative. 11; = arg:ax{ logp(wi) [ + ~~logp(xalwiEaAa)] + [ ~~logp(xvlwiEv'\v)] }. (10) The maximization with respect to the variances can be easily integrated into standard HMM packages by simply applying the EM learning algorithm (Dampster, Laird & Rubin, 1977) on the variance parameters at test time. Thus the only difference between the standard approach and competitive fusion is that we retrain the variance parameters of each HMM at test time. In practice this training takes only one or two iterations of the EM algorithm and can be done on-line. We tested this approach on the following AVSR problem. Training database We used Tulips1 (Movellan, 1995) a database consisting of 934 images of 9 male and 3 female undergraduate students from the Cognitive Science Department at the University of California, San Diego. For each of these, two samples were taken for each of the digits "one" through "four". Thus, the total database consists of 96 digit utterances. The specifics of this database are explained in (Movellan, 1995). The database is available at http://cogsci.ucsd.edu. Visual processing We have tried a wide variety of visual processing approaches on this database, including decomposition with local Gaussian templates (Movellan, 1995), PCA-based templates (Gray, Movellan & Sejnowski, 1997), and Gabor energy templates (Movellan & Prayaga, 1996). To date, best performance was achieved with the local Gaussian approach. Each frame of the video track is soft-thresholded and symmetrized along the vertical axis, and a temporal difference frame is obtained by subtracting the previous symmetrized frame from the current symmetrized frame. We calculate the inner-products between the symmetrized images and a set of basis images. Our basis images were lOx15 shifted Gaussian kernels with a standard deviation of 3 pixels. The loadings of the symmetrized image and the differential image are combined to form the final observation frame. Each of these composite frames has 300 dimensions (2xlOx15). The process is explained in more detail in Movellan (1995). Auditory processing LPC /cepstral analysis is used for the auditory front-end. First, the auditory signal is passed through a first-order emphasizer to spectrally flatten it. Then the signal is separated into non-overlapping frames at 30 frames per second. This is done so that there are an equal number of visual and auditory feature vectors for each utterance, which are then synchronized with each other. On each frame we perform the standard LPC / cepstral analysis. Each 30 msec auditory frame is characterized by 26 features: 12 cepstral coefficients, 12 delta-cepstrals, 1 log-power, and 1 delta-log-power. Each of the 26 features is encoded with 8-bit accuracy. J Movellan and P. Mineiro 746 Figure 1: Examples of the different occlusion levels, from left to right: 0%, 10%, 20%, 40%, 60%, 80%. Percentages are in terms of area. Recognition Engine In previous work (Chadderdon & Movellan, 1995) a wide variety of HMM architectures were tested on this database including architectures that did not assume conditional independence. Optimal performance was found with independent A and V modules using variance matrices of the form uI, where u is a scalar and I the identity matrix. The best A models had 5 states and 7 mixtures per state and the best V models had 3 states and 3 mixtures per state. We also determined the optimal weight of A and V modules. Optimal performance is obtained by weighting the output of V times 0.18. Factorial Contamination Experiment In this experiment we used the previously optimized architecture and compared its performance under 64 different conditions using the standard and the competitive fusion approaches. We used a 2 x 8 x 8 factorial design, the first factor being the fusion rule, and the second and third factors the context in the audio and video channels. To our knowledge this is the first time an AVSR system is tested with a factorial experimental design with both A and V contaminated at various levels. The independent variables were: 1. Fusion rule: Classical, and competitive fusion. 2. Audio Context: Inexistent, clean, or contaminated at one of the following signal to noise ratios: 12 Db, 6 Db, 0 Db, -6 Db, -12 Db and -100 Db. The contamination was done with audio digitally sampled from the interior of a car while running on a busy highway with the doors open and the radio on a talk-show station. 3. Video Context: Inexistent, clean or occluded by a grey level patch. The percentages of visual area occupied by the patch were 10%,20%,40%,60%, 80% and 100% (see Figure 1). The dependent variable was performance on the digit recognition task evaluated in terms of generalization to new speakers. In all cases training was done with clean signals and testing was done with one of the 64 contexts under study. Since the training sample is small, generalization performance was estimated using a jackknife procedure (Efron, 1982). Models were trained with 11 subjects, leaving a different subject out for generalization testing. The entire procedure was repeated 12 times, each time leaving a different subject out for testing. Statistics of generalization performance are thus based on 96 generalization trials (4 digits x 12 subjects x 2 observations per subject). Standard statistical tests were used to compare the classical and competitive context rules. The results of this experiment are displayed in Table 1. Note how the experiment replicates the phenomenon of catastrophic fusion. With the classic approach, when one of the channels is contaminated, performance after fusion can be significantly Bayesian Robustificationfor Audio Visual Fusion Video 747 one None -6 -1 95.83 95.83 90.62 80.20 67.70 42 . 70 Clean 84 .37 97.92 97.92 94.80 90.62 89.58 81.25 10% 73.95 93.75 93.75 94 .79 87 .50 80 .20 71.87 19 .80 82 20 1 64.58 . I 20% 62.50 96.87 96 .87 94.79 89.58 80.20 1 6 2 .501 41.66 40% 37.50 93 .75 89.58 87.50 83.30 70.83 43 .75 30.20 60% 34.37 93.75 91.66 82.29 65.62 42.70 26.04 80% 27.00 95.83 90.62 79.16 64.58 146.871 25 .00 100% 25.00 93.75 92.71 84.37 78.12 63.54 Performance wIth C/a .. 1e Fu" on Audio 44 .79 26.04 Video one 20 .83 None ean 95 .83 94.79 89.58 79. 16 65 .62 40 .62 Clean 86 .45 98.95 96.87 95.83 93.75 87.50 79 . 16 10% 73.95 93.75 93 .75 93.75 89.58 79 .16 70 .83 70 83 1 52 ..58 / 20% 54.16 89.58 84.37 84.37 75.00 29.16 ~ 43 .00 40% 81.25 78.12 67.20 52 .08 38.54 34.37 60% 32.29 77 .08 72.91 62 .50 47.91 37.50 29 . 16 80% 29 . 16 70.83 68.75 54.16 44 .79 1 33 .831 28.12 100% 25.00 61.46 58.33 51.04 42.70 38.54 29 . 16 61.45 Table 1: Average generalization performance with standard and competitive fusion. Boxed cells indicate a statistically significant difference a = 0.05 between the two fusion approaches. worse than performance with the clean channel alone. For example, when the audio is clean, the performance of the audio-only system is 95.83%. When combined with bad video (100% occlusion), this performance drops down to 61.46%, a statistically significant difference, F(l,ll) = 132.0, p < 10- 6 . Using competitive fusion, the performance of the joint system is 93.75%, which is not significantly different from the performance of the A system only, F(l,ll) = 2.4, p= 0.15. The table shows in boxes the regions for which the classic and competitive fusion approaches were significantly different (a = 0.05). Contrary to the classic approach, the competitive approach behaves robustly in all tested conditions. 4 Discussion Catastrophic fusion may occur when the environment is non-stationary forcing modules to operate outside their assumed context. The reason for this problem is that in the absence of a context model, deviations from the expected context are interpreted as information about the different perceptual interpretations instead of information about contextual changes. We explored a principled solution to this problem inspired by the Bayesian ideas of robustification (Box, 1980) and competitive models (Clark & Yuille, 1990). Each module was provided with simple white-noise context models and the most probable context and perceptual hypothesis were jointly estimated. Consequently, context deviations are interpreted as changes in the white noise contamination strength, automatically adjusting the influence of the module. The approach worked very well on a fixed lexicon AVSR problem. References Adjondani, A. & Benoit, C. (1996). On the Integration of Auditory and Visual Parameters in an HMM-based ASR. In D. G. Stork & M. E. Hennecke (Eds.), Speechreading by Humans and Machines: Models, Systems, and Applications, pages 461-471. New York: NATO/Springer-Verlag. Bernstein, L. & Benoit, C. (1996). For Speech Perception Three Senses are Bettern 748 1. Movellan and P. Mineiro than One. In Proc. of the 4th Int. Conf. on Spoken Language Processing, Philadelphia, PA., USA. Box, G. E. P. (1980). Sampling and Bayes inference in scientific modeling. J. Roy. Stat. Soc., A., 143, 383-430. Bregler, C., Hild, H., Manke, S., & Waibel, A. (1993). Improving Connected Letter Recognition by Lipreading. In Proc. Int. Conf. on Acoust., Speech, and Signal Processing, volume 1, pages 557-560, Minneapolis. IEEE. Biilt hoff, H. H. & Yuille, A. L. (1996). A Bayesian framework for the integration of visual modules. In T. Inui & J. L. McClelland (Eds.), Attention and performance XVI: Information integmtion in perception and communication, pages 49-70. Cambridge, MA: MIT Press. Chadderdon, G. & Movellan, J. (1995). Testing for Channel Independence in Bimodal Speech Recognition. In Proceedings of 2nd Joint Symposium on Neuml Computation, pages 84-90. Clark, J. J. & Yuille, A. L. (1990). Data Fusion for Sensory Information Processing Systems. Boston: Kluwer Academic Publishers. Dampster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc., 39, 1-38. Efron, A. (1982). The jacknife, the bootstmp and other resampling plans. Philadelphia, Pennsylvania: SIAM. Gray, M. S., Movellan, J. R., & Sejnowski, T. (1997). Dynamic features for visual speechreading: A systematic comparison. In Mozer, Jordan, & Petsche (Eds.), Advances in Neuml Information Processing Syste"ms, volume 9. MIT Press. Movellan, J. R. (1995). Visual speech recognition with stochastic neural networks. In G. Tesauro, D. Touretzky, & T. Leen (Eds.), Advances in neuml information processing systems. Cambridge,Massacusetts: MIT Press. Movellan, J. R. & Chadderdon, G. (1996). Channel Separability in the Audio Visual Integration of Speech: A Bayesian Approach. In D. G. Stork & M. E. Hennecke (Eds.), Speechreading by Humans and Machines: Models, Systems, and Applications, pages 473-487. New York: NATO/Springer-Verlag. Movellan, J. R. & Prayaga, R. S. (1996). Gabor Mosaics: A description of Local Orientation Statistics with Applications to Machine Perception. In G. W. Cottrell (Ed.), proceedings of the Eight Annual Conference of the Cognitive Science Society, page 817. Mahwah, New Jersey: LEA. O'Hagan, A. (1994). Kendall's Advanced Theory of Statistics: Volume 2B, Bayesian Inference. volume 2B. Cambridge University Press. Wolff, G. J., Prasad, K. V., Stork, D. G., & Hennecke, M. E. (1994). Lipreading by Neural Networks: Visual Preprocessing, Learning and Sensory Integration. In J. D. Cowan, G. Tesauro, & J. Alspector (Eds.), Advances in Neuml Information Processing Systems, volume 6, pages 1027-1034. Morgan Kaufmann. 

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Reinforcement Learning for Continuous Stochastic Control Problems Remi Munos CEMAGREF, LISC, Pare de Tourvoie, BP 121, 92185 Antony Cedex, FRANCE. Rerni.Munos@cemagref.fr Paul Bourgine Ecole Polyteclmique, CREA, 91128 Palaiseau Cedex, FRANCE. Bourgine@poly.polytechnique.fr Abstract This paper is concerned with the problem of Reinforcement Learning (RL) for continuous state space and time stocha.stic control problems. We state the Harnilton-Jacobi-Bellman equation satisfied by the value function and use a Finite-Difference method for designing a convergent approximation scheme. Then we propose a RL algorithm based on this scheme and prove its convergence to the optimal solution. 1 Introduction to RL in the continuous, stochastic case The objective of RL is to find -thanks to a reinforcement signal- an optimal strategy for solving a dynamical control problem. Here we sudy the continuous time, continuous state-space stochastic case, which covers a wide variety of control problems including target, viability, optimization problems (see [FS93], [KP95])}or which a formalism is the following. The evolution of the current state x(t) E 0 (the statespace, with 0 open subset of IRd ), depends on the control u(t) E U (compact subset) by a stochastic differential equation, called the state dynamics: (1) dx = f(x(t), u(t))dt + a(x(t), u(t))dw where f is the local drift and a .dw (with w a brownian motion of dimension rand (j a d x r-matrix) the stochastic part (which appears for several reasons such as lake of precision, noisy influence, random fluctuations) of the diffusion process. For initial state x and control u(t), (1) leads to an infinity of possible traj~tories x(t). For some trajectory x(t) (see figure I)., let T be its exit time from 0 (with the convention that if x(t) always stays in 0, then T = 00). Then, we define the functional J of initial state x and control u(.) as the expectation for all trajectories of the discounted cumulative reinforcement : J(x; u(.)) = Ex,u( .) {loT '/r(x(t), u(t))dt +,,{ R(X(T))} R. Munos and P. Bourgine 1030 where rex, u) is the running reinforcement and R(x) the boundary reinforcement. 'Y is the discount factor (0 :S 'Y < 1). In the following, we assume that J, a are of class C2 , rand Rare Lipschitzian (with constants Lr and LR) and the boundary 80 is C2 . ? ? all ? II ? ? xirJ ? ? ? ? ? ? ? ? Figure 1: The state space, the discretized ~6 (the square dots) and its frontier 8~6 (the round ones). A trajectory Xk(t) goes through the neighbourhood of state ~. RL uses the method of Dynamic Program~ing (DP) which generates an optimal (feed-back) control u*(x) by estimating the value function (VF), defined as the maximal value of the functional J as a function of initial state x : (2) Vex) = sup J(x; u(.). u( .) In the RL approach, the state dynamics is unknown from the system ; the only available information for learning the optimal control is the reinforcement obtained at the current state. Here we propose a model-based algorithm, i.e. that learns on-line a model of the dynamics and approximates the value function by successive iterations. Section 2 states the Hamilton-Jacobi-Bellman equation and use a Finite-Difference (FD) method derived from Kushner [Kus90] for generating a convergent approximation scheme. In section 3, we propose a RL algorithm based on this scheme and prove its convergence to the VF in appendix A. 2 A Finite Difference scheme Here, we state a second-order nonlinear differential equation (obtained from the DP principle, see [FS93J) satisfied by the value function, called the Hamilton-JacobiBellman equation. Let the d x d matrix a = a.a' (with' the transpose of the matrix). We consider the uniformly pambolic case, Le. we assume that there exists c > 0 such that V$ E 0, Vu E U, Vy E IR d ,2:t,j=l aij(x, U)YiYj 2: c1lY112. Then V is C2 (see [Kry80J). Let Vx be the gradient of V and VXiXj its second-order partial derivatives. Theorem 1 (Hamilton-Jacohi-Bellman) The following HJB equation holds : Vex) In 'Y + sup [rex, u) uEU + Vx(x).J(x, u) + ! 2:~j=l aij V XiXj (x)] = 0 for x E 0 Besides, V satisfies the following boundary condition: Vex) = R(x) for x E 80. 1031 Reinforcement Learningfor Continuous Stochastic Control Problems Remark 1 The challenge of learning the VF is motivated by the fact that from V, we can deduce the following optimal feed-back control policy: u*(x) E arg sup [r(x, u) uEU + Vx(x).f(x, u) + ! L:7,j=l aij VXiXj (x)] In the following, we assume that 0 is bounded. Let eI, ... , ed be a basis for JRd. Let the positive and negative parts of a function 4> be : 4>+ = ma.x(4),O) and 4>- = ma.x( -4>,0). For any discretization step 8, let us consider the lattices: 8Zd = {8. L:~=1 jiei} where j}, ... ,jd are any integers, and ~6 = 8Z d n O. Let 8~6, the frontier of ~6 denote the set of points {~ E 8Zd \ 0 such that at least one adjacent point ~ ? 8ei E ~6} (see figure 1). Let U 6 cUbe a finite control set that approximates U in the sense: 8 ~ 8' => U6' C U6 and U6U 6 = U. Besides, we assume that: Vi = l..d, (3) By replacing the gradient Vx(~) by the forward and backward first-order finitedifference quotients: ~;, V(~) = l [V(~ ? 8ei) - V(~)l and VXiXj (~) by the secondorder finite-difference quotients: -b [V(~ + 8ei) + V(,' ~XiXi V(~) ~;iXj V(~) - 8ei) - 2V(O] = 2P[V(~ + 8ei ? 8ej) + V(~ - 8ei =F 8ej) -V(~ + 8ei) - V(~ - 8ei) - V(~ + 8ej) - V(~ - 8ej) + 2V(~)] in the HJB equation, we obtain the following : for ~ E :?6, V6(~)In,+SUPUEUh {r(~,u) + L:~=1 [f:(~,u)'~~iV6(~) - fi-(~,U)'~;iV6(~) + aii (~.u) ~ X,X,'" V(C) + " . . 2 . . wJ'l=~ (at; (~,'U) ~ +x,x.1'" .V(C) _ a~ (~,'U) ~ x,x V(C))] } = 0 2 2 -. . J '" Knowing that (~t In,) is an approximation of (,l:l.t -1) as ~t tends to 0, we deduce: V6(~) with T(~, SUPuEUh [,"'(~'U)L(EEbP(~,U,()V6?()+T(~,u)r(~,u)] (4) (5) u) which appears as a DP equation for some finite Markovian Decision Process (see [Ber87]) whose state space is ~6 and probabilities of transition: p(~,u,~ p(~, ? 8ei) u, ~ + 8ei ? 8ej) p(~,u,~ - 8ei ? 8ej) p(~,u,() "'~~r) [28Ift(~, u)1 + aii(~' u) - Lj=l=i laij(~, u)l] , "'~~r)a~(~,u)fori=f:j, (6) "'~~r)a~(~,u) for i =f: j, o otherwise. Thanks to a contraction property due to the discount factor" there exists a unique solution (the fixed-point) V to equation (4) for ~ E :?6 with the boundary condition V6(~) = R(~) for ~ E 8:?6. The following theorem (see [Kus90] or [FS93]) insures that V 6 is a convergent approximation scheme. R. Munos and P. Bourgine 1032 Theorem 2 (Convergence of the FD scheme) V D converges to V as 8 1 0 : lim VD(~) /)10 ~-x = Vex) un~formly on 0 Remark 2 Condition (3) insures that the p(~, u, () are positive. If this condition does not hold, several possibilities to overcome this are described in [Kus90j. 3 The reinforcement learning algorithm Here we assume that f is bounded from below. As the state dynami,:s (J and a) is unknown from the system, we approximate it by building a model f and from samples of trajectories Xk(t) : we consider series of successive states Xk = Xk(tk) and Yk = Xk(tk + Tk) such that: a - "It E [tk, tk + Tk], x(t) E N(~) neighbourhood of ~ whose diameter is inferior to kN.8 for some positive constant kN, - the control u is constant for t E [tk, tk - Tk + Tk], satisfies for some positive kl and k2, (7) Then incrementally update the model : .1 ",n n ~k=l n (Yk - Xk -;;; Lk=l 1 an(~,u) Yk - Xk Tk Tk.fn(~, u)) and compute the approximated time T( x, u) transition p(~, u, () by replacing f and a by (Yk - Xk Tk Tk?fn(~, u))' (8) ~d f the approximated probabilities of and a in (5) and (6). We obtain the following updating rule of the V D-value of state ~ : V~+l (~) = sUPuEU/) [,~/:(x,u) L( p(~, u, ()V~(() + T(x, u)r(~, u)] (9) which can be used as an off-line (synchronous, Gauss-Seidel, asynchronous) or ontime (for example by updating V~(~) as soon as a trajectory exits from the neighbourood of ~) DP algorithm (see [BBS95]). Besides, when a trajectory hits the boundary [JO at some exit point Xk(T) then update the closest state ~ E [JED with: (10) Theorem 3 (Convergence of the algorithm) Suppose that the model as well as the V D-value of every state ~ E :ED and control u E U D are regularly updated (respectively with (8) and (9)) and that every state ~ E [JED are updated with (10) at least once. Then "Ie> 0, :3~ such that "18 ~ ~, :3N, "In 2: N, sUP~EE/) IV~(~) - V(~)I ~ e with probability 1 1033 Reinforcement Learningfor Continuous Stochastic Control Problems 4 Conclusion This paper presents a model-based RL algorithm for continuous stochastic control problems. A model of the dynamics is approximated by the mean and the covariance of successive states. Then, a RL updating rule based on a convergent FD scheme is deduced and in the hypothesis of an adequate exploration, the convergence to the optimal solution is proved as the discretization step 8 tends to 0 and the number of iteration tends to infinity. This result is to be compared to the model-free RL algorithm for the deterministic case in [Mun97]. An interesting possible future work should be to consider model-free algorithms in the stochastic case for which a Q-Iearning rule (see [Wat89]) could be relevant. A Appendix: proof of the convergence Let M f ' M a, M fr. and Ma .? be the upper bounds of j, a, f x and 0' x and m f the lower bound of f. Let EO = SUP?EI:h !V0 (';) - V(';)I and E! = SUP?EI:b \V~(';) - VO(.;)\. Estimation error of the model fn and an and the probabilities Pn A.I Suppose that the trajectory Xk(t) occured for some occurence Wk(t) of the brownian motion: Xk(t) = Xk + f!k f(Xk(t),u)dt + f!" a(xk(t),U)dwk. Then we consider a trajectory Zk (t) starting from .; at tk and following the same brownian motion: Zk(t) ='; + fttk. f(Zk(t), u)dt + fttk a(zk(t), U)dWk' Let Zk = Zk(tk + Tk). Then (Yk - Xk) - (Zk -.;) = ftk [f(Xk(t), u) - f(Zk(t), u)] dt + t Tk ft [a(xk(t), u) - a(zk(t), u)J dWk. Thus, from the C1 property of f and a, :.+ II(Yk - Xk) - (Zk - ';)11 ~ (Mf'" + M aJ.kN.Tk. 8. (11) The diffusion processes has the following property ~ee for example the ItO- Taylor majoration in [KP95j) : Ex [ZkJ = ';+Tk.f(';, U)+O(Tk) which, from (7), is equivalent to: Ex [z~:g] = j(';,u) + 0(8). Thus from the law of large numbers and (11): li~-!~p Ilfn(';, u) - f(';, u)11 - li;;:s~p II~ L~=l [Yk;kX & - (Mf:r: ?.] I + 0(8) + M aJ?kN?8 + 0(8) = 0(8) w.p. 1 (12) Besides, diffusion processes have the following property (again see [KP95J): Ex [(Zk -.;) (Zk - .;)'] = a(';, U)Tk + f(';, u).f(';, U)'.T~ + 0(T2) which, from (7), is equivalent to: Ex [(Zk-?-Tkf(S'U)~(kZk-S-Tkf(S'U?/] = a(';, u) + 0(82). Let rk = = Yk - Xk - Tkfn(';, u) which satisfy (from (11) and (12? : Ilrk - ikll = (Mf:r: + M aJ.Tk.kN.8 + Tk.o(8) (13) Zk -.; - Tkf(';, u) and ik From the definition of Ci;;(';,u), we have: Ci;;(';,u) - a(';,u) Ex [r~':k] + 0(8 2 ) li~~~p = ~L~=l '\:1.' - and from the law of large numbers, (12) and (13), we have: 11~(';,u) - a(';,u)11 = Ilik -rkllli:!s!p~ li~-!~p II~ L~=l rJ./Y fl (II~II + II~II) - r~':k I + 0(8 +0(82 ) = 0(8 2 ) 2) R. Munos and P. Bourgine 1034 "In(';, u) - I(';, u)" ~ 1Ill;;(';, u) - a(';, u)1I kf?8 w.p. 1 ~ ka .82 w.p. 1 (14) Besides, from (5) and (14), we have: 2 2 c ) _ -Tn (cr.",U )1 _ < d.(k[.6 kT'UJ:2 1T(r.",U (d.m,+d.k,,6 .6)2 ) UJ:2 < _ (15) and from a property of exponential function, I,T(~.u) _ ,7' . .1?) I= kT.In ~ .8 (? 2. (16) We can deduce from (14) that: . 1( ) -( limsupp';,u,( -Pn';,u,( )1 ~ n-+oo (2.6.Mt+d.Ma)(2.kt+d.k,,)62 k J: 6mr(2.k,+d.ka)62 S; puw.p.l Estimation of IV~+l(';) - V 6(.;) 1 A.2 Mter having updated V~(';) with rule (9), IV~+l(';) - V 6(.;) From (4), (9) and (8), I. A (17) let A denote the difference < ,T(?.U) L: [P(';, u, () - p(.;, u, ()] V 6(() + (,T(?.1?) - ,7'(~'1??) L p(.;, u, () V 6 (() ( ( +,7' (?.u) . L:p(.;, u, () [V6(() - V~(()] + L:p(.;, u, ().T(';, u) [r(';, u) - F(';, u)] ( ( + L:( p(.;, u, () [T(';, u) - T(';, u)] r(';, u) for all u E U6 As V is differentiable we have : Vee) = V(';) + VX ' ( ( - . ; ) + 0(1I( - ';11). Let us define a linear function V such that: Vex) = V(';) + VX ' (x - ';). Then we have: [P(';, u, () - p(.;, u, ()] V 6(() = [P(';, u, () - p(.;, u, ()] . [V6(() - V(()] + [P(';,u,()-p(';,u,()]V((), thus: L:([p(';,u,()-p(';,u,()]V6(() = kp .E6.8 + [V(() +0(8)] = [V(7J)-VUD] + kp .E6.8 + 0(8) with: 7J = L:( p(';, u, () (( -.;) and 1j = L:( p(.;, u, () (( - .;). Besides, from the convergence of the scheme (theorem 2), we have E6.8 = L([P(';,U,()-p(.;,u,()] [V(7J) - V(1j)] + 0(8) 0(8). From the linearity of V, IL( [P(';, u, () - p(.;, u, ()] V6 (() I IV(() - I II( - ZII?Mv", V(Z) ~ = S; 2kp 82 . Thus = 0(8) and from (15), (16) and the Lipschitz prop- erty of r, A= 1'l'(?'U), L:( p(.;, u, () [V6(() - V~ (()] 1+ 0(8). As ,..,.7'(?.u) < 1 - 7'(?.U) In 1 < 1 _ T(?.u)-k.,.6 2 In 1 < 1 _ ( I - 2 'Y - 2 we have: A with k = 2d(M[~d.M,,). = (1 - k.8)E~ 'Y - + 0(8) 6 2d(M[+d.M,,) _ !ix..82) In 1 2 'Y ' (18) 1035 Reinforcement Learning for Continuous Stochastic Control Problems A.3 A sufficient condition for sUP?EE~ IV~(~) - V6(~)1 :S C2 Let us suppose that for all ~ E ~6, the following conditions hold for some a > 0 E~ > C2 =} IV~+I(O - V6(~)1 :S E~ - a (19) E~ :S c2=}IV~+I(~)_V6(~)I:Sc2 (20) From the hypothesis that all states ~ E ~6 are regularly updated, there exists an integer m such that at stage n + m all the ~ E ~6 have been updated at least once since stage n. Besides, since all ~ E 8C 6 are updated at least once with rule (10), V~ E 8C6, IV~(~) - V6(~)1 = IR(Xk(T)) - R(~)I :S 2.LR.8 :S C2 for any 8 :S ~3 = 2~lR' Thus, from (19) and (20) we have: E! > E! :S C2 =} E!+m :S E! - a C2 =} E!+m :S Thus there exists N such that: Vn ~ N, E~ :S A.4 C2 C2. Convergence of the algorithm Let us prove theorem 3. For any c > 0, let us consider Cl > 0 and C2 > 0 such that = c. Assume E~ > ?2, then from (18), A = E! - k.8'?2+0(8) :S E~ -k.8.~ for 8 :S ~3. Thus (19) holds for a = k.8.~. Suppose now that E~ :S ?2. From (18), A :S (1 - k.8)?2 + 0(8) :S ?2 for 8 :S ~3 and condition (20) is true. Cl +C2 Thus for 8 :S min {~1, ~2, ~3}, the sufficient conditions (19) and (20) are satisfied. So there exists N, for all n ~ N, E~ :S ?2. Besides, from the convergence of the scheme (theorem 2), there exists ~o st. V8:S ~o, sUP?EE~ 1V6(~) - V(~)I :S ?1? Thus for 8 :S min{~o, ~1, ~2, ~3}, "3N, Vn ~ sup IV~(~) - V(~)I :S sup IV~(~) - V6(~)1 ?EE6 ?EEh N, + sup ?EE6 1V6(~) - V(~)I :S ?1 + c2 = ?. References [BBS95j Andrew G. Barto, Steven J. Bradtke, and Satinder P. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence, (72):81138, 1995. [Ber87j Dimitri P. Bertsekas. Dynamic Programming: Deterministic and Stochastic Models. Prentice Hall, 1987. [FS93j Wendell H. Fleming and H. Mete Soner. Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics. Springer-Verlag, 1993. [KP95j Peter E. Kloeden and Eckhard Platen. Numerical Solutions of Stochastic Differential Equations. Springer-Verlag, 1995. [Kry80j N.V. Krylov. Controlled Diffusion Processes. Springer-Verlag, New York, 1980. [Kus90j Harold J. Kushner. Numerical methods for stochastic control problems in continuous time. SIAM J. Control and Optimization, 28:999-1048, 1990. [Mun97j Remi Munos. A convergent reinforcement learning algorithm in the continuous case based on a finite difference method. International Joint Conference on Art~ficial Intelligence, 1997. [Wat89j Christopher J.C.H. Watkins. Learning from delayed reward. PhD thesis, Cambridge University, 1989. Use of a Multi-Layer Percept ron to Predict Malignancy in Ovarian Tumors Herman Verrelst, Yves Moreau and Joos Vandewalle Dept. of Electrical Engineering Katholieke Universiteit Leuven Kard. Mercierlaan 94 B-3000 Leuven, Belgium Dirk Timmerman Dept. of Obst. and Gynaec. University Hospitals Leuven Herestraat 49 B-3000 Leuven, Belgium Abstract We discuss the development of a Multi-Layer Percept ron neural network classifier for use in preoperative differentiation between benign and malignant ovarian tumors. As the Mean Squared classification Error is not sufficient to make correct and objective assessments about the performance of the neural classifier, the concepts of sensitivity and specificity are introduced and combined in Receiver Operating Characteristic curves. Based on objective observations such as sonomorphologic criteria, color Doppler imaging and results from serum tumor markers, the neural network is able to make reliable predictions with a discriminating performance comparable to that of experienced gynecologists. 1 Introd uction A reliable test for preoperative discrimination between benign and malignant ovarian tumors would be of considerable help to clinicians. It would assist them to select patients for whom minimally invasive surgery or conservative management suffices versus those for whom urgent referral to a gynecologic oncologist is needed. We discuss the development of a neural network classifier/diagnostic tool. The neural network was trained by supervised learning, based on data from 191 thoroughly examined patients presenting with ovarian tumors of which 140 were benign and 51 malignant. As inputs to the network we chose indicators that in recent studies have proven their high predictive value [1, 2, 3]. Moreover, we gave preference to those indicators that can be obtained in an objective way by any gynecologist. Some of these indicators have already been used in attempts to make one single protocol or decision algorithm [3, 4]. Use of a MLP to Predict Malignancy in Ovarian Tumors 979 In order to make reliable assessments on the practical performance of the classifier, it is necessary to work with other concepts than Mean Squared classification Error (MSE), which is traditionally used as a measure of goodness in the training of a neural network. We will introduce notions as specificity and sensitivity and combine them into Receiver Operating Characteristic (ROC) curves. The use of ROC-curves is motivated by the fact that they are independent of the relative proportion of the various output classes in the sample population. This enables an objective validation of the performance of the classifier. We will also show how, in the training of the neural network, MSE optimization with gradient methods can be refined and/or replaced with the help of ROC-curves and simulated annealing techniques. The paper is organized as follows. In Section 2 we give a brief description of the selected input features. In Section 3 we state some drawbacks to the MSE criterion and introduce the concepts of sensitivity, specificity and ROC-curves. Section 4 then deals with the technicalities of training the neural network. In Section 5 we show the results and compare them to human performance. 2 Data acquisition and feature selection The data were derived from a study group of 191 consecutive patients who were referred to a single institution (University Hospitals Leuven, Belgium) from August 1994 to August 1996. Table 1 lists the different indicators which were considered, together with their mean value and standard deviations or together with the relative presence in cases of benign and malignant tumors. Table 1 Indicator Demographic Age Postmenopausal CA 125 (log) Blood flow present Abdominal fluid Bilateral mass Unilocular cyst Multiloc/solid cyst Smooth wall Irregular wall Papillations Serum marker CD! Morphologic Benign Malignant 49.3 ? 16.0 40% 2.8?1.1 72.9% 12.1% 11.4% 42.1% 16.4% 58.6% 32.1% 7.9% 58.3 ? 14.3 70.6% 5.2 ? 1.9 100% 52.9% 35.3% 5.9% 49.0% 2.0% 76.5% 74.5% Table 1: Demographic, serum marker, color Doppler imaging and morphologic indicators. For the continuous valued features the mean and standard deviation for each class are reported. For binary valued indicators, the last two columns give the presence of the feature in both classes e.g. only 2% of malignant tumors had smooth walls. First, all patients were scanned with ultrasonography to obtain detailed gray-scale images of the tumors. Every tumor was extensively examined for its morphologic characteristics. Table 1 lists the selected morphologic features: presence of abdominal fluid collection, papillary structures (> 3mm), smooth internal walls, wall irregularities, whether the cysts were unilocular, multilocular-solid and/or present on both pelvic sides. All outcomes are binary valued: every observation relates to the presence (1) or absence (0) of these characteristics. Secondly, all tumors were entirely surveyed by color Doppler imaging to detect presence or absence of blood flow within the septa, cyst walls, solid tumor areas or ovarian tissue. The outcome is also binary valued (1/0). H. Verrelst, Y. Moreau, 1. Vandewalle and D. TImmennan 980 Thirdly, in 173 out of the total of 191 patients, serum CA 125 levels were measured, using CA 125 II immunoradiometric assays (Centocor, Malvern, PA). The CA 125 antigen is a glycoprotein that is expressed by most epithelial ovarian cancers. The numerical value gives the concentration in U Iml. Because almost all values were situated in a small interval between 0 and 100, and because a small portion took values up to 30,000, this variable was rescaled by taking its logarithm. Since age and menopausal status of the patient are considered to be highly relevant, these are also included. The menopausal score is -1 for premenopausal, + 1 for postmenopausal. A third class of patients were assigned a 0 value. These patients had had an hysterectomy, so no menopausal status could be appointed to them. It is beyond the scope of this paper to give a complete account of the meaning of the different features that are used or the way in which the data were acquired. We will limit ourselves to this short description and refer the reader to [2, 3] and gynecological textbooks for a more detailed explanation. 3 3.1 Receiver Operating Characteristics Drawbacks to Mean Squared classification Error Let us assume that we use a one-hidden-Iayer feed-forward NN with m inputs nh hidden neurons with the tanh(.) as activation function, and one output i1k, m nh Yk(B) = L xl, Wj tanh(L VijX~ j=l + {3j)' (1) i=l parameterized by the vector 0 consisting of the network's weights Wj and Vij and bias terms {3j. The cost function is often chosen to be the squared difference between the desired d k and the actual response Yk. averaged over all N samples [12], 1 N J(O) = N 2:)dk - Yk(9))2. (2) k=l This type of cost function is continuous and differentiable, so it can be used in gradient based optimization techniques such as steepest descent (back-propagation), quasi-Newton or Levenberg-Marquardt methods [8, 9, 11, 12]. However there are some drawbacks to the use of this type of cost function. First of all, the MSE is heavily dependent on the relative proportion of the different output classes in the training set. In our dichotomic case this can easily be demonstrated by writing the cost function, with superscripts b and m respectively meaning benign and malignant, as J(O) - Nb Nb + N m 1 ""Nb (db )2 Nb wk=l k - Yk Nm + Nb + N m ~ ~ ,\ (1-,\) 1 ""Nm N m wk=l (dm k - Yk )2 (3) If the relative proportion in the sample population is not representative for reality, the .x parameter should be adjusted accordingly. In practice this real proportion is often not known accurately or one simply ignores the meaning of .x and uses it as a design parameter in order to bias the accuracy towards one of the output classes. A second drawback of the MSE cost function is that it is not very in- formative towards practical usage of the classification tool. A clinician is not interested in the averaged deviation from desired numbers, but thinks in terms of percentages found, missed or misclassified. In the next section we will introduce the concepts of sensitivity and specificity to express these more practical measures. Use of a MLP to Predict Malignancy in Ovarian Tumors 3.2 981 Sensitivity, specificity and ROC-curves If we take the desired response to be 0 for benign and 1 for malignant cases, the way to make clear cut (dichotomic) decisions is to compare the numerical outcome of the neural network to a certain threshold value T between 0 and 1. When the outcome is above the threshold T, the prediction is said to be positive. Otherwise the prediction is said to be negative. With this convention, we say that the prediction was True Positive (TP) if the prediction was positive when the sample was malignant. True Negative (TN) if the prediction was negative when the sample was benign. False Positive (FP) if the prediction was positive when the sample was benign. False Negative (FN) if the prediction was negative when the sample was malignant. To every of the just defined terms T P, TN, F P and F N, a certain subregion of the total sample space can be associated, as depicted in Figure 1. In the same sense, we can associate to them a certain number counting the samples in each subregion. We can then define sensitivity as the proportion of malignant cases that Tl:FN' Total opulation ?~li~nant .... . " ... , \ TP - '- ,, " Figure 1: The concepts of true and false positive and negative illustrated. The dashed area indicates the malignant cases in the total sample population. The positive prediction of an imperfect classification (dotted area) does not fully coincide with this sub area. F::rN' are predicted to be malignant and specificity as the proportion of benign cases that are predicted to be benign. The false positive rate is I-specificity. When varying the threshold T, the values of T P, TN, F P, F N and therefore also sensitivity and specificity, will change. A low threshold will detect almost all malignant cases at the cost of many false positives. A high threshold will give less false positives, but will also detect less malignant cases. Receiver Operating Characteristic (ROC) curves are a way to visualize this relationship. The plot gives the sensitivity versus false positive rate for varying thresholds T (e.g. Figure 2). The ROC-curve is useful and widely used device for assessing and comparing the value of tests [5, 7]. The proportion of the whole area of the graph which lies below the ROC-curve is a one-value measure of the accuracy of a test [6]. The higher this proportion, the better the test. Figure 2 shows the ROC-curves for two simple classifiers that use only one single indicator. (Which means that we classify a tumor being malignant when the value of the indicator rises above a certain value.) It is seen that the CA 125 level has high predictive power as its ROC-curve spans 87.5% of the total area (left Figure 2). For the age parameter, the ROC-curve spans only 65.6% (right Figure 2). As indicated by the horizontal line in the plot, a CA 125 level classification will only misclassify 15% of all benign cases to reach a 80% sensitivity, whereas using only age, one would then misclassify up to 50% of them. H. Verrelst, Y. Moreau, 1. Vandewalle and D. Timmennan 982 :.... I( 'f ., ,r o? ? ? ,J " . , .. t f 1 t, '1 U U ?? ,.I .. 0' ?? " 1 Figure 2: The Receiver Operating Characteristic (ROC) curve is the plot of the sensitivity versus the false positive rate of a classifier for varying thresholds used. Only single indicators (left: CA 125, right: age) are used for these ROC-curlVes. The horizontal line marks the 80% specificity level. Since for every set of parameters of the neural network the area under the ROCcurve can be calculated numerically, this one-value measure can also be used for supervised training, as will be shown in the next Section. 4 4.1 Simulation results Inputs and architecture The continuous inputs were standardized by subtracting their mean and dividing by their standard deviation (both calculated over the entire population). Binary valued inputs were left unchanged. The desired outputs were labeled 0 for benign examples, 1 for malignant cases. The data set was split up: 2/3 of both benign and malignant samples were randomly selected to form the training set. The remaining examples formed the test set. The ratio of benign to all examples is >. ~ j. Since the training set is not large, there is a risk of overtraining when too many parameters are used. We will limit the number of hidden neurons to nh = 3 or 5. As the CA 125 level measurement is more expensive and time consuming, we will investigate two different classifiers: one which does use the CA 125 level and one which does not. The one-hidden-Iayer MLP architectures that are used, are 11-3-1 and 10-5-1. A tanh(.) is taken for the activation function in the hidden layer. 4.2 Training A first way of training was MSE optimization using the cost function (3) . By taking ~ in this expression, the role of malignant examples is more heavily weighted. The parameter vector e was randomly initialized (zero mean Gaussian distribution, standard deviation a = 0.01). Training was done using a quasi-Newton method with BFGS-update of the Hessian (fminu in Matlab) [8, 9]. To prevent overtraining, the training was stopped before the MSE on the test set started to rise. Only few iterations (~ 100) were needed. >. = A second way of training was through the use of the area spanned by the ROC-curve of the classifier and simulated annealing techniques [10]. The area-measure AROC was numerically calculated for every set of trial parameters: first the sensitivity and false positive rate were calculated for 1000 increasing values of the threshold T between 0 and 1, which gave the ROC-curve; secondly the area AROC under the curve was numerically calculated with the trapezoidal integration rule. 983 Use of a MLP to Predict Malignancy in Ovarian Tumors We used Boltzmann Simulated Annealing to maximize the ROC-area. At time k a trial parameter set of the neural network OHl is randomly generated in the neighborhood of the present set Ok (Gaussian distribution, a = O.OO~. The trial set 8H1 is always accepted if the area Af.2? 2: Afoc. If Af'?? < Ak OC, Ok+! is accepted if A r:g? - A r;oc ( ROC )/T. e Ak >Q with Q a uniformly distributed random variable E [0,1] and Te the temperature. As cooling schedule we took Te = 1/(100 + 10k), so that the annealing was low-temperature and fast-cooling. The optimization was stopped before the ROC-area calculated for the test set started to decrease. Only a few hundred annealing epochs were allowed. 4.3 Results Table 2 states the results for the different approaches. One can see that adding the CA 125 serum level clearly improves the classifier's performance. Without it, the ROC-curve spans about 96.5% of the total square area of the plot, whereas with the CA 125 indicator it spans almost 98%. Also, the two training methods are seen to give comparable results. Figure 3 shows the ROC-curve calculated for the total population for the 11-3-1 MLP case, trained with simulated annealing Table 2 MLP, MLP, MLP, MLP, 10-5-1 10-5-1 11-3-1 11-3-1 Training set 96.7% 96.6% 97.9% 97.9% MSE SA MSE SA Test set 96.4% 96.2% 97.4% 97.5% Total population 96.5% 96.4% 97.7% 97.8% Table 2: For the two architectures (10-5-1 and 11-3-1) of the MLP and for the gradient (MSE) and the simulated annealing (SA) optimization techniques, this table gives the resulting areas under the ROC-curves . .. 07 ?0 0. 1 G.2 ":I 04 0.' 0, 07 01 0, , Figure 3: ROC-curves of 11-3-1 MLP (with CA 125 level indicator), trained with simulated annealing. The curve, calculated for the total population, spans 97.8% of the total region. All patients were examined by two gynecologists, who gave their subjective impressions and also classified the ovarian tumors into (probably) benign and malignant. Histopathological examinations of the tumors afterwards showed these gynecologists H. Vendst, Y. Moreau, 1. Vandewalle and D. Timmerman 984 to have a sensitivity up to 98% and a false positive rate of 13% and 12% respectively. As can be seen in Figure 3, the 11-3-1 MLP has a similar performance. For a sensitivity of 98%, its false positive rate is between 10% and 15%. 5 Conclusion In this paper we have discussed the development of a Multi-Layer Perceptron neural network classifier for use in preoperative differentiation between benign and malignant ovarian tumors. To assess the performance and for training the classifiers, the concepts of sensitivity and specificity were introduced and combined in Receiver Operating Characteristic curves. Based on objective observations available to every gynecologist, the neural network is able to make reliable predictions with a discriminating performance comparable to that of experienced gynecologists. Acknowledgments This research work was carri e d out at the ESAT laboratory and the Interdisciplinary Center of Neural Networks ICNN of the Katholieke Universiteit Leuven, in the following frameworks : the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister 's Office for Science, Technology and Culture (IUAP P4-02 and IUAP P4-24), a Concerted Action Project MIPS (Model based Information Processing Systems) of the Flemish Community and the FWO (Fund for Scientific Research - Flanders) project G.0262.97 : Learning and Optimization: an Interdisciplinary Approach . The scientific responsibility rests with its authors . References [1] Bast R. C., Jr., Klug T .L. St. John E., et aI, "A radioimmunoassay using a monoclonal antibody to monitor the course of epithelial ovarian cancer," N. Engl. J . Med., Vol. 309, pp. 883-888, 1983 [2] Timmerman D ., Bourne T ., Tailor A., Van Assche F .A., Vergote 1., "Preoperative differentiation between benign and malignant adnexal masses," submitted [3] Tailor A., Jurkovic D ., Bourne T.H., Collins W.P., Campbell S., "Sonographic prediction of malignancy in adnexal masses using multivariate logistic regression analysis," Ultrasound Obstet. Gynaecol. in press, 1997 [4] Jacobs 1., Oram D., Fairbanks J ., et aI. , "A risk of malignancy index incorporating CA 125, ultrasound and menopausal status for the accurate preoperative diagnosis of ovarian cancer," Br. J. Obstet. Gynaecol., Vol. 97, pp. 922-929, 1990 [5] Hanley J.A., McNeil B., "A method of comparing the areas under the receiver operating characteristics curves derived from the same cases," Radiology, Vol. 148, pp. 839-843, 1983 [6J Swets J.A., "Measuring the accuracy of diagnostic systems," Science, Vol. 240, pp. 1285-1293, 1988 [7J Galen R.S., Gambino S., Beyond normality: the predictive value and efficiency of medical diagnosis, John Wiley, New York, 1975. [8J Gill P., Murray W ., Wright M., Practical Optimization, Acad. Press, New York, 1981 [9] Fletcher R., Practical methods of optimization, 2nd ed., John Wiley, New York, 1987. [10J Kirkpatrick S., Gelatt C.D., Vecchi M., "Optimization by simulated annealing," Science, Vol. 220, pp . 621-680, 1983. [11] Rumelhart D.E., Hinton G.E., Williams R.J ., "Learning representations by backpropagating errors," Nature, Vol. 323, pp. 533-536, 1986. [12] Bishop C., Artificial Neural Networks for Pattern Recognition, OUP, Oxford, 1996 

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New Approximations of Differential Entropy for Independent Component Analysis and Projection Pursuit Aapo Hyvarinen Helsinki University of Technology Laboratory of Computer and Information Science P.O. Box 2200, FIN-02015 HUT, Finland Email: aapo.hyvarinen<Ohut.fi Abstract We derive a first-order approximation of the density of maximum entropy for a continuous 1-D random variable, given a number of simple constraints. This results in a density expansion which is somewhat similar to the classical polynomial density expansions by Gram-Charlier and Edgeworth. Using this approximation of density, an approximation of 1-D differential entropy is derived. The approximation of entropy is both more exact and more robust against outliers than the classical approximation based on the polynomial density expansions, without being computationally more expensive. The approximation has applications, for example, in independent component analysis and projection pursuit. 1 Introduction The basic information-theoretic quantity for continuous one-dimensional random variables is differential entropy. The differential entropy H of a scalar random variable X with density f(x) is defined as H(X) = - / f(x) log f(x)dx. (1) The 1-D differential entropy, henceforth called simply entropy, has important applications such areas as independent component analysis [2, 10] and projection pursuit [5, 6]. Indeed, both of these methods can be considered as a search for directions in which entropy is minimal, for constant variance. Unfortunately, the estimation of entropy is quite difficult in practice. Using definition (1) requires estimation of the density of X, which is recognized to be both A Hyviirinen 274 theoretically difficult and computationally demanding. Simpler approximations of entropy have been proposed both in the context of projection pursuit [9] and independent component analysis [1, 2]. These approximations are usually based on approximating the density f(x) using the polynomial expansions of Gram-Charlier or Edgeworth [11]. This construction leads to the use of higher-order cumulants, like kurtosis. However, such cumulant-based methods often provide a rather poor approximation of entropy. There are two main reasons for this. Firstly, finitesample estimators of higher-order cumulants are highly sensitive to outliers: their values may depend on only a few, possibly erroneous, observations with large values [6]. This means that outliers may completely determine the estimates of cumulants, thus making them useless. Secondly, even if the cumulants were estimated perfectly, they measure mainly the tails of the distribution, and are largely unaffected by structure near the centre of the distribution [5]. Therefore, better approximations of entropy are needed. To this end, we introduce in this paper approximations of entropy that are both more exact in the expectation and have better finite-sample statistical properties, when compared to the cumulantbased approximations. Nevertheless, they retain the computational and conceptual simplicity of the cumulant-based approach. Our approximations are based on an approximative maximum entropy method. This means that we approximate the maximum entropy that is compatible with our measurements of the random variable X. This maximum entropy, or further approximations thereof, can then be used as a meaningful approximation of the entropy of X. To accomplish this, we derive a first-order approximation of the density that has the maximum entropy given a set of constraints, and then use it to derive approximations of the differential entropy of X. 2 Applications of Differential Entropy First, we discuss some applications of the approximations introduced in this paper. Two important applications of differential entropy are independent component analysis (ICA) and projection pursuit. 'In the general formulation of ICA [2], the purpose is to transform an observed random vector x = (Xl, ... , Xm)T linearly into a random vector s = (81, ... , 8m )T whose components are statistically as independent from each other as possible. The mutual dependence of the 8i is classically measured by mutual information. Assuming that the linear transformation is invertible, the mutual information 1(81, ... , 8 m ) can be expressed as 1(81, ... , 8m ) = 2:i H(8i) - H(Xl, ... , Xm) -log Idet MI where M is the matrix defining the transformation s = Mx. The second term on the right-hand side does not depend on M, and the minimization of the last term is a simple matter of differential calculus. Therefore, the critical part is the estimation of the 1-D entropies H(8i}: finding an efficient and reliable estimator or approximation of entropy enables an efficient and reliable estimation of the ICA decomposition . In projection pursuit, the purpose is to search for projections of multivariate data which have 'interesting' distributions [5, 6, 9]. Typically, interestingness is considered equivalent with non-Gaussianity. A natural criterion of non-Gaussianity is entropy [6, 9], which attains its maximum (for constant variance) when the distribution is Gaussian, and all other distributions have smaller entropies. Because of the difficulties encountered in the estimation of entropy, many authors have considered other measures of non-Gaussianity (see [3]) but entropy remains, in our view, the best choice of a projection pursuit index, especially because it provides a simple connection to ICA. Indeed, it can be shown [2] that in ICA as well as in projection pursuit, the basic problem is to find directions in which entropy is minimized for New Approximations of Differential Entropy 275 constant variance. 3 Why maximum entropy? Assume that the information available on the density f(x) of the scalar random variable X is of the form J f(x)Gi(x)dx = Ci, for i = 1, ... , n, (2) which means in practice that we have estimated the expectations E{Gi(X)} of n different functions of X. Since we are not assuming any model for the random variable X, the estimation of the entropy of X using this information is not a well-defined problem: there exist an infinite number of distributions for which the constraints in (2) are fulfilled, but whose entropies are very different from each other. In particular, the differential entropy reaches -00 in the limit where X takes only a finite number of values. A simple solution to this dilemma is the maximum entropy method. This means that we compute the maximum entropy that is compatible with our constraints or measurements in (2), which is a well-defined problem. This maximum entropy, or further approximations thereof, can then be used as an approximation of the entropy of X. Our approach thus is very different from the asymptotic approach often used in projection pursuit [3, 5]. In the asymptotic approach, one establishes a sequence of functions G i so that when n goes to infinity, the information in (2) gives an asymptotically convergent approximation of some theoretical projection pursuit index. We avoid in this paper any asymptotic considerations, and consider directly the case of finite information, i.e., finite n. This non-asymptotic approach is justified by the fact that often in practice, only a small number of measurements of the form (2) are used, for computational or other reasons. 4 Approximating the maximum entropy density In this section, we shall derive an approximation of the density of maximum entropy compatible with the measurements in (2). The basic results of the maximum entropy method tell us [4] that under some regularity conditions, the density fo(x) which satisfies the constraints (2) and has maximum entropy among all such densities, is of the form (3) where A and ai are constants that are determined from the Ci, using the constraints in (2) (i.e., by substituting the right-hand side of (3) for fin (2)), and the constraint J fo(x)dx = 1. This leads in general to a system of n+ 1 non-linear equations which is difficult to solve. Therefore, we decide to make a simple approximation of fo. This is based on the assumption that the density f(x) is not very far from a Gaussian distribution of the same mean and variance. Such an assumption, though perhaps counterintuitive, is justified because we shall construct a density expansion (not unlike a Taylor expansion) in the vicinity of the Gaussian density. In addition, we can make the technical assumption that f(x) is near the standardized Gaussian density 'P(x) = exp(-x 2 /2)/..;2ii, since this amounts simply to making X zeromean and of unit variance. Therefore we put two additional constraints in (2), defined by G n+1 (x) = x, Cn+l = 0 and G n+2(x) = x 2, Cn+2 = 1. To further simplify A. HyvlJrinen 276 the calculations, let us make another, purely technical assumption: The functions G i , i = 1, ... , n, form an orthonormal system according to the metric defined by cp, and are orthogonal to all polynomials of second degree. In other words, for all i,j=1, ... ,n ! cp(x)Gi(x)Gj(x)dx = { 1, if i = j 0, if i i- j (4) , For any linearly independent functions G i , this assumption can always be made true by ordinary Gram-Schmidt orthonormalization. Now, note that the assumption of near-Gaussianity implies that all the other ai in (3) are very small compared to a n +2 ~ -1/2, since the exponential in (3) is not far from exp( _x 2 /2). Thus we can make a first-order approximation of the exponential function (detailed derivations can be found in [8]). This allows for simple solutions for the constants in (3), and we obtain the approximative maximum entropy density, which we denote by j(x): n j(x) = cp(x)(1 + L CiGi(X)) (5) i=l where Ci = E{ G i (X)}. To estimate this density in practice, the Ci are estimated, for example, as the corresponding sample averages of the Gi(X). The density expansion in (5) is somewhat similar to the Gram-Charlier and Edgeworth expansions [11]. 5 Approximating the differential entropy An important application of the approximation of density shown in (5) is in approximation of entropy. A simple approximation of entropy can be found by approximating both occurences of f in the definition (1) by j as defined in Eq. (5), and using a Taylor approximation of the logarithmic function, which yields (1 + ?) log(1 + ?) ~ ? + ?2/2. Thus one obtains after some algebraic manipulations [8] H(X) ~- ! j(x) log j(x)dx ~ H(v) - ~ t (6) c; i=l where H(v) = ~(1 +log(27r)) means the entropy of a standardized Gaussian variable, and Ci = E{ Gi(X)} as above. Note that even in cases where this approximation is not very accurate, (6) can be used to construct a projection pursuit index (or a measure of non-Gaussianity) that is consistent in the sense that (6) obtains its maximum value, H(v), when X has a Gaussian distribution. 6 Choosing the measuring functions Now it remains to choose the 'measuring' functions G i that define the information given in (2). As noted in Section 4, one can take practically any set of linearly independent functions, say Ch i = 1, ... , n, and then apply Gram-Schmidt orthonormalization on the set containing those functions and the monomials xk, k 0,1,2, so as to obtain the set G i that fulfills the orthogonality assumptions in (4). This can be done, in general, by numerical integration. In the practical choice of the functions Gi , the following criteria must be emphasized: First, the practical estimation of E{Gi(x)} should not be statistically difficult. In particular, this estimation should not be too sensitive to outliers. Second, the maximum entropy method assumes = 277 New Approximations of Differential Entropy that the function fo in (3) is integrable. Therefore, to ensure that the maximum entropy distribution exists in the first place, the Gi(x) must not grow faster than quadratically as a function of Ixl, because a function growing faster might lead to non-integrability of fo [4]. Finally, the Gi must capture aspects of the distribution of X that are pertinent in the computation of entropy. In particular, if the density f(x) were known, the optimal function GoPt would clearly be -logf(x), because -E{log f(X)} gives directly the entropy. Thus, one might use the log-densities of some known important densities as Gi . The first two criteria are met if the Gi(x) are functions that do not grow too fast (not faster than quadratically) when Ixl grows. This excludes, for example, the use of higher-order polynomials, as are used in the Gram-Charlier and Edgeworth expansions. One might then search, according to the last criterion above, for logdensities of some well-known distributions that also fulfill the first two conditions. Examples will be given in the next section. It should be noted, however, that the criteria above only delimit the space of function that can be used. Our framework enables the use of very different functions (or just one) as Ch The choice is not restricted to some well-known basis of a functional space, as in most approaches [1,2,9]. However, if prior knowledge is available on the distributions whose entropy is to estimated, the above consideration shows how to choose the optimal function. 7 A simple special case A simple special case of (5) is obtained if one uses two functions G1 and G2 , which are chosen so that G1 is odd and (h is even. Such a system of two functions can measure the two most important features of non-Gaussian 1-D distributions. The odd function measures the asymmetry, and the even function measures the bimodality /sparsity dimension (called central hole/central mass concentration in [3)). After extensive experiments, Cook et al [3] also came to the conclusion that two such measures (or two terms in their projection pursuit index) are enough for projection pursuit in most cases. Classically, these features have been measured by skewness and kurtosis, which correspond to G 1 (x) = x 3 and G2 (x) = X4, but we do not use these functions for the reasons explained in Section 6. In this special case, the approximation in (6) simplifies to where k1 and k2 are positive constants (see [8]), and v is a Gaussian random variable of zero mean and unit variance. Practical examples of choices of Gi that are consistent with the requirements in Section 6 are the following. First, for measuring bimodality /sparsity, one might use, according to the recommendations of Section 6, the log-density of the double exponential (or Laplace) distribution: G2a (x) = Ixl. For computational reasons, a smoother version of G2a might also be used. Another choice would be the Gaussian function, which may be considered as the log-density of a distribution with infinitely heavy tails: G2b (X) = exp( -x 2/2). For measuring asymmetry, one might use, on more heuristic grounds, the following function: G1 (x) = x exp (- x 2 /2). which corresponds to the second term in the projection pursuit index proposed in [3]. Using the above examples one obtains two practical examples of (7): Ha(X) = H(v) - [k1 (E{X exp( _X2 /2)})2 Hb(X) = H(v) - [k1 (E{X exp( _X2 /2)})2 + k2(E{IXI} - }2/71")2], + k~(E{exp( _X2 /2)} (8) - Ji72)2], (9) A. Hyvllrinen 278 with kl = 36/(8V3 - 9) , k~ = 1/(2 - 6/'rr), and k~ = 24/(16V3 - 27). As above, H(v) = + log(27r)) means the entropy of a standardized Gaussian variable. These approximations Ha(X) and Hb(X) can be considered more robust and accurate generalizations of the approximation derLved using the Gr~m-Charlier expansion in [9] . Indeed, using the polynomials G 1 (x) = x 3 and G 2 (x) = x4 one obtains the approximation of entropy in (9), which is in practice almost identical to those proposed in [1, 2]. Finally, note that the approximation in (9) is very similar to the first two terms of the projection pursuit index in [3] . Algorithms for independent component analysis and projection pursuit can be derived from these approximations, see [7]. !(1 8 Simulation results To show the validity of our approximations of differential entropy we compared the approximations Ha and Hb in Eqs (8) and (9) in Section 7, with the one offered by higher-order cumulants as given in [9]. The expectations were here evaluated exactly, ignoring finite-sample effects. First, we used a family of Gaussian mixture densities, defined by f(x) = J.tcp(x) + (1 - J.t)2cp(2(x - 1)) (10) where J.t is a parameter that takes all the values in the interval 0 ::; J.t ::; 1. This family includes asymmetric densities of both negative and positive kurtosis. The results are depicted in Fig. 1. Note that the plots show approximations of negentropies: the negentropy of X equals H(v) -H(X), where v is again a standardized Gaussian variable. One can see that both of the approximations Ha and Hb introduced in Section 7 were considerably more accurate than the cumulant-based approximation. Second, we considered the following family of density functions : (11) where a is a positive constant : and C 1 , C2 are normalization constants that make f Ot a probability density of unit variance. For different values of a , the densities in this family exhibit different shapes. For a < 2, one obtains (sparse) densities of positive kurtosis. For a = 2, one obtains the Gaussian density, and for a > 2, a density of negative kurtosis. Thus the densities in this family can be used as examples of different symmetric non-Gaussian densities. In Figure 2, the different approximations are plotted for this family, using parameter values .5 ::; a ~ 3. Since the densities used are all symmetric, the first terms in the approximations were neglected. Again, it is clear that both of the approximations Ha and Hb introduced in Section 7 were much more accurate than the cumulant-based approximation in [2, 9]. (In the case of symmetric densities, these two cumulant-based approximations are identical). Especially in the case of sparse densities (or densities of positive kurtosis), the cumulant-based approximations performed very poorly; this is probably because it gives too much weight to the tails of the distribution. References [1] S. Amari, A. Cichocki, and H.H. Yang. A new learning algorithm for blind source separation. In D . S. Touretzky, M. C. Mozer, and M. E . Hasselmo, editors, Advances in Neural Information Processing 8 (Proc. NIPS '95), pages 757- 763. MIT Press, Cambridge, MA, 1996. 12] P. Comon. Independent component analysis - a new concept? 36:287- 314, 1994. Signal Processing, New Approximationr of Differential Entropy 0025 ... :-: "" - .... "', "- ... ", 0 01 $ , \' " " o. 0 6 279 Figure 1: Comparison of different approximations of negentropy, for the family of mixture densities in (10) parametrized by JL ranging from 0 to 1. Solid curve: true negentropy. Dotted curve: cumulantbased approximation. Dashed curve: approximation Ha in (8). Dot-dashed Cu:Lve: approximation Hb in (9). Our two approximations were clearly better than the cumulant-based one. , . '. " , ' , -. , '. , . .' , ' "':" ,'...... -- ~~ . ""- .... " . -:- ~ ,~--~-.--- Figure 2: Comparison of different approximations of negentropy, for the family of densities (11) parametrized by Q. On the left, approximations for densities of positive kurtosis (.5 ~ Q < 2) are depicted, and on the right, approximations for densities of negative kurtosis (2 < Q ~ 3). Solid curve: true negentropy. Dotted curve: cumulant-based approximation. Dashed curve: approximation Ha in (8). Dot-dashed curve: approximation Hb in (9) . Clearly, our two approximations were much better than the cumulant-based one, especially in the case of densities of positive kurtosis. [3) D. Cook, A. Buja, and J. Cabrera. Projection pursuit indexes based on orthonormal function expansions. J. of Computational and Graphical Statistics, 2(3) :225-250, 1993. [4) T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, 1991. [5) J.H. Friedman. Exploratory projection pursuit. J. of the American Statistical Association, 82(397):249-266, 1987. [6) P.J. Huber. Projection pursuit. The Annals of Statistics, 13(2):435-475, 1985. [7) A. Hyvarinen. Independent component analysis by minimization of mutual information. Technical Report A46, Helsinki University of Technology, Laboratory of Computer and Information Science, 1997. [8) A. Hyviirinen. New approximations of differential entropy for independent component analysis and projection pursuit. Technical Report A47, Helsinki University of Technology, Laboratory of Computer and Information Science, 1997. Available at http://www.cis.hut.fi;-aapo. [9) M.C. Jones and R. Sibson. What is projection pursuit ? J. of the Royal Statistical Society, ser. A , 150:1-36, 1987. [10) C. Jutten and J. Herault. Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture. Signal Processing, 24:1-10, 1991. [11) M. Kendall and A. Stuart. The Advanced Theory of Statistics. Charles Griffin & Company, 1958. 

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Generalized Prioritized Sweeping David Andre Nir Friedman Ronald Parr Computer Science Division, 387 Soda Hall University of California, Berkeley, CA 94720 {dandre,nir,parr}@cs.berkeley.edu Abstract Prioritized sweeping is a model-based reinforcement learning method that attempts to focus an agent's limited computational resources to achieve a good estimate of the value of environment states. To choose effectively where to spend a costly planning step, classic prioritized sweeping uses a simple heuristic to focus computation on the states that are likely to have the largest errors. In this paper, we introduce generalized prioritized sweeping, a principled method for generating such estimates in a representation-specific manner. This allows us to extend prioritized sweeping beyond an explicit, state-based representation to deal with compact representations that are necessary for dealing with large state spaces. We apply this method for generalized model approximators (such as Bayesian networks), and describe preliminary experiments that compare our approach with classical prioritized sweeping. 1 Introduction In reinforcement learning, there is a tradeoff between spending time acting in the environment and spending time planning what actions are best. Model-free methods take one extreme on this question- the agent updates only the state most recently visited. On the other end of the spectrum lie classical dynamic programming methods that reevaluate the utility of every state in the environment after every experiment. Prioritized sweeping (PS) [6] provides a middle ground in that only the most "important" states are updated, according to a priority metric that attempts to measure the anticipated size of the update for each state. Roughly speaking, PS interleaves perfonning actions in the environment with propagating the values of states. After updating the value of state s, PS examines all states t from which the agent might reach s in one step and assigns them priority based on the expected size of the change in their value. A crucial desideratum for reinforcement learning is the ability to scale-up to complex domains. For this, we need to use compact (or generalizing) representations of the model and the value function. While it is possible to apply PS in the presence of such representations (e.g., see [1]), we claim that classic PS is ill-suited in this case. With a generalizing model, a single experience may affect our estimation of the dynamics of many other states. Thus, we might want to update the value of states that are similar, in some appropriate sense, to s since we have a new estimate of the system dynamics at these states. Note that some of these states might never have been reached before and standard PS will not assign them a priority at all. D. Andre, N. Friedman and R Parr 1002 In this paper, we present generalized prioritized sweeping (GenPS), a method that utilizes a fonnal principle to understand and extend PS and extend it to deal with parametric representations for both the model and the value function. If GenPS is used with an explicit state-space model and value function representation, an algorithm similar to the original (classic) PS results. When a model approximator (such as a dynamic Bayesian network [2]) is used, the resulting algorithm prioritizes the states of the environment using the generalizations inherent in the model representation. 2 The Basic Principle We assume the reader is familiar with the basic concepts of Markov Decision Processes (MDPs); see, for example, [5]. We use the following notation: A MDP is a 4-tuple, (S,A,p,r) where S is a set of states, A is a set of actions, p(t I s,a) is a transition model that captures the probability of reaching state t after we execute action a at state s, and r( s) is a reward function mapping S into real-valued rewards. In this paper, we focus on infinite-horizon MDPs with a discount factor ,. The agent's aim is to maximize the expected discounted total reward it will receive. Reinforcement learning procedures attempt to achieve this objective when the agent does not know p and r. A standard problem in model-based reinforcement learning is one of balancing between planning (Le., choosing a policy) and execution. Ideally, the agent would compute the optimal value function for its model of the environment each time the model changes. This scheme is unrealistic since finding the optimal policy for a given model is computationally non-trivial. Fortunately, we can approximate this scheme if we notice that the approximate model changes only slightly at each step. Thus, we can assume that the value function from the previous model can be easily "repaired" to reflect these changes. This approach was pursued in the DYNA [7] framework, where after the execution of an action, the agent updates its model of the environment, and then performs some bounded number of value propagation steps to update its approximation of the value function . Each vaiuepropagation step locally enforces the Bellman equation by setting V(s) ~ maxaEA Q(s, a), where Q(s,a) = f(s) + ,Ls'ESP(s' I s,a)V(s'), p(s' I s,a) and f(s) are the agent's approximation of the MDP, and V is the agent's approximation of the value function. This raises the question of which states should be updated. In this paper we propose the following general principle: GenPS Principle: Update states where the approximation of the value function will change the most. That is, update the states with the largest Bellmanerror,E(s) IV(s) -maxaEAQ(s,a)l. The motivation for this principle is straightforward. The maximum Bellman error can be used to bound the maximum difference between the current value function, V(s) and the optimal value function, V"'(s) [9]. This difference bounds the policy loss, the difference between the expected discounted reward received under the agent's current policy and the expected discounted reward received under the optimal policy. To carry out this principle we have to recognize when the Bellman error at a state changes. This can happen at two different stages. First, after the agent updates its model of the world, new discrepancies between V (s) and max a Q( s, a) might be introduced, which can increase the Bellman error at s. Second, after the agent performs some value propagations, V is changed, which may introduce new discrepancies. We assume that the agent maintains a value function and a model that are parameterized by Dv and DM . (We will sometimes refer to the vector that concatenates these vectors together into a single, larger vector simply as D.) When the agent observes a transition from state s to s' under action a, the agent updates its environment model by adjusting some of the parameters in DM. When perfonning value-propagations, the agent updates V by updating parameters in Dv. A change in any of these parameters may change the Bellman error at other states in the model. We want to recognize these states without explicitly = Generalized Prioritized Sweeping 1003 computing the Bellman error at each one. Formally, we wish to estimate the change in error, I~E(B) I, due to the most recent change ~() in the parameters. We propose approximating I~E(8) 1 by using the gradient of the right hand side of the Bellman equation (i.e. max a Q(8,a). Thus, we have: I~E(s)1 ~ lV'max a Q(8,a) . ~()I which estimates the change in the Bellman error at state 8 as a function of the change in Q( 8, a). The above still requires us to differentiate over a max, which is not differentiable. In general, we want to to overestimate the change, to avoid "starving" states with nonnegligible error. Thus, we use the following upper bound: 1V'(max a Q(8, a)) . ~81 ~ max a IV'Q(s,a). ~81? We now define the generalized prioritized sweeping procedure. The procedure maintains a priority queue that assigns to each state 8 a priority,pri( 8). After making some changes, we can reassign priorities by computing an approximation of the change in the value function. Ideally, this is done using a procedure that implements the following steps: procedure update-priorities (&) for all s E S pri(s) +- pri(s) + max a IV'Q(s, a) . &1. Note that when the above procedure updates the priority for a state that has an existing priority, the priorities are added together. This ensures that the priority being kept is an overestimate of the priority of each state, and thus, the procedure will eventually visit all states that require updating. Also, in practice we would not want to reconsider the priority of all states after an update (we return to this issue below). Using this procedure, we can now state the general learning procedure: procedure GenPS 0 loop perform an action in the environment update the model; let & be the change in () call update-priorities( &) while there is available computation time arg max s pri( s) let smax perform value-propagation for V(smax); let & be the change in () call update-priorities( &) pri(smax) +- W(smax) - max a Q(smax,a)11 = Note that the GenPS procedure does not determine how actions are selected. This issue, which involves the problem of exploration, is orthogonal to the our main topic. Standard approache!;, such as those described in [5, 6, 7], can be used with our procedure. This abstract description specifies neither how to update the model, nor how to update the value function in the value-propagation steps. Both of these depend on the choices made in the corresponding representation of the model and the value function. Moreover, it is clear that in problems that involve a large state space, we cannot afford to recompute the priority of every state in update-priorities. However, we can simplify this computation by exploiting sparseness in the model and in the worst case we may resort to approximate methods for finding the states that receive high priority after each change. 3 Explicit, State-based Representation In this section we briefly describe the instantiation of the generalized procedure when the rewards, values, and transition probabilities are explicitly modeled using lookup tables. In this representation, for each state 8, we store the expected reward at 8, denoted by Of(s)' the estimated value at 8, denoted by Bv (s), and for each action a and state t the number of times the execution of a at 8 lead to state t, denoted NS,Q.,t. From these transition counts we can I In general, this will assign the state a new priority of 0, unless there is a self loop. In this case it will easy to compute the new Bellman error as a by-product of the value propagation step. D. Andre, N. Friedman and R. Parr 1004 reconstruct the transition probabilities pACt I 8, a) = N?. a.! +N~! a .t O W t' N ",a ,t' +N" , a , t l " ' where NO 8,a,t are fictional counts that capture our prior information about the system's dynamics. 2 After each step in the worJd, these reward and probability parameters are updated in the straightforward manner. Value propagation steps in this representation set 8Y (t) to the right hand side of the Bellman equation. To apply the GenPS procedure we need to derive the gradient of the Bellman equation for two situations: (a) after a single step in the environment, and (b) after a value update. In case (a), the model changes after performing action 8~t . In this case, it is easy to verify that V'Q(s,a) 'do = dOr(t) + N.}t+N~.a.t (V(t) - 2:~p(t' I s,a)V(t')), and tt that V' Q( s', a') . do = 0 if s' =1= s or a' =1= a. Thus, s is the only state whose priority changes. In case (b), the value function changes after updating the value of a state t. In this case, V'Q(s, a) ?do = ,pet I s, a)l1o v (t)' It is easy to see that this is nonzero only ift is reachable from 8. In both cases, it is straightforward to locate the states where the Bellman error might have have changed, and the computation of the new priority is more efficient than computing the Bellman-error. 3 Now we can relate GenPS to standard prioritized sweeping. The PS procedure has the general form of this application of GenPS with three minor differences. First, after performing a transition s~t in the environment, PS immediately performs a value propagation for state s, while GenPS increments the priority of s. Second, after performing a value propagation for state t, PS updates the priority of states s that can reach t with the value max a p( tis, a) ./1y( t). The priority assigned by GenPS is the same quantity multiplied by ,. Since PS does not introduce priorities after model changes, this multiplicative constant does not change the order of states in the queue. Thirdly, GenPS uses addition to combine the old priority of a state with a new one, which ensures that the priority is indeed an upper bound. In contrast, PS uses max to combine priorities. This discussion shows that PS can be thought of as a special case of GenPS when the agent uses an explicit, state-based representation. As we show in the next section, when the agent uses more compact representations, we get procedures where the prioritization strategy is quite different from that used in PS. Thus, we claim that classic PS is desirable primarily when explicit representations are used. 4 Factored Representation We now examine a compact representation of p( s' I s, a) that is based on dynamic Bayesian networks (DBNs) [2]. DBNs have been combined with reinforcement learning before in [8], where they were used primarily as a means getting better generalization while learning. We will show that they also can be used with prioritized sweeping to focus the agent's attention on groups of states that are affected as the agent refines its environment model. We start by assuming that the environment state is described by a set of random variables, XI, . .. , X n ? For now, we assume that each variable can take values from a finite set Val(Xi). An assignment of values XI, .? ? , Xn to these variables describes a particular environment state. Similarly, we assume that the agent's action is described by random variables AI, ... ,A k . To model the system dynamics, we have to represent the probability of transitions s~t, where sand t are two assignments to XI, .. . , Xn and a is an assignment to AI, ... ,A k ? To simplify the discussion, we denote by Yi, .. . , Yn the agent's state after 2Formally, we are using multinomial Dirichlet priors. See, for example, [4] for an introduction to these Bayesian methods. 3 Although ~~(s .a ) involves a summation over all states, it can be computed efficiently. To see .. ,a ,t this, note that the summation is essentially the old value of Q( s, a) (minus the immediate reward) which can be retained in memory. Generalized Prioritized Sweeping 1005 the action is executed (e.g., the state t). Thus, p(t I s,a) is represented as a conditional probability P(YI , ... , Y n I XI, ... ,Xn , AI , ... ,Ak). A DBN model for such a conditional distribution consists of two components. The first is a directed acyclic graph where each vertex is labeled by a random variable and in which the vertices labeled XI, . .. ,Xn and AI, .. . , Ak are roots. This graph speoifies the factorization of the conditional distribution: n P(Yi, ... , Y n I XI'? .. ' X n , AI,???, Ak) = II P(Yi I Pai), (1) i=I where Pai are the parents of Yi in the graph. The second component of the DBN model is a description of the conditional probabilities P(Yi I Pai). Together, these two components describe a unique conditional distribution. The simplest representation of P(Yi I Pai) is a table that contains a parameter (}i ,y,z = P(Yi = y I Pai = z) for each possible combination of y E Val(Yi) and z E Val(Pai) (note that z is a joint assignment to several random variables). It is easy to see that the "density" of the DBN graph determines the number of parameters needed. In particular, a complete graph, to which we cannot add an arc without violating the constraints, is equivalent to a state-based representation in terms of the number of parameters needed. On the other hand, a sparse graph requires few parameters. In this paper, we assume that the learneris supplied with the DBN structure and only has to learn the conditional probability entries. It is often easy to assess structure information from experts even when precise probabilities are not available. As in the state-based representation, we learn the parameters using Dirichlet priors for each multinomial distribution [4]. In this method, we assess the conditional probability (}i,y,z using prior knowledge and the frequency of transitions observed in the past where Yi = y among those transitions where Pai = z. Learning amounts to keeping counts Ni ,y,z that record the number of transitions where Yi = y and Pai = z for each variable Yi and values y E Val(Yi) and z E Val(Pai). Our prior knowledge is represented by fictional counts Np,y,z. Then we estimate probabil. . USIng . the 10rmu ~ Ia Ui,y,z LI , z+N?,y , z h N?~, ' ,z ? ltles - N; ,1I N;,. ,z ' W ere - " L....,y' N t,y',z + NOi,y' ,z? We now identify which states should be reconsidered after we update the DBN parameters. Recall that this requires estimating the term V Q( s, a) ./1n. Since!!:.o is sparse, after making the transition s* ~t*, we have that VQ(s, a) . !!:.O = 2:i a~Q(s:a). ' where yi and zi are the I,V i '%i assignments to Yi and Pai, respectively, in s* ~t*. (Recall that s*, a* and t* jointly assign values to all the variables in the DBN.) We say that a transition s~t is consistent with an assignment X = x for a vector of random variables X, denoted (s,a, t) F= (X = x), if X is assigned the value x in s~t. We also need a similar notion for a partial description of a transition. We say that s and a are consistent with X = x, denoted (s,a,?) F= (X = x), if there is a t such that (s, a, t) F= (X = x). Using this notation, we can show that if (s, a, .) F (Pai = zi), then 8Q(s, a) = -: Nt'~-''-':- [8,.?:.,: t(,.J~.: .': p(t I s, a)1l(t) - t(,. ~I=': p(t I s, a)1l(t)] and if s, a are inconsistent with Pai = zi, then aa:.(8:a). ""i = o. 'Z i This expression shows that if s is similar to s* in that both agree on the values they assign to the parents of some Yi (i.e., (s, a*) is consistent with zi), then the priority of s would change after we update the model. The magnitude of the priority change depends upon both the similarity of sand s* (i.e. how many of the terms in VQ(s, a) ? !!:'o will be non-zero), and the value of the states that can be reached from s. D. Andre, N. Friedman and R. Parr 1006 PSPS+fact<X'ed --- .- G:nps ??..:.~.._~_..::.:.;. ::::.:....~..:.....:._--I---...?--- 2.S ?? oX-'-- ~ '3 " 0 ~ I.S ? 005 ./ ,.l ___ /~ o~~~--~~--~~~--~~ 1000 2000 3000 4000 sooo 6000 7000 8000 9000 10000 Number of iterations (b) (a) (c) Figure 1: (a) The maze used in the experiment. S marks the start space, G the goal state, and 1, 2 and 3 are the three flags the agent has to set to receive the reward. (b) The DBN structure that captures the independencies in this domain. (c) A graph showing the performance of the three procedures on this example. PS is GenPS with a state-based model, PS+factored is the same procedure but with a factored model, and GenPS exploits the factored model in prioritization. Each curve is the average of 5 runs. The evaluation of 8~(8:a). requires us to sum over a subset of the states -namely, those 1,1Ii .z, states t that are consistent with zi. Unfortunately, in the worst case this will be a large fragment of the state space. If the number of environment states is not large, then this might be a reasonable cost to pay for the additional benefits of GenPS. However, this might be a burdensome when we have a large state space, which are the cases where we expect to gain the most benefit from using generalized representations such as DBN. In these situations, we propose a heuristic approach for estimating V'Q(s, a)~ without summing over large numbers of states for computing the change of priority for each possible state. This can be done by finding upper bounds on or estimates of 8~( 8:a ).. Once we I,ll, ' %i have computed these estimates, we can estimate the priority change for each state s. We use the notation s '"Vi s* if sand s* both agree on the assignment to Pai . If Ci is an upper bound on (or an estimate of) 18~~~:t:: I, we have that IV'Q(8, a)/loM ~ Li:s~iS. C i . 1 Thus, to evaluate the priority of state s, we simply find how "similar" it is to s*. Note that it is relatively straightforward to use this equation to enumerate all the states where the priority change might be large. Finally, we note that the use of a DBN as a model does not change the way we update priorities after a value propagation step. If we use an explicit table of values, then we would update priorities as in the previous section. If we use a compact description of the value function, then we can apply GenPS to get the appropriate update rule. S An Experiment We conducted an experiment to evaluate the effect of using GenPS with a generalizing model. We used a maze domain similar to the one described in [6]. The maze, shown in Figure 1(a), contains 59 cells, and 3 binary flags, resulting in 59 x 2 3 = 472 possible states. Initially the agent is at the start cell (marked by S) and the flags are reset. The agent has four possible actions, up, down, left, and right, that succeed 80% of the time, and 20% of the time the agent moves in an unintended perpendicular direction. The i'th flag is set when the agent leaves the cell marked by i. The agent receives a reward when it arrives at the goal cell (marked by G) and all of the flags are set. In this situation, any action resets the game. As noted in [6], this environment exhibits independencies. Namely, the probability of transition from one cell to another does not depend on the flag settings. Generalized Prioritized Sweeping 1007 These independencies can be captured easily by the simple DBN shown in Figure I(b) Our experiment is designed to test the extent to which GenPS exploits the knowledge of these independencies for faster learning. We tested three procedures. The first is GenPS, which uses an explicit state-based model. As explained above, this variant is essentially PS. The second procedure uses a factored model of the environment for learning the model parameters, but uses the same prioritization strategy as the first one. The third procedure uses the GenPS prioritization strategy we describe in Section 4. All three procedures use the Boltzman exploration strategy (see for example [5]). Finally, in each iteration these procedures process at most 10 states from the priority queue. The results are shown in Figure l(c). As we can see, the GenPS procedure converged faster than the procedures that used classic PS. As we can see, by using the factored model we get two improvements. The first improvement is due to generalization in the model. This allows the agent to learn a good model of its environment after fewer iterations. This explains why PS+factored converges faster than PS. The second improvement is due to the better prioritization strategy. This explains the faster convergence of GenPS. 6 Discussion We have presented a general method for approximating the optimal use of computational resources during reinforcement learning . Like classic prioritized sweeping, our method aims to perform only the most beneficial value propagations. By using the gradient of the Bellman equation our method generalizes the underlying principle in prioritized sweeping. The generalized procedure can then be applied not only in the explicit, state-based case, but in cases where approximators are used for the model. The generalized procedure also extends to cases where a function approximator (such as that discussed in [3]) is used for the value function, and future work will empirically test this application of GenPS. We are currently working on applying GenPS to other types of model and function approximators. Acknowledgments We are grateful to Geoff Gordon, Daishi Harada, Kevin Murphy, and Stuart Russell for discussions related to this work and comments on earlier versions of this paper. This research was supported in part by ARO under the MURI program "Integrated Approach to Intelligent Systems," grant number DAAH04-96-I-0341. The first author is supported by a National Defense Science and Engineering Graduate Fellowship. References [I] S. Davies. Multidimensional triangulation and interpolation for reinforcement learning. In Advances in Neurallnfonnation Processing Systems 9. 1996. [2] T. Dean and K. Kanazawa. A model for reasoning about persistence and causation. Computationallntelligence, 5:142-150, 1989. [3] G. J. Gordon. Stable function approximation in dynamic programming. In Proc. 12th Int. Con! on Machine Learning, 1995. [4] D. Heckerman. A tutorial on learning with Bayesian networks. Technical Report MSR-TR-9506, Microsoft Research, 1995. Revised November 1996. [5] L. P. Kaelbling, M. L. Littman and A. W. Moore. Reinforcement learning: A survey. Journal of Artificial Intelligence Research, 4:237-285, 1996. [6] A. W. Moore and C. G. Atkeson. Prioritized sweeping-reinforcement learning with less data and less time. Machine Learning, 13:103-130, 1993. [7] R. S. Sutton. Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. In Machine Learning: Proc. 7th Int. Con!, 1990. [8] P. Tadepalli and D. Ok. Scaling up average reward reinforcement learning by approximating the domain models and the value function. In Proc. 13th Int. Con! on Machine Learning, 1996. [9] R. J. Williams and L. C. III Baird. Tight performance bounds on greedy policies based on imperfect value functions. Technical report, Computer Science, Northeastern University. 1993. 

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232 SPEECH PRODUCTION USING A NEURAL NETWORK WITH A COOPERATIVE LEARNING MECHANISM Mitsuo Komura Akio Tanaka International Institute for Advanced Study of Social Information Science, Fujitsu Limited 140 Miyamoto, Numazu-shi Shizuoka, 410-03 Japan ABSTRACT We propose a new neural network model and its learning algorithm. The proposed neural network consists of four layers - input, hidden, output and final output layers. The hidden and output layers are multiple. Using the proposed SICL(Spread Pattern Information and Cooperative Learning) algorithm, it is possible to learn analog data accurately and to obtain smooth outputs. Using this neural network, we have developed a speech production system consisting of a phonemic symbol production subsystem and a speech parameter production subsystem. We have succeeded in producing natural speech waves with high accuracy. INTRODUCTION Our purpose is to produce natural speech waves. In general, speech synthesis by rule is used for producing speech waves. However, there are some difficulties in speech synthesis by rule. First, the rules are very complicated. Second, extracting a generalized rule is difficult. Therefore, it is hard to synthesize a natural speech wave by using rules. We use a neural network for producing speech waves. Using a neural network, it is possible to learn speech parameters without rules. (Instead of describing rules explicitly, selecting a training data set becomes an important subject.) In this paper, we propose a new neural network model and its learning algorithm. Using the proposed neural network, it is possible to learn and produce analog data accurately. We apply the network to a speech production system and examine the system performance. PROPOSED NEURAL NETWORK AND ITS LEARNING ALGORITHM We use an analog neuron-like element in a neural network. The element has a logistic activation function presented by equation (3). As a learning algorithm, Speech Production Using A Neural Network the BP(Back Propagation) method is widely used. By using this method it is possible to learn the weighting coefficients of the units whose target values are not given directly. However, there are disadvantages. First, there are singular points at 0 and 1 (outputs of the neuron-like element). Second, finding the optimum values of learning constants is not easy. We have proposed a new neural network model and its learning algorithm to solve this problem. The proposed SICL(Spread Pattern Information and Cooperative Learning) method has the following features. (a)The singular points of the BP method are removed. (Outputs are not simply oor 1.) This improves the convergence rate. (b)A spread pattern information(SI) learning algorithm is proposed. In the SI learning algorithm, the weighting coefficients from the hidden layers to the output layers are fixed to random values. Pattern information is spread over the memory space of the weighting coefficients. As a result, the network can learn analog data accurately. (c)A cooperative learning(CL) algorithm is proposed. This algorithm makes it possible to obtain smooth and stable output. The CL system is shown in Fig.1 where D(L) is a delay line which delays L time units. In the following sections, we define a three-layer network, introduce the BP method, and propose the SICL method . .. ' .' k=K ", .....,..?.?. .... ".'. -'. ~,;" ,:yi I';; ,/ Input layar Hlddan layar. Output layar. Final output layar Figure 1. Cooperative Learning System (Speech Parameter I Phonemic Symbol Production Subsystem) 233 234 Komura and Tanaka THREE?LAYER NETWORK We define a three-layer network that has an input layer, a hidden layer, and an output layer. The propagation of a signal from the input layer to the hidden layer is represented as follows. Uj = 'EiwIHijXi, Yj = f(Uj -OJ) , (1) where i= 1,2,... ,1; j= 1,2, ... ,J and Xi is an input, Yj the output of the hidden layer, and OJ a threshold value. The propagation of a signal from the hidden layer to the output layer is represented as follows. (2) where Zk is the output of the output layer and k= 1,2, ... ,K. The activation function f(u) is defined by f(u) = (1+exp(-u+O?-l. Settingy (3) = f(u), the derivation off(u) is then given by f' (u) =y(1-y). BACK PROPAGATION (BP) METHOD The three-layer BP algorithm is shown as follows. The back propagation error, 80 k(n) , for an output unit is given by (4) where n is an iteration number. The back propagation error, 8Hin), for a hidden unit is given by 8Hin) = ('Ek80k(n) WHOjk) f' (u)<n? . (5) The change to the weight from the i-th to thej-th unit, flwIH;,in) is given by flwIHi/n) =a8H/n)xi(n) + jJflwIH;,in-l) . (6) The change to the weight from thej-th to the k-th unit, flwH0jk(n) is given by flwHOjk(n) = a80k(n) Yin) + jJflwHOjk(n-l) , (7) where a and jJ are learning constants, and have positive values. SPREAD PATTERN INFORMATION AND COOPERATIVE LEARNING (SICL) METHOD The proposed learning algorithm - SICL method is shown as follows. The propagation of a signal from the input layer to the hidden layer is given by u/l,n) = 'EiwlHij(l,n-l) xi(n) , y/l,n) =f(u/l,n) -oin) , (8) where l is a stage number (l =-LI2, ... ,LI2). The propagation of a signal from the hidden layer to the output layer is given by vk(l,n) ='EjwH0jk(n Yin) , zk(l,n) = f(vk(l,n) -Ok(n) . The back propagation error, 80 k(n), for an output unit is given by (9) Speech Production Using A Neural Network (10) where y is a constant for removing singular points. The back propagation error, OHj(l,n), for a hidden unit is given by 8HjCI,n) = (EkoOk(l,n) wH0jk(l) (Yil,n)(1-yjCl,n? + y) . (11) The change to the weight from the i-th to thej-th unit, ll.wIHi/l,n) is given by AwIHij(l,n) =aOHjCl,n)xln) + jJAwIHij(l,n-l) . (12) The weight from thej-th to the k-th unit,wHOjk(l) is given by wH?jk(l) = Cjk(l), (13) where Cjk(l) is a fixed value and a random number with normal distribution. A final output is a weighting sum of outputs zk(l,n) ,and is given by ZFk( n) =E,Wl zk(l,n). (14) A SPEECH PRODUCTION SYSTEM USING THE PROPOSED NEURAL NETWORK The block diagram of a speech production system is shown in Fig.2. The system consists of a phonemic symbol production subsystem and a speech parameter production subsystem using the proposed neural network. Phonemic Symbols Input String Speech ParameterslAnalog detal =! ? Speech Parameter Production Subsystern Phonemic SYlnbol I'Iaduclim SiRy.1n _ @ 1 """"?~-=0--1 Synthesized Speech Automatic Input Speech S~lg...Lnll~I-------li Speech Psremeter ExtlClIon S.mystem Figure 2. The Speech Production System Using the Proposed Neural Network In the automatic speech parameter extraction subsystem@, speech parameters are extracted automatically from an input speech signal. Speech parameters are composed of source parameters( voiced/unvoiced ratio, pitch and source power) and a vocal tract area function(PARCOR coefficients). The extracted speech parameters are used as training data of the speech parameter production subsystem. In the speech parameter production subsystem?, input is the string of phonemic symbols. (In the training stage, phonemic symbols are decided by the teacher using input speech data. After training, phonemic symbols are given by phonemic symbol production subsystem.) Targets are speech parameters extracted in @ . The phonemic symbol production subsystem CD consists of a preprocessor and a learning system using the proposed neural network. In the preprocessor, a 235 236 Komura and Tanaka string ofinput characters is converted to a string of phonemic symbols with the mean length of utterance. The input is the string of phonemic symbols converted by the preprocessor. In the training stage, the targets are actual phonemic symbols, namely, the inputs of subsystem @ which are decided by the teacher. The output speech parameters are converted into the synthesized speech wave using the PARCOR synthesizer circuit~. EXPERIMENTS We performed two separate experiments. Experiment I Automatic Speech Parameter Extraction Subsystem@ Sampling frequency: 8 KHz Frame length : 20 ms Frame period: 10 ms Speech Parameter Production Subsystem @ Input layer: 1,044(36 X 29) units Hidden layers: 80 units X 9 stages Output layers: 13 units X 9 stages Final output layer: 13 units For source parameters, 3(VIUV, pitch, source power) units are assigned. For a vocal tract area function, 10(PARCOR coefficients) units are assigned. This system has an input layer, 9 hidden layers, 9 output layers, and a final output layer. For each output layer, different target data are assigned. In the final output layer, these outputs are summed with weighting coefficients. Phonemic Symbol Production SubsystemCD Input layer: 1,044(36 X 29) units Hidden layers: 80 units X 9 stages Output layers: 36 units X 9 stages Final output layer: 36 units Input Speech Signal No.1 The input speech signal No.1 is r ASAHA YAKU BANGARONI DENPOGA TODOITA J, that is, Japanese sentence which means that "A telegram was sent to the bungalow early in the morning.". This signal is a 408 frame (4.08s) sequence. Experimental Results In Fig.3, the targets and the outputs of the speech parameter production subsystem are shown. The dotted line is the sequence of target values. The solid line is the sequence of output values after the training stage. In this case, the learning constant a is equal to 0.7 and jJ to 0.2. After training, the actual outputs produced by SICL system agree well with the targets. Fig.4 (a) shows the learning behaviors of the speech parameter production subsystem. In this case, the input speech signal is No.1. The learning constant a is equal to 0.7 and P to 0.2. The learning curve based on the SICL and SI Speech Production Using A Neural Network methods converged. However, the learning curve based on the BP method did not converge. Fig.4 (b) also shows the learning behaviors. The learning constant a is equal to 0.07 and jJ to 0.02. In this case, all of the learning curves converged. These results show that if we use the SICL or SI method instead of the BP method, we can obtain better results. 1.0 .\ ~. If '. I I 0.5 I 0.0 o 1.0 2.0 3.0 4.0 s a = 0.7,/1 = 0.2 Input speech signal : No.1 Source Power Figure 3. Targets and Outputs of Speech Parameter Production Subsystem MSE(log) MSE(log) 1 1 10-1 10-1 10-2 10-2 10-3 10-3 10- 4 10-4 10-5 10.5 1 10 102 103 (a) a = 0.7 ,jJ = 0.2 (NNI) 1 10 (b) a 102 103 = 0.07 ,p = 0.02 MSE: Mean square error NNI : Normalized number of iteration G-?J : SICL method (9stage) G-B: SI method (l = 0) BP method Figure 4. Learning Behavior of Speech Parameter Production Subsystem * *: Table 1 shows mean square errors of the system using SICL method for a, p. It should be noted that the domain of convergence is very wide. From these 237 238 Komura and Tanaka experiments, it is seen that the SICL method almost always allows stable and smooth output to be obtained. We also examined the system performance for another input speech signal which is a 1,60S frame (16.0S s) sequence. The learning curves converged and were similar to those for the input speech signal No.1 0.01 0.02 0.1 0.2 0.35 0.7 0.07 0.20 0.7 7.9S X 10-5 7.97X10-5 7.8SX 10-5 7.85 X 10-5 7.82x 10-5 7.80X 10-5 7.S2X 10-5 7.S2X10-5 7.S1 X 10- 5 7.S0X 10-5 7.S1 X 10-5 8_26X 10-5 1.03 X 10-4 1.09X 10-4 1.26 X 10-4 1.72 X 10- 4 2.S6X 10- 4 1.72x 10-3 Input speech signal: No.1. TABLE 1. Mean Square Errors for a, p Experiment II Speech Parameter Production Subsystem @ Input layer: 665 (35X 19)units Hidden layers: 400 units X 9 stages Output layers: 14 (12 + 2)units X 9 stages For the first target data(VIUV, power ratio, 10 PARCOR coefficients), 12 units are . assigned. For the second target data(pitch, source power), 2 units are assigned. . Input Speech Signal No.2 This signal is a 1:),700 frame (137s) sequence shown as follows. IARAARN,IARAIRN, .... ,1ARAKARN, I ARAKIRN, ..... ,1ARASARN, ..... . I ARARAN,I!RARA!/, ..... ,1KARARAKN ,1KIRARAKI/, ... ,1SARARASN, .. ... In experiment II , a learning test for producing an arbitrary combination of the input phonemes (ai) was done. The learning behavior of subsystem? is shown in Fig.5. In this case, the fIrst 12 target data items are trained. The other 2 target data it~ms should be trained using another network. The upper line is a learning curve for the SI method, and the lower line for the SICL method. Some of the learning curves for the SI method didn't converge. However, the learning curve for the SICL method did. We can thus say that the SICL method is very powerful for actual use. (In this case, 400 hidden units is not sufficient for the size of input data. If more hidden units are used, the MSE will be small.) CONCLUSION Speech Production Using A Neural Network MSE (log) ~---------..., G-B : SICL method (9stage) 10-1 G-E) : SI method (l= 0) 10-2 a =0.2, 10-3 11 =0.035 Input speech signal: No.2 10-4 1 10 102 NNI Figure 5. Learning Behavior of Speech Parameter Production Subsystem From the experiments shown in previous section, it should be noted that using the SICL (Spread Pattern Information and Cooperative Learning) method makes it possible to learn speech parameters or phonemic symbols stably and to produce more natural speech waves than those synthesized by rules. If the input is the combination of a word and postpositional particle, it is easy to produce sound for unknown input data using the proposed speech production system. However, the number of the hidden units and training data will be great. Therefore we have to make the system learn phonemes among phoneme sequence. Using this training data makes it possible to produce an arbitrary sequence of phonemes. In experiment II , phonemic information (VIUV power ratio, PARCOR coefficients) is trained. Then the range of input window was set to be 190 ms. For prosodic information (pitch, source power), we must use another network. Because, if we want to make the system learn prosodic information, we must set the range of input data wider than that of words. U sing these strategies, it is possible to produce arbitrary natural speech. I Acknowledgments The authors thank Dr. Tosio Kitagawa, the president of liAS-SIS for his encouragement. References D. E. Rumelhart, J. L. McClelland and PDP Research Group Parallel Distributed Processing. VOL.l The MIT Press (1987). T.J.Sejnowski and C.R.Rosenberg Parallel Networks that Learn to Pronounce English Text. Complex Systems, 1, pp.145-168 (1987). M.Komura and A.Tanaka Speech Synthesis Using a Neural Network with a Cooperative Learning Mechanism. IEICE Tech. Rep. MBE88-8 (1988) (in Japanese). 239 

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A mathematical model of axon guidance by diffusible factors Geoffrey J. Goodhill Georgetown Institute for Cognitive and Computational Sciences Georgetown University Medical Center 3970 Reservoir Road Washington DC 20007 geoff@giccs.georgetown.edu Abstract In the developing nervous system, gradients of target-derived diffusible factors play an important role in guiding axons to appropriate targets. In this paper, the shape that such a gradient might have is calculated as a function of distance from the target and the time since the start of factor production. Using estimates of the relevant parameter values from the experimental literature, the spatiotemporal domain in which a growth cone could detect such a gradient is derived. For large times, a value for the maximum guidance range of about 1 mm is obtained. This value fits well with experimental data. For smaller times, the analysis predicts that guidance over longer ranges may be possible. This prediction remains to be tested. 1 Introduction In the developing nervous system, growing axons are guided to targets that may be some distance away. Several mechanisms contribute to this (reviewed in TessierLavigne & Goodman (1996?. One such mechanism is the diffusion of a factor from the target through the extracellular space, creating a gradient of increasing concentration that axons can sense and follow. In the central nervous system, such a process seems to occur in at least three cases: the guidance ofaxons from the trigeminal ganglion to the maxillary process in the mouse (Lumsden & Davies, 1983, 1986), of commissural axons in the spinal cord to the floor plate (TessierLavigne et al., 1988), and ofaxons and axonal branches from the corticospinal tract to the basilar pons (Heffner et al., 1990). The evidence for this comes from both in vivo and in vitro experiments. For the latter, a piece of target tissue is embedded in a three dimensional collagen gel near to a piece of tissue containing the appropriate G. J Goodhill 160 population of neurons. Axon growth is then observed directed towards the target, implicating a target-derived diffusible signal. In vivo, for the systems described, the target is always less than 500 J..lm from the population ofaxons. In vitro, where the distance between axons and target can readily be varied, guidance is generally not seen for distances greater than 500 - 1000 J..lm. Can such a limit be explained in terms of the mathematics of diffusion? There are two related constraints that the distribution of a diffusible factor must satisfy to provide an effective guidance cue at a point. Firstly, the absolute concentration of factor must not be too small or too large. Secondly, the fractional change in concentration of factor across the width of the gradient-sensing apparatus, generally assumed to be the growth cone, must be sufficiently large. These constraints are related because in both cases the problem is to overcome statistical noise. At very low concentrations, noise exists due to thermal fluctuations in the number of molecules of factor in the vicinity of the growth cone (analyzed in Berg & Purcell (1977?. At higher concentrations, the limiting source of noise is stochastic variation in the amount of binding of the factor to receptors distributed over the growth cone. At very high concentrations, all receptors will be saturated and no gradient will be apparent. The closer the concentration is to the upper or lower limits, the higher the gradient that is needed to ensure detection (Devreotes & Zigmond, 1988; Tessier-Lavigne & Placzek, 1991). The limitations these constraints impose on the guidance range of a diffusible factor are now investigated. For further discussion see Goodhill (1997; 1998). 2 Mathematical model Consider a source releasing factor with diffusion constant D cm2 /sec, at rate q moles/sec, into an infinite, spatially uniform three-dimensional volume. Initially, zero decay of the factor is assumed. For radially symmetric Fickian diffusion in three dimensions, the concentration C(r, t) at distance r from the source at time t is given by q r (1) C(r, t) = - 4 D erfc r;-r:;-; 7r r v4Dt (see e.g. Crank (1975?, where erfc is the complementary error function. The percentage change in concentration p across a small distance D.r (the width of the growth cone) is given by D.r [ p = --;:- r e-r2/4Dt 1 + J7rDt erfc(r/J4Dt) 1 This function has two perhaps surprising characteristics. Firstly, for fixed r, (2) Ipi decreases with t. That is, the largest gradient at any distance occurs immediately after the source starts releaSing factor. For large t, Ipi asymptotes at D.r Jr. Secondly, for fixed t < 00, numerical results show that p is nonmonotonic with r. In particular it decreases with distance, reaches a minimum, then increases again. The position of this minimum moves to larger distances as t increases. The general characteristics of the above constraints can be summarized as follows. (1) At small times after the start of production the factor is very unevenly distributed. The concentration C falls quickly to almost zero moving away from the source, the gradient is steep, and the percentage change across the growth cone p is everywhere large. (2) As time proceeds the factor becomes more evenly distributed. C everywhere increases, but p everywhere decreases. (3) For large times, C tends to an inverse variation with the distance from the source r, while Ipi tends A Mathematical Model ofAxon Guidance by Diffusible Factors 161 to 6.r/r independent of all other parameters. This means that, for large times, the maximum distance over which guidance by diffusible factors is possible scales linearly with growth cone diameter 6.r. 3 Parameter values Diffusion constant, D. Crick (1970) estimated the diffusion constant in cytoplasm for a molecule of mass 0.3 - 0.5 kDa to be about 10- 6 cm2 /sec. Subsequently, a direct determination of the diffusion constant for a molecule of mass 0.17 kDa in the aqueous cytoplasm of mammalian cells yielded a value of about 3.3 x 10- 6 cm2 / sec (Mastro et aL, 1984). By fitting a particular solution of the diffusion equation to their data on limb bud determination by gradients of a morphogenetically active retinoid, Eichele & Thaller (1987) calculated a value of 10- 7 cm2 /sec for this molecule (mass 348.5 kDa) in embryonic limb tissue. One chemically identified diffusible factor known to be involved in axon guidance is the protein netrin-1, which has a molecular mass of about 75 kDa (Kennedy et al., 1994). D should scale roughly inversely with the radius of a molecule, Le. with the cube root of its mass. Taking the value of 3.3 x 10- 6 cm2 /sec and scaling it by (170/75,000)1/3 yields 4.0 x 10- 7 cm2 /sec. This paper therefore considers D = 10- 6 cm2 /sec and D = 10- 7 cm2 /sec. Rate of production of factor q. This is hard to estimate in vivo: some insight can be gained by considering in vitro experiments. Gundersen & Barrett (1979) found a turning response in chick spinal sensory axons towards a nearby pipette filled with a solution of NGF. They estimated the rate of outflow from their pipette to be 1 /LI/hour, and found an effect when the concentration in the pipette was as low as 0.1 nM NGF (Tessier-Lavigne & Placzek, 1991). This corresponds to a q of 3 x 1O- 11 nM/sec. Lohof et al. (1992) studied growth cone turning induced by a gradient of cell-membrane permeant cAMP from a pipette containing a 20 mM solution and a release rate of the order of 0.5 pI/sec: q = 10- 5 nM/sec. Below a further calculation for q is performed, which suggests an appropriate value may be q = 10- 7 nM/sec. Growth cone diameter, 6.r. For the three systems mentioned above, the diameter of the main body of the growth cone is less than 10 /Lm. However, this ignores filopodia, which can increase the effective width for gradient sensing purposes. The values of 10 /Lm and 20 /Lm are considered below. Minimum concentration for gradient detection. Studies of leukocyte chemotaxis suggest that when gradient detection is limited by the dynamics of receptor binding rather than physical limits due to a lack of molecules of factor, optimal detection occurs when the concentration at the growth cone is equal to the dissociation constant for the receptor (Zigmond, 1981; Devreotes & Zigmond, 1988). Such studies also suggest that the low concentration limit is about 1% of the dissociation constant (Zigmond, 1981). The transmembrane protein Deleted in Colorectal Cancer (DeC) has recently been shown to possess netrin-1 binding activity, with an order of magnitude estimate for the dissociation constant of 10 nM (Keino-Masu et aI, 1996). For comparison, the dissociation constant of the low-affinity NGF receptor P75 is about 1 nM (Meakin & Shooter, 1992). Therefore, low concentration limits of both 10- 1 nM and 10- 2 nM will be considered. Maximum concentration for gradient detection. Theoretical considerations suggest that, for leukocyte chemotaxis, sensitivity to a fixed gradient should fall off symmetrically in a plot against the log of background concentration, with the peak at the dissociation constant for the receptor (Zigmond, 1981). Raising the con- 162 G. 1. Goodhill centration to several hundred times the dissociation constant appears to prevent axon guidance (discussed in Tessier-Lavigne & Placzek (1991?. At concentrations very much greater than the dissociation constant, the number of receptors may be downregulated, reducing sensitivity (Zigm0nd, 1981). Given the dissociation constants above, 100 nM thus constitutes a reasonable upper bound on concentration. Minimum percentage change detectable by a growth cone, p. By establishing gradients of a repellent, membrane-bound factor directly on a substrate and measuring the response of chick retinal axons, Baier & Bonhoeffer (1992) estimated p to be about 1%. Studies of cell chemotaxis in various systems have suggested optimal values of 2%: for concentrations far from the dissociation constant for the receptor, p is expected to be larger (Devreotes & Zigmond, 1988). Both p = 1% and p = 2% are considered below. 4 Results In order to estimate bounds for the rate of production of factor q for biological tissue, the empirical observation is used that, for collagen gel assays lasting of the order of one day, guidance is generally seen over distances of at most 500 I'm (Lumsden & Davies, 1983,1986; Tessier-Lavigne et al., 1988). Assume first that this is constrained by the low concentration limit. Substituting the above parameters (with D = 10- 7 cm2/sec) into equation 1 and specifying that C(500pm, 1 day) = 0.01 nM gives q :::::: 10- 9 nM/ sec. On the other hand, assuming constraint by the high concentration limit, i.e. C(500pm, 1 day) = 100 nM, gives q :::::: 10- 5 nM/sec. Thus it is reasonable to assume that, roughly, 10- 9 nM/ sec < q < 10- 5 nM/ sec. The results discussed below use a value in between, namely q = 10- 7 nM/sec. The constraints arising from equations 1 and 2 are plotted in figure 1. The cases of D = 10- 6 cm2/sec and D = 10- 7 cm2/sec are shown in (A,C) and (B,D) respectively. In all four pictures the constraints C = 0.01 nM and C = 0.1 nM are plotted. In (A,B) the gradient constraint p = 1% is shown, whereas in (C,D) p = 2% is shown. These are for a growth cone diameter of 10 I'm. The graph for a 2% change and a growth cone diameter of 20 I'm is identical to that for a 1% change and a diameter of 10 I'm. Each constraint is satisfied for regions to the left of the relevant line. The line C = 100 nM is approximately coincident with the vertical axis in all cases. For these parameters, the high concentration limit does not therefore prevent gradient detection until the axons are within a few microns of the source, and it is thus assumed that it is not an important constraint. As expected, for large t the gradient constraint asymptotes at D.r Ir = p, i.e. r = 1000 I'm for p = 1% and r = 500 I'm for p = 2% and a 10 I'm growth cone. That is, the gradient constraint is satisfied at all times when the distance from the source is less than 500 I'm for p = 2% and D.r = 10 I'm. The gradient constraint lines end to the right because at earlier times p exceeds the critical value over all distances (since the formula for p is non-monotonic with r, there is sometimes another branch of each p curve (not shown) off the graph to the right). As t increases from zero, guidance is initially limited only by the concentration constraint. The maximum distance over which guidance can occur increases smoothly with t, reaching for instance 1500 I'm (assuming a concentration limit of 0.01 nM) after about 2 hours for D = 10- 6 cm2/sec and about 6 hours for D = 10- 7 cm2/sec. However at a particular time, the gradient constraint starts to take effect and rapidly reduces the maximum range of guidance towards the asymptotic value as t increases. This time (for p = 2%) is about 2 hours for D = 10- 6 cm 2/sec, and about one day for D = 10- 7 cm2 / sec. It is clear from these pictures that although the exact size of A Mathematical Model ofAxon Guidance by Diffusible Factors A 163 B Ci) >- '" Ci) >- 4.0 ~ CD E '" ... C=O.OlnM . C=O.lnM 3.5 i= CD E - 3.5 i= " 3.0 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 500 1000 ... 3.0 p=O.OI 2.5 0 "'-J 4.0 ~ 1500 2000 2500 Distance (microns) ... - C=O.OlnM C = O.lnM p= 0.01 ....... " 0 500 1000 . .. " , ", " 1500 2000 2500 Distance (microns) D C ~ Ci) >- '" ~ 4.0 ... C = O.OlnM ., . C = O.lnM CD E 3.5 i= . 3.0 - p=0.02 '" 4.0 CD E 3.5 i= 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 .. ' ., ' 0.5 0.0 0 1500 2000 2500 Distance (microns) ... C = O.OlnM ., . C=O.lnM - P =0.02 ~ 0.5 !~ 0.0 0 500 ......... 1000 .. ' ." .. .' " " " 1500 2000 2500 Distance (microns) Figure 1: Graphs showing how the gradient constraint (solid line) interacts with the minimum concentration constraint (dashed/dotted lines) to limit guidance range, and how these constraints evolve over time. The top row (A,B) is for p = 1%, the bottom row (C,D) for p = 2%. The left column (A,C) is for D = 10- 6 cm2 / sec, the right column (B,D) for D = 10- 7 cm2 /sec. Each constraint is satisfied to the left of the appropriate curve. It can be seen that for D = 1O- 6 cm2 / s~c the gradient limit quickly becomes the dominant constraint on maximum guidance range. In contrast for D = 10- 7 cm2 / sec, the concentration limit is the dominant constraint at times up to several days. However after this the gradient constraint starts to take effect and rapidly reduces the maximum guidance range. 164 G. 1. Goodhill the diffusion constant does not affect the position of the asymptote for the gradient constraint, it does play an important role in the interplay of constraints while the gradient is evolving. The effect is however subtle: reducing D from 10- 6 cm2 /sec to 10- 7 cm 2 /sec increases the time for the C = 0.01 nM limit to reach 2000 J..tm, but decreases the time for the C = 0.1 nM limit to reach 2000 f..lm. 5 Discussion Taking the gradient constraint to be a fractional change of at least 2% across a growth cone of width of 10 f..lm or 20 J..tm yields asymptotic values for the maximum distance over which guidance can occur once the gradient has stabilized of 500 f..lm and 1000 f..lm respectively. This fits well with both in vitro data, and the fact that for the systems mentioned in the introduction, the growing axons are always less than 500 f..lm from the target in vivo. The concentration limits seem to provide a weaker constraint than the gradient limit on the maximum distances possible. However, this is very dependent on the value of q, which has only been very roughly estimated: if q is significantly less than 10- 7 nM/sec, the low concentration limits will provide more restrictive constraints (q may well have different values in different target tissues). The gradient constraint curves are independent of q. The gradient constraint therefore provides the most robust explanation for the observed guidance limit. The model makes the prediction that guidance over longer distances than have hitherto been observed may be possible before the gradient has stabilized. In the early stages following the start of factor production the concentration falls off more steeply, providing more effective guidance. The time at which guidance range is a maximum defends on the diffusion constant D. For a rapidly diffusing molecule (D >::::: 1O- 6 cm jsec) this occurs after only a few hours. For a more slowly diffusing molecule however (D >::::: 10- 7 cm2 jsec) this occurs after a few days, which would be easier to investigate in vitro. In vivo, molecules such as netrin-1 may thus be large because, during times immediately following the start of production by the source, there could be a definite benefit (i.e. steep gradient) to a slowly-diffusing molecule. Also, it is conceivable that Nature has optimized the start of production of factor relative to the time that guidance is required in order to exploit an evolving gradient for extended range. This could be especially important in larger animals, where axons may need to be guided over longer distances in the developing embryo. Bibliography Baier, H. & Bonhoeffer, F. (1992). Axon guidance by gradients of a target-derived component. Science, 255,472-475. Berg, H.C. and Purcell, E.M. (1977). Physics of chemoreception. Biophysical Journal, 20,193-219. Crick, F. (1970). Diffusion in embryogenesis. Nature, 255,420-422. Crank, J. (1975). The mathematics of diffusion, Second edition. Oxford, Clarendon. Devreotes, P.N. & Zigmond, S.H. (1988). Chemotaxis in eukaryotic cells: a focus on leukocytes and Dictyostelium. Ann. Rev. Cell. Bioi., 4, 649-686. Eichele, G. & Thaller, C. (1987). Characterization of concentration gradients of a morphogenetically active retinoid in the chick limb bud. J. Cell. Bioi., 105, 19171923. A Mathematical Model ofAxon Guidance by Diffusible Factors 165 Goodhill, G.J. (1997). Diffusion in axon guidance. Eur. J. Neurosci., 9,1414-1421. Goodhill, G.J. (1998). Mathematical guidance for axons. Trends. Neurosci., in press. Gundersen, R.W. & Barrett, J.N. (1979). Neuronal chemotaxis: chick dorsal-root axons tum toward high concentrations of nerve growth factor. Science, 206, 1079-1080. Heffner, C.D., Lumsden, AG.5. & O'Leary, D.D.M. (1990). Target control of collateral extension and directional growth in the mammalian brain. Science, 247, 217-220. Keino-Masu, K., Masu, M., Hinck, L., Leonardo, E.D., Chan, S.5.-Y., Culotti, J.G. & Tessier-Lavigne, M. (1996). Deleted in Colorectal Cancer (DCC) encodes a netrin receptor. Cell, 87, 175-185. Kennedy, T.E., Serafini, T., de al Torre, J.R. & Tessier-Lavigne, M. (1994). Netrins are diffusible chemotropic factors for commissural axons in the embryonic spinal cord. Cell, 78,425-435. Lohof, A.M., Quillan, M., Dan, Y, & Poo, M-m. (1992). Asymmetric modulation of cytosolic cAMP activity induces growth cone turning. J. Neurosci., 12, 12531261. Lumsden, AG.s. & Davies, A.M. (1983). Earliest sensory nerve fibres are guided to peripheral targets by attractants other than nerve growth factor. Nature, 306, 786-788. Lumsden, A.G.5. & Davies, AM. (1986). Chemotropic effect of specific target epithelium in the developing mammalian nervous system. Nature, 323, 538-539. Mastro, AM., Babich, M.A, Taylor, W.D. & Keith, AD. (1984). Diffusion of a small molecule in the cytoplasm of mammalian cells. Proc. Nat. Acad. Sci. USA, 81, 3414-3418. Meakin, S.O. & Shooter, E.M. (1992). The nerve growth family of receptors. Trends. Neurosci., 15, 323-331. Tessier-Lavigne, M. & Placzek, M. (1991). Target attraction: are developing axons guided by chemotropism? Trends Neurosci., 14,303-310. Tessier-Lavigne, M. & Goodman, C.S. (1996). The molecular biology of axon guidance. Science, 274, 1123-1133. Tessier-Lavigne, M., Placzek, M., Lumsden, A.G.5., Dodd, J. & Jessell, T.M. (1988). Ch~motropic guidance of developing axons in the mammalian central nervous system. Nature, 336, 775-778. Tranquillo, R.T. & Lauffenburger, D.A. (1987). Stochastic model of leukocyte chemosensory movement. J. Math. BioI., 25,229-262. Zigmond, S.H. (1981). Consequences of chemotactic peptide receptor modulation for leukocyte orientation. J. Cell. BioI., 88, 644-647. 

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Instabilities in Eye Movement Control: A Model of Periodic Alternating Nystagmus ErnstR. Dow Center for Biophysics and Computational Biology, Beckman Institute University of Illinois at UrbanaChampaign,Urbana, IL 61801. edow@uiuc.edu Thomas J. Anastasio Department of Molecular and Integrative Physiology, Center for Biophysics and Computational Biology, Beckman Institute University of Illinois at UrbanaChampaign, Urbana, IL 61801. tstasio@uiuc.edu Abstract Nystagmus is a pattern of eye movement characterized by smooth rotations of the eye in one direction and rapid rotations in the opposite direction that reset eye position. Periodic alternating nystagmus (PAN) is a form of uncontrollable nystagmus that has been described as an unstable but amplitude-limited oscillation. PAN has been observed previously only in subjects with vestibulo-cerebellar damage. We describe results in which PAN can be produced in normal subjects by prolonged rotation in darkness. We propose a new model in which the neural circuits that control eye movement are inherently unstable, but this instability is kept in check under normal circumstances by the cerebellum. Circumstances which alter this cerebellar restraint, such as vestibulocerebellar damage or plasticity due to rotation in darkness, can lead to PAN. 1 INTRODUCTION Visual perception involves not only an operating visual sensory system, but also the ability to control eye movements. The oculomotor subsystems provide eye movement control. For example, the vestibulo-ocular reflex (VOR) maintains retinal image stability by making slow-phase eye rotations that counterbalance head rotations, making it possible to move and see at the same time (Wilson and Melvill Jones, 1979). The VOR makes slowphase eye rotations that are directed opposite to head rotations. When these ongoing slow-phase eye rotations are interrupted by fast-phase eye rotations that reset eye position, the resulting eye movement pattern is called nystagmus. Periodic alternating nys- A Model of Periodic Alternating Nystagmus 139 tagmus (PAN) is a congenital or acquired eye movement disorder characterized by uncontrollable nystagmus that alternates direction roughly sinusoidally with a period of 200 s to 400 s (Baloh et al., 1976; Leigh et al., 1981; Furman et aI., 1990). Furman and colleagues (1990) have determined that PAN in humans is caused by lesions of parts of the vestibulo-cerebellum known as the nodulus and uvula (NU). Lesions to the NU cause PAN in the dark (Waespe et aI., 1985; Angelaki and Hess, 1995). NU lesions also prevent habituation (Singleton, 1967; Waespe et aI, 1985; Torte et al., 1994), which is a semi-permanent decrease in the gain (eye velocity / head velocity) of the VOR response that can be brought about by prolonged low-frequency rotational stimulation in the dark. Vestibulo-cerebellectomy in habituated goldfish causes VOR dishabituation (Dow and Anastasio, 1996). Temporary inactivation of the vestibulo-cerebellum in habituated goldfish causes temporary dishabituation and can result in a temporary PAN (Dow and Anastasio, in press). Stimulation of the NU temporarily abolish the VOR response (Fernandez and Fredrickson, 1964). Cerebellar influence on the VOR may be mediated by connections between the NU and vestibular nucleus neurons, which have been demonstrated in many species (Dow, 1936; 1938). We have previously shown that intact goldfish habituate to prolonged low-frequency (0.01 Hz) rotation (Dow and Anastasio, 1996) and that rotation at higher frequencies (0.05-0.1 Hz) causes PAN (Dow and Anastasio, 1997). We also proposed a limit-cycle model of PAN in which habituation or PAN result from an increase or decrease, respectively, of the inhibition of the vestibular nuclei by the NU. This model suggested that velocity storage, which functions to increase low-frequency VOR gain above the biophysical limits of the semicircular canals (Robinson, 1977;1981), is mediated by a potentially unstable low-frequency resonance. This instability is normally kept in check by constant suppression by the NU. 2 METHODS PAN was studied in intact, experimentally naive, comet goldfish (carassius auratus). Each goldfish was restrained horizontally underwater with the head at the center of a cylindrical tank. Eye movements were measured using the magnetic search coil technique (Robinson, 1963). For technical details see Dow and Anastasio (1996). The tank was centered on a horizontal rotating platform. Goldfish were rotated continuously for various durations (30 min to 2 h) in darkness at various single frequencies (0.03 - 0.17 Hz). Some data have been previously reported (Dow and Anastasio, 1997). All stimuli had peak rotational velocities of 60 degls. Eye position and rotator (i.e. head) velocity signals were digitiZed for analysis. Eye position data were digitally differentiated to compute eye velocity and fast-phases were removed. Data were analyzed and simulated using MATLAB and SIMULINK (The Mathworks, Inc.). 3 RESULTS Prolonged rotation in darkness at frequencies which produced some habituation in naive goldfish (0.03-0.17 Hz) could produce a lower-frequency oscillation in slow-phase eye velocity that was superimposed on the normal VOR response (fig 1). This lowerfrequency oscillation produced a periodic alternating nystagmus (PAN). When PAN occurred, it was roughly sinusoidal and varied in period, amplitude, and onset-time. Habituation could occur simultaneously with PAN (fig IB) or habituation could completely E. R. Dow and T. J. Anastasio 140 60 A Initial response ~ 40 -'g ~ 20 ~ 0 g! -20 Q) >Q) -40 ~-60~------~--------~~--~--~------~--------~ Q) ; -:8 ~JJJvAvAJ~\VV~AvA~?~AvA~\AvAJv\Av?v?~AJfv\?M~?v?JvA~ i .r; 60 B Response after 1 h. rotation ~ 40 -'8 ~ 20 ~ Q) > g!. Q) -20 ,A iJ 0 ~ ~ V ~ v~ A I? A v ~j y~ l~J v~ -40 tn-60 t ~ 50~A66A6GGGA66666AGGAAG6A666A6GA6AAAA666AGA666GA6A6 -58 VVvv rv VVIlVlJlJ VVVVVVUVVVrv VVVlJ uvrvvrvwvlv vvrVV VVVt al 0 200 400 600 800 1000 1 '0 .r; time (s) Figure 1: Initial 1000 s 0.05 Hz rotation showing PAN (A). Slow-phase eye velocity shows that PAN starts almost immediately and there is a slight reduction in VOR gain after 1000 s. Following 1 h continuous rotation in the same goldfish (B), VOR gain has decreased. suppress PAN (fig 4). PAN observed at lower frequencies (0.03 and 0.05 Hz) typically decreased in amplitude as rotation continued. Previous work has shown that PAN was most likely to occur during prolonged rotations at frequencies between 0.05 and 0.1 Hz (Dow and Anastasio, 1997). At these frequencies, habituation also caused a slight decrease in VOR gain (1.3 to 1.8 times, initial gain I final gain) following 1 h of rotation. At higher frequencies, neither habituation nor PAN were observed. At lower frequencies (0.03 Hz) PAN could occur before habituation substantially reduced VOR gain (fig 4). PAN, was not observed in naIve goldfish rotated a lower frequency (0.01 Hz) where VOR gain fell by 22 times due to habituation (Dow and Anastasio, 1997). A Model of Periodic Alternating Nystagmus 141 4 MODEL Previously, a non-linear limit cycle model was constructed by Leigh, Robinson, and Zee (1981; see also Furman, 1989) to simulate PAN in humans. This model included a velocity storage loop with saturation, and a central adaptation loop. This second order system would spontaneously oscillate, producing PAN, if the gain of the velocity storage loop was greater than 1. We adjusted Robinson's model to simulate rotation inducible PAN and habituation in the goldfish. Input to and output from the model (fig 2) represent head and slow-phase eye velocity, respectively. The time constants of the canal (S'tc/(S'tc+ 1? and velocity-storage (g.l(S'ts+l? elements were set to the value of the canal time constant as determined experimentally in goldfish ('Cc = '1:s = 3 s) (Hartman and Klinke, 1980). The time constant of the central adaptation element (l/S'ta) was 10 times longer ('1: a = 30 s). The Laplace variable (s) is complex frequency (s = jro where / is -1 and ro is frequency in rad/s). The gain of the velocity-storage loop (gs) is 1.05 while that of the central adaptation loop (ga) is 1. The central adaptation loop represents in part a negative feedback loop onto vestibular nucleus neurons through inhibitory Purkinje cells of the NU. The vestibulo-cerebellum is known to modulate the gain of the VOR (Wilson and Melvill Jones, 1979). The static nonlinearity in the velocity storage loop consists of a threshold (? 0.0225) and a saturation (? 1.25). The threshold was added to model the decay in PAN following termination of rotation (Dow and Anastasio, 1997), which is not modeled here. Increases or decreases in the absolute value of ga will cause VOR habituation or PAN, respectively. However, it was more common for VOR habituation and PAN to occur simultaneously (fig tB). This behavior could not be reproduced with the lumped model (fig 2). It would be necessary on one hand to increase ga to decrease overall VOR gain while, on the other hand, decrease ga to produce PAN. A distributed system would address this problem, with multiple parallel pathways, each having velocity-storage and adaptive control through the NU. The idea can head velocity be illustrated using the simplest distributed syscanal tem which has 2 lumped models in parallel (not transfer shown), each having an independently adjustable function L...--'r---' gao The results from such a two parallel pathway model are shown in fig 3. In one pathway, ga(h) was increased to model habituation, and gio) was decreased to start oscillations. Paradoxically, although the ultimate effect of increasing ga(h) is to decrease VOR gain, the initial effect as ga(h) is increased is to increase gain. This is due to the resonant frequency of the system concentral tinuously shifting to higher frequencies and temadaptation porarily matching the frequency of rotation (see eye velocity DISCUSSION). Conversely, when ga(o) is deFigure 2: Model used creased, there is a temporary decrease in gain as to simulation PAN the resonant frequency moves away from the (Dow and Anastasio frequency of rotation. The two results are com1997). Used with bined after the gain is reduced by half (fig 3B). permission. E. R Dow and T. 1. Anastasio 142 The combined result shows a continual decrease in VOR gain with the oscillations superimposed. 5 DISCUSSION If the nonlinearities (i.e. threshold and saturation) in the model are ignored, linear analysis shows that the model will be unstable when [(1 - gs)/ts + gJ'tal is negative, and will oscillate with a period of [2m1(tstJga)]. With the initial Q) parameters, the model is stable 8because the central adaptation ~ loop can compensate for the unstable gain of the velocity storage loop. The natural frequency of the system, calculated from the Q) -2L---------------~------~--------~---above equation, is 0.017 Hz. This ~ resonance, which peaks at the al 4 :::l . . ?. resonant frequency but is stiII > ..... < .. ... pronounced at nearby frequencies, produces an enhancement of the c / _:_ :_ : _ _ _ _ _ _ _ _ _ _........_ _ VOR response. The hypothesis I:: 0 L_ _ _ _--=::. that low frequency VOR gain eno 1000 2000 3000 hancement is produced by a potime (sec) tentially unstable resonance is a Figure 3: Model at 0.03 Hz. Two novel feature of our model. The simulations with differing values of natural frequency increases with ga (A) are combined in (B) with the increases in ga and can alter the values of ga in (C). frequency specific enhancement. Decreases in ga, in addition to decreasing the natural frequency, also cause the model to become unstable. (If ga is reduced to zero, the model becomes first order and the equations are no longer valid). The ability to get either habituation or PAN by varying only one parameter suggests that habituation and PAN are a related phenomena 2[--- B il=--~ ~2 k .. Through the process of habituation, prolonged low frequency rotation (0.01 Hz) in goldfish severely decreased VOR gain, often abruptly and unilaterally (Dow and Anastasio, in press). The decrease in gain due to habituation can effectively eliminate PAN at the lowest frequency at which PAN was observed (0.03 Hz) as shown in fig 4. In this example the naive VOR responds symmetrically for the first cycle of rotation. It then becomes markedly asymmetrical, with a strong, unilateral response in one direction for -10 cycles followed by another in the opposite direction for -17 cycles. The VOR response abruptly habituates after that with no PAN. Complete habituation can be simulated by further increases in the value of ga' in the limit-cycle model (fig 2). Unilateral habituation has been simulated previously with a bilateral network model of the VOR in which the cerebellum inhibits the vestibular nuclei unilaterally (Dow and Anastasio, in press). A Model of Periodic Alternating Nystagmus 143 60 40 ~ Q) --- 20 "0 ~ ?0 o 0 .l bJ ~ A ~tLL r~ I'~ "'1 CD > ~-20 Q) -40 o 200 400 600 time (5) 800 Figure 4: PAN superimposed on the VOR response to continuous rotation at 0.03 Hz. Upper trace, slow-phase eye velocity (fast phases removed); lower trace, head velocity (not to scale). The cerebellum has several circuits which could provide an increase in firing rate of some Purkinje cells with a concurrent decrease in the firing rate of other Purkinje cells suggested by the model. There are many lateral inhibitory pathways including inhibition of Purkinje cells to neighboring Purkinje cells (Llimis and Walton, 1990). Therefore, if one Purkinje cell were to increase its firing rate, this circuitry suggests that neighboring Purkinje cells would decrease their firing rates. Also, experimental evidence shows that during habituation, not all vestibular nuclei gradually decrease their firing rate as might be expected. Kileny and colleagues (1980) recorded from vestibular nucleus neurons during habituation. They could divide the neurons into 3 roughly equal groups based on response over time: continual decrease, constant followed by a decrease, and increase followed by decrease. The cerebellar circuitry and the single-unit recording data support multiple, variable levels of inhibition from the NU. How this mechanism may work is being explored with a more biologically realistic distributed model. 6 CONCLUSION Our experimental results are consistent with a multi-parallel pathway model of the VOR. In each pathway an unstable, positive velocity-storage loop is stabilized by an inhibitory, central adaptation loop, and their interaction produces a low-frequency resonance that enhances the low-frequency response of the VOR. Prolonged rotation at specific frequencies could produce a decrease in central adaptation VOR gain in some pathways resulting in an unstable, low-frequency oscillation resembling PAN in these pathways. An 144 E. R. Dow and T. J. Anastasio increase in adaptation loop gain in the other pathways would result in a decrease in VOR gain resembling habituation. The sum over the VOR pathways would show PAN and habituation occurring together. We suggest that resonance enhancement and mUltiple parallel (i.e. distributed) pathways are necessary to model the interrelationship between PAN and habituation. Acknowledgments The work was supported by grant MH50577 from the National Institutes of Health. We thank M. Zelaya and X. Feng for experimental assistance. References Angelaki DE and Hess BJM. J Neurophysl73 1729-1751 (1995). Baloh RW, Honrubia V and Konrad HR. Brain 99 11-26 (1976). Dow ER and Anastasio TJ. NeuroReport 71305-1309 (1996). Dow ER and Anastasio TJ. NeuroReport 82755-2759 (1997). Dow ER and Anastasio TJ. J. Computat. Neuro. in press. Dow RS. J Comp Neurol63 527-548 (1936). Dow RS. J Comp Neurol68 297-305 (1938). Fernandez C and Fredrickson lM. Acta Otolaryngol Suppl192 52-62 (1964). Furman JMR, Wall C and Pang D. Brain 113 1425-1439 (1990). Furman lMR, Hain TC and Paige GO. Biol Cybern 61255-264 (1989). Hartmann R and Klinke R. Pflugers Archiv 388 111-121 (1980). Kileny P, Ryu JH, McCabe BF and Abbas PJ. Acta Otolaryngol90 175-183 (1980). Leigh RJ, Robinson DA and Zee OS. Ann NY A cad Sci 374619-635 (1981). LIinas RR and Walton KD. Cerebellum. In: Shepherd GM ed. The Synaptic Organization o/the Brain. Oxford: Oxford University Press, 1990: 214-245. Remmel RS. IEEE Trans Biomed Eng 31 388-390 (1984). Robinson DA. IEEE Trans Biomed Eng 10 137-145 (1963). Robinson DA. Exp Brain Res 30447-450 (1977). Robinson DA. Ann Rev Neurosci 4463-503 (1981). Singleton GT. Laryngoscope 77 1579-1620 (1967). Torte MP, Courjon JH, Flandrin JM, et al. Exp Brain Res 99 441-454 (1994). Waespe W, Cohen Band Raphan T. Science 228 199-202 (1985). Wilson V and Melvill Jones G. Mammalian Vestibular Physiology, New York: Plenum Press, 1979. 

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Boltzmann Machine learning using mean field theory and linear response correction H.J. Kappen Department of Biophysics University of Nijmegen, Geert Grooteplein 21 NL 6525 EZ Nijmegen, The Netherlands F. B. Rodriguez Instituto de Ingenieria del Conocimiento & Departamento de Ingenieria Informatica. Universidad Aut6noma de Madrid, Canto Blanco,28049 Madrid, Spain Abstract We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons. We conclude that 1) the method works well for paramagnetic problems 2) the TAP correction gives a significant improvement over the Weiss mean field theory, both for paramagnetic and spin glass problems and 3) that the inclusion of diagonal weights improves the Weiss approximation for paramagnetic problems , but not for spin glass problems. 1 Introduction Boltzmann Machines (BMs) [1], are networks of binary neurons with a stochastic neuron dynamics, known as Glauber dynamics. Assuming symmetric connections between neurons, the probability distribution over neuron states s will become stationary and is given by the Boltzmann-Gibbs distribution P( S). The Boltzmann distribution is a known function of the weights and thresholds of the network. However, computation of P(Sj or any statistics involving P(S), such as mean firing rates or correlations, requires exponential time in the number of neurons. This is Boltzmann Machine Learning Using Mean Field Theory 281 due to the fact that P(S) contains a normalization term Z, which involves a sum over all states in the network, of which there are exponentially many. This problem is particularly important for BM learning. Using statistical sampling techiques [2], learning can be significantly improved [1]. However, the method has rather poor convergence and can only be applied to small networks. In [3 , 4], an acceleration method for learning in BMs is proposed using mean field theory by replacing (SjSj) by mimj in the learning rule. It can be shown [5] that such a naive mean field approximation of the learning rules does not converge in general. Furthermore, we argue that the correlations can be computed using the linear response theorem [6]. In [7, 5] the mean field approximation is derived by making use of the properties of convex functions (Jensen's inequality and tangential bounds). In this paper we present an alternative derivation which uses a Legendre transformation and a small coupling expansion [8] . It has the advantage that higher order contributions (TAP and higher) can be computed in a systematic manner and that it may be applicable to arbitrary graphical models. 2 Boltzmann Machine learning The Boltzmann Machine is defined as follows. The possible configurations of the network can be characterized by a vector s = (S1, .. , Si, .. , sn), where Si = ?1 is the state of the neuron i, and n the total number of the neurons. Neurons are updated using Glauber dynamics. Let us define the energy of a configuration -E(S) = s as 1 2L WijSiSj + i,j L SiOi. i After long times, the probability to find the network in a state s becomes independent of time (thermal equilibrium) and is given by the Boltzmann distribution 1 p(S) = Z exp{ -E(S)}. (1) Z = L; exp{ - E( S)} is the partition function which normalizes the probability distribution. Learning [1] consists of adjusting the weights and thresholds in such a way that the Boltzmann distribution approximates a target distribution q(S) as closely as possible. A suitable measure of the difference between the distributions p(S) and q(S) is the K ullback divergence [9] = J{ ~ q(S) log ;~~. (2) s Learning consists of minimizing ~Wij J{ = 17( (SiSj)c - using gradient descent [1] (SiSj) ), ~()i = 17( (Si)c - (Si) ). The parameter 17 is the learning rate. The brackets (-) and (-) c denote the 'free' and 'clamped' expectation values, respectively. H. 1. Kappen and F. B. Rodr(guez 282 The computation of both the free and the clamped expectation values is intractible, because it consists of a sum over all unclamped states. As a result, the BM learning algorithm can not be applied to practical problems. 3 The mean field approximation We derive the mean field free energy using the small, expansion as introduced by Plefka [8]. The energy of the network is given by E(s, w, h, ,) for, = 1. The free energy is given by F(w, (), ,) = -logTr s e- E (s ,w,8,),) and is a function of the independent variables Wij, ()j and ,. We perform a Legendre transformation on the variables (}i by introducing mj = - ~:.. The Gibbs free energy G(w, m, ,) = F(w, (), ,) + L ()jmj is now a function of the independent variables mj and Wij, and ()i is implicitly given by (Si) )' = mi. The expectation 0" is with respect to the full model with interaction , . We expand = G(O) + ,G' (0) + ~,2G"(O) + 0(,3) G(,) We directly obtain from [8] G'(,) (Eint)" GUb) (E;n')~ - {Ei:,,), + (E;n,;;= ~ (s, - For, = 0 the expectation values can directly compute: G(O) 0 )' m;)), become the mean field expectations which we 1 21 ~ ( (1 + mi) log 2(1 + mi) + (1 - ) md log 1 2(1- md J G'(O) -~ L Wjjmjmj ij G"(O) Thus G(l) 1 -L 2 . ( 1 1md ) log-(l + md + (1- mdlog-(l(1 + md 2 I "'w . -~2 L- IJ 2 m I?m?J ij -~ L ij w;j(l - m;)(l- m]) + 0(w 3 f(m)) (3) Boltvnann Machine Learning Using Mean Field Theory 283 where f(m) is some unknown function of m. The mean field equations are given by the inverse Legendre transformation e. - ami ae _ - tan h - 1 ( mi ) 1 - ""' L- Wijmj 2 1+ ""' ~ Wjjmd J m j2 ), (4) J which we recognize as the mean field equations. The correlations are given by a 2F (SiSj) - (Si) (Sj) ami = - oeiooj = oej = ( ao ) -1 am ij ( 02~) -1 am ij We therefore obtain from Eq. 3 (Si Sj ) - with (A-')oj = Jij ( 1 _1 ml (Si) (s j) = Aij + ~ wi.(1 - ml )) - Wij - 2mimjW;j (5) Thus, for given Wij and OJ, we obtain the approximate mean firing rates mj by solving Eqs . 4 and the correlations by their linear response approximations Eqs. 5. The inclusion of hidden units is straigthforward . One applies the above approximations in the free and the clamped phase separately [5]. The complexity of the method is O(n 3 ), due to the matrix inversion. 4 Learning without hidden units We will assess the accuracy of the above method for networks without hidden units. Let us define Cij = (SjSj)c - (Si)c (Sj)c' which can be directly computed from the data. The fixed point equation for D..Oj gives D..Oi = 0 {:} mj = (Si)c . (6) The fixed point equation for D..wij, using Eq. 6, gives D..wij 0 {:} Aij Cij ' i =F j. (7) From Eq. 7 and Eq. 5 we can solve for Wij, using a standard least squares method. In our case, we used fsolve from Matlab. Subsequently, we obtain ei from Eq. 4. We refer to this method as the TAP approximation. = = In order to assess the effect of the TAP term, we also computed the weights and thresholds in the same way as described above, but without the terms of order w 2 in Eqs. 5 and 4. Since this is the standard Weiss mean field expression, we refer to this method as the Weiss approximation. The fixed point equations are only imposed for the off-diagonal elements of D..Wjj because the Boltzmann distribution Eq. 1 does not depend on the diagonal elements Wij. In [5], we explored a variant of the Weiss approximation, where we included diagonal weight terms . As is discussed there, if we were to impose Eq. 7 for i = j as well, we have A = C. HC is invertible, we therefore have A-I = C- 1 . However, we now have more constraints than variables. Therefore, we introduce diagonal weights Wii by adding the term Wiimi to the righthandside of Eq. 4 in the Weiss approximation. Thus, Wij ,sij = 1 _ m? _ (C- 1 ) .. lJ I and OJ is given by Eq. 4 in the Weiss approximation. Clearly, this method is computationally simpler because it gives an explicit expression for the solution of the weights involving only one matrix inversion. 284 5 H. 1. Kappen and F. B. Rodr(guez Numerical results For the target distribution q(s) in Eq. 2 we chose a fully connected Ising spin glass model with equilibrium distribution with lij i.i.d. Gaussian variables with mean n~l and variance /~1 ' This model is known as the Sherrington-Kirkpatrick (SK) model [10]. Depending on the values of 1 and 1 0 , the model displays a para-magnetic (unordered), ferro-magnetic (ordered) and a spin-glass (frustrated) phase. For 10 = 0, the para-magnetic (spinglass) phase is obtained for 1 < 1 (1 > 1). We will assess the effectiveness of our approximations for finite n, for 10 = 0 and for various values of 1. Since this is a realizable task, the optimal KL divergence is zero, which is indeed observed in our simulations. We measure the quality of the solutions by means ofthe Kullback divergence. Therefore, this comparison is only feasible for small networks . The reason is that the computation of the Kullback divergence requires the computation of the Boltzmann distribution, Eq. 1, which requires exponential time due to the partition function Z. We present results for a network of n = 10 neurons . For 10 = 0, we generated for each value of 0.1 < 1 < 3, 10 random weight matrices l i j. For each weight matrix, we computed the q(S) on all 2n states. For each of the 10 problems, we applied the TAP method, the Weiss method and the Weiss method with diagonal weights. In addition, we applied the exact Boltzmann Machine learning algorithm using conjugate gradient descent and verified that it gives KL diver?ence equal to zero, as it should. We also applied a factorized model p(S) = Ili ?"(1 + misd with mi = (Si)c to assess the importance of correlations in the target distribution. In Fig. la, we show for each 1 the average KL divergence over the 10 problem instances as a function of 1 for the TAP method, the Weiss method, the Weiss method with diagonal weights and the factorized model. We observe that the TAP method gives the best results, but that its performance deteriorates in the spin-glass phase (1) 1) . The behaviour of all approximate methods is highly dependent on the individual problem instance. In Fig. Ib, we show the mean value of the KL divergence of the TAP solution, together with the minimum and maximum values obtained on the 10 problem instances. Despite these large fluctuations , the quality of the TAP solution is consistently better than the Weiss solution. In Fig. lc, we plot the difference between the TAP and Weiss solution, averaged over the 10 problem instances. In [5] we concluded that the Weiss solution with diagonal weights is better than the standard Weiss solution when learning a finite number of randomly generated patterns. In Fig. Id we plot the difference between the Weiss solution with and without diagonal weights. We observe again that the inclusion of diagonal weights leads to better results in the paramagnetic phase (1 < 1), but leads to worse results in the spin-glass phase. For 1 > 2, we encountered problem instances for which either the matrix C is not invertible or the KL divergence is infinite. This problem becomes more and more severe for increasing 1 . We therefore have not presented results for the Weiss approximation with diagonal weigths for 1 > 2. BoltvnannMachine Learning Using Mean Field Theory 285 Comparison mean values TAP 5r-------~------~------. 4 2 fact weiss+d weiss tap Q) 1.5 u c: mean min max Q) e> Q) 1 I > / ", .... '" 1_'" '5 ....J :.:: -- 0.5 J..:y '", o~ o .... ; ./ __~...=?~J~______~____~ 2 o~-=~~------~----~ o 3 2 J 3 J Difference WEISS and TAP Difference WEISS+D and WEISS 1.5 en ':!? n. .1 0.5 w J :.:: I :.:: I en en J V 0.5 J :.:: ~ _ _ _ _~~_ _ _ _----.J o 2 J 3 0 -0.5 v / W 0 -0.5 L-._ _ _ _ _ _ '? en en W :.:: 1 ~ 1 o 2 3 J Figure 1: Mean field learning of paramagnetic (J < 1) and spin glass (J > 1) problems for a network of 10 neurons. a) Comparison of mean KL divergences for the factorized model (fact), the Weiss mean field approximation with and without diagonal weights (weiss+d and weiss), and the TAP approximation, as a function of J. The exact method yields zero KL divergence for all J. b) The mean, minimum and maximum KL divergence of the TAP approximation for the 10 problem instances , as a function of J. c) The mean difference between the KL divergence for the Weiss approximation and the TAP approximation, as a function of J. d) The mean difference between the KL divergence for the Weiss approximation with and without diagonal weights, as a function of J . 6 Discussion We have presented a derivation of mean field theory and the linear response correction based on a small coupling expansion of the Gibbs free energy. This expansion can in principle be computed to arbitrary order. However , one should expect that the solution of the resulting mean field and linear response equations will become more and more difficult to solve numerically. The small coupling expansion should be applicable to other network models such as the sigmoid belief network , Potts networks and higher order Boltzmann Machines . The numerical results show that the method is applicable to paramagnetic problems. This is intuitively clear, since paramagnetic problems have a unimodal probability distribution, which can be approximated by a mean and correlations around the mean. The method performs worse for spin glass problems. However , it still gives a useful approximation of the correlations when compared to the factorized model which ignores all correlations. In this regime, the TAP approximation improves 286 H. 1. Kappen and F. B. Rodr(guez significantly on the Weiss approximation. One may therefore hope that higher order approximation may further improve the method for spin glass problems. Therefore. we cannot conclude at this point whether mean field methods are restricted to unimodal distributions. In order to further investigate this issue, one should also study the ferromagnetic case (Jo > 1, J > 1), which is multimodal as well but less challenging than the spin glass case. It is interesting to note that the performance of the exact method is absolutely insensitive to the value of J. Naively, one might have thought that for highly multi-modal target distributions, any gradient based learning method will suffer from local minima. Apparently, this is not the case: the exact KL divergence has just one minimum, but the mean field approximations of the gradients may have multiple solutions. Acknowledgement This research is supported by the Technology Foundation STW, applied science division of NWO and the techology programme of the Ministry of Economic Affairs. References [1] D. Ackley, G. Hinton, and T. Sejnowski. A learning algorithm for Boltzmann Machines. Cognitive Science, 9:147-169, 1985. [2] C. Itzykson and J-M. Drouffe. Statistical Field Theory. Cambridge monographs on mathematical physics. Cambridge University Press. Cambridge, UK, 1989. [3] C. Peterson and J.R. Anderson. A mean field theory learning algorithm for neural networks. Complex Systems, 1:995-1019, 1987. [4] G.E. Hinton. Deterministic Boltzmann learning performs steepest descent in weightspace. Neural Computation, 1:143-150. 1989. [5] H.J. Kappen and F.B. Rodriguez. Efficient learning in Boltzmann Machines using linear response theory. Neural Computation, 1997. In press. [6] G. Parisi. Statistical Field Theory. Frontiers in Physics. Addison-Wesley, 1988. [7] L.K. Saul, T. Jaakkola, and M.1. Jordan. Mean field theory for sigmoid belief networks. Journal of artificial intelligence research, 4:61-76, 1996. [8] T. Plefka. Convergence condition of the TAP equation for the infinite-range Ising spin glass model. Journal of Physics A, 15:1971-1978, 1982. [9] S. Kullback. Information Theory and Statistics. Wiley, New York, 1959. [10] D. Sherrington and S. Kirkpatrick. Solvable model of Spin-Glass. letters, 35:1792-1796, 1975. Physical review 

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Selecting weighting factors in logarithmic opinion pools Tom Heskes Foundation for Neural Networks, University of Nijmegen Geert Grooteplein 21, 6525 EZ Nijmegen, The Netherlands tom@mbfys.kun.nl Abstract A simple linear averaging of the outputs of several networks as e.g. in bagging [3], seems to follow naturally from a bias/variance decomposition of the sum-squared error. The sum-squared error of the average model is a quadratic function of the weighting factors assigned to the networks in the ensemble [7], suggesting a quadratic programming algorithm for finding the "optimal" weighting factors. If we interpret the output of a network as a probability statement, the sum-squared error corresponds to minus the loglikelihood or the Kullback-Leibler divergence, and linear averaging of the outputs to logarithmic averaging of the probability statements: the logarithmic opinion pool. The crux of this paper is that this whole story about model averaging, bias/variance decompositions, and quadratic programming to find the optimal weighting factors, is not specific for the sumsquared error, but applies to the combination of probability statements of any kind in a logarithmic opinion pool, as long as the Kullback-Leibler divergence plays the role of the error measure. As examples we treat model averaging for classification models under a cross-entropy error measure and models for estimating variances. 1 INTRODUCTION In many simulation studies it has been shown that combining the outputs of several trained neural networks yields better results than relying on a single model. For regression problems, the most obvious combination seems to be a simple linear Selecting Weighting Factors in Logarithmic Opinion Pools 267 averaging of the network outputs. From a bias/variance decomposition of the sumsquared error it follows that the error of the so obtained average model is always smaller or equal than the average error of the individual models. In [7] simple linear averaging is generalized to weighted linear averaging, with different weighting factors for the different networks in the ensemble . A slightly more involved bias/variance decomposition suggests a rather straightforward procedure for finding "optimal" weighting factors. Minimizing the sum-squared error is equivalent to maximizing the loglikelihood of the training data under the assumption that a network output can be interpreted as an estimate of the mean of a Gaussian distribution with fixed variance. In these probabilistic terms, a linear averaging of network outputs corresponds to a logarithmic rather than linear averaging of probability statements. In this paper, we generalize the regression case to the combination of probability statements of any kind. Using the Kullback-Leibler divergence as the error measure, we naturally arrive at the so-called logarithmic opinion pool. A bias/variance decomposition similar to the one for sum-squared error then leads to an objective method for selecting weighting factors. Selecting weighting factors in any combination of probability statements is known to be a difficult problem for which several suggestions have been made. These suggestions range from rather involved supra-Bayesian methods to simple heuristics (see e.g. [1, 6] and references therein). The method that follows from our analysis is probably somewhere in the middle: easier to compute than the supra-Bayesian methods and more elegant than simple heuristics. To stress the generality of our results, the presentation in the next section will be rather formal. Some examples will be given in Section 3. Section 4 discusses how the theory can be transformed into a practical procedure. 2 LOGARITHMIC OPINION POOLS Let us consider the general problem of building a probability model of a variable y given a particular input x. The "output" y may be continuous, as for example in regression analysis, or discrete, as for example in classification. In the latter case integrals over y should be replaced by summations over all possible values of y. Both x and y may be vectors of several elements; the one-dimensional notation is chosen for convenience. We suppose that there is a "true" conditional probability model q(ylx) and have a whole ensemble (also called pool or committee) of experts, each supplying a probability model Pa(Ylx). p{x) is the unconditional probability distribution of inputs. An unsupervised scenario, as for example treated in [8], is obtained if we simply neglect the inputs x or consider them constant. We define the distance between the true probability q(ylx) and an estimate p(ylx) to be the Kullback-Leibler divergence , J\. (q, p) == - J J dx p(x) dy q(ylx)log [p(Y1x)] q(ylx) If the densities p(x) and q(ylx) correspond to a data set containing a finite number p of combinations {xlJ , ylJ}, minus the Kullback divergence is, up to an irrelevant T. Heskes 268 constant, equivalent to the loglikelihood defined as L(p, {i, Y}) == ~ L logp(y~ Ix~) . ~ The more formal use of the Kullback-Leibler divergence instead of the loglikelihood is convenient in the derivations that follow. Weighting factors Wa are introduced to indicate the reliability of each ofthe experts fr. In the following we will work with the constraints La Wa = 1, which is used in some of the proofs, and Wa ? 0 for all experts fr, which is not strictly necessary, but makes it easier to interpret the weighting factors and helps to prevent overfitting when weighting factors are optimized (see details below). We define the average model j)(y/x) to be the one that is closest to the given set of models: j)(y/x) == argmin waJ{(P,Pa) . p(ylx) L a Introducing a Lagrange mUltiplier for the constraint find the solution J dxp(y/x) = 1, we immediately (1) with normalization constant Z(x) = J dy II[Pa(Y/X)]W Q ? (2) a This is the logarithmic opinion pool, to be contrasted with the linear opinion pool, which is a linear average'ofthe probabilities. In fact, logarithmic opinion pools have been proposed to overcome some of the weaknesses of the linear opinion pool. For example, the logarithmic opinion pool is "externally Bayesian" , i.e., can be derived from joint probabilities using Bayes' rule [2]. A drawback of the logarithmic opinion pool is that if any of the experts assigns probability zero to a particular outcome, the complete pool assigns probability zero, no matter what the other experts claim. This property of the logarithmic opinion pool, however, is only a drawback if the individual density functions are not carefully estimated. The main problem for both linear and logarithmic opinion pools is how to choose the weighting factors Wa. The Kullback-Leibler divergence of the opinion pool p(y/x) can be decomposed into a term containing the Kullback-Leibler divergences of individual models and an "ambiguity" term: (3) Proof: The first term in (3) follows immediately from the numerator in (1), the second term is minus the logarithm of the normalization constant Z (x) in (2) which can, using (1), be rewritten as Selecting Weighting Factors in Logarithmic Opinion Pools 269 for any choice of y' for which p(y'lx) is nonzero. Integration over y' with probability measure p(y'lx) then yields (3) . Since the ambiguity A is always larger than or equal to zero, we conclude that the Kullback-Leibler divergence of the logarithmic opinion pool is never larger than the average Kullback-Leibler divergences of individual experts. The larger the ambiguity, the larger the benefit of combining the experts' probability assessments. Note that by using Jensen's inequality, it is also possible to show that the Kullback-Leibler divergence of the linear opinion pool is smaller or equal to the average KullbackLeibler divergences of individual experts. The expression for the ambiguity, defined as the difference between these two, is much more involved and more difficult to interpret (see e.g. [10]). The ambiguity of the logarithmic opinion pool depends on the weighting factors Wa , not only directly as expressed in (3), but also through p(ylx). We can make this dependency somewhat more explicit by writing A = ~ ~ waw/3K(Pa ,P/3) + ~ ~ Wa [K(P,Pa) a/3 K(Pa,p)] . (4) a = Proof: Equation (3) is valid for any choice of q(ylx). Substitute q(ylx) p/3(ylx), multiply left- and righthand side by w/3, and sum over {3. Simple manipulation of terms than yields the result. Alas, the Kullback-Leibler divergence is not necessarily symmetric, i.e., in general K (PI, P2) # K (P2, pd . However, the difference K (PI, P2) - K (p2, pd is an order of magnitude smaller than the divergence K(PI,P2) itself. More formally, writing PI (ylx) = [1 +?(ylx )]p2(ylx) with ?(ylx) small, we can easily show that K (PI, P2) is of order (some integral over) ?2(ylx) whereas K(PI,P2) - K(p2,pd is of order ?3(ylx). Therefore, if we have reason to assume that the different models are reasonably close together, we can, in a first approximation, and will, to make things tractable, neglect the second term in (4) to arrive at K(q,p) ~ ~ waK(q,Pa) - ~ L a wa w/3 [K(Pa,P/3) + K(P/3,Pa)] . The righthand side of this expression is quadratic in the weighting factors property which will be very convenient later on. 3 (5) a ,/3 Wa , a EXAMPLES Regression. The usual assumption in regression analysis is that the output functionally depends on the input x, but is blurred by Gaussian noise with standard deviation (j. In other words, the probability model of an expert a can be written Pa(ylx) = 1 [-(y - fa(x))2] Vj~ exp 2(j2 . (6) The function fa(x) corresponds to the network's estimate of the "true" regression given input x. The logarithmic opinion pool (1) also leads to a normal distribution with the same standard deviation (j and with regression estimate T. Heskes 270 In this case the Kullback-Leibler divergence is symmetric, which makes (5) exact instead of an approximation. In [7], this has all been derived starting from a sum-squared error measure. Variance estimation. There has been some recent interest in using neural networks not only to estimate the mean of the target distribution, but also its variance (see e.g. [9] and references therein). In fact, one can use the probability density (6) with input-dependent <7( x). We will consider the simpler situation in which an input-dependent model is fitted to residuals y, after a regression model has been fitted to estimate the mean (see also [5]). The probability model of expert 0' can be written _ ~ [za(x)y2] Pa (YjX ) ~ exp 2 ' V where l/za(x) is the experts' estimate of the residual variance given input x. The logarithmic opinion pool is of the same form with za(x) replaced by z(x) = L waza(x) . Here the Kullback-Leibler divergence " - I\ (P,Pa) = 21 J [ z(x) z(x)] dx p(x) za(x) -log za(x) - 1 is asymmetric. We can use (3) to write the Kullback-Leibler divergence of the opinion pool explicitly in terms of the weighting factors Wa. The approximation (5), with ? Ii(Pa,PP) ? - 1 + Ii(Pp,Pa) -"2 J dx p(x) [za(x) - zp(x)]2 za(x)zp(x) , is much more appealing and easier to handle. Classification. In a two-class classification problem, we can treat y as a discrete variable having two possible realizations, e.g., y E {-I, I}. A convenient representation for a properly normalized probability distribution is 1 Pa(yjX) = 1 + exp[-2h a (x)y] . In this logistic representation, the logarithmic opinion pool has the same form with The Kullback-Leibler divergence is asymmetric, but yields the simpler form to be used in the approximation (5). For a finite set of patterns, minus the loglikelihood yields the well-known cross-entropy error. Selecting Weighting Factors in Logarithmic Opinion Pools 271 The probability models in these three examples are part of the exponential family. The mean f 0:, inverse variance zo:, and logit ho: are the canonical parameters. It is straightforward to show that, with constant dispersion across the various experts, the canonical parameter of the logarithmic opinion pool is always a weighted average of the canonical parameters of the individual experts. Slightly more complicated expressions arise when the experts are allowed to have different estimates for the dispersion or for probability models that do not belong to the exponential family. 4 SELECTING WEIGHTING FACTORS The decomposition (3) and approximation (5) suggest an objective method for selecting weighting factors in logarithmic opinion pools. We will sketch this method for an ensemble of models belonging to the same class, say feedforward neural networks with a fixed number of hidden units, where each model is optimized on a different bootstrap replicate of the available data set. Su ppose that we have available a data set consisting of P combinations {x/J, y/J }. As suggested in [3], we construct different models by training them on different bootstrap replicates of the available data set. Optimizing nonlinear models is often an unstable process: small differences in initial parameter settings or two almost equivalent bootstrap replicates can result in completely different models. Neural networks, for example, are notorious for local minima and plateaus in weight space where models might get stuck. Therefore, the incorporation of weighting factors, even when models are constructed using the same pro?cedure, can yield a better generalizing opinion pool. In [4] good results have been reported on several regression lin problems. Balancing clearly outperformed bagging, which corresponds to Wo: with n the number of experts, and bumping, which proposes to keep a single expert. = Each example in the available data set can be viewed as a realization of an unknown probability density characterized by p(x) and q(ylx). We would like to choose the weighting factors Wo: such as to minimize the Kullback-Leibler divergence K(q, p) of the opinion pool. If we accept the approximation (5), we can compute the optimal weighting factors once we know the individual Kullbacks K(q,po:) and the Kullbacks between different models K(po:, PI3). Of course, both q(ylx) and p(x) are unknown, and thus we have to settle for estimates. In an estimate for K(po:,PI3) we can simply replace the average over p(x) by an average over all inputs x/J observed in the data set: A similar straightforward replacement for q(ylx) in an estimate for K(q, Po:) is biased, since each expert has, at least to some extent, been overfitted on the data set. In [4] we suggest how to remove this bias for regression models minimizing sumsquared errors. Similar compensations can be found for other probability models. Having estimates for both the individual Kullback-Leibler divergences K(q,po:) and the cross terms K (Po:, PI3), we can optimize for the weighting factors Wo:. Under the constraints 2:0: Wo: = 1 and Wo: 2: 0 the approximation (5) leads to a quadratic programming problem. Without this approximation, optimizing the weighting factors becomes a nasty exercise in nonlinear programming. 272 T. Heskes The solution of the quadratic programming problem usually ends up at the edge of the unit cube with many weighting factors equal to zero. On the one hand, this is a beneficial property, since it implies that we only have to keep a relatively small number of models for later processing. On the other hand, the obtained weighting factors may depend too strongly on our estimates of the individual Kullbacks K (q, Pet). The following version prohibits this type of overfitting. Using simple statistics, we obtain a rough indication for the accuracy of our estimates K(q,pet). This we use to generate several, say on the order of 20, different samples with estimates {K (q, pI), .. . , K (q, Pn)}. For each of these samples we solve the corresponding quadratic programming problem and obtain a set of weighting factors. The final weighting factors are obtained by averaging. In the end, there are less experts with zero weighting factors, at the advantage of a more robust procedure. Acknowledgements I would like to thank David Tax, Bert Kappen, Pierre van de Laar, Wim Wiegerinck, and the anonymous referees for helpful suggestions. This research was supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs. References [1] J. Benediktsson and P. Swain. Consensus theoretic classification methods. IEEE Transactions on Systems, Man, and Cybernetics, 22:688-704, 1992. [2] R. Bordley. A multiplicative formula for aggregating probability assessments. Management Science, 28:1137-1148,1982. [3] L. Breiman. Bagging predictors. Machine Learning, 24:123-140, 1996. [4] T. Heskes. Balancing between bagging and bumping. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Eystems 9, pages 466-472, Cambridge, 1997. MIT Press. [5] T. Heskes. Practical confidence and prediction intervals. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Eystems 9, pages 176-182, Cambridge, 1997. MIT Press. [6] R. Jacobs. Methods for combining experts' probability assessments. Neural Computation, 7:867-888, 1995. [7] A. Krogh and J. Vedelsby. Neural network ensembles, cross validation, and active learning. In G. Tesauro, D. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Eystems 7, pages 231-238, Cambridge, 1995. MIT Press. [8] P. Smyth and D. Wolpert. Stacked density estimation. These proceedings, 1998. [9] P. Williams. Using neural networks to model conditional multivariate densities. Neural Computation, 8:843-854, 1996. [10] D. Wolpert. On bias plus variance. Neural Computation, 9:1211-1243, 1997. 

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Visual Navigation in a Robot using Zig-Zag Behavior M. Anthony Lewis Beckman Institute 405 N. Mathews Avenue University of Illinois Urbana, lllinois 61801 Abstract We implement a model of obstacle avoidance in flying insects on a small, monocular robot. The result is a system that is capable of rapid navigation through a dense obstacle field. The key to the system is the use of zigzag behavior to articulate the body during movement. It is shown that this behavior compensates for a parallax blind spot surrounding the focus of expansion normally found in systems without parallax behavior. The system models the cooperation of several behaviors: halteres-ocular response (similar to VOR), optomotor response, and the parallax field computation and mapping to motor system. The resulting system is neurally plausible, very simple, and should be easily hosted on aVLSI hardware. 1 INTRODUCTION Srinivasan and Zhang (1993) describe behavioral evidence for two distinct movement detecting systems in bee: (1) A direction selective pathway with low frequency response characteristics serving the optomotor response and (2) A non-direction selective movement system with higher frequency response serving functions of obstacle avoidance and the 'tunnel centering' response where the animal seeks a flight path along the centerline of a narrow corridor. Recently, this parallel movement detector view has received support from anatomical evidence in fly (Douglass and Strausfeld, 1996). We are concerned here with the implications of using non-direction selective movement detectors for tasks such as obstacle avoidance. A reasonable model of a non-direction selective pathwa:? would be that this pathway is computing the absolute value of the optic flow, i.e. s = II[x, Y111 where y are the components of the optic flow field on the retina at the point [x, y 1 . x, What is the effect of using the absolute value of the flow field and throwing away direction information? In section 2 we analyze the effect of a non-direction selective movement field. We understand from this analysis that rotational information, and the limited dynamic range of real sensors contaminates the non-direction selective field and 823 Visual Navigation in a Robot Using Zig-Zag Behavior probably prevents the use of this technique in an area around the direction heading of the observer. One technique to compensate for this 'parallax blind spot' is by periodically changing the direction of the observer. Such periodic movements are seen in insects as well as lower vertebrates and it is suggestive that these movements may compensate for this basic problem. In Section 3, we describe a robotic implementation using a crude non-direction selective movement detector based on a rectified temporal derivative of luminosity. Each ' neuron' in the model retina issues a vote to control the motors of the robot. This system, though seemingly naively simple, compares favorably with other robotic implementations that rely on the optic flow or a function of the optic flow (divergence). These techniques typically require a large degree of spatial temporal averaging and seem computationally complex. In addition, our model agrees better with with the biological evidence. Finally, the technique presented here is amenable to implementation in custom aVLSI or mixed aVLSIIdVLSI chips. Thus it should be possible to build a subminiature visually guided navigation system with several (one?) low-power simple custom chips. 2 ANALYSIS OF NON-DIRECTION SELECTIVE MOVEMENT DETECTION SYSTEM Let us assume a perspective projection (1) where A. is the focal length of the lens, X, Y, Z is the position of a point in the environment, and x, y is the projection of that point on the retinal plane. The velocity of the image of a moving point in the world can be found by differentiating (1) with respect to time: (2) If we assume that objects in the environment are fixed in relation to one-a~-other and that and relative the observer is moving with relativTtranslational velocity Cv = [vx Vy v to the environment given in frame c, a point in the rotational velocity cn = rrox roy roJ environment has relati~e v~loclty: J (3) Now substituting in (2): ~ ~l: ~Jv.+{:> -~~;' j'n. = (4) and taking the absolute value of the optic flow : r;; '~l"x- ~(Xy... +..y,x' +1)+ y",l)' +~+y+ ~(-..p' +\)+xY"y+x",l]j = where we have made the substitution: [X/2 Y/~ [(1 IlJ -t (5) (that is, the heading direction). We can see that the terms involving [rox ro vrozl cannot be separated from the x, y terms. If we assume that [rox roy roJ = 0 then we can r~arrange the equation as: M. A. Lewis 824 .-l() L> s in the case of Z translation. If ITzl _ITZI_ -1Zf- =0 ~ -1(s) = lsi AJ[(X-a)2 + (y_~)2J (6) then we have: 1 IZI = lsi AJTx 2 + Ty2 (7) this corresponds to the case of pure lateral translations. Locusts (as well as some vertebrates) use peering or side to side movements to gauge distances before jumping. We call the quantity in (6) inverse relative depth. Under the correct circumstances it is equivalent to the reciprocal of time to contact. Equation (6) can be restated as: ~-l(s) = gjS where g is a gain factor that depends on the current direction heading and the position in the retina. This gain factor can be implemented neurally as a shunting inhibition, for example. This has the following implications. If the observer is using a non-direction sensitive movement detector then (A) it must rotationally stabilize its eyes (B) it must dynamically alter the gain of this infonnation in a pathway between the retinal input and motor output or it must always have a constant direction heading and use constant gain factors. In real systems there is likely to be imperfection in rotational stabilization of the observer as well as sensors with limited dynamic range. To understand the effect of these, let us assume that there is a base-line noise level 0 and we assume that this defines a minimum threshold substituting s = 0, we can find a level curve for the minimum detectability of an object, i.e.: (8) Thus, for constant depth and for 0 independent of the spatial position on the retina, the level curve is a circle. The circle increases in radius with increasing distance, and noise, and decreases with increasing speed. The circle is centered around the direction heading. The solution to the problem of a 'parallax blind spot' is to make periodic changes of direction. This can be accomplished in an open loop fashion or, perhaps, in an image driven fashion as suggested by Sobey (1994). 3 ROBOT MODEL Figure la is a photograph of the robot model. The robot's base is a Khepera Robot. The Khepera is a small wheeled robot a little over 2" in diameter and uses differential drive motors. The robot has been fitted with a solid-state gyro attached to its body. This gyroscope senses angular velocities about the body axis and is aligned with the axis of the camera joint. A camera, capable of rotation about an axis perpendicular to the ground plane, is also attached. The camera has a field of view of about 90 0 and can swing of ?90? . The angle of the head rotation is sensed by a small potentiometer. For convenience, each visual process is implemented on a separate Workstation (SGI Indy) as a heavyweight process. Interprocess communication is via PVM distributed computing library. Using a distributed processing model, behaviors can be dynamically added and deleted facilitating analysis and debugging. 3.1 ROBOT CONTROL SYSTEM The control is divided into control modules as illustrated in Fig 2. At the top of the drawing we see a gaze stabilization pathway. This uses a gyro (imitating a halteres organ) for stabilization of rapid head movements. In addition, a visual pathway, using direction selective movement detector (DSMD) maps is used for slower optomotor response. Each of the six maps uses correlation type detectors (Borst and Egelhaaf, 1989). Each map is Visual Navigation in a Robot Using Zig-Zag Behavior 825 Figure 1. Physical setup. (A) Modified Khepera Robot with camera and gyro mounted. (B) Typical obstacle field run experiment. tuned to a different horizontal velocity (three for left image translations and three for right image translations). The lower half of the drawing shows the obstacle avoidance pathway. A crude nondirection selective movement detector is created using a simple temporal derivative. The use of this as a movement detector was motivated by the desire to eventually replace the camera front end with a Neuromorphic chip. Temporal derivative chips are readily available (Delbrtick and Mead, 1991). Next we reason that the temporal derivative gives a crude estimate of the ~bsolute value of the optic flow. For example if we expect only horizontal flows then: E xX = - E t (Horn and Shunck, 1981). Here E, is the temporal derivative of the luminosity and gA; is the spatial derivative. If we sample over a patch of the image, Ex will take on a range of values. If we take the average rectified temporal derivative over a patch then Ixl = I-E,I/IExl. Thus the average rectified temporal derivative over a patch will give a velocity proportional the absolute value of the optic flow. In order to make the change to motor commands, we use a voting scheme. Each pixel in the nondirection selective movement detector field (NDSMD) votes on a direction for the robot. The left pixels for a right turn and the right pixels vote for a left turn. The left and right votes are summed. In certain experiments described below the difference of the left and right votes was used to drive the rotation of the robot. In others a symmetry breaking scheme was used. It was observed that with an object dead ahead of the robot, often the left and right activation would have high but nearly equal activation. In the symmetry breaking scheme, the side with the lower activation was further decrease by a factor of 50%. This admittedly ad hoc solution remarkably improved the performance in the nonzig-zag case as noted below. The zig-zag behavior is implemented as a feedforward command to the motor system and is modeled as: Khepera COZigZag . = stn(cot)K Finally, a constant forward bias is added to each wheel so the robot makes constant progress. K is chosen empirically but in principle one should be able to derive it using the analysis in section 2. As described above, the gaze stabilization module has control of head rotation and the zigzag behavior and the depth from parallax behavior control the movement of the robot's body. During normal operation, the head may exceed the ?90? envelope defined by the mechanical system. This problem can be addressed in several ways among them are by making a body saccade to bring the body under the head or making a head saccade to align the head with the body. We choose the later approach solely because it seemed to work better in practice. 826 M. A. Lewis c J -k e Gyro Quick Phase Nystagmus Gaze Stabilillltion Cmd Camera 60x80image l -_ _--:-_ _ _ L-_~-:_;: a,-l(s) Motor Map f--__~ NDSMD Figure 2. ZigZag Navigation model is composed of a gaze stabilization system (top) and an obstacle avoidance system (bottom). See text. 3.2 BIOLOGICAL INSPIRATION FOR MODEL Course-grained visual pathways are modeled using inspiration from insect neurobiology. The model of depth from parallax is inspired by details given in Srinivasan & Zhang (1993) on work done in bees. Gaze stabilization using a fast channel, mediated by the halteres organs, and a slow optomotor response is inspired by a description of the blowfly Calliphora as reviewed by Hengstenberg (1991). 4 EXPERIMENTS Four types of experimental setups were used. These are illustrated in Fig 3. In setup 1 the robot must avoid a dense field of obstacles (empty soda cans). This is designed to test the basic competence of this technique. In setup 2, thin dowels are place in the robot's path. This tests the spatial resolving capability of the robot. Likewise setup 3 uses a dense obstacle field with one opening replaced by a lightly textured surface. Finally, experimental setup 4 uses a single small object (1cm black patch) and tests the distance at which the robot can 'lock-on' to a target. In this experiment, the avoidance field is sorted for a maximal element over a given threshold. A target cross is placed at this maximal element. The closest object should correspond with this maximal element. If a maximal element over a threshold is identified for a continuous 300ms and the target cross is on the correct target, the robot is stopped and its distance to the object is measured. The larger the distance, the better. 5 RESULTS The results are described briefly here. In the setup 1 without the use of symmetry breaking, the scores were ZigZag: 10 Success, 0 Failures and the non-ZigZag: 4 Success and 6 Failures. With Symmetry Breaking installed the results were: ZigZag: 49 Success, 3 Failures and the non-ZigZag: 44 Success and 9 failures. In the case palisades test: ZigZag: 22 Success, 4 Failures and the non-ZigZag: 14 Success and 11 failures. In the false opening case: ZigZag: 8 Success, 2 Failures and the non-ZigZag: 6 Success and 4 Failures. Finally, in the distance-to-Iock setup, a lock was achieved at an average distance 21.3 CM (15 data points) for zigzag and 9.6 cm (15 data points) for the non-zigzag case. Visual Navigation in a Robot Using Zig-ZAg Behavior Dense Obstacle Field A voidance ~ Palisades Test ?? ? ~o @ False Opening A voidance 827 Distance-To-Lock ? Thin Dowels lightly textured barrier 8 Figure 3. Illustrations of the four experimental setups. We tentatively conclude that zig-zag behavior should improve performance in robot and in animal navigation. 6 DISCUSSION In addition to the robotic implementation presented here, there have been many other techniques presented in the literature. Most relevant is Sobey (1994) who uses a zigzag behavior for obstacle avoidance. In this work, optic flow is computed through a process of discrete movements where 16 frames are collected, the robot stops, and the stored frames are analyzed for optic flow. The basic strategy is very clever: Always choose the next move in the direction of an identified object. The reasoning is that since we know the distance to the object in this direction, we can confidently move toward the object, stopping before collision. The basic idea of using zig-zag behavior is similar except that the zig-zagging is driven by perceptual input. In addition, the implementation requires estimation of the flow field requiring smoothing over numerous images. Finally, Sobey uses Optic Flow and we use the absolute value of the Optic Flow as suggested by biology. Franceschini et al (1992) reports an analog implementation that uses elementary movement detectors. A unique feature is the non-uniform sampling and the use of three separate arrays. One array uses a sampling around the circumference. The other two sampling systems are mounted laterally on the robot and concentrate in the 'blind-spot' immediately in front of the robot. It is not clear that the strategy of using three sensor arrays, spatially separated, and direction selective movement detectors is in accord with the biological constraints. Santos-Victor et al (1995) reports a system using optic flow and having lateral facing cameras. Here the authors were reproducing the centering reflex and did not focus on avoiding obstacles in front of the robot. Coombs and Roberts (1992,1993) use a similar technique. Weber et al (1996) describe wall following and stopping in front of an obstacle using an optic flow measure. Finally, a number of papers report the use of flow field divergence, apparently first suggested by Nelson and Aloimonos (1989). This requires the computation of higher derivatives and requires significant smoothing. Even in this case, there is a problem of a 'parallax hole.' See Fig. 3 of that article, for example. In any case they did not implement their idea on a mobile robot. However, this approach has been followed up with an implementation in a robot by Camus et al (1996) reporting good results. The system described here presents a physical model of insect like behavior integrated on a small robotic platform. Using results derived from an analysis of optic flow, we concluded that a zig-zag behavior in the robot would allow it to detect obstacles in front of the robot by periodically articulating the blind spot. The complexity of the observed behavior and the simplicity of the control is striking. The robot is able to navigate through a field of obstacles, always searching out a freeway for movement. 828 M. A. Lewis The integrated behavior outlined here should be a good candidate for a neuromorphic implementation. A multichip (or single chip?) system could be envisioned using a relatively simple non-directional 2-d movement detector. Two arrays of perpendicular I-d array of movement detectors should be sufficient for the optomotor response. This information could then be mapped to a circuit comprised of a few op-amp adder circuits and then sent to the head and body motors. Even the hal teres organ could be simulated with a silicon fabricated gyroscope. The results would be an extremely compact robot capable of autonomous, visually guided navigation. Finally, from our analysis of optic flow, we can make a reasonable prediction abuut the neural wiring in flying insects. The estimated depth of objects in the environment depends on where the object falls on the optic array as well as the ratio of translation to forward movement. Thus a bee or a fly should probably modulate its incoming visual signal to account for this time varying interpretation of the scene. We would predict that there should be motor information, related to the ratio of forward to lateral velocities would be projected to the non-directional selective motion detector array. This would allow a valid time varying interpretation of the scene in a zig-zagging animal. Acknowledgments The author acknowledges the useful critique of this work by Narendra Ahuja, Mark Nelson, John Hart and Lucia Simo. Special thanks to Garrick Kremesic and Barry Stout who assisted with the experimental setup and the modification of the Khepera. The author acknowledges the support of ONR grant NOOOI49610657. The author also acknowledges the loan of the Khepera from UCLA (NSF grant CDA-9303148). References A. Borst and M. Egelhaaf (1989), Principles of Visual Motion Detection, Trends in Neurosciences, 12(8):297-306 T. Camus, D. Coombs, M. Herman, and T.-H. Hong (1996), "Real-time Single-Workstation Obstacle Avoidance Using Only Wide-Field Flow Divergence", Proceedings of the 13th International Conference on Pattern Recognition. pp. 323-30 vol.3 D. Coombs and K. Roberts (1992), '''Bee-Bot': Using Peripheral Optical Flow to A void Obstacles" SPIE Vol 1825, Intelligent Robots and Computer Vision XI, pp 714-721. I D. Coombs and K. Roberts (1993), "Centering behavior using peripheral vision", Proc. 1993 IEEE Computer Society Conf. CVPR pp. 440-5, 16 refs. 1993 T. Delbriick and C. A. Mead (1991), Time-derivative adaptive silicon photoreceptor array. Proc. SPIE - Int. Soc. Opt. Eng. (USA). vol 1541, pp. 92-9. J. K. Douglass and N. 1. Strausfeld (1996), Visual Motion-Detection Circuits in Flies: Parallel Direction- and Non-Direction-Sensitive Pathways between the Medulla and Lobula Plate, J. of Neuroscience 16(15):4551-4562. N. Franceschini, J. M. Pichon and C. Blanes (1992), "From Insect Vision to Robot Vision", Phil. Trans. R. Soc Lond. B. 337, pp 283-294. R. Hengstenberg (1991), Gaze Control in the Blowfly Calliphora: a Multisensory, Two-Stage Integration Process, Seminars, in the Neurosciences, Vol3,pp 19-29. B. K. P. Hom and B. G. Shunck (1981), "Determining Optic Flow", Artificial Intelligence, 17(13):185-204. R. C. Nelson and J. Y. Aloimonos (1989) Obstacle Avoidance Using Flow Field Divergence, IEEE Trans. on Pattern Anal. and Mach. Intel. 11(10):1102-1106. 1. Santos-Victor, G. Sandini, F. Curotto and S. Garibaldi (1995), "Divergent Stereo in Autonomous Navigation: From Bees to Robots," Int. J. of Compo Vis. 14, pp 159-177. P. J. Sobey (1994), "Active Navigation With a Monocular Robot" BioI. Cybern, 71:433-440 M. V. Srinivasan and S. W. Zhang (1993), Evidence for Two Distinct Movement-Detecting Mechanisms in Insect Vision, Naturwissenschaften, 80, pp 38-41. K. Weber, S. Venkatash and M.V. Srinivasan (1996), "Insect Inspired Behaviours for the Autonomous Control of Mobile Robots" Proc. of ICPR'96, pp 156-160. 

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A Hippocampal Model of Recognition Memory Randall C. O'Reilly Department of Psychology University of Colorado at Boulder Campus Box 345 Boulder, CO 80309-0345 oreilly@psych.colorado.edu Kenneth A. Norman Department of Psychology Harvard University 33 Kirkland Street Cambridge, MA 02138 nonnan@wjh.harvard.edu James L. McClelland Department of Psychology and Center for the Neural Basis of Cognition Carnegie Mellon University Pittsburgh, PA 15213 jlm@cnbc.cmu.edu Abstract A rich body of data exists showing that recollection of specific information makes an important contribution to recognition memory, which is distinct from the contribution of familiarity, and is not adequately captured by existing unitary memory models. Furthennore, neuropsychological evidence indicates that recollection is sub served by the hippocampus. We present a model, based largely on known features of hippocampal anatomy and physiology, that accounts for the following key characteristics of recollection: 1) false recollection is rare (i.e., participants rarely claim to recollect having studied nonstudied items), and 2) increasing interference leads to less recollection but apparently does not compromise the quality of recollection (i.e., the extent to which recollected infonnation veridically reflects events that occurred at study). 1 Introduction For nearly 50 years, memory researchers have known that our ability to remember specific past episodes depends critically on the hippocampus. In this paper, we describe our initial attempt to use a mechanistically explicit model of hippocampal function to explain a wide range of human memory data. Our understanding of hippocampal function from a computational and biological perspec- 74 R. C. 0 'Reilly, K. A. Norman and 1. L McClelland tive is based on our prior work (McClelland, McNaughton, & O'Reilly, 1995; O'Reilly & McClelland, 1994). At the broadest level, we think that the hippocampus exists in part to provide a memory system which can learn arbitrary information rapidly without suffering undue amounts of interference. This memory system sits on top of, and works in conjunction with, the neocortex, which learns slowly over many experiences, producing integrative representations of the relevant statistical features of the environment. The hippocampus accomplishes rapid, relatively interference-free learning by using relatively non-overlapping (pattern separated) representations. Pattern separation occurs as a result of 1) the sparseness of hippocampal representations (relative to cortical representations), and 2) the fact that hippocampal units are sensitive to conjunctions of cortical features - given two cortical patterns with 50% feature overlap, the probability that a particular conjunction of features will be present in both patterns is much less than 50%. We propose that the hippocampus produces a relatively high-threshold, high-quality recollective response to test items. The response is "high-threshold" in the sense that studied items sometimes trigger rich recollection (defined as "retrieval of most or all of the test probe's features from memory") but lures never trigger rich recollection. The response is "high-quality" in the sense that, most of the time, the recollection signal consists of part or all of a single studied pattern, as opposed to a blend of studied patterns. The highthreshold, high-quality nature of recollection can be explained in terms of the conjunctivity of hippocampal representations: Insofar as recollection is a function of whether the features of the test probe were encountered together at study, lures (which contain many novel feature conjunctions, even if their constituent features are familiar) are unlikely to trigger rich recollection; also, insofar as the hippocampus stores feature conjunctions (as opposed to individual features), features which appeared together at study are likely to appear together at test. Importantly, in accordance with dual-process accounts of recognition memory (Yonelinas, 1994; Jacoby, Yonelinas, & Jennings, 1996), we believe that hippocampally-driven recollection is not the sole contributor to recognition memory performance. Rather. extensive evidence exists that recollection is complemented by a "fallback" familiarity signal which participants consult when rich recollection does not occur. The familiarity signal is mediated by as-yet unspecified areas (likely including the parahippocampal temporal cortex: Aggleton & Shaw, 1996; Miller & Desimone, 1994). Our account differs substantially from most other computational and mathematical models of recognition memory. Most of these models compute the "global match" between the test probe and stored memories (e.g .? Hintzman, 1988; Gillund & Shiffrin, 1984); recollection in these models involves computing a similarity-weighted average of stored memory patterns. In other memory models, recollection of an item depends critically on the extent to which the components of the item's representation were linked with that of the study context (e.g., Chappell & Humphreys, 1994). Critically, recollection in all of these models lacks the high-threshold, high-quality character of recollection in our model. This is most evident when we consider the effects of manipulations which increase interference (e.g., increasing the length of the study list. or increasing inter-item similarity). As interference increases, global matching models predict increasingly "blurry" recollection (reflecting the contribution of more items to the composite output vector), while the other models predict that false recollection of lures will increase. In contrast, our model predicts that increasing interference should lead to decreased correct recollection of studied test probes, but there should be no concomitant increase in "erroneous" types of recollection (i.e., recollection of details which mismatch studied test probes, or rich recollection of lures). This prediction is consistent with the recent finding that correct recollection of studied items decreases with increasing list length (Yonelinas, 1994). Lastly, although extant data certainly do not contradict the claim that the veridicality of recollection is robust to interference, we acknowledge that additional, focused experimentation is needed to definitively resolve this issue. A Hippocampal Model ofRecognition Memory /" - ,\ 75 / I '---;----; - -- -C~-:;~ _ 1 L __ oL::..J' '- Figure I: The model. a) Shows the areas and connectivity, and the corresponding columns within the Input, EC. and CAl (see text). b) Shows an example activity pattern. Note the sparse activity in the DG and CA3, and intermediate sparseness of the CAL 2 Architecture and Overall Behavior Figure I shows a diagram of our model, which contains the basic anatomical regions of the hippocampal formation, as well as the entorhinal cortex (EC), which serves as the primary cortical input/output pathway for the hippocampus. The model as described below instantiates a series of hypotheses about the structure and function of the hippocampus and associated cortical areas, which are based on anatomical and physiological data and other models as described in O'Reilly and McClelland (1994) and McClelland et al. (1995), but not elaborated upon significantly here. The Input layer activity pattern represents the state of the EC resulting from the presentation of a given item. We assume that the hippocampus stores and retrieves memories by way of reduced representations in the EC, which have a correspondence with more elaborated representations in other areas of cortex that is developed via long-term cortical learning. We further assume that there is a rough topology to the organization of EC, with different cortical areas and/or sub-areas represented by different slots, which can be thought of as representing different feature dimensions of the input (e.g_, color, font, semantic features, etc.). Our EC has 36 slots with four units per slot; one unit per slot was active (with each unit representing a particular "feature value"). Input patterns were constructed from prototypes by randomly selecting different feature values for a random subset of slots. There are two functionally distinct layers of the EC, one which receives input from cortical areas and projects into the hippocampus (superficial or ECin ), and another which receives projections from the CAl and projects back out to the cortex (deep or ECout ). While the representations in these layers are probably different in their details, we assume that they are functionally equivalent, and use the same representations across both for convenience. ECin projects to three areas of the hippocampus: the dentate gyrus (DO), area CA3, and area CAL The storage of the input pattern occurs through weight changes in the feedforward and recurrent projections into the CA3, and the CA3 to CAl connections. The CA3 and CAl contain the two primary representations of the input pattern, while the DO plays an important but secondary role as a pattern-separation enhancer for the CA3. The CA3 provides the primary sparse, pattern-separated, conjunctive representation described above. This is achieved by random, partial connectivity between the EC and CA3, and a high threshold for activation (i.e., sparseness), such that the few units which are activated in the CA3 (5% in our model) are those which have the most inputs from active EC units. The odds of a unit having such a high proportion of inputs from even two relatively similar EC patterns is low, resulting in pattern separation (see O'Reilly & McClelland, 76 R. C. O'Reilly, K. A. Norman and 1. L. McClelland 1994 for a much more detailed and precise treatment of this issue, and the role of the DO in facilitating pattern separation). While these CA3 representations are useful for allowing rapid learning without undue interference, the pattern-separation process eliminates any systematic relationship between the CA3 pattern and the original EC pattern that gave rise to it. Thus, there must be some means of translating the CA3 pattern back into the language of the EC. The simple solution of directly associating the CA3 pattern with the corresponding EC pattern is problematic due to the interference caused by the relatively high activity levels in the EC (around 15%, and 25% in our model). For this reason, we think that the translation is formed via the CAl, which (as a result of long-term learning) is capable of expanding EC representations into sparser patterns that are more easily linked to CA3, and then mapping these sparser patterns back onto the EC. Our CAl has separate representations of small combinations of slots (labeled columns); columns can be arbitrarily combined to reproduce any valid EC representation. Thus, representations in CAl are intermediate between the fully conjunctive CA3, and the fully combinatorial EC. This is achieved in our model by training a single CAl column of 32 units with slightly less than 10% activity levels to be able to reproduce any combination of patterns over 3 ECin slots (64 different combinations) in a corresponding set of3 ECout slots. The resulting weights are replicated across columns covering the entire EC (see Figure la). The cost of this scheme is that more CAl units are required (32 vs 12 per column in the EC), which is nonetheless consistent with the relatively greater expansion of this area relative to other hippocampal areas as a function of cortical size. After learning, our model recollects studied items by simply reactivating the original CA3, CAl and ECout patterns via facilitated weights. With partial or noisy input patterns (and with interference), these weights and two forms of recurrence (the "short loop" within CA3, and the "big loop" out to the EC and back through the entire hippocampus) allow the hippocampus to bootstrap its way into recalling the complete original pattern (pattern completion). If the EC input pattern corresponds to a nonstudied pattern, then the weights will not have been facilitated for this particular activity pattern, and the CAl will not be strongly driven by the CA3. Even if the ECin activity pattern corresponds to two components that were previously studied, but not together (see below), the conjunctive nature of the CA3 representations will minimize the extent to which recall occurs. Recollection is operationalized as successful recall of the test probe. This raises the basic problem that the system needs to be able to distinguish between the ECout activation due to the item input on ECin (either directly or via the CAl), and that which is due to activation coming from recall in the CA3-CAl pathway. One solution to this problem, which is suggested by autocorrelation histograms during reversible CA3 lesions (Mizumori et aI., 1989), is that the CA3 and CAl are 180 0 out of phase with respect to the theta rhythm. Thus, when the CA3 drives the CAl, it does so at a point when the CAl units would otherwise be silent, providing a means for distinguishing between EC and CA3 driven CA 1 activation. We approximate something like this mechanism by simply turning off the ECin inputs to CAl during testing. We assess the quality of hippocampal recall by comparing the resulting ECout pattern with the ECin cue. The number of active units that match between ECin and ECout (labeled C) indicates how much of the test probe was recollected. The number of units that are active in ECout but not in ECin (labeled E) indicates the extent to which the model recollected an item other than the test probe. 3 Activation and Learning Dynamics Our model is implemented using the Leabra framework, which provides a robust mechanism for producing controlled levels of sparse activation in the presence of recurrent activa- A Hippocampal Model ofRecognition Memory 77 tion dynamics, and a simple, effective Hebbian learning rule (O'Reilly, 1996)1. The activation function is a simple thresholded single-compartment neuron model with continuousvalued spike rate output. Membrane potential is updated by dVd't(t) = T L:c gc (t)gc (Ec Vm(t)), with 3 channels (c) corresponding to: e excitatory input; lleak current; and i inhibitory input. Activation communicated to other cells is a simple thresholded function of the membrane potential: Yj(t) = 1/ (1 + 'Y[v>n(:)-9J+)' As in the hippocampus (and cortex), all principal weights (synaptic efficacies) are excitatory, while the local-circuit inhibition controls positive feedback loops (i.e., preventing epileptiform activity) and produces sparse representations. Leabra assumes that the inhibitory feedback has an approximate set-point (i.e., strong activity creates compensatorially stronger inhibition, and vice-versa), resulting in roughly constant overall activity levels, with a firm upper bound. Inhibitory current is given by gi = g~+l + q(gr - g~+l)' where 0 < q < 1 is typically .25, and 8 L:. 9c9c(Ec-8) .. . ' . 9 = ct? 8-Ei for the UnIts With the k th and k + 1 th highest excitatory mputs. A simple, appropriately normalized Hebbian rule is used in Leabra: f).wij = XiYj - YjWij, which can be seen as computing the expected value of the sending unit's activity conditional on the receiver's activity (if treated like a binary variable active with probability Yj): Wij ~ (xiIYj}p' This is essentially the same rule used in standard competitive learning or mixtures-of-Gaussians. 4 Interference and List-Length, Item Similarity Here, we demonstrate that the hippocampal recollection system degrades with increasing interference in a way that preserves its essential high-threshold, high-quality nature. Figure 2 shows the effects of list length and item similarity on our C and E measures. Only studied items appear in the high C, low E comer representing rich recollection. As length and similarity increase, interference results in decreased C for studied items (without increased E), but critically there is no change in responding to new items. Interference in our model arises from the reduced but nevertheless extant overlap between representations in the hippocampal system as a function of item similarity and number of items stored. To the extent that increasing numbers of individual CA3 units are linked to mUltiple contradictory CAl representations, their contribution is reduced, and eventually recollection fails. As for the frequently obtained finding that decreased recollection of studied items is accompanied by an increase in overall false alarms, we think this results from subjects being forced to rely more on the (less reliable) fallback familiarity mechanism. 5 Conjunctivity and Associative Recognition Now, we consider what happens when lures are constructed by recombining elements of studied patterns (e.g., study ''window-reason'' and "car-oyster", and test with "windowoyster"). One recent study found that participants are much more likely to claim to recollect studied pairs than re-paired lures (Yonelinas, 1997). Furthermore, data from this study is consistent with the idea that re-paired lures sometimes trigger recollection of the studied word pairs that were re-combined to generate the lure; when this happens (assuming that each word occurred in only one pair), the participant can confidently reject the lure. Our simulation data is consistent with these findings: For studied word pairs, the model (richly) recollected both pair components 86% of the time. As for re-paired lures, both pair components were never recalled together, but 16% of the time the model recollected one of the pair components, along with the component that it was paired with at study. The I Note that the version of Leabra described here is an update to the cited version, which is currently being prepared for publication. 78 R. C. O'Reilly, K. A. Nonnan and 1. L McClelland Figure 2: Effects of list length and similarity on recollection perfonnance. Responses can be categorized according to the thresholds shown, producing three regions: rich recollection (RR), weak recollection (WR), and misrecollection (MR). Increasing list length and similarity lead to less rich recollection of studied items (without increasing misrecollection for these items), and do not significantly affect the model's responding to lures. model responded in a similar fashion to pairs consisting of one studied word and a new word (never recollecting both pair components together, but recollecting the old item and the item it was paired with at study 13% of the time). Word pairs consisting of two new items failed to trigger recollection of even a single pair component. Similar findings were obtained in our simulation of the (Hintzman, Curran, & Oppy, 1992) experiment involving recombinations of word and plurality cues. 6 Discussion While the results presented above have dealt with the presentation of complete probe stimuli for recognition memory tests, our model is obviously capable of explaining cued recall and related phenomena such as source or context memory by virtue of its pattern completion abilities. There are a number of interesting issues that this raises. For example, we predict that successful item recollection will be highly correlated with the ability to recall additional information from the study episode, since both rely on the same underlying memory. Further, to the extent that elderly adults form less distinct encodings of stimuli (Rabinowitz & Ackerman, 1982), this explains both their impaired recollection on recognition tests (Parkin & Walter, 1992) and their impaired memory for contextual ("source") details (Schacter et aI., 1991). In summary, existing mathematical models of recognition memory are most likely incorrect in assuming that recognition is performed with one memory system. Global matching models may provide a good account of familiarity-based recognition, but they fail to account for the contributions of recollection to recognition, as discussed above. For example, global matchil).g models predict that lures which are similar to studied items will always trigger a stronger signal than dissimilar lures; as such, these models can not account for the fact that sometimes subjects can reject similar lures with high levels of confidence (due, in our model, to recollection ofa similar studied item; Brainerd, Reyna, & Kneer, 1995; Hintzman et aI., 1992). Further, global matching models confound the signal for the extent to which individual components of the test probe were present at all during study, and signal for the A Hippocampal Model ofRecognition Memory 79 extent to which they occurred together. We believe that these signals may be separable, with recollection (implemented by the hippocampus) showing sensitivity to conjunctions of features, but not the occurrence of individual features, and familiarity (implemented by cortical regions) showing sensitivity to component occurrence but not co-occurence. This division of labor is consistent with recent data showing that familiarity does not discriminate well between studied item pairs and lures constructed by conjoining items from two different studied pairs (so long as the pairings are truly novel) (Yonelinas, 1997), and with the point, set forth by (McClelland et aI., 1995), that catastrophic interference would occur if rapid learning (required to learn feature co-occurrences) took place in the neocortical structures which generate the familiarity signal. 7 References Aggleton, J. P., & Shaw, C. (1996). Amnesia and recognition memory: are-analysis of psychometric data. Neuropsychologia, 34, 51. Brainerd, C. J., Reyna, V. F., & Kneer, R. (1995). False-recognition reversal: When similarity is distinctive. Journal ofMemory and Language, 34, 157-185. Chappell, M., & Humphreys, M. S. (1994). An auto-associative neural network for sparse representations: Analysis and application to models of recognition and cued recall. Psychological Review, 101, 103-128. Gillund, G., & Shiffrin, R. M. (1984). A retrieval model for both recognition and recall. Psychological Review, 91, 1-67. Hintzman, D. L. (1988). Judgments of frequency and recognition memory in a multiple-trace memory model. Psychological Review, 95, 528-551. Hintzman, D. L., Curran, T., & Oppy, B. (1992). Effects of similiarity and repetition on memory: Registration without learning. Journal ofExperimental Psychology: Learning. Memory. and Cognition, 18, 667-680. Jacoby, L. L., Yonelinas, A. P., & Jennings, J. M. (1996). The relation between conscious and unconscious (automatic) influences: A declaration of independence. In J. D. Cohen, & J. W. Schooler (Eds.), Scientific approaches to the question ofconsciousness (pp. 13-47). Hi1lsdale, NJ: Lawrence Erbaum Associates. McClelland, J. L., MCNaughton, B. L., & O'Reilly, R. C. (1995). Why there are complementary learning systems in the hippocampus and neocortex: Insights from the successes and failures of connectionst models of learning and memory. Psychological Review, 102,419-457. Miller, E. K., & Desimone, R. (1994). Parallel neuronal mechanisms for short-term memory. Science, 263, 520--522. Mizumori, S. J. Y., McNaughton, B. L., Barnes, C. A., & Fox, K. B. (1989). Preserved spatial coding in hippocampal CAl pyramidal cells during reversible suppression ofCA3c output: Evidence for pattern completion in hippocampus. Journal of NeuroSCience, 9( II), 3915-3928. O'Reilly, R. C. (1996). The leabra model of neural interactions and learning in the neocortex. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, USA. O'Reilly, R. C., & McClelland, J. L. (1994). Hippocampal conjunctive encoding, storage, and recall: Avoiding a tradeoff. Hippocampus, 4(6), 661-682. Parkin, A. J., & Walter, B. M. (1992). Recollective experience, normal aging, and frontal dysfunction. Psychology and Aging, 7,290--298. Rabinowitz, J. C., & Ackerman, B. P. (1982). General encoding of episodic events by elderly adults. In F. I. M. C. S. Trehub (Ed.), Aging and cognitive processes. Plenum Publishing Corporation. Schacter, D. L., Kaszniak, A. W., Kihlstrom, J. F., & Valdiserri, M. (1991). The relation between source memory and aging. Psychology and Aging, 6, 559-568. Yonelinas, A. P. (1994). Receiver-operating characteristics in recognition memory: Evidence for a dual-process model. Journal ofExperimental Psychology: Learning. Memory. and Cognition, 20, 1341-1354. Yonelinas, A. P. (1997). Recognition memory ROCs for item and associative information: The contribution of recollection and familiarity. Memory and Cognition, 25,747-763. 

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Correlates of Attention in a Model of Dynamic Visual Recognition* . Rajesh P. N. Rao Department of Computer Science University of Rochester Rochester, NY 14627 rao@cs.rochester.edu Abstract Given a set of objects in the visual field, how does the the visual system learn to attend to a particular object of interest while ignoring the rest? How are occlusions and background clutter so effortlessly discounted for when recognizing a familiar object? In this paper, we attempt to answer these questions in the context of a Kalman filter-based model of visual recognition that has previously proved useful in explaining certain neurophysiological phenomena such as endstopping and related extra-classical receptive field effects in the visual cortex. By using results from the field of robust statistics, we describe an extension of the Kalman filter model that can handle multiple objects in the visual field. The resulting robust Kalman filter model demonstrates how certain forms of attention can be viewed as an emergent property of the interaction between top-down expectations and bottom-up signals. The model also suggests functional interpretations of certain attentionrelated effects that have been observed in visual cortical neurons. Experimental results are provided to help demonstrate the ability of the model to perform robust segmentation and recognition of objects and image sequences in the presence of varying degrees of occlusions and clutter. 1 INTRODUCTION The human visual system possesses the remarkable ability to recognize objects despite the presence of distractors and occluders in the field of view. A popular suggestion is that an "attentional spotlight" mediates this ability to preferentially process a relevant object in a given scene (see [5, 9] for reviews). Numerous models have been proposed to simulate the control of this ..focus of attention" [10, 11, 15]. Unfortunately, there is inconclusive evidence for the existence of an explicit neural mechanism for implementing an attentional spotlight in the visual *This research was supported by NIH/PHS research grant 1-P41-RR09283. I am grateful to Dana Ballard for many useful discussions and suggestions. Author's current address: The Salk Institute, CNL, 10010 N. Torrey Pines Road, La Jolla, CA 92037. E-mail: rao@salk. edu . .??..\.'.. ' Correlates ofAttention in a Model ofDynamic Visual Recognition 81 cortex. Thus, an important question is whether there are alternate neural mechanisms which don't explicitly use a spotlight but whose effects can nevertheless be interpreted as attention. In other words, can attention be viewed as an emergent property of a distributed network of neurons whose primary goal is visual recognition? In this paper, we extend a previously proposed Kalman filter-based model of visual recognition [13, 12] to handle the case of multiple objects, occlusions, and clutter in the visual field. We provide simulation results suggesting that certain forms of attention can be viewed as an emergent property of the interaction between bottom-up signals and top-down expectations during visual recognition. The simulation results demonstrate how "attention" can be switched between different objects in a visual scene without using an explicit spotlight of attention. 2 A KALMAN FILTER MODEL OF VISUAL RECOGNITION We have previously introduced a hierarchical Kalman filter-based model of visual recognition and have shown how this model can be used to explain neurophysiological effects such as endstopping and neural response suppression during free-viewing of natural images [ 12, 13 ]. The Kalman filter [7] is essentially a linear dynamical system that attempts to mimic the behavior of an observed natural process. At any time instant t, the filter assumes that the internal state of the given natural process can be represented as a k x 1 vector r(t). Although not directly accessible, this internal state vector is assumed to generate ann x 1 measurable and observable output vector I(t) (for example, an image) according to: I(t) = Ur(t) + n(t) (1) where U is ann x k generative (or measurement) matrix, and n(t) is a Gaussian stochastic noise process with mean zero and a covariance matrix given by E = E[nilT] (E denotes the expectation operator and T denotes transpose). In order to specify how the internal state r changes with time, the Kalman filter assumes that the process of interest can be modeled as a Gauss-Markov random process [1]. Thus, given the state r(t- 1) at time instant t - 1, the next state r(t) is given by: r(t) = Vr(t- 1) + m(t- 1) (2) where Vis the state transition (or prediction) matrix and m is white Gaussian noise with mean m = E[m] and covariance II= E[(m- m)(m- m)T]. Given the generative model in Equation 1 and the dynamics in Equation 2, the goal is to optimally estimate the current internal state r( t) using only the measurable inputs I( t). An optimization function whose minimization yields an estimate of r is the weighted least-squares criterion: = J (I- Ur)TE- 1 (I- Ur) + (r- r f M- 1 (r- r) (3) where r(t) is the mean of the state vector before measurement of the input data I(t) and M = E[(r - r)(r - r)T] is the corresponding covariance matrix. It is easy to show [1] that J is simply the sum of the negative log-likelihood of generating the data I given the stater, and the negative log of the prior probability of the stater. Thus, minimizing J is equivalent to maximizing the posterior probability p(rji) of the stater given the input data. = 0 and solving for the minimum The optimization function J can be minimized by setting ~J valuer of the stater (note thatr equals the mean ofr afte~ measurement of I). The resultant Kalman .filter equation is given by: ? = + N(t)UTE(t)- 1 (I(t)- Ur(t)) (4) = W(t- 1) + m(t- 1) (5) = (UTE(t)- 1 U + M(t)- 1 )-1 is a "normalization" matrix that maintains the r(t) r(t) r(t) where N(t) covariance of the stater after measurement ofl. The matrix M, which is the covariance before R. P. N. Rao 82 measurement of I, is updated as M (t) = V N (t - 1) VT +II (t - 1). Thus, the Kalman filter predicts one step into the future using Equation 5, obtains the next sensory input I(t), and then co~ects its Jj.rediction r(9 u~ing the sensory resi~ual e~or (I(t) --: Ur(t)) a~d ~e Kalman gam N(t)U ~(t)- 1 ? This yields the corrected est1mate r(t) (Equation 4), wh1ch IS then used to make the next state predictionf(t + 1). The measurement (or generative) matrix U and the state transition (or prediction) matrix V used by the Kalman filter together encode an internal nwdel of the observed dynamic process. As suggested in [13], it is possible to learn an internal model of the input dynamics from observed data. Let u and v denote the vectorized forms of the matrices U and V respectively. For exam,fle, the n ~ k generative matrix U can be collapsed into an nk x 1 vector u [U1 U 2 ??. un] where U' denotes the ith row of U. Note that (I- Ur) (I- Ru) where R is then x nk matrix given by: = R =[ r;.. r~ ~... . 0 0 = l (6) rT By minimizing an optimization function similar to J [13], one can derive a Kalman filter-like ..learning rule" for the generative matrix U: u(t) = ii(t) + Nu(t)R(t)T~(t)- 1 (I(t)- R(t)ii(t))- aNu(t)ii(t) (7) where ii(t) = u(t- 1), Nu(t) = (Nu(t- 1)- 1 + R(t)T~(t)- 1 R(t) + ai)- 1 , and I is the nk x nk identity matrix. The constant a determines the decay rate of ii. As in the case of U, an estimate of the prediction matrix V can be obtained via the following learning rule for v [13]: v(t) = = v(t) + Nv(t)R(tf M(t)- 1 [r(t + 1)- r(t + 1)]- f3Nv(t)v(t) = (8) wherev(t) v(t-1), Nv(t) (Nv(t-1)- 1 +R(t)T M(t)- 1 R(t)+f3I)- 1 andRisak X k 2 matrix analogous to R (Equation 6) but with rT = The constant /3 determines the decay rate for v while I denotes the k 2 x k 2 identity matrix. Note that in this case, the estimate ofV is corrected using the prediction residual error (r( t + 1) - r( t + 1)), which denotes the difference between the actual state and the predicted state. One unresolved issue is the specification of values for r(t) (comprising R(t)) in Equation 7 and r(t + 1) in Equation 8. The ExpectationMaximization (EM) algorithm [4] suggests that in the case of static stimuli (f(t) = r(t 1)), one may use r(t) = r which is the converged optimal state estimate for the given static r( tl N), which is input. In the case of dynamic stimuli, the EM algorithm prescribes r( t) the optimal temporally snwothed state estimate [1] for timet (5 N), given input data for each of the time instants 1, ... , N. Unfortunately, the smoothed estimate requires knowledge of future inputs and is computationally quite expensive. For the experimental results, we used the on-line estimates r(t) when updating the matrices and during training. rr. = u v 3 ROBUST KALMAN FILTERING The standard derivation of the Kalman filter minimizes Equation 3 but unfortunately does not specify how the covariance ~ is to be obtained. A common choice is to use a constant matrix or even a constant scalar. Making ~ constant however reduces the Kalman filter estimates to standard least-squares estimates,? which are highly susceptible to outliers or gross errors i.e. data points that lie far away from the bulk of the observed or predicted data [6]. For example, in the case where I represents an input image, occlusions and clutter will cause many pixels in I to deviate significantly from corresponding pixels in the predicted image Ur. The problem 83 Correlates ofAJtention in a Model ofDynamic Visual Recognition Gating Matrix Sensory Residual G - Feedforward Matrix uT I- ltd Input I - Nonnalization Inhibition Itd=Ur Top-Down Prediction of Expeeted I nput N Robust Kalman Filter Estimate Prediction r Matrix r0 " v - ~ Feedback Matrix Predicted State r u Figure 1: Recurrent Network Implementation of the Robust Kalman Filter. The gating matrix G is a non-linear function of the current residual error between the input I and its top-down prediction ur. G effectively filters out any high residuals, thereby preventing outliers in input data I from influencing the robust Kalman filter estimate r. Note that the entire filter can be implemented in a recurrent neural network, with U, UT, and V represented by the synaptic weights of neurons with linear activation functions and G being implemented by a set of threshold non-linear neurons with binary outputs. of outliers can be tackled using robust estimation procedures [6] such as M-estimation, which involves minimizing a function of the form: n (9) i=1 where Ii and Ui are the ith pixel and ith row of I and U respectively, and p is a function that increases less rapidly than the square. This reduces the influence oflarge residual errors (which correspond to outliers) on the optimization of J', thereby "rejecting" the outliers. A special case of the above function is the following weighted least squares criterion: J' =(I-:- Urf S(I- Ur) (10) where Sis a diagonal matrix whose diagonal entries Si,i determine the weight accorded to the corresponding pixel error (Ii - Uir). A simple but attractive choice for these weights is the non-linear function given by Si,i =min {1, c/(Ii- Uir) 2 }, wherecisa threshold parameter. To understand the behavior of this function, note that S effectively clips the ith summand in J' (Equation 10 above) to a constant value c whenever the ith squared residual (Ii - Uir? exceeds the threshold c; otherwise, the summand is set equal to the squared residual. By substituting E- 1 = Sin the optimization function J (Equation 3), we can rederive the following robust Kalman filter equation: r(t) = r(t) + N(t)UTG(t)(I- Ur(t)) (11) where r(t) = W(t- 1)) + iii(t- 1), N(t) = (UTG(t)U + M(t)- 1 ) - 1 , M(t) = V N(t1)VT + II(t -1), and G(t) is ann x n diagonal matrix whose diagonal entries at time instant t are given by: Gi,i _ { 0 if (Ii(t) - Uir(t)) 2 > c(t) 1 otherwise G can be regarded as the sensory residual gain or "gating" matrix, which determines the (binary) gain on the various components of the incoming sensory residual error vector. By effectively filtering out any high residuals, G allows the Kalman filter to ignore the corresponding outliers in the input I, thereby enabling it to robustly estimate the stater. Figure 1 depicts an implementation of the robust Kalman filter in the form of a recurrent network of linear and threshold non-linear neurons. In particular, the feedforward, feedback and prediction neurons possess linear activation functions while the gating neurons implementing G compute binary outputs based on a threshold non-linearity. R. P.N.Rao 84 Training Objects Robust Estimate Input Image (b) (a) Input Image Outliers Robust Estimate I Robust Estimate 2 Outliers I Least Squares Estimate I. ' (c) Figure 2: Correlates of Attention during Static Recognition. (a) Images of size 105 x 65 used to train a robust Kalman filter network. The generative matrix U was 6825 x 5. (b) Occlusions and background clutter are treated as outliers (white regions in the third image, depicting the diagonal of the gating matrix G). This allows the network to "attend to" and recognize the training object, as indicated by the accurate reconstruction (middle image) of the training image based on the final robust state estimate. (c) In the more interesting case of the training objects occluding each other, the network converges to one of the objects (the "dominant" one in the image - in this case, the object in the foreground). Having recognized one object, the second object is attended to and recognized by taking the complement of the outliers (diagonal of G) and repeating the robust filtering process (third and fourth images). The fifth image is the image reconstruction obtained from the standard (least squares derived) Kalman filter estimate, showing an inability to resolve or recognize either of the two objects. 4 VISUAL ATIENTION IN A SIMULATED NETWORK The gating matrix G allows the Kalman filter network to "selectively attend.. to an object while treating the remaining components of the sensory input as outliers. We demonstrate this capability of the network using three different examples. In the first example, a network was trained on static grayscale images of a pair of 3D objects (Figure 2 (a)}. For learning static inputs, the prediction matrix Vis unnecessary since we may use r(t) = r(t -1) and M(t) = N(t -1). After training, the network was tested on images containing the training objects with varying degrees of occlusion and clutter (Figure 2 (b) and (c)). The outlier threshold c was initialized to the sum of the mean plus k standard deviations of the current distribution of squared residual errors (Ii - Uir )2 ? The value of k was gradually decreased during each iteration in order to allow the network to refine its robust estimate by gradually pruning away the outliers as it converges to a single object estimate. After convergence, the diagonal of the matrix G contains zeros in the image locations containing the outliers and ones in the remaining locations. As shown in Figure 2 (b), the network was successful in recognizing the training object despite occlusion and background clutter. More interestingly, the outliers (white) produce a crude segmentation of the occluder and background clutter, which can subsequently .be used to focus "attention" on these previously ignored objects and recover their identity. In particular, an outlier mask m can be defined by taking the complement of the diagonal of G (i.e. m i = 1- Gi,i). By replacing the diagonal of G with m in Equation tt 1 and repeating the estimation process, the network can "attend to" 1 Although not implemented here, this "shifting of attentional focus" can be automated using a model of neuronal fatigue and active decay (see, for example, [3)). Correlates ofAUention in a Model of Dynamic Visual Recognition laput 85 ~ ~~~~~ .. J~r5' ~ -~ . Oulllen ----- ~/' ~ ---?--~- (a) ??Ill . . . I I??? ?- ~ (b) Inputs Predictions Predictions Outliers Outliers (c) (d) Figure 3: Correlates of Attention during Dynamic Recognition. (a) A network was trained on a cyclic image sequence of gestures (top), each image of size 75 x 75, with U and V of size 5625 x 15 and 15 x 15 respectively. The panels below show how the network can ignore various fonns of occlusion and clutter (outliers}, "attending to" the sequence of gestures that it has been trained on. The outlier threshold c was computed as the mean plus 0.3 standard deviati9ns of the current distribution of squared residual errors. Results shown are those obtained after 5 cycles of exposure to the occluded images. (b) Three image sequences used to train a network. (c) and (d) show the response of the network to ambiguous stimuli comprised of images containing both a horizontal and a vertical bar. Note that the network was trained on a horizontal bar moving downwards and?a vertical bar moving rightwards (see (b)) but not both simultaneously. Given ambiguous stimuli containing both these stimuli, the network interprets the input differently depending on the initial "priming" input. When the initial input is a vertical bar as in (c), the network interprets the sequence as a vertical bar moving rightwards (with some minor artifacts due to the other training sequences). On the other hand, when the initial input is a horizontal bar as in (d), the sequence is interpreted as a horizontal bar moving downwards, not paying "attention" to the extraneous vertical bars, which are now treated as outliers. the image region(s) that were previously ignored as outliers. Such a two-step serial recognition process is depicted in Figure 2 (c). The network first recognizes the "doniinant" object, which was generally observed to be the object occupying a larger area of the input image or possessing regions with higher contrast.? The outlier mask m is subsequently used for "switching attention" and extracting the identity of the second object (lower arrow). Figure 3 shows examples of attention during recognition of dynamic stimuli. In particular, Figure 3 (c) and (d) show how the same image sequence can be interpreted in two different ways depending on which part of the stimulus is "attended to," which in tum depends on the initial priming input. 5 CONCLUSIONS The simulation results indicate that certain? experimental observations that have previously been interpreted using the metaphor of an attentional spotlight can also arise as a result of competition and cooperation during visual recognition within networks of linear and non-linear 86 R. P. N. Rao neurons. Although not explicitly designed to simulate attention, the robust Kalman filter networks nevertheless display some of the essential characteristics of visual attention, such as the preferential processing of a subset of the input signals and the consequent "switching" of attention to previously ignored stimuli. Given multiple objects or conflicting stimuli in their receptive fields (Figures 2 and 3), the responses of the feedforward, .feedback, and prediction neurons in the simulated network were modulated according to the current object being "attended to." The modulation in responses was mediated by the non-linear gating neurons G, taking into account both bottom-up signals as well top-down feedback signals. This suggests a network-level interpretation of similar forms of attentional response modulation in the primate visual cortex [2, 8, 14], with the consequent prediction that the genesis of attentional modulation in such cases may not necessarily lie within the recorded neurons themselves but within the distributed circuitry that these neurons are an integral part of. References [1] A.B. Bryson and Y.-C. Ho. Applied Optimtzl Control. New York: John Wiley, 1975. [2] L. Chelazzi, E.K. Miller, J. Duncan, and R. Desimone. A neural basis for visual search in inferior temporal cortex. Nature, 363:345-347, 1993. [3] P. Dayan. An hierarchical model of visual rivalry. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 48-54. Cambridge, MA: MIT Press, 1997. [4] A.P. Dempster, N.M. Laird, andD.B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39:1-38, 1977. [5] R. Desimone and J. Duncan. Neural mechanisms of selective visual attention. Annual Review of Neuroscience, 18:193-222,1995. [6] P.J. Huber. Robust Statistics. New York: John Wiley, 1981. [7] R.E. Kalman. A new approach to linear filtering and prediction theory. Trans. ASME J. Basic Eng., 82:35-45, 1960. [8] J. Moran and R. Desimone. Selective attention gates visual processing in the extrastriate cortex. Science, 229:782-784, 1985. [9] W.T. Newsome. Spotlights, highlights and visual awareness. Current Biology, 6(4):357360, 1996. [10] E. Niebur and C. Koch. Control of selective visual attention: Modeling the "where" pathway. In D. Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural lnformtztion Processing Systems 8, pages 802-808. Cambridge, MA: MIT Press, 1996. [11] B.A. Olshausen, D.C. Van Essen, and C.H. Anderson. A neurobiological model of visual attention and invariant pattern recognition based on dynamic routing of information. Journal of Neuroscience, 13:4700-4719, 1993. [12] R.P.N. Rao and D.H. Ballard. The visual cortex as a hierarchical predictor. Technical Report 96.4, National Resource Laboratory for the Study of Brain and Behavior, Department of Computer Science, University of Rochester, September 1996. [13] R.P.N. Rao and D.H. Ballard. Dynamic model of visual recognition predicts neural response properties in the visual cortex. Neural Computation, 9(4):721-763, 1997. [14] S. Treue and J.H.R. Maunsell. Attentional modulation of visual motion processing in cortical areas MT and MST. Nature, 382:539-541, 1996. [15] J .K. Tsotsos, S.M. Culhane, W. Y.K. Wai, Y. Lai, N. Davis, and F. Nuflo. Modeling visual attention via selective tuning. Artificial Intelligence, 78:507~545, 1995. 

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S-Map: A network with a simple self-organization algorithm for generative topographic mappings Kimmo Kiviluoto Laboratory of Computer and Information Science Helsinki University of Technology P.O. Box 2200 FIN-02015 HUT, Espoo, Finland Kimmo.KiviluotoChut.fi Erkki Oja Laboratory of Computer and Information Science Helsinki University of Technology P.O. Box 2200 FIN-02015 HUT, Espoo, Finland Erkki.OjaChut.fi Abstract The S-Map is a network with a simple learning algorithm that combines the self-organization capability of the Self-Organizing Map (SOM) and the probabilistic interpretability of the Generative Topographic Mapping (GTM). The simulations suggest that the SMap algorithm has a stronger tendency to self-organize from random initial configuration than the GTM. The S-Map algorithm can be further simplified to employ pure Hebbian learning, without changing the qualitative behaviour of the network. 1 Introduction The self-organizing map (SOM; for a review, see [1]) forms a topographic mapping from the data space onto a (usually two-dimensional) output space. The SOM has been succesfully used in a large number of applications [2]; nevertheless, there are some open theoretical questions, as discussed in [1, 3]. Most of these questions arise because of the following two facts: the SOM is not a generative model, i.e. it does not generate a density in the data space, and it does not have a well-defined objective function that the training process would strictly minimize. Bishop et al. [3] introduced the generative topographic mapping (GTM) as a solution to these problems. However, it seems that the GTM requires a careful initialization to self-organize. Although this can be done in many practical applications, from a theoretical point of view the GTM does not yet offer a fully satisfactory model for natural or artificial self-organizing systems. K. Kiviluoto and E. Oja 550 In this paper, we first briefly review the SOM and GTM algorithms (section 2); then we introduce the S-Map, which may be regarded as a crossbreed of SOM and GTM (section 3); finally, we present some simulation results with the three algorithms (section 4), showing that the S-Map manages to combine the computational simplicity and the ability to self-organize of the SOM with the probabilistic framework of the GTM. 2 2.1 SOM and GTM The SOM algorithm e The self-organizing map associates each data vector t with that map unit that has its weight vector closest to the data vector. The activations 11! of the map units are given by ~ = {I, when IIJLi 11, 0, otherwise - etll < IIJLj - etll, 'Vj #i (1) where JLi is the weight vector of the ith map unit Ci , i = 1, . . . , K. Using these activations, the SOM weight vector update rule can be written as K JLj := JLj + tSt L 11: h(Ci' Cj ; j3t)(e t - JLj) (2) i=l Here parameter tSt is a learning rate parameter that decreases with time. The neighborhood function h(Ci' Cj ; j3t) is a decreasing function of the distance between map units Ci and Cj ; j3t is a width parameter that makes the neighborhood function get narrower as learning proceeds. One popular choice for the neighborhood function is a Gaussian with inverse variance j3t. 2.2 The GTM algorithm In the GTM algorithm, the map is considered as a latent space, from which a nonlinear mapping to the data space is first defined. Specifically, a point C in the latent space is mapped to the point v in the data space according to the formula L v(C; M) = MtP(C) = L ?j(C)JLj (3) j=l where tP is a vector consisting of L Gaussian basis functions, and M is a D x L matrix that has vectors ILj as its columns, D being the dimension of the data space. The probability density p( C) in the latent space generates a density to the manifold that lies in the data space and is defined by (3). IT the latent space is of lower dimension than the data space, the manifold would be singular, so a Gaussian noise model is added. A single point in the latent space generates thus the following density in the data space: j3)D/2 exp [j3 p(eIC;M,j3)= ( 211' -2'llv(C;M)-eW ] (4) where j3 is the inverse of the variance of the noise. The key point of the GTM is to approximate the density in the data space by assuming the latent space prior p( C) to consist of equiprobable delta functions that S-Map 551 form a regular lattice in the latent space. The centers Ci of the delta functions are called the latent vectors of the GTM, and they are the GTM equivalent to the SOM map units. The approximation of the density generated in the data space is thus given by 1 K p(eIM,m = K (5) LP(elCi;M,,8) i=l The parameters of the GTM are determined by minimizing the negative log likelihood error f(M,,8) = - t. [~ t. ln (6) P ({'IC,;M,,8)] over the set of sample vectors {e t }. The batch version of the GTM uses the EM algorithm [4]; for details, see [3]. One may also resort to an on-line gradient descent procedure that yields the GTM update steps K 1L~+1 := IL~ + 6t ,8t L lI:(M t ,,8t)<pj(Ci)[et - v(Ci;M t )] (7) i=l fJH' := i3' + 6' [ ~ t. ~hM', (J')lle' - v(C,; MI)II' - 2~ ] (8) where 11: (M,,8) is the GTM counterpart to the SOM unit activation, the posterior probability p(Cilet ; M,,8) of the latent vector Ci given data vector et : lIf(M,,8) = p(Cilet ;M,,8) p(etICi; M,,8) - L:f.=1 p(etICi'; M,,8) (9) exp[-~IIV(Ci;M) 2.3 _e t Il 2 ] Connections between SOM and GTM Let us consider a GTM that has an equal number of latent vectors and basis functions!, each latent vector Ci being the center for one Gaussian basis function <Pi(C). Latent vector locations may be viewed as units of the SOM, and consequently the basis functions may be interpreted as connection strengths between the units. Let us use the shorthand notation <P~ == <Pj(Ci). Note that <P~ = <frt, and assume that the basis functions be normalized so that L:~l <P~ At the zero-noise limit, or when ,8 ~ 00, = L:~1 <P~ = 1. the softmax activations of the GTM given in (9) approach the winner-take-all function (1) of the SOM. The winner unit Cc(t) e for the data vector t is the map unit that has its image closest to the data vector, so that the index c( t) is given by c(t) = ar~in Ilv(C,) - e'll = ~in (t, 4>;1';) - {t (10) INote that this choice serves the purpose of illustration only; to use GTM properly, one should choose much more latent vectors than basis functions. 552 K. Kiviluoto and E. Oja The GTM weight update step (7) then becomes IL~+1 := IL~ + ~t?j(t) ret - v(c(t); Mt)] (11) This resembles the variant of SOM, in which the winner is searched with the rule (10) and weights are updated as IL~+1 := ILj +~t?j(t)(et - IL;) (12) Unlike the original SOM rules (1) and (2), the modified SOM with rules (10) and (12) does minimize a well-defined objective function: the SOM distortion measure [5, 6, 7, 1]. However, there is a difference between GTM and SOM learning rules (11) and (12). With SOM, each individual weight vector moves towards the data vector, but with GTM, the image ofthe winnerlatent vector v(c(t); M) moves towards the data vector, and all weight vectors ILj move to the same direction. For nonzero noise, when 0 < f3 < 00, there is more difference between GTM and SOM: with GTM, not only the winner unit but activations from other units as well contribute to the weight update. 3 S-Map Combining the softmax activations of the GTM and the learning rule of the SOM, we arrive at a new algorithm: the S-Map. 3.1 The S-Map algorithm The S-Map resembles a GTM with an equal number of latent vectors and basis functions. The position of the ith unit on the map is is given by the latent vector (i; the connection strength of the unit to another unit j is ?), and a weight vector ILi is associated with the unit. The activation of the unit is obtained using rule (9). The S-Map weights learn proportionally to the activation of the unit that the weight is associated with, and the activations of the neighboring units: 1'1+1 ,= 1'; +6' (t. ?iij:) (e' -1';) (13) which can be further simplified to a fully Hebbian rule, updating each weight proportionally to the activation of the corresponding unit only, so that IL}+l := ILj + ~t7]~ (e t - ILj) (14) The parameter f3 value may be adjusted in the following way: start with a small value, slowly increase it so that the map unfolds and spreads out, and then keep increasing the value as long as the error (6) decreases. The parameter adjustment scheme could also be connected with the topographic error of the mapping, as proposed in [9] for the SOM. Assuming normalized input and weight vectors, the "dot-product metric" form of the learning rules (13) and (14) may be written as 1')+1 ,= 1'] + 6' (t. ?;ij:) (I -1']1'?)e' (15) S-Map 553 and JL~+1 := 1') + 6t1Jj(I - JL)JLf)e t (16) respectively; the matrix in the second parenthesis keeps the weight vectors normalized to unit length, assuming a small value for the learning rate parameter 6t [8]. The dot-product metric form of a unit activity is ~' = (L:~1 ?)I't {t] L: exp [13 (Lf=1 1>;' JLj) et ] exp [,8 T ==1 l (17) which approximates the posterior probability p('ile t ;M, 13) that the data vector were generated by that specific unit. This is based on the observation that if the data vectors {e t } are normalized to unit length, the density generated in the data space (unit sphere in RD) becomes .. -1 normallzmg p({IC,; M,,8) = ( constant) x exp [ ,8 3.2 K f;. ( i ?jl'j ) T { ] (18) S-Map algorithm minimizes the GTM error function in dot-product metric The GTM error function is the negative log likelihood, which is given by (6) and is reproduced here: (19) When the weights are updated using a batch version of (15), accumulating the updates for one epoch, the expected value of the error [4] for the unit (i is T = - LpOld(ilet;M,m In[p?ewU:i)pOeW(e t l(i;M,!3)] E(?rW) t=1 ' T =- L '" T/~Id.t 1J~ld,t 13 ( t==1 K -'---==1!K 1>] JLjew L ) T (20) e t + terms not involving the weight vectors j==1 The change of the error for the whole map after one epoch is thus K E(?oew - fOld) =- L T K L L 1J~ld,t131>;(JLjew - JLjldfe t i=1 t=1 j=1 = - p.s t, (~ ~ q~ld" ?;{t) , '" T (I - I'lld 1''Jld T) -' aT J (t, t, q~ld,t' ?j'(" ) '", ~ K = -136 L[uf Uj - (uf JLjld)2] ~ 0 j=1 with equality only when the weights are already in the error minimum. -'(21) 554 4 K. Kiviluoto and E. Oja Experimental results The self-organization ability of the SOM, the GTM, and the S-Map was tested on an artificial data set: 500 points from a uniform random distribution in the unit square. The initial weight vectors for all models were set to random values, and the final configuration of the map was plotted on top of the data (figure 1). For all the algorithms, the batch version was used. The SOM was trained as recommended in [1] in two phases, the first starting with a wide neighborhood function, the second with a narrow neighborhood. The GTM was trained using the Matlab implementation by Svensen, following the recommendations given in [to]. The S-Map was trained in two ways: using the "full" rule (13), and the simplified rule (14) . In both cases, the parameter {3 value was slowly increased every epoch; by monitoring the error (6) of the S-Map (see the error plot in the figure) the suitable value for {3 can be found. In the GTM simulations, we experimented with many different choices for basis function width and their number, both with normalized and unnormalized basis functions. It turned out that GTM is somewhat sensitive to these choices: it had difficulties to unfold after a random initialization, unless the basis functions were set so wide (with respect to the weight matrix prior) that the map was well-organized already in its initial configuration. On the other hand, using very wide basis functions with the GTM resulted in a map that was too rigid to adapt well to the data. We also tried to update the parameter {3 according to an annealing schedule, as with the S-Map, but this did not seem to solve the problem. r ." " . ..... : , .. :. f . ~: ~. . .. , n' ... . . 7 ' .. " ~. r? ? . .. .. ': l~ ~..:. ' ' ~'. ,':' [ . " ,'-. .. . I' .' . .: :. : .. ' :. :t;jj ,{ .~.. ".. :. .' f .f.:t+1. ..... . il' .t}'. hS* ~:_I;r7. .. r?? I. .~ ,. .' : ~"" " ..t'>i ~: . : H ~' ? ~? . ';I..r.' '~.' .~ . ,.-' ?I.n ~. Figure 1: Random initialization (top left), SOM (top middle), GTM (top right), "full" S-Map (bottom left), simplified S-Map (bottom middle), On bottom right, the S-Map error as a function of epochs is displayed; the parameter {3 was slightly increased every epoch, which causes the error to increase in the early (unfolding) phase of the learning, as the weight update only minimizes the error for a given {3. 5 Conclusions The S-Map and SOM seem to have a stronger tendency to self-organize from random initialization than the GTM. In data analysis applications, when the GTM can S-Map 555 be properly initialized, SOM, S-Map, and GTM yield comparable results; those obtained using the latter two algorithms are also straightforward to interpret in probabilistic terms. In Euclidean metric, the GTM has the additional advantage of guaranteed convergence to some error minimum; the convergence of the S-Map in Euclidean metric is still an open question. On the other hand, the batch G TM is computationally clearly heavier per epoch than the S-Map, while the S-Map is somewhat heavier than the SOM. The SOM has an impressive record of proven applications in a variety of different tasks, and much more experimenting is needed for any alternative method to reach the same level of practicality. SOM is also the basic bottom-up procedure of selforganization in the sense that it starts from a minimum of functional principles realizable in parallel neural networks. This makes it hard to analyze, however. A probabilistic approach like the GTM stems from the opposite point of view by emphasizing the statistical model, but as a trade-off, the resulting algorithm may not share all the desirable properties of the SOM. Our new approach, the S-map, seems to have succeeded in inheriting the strong self-organization capability of the SOM, while offering a sound probabilistic interpretation like the GTM. References [1] T. Kohonen, Self-Organizing Maps. Springer Series in Information Sciences 30, Berlin Heidelberg New York: Springer, 1995. [2] T. Kohonen, E. Oja, O. Simula, A. Visa, and J. Kangas, "Engineering applications of the self-organizing map," Proceedings of the IEEE, vol. 84, pp. 13581384, Oct. 1996. [3] C. M. Bishop, M. Svensen, and C. K. I. Williams, "GTM: A principled alternative to the self-organizing map," in Advances in Neural Information Processing Systems (to appear) (M. C. Mozer, M. I. Jordan, and T . Petche, eds.), vol. 9, MIT Press, 1997. [4] A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," Journal of the Royal Statistical Society, vol. B 39, no. 1, pp. 1-38, 1977. [5] S. P. Luttrell, "Code vector density in topographic mappings," Memorandum 4669, Defense Research Agency, Malvern, UK, 1992. [6] T. M. Heskes and B. Kappen, "Error potentials for self-organization," in Proceedings of the International Conference on Neural Networks (ICNN'99), vol. 3, (Piscataway, New Jersey, USA), pp. 1219-1223, IEEE Neural Networks Council, Apr. 1993. [7] S. P. Luttrell, "A Bayesian analysis of self-organising maps," Neural Computation, vol. 6, pp. 767-794, 1994. [8] E. Oja, "A simplified neuron model as a principal component analyzer," Journal of Mathematical Biology, vol. 15, pp. 267-273, 1982. [9] K. Kiviluoto, "Topology preservation in self-organizing maps," in Proceedings of the International Conference on Neural Networks (ICNN'96), vol. 1, (Piscataway, New Jersey, USA), pp. 294-299, IEEE Neural Networks Council, June 1996. [10] M. Svensen, The GTM toolbox - user's guide. Neural Computing Research Group / Aston University, Birmingham, UK, 1.0 ed., Oct. 1996. Available at URL http://neural-server . aston. ac. uk/GTM/MATLAB..Impl. html. 

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An Incremental Nearest Neighbor Algorithm with Queries Joel Ratsaby? N.A.P. Inc. Hollis, New York Abstract We consider the general problem of learning multi-category classification from labeled examples. We present experimental results for a nearest neighbor algorithm which actively selects samples from different pattern classes according to a querying rule instead of the a priori class probabilities. The amount of improvement of this query-based approach over the passive batch approach depends on the complexity of the Bayes rule. The principle on which this algorithm is based is general enough to be used in any learning algorithm which permits a model-selection criterion and for which the error rate of the classifier is calculable in terms of the complexity of the model. 1 INTRODUCTION We consider the general problem of learning multi-category classification from labeled examples. In many practical learning settings the time or sample size available for training are limited. This may have adverse effects on the accuracy of the resulting classifier. For instance, in learning to recognize handwritten characters typical time limitation confines the training sample size to be of the order of a few hundred examples. It is important to make learning more efficient by obtaining only training data which contains significant information about the separability of the pattern classes thereby letting the learning algorithm participate actively in the sampling process. Querying for the class labels of specificly selected examples in the input space may lead to significant improvements in the generalization error (cf. Cohn, Atlas & Ladner, 1994, Cohn, 1996). However in learning pattern recognition this is not always useful or possible. In the handwritten recognition problem, the computer could ask the user for labels of selected patterns generated by the computer -The author's coordinates are: Address: Hamered St. #2, Ra'anana, ISRAEL. Eaail: jer@ee. technion. ac.il An Incremental Nearest Neighbor Algorithm with Queries 613 however labeling such patterns are not necessarily representative of his handwriting style but rather of his reading recognition ability. On the other hand it is possible to let the computer (learner) select particular pattern classes, not necessarily according to their a priori probabilities, and then obtain randomly drawn patterns according to the underlying unknown class-conditional probability distribution. We refer to such selective sampling as sample querying. Recent theory (cf. Ratsaby, 1997) indicates that such freedom to select different classes at any time during the training stage is beneficial to the accuracy of the classifier learnt. In the current paper we report on experimental results for an incremental algorithm which utilizes this sample-querying procedure . 2 THEORETICAL BACKGROUND We use the following setting: Given M distinct pattern classes each with a class conditional probability density fi (x), 1 ::; i ::; M, x E IRd, and a priori probabilities Pi, 1 ::; i ::; M. The functions fi (x), 1 ::; i ::; M, are assumed to be unknown while the Pi are assumed to be known or easily estimable as is the case oflearning character recognition. For a sample-size vector m = [m1' ... , mM] where L~1 mj = m denote by (m = {( x j , Yj ) } 1 a sample of labeled examples consisting of mi example from pattern class i where Yj, 1 ::; j ::; m, are chosen not necessarily at random from {I, 2 .... , M}, and the corresponding Xj are drawn at random i.i.d. according to the class conditional probability density fy) (x). The expected misclassification error of a classifier c is referred to as the loss of c and is denoted by L( c ). It is defined as the probability of miselassification of a randomly drawn x with respect to the underlying mixture probability density function f(x) = L~l pdi(X). The loss is commonly represented as L(c) = El{x :c(x);iy(x)}, where l{xEA} is the indicator function of a set A, expectation is taken with respect to the joint probability distribution fy (x )p(y) where p(y) is a discrete probability distribution taking values Pi over 1 ::; i ::; M, while y denotes the label of the class whose distribution fy(x) was used to draw x. T= The loss L(e) may also be written as L(e) = Lf!l PiEi1{c(x);ii} where Ei denotes expectation with respect to fi(X) . The pattern recognition problem is to learn based on the optimal classifier, also known as the Bayes classifier, which by definition has minimum loss whkh we denote by L * . em A multi-category classifier c is represented as a vector e(x) = [el(x), . .. , CM(X)] of boolean classifiers, where Ci (x) = 1 if e( x) = i, and Ci (x) = 0 otherwise, 1 ::; i ::; M. The loss L(e) of a multi-category classifier c may then be expressed as the average of the losses of its component classifiers, i.e., L(e) = L~l PiL(ei) where for a boolean classifier ei the loss is defined as L(ed = Ed{c.(x);il}' As an estimate of L(e) we define the empirical loss Lm(c} = L~l p;Lm.(e) where Lm,(c) = ~, Lj :Y1=i l{c(x);ii} which may also can be expressed as Lm,(ci) = ~, Lj:YJ=i l{c.(x);il}' The family of all classifiers is assumed to be decomposed into a multi-structure 5 51 X 52 X .. , X 5 M , where 5 j is a nested structure (cf. Vapnik, 1982) of boolean families Bk). ' ji = 1,2, ... , for 1 ::; i ::; M, i.e., 51 Bkl , Bk 2 , ?? ? ,Bk)1 ' ... , 52 = BkllBk2,???,Bk12'oo" up to 5M = Bk1 ,Bk2 ,? .. ,Bk) M ' ? ' " where ki, E71+ denotes the VC-dimension of BkJ , and Bk 1 , ~ Bk1,+I' 1 ::; i ::; M . For any fixed positive integer vector j E 7l ~ consider the class of vector classifiers 1i k(j) = Bk1 1 x .Bk x? .. X Bk 1M == 1ik where we take the liberty in dropping the multil2 index j and write k instead of k(j) . Define by (h the subfamily of 1ik consisting = = J. Ratsaby 614 of classifiers C that are well-defined , i.e., ones whose components Ci, 1 S; i S; M satisfy Ut!I{X: Ci(X) I} IRd and {x : Ci(X) l}n{x: Cj(x) I} 0, for 1 S; i =1= j S; M . = = = = = From the Vapnik-Chervonenkis theory (cf. Vapnik , 1982, Devroye, Gyorfi & Lugosi , 1996) it follows that the loss of any boolean classifier Ci E Bk ], is , with high confidence, related to its empirical loss as L( Ci ) S Lm. (Ci) + f( m i, kj ,) where f(mi' kj ,) = const Jkj,ln mi!mi , 1 S; i S; M , where henceforth we denote by const any constant which does not depend on the relevant variables in the expression. Let the vectors m = [ml, "" mM] and k == k(j) = [kil , . ??, kjM] in 'lh~. Define f(m, k) = 2:f'!1 Pif(mi ,kj,) . It follows that the deviation between the empirical loss and the loss is bounded uniformly over all multi-category classifiers in a class (}k by f(m, k) . We henceforth denote by c k the optimal classifier in (}k , i.e., ck = argmincE~h L( c) and Ck = argmincEQk Lm (c) is the empirical loss minimizer over the class (}k. The above implies that the classifier Ck has a loss which is no more than L( c~) + f(m, k) . Denote by k* the minimal complexity of a class (}k which contains the Bayes classifier. We refer to it as the Bayes complexity and henceforth assume k: < 00, 1 S; i S; M. If k* was known then based on a sample of size m with a sample size vector m = [ml , "" mM] a classifier Ck o whose loss is bounded from above by L * + f( m, k*) may be determined where L * = L( c~o) is the Bayes loss. This bound is minimal with respect to k by definition of k* and we refer to it as the minimal criterion. It can be further minimized by selecting a sample of size vector m* = argmin{ 'WM . "\,,,M _ _ }f(m,k*). This basically says that more examples mEaJ+ '6,=1 m,_m should be queried from pattern classes which require more complex discriminating rules within the Bayes classifier. Thus sample-querying via minimization of the minimal criterion makes learning more efficient through tuning the subsample sizes to the complexity of the Bayes classifier. However the Bayes classifier depends on the underlying probability distributions which in most interesting scenarios are unknown thus k* should be assumed unknown . In (Ratsaby, 1997) an incremental learning algorithm, based on Vapnik's structural risk minimization, generates a random complexity sequence ken) , corresponding to a sequence of empirical loss minimizers ck(n) over (}k(n), which converges to k* with increasing time n for learning problems with a zero Bayes loss. Based on this, a sample-query rule which achieves the same minimization is defined without the need to know k*. We briefly describe the main ideas next. At any time n, the criterion function is c(-, ken)) and is defined over the m-domain 'lhtt? A gradient descent step of a fixed size is taken to minimize the current criterion. After a step is taken , a new sample-size vector men + 1) is obtained and the difference m( n + 1) - m( n) dictates the sample-query at time n, namely, the increment in subsample size for each of the M pattern classes. With increasing n the vector sequence men) gets closer to an optimal path defined as the set which is comprised of the solutions to the minimization of f( m, k*) under all different constraints of 2:~1 mi = m, where m runs over the positive integers. Thus for all large n the sample-size vector m( n) is optimal in that it minimizes the minimal cri terion f(', k*) for the current total sample size m( n). This consti tutes the samplequerying procedure of the learning algorithm. The remaining part does empirical loss minimization over the current class (}k(n) and outputs ck(n)" By assumption, since the Bayes classifier is contained in (}k o , it follows that for all large n, the loss L(ck(n? S; L* + min{mE~ :2::1 m,=m'(n)} f(m, k*), which is basically the minimal criterion mentioned above. Thus the algorithm produces a classifier ck(n) with a An Incremental Nearest Neighbor Algorithm with Queries 615 minimal loss even when the Bayes complexity k* is unknown. In the next section we consider specific modf'l classes consisting of nearest-nf'ighbor classifiers on which we implement this incremental learning approach. 3 INCREMENTAL NEAREST-NEIGHBOR ALGORITHM Fix and Hodges, cf. Silverman & Jones (1989). introduced the simple but powerful nearest-neighbor classifier which based on a labeled training sample {(Xj,yd}i!:I' Xi E m,d, Yi E {I, 2, ... , M}, when given a pattern x, it outputs the label Yj corresponding to the example whose x j is closest to x. Every example in the training sample is used for this decision (we denote such an example as a prototype) thus the empirical loss is zero. The condensed nearest-neighbor algorithm (Hart, 1968) and the reduced nearest neighbor algorithm (G ates, 1972) are procedures which aim at reducing the number of prototypes while maintaining a zero empirical loss. Thus given a training sample of size m, after running either of these procedures, a nearest neighbor classifier having a zero empirical loss is generated based on s ~ m prototypes. Learning in this manner may be viewed as a form of empirical loss minimization with a complexity regularization component which puts a penalty proportional to the number of prototypes. A cell boundary ej,j of the voronoi diagram (cf. Preparata & Shamos, 1985) corresponding to a multi-category nearest-neighbor classifier c is defined as the (d - 1)-dimensional perpendicular-bisector hyperplane <?f the line connecting the x-component of two prototypes Xi and Xj. For a fixed I E {1, ... ,M}, the collection of voronoi cell-boundaries based on pairs of prototypes of the form (xi,/), (Xj,q) where q =1= I, forms the boundary which separates the decision region labeled I from its complement and represents the boolean nearest-neighbor classifier CI. Denote by kl the number of suc.h cell-boundaries and denote by SI the number of prototypes from a total of ml examples from pattern class t. The value of kl may be calculated directly from the knowledge of the SI prototypes, 1 ~ I ~ M, using various algorithms. The boolean classifier Cl is an element of an infinite class of boolean classifiers based on partitions of m,d by arrangements of kl hyperplanes of dimensionality d - 1 where each of the cells of a partition is labeled either 0 or 1. It follows, cf. Devroye et. al. (1996), that the loss of a multi-category nearestneighbor classifier C which consists of 81 prototypes out of ml examples, 1 ~ I ~ M, is bounded as L(c) ~ Lm(c) + f(m, k), where the a priori probabilities are taken as known, m [mI, ... ,mM)' k [k I , ... ,kM] and f(m,k) E~lPlf(ml,kl)' where f( ml, kz) = const ? d + 1 )kl In ml + (ekd d)d) / mi . Letting k* denote the Bayes complexity then f(-, k*) represents the minimal criterion. = J = = The next algorithm uses the Condense and Reduce procedures in order to generate a sequence of classifiers ck(n) with a complexity vector k( n) which tends t.o k* as n --+ 00. A sample-querying procedure referred to as Greedy Query (GQ) chooses at any time n to increment the single subsample of pattern class j*(n) where mjO(n) is the direction of maximum descent of the criterion f(', k( n)) at the current sample-size vector m( n). For the part of the algorithm which utilizes a Delaunay-Triangulation procedure we use the fast Fortune's algorithm (cf. 0 'Rourke) which can be used only for dimensionality d = 2. Since all we are interested is in counting Voronoi borders between all adjacent Voronoi cells then an efficient computation is possible also for dimensions d > 2 by resorting to linear programming for computing the adjacencies of facets of a polyhedron, d. Fukuda (1997). 1. Ratsaby 616 Incremental Nearest Neighbor (INN) Algorithm Initialization: (Time n = 0) Let increment-size t::. be a fixed small positive integer. Start with m(O) = [e, . .. , e], where e is a small posit.ive integer. Draw (m(o) = {(m](o)}?";l where (m)(O ) consists of mJ(O) randomly drawn i.i.d. examples from pattern class j. While (number of available examples 2: t::.) Do: 1. Call Procedure CR: chIn ) = CR?(m(n? . 2. Call Procedure GQ: m(n 3. n:= n + 1) = GQ(n). + 1. End While //Used up all examples . Output: NN-cIassifier ck(n). Procedure Condense-Reduce (CR) Input: Sample (m(n) stored in an array A[] of size m(n). Initialize: Make only the first example A[I] be a prototype. //Condense Do: ChangeOccl.lred := FALSE. For i= 1, . .. , m(n): ? Classify A[i] based on available prototypes using the NN-Rule. ? If not correct then - Let A[i] be a prototype. - ChangeOcel.lred:= TRUE. ? End If End For While ( ChangeOecl.lred). //Reduce Do: ChangeOccl.lred := FALSE. For i = 1, ... . m(n): ? If A[ i] is a prototype then classify it using the remaining prototypes by the NN-Rule. ? If correct then - Make A[i] be not a prototype. - ChangeOccl.lred := TRUE. ? End If End For While ( ChangeOec'Ured) . Run Delaunay-Triangulation Let k(n) = [k~, ... , kM ], k. denotes the number of Voronoi-cell boundaries associated with the s, prototypes. Return (NN-classifier with complexity vector k(n?. Procedure Greedy-Query (GQ) Input: Time n. j*(n) := I argmaxl~J~M a!) f(m, k(n?1 Im(n) Draw: t::. new i.i.d. examples from class j?(n). Denote them by ( . Update Sample: (m]?(n)(n+l) := Cn)?(n)Cn) U (, while (m,Cn+l) := (m,Cn), for 1 :S i:;i: j?(n) :S M . Return: (m(n)+ t::. eJ.Cn?, where eJ is an all zero vector except 1 at jth element. An Incremental Nearest Neighbor Algorithm with Queries 3.1 617 EXPERIMENTAL RESULTS We ran algorithm INN on several two-dimensional (d = 2) multi-category classification problems and compared its generalization error versus total sample size m with that of batch learning, the latter uses Procedure CR (but not Procedure GQ) with uniform subsample proportions, i.e., mi = ~, 1 ~ i ~ M. We ran three classification problems consisting of 4 equiprobable pattern classes with a zero Bayes loss. The generalization curves represent the average of 15 independent learning runs of the empirical error on a fixed size test set. Each run (both for INN and Batch learning) consists of 80 independent experiments where each differs by 10 in the sample size used for training where the maximum sample size is 800. We call an experiment a success if INN results in a lower generalization error than Batch. Let p be the probability of INN beating Batch. We wish to reject the hypothesis H that p ~ which says that INN and Batch are approximately equal in performance. The results are displayed in Figure 1 as a series of pairs, the first picture showing the pattern classes of the specific problem while the second shows the learning curves for the two learning algorithms. Algorithm INN outperformed the simple Batch approach with a reject level of less than 1%, the latter ignoring the inherent Bayes complexity and using an equal subsample size for each of the pattern classes. In contrast, the INN algorithm learns, incrementally over time, which of the classes are harder to separate and queries more from these pattern classes. = References Cohn D., Atlas L., Ladner R. (1994), Improving Generalization with Active Learning. Machine Learning, Vol 15, p.201-221. Devroye L., Gyorfi L. Lugosi G. (1996). "A Probabilistic Theory of Pattern Recognition", Springer Verlag. Fukuda K. (1997). Frequently Asked Questions in Geometric Computation. Technical report, Swiss Federal Institute of technology, Lausanne. Available at ftp://ftp.ifor.ethz.ch/pub/fukuda/reports. Gates, G. W. (1972) The Reduced Nearest Neighbor Rule. IEEE Trans. Theo., p.431-433. Info. Hart P. E. (1968) The Condensed Nearest Neighbor Rule. IEEE Trans. on Info. Thea., Vol. IT-14, No.3. O'rourke J . (1994). "Computational Geometry in C". Cambridge University Press. Ratsaby, J. (1997) Learning Classification with Sample Queries. Electrical Engineering Dept., Technion, CC PUB #196. Available at URL http://www.ee.technion.ac.il/jer/iandc.ps. Rivest R. L., Eisenberg B. (1990), On the sample complexity of pac-learning using random and chosen examples. Proceedings of the 1990 Workshop on Computational Learning Theory, p. 154-162, Morgan Kaufmann, San Maeto, CA. B. W. Silverman and M. C. Jones . E. Fix and J. 1. Hodges (1951): An important contribution to nonparametric discriminant analysis and density estimationcommentary on Fix and Hodges (1951). International statistical review, 57(3), p.233-247, 1989. Vapnik V.N., (1982), "Estimation of Dependences Based on Empirical Data", Springer-Verlag. Berlin. 618 J. Ratsaby n.2 ~" __- - - I ' - - - - " " I - - - - r - - - - ' n.1 ~, il ' \ . ~ 0.1 \ , .~ ~. I ~ \. ',. 8 o.0K ~":-. !t~ 0.06 0.04 I I 100 1:-0 0.0 PoltcmCI3.~S I 0 100 PaltcmCla.... 2 PoltcmCl3.'i.'i.l '..00 200 fb- '""'" ~OO ", 400 ..: 500 ' . 600 "'-, ,' . 700 ROO Tolal number of """"",I.... Balch l'a"cmCJa.~s4 I(ICl( . ..... "<.., . :!OO ....... INN 0.25 O. ~ 0.17 ..,; .~ 0.15 .~ O. IJ ;; Iic 0.1 C 0.011 -~""''' ............. ....... -.....:""," . 0.05 (.<>,) l'aU<:m('I_1 PaucmCl_2 0.03 l'aUc:mCIass.~ Cit( 0 l'aUcmCl-.l 100 2110 JOn 400 SIlO TnUl numhcr of """"",lea Balch -(r CoX w'Ic .?~1~< : ., I( IC( . , ';;'{ .Itc>fC" i.8cl"? 0.26 \?qllc . ~ 0.24 , ~ I(~ ? I ~PC( icc ~ ,(. xi/(o( I( 0 , ><'V-'~t. 'S .:: (# ( ~"'x 0 O.J 0.2 ., .... '~ 'J 1 - (II, INN ~- t.f. ~::?.t"<i .... ~:I( _-.:-c ~ 100 PanemCI ..... 1 ",,,,. PanemClass2 PananCI.. 1(( PanemCl.....4 ....u ~.>...-,...". ~ I~ . ~ 0.22 ."? J 0.2 "ii 0.1 ? 200 0. 16 0. 14 0.12 0.1 0 400 ToI.tl numher of cxamplCli Batch -- INN Figure 1. Three different Pattern Classification Problems and Learning Curves of the INN-Algorithm compared to Batch Learning. (1110 70n 1100 

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Competitive On-Line Linear Regression V. Vovk pepartment of Computer Science Royal Holloway, University of London Egham, Surrey TW20 OEX, UK vovkGdcs.rhbnc.ac.uk Abstract We apply a general algorithm for merging prediction strategies (the Aggregating Algorithm) to the problem of linear regression with the square loss; our main assumption is that the response variable is bounded. It turns out that for this particular problem the Aggregating Algorithm resembles, but is slightly different from, the wellknown ridge estimation procedure. From general results about the Aggregating Algorithm we deduce a guaranteed bound on the difference between our algorithm's performance and the best, in some sense, linear regression function's performance. We show that the AA attains the optimal constant in our bound, whereas the constant attained by the ridge regression procedure in general can be 4 times worse. 1 INTRODUCTION The usual approach to regression problems is to assume that the data are generated by some stochastic mechanism and make some, typically very restrictive, assumptions about that stochastic mechanism. In recent years, however, a different approach to this kind of problems was developed (see, e.g., DeSantis et al. [2], Littlestone and Warmuth [7]): in our context, that approach sets the goal of finding an on-line algorithm that performs not much worse than the best regression function found off-line; in other words, it replaces the usual statistical analyses by the competitive analysis of on-line algorithms. DeSantis et al. [2] performed a competitive analysis of the Bayesian merging scheme for the log-loss prediction game; later Littlestone and Warmuth [7] and Vovk [10] introduced an on-line algorithm (called the Weighted Majority Algorithm by the Competitive On-line Linear Regression 365 former authors) for the simple binary prediction game. These two algorithms (the Bayesian merging scheme and the Weighted Majority Algorithm) are special cases of the Aggregating Algorithm (AA) proposed in [9, 11]. The AA is a member of a wide family of algorithms called "multiplicative weight" or "exponential weight" algorithms. Closer to the topic of this paper, Cesa-Bianchi et al. [1) performed a competitive analysis, under the square loss, of the standard Gradient Descent Algorithm and Kivinen and Warmuth [6] complemented it by a competitive analysis of a modification of the Gradient Descent, which they call the Exponentiated Gradient Algorithm. The bounds obtained in [1, 6] are of the following type: at every trial T, (1) where LT is the loss (over the first T trials) of the on-line algorithm, LT is the loss of the best (by trial T) linear regression function, and c is a constant, c > 1; specifically, c = 2 for the Gradient Descent and c = 3 for the Exponentiated Gradient. These bounds hold under the following assumptions: for the Gradient Descent, it is assumed that the L2 norm of the weights and of all data items are bounded by constant 1; for the Exponentiated Gradient, that the Ll norm of the weights and the Loo norm of all data items are bounded by 1. In many interesting cases bound (1) is weak. For example, suppose that our comparison class contains a "true" regression function, but its values are corrupted by an Li.d. noise. Then, under reasonable assumptions about the noise, LT will grow linearly in T, and inequality (1) will only bound the difference LT - LT by a linear function of T. (Though in other situations bound (1) can be better than our bound (2), see below. For example, in the case of the Exponentiated Gradient, the 0(1) in (1) depends on the number of parameters n logarithmically whereas our bound depends on n linearly.) In this paper we will apply the AA to the problem of linear regression. The AA has been proven to be optimal in some simple cases [5, 11], so we can also expect good performance in the problem of linear regression. The following is a typical result that can be obtained using the AA: Learner has a strategy which ensures that always (2) LT ~ LT + nIn(T + 1) + 1 (n is the number of predictor variables). It is interesting that the assumptions for the last inequality are weaker than those for both the Gradient Descent and Exponentiated Gradient: we only assume that the L2 norm of the weights and the Loo norm of all data items are bounded by constant 1 (these assumptions will be further relaxed later on). The norms L2 and Loo are not dual, which casts doubt on the accepted intuition that the weights and data items should be measured by dual norms (such as Ll-Loo or L 2-L2). Notice that the logarithmic term nln(T + 1) of (2) is similar to the term ~ In T occurring in the analysis of the log-loss game and its generalizations, in particular in Wallace's theory of minimum message length, Rissanen's theory of stochastic complexity, minimax regret analysis. In the case n = 1 and Xt = 1, Vt, inequality (2) differs from Freund's [4] Theorem 4 only in the additive constant. In this paper we will see another manifestation of a phenomenon noticed by Freund [4]: for some important problems, the adversarial bounds of on-line competitive learning theory V. Vovk 366 are only a tiny amount worse than the average-case bounds for some stochastic strategies for Nature. A weaker variant of inequality (2) can be deduced from Foster's [3] Theorem 1 (if we additionally assume that the response variable take only two values, -1 or 1): Foster's result implies LT ~ LT + 8n In(2n(T + 1)) + 8 (a multiple of 4 arises from replacing Foster's set {O, 1} of possible values of the response variable by our {-1, 1}j we also replaced Foster's d by 2n: to span our set of possible weights we need 2n Foster's predictors). Inequality (2) is also similar to Yamanishi's [12] resultj in that paper, he considers a more general framework than ours but does not attempt to find optimal constants. 2 ALGORITHM We consider the following protocol of interaction between Learner and Nature: FOR t = 1,2, ... Nature chooses Xt ? m.n Learner chooses prediction Pt E Nature chooses Yt E [-Y, Y] END FOR. m. This is a "perfect-information" protocol: either player can see the other player's moves. The parameters of our protocol are: a fixed positive number n (the dimensionality of our regression problem) and an upper bound Y > 0 on the value Yt returned by Nature. It is important, however, that our algorithm for playing this game (on the part of Learner) does not need to know Y. We will only give a description of our regression algorithmj its derivation from the general AA will be given in the future full version of this paper. (It is usually a nontrivial task to represent the AA in a computationally efficient form, and the case of on-line linear regression is not an exception.) Fix n and a > O. The algorithm is as follows: A :=alj b:=O FOR TRlAL t = 1,2, ... : read new Xt E m.n A:= A +XtX~ output prediction Pt := b' A -1 Xt read new Yt E m. b:= b+YtXt END FOR. In this description, A is an n x n matrix (which is always symmetrical and positive definite), bE mn , I is the unit n x n matrix, and 0 is the all-O vector. The naive implementation of this algorithm would require O(n3) arithmetic operations at every trial, but the standard recursive technique allows us to spend only O(n2 ) arithmetic operations per trial. This is still not as good as for the Gradient Descent Algorithm and Exponentiated Gradient Algorithm (they require only O(n) Competitive On-line Linear Regression 367 operations per trial); we seem to have a trade-off between the quality of bounds on predictive performance and computational efficiency. In the rest of the paper "AA" will mean the algorithm. described in the previous paragraph (which is the Aggregating Algorithm applied to a particular uncountable pool of experts with a particular Gaussian prior). 3 BOUNDS In this section we state, without proof, results describing the predictive performance of our algorithm. Our comparison class consists of the linear functions Yt = W? Xt, where W E m.n ? We will call the possible weights w "experts" (imagine that we have continuously many experts indexed by W E m.n ; Expert w always recommends prediction w . Xt to Learner). At every trial t Expert w and Learner suffer loss (Yt - w . Xt)2 and (Yt - Pt)2, respectively. Our notation for the total loss suffered by Expert w and Learner over the first T trials will be T LT(W) := L(Yt - W? Xt)2 t=1 and T LT(Learner) := L(Yt - Pt)2, t=1 respectively. For compact pools of experts (which, in our setting, corresponds to the set of possible weights w being bounded and closed) it is usually possible to derive bounds (such as (2? where the learner's loss is compared to the best expert's loss. In our case of non-compact pool, however, we need to give the learner a start on remote experts. Specifically, instead of comparing Learner's performance to infw LT(W), we compare it to infw (LT(W) + allwlI 2 ) (thus giving ~arner a start of allwII 2 on Expert w), where a > 0 is a constant reflecting our prior expectations about the "complexity" IIwll := of successful experts. -IE:=1 w; This idea of giving a start to experts allows us to prove stronger results; e.g., the following elaboration of (2) holds: (3) (this inequality still assumes that IIXtiloo ~ 1 for all t but w is unbounded). Our notation for the transpose of matrix A will be A'; as usual, vectors are identified with one-column matrices. Theorem 1 For any fi:ted n, Leamer has a strategy which ensures that always V. Vovk 368 II, in addition, IIxt II 00 $ X, \It, (4) The last inequality of this theorem implies inequality (3): it suffices to put X = Y=a=1. The term lndet (1 +; t,x.x:) in Theorem 1 might be difficult to interpret. Notice that it can be rewritten as nlnT + lndet (~I + ~COV(Xl' ... ,Xn)) , where cov(Xl , ... , Xn) is the empirical covariance matrix of the predictor variables (in other words, cov(Xl , ... ,Xn) is the covariance matrix of the random vector which takes the values Xl, ... ,XT with equal probability ~). We can see that this term is typically close to n In T. Using standard transformations, it is easy to deduce from Theorem 1, e.g., the following results (for simplicity we assume n = 1 and Xt,Yt E [-1,1], 'It): ? if the pool of experts consists of all polynomials of degree d, Learner has a strategy guaranteeing ? if the pool of experts consists of all splines of degree d with k nodes (chosen a priori), Learner has a strategy guaranteeing LT(Learner) S inf (LT(W) w + Ilw1l 2 ) + (d + k + 1) In(T + 1). The following theorem shows that the constant n in inequality (4) cannot be improved. Theorem 2 Fix n (the number 01 attributes) and Y (the upper bound on IlItl). For any f > 0 there exist a constant C and a stochastic strategy lor Nature such that IIxtiloo = 1 and Illtl = Y, lor all t, and, lor any stochastic strategy lor Learner, E (LT(Learner) - 4 inf w:llwll:SY LT(W)) ~ (n - f)y21nT - C, 'IT. COMPARISONS It is easy to see that the ridge regression procedure sometimes gives results that are not sensible in our framework where lit E [- Y, Y] and the goal is to compete Competitive On-line Linear Regression 369 against the best linear regression function. For example, suppose n and Nature generates outcomes (Xt, Yt), t = 1,2, ... , where a ? Xl ? X2 ? ... , = 1, Y = 1, _ { 1, if todd, Yt -1, if t even. At trial t = 2,3, ... the ridge regression procedure (more accurately, its natural modification which truncates its predictions to [-1, 1]) will give prediction Pt = Yt-l equal to the previous response, and so will suffer a loss of about 4T over T trials. On the other hand, the AA's prediction will be close to 0, and so the cumulative loss of the AA over the first T trials will be about T, which is close to the best expert's loss. We can see that the ridge regression procedure in this situation is forced to suffer a loss 4 times as big as the AA's loss. The lower bound stated in Theorem 2 does not imply that our regression algorithm is better than the ridge regression procedure in our adversarial framework. (Moreover, the idea of our proof of Theorem 2 is to lower bound the performance of the ridge regression procedure in the situation where the expected loss of the ridge regression procedure is optimal.) Theorem 1 asserts that LT(Leamer) :S ~ (LT( w) + allwl12) + y2 t. In ( 1 + t. ~ X~.i) (5) when Learner follows the AA. The next theorem shows that the ridge regression procedure sometimes violates this inequality. Theorem 3 Let n = 1 (the number 0/ attributes) and Y = 1 (the upper bound on IYtl); fix a > O. Nature has a strategy such that, when Learner plays the ridge regression strategy, LT(Learner) = 4T + 0(1), (6) inf (LT(w) UI In as T 5 -4 00 + allwll2) = T + 0(1), (1+ ~ t.x~) = TIn2+ 0(1) (7) (8) (and, there/ore, (5) is violated). CONCLUSION A distinctive feature of our approach to linear regression is that our only assumption about the data is that IYt I ~ Y, 'tit; we do not make any assumptions about stochastic properties of the data-generating mechanism. In some situations (if the data were generated by a partially known stochastic mechanism) this feature is a disadvantage, but often it will be an advantage. This paper was greatly influenced by Vapnik's [8] idea of transductive inference. The algorithm analyzed in this paper is "transductive", in the sense that it outputs some prediction Pt for Yt after being given Xt, rather than to output a general rule for mapping Xt into Ptj in particular, Pt may depend non-linearly on Xt. (It is easy, however, to extract such a rule from the description of the algorithm once it is found.) V. Vovk 370 Acknowledgments Kostas Skouras and Philip Dawid noticed that our regression algorithm is different from the ridge regression and that in some situations it behaves very differently. Manfred Warmuth's advice about relevant literature is also gratefully appreciated. References [11 N. Cesa-Bianchi, P. M. Long, and M. K. Warmuth (1996), Worst-case quadratic loss bounds for on-line prediction of linear functions by gradient descent, IEEE 7rons. Neural Networks 7:604-619. [21 A. DeSantis, G. Markowsky, and M. N. Wegman (1988), Learning probabilistic prediction functions, in "Proceedings, 29th Annual IEEE Symposium on Foundations of Computer Science," pp. 110-119, Los Alamitos, CA: IEEE Comput. Soc. [31 D. P. Foster (1991), Prediction in the worst case, Ann. Statist. 19:1084-1090. [4] Y. Freund (1996), Predicting a binary sequence almost as well as the optimal biased coin, in "Proceedings, 9th Annual ACM Conference on Computational Learning Theory" , pp. 89-98, New York: Assoc. Comput. Mach. [5] D. Haussler, J. Kivinen, and M. K. Warmuth (1994), Tight worst-case loss bounds for predicting with expert advice, University of California at Santa Cruz, Technical Report UCSC-CRL-94-36, revised December. Short version in "Computational Learning Theory" (P. Vitanyi, Ed.), Lecture Notes in Computer Science, Vol. 904, pp. 69-83, Berlin: Springer, 1995. [6] J. Kivinen and M. K. Warmuth (1997), Exponential Gradient versus Gradient Descent for linear predictors, Inform. Computation 132:1-63. [7] N. Littlestone and M. K. Warmuth (1994), The Weighted Majority Algorithm, Inform. Computation 108:212-261. [8] V. N. Vapnik (1995), The Nature of Statistical Learning Theory, New York: Springer. [9] V. Vovk (1990), Aggregating strategies, in "Proceedings, 3rd Annual Workshop on Computational Learning Theory" (M. Fulk and J. Case, Eds.), pp. 371-383, San Mateo, CA: Morgan Kaufmann. [10] V. Vovk (1992), Universal forecasting algorithms, In/orm. Computation 96:245-277. [11] V. Vovk (1997), A game of prediction with expert advice, to appear in J. Comput. In/orm. Syst. Short version in "Proceedings, 8th Annual ACM Conference on Computational Learning Theory," pp. 51-60, New York: Assoc. Comput. Mach., 1995. [12] K. Yamanishi (1997), A decision-theoretic extension of stochastic complexity and its applications to learning, submitted to IEEE 7rons. In/orm. Theory. 

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99 Connectionist Learning of Expert Preferences by Comparison Training Gerald Tesauro IBl\f Thomas.1. '''atson Rcsearc11 Centcr PO Box 704, Yorktown Heights, NY 10598 USA Abstract A new training paradigm, caned the "eomparison pa.radigm," is introduced for tasks in which a. network must learn to choose a prdcrred pattern from a set of n alternatives, based on examplcs of Imma.n expert prderences. In this pa.radigm, the inpu t to the network consists of t.wo uf the n alterna tives, and the trained output is the expert's judgement of which pa.ttern is better. This para.digm is applied to the lea,rning of hackgammon, a difficult board ga.me in wllieh the expert selects a move from a. set, of legal mm?es. \Vith compa.rison training, much higher levels of performance can hc a.chiew~d, with networks that are much smaller, and with coding sehemes t.hat are much simpler and easier to understand. Furthermorf', it is possible to set up the network so tha.t it always produces consisten t rank-orderings . 1. Introduction There is now widespread interest, in tlH~ use of conncctillnist networks fllr realworld practical problem solving. The principal areas of applica.tion which have been studied so far involvc rela tiv('ly low-level signal processing and pattern recognition t.asks. However, eOllllectionist networks might also he useful in higher-level tasks whkh a.re curr('nt.l~? tackled hy cxprrt systems a.nd knowledge engineering approadl(,s [2]. In this pa per, 'vc considcr problem domains in which tlte expert is givcn a s('t of 71. alt.enulti,?es as input (71. may be either sma.II or large), and mnst select. t.}l<' m()st dC'sirtlhl(' ()T most prderable alternative. This type of task occurs rep('atccUy throughout the domains of politics, business, eeonomics, mcdicine, a.nd many ot hers. \Vhcthcr it is choosing a fureign-policy option, a. w('apons cont,rador, a course of trea.tment for a disease, or simply what to have for dinner, prohlems requiring choice are constan tly being faced and 801\'(\(1 by 111lma.n experts. How might a. learning system snch as a. cOlllH,ctionist network be set up to learn to ma.ke such choices from human expert cxampI~s? The immediately obvious a,pproa,c h is to train the network to produce a numerical output 100 Tesauro "score" for ea,ch input alterna.tive. To make a, choice, then, une would have the network score each a.lterna,tive, and select t.he alterna tive with the highest score. Since the lea,rning system learns from examples, it. seems logical to train the network on a da.ta base of examples in which a, human expert has entered a numerical score for ea,ch possible choice. Howc\'cr, there are two major problems with such a,n a,pproa,c h. First, in many domains in which n is large, it would be tremendously time-consuming for the expert to create a, data base in which each individual a.lternative has been painstaking evalua,ted, even the vast number of obviously ba,d alterna.tives which are not even worth considering. (It is importa.nt for the network t,o sec examples of ba,d alternatives, otherwise it would tend t.o produce high scores for everything.) l\1:ore importa.ntly, in ma,n y domains human experts do not think in terms of absolute scoring functions, and it would thus be extremely difficult to create training data containing a,hsolute scores, because such scoring is alien to the expert's wa,y of thinking a,h out the problem. Inst.cad, the most natural way to make training data is simply tu record the expcrt in adinn, i.e., for ea,eh problem situation, record each of the tlltcrna,th'cs hc htld t.o choose from, a,n d record which one he actually selected. . For these reasons, we advoca.te teaching the network to compare pairs of alterna.tives, rather than scoring individual aIterna.tivcs. In ot.her words, the input should be two of the set of n alternatives, and thc output should be a 1 or 0 depending on which of the two alterna.tives is hetter. From a set of recorded huma,n expert preferences, one can then teach thc network that the expert's choice is better than all other a.ltcrna tivcs. One potential concern raised by this approach is tha t, in performance mode a,fter the network is trained, it might he neccssary t.o mtlke n 2 comparisons to select the best alterna.tive, whereas only n individual scor('s tlrc needed in the other approa,ch. However, the network can select the hcst alterna.tive with only n compa.risons by going through the list of altern a tives in ordcr, a.nd compa,ring the current alternative wHh thc b('st ait.erntltive secn so fa.r. If the current alterna.tive is better, it becomcs the ncw 1>('st altcrnative, and if it is worse, it is rejected. Anot.hcr poten tial COIlcern is tha t tI. network which only knows how t.o compare might not produce a consist('nt rank-ordering, i.e., it might sa.y that alternativc a is better t.han h, b is hdtN t.han c, and c is better than a, and then one do('s not know which alt('rnative to select. However, we shall see la.ter that it is possiblc to gllartlutc'c (,(lllsist('ncy with a constrained a.rchitecture which forccs t.he network to c:()mc~ IIp wit.h absolute numerical scores for individual alterna tives. In the following, we shall exa.minc the applic~tI. ti()n of t.he comparison training pa.radigm to the ga.me of backgammon, as considC'ra hIe cxperience ha.s already been obtained in this domain. In previous papC'rs [7,6]' a network was described which lea.rned to pla.y ba,c kgammon from an expert data, base, using the so-called "back-proptl,ga.tion" learning rule [5J. In that system, the network was trained to score individual moyes. In other words, the input Connectionist Learning of Expert Preferences consists of a move (drfined by the initi::ll position bdnre the move a.nd the final position after the move), and the desired output is a. real number indica.ting the strength of the move. Henceforth we shall refer to this training paradigm as the "rcla.t.ive score" par::ldigm. 'Vhilc this approach produced considerable success, it had a. number of serious limitations. \Ve sha.ll see that the compa.rison paradigm solyes one of the most important limita.tions of t.he previous a.pproach, with the rrsult tha t the overall performance of Ute network is much bet ter, the llum her of connedions required is greatly reduced, a.nd. the network's input coding scheme is much simpler and easier to understa.nd. 2. Previous backgammon networks In [7], a network was descrihed which learnrd tl) pla~? fairly good ba.ckga.mmon by ba.ck-propa.gation learning of a. large expert t.raining set, using the rcla,tive score paradigm describc(l pr(,yiollsly. After tr::lining, the network was tested both by measuring its prrformance on ::I t('st sct. of positions not used in training, and hy a,ctual game pl::l)' against hunums ::Ind cOllventional computer programs. The best net.work was ablr to drfeat. Sun :Microsysterns' Gammontool, Ute best available commercial program, hy a sllbsta.ntial ma.rgin, but it was still fa.r from human expert-level performance. The basic conclusion of [7] was that it was possihle to achieve decent levels of performance by this network learning procedure, but it, was not a.n ea.sy matter, and required suhst.antial hum::ln intervention. The choice of a. coding scheme for the inpnt informa.t.ion, for rxamplr, was fonn(l to be an extremely important issue. The hest coding SclH'IrleS contained a. great deal of doma.inspecific information. The best, encoding of the "raw" board iuforma.tion was in terms of concepts that 111lman exprrts use to describe loc::II tra.llsitions, such as "slotting," "stripping," et.c.. Also, a few "pre-colllputrd features" were required in addition t.o the raw board inform::lt.ion. Thus it ,,?as necessary to be a domain expert in order to clpsign a suit,a hIe ndwork I.oding sdteme, a.nd it seemed tha.t the only way t.o discov('r thr hrst cOlling scheme was by painstaking trial and error. This "'::IS som('\dtat. disappoiut,ing, as it was hoped tha.t the net\vork lea rIling l)foc('(illre \y()ul<l alit lima t ically produl.e an expert ba.ckgammon net.work with little or no human ('ffort. 3. Comparison paradigul network set-up 111 the sta.nd.ard pra.ctice of ha.c:k-propag::lt.inn, a compfHisoIt paradigm network would ha.ve a.n input layer, (lIlC ()r morr layrTs of hidden nnits, and a.n output layer, wit.h full connectivity hdwcrn adjacell t l::lyers. The input. layer would represent, two final board posit.ions a and '" and the out.put layer would ha.ve just a single unit t.o represent which board position was better. The 101 102 Tesauro teacher signal for the output unit would be a, 1 if hoard position than b, a,nd a 0 if b was better tha.n a. (l was better The proposed compa,rison pa.r a.digm network "'ould m;ercome the limita,t.ion of only being able to consider individual moves in isolation, without knO\vledge of what other a.lt.erna,tives arc available. In addit.ion, t.he sophisticated coding scheme that was developed to encode transition informa tion would not be needed, since compa,risons could be based solely on the final board sta.tes. The comparison approa.ch offers grca ter sensitivity in dist.inguishing between close alternatives, a,nd as stated prc"iously, it corrcsp onds more closely to the actual form of human expert knowledge. These advantages a,re formida,b le, hut. t.here are some import.ant. problems with the a.pproach as curren t.ly descrihed. One technical prohlem is tlta,t the learning is significantly slower. This is beca,use 271, comparisons per t.ra.ining position a,re prcsen ted to the network, where 11. '" 20, wh('rC'tls in tllC relative score approach, only about 3-4 moves per position would he presented. It was therefore necessary to develop a, number of technical tricks to increase the speed of the simulator code for t.hi.s specific a.pplica tion (to he described in a future publication). A more fundamental problem with the a.pproa("h, however, is the issue of consistency of network comparisons . Two properties arc required for complete consistency: (1) The comparison between any two positions mllst be unambiguous, i.e., if the network says that. a is bett('r t.han b when a is presented on the left a,nd b on the right, it !tact hetter say that. a is hetter than b if a is on the right and b is on the left. One ca.n show tha t t.his requircs t.he network's output to exaetly invert whenever Ule input. hO(lrd positions are swapped. (2) The compa,risons must he transitire, as alluded to previously, i.e., if a is judged better than h, and 1J is judged better than (', the network lta,d better judge a to be better than c. Sta.ndard unconstrained networks hayc no gUll ra n tee flf sa t.isfying eit.her of these properties. After some thought, howe\"('I, one realizc's that the output inversion symmetry can be enforcC'd by a symmetry rc'ltl tion amongst the weight.s in the network, and that the transiti"ity ancl rank-OHler consist.ency can be guaranteed by separability in the ardtitC'dure, as illllstratC'd in Figurc 1. Here we see tlla,t t.his network really consists of t.wo half-networks, one of which is only conceIllcd with t}w evaluat.ion "fhoard positi.on (I, and the ot.her of which is concerned only wit.h t.h~ C',"alua t.iol\ of hOH.T() posit.inn b. (Duc to the indicated symmetry relation, nne neC'ds only store oue half-network in the simulator code.) Each half-network mrly have on(' or morc' lay('rs of hidden units, but as long as they a,re not cross-coupled, t.he C'valuat.ion of each of the two input boa,rd positions is hoilC'd down to a single rC'al number. Since real numbers always ra,nk-order consist.en tly, t.he ll('t\,'ork's compa,r isons a,re al ways com;is ten t. Connectionist Learning of Expert Preferences final position (a) final position (b) Figure 1: A network design for cOlTlpari~()n trailling wit h gnarallteed consistency of comparisons. \Veight groHps han~ ~YlTlmet.ry rc1ati()ns W 1 W2 and W 3 == - W 4, which ensures I hat the outPllt cxact.Jy in\Trts upon swapping positions in the input arraJ'. Separation or the hidden unils cOllrlenses the evaluation of each final hoard position into a single real IIIIJTlhcr, thus ensuring transitivity. = An importa.nt a.dded benefit of t.his scheme is that an a.hsoillte hoard evalua.tion fundion is obtained in each half-network. This means t.hat the netwurk, to the extcnt that it.s cvaluation fllnct.ioll is ac-cnrat.e, has an intrinsic understanding of a. given posit.ion, as opposed to HH'rely heing a hIe to detect features which correspond t.o good moves. As has heen emphasil':cd by Berliner [1], an intrinsic uudr.rst.a.llding of the position is crucial for play at the highest levels, a.n<l for use of t,he dOllhling cnl)('. Thnf-:, this a.pproach can serve as the ba.sis for fut.ure progress, whNeas thr previous approach of scoring moves was doomed eventually to run into a cirad rud. 4. Results of comparison training The training procedure for the comparison paradigm nC't.work was as follows: Networks ""'ere set up with 289 input. units which enc(l(le a (ic'scription of single final boa.rd position, varying nnmlH'rs (If hidden units, and a single output unit. The training data was taken from a set (If 40n gamrs ill which the author played both sides. This <ia ta sd con hlins a recording of t.he a.n t,hor's preferred move for each position, and no ot.her comments. The engaged positions in the da.ta set weIe selected out, (discngflged ra.cing I)()sit.ions were not studied) and divided int.o five cat<'gories: hearoff, bearin, opponent bearoff, opponent b~a.rin, a.nd a default ca t.egnry ('overing eVPlything else. In each 103 104 Tesauro Type of n.SP net, CP net _t~e_st_s~e~t_____(~6_5_1-_1_2-_1~)~(_289-1)_ bcaroff .82 .83 hearin .54 .60 opp. bea.roff? .56 .54 opp. bearin .60 .66 other .58 .65 Table 1: Performance of neLs of indicated size on respedin~ test. sets, as measllred by fraction of positions for which TIel agrees wit h lmTllrlll expert choice of best move. HSP: relative score paradigm, CP: comparison paradigm. category, 200 positions chosen (I.t. r(1UdOIU were set. (lside to he us('d as testing <lat(l.; tIte remaining da ta. (a bon t. 1000 positions in ('(I eh category except t.he dcfa.lllt category, for which a.bout 4000 positions were used) was used to tra.iu networks which spedalil':ed in each category. The learning algorithm llsed was st.a.nd(l.rd back-propagation with IIWInelltllIn a.nd without weight decay. Performa,nce after tr(lining is summaril':ed in Ta.hl('s 1 and 2. Ta.hle 1 gives the performance of each specialist network on the appropriate set of test positions. Results for the comparison par(ldigm nebYllrk are shown for networks without hidden units, because it was fouud that the (lddition of hidden units did not improve the performance. (This is discussed in the following section.) \Ve contra.st. these results with result.s of tr(lining networks in the rela.tive score pa,radigm on the same tr(lilling da.t(l sds. "~e see in Ta hIe 1 tha.t for the bearoff and oppon('nt. hearofi' speeialists, there is only a small cha.nge in perform(lnce under t.he contpa.rison p(lradigm. Fllr the hea.rin and opponcnt. bcarin specialists, t,herr is an improv('ment. in IH'rformance of a.bout 6 percenta.ge poin ts in each casco For t.his pa.rtkular applica tion, this is a. "ery substantial improvement in perfoIIuance. How('y<'I, th(~ most, import(lnt finding is for the default ca tegory, which is much l(lrgcr and mor(' difficult than any of the specialist ea.t.egories. Th(' d('f(lult network's prrformance is the key factor in determining the syst('m's IIvrr(lll g(lme prrfOrIll(lllCe. 'Vit.h cumpa.rison t.rainillg, we find an improvc'Ineur. in perform(lll('e from 58% t.o 65%. Given the sil':e and diffkllIt.y of t,his ca t('gory, t his ('(Ill onl)- he desnibcd as a. huge improvement, in performance, and is all t.h(' mllre rem ark(l hIe when nne considers th(lt. t,he comp(lrison p(I.radigm net has only 30() w('ights, (IS opposed to 8000 weight,s for the relative score paradigm net. Next, a combined game-playing s)-stC'ln was s('t. up llsing t.hr the specialist. nets for all engaged posit.ions. (The G(llTlmontool evaluat.ion function was called for ra.cing posit.ions.) Results are given in T(lble 2. Ag(linst, Gammontool itself, the pcrforma.nce und('r thr comparison p(lr(l(ligm improves from 59% to 64%. Against the a.uthor (and tcachE'r), the pC'rforma.nce improves from a.n estimat,ed 35% (since the nsp net.s (Ire so hig and slow, a.ccnra,te Connectionist Learning of Expert Preferences Opponent Ga,mmon tool TesauIO RSP nets .59 (500 games) .35 (l00 ga,mes) CP nets .64 (2000 games) .42 (400 games) Table 2: Game-playing performance of composite network systems against Gammontool and against the author, as measured by fractioTl of games won, without counting gammons or backgammons. statistics could not be obtained) to about 42%. Qualitatively, one notices a subst.a,ntial overall improvement in the new network's level of pla,y. But what, is most, striking is the nct.work's In)lst case beha,vior. The previous relative-score network ha.d particularly bad worst-case beha,vior: about once every other ga,me, the network would make an atrocious blunder wltich would seriously jeopardize its dlances of winning that ga,me [6]. An alarming fradion of these blunders ,,,,ere seemingly random and could not be logically explained. The new comparison para,digm network's wOIst-ca.se behavior is vastly improvcd in this rega,rd. The frcquency and severity of its mistakes are significantly reducE!d, hut more imp(lrtantly, its mista,kes a.re understandable. (Some of the improvement in this respect may be dne to the elimination of the noisy teacher signal descrihed in [7].) 5. Conclusions \Ve have seen that, in the domain of backgammon, the int.roduct.ion of the comparison training pa.ra,digm has result.ed in networks whkh perform much better, with vastly reduced numbers Ilfweights, a.ud with input. coding schemes tha.t a.r e much simpler and easier t.o understand. It was surprising that snch high performa.nce could be obtained in "perceptrou" networks, i.E!., networks withou t hidden units. This reminds us t.hat. one should not. summarily dismiss perceptrons as uninteresting or unworthy of study hecause they arc only ca.pable of lea.rning linearly separable funct.ions [3]. A su hst.an tial component of many difficult real-world problems may lie in the liu('arly separable spectrum, and thus it makes sense to try perceptIons at least as a first attempt. It was also surprising tha.t the use (If hidden units in the comparison- trained networks does not improvc the pcrformanc('. This is nn('xplained, a.nd is t.he subject of current resea.rch. It is, however, not without precedent: in at least one other real-world application [4], it has heen found tha.t networks with hidden units do not pcrform any bettE'r than netwllrks without hidden units. ?vIore generally, one might conclude t.ha.t, in training a. neural network (or indeed a.ny learning system) from human expert. E'xamples in a cumplex domain, there should he a. good match hetween Ute na t,ural form of the expert's knowledge and the method by which the net.work is trained. For domains in which the expert must seh~ct a preferred alt.crnative from a set of alternatives, 105 106 Tesauro the expert naturally thinks in terms of comparisons a.mongst the top few a.lterna.tives, and the compa.rison paradigm proposed here takes advantage of that fact. It would be possihle in principle to train a network using absolute evaluations, hut the crea.tion of sueh a. training set might he too difficult to underta.ke on a large scale. If the above discussion is coned, then the comparisou pa.ra.digm should be useful in ot,her applications involving expert choice, and in other lea.rning syst,ems besides connectionist networks. Typically expert systems a.re handcrafted by knowledge engineers, ra.ther than learned from human expert examples; however, there has recently been some interest in sl1pervis(~d lea.rning approa.ches. It will be interesting to see if the compa.rison paradigm proves to be useful when supervised lea.rning procedures are applied t,o otller domains involving expert choice. In using the compa.rison paradigm, it. will be important to ha.ve some way to gua.ra.n t,ee that the syst,em's comparisons will be unambiguous and t,ra.nsitive. For feed-forward networks, it was shown in this pa.per how to gua.rantee this using symmetric, separa.t.ed nrtworks; it should be possible to impose similar constraints Oil otll<'r learning systems to enforce consistency. References [l} H. Berliner, "On the construction Proc. of IleA T (1979) 53--55. or c\"aluation runctions for large domains," [2] S. 1. Gallant, "Conncctionist r.xpert systems," Comm. l\eM 31, 152 -- 169 (1988). P} M. Minsky and S. Papcrt, Pcrceptrons, MIT Press, Camhridge ~fA (1969). [,1] N. Qian and T. J. Sejnowski, "Predicting the' secondary slrucl,nre of globular proteins using neural nctwork models," .1. Mol. Bioi. 202, R(jfi 8R,1 (1988). [5] D. E. Rumelarl and J. L. ~lcClellanrl (cds.), Parallel J>i.<;frifllltecl Processing: Explorations in I.Ire Microsirucl.lIl'r Cogn;t,;on, Vt)ls. I and 2, l\f1T Prrss, Cambrioge MA (1986). or [6} G. Tesauro, "Neural network ddral.s creal.or in hack~annl1(ln match." Univ. of ll1inoiR, Center for Complex S~' strll1s Technical Heporl. CCSl1-8R-6 (1988). [71 G. Tesauro and T. J. Sejnowski, "/\ parallel nei.work that ]rarns to play backgammon," A rtificialTntell;gmrcc, in press (H)89) . 

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How to Dynamically Merge Markov Decision Processes Satinder Singh Department of Computer Science University of Colorado Boulder, CO 80309-0430 baveja@cs.colorado.edu David Cohn Adaptive Systems Group Harlequin, Inc. Menlo Park, CA 94025 cohn@harlequin.com Abstract We are frequently called upon to perform multiple tasks that compete for our attention and resource. Often we know the optimal solution to each task in isolation; in this paper, we describe how this knowledge can be exploited to efficiently find good solutions for doing the tasks in parallel. We formulate this problem as that of dynamically merging multiple Markov decision processes (MDPs) into a composite MDP, and present a new theoretically-sound dynamic programming algorithm for finding an optimal policy for the composite MDP. We analyze various aspects of our algorithm and illustrate its use on a simple merging problem. Every day, we are faced with the problem of doing mUltiple tasks in parallel, each of which competes for our attention and resource. If we are running a job shop, we must decide which machines to allocate to which jobs, and in what order, so that no jobs miss their deadlines. If we are a mail delivery robot, we must find the intended recipients of the mail while simultaneously avoiding fixed obstacles (such as walls) and mobile obstacles (such as people), and still manage to keep ourselves sufficiently charged up. Frequently we know how to perform each task in isolation; this paper considers how we can take the information we have about the individual tasks and combine it to efficiently find an optimal solution for doing the entire set of tasks in parallel. More importantly, we describe a theoretically-sound algorithm for doing this merging dynamically; new tasks (such as a new job arrival at a job shop) can be assimilated online into the solution being found for the ongoing set of simultaneous tasks. 1058 1 S. Singh and D. Cohn The Merging Framework Many decision-making tasks in control and operations research are naturally formulated as Markov decision processes (MDPs) (e.g., Bertsekas & Tsitsikiis, 1996). Here we define MDPs and then formulate what it means to have multiple simultanous MDPs. 1.1 Markov decision processes (MDPs) An MDP is defined via its state set 8, action set A, transition probability matrices P, and payoff matrices R. On executing action a in state s the probability of transiting to state s' is denoted pa(ss') and the expected payoff associated with that transition is denoted Ra (ss'). We assume throughout that the payoffs are non-negative for all transitions. A policy assigns an action to each state of the MDP. The value of a state under a policy is the expected value of the discounted sum of payoffs obtained when the policy is followed on starting in that state. The objective is to find an optimal policy, one that maximizes the value of every state. The optimal value of state s, V* (s), is its value under the optimal policy. The optimal value function is the solution to the Bellman optimality equations: for all s E 8 , V(s) = maxaEA(E s pa(ss') [Ra (ss') +/V(s'))), where the discount factor o ~ / < 1 makes future payoffs less valuable than more immediate payoffs (e.g., Bertsekas & Tsitsiklis, 1996). It is known that the optimal policy 7r* can be determined from V* as follows: 7r*(s) = argmaxaE A(E s pa(ss')[Ra(ss') +/V*(s'))). Therefore solving an MDP is tantamount to computing its optimal value function. l l 1.2 Solving MDPs via Value Iteration Given a model (8, A, P, R) of an MDP value iteration (e.g., Bertsekas & Tsitsikiis, 1996) can be used to determine the optimal value function. Starting with an initial guess, Vo, iterate for all s Vk+1(S) = maxaEA(E sIES pa(ss')[Ra(ss') + /Vk(S'))). It is known that maxsES 1Vk+1 (s) - V*(s)1 ~ / maxsES IVk(S) - V*(s)1 and therefore Vk converges to V* as k goes to infinity. Note that a Q-value (Watkins, 1989) based version of value iteration and our algorithm presented below is also easily defined. 1.3 Multiple Simultaneous MDPs The notion of an optimal policy is well defined for a single task represented as an MDP. If, however, we have multiple tasks to do in parallel, each with its own state, action, transition probability, and payoff spaces, optimal behavior is not automatically defined. We will assume that payoffs sum across the MDPs, which means we want to select actions for each MDP at every time step so as to maximize the expected discounted value of this summed payoff over time. If actions can be chosen independently for each MDP, then the solution to this "composite" MDP is obvious - do what's optimal for each MDP. More typically, choosing an action for one MDP constrains what actions can be chosen for the others. In a job shop for example, actions correspond to assignment of resources, and the same physical resource may not be assigned to more than one job simultaneously. Formally, we can define a composite MDP as a set of N MDPs {Mi}f. We will use superscripts to distinguish the component MDPs, e.g., 8i , Ai, pi, and Ri are the state, action, transition probability and payoff parameters of MDP Mi. The state space of the composite MDP, 8, is the cross product of the state spaces of the component MDPs, i.e., 8 = 8 1 X 8 2 X ... X 8 N . The constraints on actions implies that 1059 How to Dynamically Merge Markov Decision Processes the action set of the composite MDP, A, is some proper subset of the cross product of the N component action spaces. The transition probabilities and the payoffs of the composite MDP are factorial because the following decompositions hold: for all s, s' E S and a E A, pa(ss') = nf:lpai (SiSi') and Ra(ss') = l:~l ~i (SiSi'). Singh (1997) has previously studied such factorial MDPs but only for the case of a fixed set of components. The optimal value function of a composite MDP is well defined, and satisfies the following Bellman equation: for all s E S, N V(s) = ~a:L (nf:lpa'(sisi')[LRa\sisi')+'YV(s')]). ~ES (1) i=l Note that the Bellman equation for a composite MDP assumes an identical discount factor across component MDPs and is not defined otherwise. 1.4 The Dynamic Merging Problem Given a composite MDP, and the optimal solution (e.g. the optimal value function) for each of its component MDPs, we would like to efficiently compute the optimal solution for the composite MDP. More generally, we would like to compute the optimal composite policy given only bounds on the value functions of the component MDPs (the motivation for this more general version will become clear in the next section). To the best of our knowledge, the dynamic merging question has not been studied before. Note that the traditional treatment of problems such as job-shop scheduling would formulate them as nonstationary MDPs (however, see Zhang and Dietterich, 1995 for another learning approach). This normally requires augmenting the state space to include a "time" component which indexes all possible state spaces that could arise (e.g., Bertsekas, 1995). This is inefficient, and potentially infeasible unless we know in advance all combinations of possible tasks we will be required to solve. One contribution of this paper is the observation that this type of nonstationary problem can be reformulated as one of dynamically merging (individually) stationary MDPs. 1.4.1 The naive greedy policy is suboptimal Given bounds on the value functions of the component MDPs, one heuristic composite policy is that of selecting actions according to a one-step greedy rule: N 7I"(s) = argmax(l: nf:,lpa i (Si si')[l:(Rai (si, ai ) + 'YXi(Si'))]), a 8' i=l where Xi is the upper or lower bound of the value function, or the mean of the bounds. It is fairly easy however, to demonstrate that these policies are substantially suboptimal in many common situations (see Section 3). 2 Dynamic Merging Algorithm Consider merging N MDPs; job-shop scheduling presents a special case of merging a new single MDP with an old composite MDP consisting of several factor MDPs. One obvious approach to finding the optimal composite policy would be to directly perform value iteration in the composite state and action space. A more efficient approach would make use of the solutions (bounds on optimal value functions) of the existing components; below we describe an algorithm for doing this. S. SinghandD. Cohn 1060 Our algorithm will assume that we know the optimal values, or more generally, upper and lower bounds to the optimal values of the states in each component MDP. We use the symbols Land U for the lower and upper bounds; if the optimal value function for the ith factor MDP is available then Li = U i = V?,i.l Our algorithm uses the bounds for the component MDPs to compute bounds on the values of composite states as needed and then incrementally updates and narrows these initial bounds using a form of value iteration that allows pruning of actions that are not competitive, that is, actions whose bounded values are strictly dominated by the bounded value of some other action. Initial State: The initial composite state So is composed from the start state of all the factor MOPs. In practice (e.g. in job-shop scheduling) the initial composite state is composed of the start state of the new job and whatever the current state of the set of old jobs is. Our algorithm exploits the initial state by only updating states that can occur from the initial state under competitive actions. Initial Value Step: When we need the value of a composite state S for the first time. we compute upper and lower bounds to its optimal value as follows: L(s) = max!1 Li(Si), and U(s) = E~1 Ui(S). Initial Update Step: We dynamically allocate upper and lower bound storage space for composite states as we first update them. We also create the initial set of competitive actions for S when we first update its value as A(s) = A. As successive backups narrow the upper and lower bounds of successor states, some actions will no longer be competitive, and will be eliminated from further consideration. Modified Value Iteration Algorithm: At step t if the state to be updated is St: Lt+l(St) Ut+l(St) At+l (St) = (L pa(sts')[Ra(st. s') + -yLt(s')]) s max (L pa(sts')[Ra(st, s') + -yUt(s')]) Ua E At(st) AND L pa(sts')[Ra(st, s') + -yUt(s')] s' ;::: argmax L pb(sts')[Rb(st, s') + -yLt(s')] max aEAt{st} aEAt(st} J s' bEAt(St) St+l { 8' So if s~ is terminal for all Si E s s' E S such that 3a E A t+1 (St), pa(StS') > 0 otherwise The algorithm terminates when only one competitive action remains for each state, or when the range of all competitive actions for any state are bounded by an indifference parameter ?. To elaborate, the upper and lower bounds on the value of a composite state are backed up using a form of Equation 1. The set of actions that are considered competitive in that state are culled by eliminating any action whose bounded values is strictly dominated by the bounded value of some other action in At(st). The next state to be updated is chosen randomly from all the states that have non-zero 1 Recall that unsuperscripted quantities refer to the composite MDP while superscripted quantities refer to component MDPs. Also, A is the set of actions that are available to the composite MDP after taking into account the constraints on picking actions simultaneously for the factor MDPs. How to Dynamically Merge Markov Decision Processes 1061 pro babili ty of occuring from any action in At+! (St) or, if St is the terminal state of all component MDPs, then StH is the start state again. A significant advantage of using these bounds is that we can prune actions whose upper bounds are worse than the best lower bound. Only states resulting from remaining competitive actions are backed up. When only one competitive action remains, the optimal policy for that state is known, regardless of whether its upper and lower bounds have converged. Another important aspect of our algorithm is that it focuses the backups on states that are reachable on currently competitive actions from the start state. The combined effect of only updating states that are reachable from the start state and further only those that are reachable under currently competitive actions can lead to significant computational savings. This is particularly critical in scheduling, where jobs proceed in a more or less feedforward fashion and the composite start state when a new job comes in can eliminate a large portion of the composite state space. Ideas based on Kaelbling's (1990) interval-estimation algorithm and Moore & Atkeson's (1993) prioritized sweeping algorithm could also be combined into our algorithm. The algorithm has a number of desirable "anytime" characteristics: if we have to pick an action in state So before the algorithm has converged (while multiple competitive actions remain), we pick the action with the highest lower bound. If a new MDP arrives before the algorithm converges, it can be accommodated dynamically using whatever lower and upper bounds exist at the time it arrives. 2.1 Theoretical Analysis In this section we analyze various aspects of our algorithm. UpperBound Calculation: For any composite state, the sum of the optimal values of the component states is an upper bound to the optimal value of the composite state, i.e., V*(s = SI, S2, .. . , SN) ~ 2:~1 V*,i(Si). If there were no constraints among the actions of the factor MDPs then V* (s) would equal L~l V*,i(Si) because of the additive payoffs across MDPs. The presence of constraints implies that the sum is an upper bound. Because V*,i(S') ~ Ut(Si) the result follows. LowerBound Calculation: For any composite state, the maximum of the optimal values of the component states is a lower bound to the optimal value of the composite states, i.e., V*(s = SI, S2, . .. ,SN) ~ max~1 V*,i(Si). To see this for an arbitrary composite state s, let the MDP that has the largest component optimal value for state s always choose its component-optimal action first and then assign actions to the other MDPs so as to respect the action constraints encoded in set A. This guarantees at least the value promised by that MDP because the payoffs are all non-negative. Because V*,i(Si) ~ Lt(Si) the result follows. Pruning of Actions: For any composite state, if the upper bound for any composite action, a, is lower than the lower bound for some other composite action, then action a cannot be optimal - action a can then safely be discarded from the max in value iteration. Once discarded from the competitive set, an action never needs to be reconsidered. Our algorithm maintains the upper and lower bound status of U and L as it updates them. The result follows. S. Singh and D. Cohn 1062 Convergence: Given enough time our algorithm converges to the optimal policy and optimal value function for the set of composite states reachable from the start state under the optimal policy. If every state were updated infinitely often, value iteration converges to the optimal solution for the composite problem independent of the intial guess Vo. The difference between standard value iteration and our algorithm is that we discard actions and do not update states not on the path from the start state under the continually pruned competitive actions. The actions we discard in a state are guaranteed not to be optimal and therefore cannot have any effect on the value of that state. Also states that are reachable only under discarded actions are automatically irrelevant to performing optimally from the start state. 3 An Example: Avoiding Predators and Eating Food We illustrate the use of the merging algorithm on a simple avoid-predator-andeat-food problem, depicted in Figure 1a. The component MDPs are the avoidpredator task and eat-food task; the composite MDP must solve these problems simultaneously. In isolation, the tasks avoid-predator and eat-food are fairly easy to learn. The state space of each task is of size n\ 625 states in the case illustrated. Using value iteration, the optimal solutions to both component tasks can be learned in approximately 1000 backups. Directly solving the composite problem requires n 6 states (15625 in our case), and requires roughly 1 million backups to converge. Figure 1b compares the performance of several solutions to the avoid-predatorand-eat-food task. The opt-predator and opt-food curves shows the performance of value iteration on the two component tasks in isolation; both converge qUickly to their optima. While it requires no further backups, the greedy algorithm of Section 1.4.1 falls short of optimal performance. Our merging algorithm, when initialized with solutions for the component tasks (5000 backups each) converges quickly to the optimal solution. Value iteration directly on the composite state space also finds the optimal solutions, but requires 4-5 times as many backups. Note that value iteration in composite state space also updated states on trajectories (as in Barto etal.'s, 1995 RTDP algorithm) through the state space just as in our merging algorithm, only without the benefit of the value function bounds and the pruning of non-competitive actions. 4 Conclusion The ability to perform multiple decision-making tasks simultaneously, and even to incorporate new tasks dynamically into ongoing previous tasks, is of obvious interest to both cognitive science and engineering. Using the framework of MDPs for individual decision-making tasks, we have reformulated the above problem as that of dynamically merging MDPs. We have presented a modified value iteration algorithm for dynamically merging MDPs, proved its convergence, and illustrated its use on a simple merging task. As future work we intend to apply our merging algorithm to a real-world jobshop scheduling problem, extend the algorithm into the framework of semi-Markov decision processes, and explore the performance of the algorithm in the case where a model of the MDPs is not available. How to Dynamically Merge Markov Decision Processes a) b) 0.80 1063 I f 0.70 *'~ f P Q. A & 0.60 j opt-predator 0.40 0.0 , I -- - - - - : : - :- . . . . , .- : - - -- - - ' - . , . . . , . . , . -- 500000.0 1000000.0 - ---' 1500000.0 Number 01 Backups Figure 1: a) Our agent (A) roams an n by n grid. It gets a payoff of 0.5 for every time step it avoids predator (P), and earns a payoff of 1.0 for every piece of food (f) it finds. The agent moves two steps for every step P makes, and P always moves directly toward A. When food is found , it reappears at a random location on the next time step. On every time step, A has a 10% chance of ignoring its policy and making a random move. b) The mean payoff of different learning strategies vs. number of backups. The bottom two lines show that when trained on either task in isolation, a learner reaches the optimal payoff for that task in fewer than 5000 backups. The greedy approach makes no further backups, but performs well below optimal. The optimal composite solution, trained ab initio, requires requires nearly 1 million backups. Our algorithm begins with the 5000-backup solutions for the individual tasks, and converges to the optimum 4-5 times more quickly than the ab initio solution. Acknowledgements Satinder Singh was supported by NSF grant IIS-9711753. References Barto, A. G., Bradtke, S. J., & Singh, S. (1995) . Learning to act using real-time dynamic programming. Artificial Intelligence, 72, 81-138. Bertsekas, D. P. (1995). Dynamic Programming and Optimal Control. Belmont, MA: Athena Scientific. Bertsekas, D. P. & Tsitsiklis, J. N. (1996). Neuro-Dynamic Programming. Belmont, MA: Athena Scientific. Kaelbling, L. P. (1990) . Learning in Embedded Systems. PhD thesis, Stanford University, Department of Computer Science, Stanford, CA. Technical Report TR-90-04. Moore, A. W . & Atkeson, C. G. (1993). Prioritized sweeping: Reinforcement learning with less data and less real time. Machine Learning, 19(1). Singh , S. (1997). Reinforcement learning in factorial environments. submitted. Watkins, C. J. C. H. (1989). Learning from Delayed Rewards. PhD thesis, Cambridge Univ., Cambridge, England. Zhang, W. & Dietterich, T . G. (1995). High-performance job-shop scheduling with a time delay TD(lambda) network. In NIPSystems 8. MIT Press. 

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Bayesian model of surface perception William T. Freeman MERL, Mitsubishi Electric Res. Lab . 201 Broadway Cambridge, MA 02139 Paul A. Viola Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 freeman~erl.com viola~ai.mit.edu Abstract Image intensity variations can result from several different object surface effects, including shading from 3-dimensional relief of the object, or paint on the surface itself. An essential problem in vision , which people solve naturally, is to attribute the proper physical cause, e.g. surface relief or paint, to an observed image. We addressed this problem with an approach combining psychophysical and Bayesian computational methods. We assessed human performance on a set of test images, and found that people made fairly consistent judgements of surface properties. Our computational model assigned simple prior probabilities to different relief or paint explanations for an image, and solved for the most probable interpretation in a Bayesian framework. The ratings of the test images by our algorithm compared surprisingly well with the mean ratings of our subjects. 1 Introduction When people study a picture, they can judge whether it depicts a shaded , 3dimensional surface, or simply a flat surface with markings or paint on it. The two images shown in Figure 1 illustrate this distinction [1]. To many observers Figure 1a appears to be a raised plateau lit from the left. Figure 1b is simply a re-arrangement of the local features of la, yet it does not give an impression of shape or depth. There is no simple correct answer for this problem; either of these images could be explained as marks on paper, or as illuminated shapes. Nevertheless people tend to make particular judgements of shape or reflectance . We seek an algorithm to arrive at those same judgements. There are many reasons to study this problem. Disentangling shape and reflectance is a prototypical underdetermined vision problem, which biological vision systems routinely solve. Insights into this problem may apply to other vision problems W T. Freeman and P. A. Viola 788 as well. A machine that could interpret images as people do would have many applications , such as the interactive editing and manipulation of images. Finally, there is a large body of computer vision work on "shape from shading"-inferring the 3-dimensional shape of a shaded object [4]. Virtually every algorithm assumes that all image intensity changes are caused by shading; these algorithms fail for any image with reflectance changes. To bring this body of work into practical use , we need to be able to disambiguate shading from reflectance changes. There has been very little work on this problem. Sinha and Adelson [9] examined a world of painted polyhedra, and used consistancy constraints to identify regions of shape and reflectance changes. Their consistancy constraints involved specific assumptions which need not always hold and may be better described in a probabilistic framework . In addition , we seek a solution for more general, greyscale lmages. Our approach combines psychophysics and computational modeling. First we will review the physics of image formation and describe the under-constrained surface perception problem. We then describe an experiment to measure the interpretations of surface shading and reflectance among different individuals. We will see that the judgements are fairly consistent across individuals and can be averaged to define "ground truth" for a set of test images. Our approach to modeling the human judgements is Bayesian . We begin by formulating prior probabilities for shapes and reflectance images , in the spirit of recent work on the statistical modeling of images [5 , 8, 11]. Using these priors, the algorithm then determines whether an image is more likely to have been generated by a 3D shape or as a pattern of reflectance. We compare our algorithm's performance to that of the human subjects. (a) (c ) (b) (d) Figure 1: Images (a) and (b), designed by Adelson [1] , are nearly the same everywhere, yet give different percepts of shading and reflectance . (a) looks like a plateau, lit from the left ; (b) looks like marks on paper. Illustrating the underconstrained nature of perception , both images can be explained either by reflectance changes on paper (they are), or, under appropriate lighting conditions, by the shapes (c) and (d) , respectively (vertical scale exaggerated). Bayesian Model of Surface Perception 2 789 Physics of Imaging One simple model for the generation of an image from a three dimensional shape is the Lambertian model: I(x, y) = R(x, y) (i. n(x, y)) , (1) where I (x, y) is an image indexed by pixel location , n( x, y) is the surface normal at every point on the surface conveniently indexed by the pixel to which that surface patch projects, i is a unit vector that points in the direction of the light source, and R( x, y) is the reflectance at every point on the surface 1 . A patch of surface is brighter if the light shines onto it directly and darker if the light shines on it obliquely. A patch can also be dark simply because it is painted with a darker pigment. The shape of the object is probably more easily described as a depth map z(x, y) from which n(x, y) is computed. The classical "shape from shading" task attempts to compute z from I given knowledge of i and assuming R is everywhere constant. Notice that the problem is "illposed"; while I( x, y) does constrain n( x, y) it is not sufficient to uniquely determine the surface normal at each pixel. Some assumption about global properties of z is necessary to condition the problem. If R is allowed to vary, the problem becomes even more under-constrained. For example, R = I and n( x, y) = j is a valid solution for every image. This is the "all reflectance" hypothesis, where the inferred surface is flat and? all of the image variation is due to reflectance. Interestingly there is also an "all shape" solution for every image where R =. 1 and I(x, y) = i? n(x, y) (see Figure 1 for examples of such shapes). Since the relationship between z and I is non-linear, "shape from shading" cannot be solved directly and requires a time consuming search procedure. For our computational experiments we seek a rendering model for shapes which simplifies the mathematics, yet maintains the essential ambiguities of the problem. We use the approximations of linear shading [6]. This involves two sets of approximations. First, that the rendered image I (x, y) is some function, G ( only of the surface slope at any point: g; , g; ), (2) The second approximation is that the rendering function G itself is a linear function of the surface slopes, oz oz G( ox' Oy) ~ kl oz oz + k2 ox + k3 oy . (3) Under linear shading, finding a shape which explains a given image is a trivial integration along the direction of the assumed light source. Despite this simplicity, images rendered under linear shading appear fairly realistically shaded [6]. 3 Psychophysics We used a survey to assess subjects' image judgements. We made a set of 60 test images, using Canvas and Photoshop programs to generate and manipulate the images. Our goal was to create a set of images with varying degrees of shadedness. We sought to assess to what extent each subject saw each image as created by 1 Note: we assume orthographic projection, a distant light source, and no shadowing. W T. Freeman and P. A. Vwla 790 shading changes or reflectance changes. Each of our 18 naive observers was given a 4 page survey showing the images in a different random order. To explain the problem of image interpretation quickly to naive subjects , we used a concrete story (Adelson's Theater Set Shop analogy [2] is a related didactic example). The survey instructions were as follows : Pretend that each of the following pictures is a photograph of work made by either a painter or a sculptor. The painter could use paint , markers, air brushes, computer, etc. , to make any kind of mark on a fiat canvas. The paint had no 3-dimensionality; everything was perfectly fiat. The sculptor could make 3-dimensional obJects, but could make no markings on them . She could mold, sculpt, and scrape her sculptures, but could not draw or paint. All the objects were made out of a uniform plaster material and were made visible by lighting and shading effects. The subjects used a 5-point rating scale to indicate whether each image was made by the painter (P) or sculptor (S): S, S? , ?, P?, P. 3.1 Survey Results We examined a non-parametric comparison of the image ratings, the rank order correlation (the linear correlation of image rankings in order of shapeness by each observer) [7]. Over all possible pairings of subjects, the rank order correlations ranged from 0.3 to 0.9, averaging 0.65. All of these correlations were statistically significant , most at the 0.0001 level. We concluded that for our set of test images , people do give a very similar set of interpretations of shading and reflectance. We assigned a numerical value to each of the 5 survey responses (S=2; S?=l; ?=O; P?=-l; P=-2) and found the average numerical "shadedness" score for each image. Figure 2 shows a histogram of the survey responses for each image , ordered in decreasing order of shadedness. The two images of Figure 1 had average scores of 1.7 and -1.6 , respectively, confirming the impressions of shading and reflectance . There was good consensus for the rankings of the most paint-like and most sculpturelike images; the middle images showed a higher score variance . The rankings by each individual showed a strong correlation with the rankings by the average of the remaining subjects, ranging from 0.6 to 0.9 . Figure 4 shows the histogram of those correlations. The ordering of the images by the average of the subjects ' responses provides a "ground truth" with which to compare the rankings of our algorithm. Figure 3, left , shows a randomly chosen subset of the sorted images , in decreasing order of assessed sculptureness . 4 Algorithm We will assume that people are choosing the most probable interpretation of the observed image. We will adopt a Bayesian approach and calculate the most probable interpretation for each image under a particular set of prior probabilities for images and shapes. To parallel the choices we gave our subjects , we will choose between interpretations that account for the image entirely by shape changes, or entirely by reflectance changes. Thus, our images are either a rendered shape , multiplied by a uniform reflectance image , or a flat shape , multiplied by some non-uniform reflectance image. Bayesian Model of Surface Perception 791 intensity: score frequency for each image l!!S o (,) cnp 10 20 30 40 50 60 image number (sorted by shapeness) Figure 2: Histogram of survey responses. Intensity shows the number of responses of each score (vertical scale) for each image (horizontal, sorted in increasing order of shape ness ). To find the most probable interpretation, given an image, we need to assign prior probabilities to shape and reflectance configurations. There has been recent interest in characterizing the probabilities of images by the expected distributions of subband coefficient values [5, 8, 11]. The statistical distribution of bandpass linear filter outputs, for natural images, is highly kurtotic; the output is usually small , but in rare cases it takes on very large values. This non-gaussian behavior is not a property of the filter operation, because filtered "random" images appear gaussian. Rather it is a property of the structure of natural images. An exponential distribution, P(c) ex: e- 1cl , where c is the filter coefficient value, is a reasonable model. These priors have been used in texture synthesis, noise removal, and receptive field modeling. Here, we apply them to the task of scene interpretation. We explored using a very simple image prior: P(I) ex: exp (- L r,y oI(x,y)2 + OI(x , y)2) ox oy (4) Here we treat the image derivative as an image subband corresponding to a very simple filter. We applied this image prior to both reflectance images, I(x , y), as well as range images , z(x, y). For any given picture, we seek to decide whether a shape or a reflectance explanation is more probable. The proper Bayesian approach would be to integrate the prior probabilities of all shapes which could explain the image in order to arrive at the total probability of a shape explanation. (The reflectance explanation, R is unique ; the image itself). We employed a computationally simpler procedure, a very rough approximation to the proper calculation: we evaluated the prior probability, P(S) of the single, most probable shape explanation, S, for the image. Using the ratio test of a binary hypothesis, we formed a shapeness index, J, by the ratio of the probabilities for the shape and reflectance explanations, J = ~i!~. The index J was used to rank the test images by shapeness. We need to find the most probable shape explanation. The overall log likelihood of a shape, z, given an image is, using the linear shading approximations of Eq. (3): log P(z, kl' k2, k31I) + log P(z) + c Lx,y(I - kl + k2* + k3~~)2 + Lx,y log P(Ilz, kl' k2, k3) Jg~2 + g;2 + c, (5) where c is a normalization constant. We use a multi-scale gradient descent algorithm that simultaneously determines the optimal shape and illumination parameters for an image (similar to that used by [10]). The optimization procedure has three stages starting with a quarter resolution version of I, and moving to the half and W. T. Freeman and P. A. Vwla 792 Figure 3: 28 of the 60 test images, arranged in decreasing order of subjects' shapeness ratings. Left: Subjects' rankings. Right: Algorithm 's rankings . then full resolution. The solution found at the low resolution is interpolated up to the next level and is used as a starting point for the next step in the optimization. In our experiments images are 128x128 pixels. The optimization procedure takes 4000 descent steps at each resolution level. 5 Results Surprisingly, the simple prior probability of Eq. (4) accounts for much of the ratings of shading or paint by our human subjects. Figure 3 compares the rankings (shown in raster scan order) of a subset of the test images for our algorithm and the average of our subjects. The overall agreement is good . Figure 4 compares two measures: (1) the correlations (dark bars) of the subjects' individual ratings to the mean subject rating with (2) the correlation of our algorithm's ratings to the mean subject rating. SUbjects show correlations between 0.6 and 0.9; our Bayesian algorithm showed a correlation of 0.64. Treating the mean subjects' ratings as the right answer, our algorithm did worse than most subjects but not as badly as some subjects. Figure 1 illustrates how our algorithm chooses an interpretation for an image. If a simple shape explains an image, such as the shape explanation (c) for image (a), the shape gradient penalties will be small, assigning a high prior probability to that shape. If a complicated shape (d) is required to explain a simple image (b), the Bayesian Model of Surface Perception 793 low prior probability of the shape and the high prior probability of the reflectance image will favor a "paint" explanation. We noted that many of the shapes inferred from paint-like images showed long ridges coincidently aligned with the assumed light direction. The assumption of generic light direction can be applied in a Bayesian framework [3] to penalize such coincidental alignments. We speculate that such a term would further penalize those unlikely shape interpretations and may improve algorithm performance. Rank order corralaUon with mean Imago rating 0.2 Figure 4: Correlation of individual subjects' image ratings compared with the mean rating (bars) compared with correlation of algorithm's rating with the mean rating (dashed line). Acknow ledgements We thank E. Adelson, D. Brainard, and J. Tenenbaum for helpful discussions. References [1] E. H. Adelson , 1995. personal communication. [2] E. H. Adelson and A. P. Pentland. The perception of shading and reflectance. In B. Blum, editor, Channels in the Visual Nervous System: Neurophysiology, Psychophysics, and Models, pages 195-207. Freund Publishing, London, 1991. [3] W. T. Freeman. The generic viewpoint assumption in a framework for visual perception. Nature, 368(6471):542-545, April 7 1994. [4] B. K. P. Horn and M. J. Brooks. Shape from shading. MIT Press, Cambridge, MA, 1989. [5] B. A . Olshausen and D. J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for ?natural images. Nature, 381:607-609, 1996. [6] A. P. Pentland. Linear shape from shading. Inti. J. Compo Vis., 1(4):153-162, 1990. [7] W. H. Press, S. A . Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C. Cambridge Univ. Press, 1992. [8] E. P. Simoncelli and E. H. Adelson. Noise removal via Bayesian wavelet coring. In 3rd Annual Inti. Con/. on Image Processing, Laussanne, Switzerland, 1996. IEEE. [9] P. Sinha and E. H. Adelson . Recovering reflectance and illumination in a world of painted polyhedra. In Proc. 4th Intl. Con/. Computer Vision, pages 156-163. IEEE, 1993. [10] D. Terzopoulos. Multilevel computational processes for visual surface reconstruction. Comp o Vis., Graphics, Image Proc., 24:52-96 , 1983. [11] S. C. Zhu and D. Mumford. Learning generic prior models for visual computation. Submitted to IEEE Trans. PAMI, 1997. 

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An application of Reversible-J ump MCMC to multivariate spherical Gaussian mixtures Alan D. Marrs Signal & Information Processing Dept. Defence Evaluation & Research Agency Gt. Malvern, UK WR14 3PS marrs@signal.dra.hmg.gb Abstract Applications of Gaussian mixture models occur frequently in the fields of statistics and artificial neural networks. One of the key issues arising from any mixture model application is how to estimate the optimum number of mixture components. This paper extends the Reversible-Jump Markov Chain Monte Carlo (MCMC) algorithm to the case of multivariate spherical Gaussian mixtures using a hierarchical prior model. Using this method the number of mixture components is no longer fixed but becomes a parameter of the model which we shall estimate. The Reversible-Jump MCMC algorithm is capable of moving between parameter subspaces which correspond to models with different numbers of mixture components. As a result a sample from the full joint distribution of all unknown model parameters is generated. The technique is then demonstrated on a simulated example and a well known vowel dataset. 1 Introduction Applications of Gaussian mixture models regularly appear in the neural networks literature. One of their most common roles in the field of neural networks, is in the placement of centres in a radial basis function network. In this case the basis functions are used to model the distribution of input data (Xi == [Xl, X2, .?? , Xd]T, (i = l,n)), and the problem is one of mixture density estimation. A D. Marrs 578 k p(Xi ) = L 7Ijp(X I9j), i (1) j=1 where k is the number of mixture components, 7rj the weight or mixing proportion for component j and 8 j the component parameters (mean & variance in this case). The mixture components represent the basis functions of the neural network and their parameters (centres & widths) may be estimated using the expectationmaximisation (EM) algorithm. One of the key issues arising in the use of mixture models is how to estimate the number of components. This is a model selection problem: the problem of choosing the 'correct' number of components for a mixture model. This may be thought of as one of comparing two (or more) mixture models with different components, and choosing the model that is 'best' based upon some criterion. For example, we might compare a two component model to one with a single component. (2) This may appear to be a case of testing of nested hypotheses. However, it has been noted [5] that the standard frequentist hypothesis testing theory (generalised likelihood ratio test) does not apply to this problem because the desired regularity conditions do not hold. In addition, if the models being tested have 2 and 3 components respectively, they are not strictly nested. For example, we could equate any pair of components in the three component model to the components in the two component model, yet how do we choose which component to 'leave out'? 2 Bayesian approach to Gaussian mixture models A full Bayesian analysis treats the number of mixture components as one of the parameters of the model for which we wish to find the conditional distribution. In this case we would represent the joint distribution as a hierarchical model where we may introduce prior distributions for the model parameters, ie. p(k, 7r, Z, 9, X) = p(k)p(7rlk)p(zl7r, k)p(9Iz, 7r, k)p(XI9, z, 7r, k), (3) where7r = (7rj)J=1, 9 = (9 j )J=1 and z = (Zi)f::l are allocation variables introduced by treating mixture estimation as a hidden data problem with Zi allocating the ith observation to a particular component. A simplified version of this model can be derived by imposing further conditional independencies, leading to the following expression for the joint distribution p(k, 7r, Z, 9, X) = p(k)p(7rlk)p(zl7r, k)p(9Ik)p(XI9, z). (4) In addition, we add an extra layer to the hierarchy representing priors on the model parameters giving the final form for the joint distribution peA, 6, T}, k, 7r, Z, 9, X) = p(A)p(6)p(T})p(kIA)p(7rlk, 6)p(zl7r, k) x p(9Ik, T})p(XI9, z). (5) Until recently a full Bayesian analysis has been mathematically intractable. Model comparison was carried out by conducting an extensive search over all possible Reversible-Jump MCMC for Multivariate Spherical Gaussian Mixtures 579 model orders comparing Bayes factors for all possible pairs of models. What we really desire is a method which will estimate the model order along with the other model parameters. Two such methods based upon Markov Chain Monte Carlo (MCMC) techniques are reversible-jump MCMC [2] and jump-diffusion [3]. In the following sections, we extend the reversible-jump MCMC technique to multivariate spherical Gaussian mixture models. Results are then shown for a simulated example and an example using the Peterson-Barney vowel data. 3 Reversible-jump MCMC algorithm Following [4) we define the priors for our hierarchical model and derive a set of 5 move types for the reversible jump MCMC sampling scheme. To simplify some of the MCMC steps we choose a prior model where the prior on the weights is is that they are Dirichlet and the prior model for IJ.j = [JLji' .. .,JLjclV and drawn independently with normal and gamma priors, U;2 (6) where for the purposes of this study we follow[4] and define the hyper-parameters thus: 6 = 1.0; 'TJ is set to be the mean of the data; A is the diagonal precision matrix for the prior on IJ.j with components aj which are taken to be liT] where Tj is the data range in dimension j; a = 2.0 and (3 is some small multiple of liT;' The moves then consist of: I: updating the weights; II: updating the parameters (IJ., u); III: updating the allocation; IV: updating the hyper-parameters; V: splitting one component into two, or combining two into one. The first 4 moves are relatively simple to define, since the conjugate nature of the priors leads to relatively simple forms for the full conditional distribution of the desired parameter. Thus the first 4 moves are Gibbs sampling moves and the full conditional distributions for the weights 1rj, means Jij, variances Uj and allocation variables Zi are given by: (7) where nk is the number of observations allocated to component k; d p(ltjl .. ?) = -2 - II P(JLjml .. ?) : p(JLj..'!-.. ) '" N( njXimUj -2 m=1 (njuj + am'f/m -2 -1 ,(njuj + am) ), + am) (8) where we recognise that IJ.j is an d dimensional vector with components JL;m (m = 1, d), 'f/m are the components of the Itj prior mean and am represent the diagonal components of A. -2 p(u j \... ) 1 == r(lI + nj - 1, '2 n ~ L- (9) i=I :Zi;=l and (10) 580 A. D. Marrs The final move involves splitting/combining model components. The main criteria which need to be met when designing these moves are that they are irreducible, aperiodic, form a reversible pair and satisfy detailed balance [1]. The MCMC step for this move takes the form of a Metropolis-Hastings step where a move from state y to state y' is proposed, with 1r(Y) the target probability distribution and qm(Y, Y') the proposal distribution for the move m. The resulting move is then accepted with probability am 1r(Y')qm(y/,y)} (11) am - mtn , 1r() Y qm (y, Y') . _ . {I In the case of a move from state Y to a state y' which lies in a higher dimensional space, the move may be implemented by drawing a vector of continuous random variables u, independent of y. The new state y' is then set using an invertible deterministic function of x and u. It can be shown [2] that the acceptance probability is then given by . { 1r(y')Tm{y') 8y' } (12) am=mm 1'1r(y)Tm (y)q{u)1 8 (y,u)1 , where Tm(Y) is the probability of choosing move type m when in state y, and q(u) is the density function of u. The initial application of the reversible jump MCMC technique to normal mixtures [4J was limited to the univariate case. This yielded relatively simple expressions for the split/combine moves, and, most importantly, the determinant ofthe Jacobian of the tra~formation from a model with k components to one with k + 1 components was simple to derive. In the more general case of multivariate normal models care must be taken in prescribing move transformations. A complicated transformation will lead to problems when the !Jacobian I for a d-dimensional model is required. For multivariate spherical Gaussian models, we randomly choose a model component from the current k component model. The decision is then made to split or combine with one of its neighbours with probability P'k and PCIr respectively (where PCk = 1- Pile)' If the choice is to combine the component, we label the chosen component Zl, and choose Z2 to be a neighbouring component i with probability Q( l/T; where Tj is the distance from the component Zl. The new component resulting from the combination of Zl and Z2 is labelled Zc and its parameters are calculated from: (13) If the decision is to split, the chosen component is labelled Zc and it is used to define two new model components Zl and Z2 with weights and parameters conforming to (13). In making this transformation there are 2 + d degrees of freedom, so we need to generate 2 + d random numbers to enable the specification of the new component parameters. The random numbers are denoted u}, U2 = [U211 ... , u2dlT and U3. All are drawn from Beta{2,2) distributions while the components of U2 each have probability 0.5 of being negative. The split transformation is then defined by: Reversible-Jwnp MCMC for Multivariate Spherical Gaussian Mixtures 581 2 7r ZI (14) , UZ2 = (1 - U3 )U2Zc . 7r Z2 Once the new components have been defined it is necessary to evaluate the probability of choosing to combine component ZI with component Z2 in this new model. Having proposed the split/combine move all that remains is to calculate the Metropolis-Hastings acceptance probability (t, where (t = min(I, R) for the split move and (t = min(I, 1/R) for the combine move. Where in the case of a split move from a model with k components to one with k+ 1 components, or a combine move from k + 1 to k, R is given by: n~ p(X, le,e) n~ n ._l:?ij-?h;-I:?,,-.c p~~I,:~e,:;2 R= p(X,le,e) X o-l+nl 0-I+n2 11' '" 1r "2 'II"!c 1+nl +n2 B(6,k6) x n~;::1 J(~;) exp ( -~am ((/LZl m -11m)2 + (/LZ2 m - 11m? - (JLzcm - 11m)) ) X &(c7(';~2) (a-I) exp ( -f3(u;..2 +u~2 -u;:2)) X n;=1 g2,2(U2,)) X c7d+1 p:c::;;oc (g2,2(Ut)gl,1 (U3) 'II" ?c ?c (2?I-uI)uI)(d+ 1 )/2 J(I- u 3)u3) , (15) where g2,20 denotes a Beta(2,2) density function. The first line on the R.H.S is due to the ratio of likelihoods for those observations assigned to the components in question, the subsequent three lines are due to the prior ratios, the fifth line is due to the the proposal ratio and the last line due to the IJacobian I of the transformation. The term Palloe represents a combination of the probability of obtaining the current allocation of data to the components in question and the probability of choosing to combine components Zl and Z2. 4 Results To assess this approach to the estimation of multivariate spherical Gaussian mixture models, we firstly consider a toy problem where 1000 bivariate samples were generated from a known 20 component mixture model. This is followed by an analysis of the Peterson-Barney vowel data set comprising 780 samples of the measured amplitUde of four formant frequencies for 10 utterances. For this mixture estimation example, we ignore the class labels and consider the straight forward density estimation problem. 4.1 Simulated data The resulting reversible-jump MCMC chain of model order can be seen in figure 1, along with the resulting histogram (after rejecting the first 2000 MCMC sampies). The histogram shows that the maximum a posteriori value for model order is 17. The MAP estimate of model parameters was obtained by averaging all the 17 component model samples, the estimated model is shown in figure 2 alongside the original generating model. The results are rather encouraging given the large number of model components and the relatively small number of samples. A. D. Marrs 582 200 100 19 20 {k} Iteration Figure 1: Reversible-jump MCMC chain and histogram of model order for simulated data. ]0 20 . ',a,.1 .' . .......~~ \.::0:''.~. '?".' .' ~.....? -. \ .,:, ?? .' . '. \0" ..... 10.?.~???? ? ? ? ? ~'&."" .. w: " :. . -20 itt . ..... .... S,.' .;~:! : . :,~' . ,.'~, .~.:' -30 Cenerating Modal 10 -20 -10 .~. /I /1.\ .,. ? ~ -10 : ... . . ~. .~~ , '.l' . ?? -]0 " ,.' ?. .. .yt.' . -. ~~:! -20 ',~ ? ? . ie'';':', , ,T...,. ~ : I. -]0 :.'0 .~. ~. tI.-' ; . ,,;,~., .1....... \... _ .. /I. ? ,:~:. ~ .. . .. .... .. "'~" ~"" ~:':;.,., ~ ? ? . ' . tI? ?:_ ~. ~ ' .,. . ., .,' rl ? f#. 20 .."~'~:"""''''~' . ~" ,~, ' 'Ie" ,:": ".!'J" ? , .. ?' ?' - 10 ]0 ?? :' :,~' ":~.:~ MAP E.tilllllta Modal : I. 0 10 20 ]0 -]0 -20 -10 0 10 20 ]0 Figure 2: Example of model estimation for simulated data. 4.2 Vowel data The reversible-jump MCMC chain of model order for the Peterson-Barney vowel data example is shown in figure 3, alongside the resulting MAP model estimate. For ease of visualisation, the estimated model and data samples have been projected onto the first two principal components of the data. Again, the results are encouraging. 5 Conclusion One of the key problems when using Gaussian mixture models is estimation of the optimum number of components to include in the model. In this paper we extend the reversible-jump MCMC technique for estimating the parameters of Gaussian mixtures with an unknown number of components to the multivariate spherical Gaussian case. The technique is then demonstrated on a simulated data example and an example using a well known dataset. The attraction of this approach is that the number of mixture components is not fixed at the outset but becomes a parameter of the model. The reversible-jump MCMC approach is then capable of moving between parameter subspaces which Reversible-Jwnp MCMC for Multivariate Spherical Gaussian Mixtures 583 . Figure 3: Reversible-jump MCMC chain of model order and MAP estimate of model (projected onto first two principal components) for vowel data. correspond to models with different numbers of mixture components. As a result a sample of the full joint distribution is generated from which the posterior distribution for the number of model components can be derived. This information may then either be used to construct a Bayesian classifier or to define the centres in a radial basis function networ k. References [1] W.R. Gilks, S. Richardson, and D.J. Spiegelhalter Eds. Markov Chain Monte Carlo in Practice. Chapman and Hall, 1995. [2] P.J. Green. Reversible jump MCMC computation and Bayesian model determination. Boimetrika, 82:711-732, 1995. [3] D.B. Phillips and A.F.M. Smith. Bayesian model comparison via jump diffusions. In W.R. Gilks, S. Richardson, and D.J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice. Chapman and Hall, 1995. [4] S. Richardson and P. J. Green. On Bayesian analysis of mixtures with an unknown number of components. J. Royal Stat. Soc. Series B, 59(4), 1997. [5] D.M. Titterington, A.F.M. Smith, and U.E. Makov. Statistical Analysis of Finite Mixture Distributions. Wiley, 1985. ?British Crown Copyright 1998 /DERA. Published with the permission of the controller of Her Britannic Majesty's Stationary Office. 

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Gradients for retinotectal mapping Geoffrey J. Goodhill Georgetown Institute for Cognitive and Computational Sciences Georgetown University Medical Center 3970 Reservoir Road Washington IX: 20007 geoff@giccs.georgetown.edu Abstract The initial activity-independent formation of a topographic map in the retinotectal system has long been thought to rely on the matching of molecular cues expressed in gradients in the retina and the tectum. However, direct experimental evidence for the existence of such gradients has only emerged since 1995. The new data has provoked the discussion of a new set of models in the experimentalliterature. Here, the capabilities of these models are analyzed, and the gradient shapes they predict in vivo are derived. 1 Introduction During the early development of the visual system in for instance rats, fish and chickens, retinal axons grow across the surface of the optic tectum and establish connections so as to form an ordered map. Although later neural activity refines the map, it is not required to set up the initial topography (for reviews see Udin & Fawcett (1988); Goodhill (1992?. A long-standing idea is that the initial topography is formed by matching gradients of receptor expression in the retina with gradients of ligand expression in the tectum (Sperry, 1963). Particular versions of this idea have been formalized in theoretical models such as those of Prestige & Willshaw (1975), Willshaw & von der Malsburg (1979), Whitelaw & Cowan (1981), and Gierer (1983;1987). However, these models were developed in the absence of any direct experimental evidence for the existence of the necessary gradients. Since 1995, major breakthroughs have occurred in this regard in the experimental literature. These center around the Eph (Erythropoetin-producing hepatocellular) subfamily of receptor tyrosine kinases. Eph receptors and their ligands have been shown to be expressed in gradients in the developing retina and tectum respectively, and to playa role in guiding axons to appropriate positions. These exciting new developments have led experimentalists to discuss theoretical models differ- Gradients/or Retinotectal Mapping 153 ent from those previously proposed (e.g. Tessier-Lavigne (1995); Tessier-Lavigne & Goodman (1996); Nakamoto et aI, (1996)). However, the mathematical consequences of these new models, for instance the precise gradient shapes they require, have not been analyzed. In this paper, it is shown that only certain combinations of gradients produce appropriate maps in these models, and that the validity of these models is therefore experimentally testable. 2 Recent experimental data Receptor tyrosine kinases are a diverse class of membrane-spanning proteins. The Eph subfamily is the largest, with over a dozen members. Since 1990, many of the genes encoding Eph receptors and their ligands have been shown to be expressed in the developing brain (reviewed in Friedman & O'Leary, 1996). Ephrins, the ligands for Eph receptors, are all membrane anchored. This is unlike the majority of receptor tyrosine kinase ligands, which are usually soluble. The ephrins can be separated into two distinct groups A and B, based on the type of membrane anchor. These two groups bind to distinct sets of Eph receptors, which are thus also called A and B, though receptor-ligand interaction is promiscuous within each subgroup. Since many research groups discovered members of the Eph family independently, each member originally had several names. However a new standardized notation was recently introduced (Eph Nomenclature Committee, 1997), which is used in this paper. With regard to the mapping from the nasal-temporal axis of the retina to the anterior-posterior axis of the tectum (figure 1), recent studies have shown the following (see Friedman & O'Leary (1996) and Tessier-Lavigne & Goodman (1996) for reviews). ? EphA3 is expressed in an increasing nasal to temporal gradient in the retina (Cheng et aI, 1995). ? EphA4 is expressed uniformly in the retina (Holash & Pasquale, 1995). ? Ephrin-A2, a ligand of both EphA3 and EphA4, is expressed in an increasing rostral to caudal gradient in the tectum (Cheng et aI, 1995). ? Ephrin-A5, another ligand of EphA3 and EphA4, is also expressed in an increasing rostral to caudal gradient in the tectum, but at very low levels in the rostral half of the tectum (Drescher et aI, 1995). All of these interactions are repulsive. With regard to mapping along the complementary dimensions, EphB2 is expressed in a high ventral to low dorsal gradient in the retina, while its ligand ephrin-B1 is expressed in a high dorsal to low ventral gradient in the tectum (Braisted et aI, 1997). Members of the Eph family are also beginning to be implicated in the formation of topographic projections between many other pairs of structures in the brain (Renping Zhou, personal communication). For instance, EphA5 has been found in an increasing lateral to medial gradient in the hippocampus, and ephrin-A2 in an increasing dorsal to ventral gradient in the septum, consistent with a role in establishing the topography of the map between hippocampus and septum (Gao et aI, 1996). The current paper focusses just on the paradigm case of the nasal-temporal to anterior-posterior axis of the retinotectal mapping. Actual gradient shapes in this system have not yet been quantified. The analysis below will assume that certain gradients are linear, and derive the consequences for the other gradients. G. J. Goodhill 154 TECTUM RETINA c N L(y) R(x) k:==::y "'-----.......;;;~x Figure 1: This shows the mapping that is normally set up from the retina to the tectum. Distance along the nasal-temporal axis of the retina is referred to as x and receptor concentration as R( x). Distance along the rostral-caudal axis of the tectum is referred to as y and ligand concentration as L(y). 3 Mathematical models Let R be the concentration of a receptor expressed on a growth cone or axon, and L the concentration of a ligand present in the tectum. Refer to position along the nasal-temporal axis of the retina as x, and position along the rostral-caudal axis of the tectum as y, so that R = R(x) and L = L(y) (see figure 1). Gierer (1983; 1987) discusses how topographic information could be signaled by interactions between ligands and receptors. A particular type of interaction, proposed by Nakamoto et al (1996), is that the concentration of a "topographic signal", the signal that tells axons where to stop, is related to the concentration of receptor and ligand by the law of mass action: G(x, y) = kR(x)L(y) (1) where G(x, y) is the concentration of topographic signal produced within an axon originating from position x in the retina when it is at position y in the tectum, and k is a constant. In the general case of multiple receptors and ligands, with promiscuous interactions between them, this equation becomes G(x, y) = L: kijRi(X)Lj(Y) (2) i,j Whether each receptor-ligand interaction is attractive or repulsive is taken care of by the sign of the relevant k ij ? Two possibilities for how G(x, y) might produce a stop (or branch) signal in the growth cone (or axon) are that this occurs when (1) a "set point" is reached (discussed in, for example, Tessier-Lavigne & Goodman (1996); Nakamoto et al (1996? ,i.e. G (x, y) = c where c is a constant, or (2) attraction (or repulsion) reaches a local maximum (or minimum), i.e. &G~~,y) = 0 (Gierer, 1983; 1987). For a smooth, uni- Gradients for Retinotectal Mapping 155 form mapping, one of these conditions must hold along a line y ex: x. For simplicity assume the constant of proportionality is unity. 3.1 Set point rule For one gradient in the retina and one gradient in the tectum (i.e. equation 1), this requires that the ligand gradient be inversely proportional to the receptor gradient: c L(x) = R(x) If R(x) is linear (c.f. the gradient of EphA3 in the retina), the ligand concentration is required to go to infinity at one end of the tectum (see figure 2). One way round this is to assume R(x) does not go to zero at x = 0: the experimental data is not precise enough to decide on this point. However, the addition of a second receptor gradient gives c L(x) = k1R1 (x) + k2R2(X) If R1 (x) is linear and R 2(x) is flat (c.f. the gradient of EphA4 in the retina), then L (y) is no longer required to go to infinity (see figure 2). For two receptor and two ligand gradients many combinations of gradient shapes are possible. As a special case, consider R1 (x) linear, R 2(x) flat, and L 1(y) linear (c.f. the gradient of Elfl in the tectum). Then L2 is required to have the shape L ( ) = ay2 2 Y dy + by +e where a, b, d, e are constants. This shape depends on the values of the constants, which depend on the relative strengths of binding between the different receptor and ligand combinations. An interesting case is where R1 binds only to L1 and R2 binds only .to L 2 , i.e. there is no promiscuity. In this case we have L 2(y) ex: y2 (see figure 2). This function somewhat resembles the shape of the gradient that has been reported for ephrin-AS in the tectum. However, this model requires one gradient to be attractive, whereas both are repulsive. 3.2 Local optimum rule For one retinal and one tectal gradient we have the requirement R(x) aL(y) = 0 ay This is not generally true along the line y = x, therefore there is no map. The same problem arises with two receptor gradients, whatever their shapes. For two receptor and two ligand gradients many combinations of gradient shapes are possible. (Gierer (1983; 1987) investigated this case, but for a more complicated reaction law for generating the topographic signal than mass action.) For the special case introduced above, L 2 (y) is required to have the shape L2(y) = ay + blog(dy + e) + f where a, b, d, e, and f are constants as before. Considering the case of no promiscuity, we again obtain L 2(y) ex: y2 i.e. the same shape for L2 (y) as that specified by the set point rule. G. 1. Goodhill 156 A L B L c Figure 2: Three combinations of gradient shapes that are sufficient to produce a smooth mapping with the mass action rule. In the left column the horizontal axis is position in the retina while the vertical axis is the concentration of receptor. In the right column the horizontal axis is position in the tectum while the vertical axis is the concentration of ligand. Models A and B work with the set point but not the local optimum rule, while model C works with both rules. For models B and C, one gradient is negative and the other positive. Gradients for Retinotectal Mapping 157 4 Discussion For both rules, there is a set of gradient shapes for the mass-action model that is consistent with the experimental data, except for the fact that they require one gradient in the tectum to be attractive. Both ephrin-A2 and ephrin-A5 have repulsive effects on their receptors expressed in the retina, which is clearly a problem for these models. The local optimum rule is more restrictive than the set point rule, since it requires at least two ligand gradients in the tectum. However, unlike the set point rule, it supplies directional information (in terms of an appropriate gradient for the topographic signal) when the axon is not at the optimal location. In conclusion, models based on the mass action assumption in conjunction with either a "set point" or "local optimum" rule can be true only if the relevant gradients satisfy the quantitative relationships described above. A different theoretical approach, which analyzes gradients in terms of their ability to guide axons over the maximum possible distance, also makes predictions about gradient shapes in the retinotectal system (Goodhill & Baier, 1998). Advances in experimental technique should enable a more quantitative analysis of the gradients in situ to be performed shortly, allowing these predictions to be tested. In addition, analysis of particular Eph and ephrin knockout mice (for instance ephrin-A5 (Yates et aI, 1997? is now being performed, which should shed light on the role of these gradients in normal map development. Bibliography Braisted, J.E., McLaughlin, T., Wang, H.U., Friedman, G.C, Anderson, D.J. & O'Leary, D.D.M. (1997). Graded and lamina-specific distributions of ligands of EphB receptor tyrosine kinases in the developing retinotectal system. Developmental Biology, 19114-28. Cheng, H.J., Nakamoto, M., Bergemann, A.D & Flanagan, J.G. (1995). Complementary gradients in expression and binding of Elf-1 and Mek4 in development of the topographic retinotectal projection map. Cell, 82,371-381. Drescher, U., Kremoser, C, Handwerker, C, Loschinger, J., Noda, M. & Bonhoeffer, F. (1995). In-vitro guidance of retinal ganglion-cell axons by RAGS, a 25 KDa tectal protein related to ligands for Eph receptor tyrosine kinases. Cell, 82, 359370. Eph Nomenclature Committee (1997). Unified nomenclature for Eph family receptors and their ligands, the ephrins. Cell, 90, 403-404. Friedman, G.C & O'Leary, D.D.M. (1996). Eph receptor tyrosine kinases and their ligands in neural development. Curro Opin. Neurobiol., 6, 127-133. Gierer, A. (1983). Model for the retinotectal projection. Proc. Roy. Soc. Lond. B, 218, 77-93. Gierer, A. (1987). Directional cues for growing axons forming the retinotectal projection. Development, 101,479-489. Gao, P.-P., Zhang, J.-H., Yokoyama, M., Racey, R, Dreyfus, CF., Black, LR & Zhou, R. (1996). Regulation of topographic projection in the brain: Elf-1 in the hippocampalseptal system. Proc. Nat. Acad. Sci. USA, 93, 11161-11166. Goodhill, G.J. (1992). Correlations, Competition and Optimality: Modelling the Development of Topography and Ocular Dominance. Cognitive Science Research Paper CSRP 226, University of Sussex. Available from www.giccs.georgetown.edu/ "'geoff 158 G. 1. Goodhill Goodhill, G.J. & Baier, H. (1998). Axon guidance: stretching gradients to the limit. Neural Computation, in press. Holash, J.A. & Pasquale, E.B. (1995). Polarized expression of the receptor proteintyrosine kinase CekS in the developing avian visual system. Developmental Biology, 172, 683-693. Nakamoto, M., Cheng H.J., Friedman, G.C, Mclaughlin, T., Hansen, M.J., Yoon, CH., O'Leary, D.D.M. & Flanagan, J.G. (1996). Topographically specific effects of ELF-Ion retinal axon guidance in-vitro and retinal axon mapping in-vivo. Cell, 86, 755-766. Prestige, M.C & Willshaw, D.J. (1975). On a role for competition in the formation of patterned neural connexions. Proc. R. Soc. Lond. B, 190, 77-98. Sperry, RW. (1963). Chemoaffinity in the orderly growth of nerve fiber patterns and connections. Proc. Nat. Acad. Sci., U.S.A., 50, 703-710. Tessier-Lavigne, M. (1995). Eph receptor tyrosine kinases, axon repulsion, and the development of topographic maps. Cell, 82, 345-348. Tessier-Lavigne, M. and Goodman, CS. (1996). The molecular biology of axon guidance. Science, 274, 1123-1133. Udin, S.B. & Fawcett, J.W. (1988). Formation of topographic maps. Ann. Rev. Neurosci., 11,289-327. Whitelaw, V.A. & Cowan, J.D. (1981). Specificity and plasticity of retinotectal connections: a computational model. Jou. Neurosci., 1, 1369-1387. Willshaw, D.J. & Malsburg, C von der (1979). A marker induction mechanism for the establishment of ordered neural mappings: its application to the retinotectal problem. Phil. Trans. Roy. Soc. B, 287, 203-243. Yates, P.A., McLaughlin, T., Friedman, G.C, Frisen, J., Barbacid, M. & O'Leary, D.D.M. (1997). Retinal axon guidance defects in mice lacking ephrin-A5 (ALl/RAGS). Soc. Neurosci. Abstracts, 23, 324. 

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Learning nonlinear overcomplete representations for efficient coding Michael S. Lewicki Terrence J. Sejnowski lewicki~salk.edu terry~salk.edu Howard Hughes Medical Institute Computational Neurobiology Lab The Salk Institute 10010 N. Torrey Pines Rd. La Jolla, CA 92037 Abstract We derive a learning algorithm for inferring an overcomplete basis by viewing it as probabilistic model of the observed data. Overcomplete bases allow for better approximation of the underlying statistical density. Using a Laplacian prior on the basis coefficients removes redundancy and leads to representations that are sparse and are a nonlinear function of the data. This can be viewed as a generalization of the technique of independent component analysis and provides a method for blind source separation of fewer mixtures than sources. We demonstrate the utility of overcomplete representations on natural speech and show that compared to the traditional Fourier basis the inferred representations potentially have much greater coding efficiency. A traditional way to represent real-values signals is with Fourier or wavelet bases. A disadvantage of these bases, however, is that they are not specialized for any particular dataset. Principal component analysis (PCA) provides one means for finding an basis that is adapted for a dataset, but the basis vectors are restricted to be orthogonal. An extension of PCA called independent component analysis (Jutten and Herault, 1991; Comon et al., 1991; Bell and Sejnowski, 1995) allows the learning of non-orthogonal bases. All of these bases are complete in the sense that they span the input space, but they are limited in terms of how well they can approximate the dataset's statistical density. Representations that are overcomplete, i. e. more basis vectors than input variables, can provide a better representation, because the basis vectors can be specialized for Learning Nonlinear Overcomplete Representations for Efficient Coding 557 a larger variety of features present in the entire ensemble of data. A criticism of overcomplete representations is that they are redundant, i.e. a given data point may have many possible representations, but this redundancy is removed by the prior probability of the basis coefficients which specifies the probability of the alternative representations. Most of the overcomplete bases used in the literature are fixed in the sense that they are not adapted to the structure in the data. Recently Olshausen and Field (1996) presented an algorithm that allows an overcomplete basis to be learned. This algorithm relied on an approximation to the desired probabilistic objective that had several drawbacks, including tendency to breakdown in the case of low noise levels and when learning bases with higher degrees of overcompleteness. In this paper, we present an improved approximation to the desired probabilistic objective and show that this leads to a simple and robust algorithm for learning optimal overcomplete bases. 1 Inferring the representation The data, X 1 :L ' are modeled with an overcomplete linear basis plus additive noise: x = AS+i (1) where A is an L x M matrix, whose columns are the basis vectors, where M ~ L . We assume Gaussian additive noise so that 10gP(xIA, s) ()( -A(X - As)2/2, where A = 1/(12 defines the precision of the noise. The redundancy in the overcomplete representation is removed by defining a density for the basis coefficients, P(s), which specifies the probability of the alternative representations. The most probable representation, 5, is found by maximizing the posterior distribution s = maxP(sIA,x) = 8 maxP(s)P(xIA,s) (2) 8 P(s) influences how the data are fit in the presence of noise and determines the uniqueness of the representation. In this model, the data is a linear function of s, but s is not, in general, a linear function of the data. IT the basis function is complete (A is invertible) then, assuming broad priors and low noise, the most probable internal state can be computed simply by inverting A. In the case of an overcomplete basis, however, A can not be inverted. Figure 1 shows how different priors induce different representations. Unlike the Gaussian prior, the optimal representation under the Laplacian prior cannot be obtained by a simple linear operation. One approach for optimizing sis to use the gradient of the log posterior in an optimization algorithm. An alternative method for finding the most probable internal state is to view the problem as the linear program: min 1 T s such that As = x. This can be generalized to handle both positive and negative s and solved efficiently and exactly with interior point linear programming methods (Chen et al., 1996) . M. S. Lewicki and T. 1. Sejnowski 558 a b if L2 G) :::l ~ 0.1 10'05~\ L1 8 \" o ~- 20 -~ - - ~- - 80 100 120 Figure 1: Different priors induce different representations. (a) The 2D data distribution has three main axes which form an overcomplete representation. The graphs marked "L2" and "L1" show the optimal scaled basis vectors for the data point x under the Gaussian and Laplacian prior, respectively. Assuming zero noise, a Gaussian for P{s) is equivalent to finding the exact fitting s with minimum L2 norm, which is given by the pseudoinverse s = A+x. A Laplacian prior (P{Sm) ex: exp[-OlsmlJ) yields the exact fit with minimum L1 norm, which is a nonlinear operation which essentially selects a subset of the basis vectors to represent the data (Chen et al., 1996). The resulting representation is sparse. (b) A 64-sample segment of speech was fit to a 2x overcomplete Fourier representation (128 basis vectors). The plot shows rank order distribution of the coefficients of s under a Gaussian prior (dashed); and a Laplacian prior (solid). Far more significantly positive coefficients are required under the Gaussian prior than under the Laplacian prior. 2 Learning The learning objective is to adapt A to maximize the probability of the data which is computed by marginalizing over the internal states P(xIA) = J (3) ds P(s)P(xIA, s) general, this integral cannot be evaluated analytically but can be approximated with a Gaussian integral around s, yielding log P(xIA) ~ const. + log pes) - ~ (x - As)2 - ~ log det H (4) where H is the Hessian of the log posterior at S, given by )'ATA - VVlogP(s). To avoid a singularity under the Laplacian prior, we use the approximation (logP(sm?)' ~ -8tanh(,8sm) which gives the Hessian full rank and positive determinant. For large ,8 this approximates the true Laplacian prior. A learning rule can be obtained by differentiating log P(xIA) with respect to A. In the following discussion, we will present the derivations of the three terms in (4) and simplifying assumptions that lead to the following simple form of the learning rule (5) Learning Nonlinear Ollercomplete Representations for Efficient Cod(ng 2.1 559 Deriving V log pes) This term specifies how to change A so as to make the probability of the representation s more probable. IT we assume a Laplacian prior, this component changes A to make the representation more sparse. pes) = rIm P(Sm). In order to obtain 8sm/8aij, we need to describe If the basis is complete (and we assume low noise), then we may simply invert A to obtain s = A -IX. When A is overcomplete, however, there We assume s as a function of A. is no simple expression, but we may still make an approximation. Under priors, the most probable solution, s, will yield at most L non-zero elements. In effect, this selects a complete basis from A. Let A represent this reduced basis under s. We then have s = A -1(X- ?) where s is equal to s with M - L zero-valued elements removed. A-I obtained by removing the columns of A corresponding to the M - L zero-valued elements of s. This allows us to use results obtained for the case when A is invertible. Following MacKay (1996) we obtain (6) Rewriting in matrix notation we have A~ -Tv~T zs 810gP(s) _ 8A - - (7) We can use this to obtain an expression in terms of the original variables. We simply invert the mapping s ~ s to obtain Z f- z and W T f- A-T (row-wise) with Zm = 0 and row m ofWT = 0 if 8m = O. We then have 8 log P(s) 8A 2.2 =- WT zs T (8) Deriving Vex - As)2 The second term specifies how to change A so as to minimize the data misfit. Letting ek = [x - AS]k and using the results and notation from above we have: 8 A~ 2 -8 .. "2 L-ek a" k ~ ~ k I 8s, alJ (9) -aklWliSj (10) = AeiSj + AL-ek L-ak'~ = AeiSj + ALek L k = AeiSj - AeiSj I =0 (11) Thus no gradient component arises from the error term. 2.3 Deriving V log det H The third term in the learning rule specifies how to change the weights so as to minimize the width of the posterior distribution P(xIA) and thus increase the overall probability of the data. An element of H is defined by Hmn = Cmn + bmn M. S. Lewicki and T. J. Sejnowski 560 where Cmn = Ek Aakmakn and bmn = [- V'V' log P(s)]mn. This gives 8logdetH _ ""H- 1 [8e mn - ~ 8a ~?? mn nm 8a U.. + 8bmn ] 8a?? ~ (12) First considering 8Cmn/8aij, we can obtain L H~~ ~~~ = L mn ~3 Using the fact that H~; m:f.j H~;.\aim + L Hj~.\aim + Hj/ 2Aaij (13) m:f.j = Hj~ due to the symmetry of the Hessian, we have (14) Next we derive 8bmn/8aij. We have that V'V'logP(s) is diagonal, because we assume pes) = P(sm). Letting 2Ym = H~!n8bmm/8sm and using the result under the reduced representation (6) we can obtain nm (15) 2.4 Stabilizing and simplifying the learning rule Putting the terms together yields a problematic expression due to the matrix inverses. This can be alleviated by multiplying the gradient by an appropriate positive definite matrix, which rescales the gradient components but preserves a direction valid for optimization. Noting that ATW T = I we have (16) H'\ is large (low noise) then the Hessian is dominated by AATA and we have (17) The vector y hides a computation involving the inverse Hessian. IT the basis vectors in A are randomly distributed, then as the dimensionality of A increases the basis vectors become approximately orthogonal and consequently the Hessian becomes approximately diagonal. It can be shown that if log pes) and its derivatives are smooth, Ym vanishes for large A. Combining the remaining terms yields equation (5). Note that this rule contains no matrix inverses and the vector z involves only the derivative of the log prior. In the case where A is square, this form of the rule is similar to the natural gradient independent component analysis (ICA) learning rule (Amari et al., 1996). The difference in the more general case where A is rectangular is that s must maximize the posterior distribution P(slx, A) which cannot be done simply with the filter matrix as in standard ICA algorithms. Learning Nonlinear Overcomplete Representations/or Efficient Coding 3 561 Examples More sources than inputs. In these 2D examples, the bases were initialized to random, normalized vectors. The coefficients were solved using BPMPD and publicly available interior point linear programming package (Meszaros, 1997) which gives the most probable solution under the Laplacian prior assuming zero noise. The algorithm was run for 30 iterations using equation (5) with a stepsize of 0.001 and a batchsize of 200. Convergence was rapid, typically requiring less than 20 iterations. In all cases, the direction of the learned vectors matched those of the true generating distribution; the magnitude was estimated less precisely, possibly due to the approximation oflogP(xIA). This can be viewed as a source separation problem, but true separation will be limited due to the projection of the sources down to a smaller subspace which necessarily loses information . . ' '. ~ Figure 2: Examples illustrating the fitting of 2D distributions with overcomplete bases. The first example is equivalent to 3 sources mixed into 2 channels; the second to 4 sources mixed into 2 channels. The data in both examples were generated from the true basis A using x = As with the elements of s distributed according to an exponential distribution with unit mean. Identical results were obtained by drawing s from a Laplacian prior (positive and negative coefficients). The overcomplete bases allow the model to capture the true underlying statistical structure in the 2D data space. Overcomplete representations of speech. Speech data were obtained from the TIMIT database, using a single speaker was speaking ten different example sentences with no preprocessing. The basis was initialized to an overcomplete Fourier basis. A conjugate gradient routine was used to obtain the most probable basis coefficients. The stepsize was gradually reduced over 10000 iterations. Figure 3 shows that the learned basis is quite different from the Fourier representation. The power spectrum for the learned basis vectors can be multimodal and/or broadband. The learned basis achieves greater coding efficiency: 2.19 ? 0.59 bits per sample compared to 3.86 ? 0.28 bits per sample for a 2x overcomplete Fourier basis. 4 Summary Learning overcomplete representations allows a basis to better approximate the underlying statistical density of the data and consequently the learned representations have better encoding and denoising properties than generic bases. Unlike the case for complete representations and the standard ICA algorithm, the transformation 562 M S. Lewicki and T. 1. Sejnowski ..... ........... ~ ~ ~ ~ JvV ~ ~ ~ fIN ~ ~ ~ ~ ."... f\;vJ'v + ~ .. ... + ..... ..... ....... ~ ..... -JVV H ~ ~~J~~ ~ ~ ~~~ A- L -L J-.-L JL~~JLi~~L~ ~J~--L~ Figure 3: An example of fitting a 2x overcomplete representation to segments of from natural speech. Each segment consisted of 64 samples, sampled at a frequency of 8000 Hz (8 msecs). The plot shows a random sample of 30 of the 128 basis vectors (each scaled to full range). The right graph shows the corresponding power spectral densities (0 to 4000 Hz). from the data to the internal representation is non-linear. The probabilistic formulation of the basis inference problem offers the advantages that assumptions about the prior distribution on the basis coefficients are made explicit and that different models can be compared objectively using log P(xIA). References Amari, S., Cichocki, A., and Yang, H. H. (1996). A new learning algorithm for blind signal separation. In Advances in Neural and Information Processing Systems, volume 8, pages 757-763, San Mateo. Morgan Kaufmann. Bell, A. J. and Sejnowski, T. J. (1995). An information maximization approach to blind separation and blind deconvolution. Neural Computation, 7(6}:1129-1159. Chen, S., Donoho, D. L., and Saunders, M. A. (1996). Atomic decomposition by basis pursuit. Technical report, Dept. Stat., Stanford Univ., Stanford, CA. Comon, P., Jutten, C., and Herault, J. (1991). Blind separation of sources .2. problems statement. Signal Processing, 24(1}:11-20. Jutten, C. and Herault, J. (1991). Blind separation of sources .1. an adaptive algorithm based on neuromimetic architecture. Signal Processing, 24(1):1-10. MacKay, D. J. C. (1996). Maximum likelihood and covariant algorithms for independent component analysis. University of Cambridge, Cavendish Laboratory. Available at ftp: / /wol. ra. phy. cam. ac. uk/pub/mackay / ica. ps. gz. Meszaros, C. (1997). BPMPD: An interior point linear programming solver. Code available at ftp: / /ftp.netlib. org/ opt/bpmpd. tar. gz. 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. 

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? Experiences with Bayesian Learning In a Real World Application Peter Sykacek, Georg Dorffner Austrian Research Institute for Artificial Intelligence Schottengasse 3, A-10ID Vienna Austria peter, georg@ai.univie.ac.at Peter Rappelsberger Institute for Neurophysiology at the University Vienna Wahringer StraBe 17, A-lOgO Wien Peter.Rappelsberger@univie.ac.at Josef Zeitlhofer Department of Neurology at the AKH Vienna Wahringer Giirtel 18-20, A-lOgO Wien Josef.Zeitlhofer@univie.ac.at Abstract This paper reports about an application of Bayes' inferred neural network classifiers in the field of automatic sleep staging. The reason for using Bayesian learning for this task is two-fold. First, Bayesian inference is known to embody regularization automatically. Second, a side effect of Bayesian learning leads to larger variance of network outputs in regions without training data. This results in well known moderation effects, which can be used to detect outliers. In a 5 fold cross-validation experiment the full Bayesian solution found with R. Neals hybrid Monte Carlo algorithm, was not better than a single maximum a-posteriori (MAP) solution found with D.J. MacKay's evidence approximation. In a second experiment we studied the properties of both solutions in rejecting classification of movement artefacts. Experiences with Bayesian Learning in a Real World Application 1 965 Introduction Sleep staging is usually based on rules defined by Rechtschaffen and Kales (see [8]). Rechtschaffen and Kales rules define 4 sleep stages, stage one to four, as well as rapid eye movement (REM) and wakefulness. In [1] J. Bentrup and S. Ray report that every year nearly one million US citizens consulted their physicians concerning their sleep. Since sleep staging is a tedious task (one all night recording on average takes abou t 3 hours to score manually), much effort was spent in designing automatic sleep stagers. Sleep staging is a classification problem which was solved using classical statistical t.echniques or techniques emerged from the field of artificial intelligence (AI) . Among classical techniques especially the k nearest neighbor technique was used. In [1] J. Bentrup and S. Ray report that the classical technique outperformed their AI approaches. Among techniques from the field of AI, researchers used inductive learning to build tree based classifiers (e.g. ID3, C4.5) as reported by M. Kubat et. a1. in [4]. Neural networks have also been used to build a classifier from training examples. Among those who used multi layer perceptron networks to build the classifier, the work of R. Schaltenbrand et. a1. seems most interesting. In [to] they use a separate network to refuse classification of too distant input vectors. The performance usually reported is in the range of 75 to 85 percent. \Vhich enhancements to these approaches can be made to get a. reliable system wit.h hopefully better performance? According to S. Roberts et . al. in [9], outlier detection is important to get reliable results in a critical (e.g. medical) environment. To get reliable results one must refuse classification of dubious inputs. Those inputs are marked separately for further inspection by a human expert. To be able to detect such dubious inputs, we use Bayesian inference to calculate a distribution over the neural network weights. This approach automatically incorporates the calculation of confidence for each network estimate. Bayesian inference has the further advantage that regularization is part of the learning algorithm. Additional methods like weight decay penalty a.nd cross validation for decay parameter tuning are no longer needed . Bayesian inference for neural networks was among others investigated by D.J. MacKay (see [5]), Thodberg (see [11]) and Buntine and Weigend (~ee [3]). The a.im of this paper is to study how Bayesian inference leads to probabilities for classes, which together with doubt levels allow to refuse classification of outliers. As we are interested in evaluating the resulting performance, we use a comparative method on the same data set and use a significance test, such that the effect of the method can easily be evaluated. 2 Methods In this section we give a short description of the inference techniques used to perform the experiments. We have used two approaches using neural networks as classifiers and an instance based approach in order to make the performance estimates comparable to other methods . 2.1 Architecture for polychotomous classification For polychotomous classification problems usually a l-of-c target coding scheme is used. Usually it is sufficient to use a network architecture with one hidden layer. In [2] pp. 237-240, C. Bishop gives a general motivation for the softmax data model, P. Sykacek, G. Dorffner; P. Rappelsberger and 1. Zeitlhofer 966 which should be used if one wants the network outputs to be probabilities for classes. If we assume that the class conditional densities, p(? I Ck), of the hidden unit activation vector, ?, are from the general family of exponential distributions, then using t.he transformation in (1), allows to interpret the network outputs as probabilities for classes. This transformation is known as normalized exponential or softmax activation function. p(Ck 1?) = exp(ak) (1) Lkl exp(ak l ) In ! 1) t.he value ak is the value at output node k before applying softmax activation. Softmax transformation of the activations in the output layer is used for both network approaches used in this paper. 2.2 Bayesian Inference In [6] D.J. MacKay uses Bayesian inference and marginalization to get moderated probabilities for classes in regions where the network is uncertain about the class label. In conjunction with doubt levels this allows to suppress a classification of such patterns. A closer investigation of this approach showed that marginalization leads to moderated probabilities, but the degree of moderation heavily depends on the direction in which we move away from the region with sufficient training data. Therefore one has to be careful about whether the moderation effect should be used for outliers detection. A Bayesian solution for neural networks is a posterior distribution over weight space calculated via Bayes' theorem using a prior over weights. (2) In (2), w is the weight vector of the network and V represents the training data. Two different possibilities are known to calculate the posterior in (2). In [5] D.J . MacKay derives an analytical expression assuming a Gaussian distribution. In [7] R. Neal uses a hybrid Monte Carlo method to sample from the posterior. For one input pattern, the posterior over weight space will lead to a distribution of network outputs. For a classification problem, following MacKay [6], the network estimate is calculated by marginalization over the output distribution. P(C 1 I~, V) =.J P(C 1 = I~, w)p(w I V)dw Jy(~, w)p(w I V)dw (3) In general, the distribution over output activations will have small variance in regions well represented in the training data and large variance everywhere else. The reason for that is the influence of the likelihood term p(V I w), which forces the network mapping to lie close to the desired one in regions with training data, but which has no influence on the network mapping in regions without training data. At least for for generalized linear models applied to regression, this property is quantifiable. In [12] C. Williams et.al. showed that the error bar is proportional to the inverse input data density p(~)-l. A similar relation is also plausible for the output activation in classification problems. Experiences with Bayesian Learning in a Real World Application 967 Due to the nonlinearity of the softmax transformation, marginalization will moderate probabilities for classes. Moderation will be larger in regions with large variance of the output activation. Compared to a decision made with the most probable weight, the network guess for the class label will be less certain. This moderation effect allows to reject classification of outlying patterns. Since upper integral can not be solved analytically for classification problems, there are t.wo possibilities to solve it. In [6] D.J. MacKay uses an approximation. Using hybrid Monte Carlo sampling as an implementation of Bayesian inference (see R. Neal in [7]), there is no need to perform upper integration analytically. The hybrid Monte Carlo algorithm samples from the posterior and upper integral is calculated as a finite sum. 1 L P(C 1 I~, 1)) ~ L LY(~' Wi) (4) i=l Assuming, that the posterior over weights is represented exactly by the sampled weights, there is no need to limit the number of hidden units, if a correct (scaled) prior is used. Consequently in the experiments the network size was chosen to be large. We used 25 hidden units. Implementation details of the hybrid Monte Carlo algorithm may be found in [7]. 2.3 The Competitor The classifier, used to give performance estimates to compare to, is built as a two layer perceptron network with softmax transformation applied to the outputs. As an error function we use the cross entropy error including a consistent weight decay penalty, as it is e.g. proposed by C. Bishop in [2], pp. 338. The decay parameters are estimated with D.J. MacKay's evidence approximation ( see [5] for details). Note that the restriction of D.J. MacKay's implementation of Bayesian learning, which has no solution to arrive at moderated probabilities in l-of-c classification problems, do not apply here since we use only one MAP value. The key problem with this approach is the Gaussian approximation of the posterior over weights, which is used to derive the most probable decay parameters. This approximation is certainly only valid if the number of network parameters is small compared to the number of training samples. One consequence is, that the size of the network has to be restricted . Our model uses 6 hidden units. To make the performance of the Bayes inferred classifier also comparable to other methods, we decided to include performance estimates of a k nearest neighbor algorithm. This algorithm is easy to implement and from [1] we have some evidence that its performance is good. 3 Experiments and Results In this sect.ion we discuss the results of a sleep staging experiment based on the t.echniques described in the "Methods" section. 3.1 Data All experiments are performed with spectral features calculated from a database of 5 different healthy subjects. All recordings were scored according to the Rechtschaffen & Kales rules. The data pool consisted from data calculated for all electrodes 968 P. Sykacek, G. Doif.fner, P. Rappelsberger and J. Zeitlhofer available, which were horizontal eye movement, vertical eye movement and 18 EEG f'lectrodes placed with respect to the international 10-20 system. The data were transformed into the frequency domain. We used power density values as well as coherency between different electrodes, which is a correlation coefficient expressed as a function of frequency as input features. All data were transformed to zero mean and unit variance. From the resulting feature space we selected 10 features, which were used as inputs for classification. Feature selection was done with a suboptimal search algorithm which used the performance of a k nearest neighbor classifier for evaluation. We used more than 2300 samples during t.raining and about 580 for testing. 3.2 Analysis of Both Classifiers The analysis of both classifiers described in the "Methods" section should reveal whether besides good classification performance the Bayes' inferred classifier is also ,apable of refusing outlying test patterns. Increasing the doubt level should lead to better results of the classifier trained by Bayesian Inference if the test data contains out.lying patterns. We performed two experiments. During the first experiment Wf' calculated results from a 5 fold cross validation, where training is done with 4 subjects and tests are performed with one independent test person. In a second j,f'St. we examine the differences of both algorithms on patterns which are definitely outliers. We used the same classifiers as in the first experiment. Test patterns for t his experiment were classified movement artefacts, which should not be classified as one of the sleep stages. The classifier used in conjunction with Bayesian inference was a 2-layer neural net.work with 10 inputs, 25 hidden units with sigmoid activation and five output units with softmax activation. The large number of hidden units is motivated by the results reported from R. Neal in [7]. R. Neal studied the properties of neural networks in a Bayesian framework when using Gaussian priors over weights. He concluded that there is no need for limiting the complexity of the network when using a correct Bayesian approach. The standard deviation of the Gaussian prior 1S scaled by the number of hidden units. For the comparative approach we used a neural network with 10 inputs, 6 hidden units and 5 outputs with softmax activation. Optimization was done via the BFGS algorit.hm (see C. Bishop in [2]) with automatic weight decay parameter tuning (D.J. MacKay's evidence approximation). As described in the methods section, the smaller network used here is motivated by the Gaussian approximation of the posterior over weights, which is used in the expression for the most probable decay parameters. The third result is a result achieved with a k nearest neighbor classifier with k set to three. All results are summaried in table 1. Each column summarizes the results achieved with one of the algorithms and a certain doubt level during the cross validation run. As the k nearest neighbor classifier gives only coarse probability estimates, we give only the performance estimate when all test patterns are classified. An examination of table 1 shows that the differences between the MAP-solution and the Bayesian solution are extremely small. Consequently, using a t-test, the O-hypothesis could not be rejected at any reasonable significance level. On the other hand compared to the Bayesian solution, the performance of the k nearest neighbor classifier is significantly lower (the significance level is 0.001). Experiences with Bayesian Learning in a Real World Application 969 Table 1: Classification Performance Doubt Cases Mean Perf. Std. Dev. Doubt Cases Mean Perf. Std. Dev. Doubt Cases Mean Perf. Std. Dev. MAP 0 5% 100/0 78.6% 80.4% 81.6% 9.1% 9.4% 9.4% Bayes 0 5% 10% 78.4% 80.2% 82.2% 9.0% 8.6% 9.4% k nearest neighbor 0 5% 10% 74.6% 8.4% - 15% 83.2'70 9.1% 15% 83.6% 9.1% 15% - Table 2: Rejection of Movement Periods Method recognized outliers No. 0 1 2 0 1 MAP % 0% 7.7% 15.4% 0% 7.7% No. 1 6 5 5 3 Bayes % 7.1% 46.f% 38.5% 38.5% 23.1% The last experiment revealed that both training algorithms lead to comparable performance estimates, when clean data is used. When using the classifier in practice there is no guarantee that the data are clean. One common problem of all night recordings are the so called movement periods, which are periods with muscle activity due to movements of the sleeping subject. During a second experiment we tried t.o assess the robustness of both neural classifiers against such inputs. During this experiment we used a fixed doubt level, for which approximately 5% of the clean t.est. data from the last experiment were rejected. With this doubt level we classified 13 movement periods, which should not be assigned to any of the other stages. The number of correctly refused outlying patterns are shown in table 2. Analysis of the results with a t-test showed a significant higher rate of removed outliers for the full Bayesian approach. Nevertheless as the number of misclassified outliers is large, one has to be careful in using this side-effect of Bayesian inference. 4 Conclusion Using Bayesian Inference for neural network training is an approach which leads to better classification results compared with simpler training procedures. Comparing wit.h the "one MAP" solution, we observed significantly larger reliability in detecting dubious patterns. The large amount of remaining misclassified patterns, which were obviously outlying, shows that we should not rely blindly on the moderating effect of marginalization. Despite the large amount of time which is required to calculate t.he solution, Bayesian inference has relevance for practical applications. On one hand the Bayesian solution shows good performance. But the main reason is the a.bility to encode a validity region of the model into the solution. Compared to all methods which do not aim at a predictive distribution, this is a clear advantage for Bayesian inference. P. Sykacek, G. Doif.fner, P. Rappelsberger and 1. Zeitlhofer 970 Acknowledgements We want to acknowledge the work of R. Neal from the Departments of Statistics and Computer Science at the University of Toronto, who made his implementation of hybrid Monte-Carlo sampling for Bayesian inference available electronically. His software was used to calculate the full Bayes' inferred classification results. We also want to express gratitude to S. Roberts from Imperial College London, one of the partners in the ANNDEE project. His work and his consequence in insisting on confidence measures for network decisions had a large positive impact on our work. This work was sponsored by the Austrian Federal Ministry of Science and Transport. It was done in the framework of the BIOMED 1 concerted action ANNDEE, financed by the European Commission, DG. XII. References [1] J.A. Bentrup and S.R. Ray. An examination of inductive learning algorithms for the classification of sleep signals. Technical Report UIUCDCS-R-93-1792, Dept of Computer Science, University of Illinois, Urbana-Champaign, 1993. [3] C. M. Bishop. Neural Networks for Pattern Recognition. Clarendon Press, Oxford, 1995. [3] W. L. Buntine and A. S. Weigend. Bayesian back-propagation. Complex Systems, 5:603-643, 1991. [4] M. Kubat, G. Pfurtscheller, and D. Flotzinger. Discrimination and classification using bot.h binary and continuous variables. Biological Cybernetics, 70:443-448, 1994. [5] D. J. C. MacKay. Bayesian interpolation. Neural Computation, 4:415-447, 1992. [6] D. J. C. MacKay. The evidence framework applied to classification networks. Neural Computation, 4:720-736, 1992. [7] R. M. Neal. Bayesian Learning for Neural Networks. Springer, New York, 1996. [8] A. Rechtschaffen and A. Kales. A manual of standardized terminology, techniques and scoring system for sleep stages of human subjects. NIH Publication No. 204, US Government Printing Office, Washington, DC., 1968. [9] S. Roberts, L. Tarassenko, J. Pardey, and D. Siegwart. A confidence measure for artificial neural networks. In International Conference Neural Networks and Expert Systems in Medicine and Healthcare, pages 23-30, Plymouth, UK, 1994. [10] N. Schaltenbrand, R. Lengelle, and J.P. Macher. Neural network model: application to automatic analysis of human sleep. Computers and Biomedical Research, 26:157171, 1993. ~ll] H. H. Thodberg. A review of bayesian neural networks with an application to near infrared spectroscopy. IEEE Transactions on Neural Networks, 7(1):56-72, January 1996. [12] C. K. I. Williams, C. Quazaz, C. M. Bishop, and H. Zhu. On the relationship between bayesian error bars and the input data density. In Fourth International Conference on Artificial Neural Networks, Churchill Col/ege, University of Cambridge, UK. lEE Conference Publication No. 409, pages 160-165, 1995. 

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RCC Cannot Compute Certain FSA, Even with Arbitrary Transfer Functions Mark Ring RWCP Theoretical Foundation GMD Laboratory GMD - German National Research Center for Information Technology Schloss Birlinghoven D-53 754 Sankt Augustin, Germany email: Mark .Ring@GMD.de Abstract Existing proofs demonstrating the computational limitations of Recurrent Cascade Correlation and similar networks (Fahlman, 1991; Bachrach, 1988; Mozer, 1988) explicitly limit their results to units having sigmoidal or hard-threshold transfer functions (Giles et aI., 1995; and Kremer, 1996). The proof given here shows that for any finite, discrete transfer function used by the units of an RCC network, there are finite-state automata (FSA) that the network cannot model, no matter how many units are used. The proof also applies to continuous transfer functions with a finite number of fixed-points, such as sigmoid and radial-basis functions. 1 Introduction The Recurrent Cascade Correlation (RCC) network was proposed by Fahlman (1991) to offer a fast and efficient alternative to fully connected recurrent networks. The network is arranged such that each unit has only a single recurrent connection: the connection that goes from itself to itself. Networks with the same structure have been proposed by Mozer (Mozer, 1988) and Bachrach (Bachrach, 1988). This structure is intended to allow simplified training of recurrent networks in the hopes of making them computationally feasible. However, this increase in efficiency comes at the cost of computational power: the networks' computational capabilities are limited regardless of the power of their activation functions. The remaining input to each unit consists of the input to the network as a whole together with the outputs from all units lower in the RCC network. Since it is the structure of the network and not the learning algorithm that is of interest here, only the structure will be described in detail. M. Ring 620 Figure 1: This finite-state automaton was shown by Giles et al. (1995) to be unrepresentable by an Ree network whose units have hard-threshold or sigmoidal transfer functions. The arcs are labeled with transition labels of the FSA which are given as input to the Ree network. The nodes are labeled with the output values that the network is required to generate. The node with an inner circle is an accepting or halting state . Figure 2: This finite-state automaton is one of those shown by Kremer (1996) not to be representable by an Ree network whose units have a hard-threshold or sigmoidal transfer function . This FSA computes the parity of the inputs seen so far . The functionality of a network of N the following way: Ree units, Uo, .. ., UN-l can be described in 1? (1) /o([(t), Vo(t /x(i(t), Vx(t - 1), Vx-1(t), Vx-2(t), .. . , Vo(t?, (2) where Vx(t) is the output value of Ux at time step t, and l(t) is the input to the network at time step t. The value of each unit is determined from: (1) the network input at the current time step, (2) its own value at the previous time step, and (3) the output values of the units lower in the network at the current time step . Since learning is not being considered here, the weights are assumed to be constant. 2 Existing Proofs The proof of Giles, et al (1995) showed that an Ree network whose units had a hard-threshold or sigmoidal transfer function cannot produce outputs that oscillate with a period greater than two when the network input is constant. (An oscillation has a period of x if it repeats itself every x steps.) Thus, the FSA shown in Figure 1 cannot be modeled by such an Ree network, since its output (shown as node labels) oscillates at a period greater than two given constant input. Kremer (1996) refined the class of FSA representable by an Ree network showing that, if the input to the net oscillates with period p, then the output can only oscillate with a period of w, where w is one of p's factors (or of 2p's factors if p is odd). An unrepresentable example, therefore, is the parity FSA shown in Figure 2, whose output has a period of four given the following input (of period two): 0,1,0,1, .... Both proofs, that by Giles et al. and that by Kremer, are explicitly designed with RCC Cannot Compute Certain FSA, Even with Arbitrary Transfer Functions 621 *0,1 Figure 3: This finite-state automaton cannot be modeled with any RCC network whose units are capable of representing only k discrete outputs. The values within the circles are the state names and the output expected from the network. The arcs describe transitions from state to state, and their values represent the input given to the network when the transition is made. The dashed lines indicate an arbitrary number of further states between state 3 and state k which are connected in the same manner as states 1,2, and 3. (All states are halting states.) hard-threshold and sigmoidal transfer functions in mind, and can say nothing about other transfer functions. In other words, these proofs do not demonstrate the limitations of the RCC-type network structure, but about the use of threshold units within this structure. The following proof is the first that actually demonstrates the limitations of the single-recurrent-link network structure. 3 Details of the Proof This section proves that RCC networks are incapable even in principle of modeling certain kinds of FSA, regardless of the sophistication of each unit's transfer function, provided only that the transfer function be discrete and finite, meaning only that the units of the RCC network are capable of generating a fixed number, k, of distinct output values. (Since all functions implemented on a discrete computer fall into this category, this assumption is minor. Furthermore, as will be discussed in Section 4, the outputs of most interesting continuous transfer functions reduce to only a small number of distinct values.) This generalized RCC network is proven here to be incapable of modeling the finite-state automaton shown in Figure 3. MRing 622 For ease of exposition, let us call any FSA of the form shown in Figure 3 an RFk+l for Ring FSA with k + 1 states. I Further, call a unit whose output can be any of k distinct values and whose input includes its own previous output, a DRU k for Discrete Recurrent Unit. These units are a generalization ofthe units used by RCC networks in that the specific transfer function is left unspecified. By proving the network is limited when its units are DRUbs proves the limitations of the network's structure regardless of the transfer function used. Clearly, a DRUk+1 with a sufficiently sophisticated transfer function could by itself model an RFk+1 by simply allocating one of its k + 1 output values for each of the k + 1 states. At each step it would receive as input the last state of the FSA and the next transition and could therefore compute the next state. By restricting the units in the least conceivable manner, i.e., by reducing the number of distinct output values to k, the RCC network becomes incapable of modeling any RFk+1 regardless of how many DRUk's the network contains. This will now be proven. The proof is inductive and begins with the first unit in the network, which, after being given certain sequences of inputs, becomes incapable of distinguishing among any states of the FSA. The second step, the inductive step, proves that no finite number of such units can 'assist a unit hi~her in the ReC network in making a distinction between any states of the RFk+ . Lemma 1 No DR Uk whose input is the current transition of an RFk+1 can reliably distinguish among any states of the RP+I. More specifically, at least one of the DR Uk,s k output values can be generated in all of the RP+I 's k + 1 states. Proof: Let us name the DRUbs k distinct output values VO, VI, ... , Vk-I. The mapping function implemented by the DRU k can be expressed as follows: (V X , i) =} VY, which indicates that when the unit's last output was V X and its current input is i, then its next output is VY. Since an RFk is cyclical, the arithmetic in the following will also be cyclical (i.e., modular): where 0 ~ xtfJy = x8y - { x+y x+y-k x-y { x+k-y if x + y < k if x + y ~ k if x 2: y if x < y x < k and 0 ~ y < k. Since it is impossible for the DRU k to represent each of the RFk+I,s k +1 states with a distinct output value, at least two of these states must be represented ambiguously by the same value. That is, there are two RFk+l states a and b and one DRU k value V a / b such that V a / b can be generated by the unit both when the FSA is in state a and when it is in state b. Furthermore, this value will be generated by the unit given an appropriate sequence of inputs. (Otherwise the value is unreachable, serves no purpose, and can be discarded, reducing the unit to a DRU k- I .) Once the DRU k has generated V a / b , it cannot in the next step distinguish whether the FSA's current state is a or b. Since the FSA could be in either state a or b, the next state after a b transition could be either a or b tfJ 1. That is: (va/b, b) =} Va/bEl'll, (3) IThanks to Mike Mozer for suggesting this catchy name. RCC Cannot Compute Certain FSA, Even with Arbitrary Transfer Functions 623 where a e b ~ be a and k > 1. This new output value Va/b$l can therefore be generated when the FSA is in either state a or state b EB 1. By repeatedly replacing b with b EB 1 in Equation 3, all states from b to a e 1 can be shown to share output values with state a, i.e., V a / b , Va/b$l, V a / b$2, ... , va/ae2, v a / ae1 all exist. Repeatedly substituting a eland a for a and b respectively in the last paragraph produces values vx/y Vx, YEO, 1, ... , k + 1. There is, therefore, at least one value that can be generated by the unit in both states of every possible pair of states. Since there are (k! 1) distinct pairs but only kdistinct output values, and since when k > 1, then not all of these pairs can be represented by unique V values. At least two of these pairs must share the same output value, and this implies that some v a / b/ e exists that can be output by the unit in any of the three FSA states a, b, and c. Starting with (V a / b/ e , c) ::::} va/b/e$l, and following the same argument given above for V a / b , there must be a vx/y/z for all triples of states x, Y, and z. Since there are distinct output values, and since where k > 3, some v a / b/ e / d fi+ll (k ~ 1) distinct triples but only k > 1, must also exist. rer)l This argument can be followed repeatedly since: >1, + 1, including when m = k. Therefore, there is at least one that can be output by the unit in all k + 1 states of the RFk+l. Call this value and any other that can be generated in all FSA states ~,k. All Vk>s are reachable (else they could be discarded and the above proof applied for DRU I , / < k). When a Vk is output by a DRU k , it does not distinguish any states of the RFH 1 . for all m < k VO/l/2f..fk/k+l Lemma 2 Once a DRUk outputs a V k , all future outputs will a/so be Vk's. Proof: The proof is simply by inspection, and is shown in the following table: Actual State Transition Next State x xEB1 xEB2 xEB3 x x x x xEB1 x xEB2 xEB3 x82 x81 x x x82 x81 M. Ring 624 If the unit's last output value was a Vk, then the FSA might be in any of its k + 1 possible states. As can be seen, if at this point any of the possible transitions is given as input, the next state can also be any of the k + 1 possible states. Therefore, no future inp'ut can ever serve to lessen the unit's ambiguity. Theorem 1 An RGG network composed of any finite number of DR Uk 's cannot model an Rpk+l. Proof: Let us describe the transitions of an RCC network of N units by using the following notation: ((VN-I , VN-2, ... , VI, Va), i) ~ (VN- I , VN- 2, ... , V{, V~), where Vrn is the output value of the m'th unit (i.e., Urn) before the given input, i, is seen by the network, and V~ is Urn's value after i has been processed by the network. The first unit, Uo, receives only i and Va as input. Every other unit Ux receives as input i and Vx as well as v~, y < x. Lemma 1 shows that the first unit, Uo, will eventually generate a value vl, which can be generated in any of the RFk+1 states. From Lemma 2, the unit will continue to produce vl values after this point. Given any finite number N of DRUk,s, Urn-I, ... , Uo that are producing their Vk values, V~ -1' .. . , Vt, the next higher unit , UN, will be incapable of disambiguating all states by itself, i.e., at least two FSA states , a and b, will have overlapping output values, Since none of the units UN-I, ... , o can distinguish between any states (including a and b), k k k) b (a / b(JJ 1 k Vk k) ( (VNa/ b ,VN-I,?? ?, VI'VO ' )~ VN 'VN- I '?? ?, I'VO ' assuming that be a ~ ae b and k > 1. The remainder of the prooffollows identically along the lines developed for Lemmas 1 and 2. The result of this development is that UN also has a set of reachable output values V~ that can be produced in any state of the FSA. Once one such value is produced, no less-ambiguous value is ever generated. Since no RCC network containing any number of DRU k's can over time distinguish among any states of an RFHI, no such RCC network can model such an FSA. V;,p. 4 U Continuous Transfer Functions Sigmoid functions can generate a theoretically infinite number of output values; if represented with 32 bits, they can generate 232 outputs. This hardly means, however, that all such values are of use. In fact, as was shown by Giles et al. (1995), if the input remains constant for a long enough period of time (as it can in all RFHI'S) , the output of sigmoid units will converge to a constant value (a fixed point) or oscillate between two values. This means that a unit with a sigmoid transfer function is in principle a DRU 2 . Most useful continuous transfer functions (radial-basis functions, for example), exhibit the same property, reducing to only a small number of distinct output values when given the same input repeatedly. The results shown here are therefore not merely theoretical, but are of real practical significance and apply to any network whose recurrent links are restricted to self connections. 5 Concl usion No RCC network can model any FSA containing an RF k+1 (such as that shown in Figure 3), given units limited to generating k possible output values, regardless RCC Cannot Compute Certain FSA, Even with Arbitrary Transfer Functions 625 of the sophistication of the transfer function that generates these values. This places an upper bound on the computational capabilities of an RCC network. Less sophisticated transfer functions, such as the sigmoid units investigated by Giles et al. and Kremer may have even greater limitations. Figure 2, for example, could be modeled by a single sufficiently sophisticated DRU 2 , but cannot be modeled by an RCe network composed of hard-threshold or sigmoidal units (Giles et al., 1995; Kremer, 1996) because these units cannot exploit all mappings from inputs to outputs. By not assuming arbitrary transfer functions, previous proofs could not isolate the network's structure as the source of RCC's limitations. References Bachrach, J. R. (1988). Learning to represent state. Master's thesis, Department of Computer and Information Sciences, University of Massachusetts, Amherst, MA 01003. Fahlman, S. E. (1991) . The recurrent cascade-correlation architecture. In Lippmann, R. P., Moody, J. E., and Touretzky, D. S., editors, Advances in Neural Information Processing Systems 3, pages 190-196, San Mateo, California. Morgan Kaufmann Publishers. Giles, C., Chen, D., Sun, G., Chen, H., Lee, Y., and Goudreau, M. (1995). Constructive learning of recurrent neural networks: Problems with recurrent cascade correlation and a simple solution. IEEE Transactions on Neural Networks, 6(4):829. Kremer, S. C. (1996). Finite state automata that recurrent cascade-correlation cannot represent. In Touretzky, D. S., Mozer, M. C., and Hasselno, M. E., editors, Advances in Neural Information Processing Systems 8, pages 679-686. MIT Press. In Press. Mozer, M. C. (1988). A focused back-propagation algorithm for temporal pattern recognition. Technical Report CRG-TR-88-3, Department of Psychology, University of Toronto. 

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Enhancing Q-Learning for Optimal Asset Allocation Ralph Neuneier Siemens AG, Corporate Technology D-81730 MUnchen, Germany Ralph.Neuneier@mchp.siemens.de Abstract This paper enhances the Q-Iearning algorithm for optimal asset allocation proposed in (Neuneier, 1996 [6]). The new formulation simplifies the approach by using only one value-function for many assets and allows model-free policy-iteration. After testing the new algorithm on real data, the possibility of risk management within the framework of Markov decision problems is analyzed. The proposed methods allows the construction of a multi-period portfolio management system which takes into account transaction costs, the risk preferences of the investor, and several constraints on the allocation. 1 Introduction Asset allocation and portfolio management deal with the distribution of capital to various investment opportunities like stocks, bonds, foreign exchanges and others. The aim is to construct a portfolio with a maximal expected return for a given risk level and time horizon while simultaneously obeying institutional or legally required constraints. To find such an optimal portfolio the investor has to solve a difficult optimization problem consisting of two phases [4]. First, the expected yields together with a certainty measure has to be predicted. Second, based on these estimates, mean-variance techniques are typically applied to find an appropriate fund allocation. The problem is further complicated if the investor wants to revise herlhis decision at every time step and if transaction costs for changing the allocations must be considered. disturbanc ies - ,--- financial market I--- j return investmen ts '---- investor rates, prices I--- Markov Decision Problem: Xt = ($t, J(t}' state: market $t and portfolio J(t policy p, actions at = p(xt) transition probabilities p(Xt+ll x d r( Xt, at, $t+l) return function Within the framework of Markov Decision Problems, MDPs, the modeling phase and the search for an optimal portfolio can be combined (fig. above). Furthermore, transaction costs, constraints, and decision revision are naturally integrated. The theory ofMDPs formalizes control problems within stochastic environments [1]. If the discrete state space is small and if an accurate model of the system is available, MDPs can be solved by con- 937 Enhancing Q-Leaming for Optimal Asset Allocation ventional Dynamic Programming, DP. On the other extreme, reinforcement learning methods using function approximator and stochastic approximation for computing the relevant expectation values can be applied to problems with large (continuous) state spaces and without an appropriate model available [2, 10]. In [6], asset allocation is fonnalized as a MDP under the following assumptions which clarify the relationship between MDP and portfolio optimization: 1. The investor may trade at each time step for an infinite time horizon. 2. The investor is not able to influence the market by her/his trading. 3. There are only two possible assets for investing the capital. 4. The investor has no risk aversion and always invests the total amount. The reinforcement algorithm Q-Learning, QL, has been tested on the task to invest liquid capital in the Gennan stock market DAX, using neural networks as value function approximators for the Q-values Q(x, a). The resulting allocation strategy generated more profit than a heuristic benchmark policy [6]. Here, a new fonnulation of the QL algorithm is proposed which allows to relax the third assumption. Furthennore, in section 3 the possibility of risk control within the MDP framework is analyzed which relaxes assumption four. 2 Q-Learning with uncontrollable state elements This section explains how the QL algorithm can be simplified by the introduction of an artificial detenninistic transition step. Using real data, the successful application of the new algorithm is demonstrated. 2.1 Q-Leaming for asset allocation = The situation of an investor is fonnalized at time step t by the state vector Xt ($t, Kt), which consists of elements $t describing the financial market (e. g. interest rates, stock indices), and of elements K t describing the investor's current allocation of the capital (e. g. how much capital is invested in which asset). The investor's decision at for a new allocation and the dynamics on the financial market let the state switch to Xt+l = ($t+l' K t+1 ) according to the transition probability p(Xt+lIXto at). Each transition results in an immer(xt, Xt+l. at} which incorporates possible transaction costs depending diate return rt on the decision at and the change of the value of K t due to the new values of the assets at time t + 1. The aim is to maximize the expected discounted sum of the returns, V* (x) = E(2::~o It rt Ixo = x). by following an optimal stationary policy J.l. (xt) = at. For a discrete finite state space the solution can be stated as the recursive Bellman equation: = V? (xd = m:-x [L Xt+l p(xt+llxt, a)rt + ~I L p(xt+llxt. a) V* (Xt+l)] . (1) X.+l A more useful fonnulationdefines a Q-function Q?(x, a) of state-action pairs (Xt. at), to allow the application ofan iterative stochastic approximation scheme, called Q-Learning [11]. The Q-value Q*(xt,a,) quantifies the expected discounted sum of returns if one executes action at in state Xt and follows an optimal policy thereafter, i. e. V* (xt) = max a Q* (Xt, a). Observing the tuple (Xt, Xt+l, at, rd, the tabulated Q-values are updated 938 R. Neuneier in the k + 1 iteration step with learning rate 17k according to: It can be shown, that the sequence of Qk converges under certain assumptions to Q* . If the Q-values Q* (x, a) are approximated by separate neural networks with weight veCtor w a for different actions a, Q* (x, a) ~ Q(x; w a ) , the adaptations (called NN-QL) are based on the temporal differences d t : dt := r(xt, at , Xt+l) + ,),maxQ(Xt+l; w~) aEA Q(Xt; wZ t ) , Note, that although the market dependent part $t of the state vector is independent of the investor's decisions, the future wealth Kt+l and the returns rt are not. Therefore, asset allocation is a multi-stage decision problem and may not be reduced to pure prediction if transaction costs must be considered. On the other hand, the attractive feature that the decisions do not influence the market allows to approximate the Q-values using historical data of the financial market. We need not to invest real money during the training phase. 2.2 Introduction of an artificial deterministic transition Now, the Q-values are reformulated in order to make them independent of the actions chosen at the time step t. Due to assumption 2, which states that the investor can not influence the market by the trading decisions, the stochastic process of the dynamics of $t is an uncontrollable Markov chain. This allows the introduction of a deterministic intermediate step between the transition from Xt to Xt+1 (see fig. below). After the investor has "hosen an action at, the capital K t changes to K: because he/she may have paid transaction c(Kt, at) and K; reflects the new allocation whereas the state of the market, costs Ct $t, remains the same. Because the costs Ct are known in advance, this transition is deterministic and controllable. Then, the market switches stochastically to $t+1 and generates r' ($t, K:, $t+1) i.e., rt Ct + r~ . The capital changes to the immediate return r~ Kt+1 r~ + K; . This transition is uncontrollable by the investor. V* ($, K) V* (x) is now computed using the costs Ct and returns r~ (compare also eq. 1) = = = = 110<'_ ...torml.lode tn.sid.. , St Kt at t+l St St+l K't Kt+l Ct Q(SbK~) Defining Q* ($t, = r: Kn as the Q-values of the intermediate time step Q* ($t , K:) E [r' ($t , K: , $t+1) + ')'V* ($t+1 ' Kt+d] Enhancing Q-Leaming for Optimal Asset Allocation 939 gives rise to the optimal value function and policy (time indices are suppressed), V* ($, K) = max[c(K, a) + Q* ($, K')], a Jl*($, K) = argmax[c(K, a) a + Q*($, K')]. Defining the temporal differences d t for the approximation Qk as dt := r' ($t, K:, $t+1) + ,max[c(Kt+b a) + Q(k)($t+1, K:+ 1 )] a Q(k)($t, KD leads to the update equations for the Q-values represented by tables or networks: QLU: NN-QLU: Q(k+l)($t,K;) w(k+l) Q(k)($t, K:) + 1/kdt , w(k) + 1/kdtV'Q($, K'; w(k?) . The simplification is now obvious, because (NN-)QLU only needs one table or neural network no matter how many assets are concerned. This may lead to a faster convergence and better results. The training algorithm boils down to the iteration of the following steps: QLU for optimal investment decisions 1. draw randomly patterns $t, $t+ 1 from the data set, draw randomly an asset allocation K: 2. for all possible actions a: compute rf, c(Kt+b a), Q(k)($t+b K:+I) 3. compute temporal difference dt 4. compute new value Q(k+1)($t, Kn resp. Q($t, K:; w(k+1?) 5. stop, ifQ-values have converged, otherwise go to 1 Since QLU is equivalent to Q-Leaming, QLU converges to the optimal Q-values under the same conditions as QL (e. g [2]). The main advantage of (NN- )QLU is that this algorithm only needs one value function no matter how many assets are concerned and how fine the grid of actions are: Q*(($,K),a) = c(K,a) + Q*($,K'). Interestingly, the convergence to an optimal policy of QLU does not rely on an explicit in step 1 simulates a random exploration strategy because the randomly chosen capital action which was responsible for the transition from K t . In combination with the randomly chosen market state $t, a sufficient exploration of the action and state space is guaranteed. K: 2.3 M\ldel-free policy-iteration The refonnulation also allows the design of a policy iteration algorithm by alternating a policy evaluation phase (PE) and a policy improvement (PI) step. Defining the temporal differences dt for the approximation Q~I of the policy JlI in the k step ofPE dt := r' ($t, K;, $t+d + ,[c(Kt+I, JlI ($t+l, K t+1 )) + Q(k) (K:+ 1 , $t+d] - leads to the following update equation for tabulated Q-values (k+l)($ t, K') Q JJI t = Q(k)($ IJ.I t, K") + 1/k d t? t Q(k)(K;, $t} 940 R. Neuneier After convergence, one can improve the policy J-li to J-lI+l by J-l1+I($t, Kt} , a) + QJ.'I ($t, KD] . = arg max[c(Kt a By alternating the two steps PE and PI, the sequence of policies [J-l1 (x )]1=0,... converges under the typical assumptions to the optimal policy J-l* (x) [2] . Note, that policy iteration is normally not possible using classical QL, if one has not an appropriate model at hand. The introduction of the detenninistic intermediate step allows to start with an initial strategy (e. g. given by a broker), which can be subsequently optimized by model-free policy iteration trained with historical data of the financial market. Generalization to parameterized value functions is straightforward. 2.4 Experiments on the German Stock Index DAX The NN-QLU algorithm is now tested on a real world task: assume that an investor wishes to invest herihis capital into a portfolio of stocks which behaves like the German stock index DAX. Herihis alternative is to keep the capital in the certain asset cash, referred to as DM. We compare the resulting strategy with three benchmarks, namely Neuro-Fuzzy, Buy&Hold and the naive prediction. The Buy&Hold strategy invests at the first time step in the DAX and only sells at the end. The naive prediction invests if the past return of the DAX has been positive and v. v. The third is based on a Neuro-Fuzzy model which was optimized to predict the daily changes of the DAX [8]. The heuristic benchmark strategy is then constructed by taking the sign of the prediction as a trading signal, such that a positive prediction leads to an investment in stocks. The input vector of the Neuro-Fuzzy model, which consists of the DAX itself and 11 other influencing market variables, was carefully optimized for optimal prediction. These inputs also constitutes the $t part of the state vector which describes the market within the NN-QLU algorithm. The data is split into a training (from 2. Jan. 1986 to 31. Dec. 1994) and a test set (from 2. Jan. 1993 to 1. Aug. 1996). The transaction costs (Ct) are 0.2% of the invested capital if K t is changed from DM to DAX, which are realistic for financial institutions. Referring to an epoch as one loop over all training patterns, the training proceeds as outlined in the previous section for 10000 epochs with T}k = "'0 . 0.999 k with start value "'0 = 0.05. Table 1: Comparison of the profitability of the strategies, the number of position changes and investments in DAX for the test (training) data. I strategy NN-QLU N euro-Fuzzy Naive Prediction Buy&Hold profit 1.60 (3.74) 1.35 (1.98) 0.80 (1.06) 1.21 (1.46) I investments in DAX I position changes I 70 (73)% 53 (53)% 51 (51)% 100 (100)% 30 (29)% 50 (52)% 51 (48)% 0(0)% The strategy constructed with the NN-QLU algorithm, using a neural network with 8 hidden neurons and a linear output, clearly beats the benchmarks. The capital at the end of the test set (training set) exceeds the second best strategy Neuro-Fuzzy by about 18.5% (89%) (fig. 1). One reason for this success is, that QLU changes less often the position and thus, avoids expensive transaction costs. The Neuro-Fuzzy policy changes almost every second day whereas NN-QLU changes only every third day (see tab. 1). It is interesting to analyze the learning behavior during training by evaluating the strategies ofNN-QLU after each epoch. At the beginning, the policies suggest to change almost never or each time to invest in DAX. After some thousand epochs, these bang-bang strategies starts to differentiate. Simultaneously, the more complex the strategies become the more profit they generate (fig. 2). 941 Enhancing Q-Leaming for Optimal Asset Allocation the Capital de~lopment 01 3.5 NN-QLU 2.5 09 . o8 '. ' NaIVe PredlCllon 1 3.94 18.96 time lime Figure 1: Comparison of the development of the capital for the test set (left) and the training set (right). The NN-QLU strategy clearly beats all the benchmarks. DAX-mvestrnents In ". o 8000 r8ILm CNGf 60 days i 10000 2000 opoehs 4000 6000 epochs 8000 10000 Figure 2: Training course: percentage ofDAX investments (left), profitability measured as the average return over 60 days on the training set (right). 3 Controlling the Variance of the Investment Strategies 3.1 Risk-adjusted MDPs People are not only interested in maximizing the return, but also in controlling the risk of their investments. This has been formalized in the Markowitz portfolio-selection, which aims for an allocation with the maximal expected return for a given risk level [4]. Given a stationary fo1icy f..L( x) with finite state space, the associated value function V JI. (x) and its variance (T (V JI. ( X )) can be defined as V"(x) E [t. ~'r(x"I", E [ (t. ~'r(x" x'+1) xo ~ xl, p" X'+1) - V"(X)), Xo = x] . Then, an optimal strategy f..L* (x ; ,\) for a risk-adjusted MDP (see [9], S. 410 for variancepenalized MDPs) is f..L*(x;,\) = argmax[VJI.(x) JI. ,\(T2(VJI.(x))] for'\ > O. By variation of '\, one can construct so-called efficient portfolios which have minimal risk for each achievable level of expected return. But in comparison to classical portfolio theory, this approach manages multi-period portfolio management systems including transaction costs. Furthermore, typical min-max requirements on the trading volume and other allocation constraints can be easily implemented by constraining the action space. 942 3.2 R. Neuneier Non-linear Utility Functions In general, it is not possible to compute (J"2 (V If. (x)) with (approximate) dynamic programming or reinforcement techniques, because (J"2 (VJ.I (x)) can not be written in a recursive Bellman equation. One solution to this problem is the use of a return function rt, which penalizes high variance. In financial analysis, the Sharpe-ratio, which relates the mean of the single returns to their variance i. e., r/(J"(r), is often employed to describe the smoothness of an equity curve. For example, Moody has developed a Sharpe-ratio based error function and combines it with a recursive training procedure [5] (see also [3]). The limitation of the Sharpe-ratio is, that it penalizes also upside volatility. For this reason, the use of an utility function with a negative second derivative, typical for risk averse investors, seems to be more promising. For such return functions an additional unit increase is less valuable than the last unit increase [4]. An example is r = log (new portfolio value I old portfolio value) which also penalizes losses much stronger than gains. The Q-function Q(x, a) may lead to intermediate values of a* as shown in the figure below. e"I ---'--'--_~~~_ .0 I ~"7Jr ~J .. '" , 1 ' ''~ ,_l 1II?'i I --~ .1 O. ~ - ~.- -~-~ - " 01 \ .J ,. " rtMaM change 01 the pcwtIoko I4l1A ... % ---'-- 't " .- ,,----.-:;----0; --;. - :i - .?-y:- - ? ? -~ % of l'N'8Sur'8n11n UncertlWl asset 4 Conclusion and Future Work Two improvements of Q-Ieaming have been proposed to bridge the gap between classical portfolio management and asset allocation with adaptive dynamic programming. It is planned to apply these techniques within the framework of a European Community sponsored research project in order to design a decision support system for strategic asset allocation [7). Future work includes approximations and variational methods to compute explicitly the risk (J"2 (V If. (x)) of a policy. References [I J D. P. Bertsekas. Dynamic Programming and Optimal Control, vol. I. Athena Scientific, 1995. [2] D. P. Bertsekas and J. N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996. [3J M. Choey and A. S. Weigend. Nonlinear trading models through Sharpe Ratio maximization. In proc. ofNNCM'96, 1997. World Scientific. [4J E. J. Elton and M. J. Gruber. Modern Portfolio Theory and Investment Analysis. 1995. [5J J. Moody, L. Whu, Y. Liao, and M. Saffell. Performance Functions and Reinforcement Learning for Trading Systems and Portfolios. Journal of Forecasting, 1998. forthcoming, [6J R. Neuneier. Optimal asset allocation using adaptive dynamic programming. In proc. of Advances in Neural Information Processing Systems, vol. 8, 1996. [7J R. Neuneier, H. G. Zimmermann, P. Hierve, and P. Nairn. Advanced Adaptive Asset Allocation. EU Neuro-Demonstrator, 1997, [8J R. Neuneier, H. G. Zimmermann, and S. Siekmann. Advanced Neuro-Fuzzy in Finance: Predicting the German Stock Index DAX, 1996. Invited presentation at ICONIP'96, Hong Kong, availabel by email fromRalph.Neuneier@mchp.siemens.de. [9J M. L. Puterman. Markov Decision Processes. John Wiley & Sons, 1994. [IOJ S. P. Singh. Learning to Solve Markovian Decision Processes, CMPSCI TR 93-77, University of Massachusetts, November 1993. [I I J C. J. C. H. Watkins and P. Dayan. Technical Note: Q-Learning. Machine Learning: Special Issue on Reinforcement Learning, 8,3/4:279-292, May 1992. 

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Modelling Seasonality and Trends in Daily Rainfall Data Peter M Williams School of Cognitive and Computing Sciences University of Sussex Falmer, Brighton BN1 9QH, UK. email: peterw@cogs.susx.ac.uk Abstract This paper presents a new approach to the problem of modelling daily rainfall using neural networks. We first model the conditional distributions of rainfall amounts, in such a way that the model itself determines the order of the process, and the time-dependent shape and scale of the conditional distributions. After integrating over particular weather patterns, we are able to extract seasonal variations and long-term trends. 1 Introduction Analysis of rainfall data is important for many agricultural, ecological and engineering activities. Design of irrigation and drainage systems, for instance, needs to take account not only of mean expected rainfall, but also of rainfall volatility. In agricultural planning, changes in the annual cycle, e.g. advances in the onset of winter rain, are significant in determining the optimum time for planting crops. Estimates of crop yields also depend on the distribution of rainfall during the growing season, as well as on the overall amount. Such problems require the extrapolation of longer term trends as well as the provision of short or medium term forecasts. 2 Occurrence and amount processes Models of daily precipitation commonly distinguish between the occurrence process, i.e. whether or not it rains, and the amount process, i.e. how much it rains, if it does. The occurrence process is often modelled as a two-state Markov chain of first or higher order. In discussion of [12], Katz traces this approach back to Quetelet in 1852. A first order chain has been considered adequate for some weather stations, but second or higher order models may be required for others, or at different times of year. Non-stationary Markov chains have been used by a number of investigators, and several approaches have been taken P. M Williams 986 to the problem of seasonal variation, e.g. using Fourier series to model daily variation of parameters [16, 12, 15]. The amount of rain X on a given day, assuming it rains, normally has a roughly exponential distribution. Smaller amounts of rain are generally more likely than larger amounts. Several models have been used for the amount process. Katz & Parlange [9], for example, assume that \IX has a normal distribution, where n is a positive integer empirically chosen to minimise the skewness of the resulting historical distribution. But use has more commonly been made of a gamma distribution [7,8, 12] or a mixture of two exponentials [16, 15]. 3 Stochastic model The present approach is to deal with the occurrence and amount processes jointly, by assuming that the distribution of the amount of rain on a given day is a mixture of a discrete and continuous component. The discrete component relates to rainfall occurrence and the continuous component relates to rainfall amount on rainy days. We use a gamma distribution for the continuous component. l This has density proportional to x v - 1 e- x to within an adjustable scaling of the x-axis. The shape parameter v > 0 controls the ratio of standard deviation to mean. It also determines the location of the mode, which is strictly positive if v > 1. For certain patterns of past precipitation, larger amounts may be more likely on the following day than smaller amounts. Specifically the distribution of the amount X of rain on a given day is modelled by the three parameter family if x < 0 (1) if x ~ 0 where 0 ~ a ~ 1 and v,O > 0 and r(v,z) = r(V)-l 1 00 yv-l e- y dy is the incomplete gamma function. For a < 1, there is a discontinuity at x = 0 corresponding to the discrete component. Putting x = 0, it is seen that a = P(X > 0) is the probability of rain on the day in question. The mean daily rainfall amount is avO and the variance is aV{l + v(l - a)}02. 4 Modelling time dependency The parameters a, v, 0 determining the conditional distribution for a given day, are understood to depend on the preceding pattern of precipitation, the time of year etc. To model this dependency we use a neural network with inputs corresponding to the conditioning events, and three outputs corresponding to the distributional parameters. 2 Referring to the activations of the three output units as zO:, ZV and zO, we relate these to the distributional parameters by 1 a=---1 + expzO: v = expzv 0= expzo (2) in order to ensure an unconstrained parametrization with 0 < a < 1 and v,O > 0 for any real values of zO:, zV, zO. 1It would be straightforward to use a mixture of gammas, or exponentials, with time-dependent mixture components. A single gamma was chosen for simplicity to illustrate the approach. 2 A similar approach to modelling conditional distributions, by having the network output distributional parameters, is used, for example, by Ghabramani & Jordan [6], Nix & Weigend [10], Bishop & Legleye [3], Williams [14], Baldi & Chauvin [2]. Modelling Seasonality and Trends in Daily Rainfall Data 987 On the input side, we first need to make additional assumptions about the statistical properties of the process. Specifically it is assumed that the present is stochastically independent of the distant past in the sense that (t > T) (3) for a sufficiently large number of days T. In fact the stronger assumption will be made that P(Xt>X!Xt-1,,,,,XO) = P(Xt>X!Rt-1, ,, ,,Rt-T) (t>T) (4) where R t = (X t > 0) is the event of rain on day t. This assumes that today's rainfall amount depends stochastically only on the occurrence or non-occurrence of rain in the recent past, and not on the actual amounts. Such a simplification is in line with previous approaches [8, 16, 12J. For the present study T was taken to be 10. To assist in modelling seasonal variations, cyclic variables sin T and cos T were also provided as inputs, where T = 21rt/ D and D = 365.2422 is the length of the tropical year. This corresponds to using Fourier series to model seasonality [16, 12J but with the number of harmonics adaptively determined by the model. 3 To allow for non-periodic nonstationarity, the current value of t was also provided as input. 5 Model fitting Suppose we are given a sequence of daily rainfall data of length N. Equation (4) implies that the likelihood of the full data sequence (x N -1 Xo) factorises as I ??? I N-1 p(XN-1 , .. . I Xo; w) = p(XT-1I" . I Xo) II p(Xt !Tt-1 I' .. I Tt-T; w) (5) t=T where the likelihood p(XT-1I'" IXO) of the initial sequence is not modelled and can be considered as a constant (compare [14]). Our interest is in the likelihood (5) of the actual sequence of observations, which is understood to depend on the variable weights w of the neural network. Note that p(Xt !Tt-1 Tt-T; w) is computed by means of the neural network outputs zf I zf I zf, using weights wand the inputs corresponding to time t. I ' ?? I The log likelihood of the data can therefore be written, to within a constant, as N-1 logp(xN-1 I' .. IXO; w) = L logp(xt !Tt-1,? .. I Tt-T; w) t=T or, more simply, N-1 L(w) =L (6) Lt(w) t=T where from (1) L () {log(1 - at) t w = log at + (lit -1) logxt -lit logOt -logr(lIt) - xt/Ot if Xt = 0 if Xt > 0 (7) where dependence of at, lit Ot on w, and also on the data, is implicit. I To fit the model, it is useful to know the gradient 'VL(w). This can be computed using backpropagation if we know the partial derivatives of L(w) with respect to network outputs. In view of (6) we can concentrate on a single observation and perform a summation. 3Note that both sin nr and cos nr can be expressed as non-linear functions of sin r and cos r. which can be approximated by the network. P. M. Williams 988 Omitting subscript references to t for simplicity, and recalling the links between network outputs and distributional parameters given by (2), we have 8L 8z Q 8L 8z v 8L 8zo -a {1-0 = { { if x = 0 if x > 0 0 x v'I/J (v) - v log (j 0 x v-() if x = 0 ifx> 0 (8) if x = 0 if x > 0 where d r'(v) 'I/J(v) = -logr(v) = - dv r(v) is the digamma function of v. Efficient algorithms for computing log r(v) in (7) and 'I/J(v) in (8) can be found in Press et al. [11] and Amos [1]. 6 Regularization Since neural networks are universal approximators, some form of regularization is needed. As in all statistical modelling, it is important to strike the right balance between jumping to conclusions (overfitting) and refusing to learn from experience (underfitting). For this purpose, each model was fitted using the techniques of [13] which automatically adapt the complexity of the model to the information content of the data, though other comparable techniques might be used. The natural interpretation of the regularizer is as a Bayesian prior. The Bayesian analysis is completed by integration over weight space. In the present case, this was achieved by fitting several models and taking a suitable mixture as the solution. On account of the large datasets used, however, the results are not particularly sensitive to this aspect of the modelling process. 7 Results for conditional distributions The process was applied to daily rainfall data from 5 stations in south east England and 5 stations in central Italy.4 The data covered approximately 40 years providing some 15,000 observations for each station. A simple fully connected network was used with a single layer of 13 input units, 20 hidden units and 3 output units corresponding to the 3 parameters of the conditional distribution shown in (2). As a consequence of the pruning features of the regularizer, the models described here used an average of roughly 65 of the 343 available parameters. To illustrate the general nature of the results, Figure 1 shows an example from the analysis of an early part of the Falmer series. It is worth observing the succession of 16 rainy days from day 39 to day 54. The lefthand figure shows that the conditional probability of rain increases rapidly at first, and then levels out after about 5-7 days.s Similar behaviour is observed for successive dry days, for example between days 13 and 23. This suggests that the choice of 10 time lags was sufficient. Previous studies have used mainly first or second order Markov chains [16, 12]. Figure 1 confirms that conditional dependence 4The English stations were at Cromptons, FaImer, Kemsing, Petworth, Rothertield; the Italian stations were at Monte Oliveto, Pisticci, Pomarico, Siena, Taverno d' Arbia. sIn view of the number of lags used as inputs, the conditional probability would necessarily be constant after 10 days apart from seasonal effects. In fact this is the last quarter of 1951 and the incidence of rain is increasing here at that time of year. Modelling Seasonality and Trends in Daily Rainfall Data 989 FALMER: conditional probability ii II 0.8 II II I 0.6 i I 0.4 FALMER: conditional mean 20 I II 15 II 10 5 0.2 I II 0 0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 Figure 1: Results for the 10 weeks from 18 September to 27 November, 1951. The lefthand figure shows the conditional probability of rain for each day, with days on which rain occurred indicated by vertical lines. The righthand figure shows the conditional expected amount of rain in millimeters for the same period, together with the actual amount recorded. decays rapidly at this station, at this time of year, but also indicates that it can persist for up to at least 5 days (compare [5,4]). 8 Seasonality and trends Conditional probabilities and expectations displayed in Figure 1 show considerable noise since they are realisations of random variables depending on the rainfall pattern for the last 10 days. For the purpose of analysing seasonal effects and longer term trends, it is more indicative to integrate out the noise resulting from individual weather patterns as follows. Let R t denote the event (X t > 0) and let R t denote the complementary event (X t = 0). The expected value of X t can then be expressed as E(Xt ) =L E(X t I A t - 1, . .. ,At-T) P(At- 1, . .. ,At-T) (9) where each event At stands for either R t or R t , and summation is over the 2T possible combinations. Equation (9) takes the full modelled jOint distribution over the variables X N -1, .. . ,X0 and extracts the marginal distribution for X t . This should be distinguished from an unconditional distribution which might be estimated by pooling the data over all 40 years. E(Xt ) relates to a specific day t. Note that (9) also holds if X t is replaced by any integrable function of X t , in particular by the indicator function of the event (X t > 0) in which case (9) expresses the probability of rain on that day. Examining (9) we see that the conditional expectations in the first term on the right are known from the model, which supplies a conditional distribution not only for the sequence of events which actually occurred, but for any possible sequence over the previous T days. It therefore only remains to calculate the probabilities P( A t - 1, ... , At-T) of T -day sequences preceding a given day t. Note that these are again time-dependent marginal probabilities, which can be calculated recursively from P(At , .. . , At-T+t} = P(At I At- 1,? . . , A t -T+1 R t-T) P(At- 1 , . .? , At-T+1 R t-T) + P(At I At-I,?? ., At-T+IRt-T) P(At- 1, . .. , At-T+IRt-T) provided we assume a prior distribution over the 2T initial sequences (AT-I, . .. , Ao) as a base for the recursion. The conditional probabilities on the right are given by the model, 990 P. M. Williams POMARICO: mean and standard deviation POMARICO: probability of rain 0.35 0.3 .----r---..---.---~-___r-__, HHHI"*-lHI-IHHHH1?-lH?lI-H~-f+H 0.25 M-I++H++I-tI+i1~++HHI#H-fHHHHI-{HI-1HHHHI?H-II-fHli 0.2 I\I*H*I*+H1flHl-+l!#~iHHIl-Iffi~+++H,HIf~KH+IHIH?HH1ffHII 2~~~~~~~~~~~~~-~ 0.15 H+1-HrH-l1-f , I 0.1 1955 ! 1 _ __'__ 1970 1975 L--_-'--_-'--_~ 1960 1965 _'__~ 1980 1985 OL----'----'---~---'---'---~ 1955 1960 1965 1970 1975 1980 1985 Figure 2: Integrated results for Pomarico from 1955-1985. The lefthand figure shows the daily probability of rain, indicating seasonal variation from a summer minimum to a winter maximum. The righthand figure shows the daily mean (above) and standard deviation (below) of rainfall amount in millimeters. as before, and the unconditional probabilities are given by the recursion. It turns out that results are insensitive to the choice of initial distribution after about 50 iterations, verifying that the occurrence process, as modelled here, is in fact ergodic. 9 Integrated results Results for the integrated distribution at one of the Italian stations are shown in Figure 2. By integrating out the random shocks we are left with a smooth representation of time dependency alone. The annual cycles are clear. Trends are also evident over the 30 year period. The mean rainfall amount is decreasing significantly, although the probability of rain on a given day of the year remains much the same. Rain is occurring no less frequently, but it is occurring in smaller amounts. Note also that the winter rainfall (the upper envelope of the mean) is decreasing more rapidly than the summer rainfall (the lower envelope of the mean) so that the difference between the two is narrowing. 10 Conclusions This paper provides a new example of time series modelling using neural networks. The use of a mixture of a discrete distribution and a gamma distribution emphasises the general principle that the "error function" for a neural network depends on the particular statistical model used for the target data. The use of cyclic variables sin T and cos T as inputs shows how the problem of selecting the number of harmonics required for a Fourier series analysis of seasonality can be solved adaptively. Long term trends can also be modelled by the use of a linear time variable, although both this and the last feature require the presence of a suitable regularizer to avoid overfitting. Lastly we have seen how a suitable form of integration can be used to extract the underlying cycles and trends from noisy data. These techniques can be adapted to the analysis of time series drawn from other domains. Modelling Seasonality and Trends in Daily Rainfall Data 991 Acknowledgement I am indebted to Professor Helen Rendell of the School of Chemistry, Physics and Environmental Sciences, University of Sussex, for kindly supplying the rainfall data and for valuable discussions. References [1] D. E. Amos. A portable fortran subroutine for derivatives of the psi function. ACM Transactions on Mathematical Software, 9:49~502, 1983. [2] P. Baldi and Y. Chauvin. Hybrid modeling, HMM/NN architectures, and protein applications. Neural Computation, 8:1541-1565, 1996. [3] C. M. Bishop and C. Legleye. Estimating conditional probability densities for periodic variables. In G. Tesauro, D. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Systems 7, pages 641-648. The MIT Press, 1995. [4] E. H. Chin. Modelling daily precipitation occurrence process with Markov chain. Water Resources Research, 13:949-956,1977. [5] P. Gates and H. Tong. On Markov chain modelling to some weather data. Journal of AppliedMeteorology, 15:1145-1151, 1976. [6] Z. Ghahramani and M. 1. Jordan. Supervised learning from incomplete data via an EM approach. In Jack D. Cowan, Gerald Tesauro, and Joshua Alspector, editors, Advances in Neural Information Processing Systems 6, pages 120-127. Morgan Kaufmann, 1994. [7] N. T. Ison, A. M. Feyerherm, andL. D. Bark. Wet period precipitation and the gamma distribution. Journal of Applied Meteorology, 10:658-665, 1971. [8] R. W. Katz. Precipitation as a chain-dependent process. Journal of Applied Meteorology, 16:671-676,1977. [9] R. W. Katz and M. B. Parlange. Effects of an index of atmospheric circulation on stochastic properties of precipitation. Water Resources Research, 29:2335-2344, 1993. [10] D. A. Nix and A. S. Weigend. Learning local error bars for nonlinear regression. In Gerald Tesauro, David S. Touretzky, and Todd K. Leen, editors, Advances in Neural Information Processing Systems 7, pages 489-496. MIT Press, 1995. [11] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C. Cambridge University Press, 2nd edition, 1992. [12] R. D. Stern and R. Coe. A model fitting analysis of daily rainfall data, with discussion. Journal of the Royal Statistical Society A, 147(Part 1):1-34,1984. [13] P. M. Williams. Bayesian regularization and pruning using a Laplace prior. Neural Computation, 7:117-143,1995. [14] P. M. Williams. Using neural networks to model conditional multivariate densities. Neural Computation, 8:843-854, 1996. [15] D. A. Woolhiser. Modelling daily precipitation-progress and problems. In Andrew T. Walden and Peter Guttorp, editors, Statistics in the Environmental and Earth Sciences, chapter 5, pages 71-89. Edward Arnold, 1992. [16] D. A. Woolhiser and G. G. S. Pegram. Maximum likelihood estimation of Fourier coefficients to describe seasonal variation of parameters in stochastic daily precipitation models. Journal of Applied Meteorology, 18:34-42, 1979. 

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240 TEMPORAL REPRESENTATIONS IN A CONNECTIONIST SPEECH SYSTEM Erich J. Smythe 207 Greenmanville Ave, #6 Mystic, CT 06355 ABSTRACT SYREN is a connectionist model that uses temporal information in a speech signal for syllable recognition. It classifies the rates and directions of formant center transitions, and uses an adaptive method to associate transition events with each syllable. The system uses explicit spatial temporal representations through delay lines. SYREN uses implicit parametric temporal representations in formant transition classification through node activation onset, decay, and transition delays in sub-networks analogous to visual motion detector cells. SYREN recognizes 79% of six repetitions of 24 consonant-vowel syllables when tested on unseen data, and recognizes 100% of its training syllables. INTRODUCTION Living organisms exist in a dynamic environment. Problem solving systems, both natural and synthetic, must relate and interpret events that occur over time. Although connectionist models are based on metaphors from the brain, few have been designed to capture temporal and sequential information common to even the most primitive nervous systems. Yet some of the most popular areas of application of these models, including speech recognition, vision, and motor control, require some form of temporal processing. The variation in a speech signal contains considerable information. Changes in format location or other acoustic parameters (Delattre, et al., 1955; Pols and Schouten, 1982) can determine the identity of constituents of speech, even when segmentation information is obscure. Speech recognition systems have shown good results when they incorporate some temporal information (Waible, et al., 1988, Anderson, et al., 1988). Successful speech systems must incorporate temporal processing. Natural organisms have sensory organs that are continuously updated and can do only limited buffering of input stimuli. Synthetic implementations can buffer their input, transforming time into space. Often the size and complexity of the input representations place limits on the amount of input that can be buffered, especially when data is coming from hundreds or thousands of sensors, and other methods must be found to integrate temporal information. Temporal Representations in a Connectionist Speech System This paper describes SYREN (SYllable REcognition Network), a connectionist network that incorporates various temporal representations for consonant-vowel (CV) syllable recognition by the classification of formant center transitions. Input is presented sequentially, one time slice at a time. The network is described, including the temporal processing used in formant transition classification, learning, and syllable recognition. The results of syllable recognition experiments are discussed in the final section. TEMPORAL REPRESENTATIONS Various types of temporal representations may be used to incorporate time in connectionist models. They range from explicit spatial representations where time is convened into space, to implicit parametric representations where time is incorporated using network computational parameters. Spatiotemporal representations are a middle ground combining the two extremes. The categories represent a continuum rather than absolute distinctions. Several of these types are found in SYREN. EXPLICIT SPATIAL REPRESENTATIONS In a purely spatial representation temporal information is preserved by spreading time steps over space through the network topology. These representations include input buffers, delay lines, and recurrent networks. Fixed input buffers allow interaction between time slices of input. Parts of the network are copied to represent states at panicular time slices. Other methods use sliding input buffers in the form of a queue. Tapped delay lines and delay filters are means of spreading network node activations over time. Composed of chains of network nodes or delay functions, they can preserve the sequential structure of information. A value on a connection from a delay line represents events that have occurred in the past. Delay lines and filters have been used in speech recognition systems by Waible, et al. (1988), and Tank and Hopfield (1987) . Recurrent networks are similar to delay lines in that information is preserved by propagating activation through the network. They can store information indefinitely or generate potentially infinite sequences of behaviors through feedback cycles, whereas delay lines without cycles are limited by their fixed length. Recurrent networks pose problems for learning, although researchers are working on recurrent back propagation networks (Jordan, 1988). Spatial representations are good for explicitly preserving sequences of events, and can simplify the learning of temporal patterns. Resource constraints place a limit on the size of fixed length buffers and delay lines, however. Input data from thousands of sensors place limits on the length of time represented in the buffer, and may not be able to retain information long enough to be of use. Fixed input buffers may introduce edge effects. Interaction is lost between the edges of the buffer and data from preceding and succeeding buffers unless the input is properly segmented. Long delay lines may be computationally expensive as well. 241 242 Smythe SPATIOTEMPORAL REPRESENTATIONS Implicit parametric methods represent time in connectionist models by the behavior of network nodes . State information stored in individual nodes allows more complex activation functions and the accumulation of statistical information. This method may be used to regulate the flow of activation in the network, provide a trace of previous activation, and learn from data separated in time. Adjusting the parameters of functions such as the interactive activation equation of McClelland and Rumelhart (1982) can control the strength of input, affecting the rate that activation reaches saturation. This leads to pulse trains used in synchronization. Variations in decay parameters control the duration of an activation trace. State and statistical information is useful in learning. Eligibility traces from classical conditioning models provide decaying memory of past connection activation. Temporally weighted averages may be used for weight computations. Spatiotemporal representations combine implicit parametric representations with explicit spatial representations. These include the regulation of propagation time and pulse trains through parameter adjustment. Gating behavior that controls the flow of activation through a network is another spatiotemporal method. SYREN DESCRIPTION SYREN is a connectionist model that incorporates temporal processing in isolated syllable recognition using formant center transitions. Formant center tracts are presented in 5 ms time slices. Input nodes are updated once per time slice. The network classifies the rates and directions of formant transitions. Transition data are used by an adaptive network to associate transition patterns with syllables. A recognition network uses output of the adaptive network to identify a syllable. Complete details of the system maybe found in Smythe (1988). DATA CORPUS Input data consist of formant centers from five repetitions of twenty-four consonant-vowel syllables (the stop consonants Ib, d, gl paired with the vowels Iii, ey, ih, eh, ae, ah, ou, uu/), and an averaged set of each of the five repetitions from work performed by Kewley Port (1982). Each repetition is presented as a binary matrix with a row representing frequency in 20 Hz units, and a column representing time in 5 ms slices. The matrix is given to the input units one column at a time. A '1' in a cell of a matrix represents a formant center at a particular frequency during a particular time slice. FORMANT TRANSITION CLASSIFICATION In the first stage of processing SYREN determines the rate and direction of formant center transitions. Formant transition detectors are subnetworks designed to respond to transitions of one of six rates in either rising or falling directions, Temporal Representations in a Connectionist Speech System and also to steady-state events. The method used is motivated by a mechanism for visual motion detection in the retina that combines interactions between subunits of a dendritic tree and shunting, veto inhibition (Koch et ai, 1982). Formant motion is analogous to visual motion, and formant transitions are treated as a one dimensional case of visual motion. Preferred Transition I Branch Nodes ' DIsta Proximal Figure 1. Formant transition detector subnetwork and its preferred descending transition type. The vertical axis is frequency (one row for each input unit) and the horizontal axis is time in 5 ms slices. A detector subnetwork for a slow transition is shown in figure 1, along with its preferred transition. Branch nodes are analogous to dendritic subunits, and serve as activation transmission lines. Their activation is computed by the equation: af+l = af(1- 0) + netf(l - aD Where a is the activation of unit i at time t, net is the weighted input, t is an update cycle (there are 7 updates per time slice), and e is a decay constant. Input to a branch node drives the activation to a maximum value, the rate of which is determined by the strength of the input, In the absence of input the activation decays to O. For the preferred direction, input nodes are activated for two time slices (10 ms) in order from top to bottom. An input node causes the activation of the most distal branch node to rise to a maximum value. This in turn causes the next node to activate, slightly delayed with respect to the first, and so on for the rest of the branch. This results in a pulse of activation flowing along the branch with a transmission delay of roughly one time slice (7 update cycles) from the distal to the proximal end. The most proximal branch node also has a connection to the input node. This connection serves to prime the node for slower transitions. Activation from an input node that lasts for only one time slice will decay in the proximal branch node before the activation from the distal region arrives. If input is present for two time steps the extra activation from the input connection primes the node, quickly driving it to a maximal value when the distal activation arrives. 243 244 Smythe An S-node provides the output of the detector. It computes a sigmoid squash function and fires (a sudden increase in activation) when sufficient activation is in the proximal branch nodes. For this particular detector, if the transition is too fast (i. e. one time step for each input unit) the proximal nodes will not attain a high enough activation; if the transition is too slow (i.e. three time steps for each input unit) activation on proximal branch nodes from earlier time steps will have decayed before the transition is complete. This architecture is tuned to a slower transition by increasing the transmission time on the branches by varying the connection weights, and by reducing the decay rate by lowering the decay constant. This illustrates the use of parametric manipulations to control temporal behavior in for rate sensitivity. Veto inhibition is used in this detector for direction selectivity. Veto nodes provide inhibition and are activated by input nodes, and use the interactive activation equation for a decaying memory. Had the transition in figure 1. been in the opposite direction, activation from previous time slices on a veto connection would prevent the input node from activating its distal branch node, preventing the flow of activation and the firing of the S-node. Here a veto connection acts as a gate, serving to select input for processing. Detectors are constructed for faster transitions by shortening the transmission lines and by using veto connections for rate sensitivity. A transition detector for a faster transition is shown in figure 2. Here the receptive field is larger, and veto connections are used to select transitions that skip one input unit at each time slice. Veto connections are still used for direction selectivity. Detectors for even faster transitions are created by widening the receptive field and increasing the number of veto connections for rate sensitivity. Detectors are designed to respond to a specific transition type and not to respond to the transitions of other detectors. They will respond to transitions with rates between their own and the next type of detector. For slower transitions the firing of two detectors indicates an intermediate rate. For faster transitions special detectors are designed to fire for only one precise rate by eliminating some of the branches. Different firing patterns of precise and more general detectors distinguish rates. This gives a very fine rate sensitivity throughout the range of transitions. Detector networks are copied to span the entire frequency range with overlapping receptive fields. This yields an array of S-nodes for each transition type. giving excellent spatial resolution of the frequency range. There are 200 S-nodes for each detector type. each signaling a transition that starts and ends at a particular frequency unit. ADAPTIVE NE1WORK The adaptive network learns to associate patterns of formant transitions with specific syllables. To do this it must be able to store at least part of the unfolding patterns or else it is forced to respond to information from only one time slice. Temporal Representations in a Connectionist Speech System Preferred Transition Veto Nodes Branch Nodes Figure 2. Formant transition detector subnetwork for a faster transition. Only the veto connections used for rate sensitivity are shown. The learning algorithm must also deal with past activation histories of connections or else it can only learn from one time slice. The network accomplishes this through tapped delay lines and decaying eligibility traces. There are twenty-four nodes in the adaptive network, each assigned to one syllable. It is a single layer network, trained using a hybrid supervised learning algorithm that merges Widrow-Hoff type learning with a classical conditioning model (Sutton and Barto, 1987). Storage of temporal patterns Tapped delay lines are used to briefly store sequences of formant transition patterns. S-nodes from each transition detector are connected to a tapped delay line of five nodes. Each delay node simply passes on its S-node's activation value once per 5 ms time slice, allowing the delay matrix to store 25 ms (five time slices) of transition patterns. The delay matrix consists of delay lines for each transition detector at each receptive field. Adaptive nodes are connected to every node in the delay matrix. The delay lines do not perform input buffering; information in the delay matrix has been subject to one level of processing. The amount of information stored (the length of the delay line) is limited by efficiency considerations. Adaptive Algorithm Nodes in the adaptive network compute their activation using a sigmoid squash function and adjust their weights according to the equation: W~"!"l IJ = w~? + a(z~ - s~)e~ IJ I I J where w is the weight from a connection from node j to node i at time t. a is a learning constant. z is the expected value of node i, s is the weighted sum of the connections of node i. and e is the exponentially decaying canonical eligibility of 245 246 Smythe connection j. The eligibility constant gives some variation in the exact timing of transition patterns, allowing limited time warping between training and testing. FINAL RECOGNITION NETWORK The adaptive network is not perfect and results in a number of false alarm errors. Many of these are eliminated by using firing patterns of other adaptive nodes. For example, a node that consistently misfires on one syllable could be blocked by the firing of the correct node for that syllable. Adaptive nodes are connected to a veto recognition network. Since an adaptive node may fire at any time (and at different times) throughout input presentation, delay lines are used to preserve patterns of adaptive node behavior, and veto inhibition is used to block false alarms. Connections in the veto network are enabled or disabled after training. Clearly this is an ad hoc solution, but it suggests the use of representations that are distributed both spatially and temporally. RESULTS AND DISCUSSION In each experiment syllable repetitions were divided into mutually exclusive training and testing sets. A training cycle consisted of one presentation of each member of the training set. In both experiments the networks were trained until adequate performance was achieved, usually after four to ten training cycles. In the first experiment the network was trained on the five raw repetitions and tested on the averaged set. It achieved 92% recognition on the testing set and 100% recognition on the training set. The network had two miss errors on the training set. In the second experiment, the network was trained on four of the raw repetitions and tested on the fifth. Five separate training runs were performed to test the network on each repetition. The network achieved 76% recognition on the testing set for all training runs, and 100% recognition on the training set. In all experiments most of the adaptive nodes responded when there was transition information in the delay matrix. Many responded when both transition and steady-state information was present, using clues from both the consonant and the vowel. This situation occurs only briefly for each formant, since the delay matrix holds information for 5 time slices, and it takes four time slices to signal a steady-state event. Transition information will be at the end of the delay matrix while steady-state is at the beginning. Many nodes were strongly inhibited in the absence of transition information even for their correct syllable, although they had fired earlier in the data presentation. CONCLUSIONS We have shown how different temporal representations and processing methods are used in a connectionist model for syllable recognition. Hybrid connectionist architectures with only slightly more elaborate processing methods can classify acoustic motion and associate sequences of transition events with syllables. The Temporal Representations in a Connectionist Speech System system is not designed as a general speech recognition system, especially since the accurate measurement of formant center frequencies is impractical. Other signal processing techniques, such as spectral peak estimation, can be used without changes in the architecture. This could provide information to a larger speech recognition system . SYREN was influenced by a neurophysiological model for visual motion detection, and shows how knowledge from one processing modality is applied to other problems. The merging of ideas from real nervous systems with existing techniques can add to the connectionist tool kit, resulting in more powerful processing systems. Acknowledgments This research was performed at Indiana University Computer Science Department as part of the author's Ph.D. thesis. The author would like to thank committee members John Barnden and Robert Port for their help and direction, and Donald Lee and Peter Brodeur for their assistance in preparing the manuscript. References Delattre, P. C., Liberman, A. M., Cooper. F. S., 1955, "Acoustic loci and transitional cues for stop consonants," J. Acous. Soc. Am .? 27. 769-773. Jordan, M . I., 1986 "Serial order: A parallel distributed processing approach," ICS Report 8604, UCSD, San Diego. Kewley-Port, D., 1982, "Measurement of formant transitions in naturally produced consonant-vowel syllables," J. Acous. Soc. Am, 72, 379-389. Koch, C., Poggio, T., Torre, V., 1982, "Retinal ganglion cells: A functional interpretation of dendritic morphology," Phil. Trans. R. Soc. Lon.: Series B, 298, 227-264. McClelland, J. L.. Rumelhart, D. E., 1982, "An interactive activation model of context effects in letter perception," Psychological Review, 88, 375-407. Pols, L. C. W., Schouten, M. F. H., 1982, "Perceptual relevance of coarticulation, " in: Carlson, R., and Grandstrom, B., The Representation of Speech in the Peripheral Auditory System, Elsevier, 203-208. Anderson, S., Merrill, J.W.L, Port, R., 1988, "Dynamic speech characterization with recurrent networks," Indiana University Dept. of Computer Science TR. No. 258, Bloomington, In. Smythe, E. J., 1988, "Temporal computation in connectionist models," Indiana University Dept. of Computer Science TR. No. 251, Bloomington, In. Sutton, R. S., Barto, A. G., 1987, "A temporal difference model of classical conditioning," GTE TR87-509.2. Tank, D. W., Hopfield, J. J., 1987, "Concentrating information in time," Proceedings of the IEEE Conference on Neural Networks, San Diego, IV-455-468. Waible, A., Hanazawa, T., Hinton, G., Shikana, K, Lang, K, 1988, "Phoneme recognition: Neural networks vs. Hidden Markov Models," Proc. Int. Conf. Acoustics, Speech, and Signal Processing, 107-110. 247 

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Perturbative M-Sequences for Auditory Systems Identification Mark Kvale and Christoph E. Schreiner? Sloan Center for Theoretical Neurobiology, Box 0444 University of California, San Francisco 513 Parnassus Ave, San Francisco, CA 94143 Abstract In this paper we present a new method for studying auditory systems based on m-sequences. The method allows us to perturbatively study the linear response of the system in the presence of various other stimuli, such as speech or sinusoidal modulations. This allows one to construct linear kernels (receptive fields) at the same time that other stimuli are being presented. Using the method we calculate the modulation transfer function of single units in the inferior colli cui us of the cat at different operating points and discuss nonlinearities in the response. 1 Introduction A popular approach to systems identification, i.e., identifying an accurate analytical model for the system behavior, is to use Volterra or Wiener expansions to model behavior via functional Taylor or orthogonal polynomial series, respectively [Marmarelis and Marmarelis1978]. Both approaches model the response r(t) as a linear combination of small powers of the stimulus s(t). Although effective for mild nonlinearities, deriving the linear combinations becomes numerically unstable for highly nonlinear systems. A more serious problem is that many biological systems are adaptive, i.e., the system behavior is dependent on the stimulus ensemble. For instance, [Rieke et al.1995] found that in the auditory nerve of the bullfrog linearity and information rates depended sensitively on whether a white noise or naturalistic ensemble is used. One approach to handling these difficulties is to forgo the full expansion, and simply compute the linear response to small (perturbative) stimuli in the presence of various different ensembles, or operating points. By collecting linear responses ? Email: kvale@phy.ucsf.edu and chris@phy.ucsf.edu Perturbative M-Sequences for Auditory Systems Identification 181 from different operating points, one may fit nonlinear responses as one fits a nonlinear function with a piecewise linear approximation. For adaptive systems the same procedure would be applied, with different operating points corresponding to different points along the time axis. Perturbative stimuli have wide application in condensed-matter physics, where they are used to characterize linear responses such as resistance, elasticity and viscosity, and in engineering, perturbative analyses are used in circuit analysis (small signal models) and structural diagnostics (vibration analysis). In neurophysiology, however, perturbative stimuli are unknown. An effective stimulus for calculating the perturbative linear response of a system is the m-sequence. M-sequences have a long history of use in engineering and the physical sciences, with applications ranging from systems identification to cryptography and cellular communication. In physiology, m-sequences have been used primarily to compute system kernels [Marmarelis and Marmarelisl978], especially in the visual system [Pinter and Nabet1987]. In this work, we use perturbative msequences to study the linear response of single units in the inferior colli cui us of a cat to amplitude-modulated (AM) stimuli. We add a small m-sequence signal to an AM carrier, which allows us to study the linear behavior of the system near a particular operating point in a non-destructive manner, i.e., without changing the operating point. Perturbative m-sequences allow one to calculate linear responses near the particular stimuli under study with only a little extra effort, and allow us to characterize the system over a wide range of stimuli, such as sinusoidal AM and naturalistic stimuli. The auditory system we selected to study was the response of single units in the central nucleus of the inferior colliculus (IC) of an anaesthetised cat. Single unit responses were recorded extraceUularly. Action potentials were amplified and stored on DAT tape, and were discriminated offline using a commercial computer-based spike sorter (Brainwave). 20 units were recorded, of which 10 yielded sufficiently stable responses to be analyzed. 2 M-Sequences and Linear Systems A binary m-sequence is a two-level pseudo-random sequence of +1's and -1's. The sequence length is L = 2n - 1, where n is the order of the sequence. Typically, a binary m-sequence can be generated by a shift register with n bits and feedback connections derived from an irreducible polynomial over the multiplicative group Z2 [Golomb1982]. For linear systems identification, m-sequences have two important properties. The first is that m-sequences have nearly zero mean: ~~':OI m[t] = -l. The second is that the autocorrelation function takes on the impulse-like form L-l Smm(T) = ~ m[t]m[t + T] = {L-1 if T = 0 otherwise (1) Impulse stimuli also have a 8-function autocorrelation function. In the context of perturbative stimuli, the advantage of an m-sequence stimulus over an impulse stimulus is that for a given signal to noise ratio, an m-sequence perturbation stays much closer to the original signal (in the least squares sense) than an impulse perturbation. Thus the perturbed signal does not stray as far from the operating point and measurement of linear response about that operating point is more accurate. We model the IC response with a system F through which a scalar stimulus s(t) is passed to give a response r(t): r(t) = F[s(t)]. (2) M. Kvale and C. E. Schreiner 182 For the purposes of this section, the functional F is taken to be a linear functional plus a DC component. In real experiments, the input and output signal are sampled into discrete sequences with t becoming an integer indexing the sequence. Then the system can be written as the discrete convolution L-1 r[t] = ho +L (3) h[ttls[t - t1] with kernels ho and h[tt] to be determined. We assume that the system has a finite memory of M time steps (with perhaps a delay) so that at most M of the h[t] coefficients are nonzero. To determine the kernels perturbatively, we add a small amount of m-sequence to a base stimulus so: s[t] = so[t] + am[t]. (4) Cross-correlating the response with the original m-sequence yields L-l L-l t=o t=o L m[t]r[t + r] = L L-l L- l m[t]ho + LL h[ttlm[t]so[t + r - ttl t=o tl =0 L-1 L-l +L L ah[tdm[t]m[t + r - tl]' (5) t=o tl=O Using the sum formula for am -sequence above, the first sum in Eq. (5) can be simplified to -ho. Using the autocorrelation Eq. (1), the third sum in Eq. (5) simplifies, and we find L-l Rrm(r) = a(L + l)h[r] - ho - a L L-l L-l h[tl] +L tl =0 L h[tt]m[t]so[t + r - ttl (6) t=o tl =0 Although the values for the kernels h(t) are set implicitly by this equation, the terms on the right hand side of Eq. (6) are widely different in size for large Land the equation can be simplified. As is customary in auditory systems, we assume the DC response ho is small. To estimate the size of the other terms, we compute statistical estimates of their sizes and look at their scaling with the parameters. The term a L:~-==~ h[tt] is a sum of M kernel elements; they may be correlated or uncorrelated, so a conservative estimate of their size is on the order of O( aM). The last term in (6) is more subtle. We rewrite it as L-l L-l LL L-l h[tdm[t]so[t + r - h=O t=o ttl = L h[tt]p[r, ttl tl=O L-l L m[t]so[t + r - ttl (7) t=o The time series of the ambient stimulus so[t] and m-sequence m[t] are assumed to be uncorrelated. By the central limit theorem, the sum p[r, tl] will then have an average of zero with a standard deviation of 0(L 1 / 2 ). If in turn, the terms p[r, ttl are un correlated with the kernels h[tl], we have that L-l L-l L L h[tt]m[t]so[t + r - ttl '" 0(MI/2 Ll/2) tl=O t=o (8) Perturbative M-Sequences for Auditory Systems Identification 183 If N cycles of the m-sequence are performed, in which sort] is different for each cycle, all the terms in Eq. (6) scale with N as O(N), except for the double sum. By the same central limits arguments above, the double sum scales as O(Nl/2). Putting all these results together into Eq. (6) and solving for the kernels yields h(r) a(L1+ 1) Rrm(r) - 0 ( ~) + 0 (aN~~~1/2 ) M 1 a(L + 1) Rrm(r) - C L 1 Ml/2 2 + C aNI/2?1/2' . (9) with the constants C1 , C2 '" O(h[r]) depending neural firing rate, statistics, etc., determined from experiment. If we take the kernel element h(r) to be the first term in Eq. 9, then the last two terms in Eq. (9) contribute errors in determining the kernel and can be thought of as noise. Both error terms vanish as L -+ 00 and the procedure is asymptotically exact for arbitrary uncorrelated stimuli sort]. In order for the cross-correlation Ram (r) to yield a good estimate, the inequalities (10) must hold. In practice, the kernel memory is much smaller than the sequence length, and the second inequality is the stricter bound. The second inequality represents a tradeoff among sequence length, number of trials and the size of the perturbation for a given level of systematic noise in the kernel estimate. For instance, if L = 215 - 1, N = 10, M = 30, and noise floor at 10%, the perturbation should be larger than a = 0.095. If no signal sort] is present, then the O(Ml/2a- 1(NL)-1/2) term drops out and the usual m-sequence cross-correlation result is recovered. 3 M-Sequences for Modulation Response Previous work, e.g., [M011er and Rees1986, Langner and Schreinerl988] has shown that many of the cells in the inferior colliculus are tuned not only to a characteristic frequency, but are also tuned to a best frequency of modulation of the carrier. A highly simplified model of the IC unit response to sound stimuli is the Ll- N - L2 cascade filter, with L1 a linear tank circuit with a transfer function matching that of the frequency tuning curve, N a nonlinear rectifying unit, and L2 a linear circuit with a transfer function matching that of the modulation transfer function. Detecting this modulation is an inherently nonlinear operation and N is not well approximated by a linear kernel. Thus IC modulation responses will not be well characterized by ordinary m-sequence stimuli using the methods described in Section 2. A better approach is to bypass the Ll - N demodulation step entirely and concentrate on measuring L2. This can be accomplished by creating a modulation m-sequence: (11) s[t] = a (so[t] + bm[t]) sin[wet], where Iso[t]1 :::; 1 is the ambient signal, i.e., the operating point, m[t] E [-1,1] is an m-sequence added with amplitude b, and We is the carrier frequency. Demodulation gives the effective input stimulus sm[t] = a (so[t] + bm[t]) . (12) Note that there is little physiological evidence for a purely linear rectifier N. In fact, both the work of [M011er and Rees1986, Rees and M011er1987] and ours below show that there is a nonlinear modulation response. Taking a modulation transfer 184 M. Kvale and C. E. Schreiner function seriously, however, implies that one assumes that modulation response is linear, which implies that the static nonlinearity used is something like a halfwave rectifier. Linearity is used here as a convenient assumption for organizing the stimulus and asking whether nonlinearities exist. For full m-sequence modulation (so[t] = 1 and b = 1) the stimulus Sm and the neural response can be used to compute, via the Lee--Schetzen cross-correlation, the modulation transfer function for the L2 system. Alternatively, for b ? 1, the m-sequence is a perturbation on the underlying modulation envelope sort]. The derivation above shows that the linear modulation kernel can also be calculated using a Lee--Schetzen cross-correlation. M-sequences at full modulation depth were first used by [M0ller and Rees1986, Rees and M011erI987] to calculate white-noise kernels. Here, we are using m-sequence in a different way-we are calculating the small-signal properties around the stimulus sort]. The m-sequences used in this experiment were of length 215 -1 = 32,767. For each unit, 10 cycles of the m-sequence were presented back-to-back. After determining the characteristic frequency of a unit, stimuli were presented which never differed from the characteristic frequency by more than 500 Hz. Figure 1 depicts the sinusoidal and m-sequence components and their combined result. The stimuli were presented in random order so as to mitigate adaptation effects. Figure 1: A depiction of stimuli used in the experiment. The top a pure sine wave modulation at modulation depth 0.8. The middle an m-sequence modulation at depth 1.0. The bottom graph shows a m-sequence modulation at depth 0.2 added to a sinusoidal modulation 4 graph shows graph shows perturbative at depth 0.8. Results Figure 2 shows the spike rates for both the pure sinusoid and the combined sinusoid and m-sequence stimuli. Note that the rates are nearly the same, indicating that the perturbation did not have a large effect on the average response of the unit. The unit shows an adaptation in firing rate over the 10 trials, but we did not find Perturbative M-Sequences for Auditory Systems Identification 185 a statistically significant change in the kernels of different trials in any of the units. G----e sinusoid ~ sinusoid + m-sequence .-... o Q) ~ 80.0 C/) Q) ~ '5. ~ 60.0 Q) ?i "40.0 100.0 200.0 300.0 400.0 500.0 Time (sec) Figure 2: A plot of the unit firing rates for both the pure sinusoid and the sinusoid + m-sequence stimuli. The carrier frequency is 9 kHz and is close to the characteristic frequency of the neuron. The sinusoidal modulation has a frequency of 20 Hz and the m-sequence modulation has a frequency of 800 sec-I . Figure 3 shows modulation response kernels at several different values of the modulation depth. Note that if the system was a linear, superposition would cause all the kernels to be equivalent; in fact it is seen that the nonlinearities are of the same magnitude as the linear response. In this particular unit, the triphasic behavior at small modulation depths gives way to monophasic behavior at high modulation depths and an FFT of the kernel shows that the bandwidth of the modulation transfer function also broadens with increasing depth. 5 Discussion In this paper, we have introduced a new type of stimulus, perturbative m-sequences, for the study of auditory systems and derived their properties. We then applied perturbative m-sequences to the analysis of the modulation response of units in the Ie, and found the linear response at a few different operation point. We demonstrated that the nonlinear response in the presence of sinusoidal modulations are nearly as large as the linear response and thus a description of unit response with only an MTF is incomplete. We believe that perturbative stimuli can be an effective tool for the analysis of many systems whose units phase lock to a stimulus. The main limiting factor is the systematic noise discussed in section 2, but it is possible to trade off duration of measurement and size of the perturbation to achieve good results. The m-sequence stimuli also make it possible to derive higher order information [Sutter1987] and with a suitable noise floor, it may be possible to derive second-order kernels as well. This work was supported by The Sloan foundation and ONR grant number N0001494-1-0547. M. Kvale and C. E. Schreiner 186 Response vs. modulation depth sine wave @40Hz + pert. m-sequence 90.0 - 0.2 - 0.4 -0.6 -0.8 -1.0 70.0 50.0 CD :::> "C "" a. 30.0 E ns 10.0 -10.0 -30.0 -50.0 0.0 5.0 10.0 15.0 20.0 time from spike (milliseconds) Figure 3: A plot of the temporal kernels derived from perturbative m-sequence stimuli in conjunction with sinusoidal modulations at various modulation depth. The y-axis units are amplitude per spike and the x-axis is in milliseconds before the spike. References [Golomb1982] S. W. Golomb. Shift Register Sequences. Aegean Park Press, Laguna Hills, CA, 1982. [Langner and Schreiner1988] G. Langner and C. E. Schreiner. Periodicity coding in the inferior colliculus of the cat: 1. neuronal mechanisms. Journal of Neurophysiology, 60: 1799-1822, 1988. [Marmarelis and Marmarelis1978] Panos Z. Marmarelis and Vasilis Z. Marmarelis. Analysis of Physiological Systems. Plenum Press, New York, NY, 10011, 1978. [M011er and Rees1986] Aage R. M011er and Adrian Rees. Dynamic properties of single neurons in the inferior colliculus of the rat. Hearing Research, 24:203-215, 1986. [Pinter and Nabet1987] Robert B. Pinter and Bahram Nabet. Nonlinear Vision. CRC Press, Boca Raton, FL, 1987. [Rees and M011er1987] Adrian Rees and Aage R. M01ler. Stimulus properties influencing the responses of inferior colliculus neurons to amplitude-modulated sounds. Hearing Research, 27:129-143, 1987. [Rieke et al.1995] F. Rieke, D. A. Bodnar, and W. Bialek. Naturalistic stimuli increase the rate and efficiency of information transmission by primary auditory afi"erents. Proceedings of the Royal Society of London. Series B, 262:259-265, 1995. [Sutter1987] E. E. Sutter. A practical non-stochastic approach to nonlinear timedomain analysis. In Vasilis Z. Marmarelis, editor, Advanced Methods of Physiological Modeling, Vol. 1, pages 303-315. Biomedical Simulations Resource, University of Southern California, Los Angeles, CA 90089-1451, 1987. 

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Learning to Order Things William W. Cohen Robert E. Schapire Yoram Singer AT&T Labs, 180 Park Ave., Florham Park, NJ 07932 {wcohen,schapire,singer} @research.att.com Abstract There are many applications in which it is desirable to order rather than classify instances. Here we consider the problem of learning how to order, given feedback in the form of preference judgments, i.e., statements to the effect that one instance should be ranked ahead of another. We outline a two-stage approach in which one first learns by conventional means a preference Junction, of the form PREF( u, v), which indicates whether it is advisable to rank u before v. New instances are then ordered so as to maximize agreements with the learned preference function. We show that the problem of finding the ordering that agrees best with a preference function is NP-complete, even under very restrictive assumptions. Nevertheless, we describe a simple greedy algorithm that is guaranteed to find a good approximation. We then discuss an on-line learning algorithm, based on the "Hedge" algorithm, for finding a good linear combination of ranking "experts." We use the ordering algorithm combined with the on-line learning algorithm to find a combination of "search experts," each of which is a domain-specific query expansion strategy for a WWW search engine, and present experimental results that demonstrate the merits of our approach. 1 Introduction Most previous work in inductive learning has concentrated on learning to classify. However, there are many applications in which it is desirable to order rather than classify instances. An example might be a personalized email filter that gives a priority ordering to unread mail. Here we will consider the problem of learning how to construct such orderings, given feedback in the form of preference judgments, i.e., statements that one instance should be ranked ahead of another. Such orderings could be constructed based on a learned classifier or regression model, and in fact often are. For instance, it is common practice in information retrieval to rank documents according to their estimated probability of relevance to a query based on a learned classifier for the concept "relevant document." An advantage of learning orderings directly is that preference judgments can be much easier to obtain than the labels required for classification learning. For instance, in the email application mentioned above, one approach might be to rank messages according to their estimated probability of membership in the class of "urgent" messages, or by some numerical estimate of urgency obtained by regression. Suppose, however, that a user is presented with an ordered list of email messages, and elects to read the third message first. Given this election, it is not necessarily the case that message three is urgent, nor is there sufficient information to estimate any numerical urgency measures; however, it seems quite reasonable to infer that message three should have been ranked ahead of the others. Thus, in this setting, obtaining preference information may be easier and more natural than obtaining the information needed for classification or regression. w. W. Cohen, R. E. Schapire and Y. Singer 452 In the remainder of this paper, we will investigate the following two-stage approach to learning how to order. In stage one, we learn a preference junction, a two-argument function PREF( u, v) which returns a numerical measure of how certain it is that u should be ranked before v. In stage two, we use the learned preference function to order a set of new instances U; to accomplish this, we evaluate the learned function PREF( u, v) on all pairs of instances u, v E U, and choose an ordering of U that agrees, as much as possible, with these pairwise preference judgments. This general approach is novel; for related work in various fields see, for instance, references [2, 3, 1, 7, 10]. As we will see, given an appropriate feature set, learning a preference function can be reduced to a fairly conventional classification learning problem. On the other hand, finding a total order that agrees best with a preference function is NP-complete. Nevertheless, we show that there is an efficient greedy algorithm that always finds a good approximation to the best ordering. After presenting these results on the complexity of ordering instances using a preference function, we then describe a specific algorithm for learning a preference function. The algorithm is an on-line weight allocation algorithm, much like the weighted majority algorithm [9] and Winnow [8], and, more directly, Freund and Schapire's [4] "Hedge" algorithm. We then present some experimental results in which this algorithm is used to combine the results of several "search experts," each of which is a domain-specific query expansion strategy for a WWW search engine. 2 Preliminaries Let X be a set of instances (possibly infinite). A preference junction PREF is a binary function PREF : X x X ~ [0,1]. A value of PREF(u, v) which is close to 1 or a is interpreted as a strong recommendation that u should be ranked before v. A value close to 1/2 is interpreted as an abstention from making a recommendation. As noted above, the hypothesis of our learning system will be a preference function, and new instances will be ranked so as to agree as much as possible with the preferences predicted by this hypothesis. In standard classification learning, a hypothesis is constructed by combining primitive features. Similarly, in this paper, a preference function will be a combination of other preference functions. In particular, we will typically assume the availability of a set of N primitive preference functions RI , ... , RN. These can then be combined in the usual ways, e.g., with a boolean or linear combination of their values; we will be especially interested in the latter combination method. It is convenient to assume that the Ri'S are well-formed in certain ways. To this end, we introduce a special kind of preference function called a rank ordering. Let S be a totally ordered set l with' >' as the comparison operator. An ordering function into S is a function f : X ~ S. The function f induces the preference function Rj, defined as I Rj(u,v) ~ { 0 21 if f (u) > f (v) if f(u) < f(v) otherwise. We call Rf a rank ordering for X into S. If Rf(u, v) to v, or u is ranked higher than v. = I, then we say that u is preferred It is sometimes convenient to allow an ordering function to "abstain" and not give a preference for a pair u, v. Let ?> be a special symbol not in S, and let f be a function into S U {?>}. We will interpret the mapping f (u) ?> to mean that u is "unranked," and let Rf (u, v) = if either u or v is unranked. = ! To give concrete examples of rank ordering, imagine learning to order documents based on the words that they contain. To model this, let X be the set of all documents in a repository, )That is, for all pairs of distinct elements 8J, 82 E S, either 8) < 82 or 8) > 82 . Learning to Order Things 453 and for N words WI, ... , W N. let Ii (u) be the number of occurrences of Wi in u. Then Rf; will prefer u to v whenever Wi occurs more often in u than v. As a second example. consider a meta-search application in which the goal is to combine the rankings of several WWW search engines. For N search engines el, ... , eN. one might define h so that R'i prefers u to v whenever u is ranked ahead of v in the list Li produced by the corresponding search engine. To do this, one could let Ii(u) = -k for the document u appearing in the k-th position in the list L i ? and let Ii( u) = </> for any document not appearing in L i . 3 Ordering instances with a preference function We now consider the complexity of finding the total order that agrees best with a learned preference function. To analyze this. we must first quantify the notion of agreement between a preference function PREF and an ordering. One natural notion is the following: Let X be a set. PREF be a preference function. and let p be a total ordering of X. expressed again as an ordering function (i.e .? p( u) > p( v) iff u precedes v in the order). We define AGREE(p, PREF) to be the sum of PREF( u, v) over all pairs u, v such that u is ranked ahead of v by p: AGREE(p, PREF) = (1) PREF(u, v). u.v:p{ul>p{v) Ideally. one would like to find a p that maximizes AGREE(p, PREF). This general optimization problem is of little interest since in practice, there are many constraints imposed by learning: for instance PREF must be in some restricted class of functions. and will generally be a combination of relatively well-behaved preference functions R i . A more interesting question is whether the problem remains hard under such constraints. The theorem below gives such a result. showing that the problem is NP-complete even if PREF is restricted to be a linear combination of rank orderings. This holds even if all the rank orderings map into a set S with only three elements. one of which mayor may not be </>. (Clearly. if S consists of more than three elements then the problem is still hard.) Theorem 1 The following decision problem is NP-complete: Input: A rational number 1\,; a set X; a set S with lSI ~ 3; a collection of N ordering functions Ii : X -t S; and a preference function PREF defined as PREF(u, v) = L~I wiR'i (u, v) where w = (WI, ... ,WN) is a weight vector in [0, l]N with L~I Wi = 1. Question: Does there exist a total order p such that AGREE(p, PREF) ~ I\,? The proof (omitted) is by reduction from CYCLIC-ORDERING [5. 6]. Although this problem is hard when lSI ~ 3. it becomes tractable for linear combinations of rank orderings into a set S of size two. In brief. suppose one is given X, Sand PREF as in Theorem 1, save that S is a two-element set. which we assume without loss of generality to be S = {O, I}. Now define p(u) = Li Wdi(U). It can be shown that the total order defined by p maximizes AGREE(p, PREF). (In case of a tie, p( u) = p( v) for distinct u and v. p defines only a partial order. The claim still holds in this case for any total order which is consistent with this partial order.) Of course, when lSI = 2, the rank orderings are really only binary classifiers. The fact that this special case is tractable underscores the fact that manipulating orderings can be computationally more difficult than performing the corresponding operations on binary classifiers. Theorem 1 implies that we are unlikely to find an efficient algorithm that finds the optimal total order for a weighted combination of rank orderings. Fortunately. there do exist efficient algorithms for finding an approximately optimal total order. Figure 1 summarizes a greedy w. W. Cohen, R. E. Schapire and Y. Singer 454 Algorithm Order- By- Preferences Inputs: an instance set X; a preference function PREF Output: an approximately optimal ordering function p let V = X for each v E V do7l'(v) = LUEVPREF(v,u) - LUEVPREF(u,v) while V is non-empty do let t = argmaxuEv 71'(u) let pet) IVI = V=V-{t} for each v E V do 71'(v) = 71'(v) + PREF(t, v) - PREF(v, t) endwhile Figure 1: A greedy ordering algorithm algorithm that produces a good approximation to the best total order, as we will shortly demonstrate. The algorithm is easiest to describe by thinking of PREF as a directed weighted graph where, initially, the set of vertices V is equal to the set of instances X, and each edge u -t v has weight PREF( u, v). We assign to each vertex v E V a potential value 71'( v), which is the weighted sum of the outgoing edges minus the weighted sum of the ingoing edges. That is, 71'(v) = LUEV PREF(v,u) - LUEV PREF(u, v) . The greedy algorithm then picks some node t that has maximum potential, and assigns it a rank by setting pet) = lVI, effectively ordering it ahead of all the remaining nodes. This node, together with all incident edges, is then deleted from the graph, and the potential values 71' of the remaining vertices are updated appropriately: This process is repeated until the graph is empty; notice that nodes removed in subsequent iterations will have progressively smaller and smaller ranks. The next theorem shows that this greedy algorithm comes within a factor of two of optimal. Furthermore, it is relatively simple to show that the approximation factor of 2 is tight. Theorem 2 Let OPT(PREF) be the weighted agreement achieved by an optimal total orderfor the preference junction PREF and let APPROX(PREF) be the weighted agreement achieved by the greedy algorithm. Then APPROX(PREF) ;::: !OPT(PREF). 4 Learning a good weight vector In this section, we look at the problem of learning a good linear combination of a set of preference functions. Specifically, we assume access to a set of ranking experts which provide us with preference functions Ri of a set of instances. The problem, then, is to learn a preference function of the form PREF(u,v) L~I wiRi(U,V). We adopt the on-line learning framework first studied by Littlestone [8J in which the weight Wi assigned to each ranking expert Ri is updated incrementally. = Learning is assumed to take place in a sequence of rounds. On the t-th round, the learning algorithm is provided with a set X t of instances to be ranked and to a set of N preference functions R~ of these instances. The learner may compute R!( u, v) for any and all preference functions R~ and pairs u, v E X t before producing a final ordering Pt of xt. Finally, the learner receives feedback from the environment. We assume that the feedback is an arbitrary set of assertions of the form "u should be preferred to v." That is, formally we regard the feedback on the t-th round as a set Ft of pairs (u, v) indicating such preferences. The algorithm we propose for this problem is based on the "weighted majority algorithm" [9J and, more directly, on the "Hedge" algorithm [4]. We define the loss of a preference function Learning to Order Things 455 Allocate Weights for Ranking Experts Parameters: (3 E [0,1] , initial weight vector WI E [0, I]N with l:~1 N ranking experts, number of rounds T Do fort = 1,2, ... ,T wl = 1 1. Receive a set of elements X t and preference functions R~, ... , R'N. 2. Use algorithm Order-By-Preferences to compute ordering function proximatesPREFt(u,v) = E~I wiRHu,v). 3. Order X t using Pt which ap- Pt . 4. Receive feedback Ft from the user. 5. Evaluate losses Loss(RL Ft) as defined in Eq. (2). 6. Set the new weight vector w!+ 1 = w! . (3Loss(R: ,Ft) / Zt where Zt is a normalization constant, chosen so that E~I w!+1 = 1. Figure 2: The on-line weight allocation algorithm. R with respect to the user's feedback F as L oss (R F) ~ E(U,V)EF(1 - R(u,v)) IFI' , (2) This loss has a natural probabilistic interpretation. If R is viewed as a randomized prediction algorithm that predicts that u will precede v with probability R(u, v), then Loss(R, F) is the probability of R disagreeing with the feedback on a pair (u, v) chosen uniformly at random from F. We now can use the Hedge algorithm almost verbatim, as shown in Figure 2. The algorithm maintains a positive weight vector whose value at time t is denoted by w t (wf, . . . , w'N). If there is no prior knowledge about the ranking experts, we set all initial weights to be equal so that = 1/N. The weight vector w t is used to combine the preference functions of the different experts to obtain the preference function PREFt = E~ I w~ R~. This, in tum, is converted into an ordering Pt on the current set of elements Xl using the method described in Section 3. After receiving feedback pt, the loss for each preference function Loss(RL Ft) is evaluated as in Eq. (2) and the weight vector w t is updated using the mUltiplicative rule W!+I = w~ . (3LQss(R: ,Ft) / Zt where (3 E [0, 1] is a parameter, and Zt is a normalization constant, chosen so that the weights sum to one after the update. Thus, based on the feedback, the weights of the ranking experts are adjusted so that experts producing preference functions with relatively large agreement with the feedback are promoted. = wI We will briefly sketch the theoretical rationale behind this algorithm. Freund and Schapire [4] prove general results about Hedge which can be applied directly to this loss function. Their results imply almost immediately a bound on the cumulative loss of the preference function PREFt in terms of the loss of the best ranking expert, specifically T LLoss(PREFt,F t ) T ~ a,Bm~n LLoss(RLFt) +c,BlnN l t=1 t=1 where a,B = InO / (3) / (1 - (3) and C,B = 1/( I - (3). Thus, if one of the ranking experts has low loss, then so will the combined preference function PREFt . However, we are not interested in the loss ofPREFt (since it is not an ordering), but rather in the performance of the actual ordering Pt computed by the learning algorithm. Fortunately, w. W. Cohen, R. E. Schapire and y. Singer 456 the losses of these can be related using a kind of triangle inequality. It can be shown that, for any PREF, F and p: Loss(Rp, F) ~ OISAGREE(p PREF) IFI ' + Loss(PREF, F) (3) where, similar to Eq. (1), OISAGREE(p, PREF) = Lu,v :p(u?p(v)(l - PREF(u, v)). Not surprisingly, maximizing AGREE is equivalent to minimizing DISAGREE. So, in sum, we use the greedy algorithm of Section 3 to minimize (approximately) the first term on the right hand side ofEq. (3), and we use the learning algorithm Hedge to minimize the second term. 5 Experimental results for metasearch We now present some experiments in learning to combine the results of several WWW searches. We note that this problem exhibits many facets that require a general approach such as ours. For instance, approaches that learn to combine similarity scores are not applicable since the similarity scores of WWW search engines are often unavailable. We chose to simulate the problem of learning a domain-specific search engine. As test cases we picked two fairly narrow classes of queries-retrieving the home pages of machine learning researchers (ML), and retrieving the home pages of universities (UNIV). We obtained a listing of machine learning researchers, identified by name and affiliated institution, together with their home pages, and a similar list for universities, identified by name and (sometimes) geographical location. Each entry on a list was viewed as a query, with the associated URL the sole relevant document. We then constructed a series of special-purpose "search experts" for each domain. These were implemented as query expansion methods which converted a name, affiliation pair (or a name, location pair) to a likely-seeming Altavista query. For example, one expert for the ML domain was to search for all the words in the person's name plus the words "machine" and "learning," and to further enforce a strict requirement that the person's last name appear. Overall we defined 16 search experts for the ML domain and 22 for the UN IV domain. Each search expert returned the top 30 ranked documents. In the ML domain there were 210 searches for which at least one search expert returned the named home page; for the UNIV domain, there were 290 such searches. For each query t, we first constructed the set X t consisting of all documents returned by all of the expanded queries defined by the search experts. Next, each search expert i computed a preference function R~. We chose these to be rank orderings defined with respect to an ordering function If in the natural way: We assigned a rank of if = 30 to the first listed document, Ii = 29 to the second-listed document, and so on, finally assigning a rank of Ii = 0 to every document not retrieved by the expanded query associated with expert i. To encode feedback, we considered two schemes. In the first we simulated complete relevance feedback-that is, for each query, we constructed feedback in which the sole relevant document was preferred to all other documents. In the second, we simulated the sort of feedback that could be collected from "click data," i.e., from observing a user's interactions with a metasearch system. For each query, after presenting a ranked list of documents, we noted the rank of the one relevant document. We then constructed a feedback ranking in which the relevant document is preferred to all preceding documents. This would correspond to observing which link the user actually followed, and making the assumption that this link was preferred to previous links. To evaluate the expected performance of a fully-trained system on novel queries in this domain, we employed leave-one-out testing. For each query q, we removed q from the Learning to Order Things Learned System (Full Feedback) Learned System ("Click Data") Naive Best (Top 1) Best (Top 10) Best (Top 30) Best (Av. Rank) 457 ML Domain University Domain Top 1 Top lO Top 30 Av. rank Top 1 Top lO Top 30 Av. rank 225 253 7.8 114 185 198 4.9 111 4.9 93 185 198 87 229 259 7.8 165 176 7.7 79 157 14.4 89 191 184 119 170 112 221 247 8.2 6.7 114 182 190 5.3 249 8.0 111 223 194 223 8.0 97 181 5.6 111 249 190 5.3 111 223 249 8.0 114 182 Table 1: Comparison of learned systems and individual search queries query set, and recorded the rank of q after training (with (3 = 0.5) on the remaining queries. For click data feedback, we recorded the median rank over 100 randomly chosen permutations of the training queries. We the computed an approximation to average rank by artificially assigning a rank of 31 to every document that was either unranked, or ranked above rank 30. (The latter case is to be fair to the learned system, which is the only one for which a rank greater than 30 is possible.) A summary of these results is given in Table 1, together with some additional data on "top-k performance"-the number of times the correct homepage appears at rank no higher than k. In the table we give the top-k performance (for three values of k) and average rank for several ranking systems: the two learned systems, the naive query (the person or university's name), and the single search expert that performed best with respect to each performance measure. The table illustrates the robustness of the learned systems, which are nearly always competitive with the best expert for every performance measure listed; the only exception is that the system trained on click data trails the best expert in top-k performance for small values of k. It is also worth noting that in both domains, the naive query (simply the person or university's name) is not very effective. Even with the weaker click data feedback, the learned system achieves a 36% decrease in average rank over the naive query in the ML domain, and a 46% decrease in the UNIV domain. To summarize the experiments, on these domains, the learned system not only performs much better than naive search strategies; it also consistently performs at least as well as, and perhaps slightly better than, any single domain-specific search expert. Furthermore, the performance of the learned system is almost as good with the weaker "click data" training as with complete relevance feedback. References [1] D.S. Hochbaum (Ed.). Approximation Algorithms for NP-hard problems. PWS Publishing Company, 1997. [2] O. Etzioni, S. Hanks, T. Jiang, R M. Karp, O. Madani, and O. Waarts. Efficient information gathering on the internet. In 37th Ann. Symp. on Foundations of Computer Science, 1996. [3] P.C Fishburn. The Theory of Social Choice. Princeton University Press, Princeton, NJ, 1973. [4] Y. Freund and RE. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 1997. [5] Z. Galil and N. Megido. Cyclic ordering is NP-complete. Theor. Compo Sci. , 5:179-182, 1977. [6] M.R Gary and D.S. Johnson. Computers and Intractibility: A Guide to the Theory of NPcompleteness. W. H. Freeman and Company, New York, 1979. [7j P.B. Kantor. Decision level data fusion for routing of documents in the TREC3 context: a best case analysis of worste case results. In TREC-3, 1994. [8] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2(4), 1988. [9) N. Littlestone and M.K. Warmuth. The weighted majority algorithm. Infonnation and Computation, 108(2):212-261, 1994. [10] K.E. Lochbaum and L.A. Streeter. Comparing and combining the effectiveness of latent semantic indexing and the ordinary vector space model for information retrieval. Infonnation processing and management, 25(6):665-676, 1989. 

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The Storage Capacity of a Fully-Connected Committee Machine Yuansheng Xiong Department of Physics, Pohang Institute of Science and Technology, Hyoja San 31, Pohang , Kyongbuk, Korea xiongOgalaxy.postech.ac.kr Chulan Kwon Department of Physics, Myong Ji University, Yongin, Kyonggi, Korea ckwonOwh.myongji.ac.kr Jong-Hoon Oh Lucent Technologies, Bell Laboratories, 600 Mountain Ave., Murray Hill , NJ07974, U. S. A. jhohOphysics.bell-labs.com Abstract We study the storage capacity of a fully-connected committee machine with a large number K of hidden nodes. The storage capacity is obtained by analyzing the geometrical structure of the weight space related to the internal representation . By examining the asymptotic behavior of order parameters in the limit of large K, the storage capacity Q c is found to be proportional to ]{ Jln ]{ up to the leading order. This result satisfies the mathematical bound given by Mitchison and Durbin , whereas the replica-symmetric solution in a conventional Gardner 's approach violates this bound. 1 INTRODUCTION Since Gardner's pioneering work on the storage capacity of a single layer perceptron[1], there have been numerous efforts to use the statistical mechanics formulation to study feed-forward neural networks. The storage capacity of multilayer neural networks has been of particular interest, together with the generalization problem. Barkai, Hansel and Kanter[2] studied a parity machine with a The Storage Capacity of a Fully-Connected Committee Machine 379 non-overlapping receptive field of continuous weights within a one-step replica symmetry breaking (RSB) scheme, and their result agrees with a mathematical bound previously found by Mitchison and Durbin (MD)[3] . Subsequently Barkai, Hansel and Sompolinsky[4] and Engel et al.[5] have studied the committee machine , which is closer to the multi-layer perceptron architecture and is most frequently used in real-world applications. Though they have derived many interesting results , particularly for the case of a finite number of hidden units, it was found that their the replica-symmetric (RS) result violates the MD bound in the limit where the number of hidden units K is large. Recently, Monasson and O'Kane[6] proposed a new statistical mechanics formalism which can analyze the weight-space structure related to the internal representations of hidden units . It was applied to single layer perceptrons[7 , 8, 9] as well as multilayer networks[10 , 11, 12]. Monasson and Zecchina[lO] have successfully applied this formalism to the case of both committee and parity machines with non-overlapping receptive fields (NRF)[lO] . They suggested that analysis of the RS solution under this new statistical mechanics formalism can yield results just as good as the onestep RSB solution in the conventional Gardner's method . In this letter, we apply this formalism for a derivation of the storage capacity of a fully-connected committee machine, which is also called a committee machine with overlapping receptive field (ORF) and is believed to be a more relevant architecture. In particular, we obtain the value of the critical storage capacity in the limit of large K , which satisfies the MD bound. It also agrees with a recent one-step RSB calculation, using the conventional Gardner method, to within a small difference of a numerical prefactor[13]. Finally we will briefly discuss the fully-connected parity machine. 2 WEIGHT SPACE STRUCTURE OF THE COMMITTEE MACHINE We consider a fully-connected committee machine with N input units , K hidden units and one output unit , where weights between the hidden units and the output unit are set to 1. The network maps input vectors {xf}, where J1. = 1, ... , P , to output yPo as: (1) where Wji is the weight between the ith input node and the jth hidden unit. hj ~ sgn(E~l WjiXn is the jth component of the internal representation for input pattern {xn . We consider continuous weights with spherical constraint , EfWji=N . Given P = aN patterns, the learning process in a layered neural network can be interpreted as the selection of cells in the weight space corresponding to a set of suitable internal representations h = {hj}, each of which has a non-zero elementary Y. Xiong, C. Kwon and J-H. Oh 380 volume defined by: (2) where 8(x) is the Heaviside step function. The Gardner's volume VG, that is, the volume of the weight space which satisfies the given input-output relations, can be written as the sum of the cells over all internal representations: (3) VG=LVh. h The method developed by Monasson and his collaborators [6, 10] is based on analysis of the detailed internal structure, that is, how the Gardner's volume VG is decomposed into elementary volumes Vb associated with a possible internal representation. The distribution of the elementary volumes can be derived from the free energy, (4) where ((- . -?) denotes the average over patterns. The entropy N[w(r)] of the volumes whose average sizes are equal to w (r) = -1/ N In (( Vb)), can be given by the Legendre relations og(r) N[w(r)] = - o(l/r)' w(r) = = = o[rg(r)] or (5) respecti vely. = = The entropies ND N[w(r 1)] and NR N[w(r 0)) are of most importance, and will be discussed below. In the thermodynamic limit, 1/ N ((In(VG))) -g(r = 1) is dominated by elementary volumes of size w(r = 1), of which there are exp(N ND). Furthermore, the most numerous elementary volumes have the size w(r = 0) and number exp(NNR). The vanishing condition for the entropies is related to the zero volume condition for VG and thus gives the storage capacity. We focus on the entropy N D of elementary volumes dominating the weight space VG. 3 = ORDER PARAMETERS AND PHASE TRANSITION For a fully-connected machine, the overlaps between different hidden units should be taken into account, which makes this problem much more difficult than the treelike (NRF) architecture studied in Ref. [10] . The replicated partition function for the fully-connected committee machine reads: (( (~ Vh)")) = (( Thh;?ThWj? IT e (It hr") ,n. e (hr" ~ Wj~.X;) )), (6) with a = 1,???, rand 0' = 1, ? ??, n. Unlike Gardner's conventional approach, we need two sets of replica indices for the weights. We introduce the order parameters, (7) where the indices a, b originate from the integer power r of elementary volumes, and 0', {3 are the standard replica indices. The replica symmetry ansatz leads to five The Storage Capacity of a Fully-Connected Committee Machine order parameters as: I q'" QOI{3ab 'k _ - J q C d'" d = 381 = (3, a =1= b), (j k, 0: (j = k, 0: =1= (j =1= k,o: = (j =1= k,o:= (j =1= k, 0: =1= (3), (3,a = b), (3,a =1= b), (3), (8) where q'" and q are, respectively, the overlaps between the weight vectors connected to the same hidden unit of the same (0: = (3) and different (0: =1= (3) replicas corresponding to the two different internal representations. The order parameters c, d'" and d describe the overlaps between weights that are connected to different hidden units, of which c and d'" are the overlaps within the same replica whereas d correlates different replicas. Using a standard replica trick, we obtain the explicit form of g( r). One may notice that the free energy evaluated at r 1 is reduced to the RS results obtained by the conventional method on the committee machine[4, 5], which is independent of q'" and d"'. This means that the internal structure of the weight space is overlooked by conventional calculation of the Gardner's volume. When we take the limit r ~ 1, the free energy can be expanded as: = d'" d) ( d) ( '" gr,q,q,c, , =gl,q,c, Ir=l' + (r-l )og(r,q"',q,c,d"',d) or (9) As noticed, g(r, q"', q, c, d"', d) is the same as the RS free energy in the Gardner's method. From the relation: N: _ D - - og(r) o(l/r) I _og(r) I r=l - -a;:- r=l ' (10) we obtain the explicit form of N D . In the case of the NRF committee machine, where each of the hidden units is connected to different input units, we do not have a phase transition. Instead, a single solution is applicable for the whole range of 0:. In contrast, the phase-space structure of the fully-connected committee machine is more complicated than that of the NRF committee machine. When a small number of input patterns are given, the system is in the permutation-symmetry (PS) phase[4, 5, 14), where the role of each hidden unit is not specialized. In the PS phase, the Gardner's volume is a single connected region. The order parameters associated with different hidden units are equal to the corresponding ones associated with the same hidden unit. When a critical number of patterns is given, the Gardner's volume is divided into many islands, each one of which can be transformed into other ones by permutation of hidden units. This phenomenon is called permutation symmetry breaking (PSB), and is usually accompanied by a first-order phase transition. In the PSB phase, the role of each hidden unit is specialized to store a larger number of patterns effectively. A similar breaking of symmetry has been observed in the study of generalization[14, 15], where the first-order phase transition induces discontinuity of the learning curve. It was pointed out that the critical storage capacity is attained in the PSB phase[4, 5), and our recent one-step replica symmetry breaking calculation confirmed this picture[13) . Therefore, we will focus on the analysis of the PSB solution near the storage capacity, in which q*, q ~ 1, and c, d"', d are of order 11K. Y. Xiong. C. Kwon and J-H. Oh 382 4 STORAGE CAPACITY When we analyze the results for free energy, the case with q( r = 1), c( r = 1) and d(r = 1) is reduced to the usual saddle-point solutions of the replica symmetric expression of the Gardner's volume g(r = 1)[4, 5). When K is large, the trace over all allowed internal representations can be evaluated similarly to Ref.[4]. The saddle-point equations for q* and d* are derived from the derivative of the free energy in the limit r -+ 1, as in Eq. (9). The details of the self-consistent equations are not shown for space consideration. In the following, we only summarize the asymptotic behavior of the order parameters for large a: 128 K2 1 - q + d - c '" (1r _ 2)2 0'2 ' " 32 ]{ (11) 1 - q + (Ii - 1)( c - d) '" 1r _ 2 a 2 ' ( 12) 1r-2 q + (K - l)d '" - a ' (13) (14) 1 - q* where + (I{ - r = -[..j1r J duH(u)lnH(u)]-l 1r 2 r2 l)(c - d*) '" 20'2 ' ~ (15) 0.62. It is found that all the overlaps between weights connecting different hidden units have scaling of -1/ K, whereas the typical overlaps between weights connecting the same hidden unit approach one. The order parameters c, d and d* are negative, showing antiferromagnetic correlations between different hidden units, which implies that each hidden unit attempts to store patterns different from those of the others[4, 5). Finally, the asymptotic behavior of the entropy N D in the large K limit can be derived using the scaling given above. Near the storage capacity, ND can be written, up to the leading order, as: "I r N D '" - K n '\ - (1r - 2)20'2 256K (16) Being the entropy of a discrete system, ND cannot be negative. Therefore, a gives an indication of the upper bound of storage capacity, that is, a c '" 7r~2 K Vln K. The storage capacity per synapse, 7r~2 Vln K, satisfies the rigorous bound", In K derived by Mitchison and Durbin (MD)[3], whereas the conventional RS result[4, 5], which scales as .JR, violates the MD bound. ND = 5 DISCUSSIONS Recently, we have studied this problem using a conventional Gardner approach in the one-step RSB scheme[13]. The result yields the same scaling with respect to K, but a coefficient smaller by a factor v'2. In the present paper, we are dealing with the fine structure of version space related to internal representations. On the other hand, the RSB calculation seems to handle this fine structure in association with symmetry breaking between replicas. Although the physics of the two approaches seems to be somehow related, it is not clear which of the two can yield a better The Storage Capacity of a Fully-Connected Committee Machine 383 estimate of the storage capacity. It is possible that the present RS calculation does not properly handle the RSB picture of the system. Monasson and his co-workers reported that the Almeida-Thouless instability of the RS solutions decreases with increasing K, in the NRF case[1O, 11] . A similar analysis for the fully-connected case certainly deserves further research. On the other hand, the one-step RSB scheme also introduces approximation, and possibly it cannot fully explain the weight-space structure associated with internal representations. It is interesting to compare our result with that of the NRF committee machine along the same lines[10]. Based on the conventional RS calculation, Angel et al. suggested that the same storage capacity per synapse for both fully-connected and NRF committee machines will be similar, as the overlap between the hidden nodes approaches zero.[5] . While the asymptotic scaling with respect to K is the same, the storage capacity in the fully-connected committee machine is larger than in the NRF one. It is also consistent with our result from one-step RSB calculation[13]. This implies that the small, but nonzero negative correlation between the weights associated with different hidden units, enhances the storage capacity. This may be good news for those people using a fully connected multi-layer perceptron in applications. From the fact that the storage capacity of the NRF parity machine is In Kj In 2[2, 10], which saturates the MD bound, one may guess that the storage capacity of a fully-connected parity machine is also proportional to Kin K. It will be interesting to check whether the storage capacity per synapse of the fully-connected parity machine is also enhanced compared to the NRF machine[16]. Acknowledgements This work was partially supported by the Basic Science Special Program of POSTECH and the Korea Ministry of Education through the POSTECH Basic Science Research Institute(Grant No. BSRI-96-2438). It was also supported by non-directed fund from Korea Research Foundation, 1995, and by KOSEF grant 971-0202-010-2. References [1] E. Gardner, Europhys, Lett. 4(4), 481 (1987); E. Gardner, J. Phys. A21, 257 (1988); E. Gardner and B. Derrida, J . Phys. A21, 271 (1988). [2] E. Barkai, D. Hansel and 1. Kanter, Phys. Rev. Lett. V 65, N18, 2312 (1990). [3] G. J. Mitchison and R. M. Durbin, Boil. Cybern. 60,345 (1989). [4] E. Barkai, D. Hansel and H. Sompolinsky, Phys. Rev. E45, 4146 (1992) . [5] A. Engel, H. M. Kohler, F. Tschepke, H. Vollmayr, and A. Zippeelius, Phys . Rev. E45, 7590 (1992). [6] R. Monasson and D. O'Kane, Europhys. Lett. 27,85(1994). [7] B. Derrida, R. B. Griffiths and A Prugel-Bennett, J. Phys. A 24,4907 (1991). [8] M. Biehl and M. Opper, Neural Networks: The Statistical Mechanics Perspective, Jong-Hoon Oh, Chulan Kwon, and Sungzoon Cho (eds.) (World Scientific, Singapore, 1995). [9] A. Engel and M. Weigt, Phys. Rev. E53, R2064 (1996). [10] R. Monasson and R. Zecchina, Phys. Rev. Lett. 75, 2432 (1995); 76, 2205 (1996). 384 Y. Xiong, C. Kwon and J-H. Oh R. Monasson and R. Zecchina, Mod. Phys. B, Vol. 9 , 1887-1897 (1996) . S. Cocco, R. Monasson and R. Zecchina, Phys. Rev . E54, 717 (1996) . C. Kwon and J. H. Oh, J . Phys. A, in press. K. Kang, J. H. Oh, C. Kwon and Y. Park, Phys. Rev. E48, 4805 (1993). [15] H. Schwarze and J. Hertz, Europhys. Lett. 21, 785 (1993) . [16] Y. Xiong, C. Kwon and J .-H. Oh, to be published (1997). [11] [12] [13] [14] 

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A Generic Approach for Identification of Event Related Brain Potentials via a Competitive Neural Network Structure Daniel H. Lange Department of Electrical Engineering Technion - liT Haifa 32000 Israel e-mail: lange@turbo.technion.ac.il Hillel Pratt Evoked Potential Laboratory Technion - liT Haifa 32000 Israel e-mail: hillel@tx.technion.ac.il Hava T. Siegelmann Department of Industrial Engineering Technion - liT Haifa 32000 Israel e-mail: iehava@ie.technion.ac.il Gideon F. Inbar Department of Electrical Engineering Technion - liT Haifa 32000 Israel e-mail: inbar@ee.technion.ac.il Abstract We present a novel generic approach to the problem of Event Related Potential identification and classification, based on a competitive Neural Net architecture. The network weights converge to the embedded signal patterns, resulting in the formation of a matched filter bank. The network performance is analyzed via a simulation study, exploring identification robustness under low SNR conditions and compared to the expected performance from an information theoretic perspective. The classifier is applied to real event-related potential data recorded during a classic odd-ball type paradigm; for the first time, withinsession variable signal patterns are automatically identified, dismissing the strong and limiting requirement of a-priori stimulus-related selective grouping of the recorded data. D. H. Lange, H. T. Siegelmann, H. Pratt and G. F. Inbar 902 1 1.1 INTRODUCTION EVENT RELATED POTENTIALS Ever since Hans Berger's discovery that the electrical activity of the brain can be measured and recorded via surface electrodes mounted on the scalp, there has been major interest in the relationship between such recordings and brain function. The first recordings were concerned with the spontaneous electrical activity of the brain, appearing in the form of rhythmic voltage oscillations, which later received the term electroencephalogram or EEG. Subsequently, more recent research has concentrated on time-locked brain activity, related to specific events, external or internal to the subject. This time-locked activity, referred to also as Event Related Potentials (ERP's), is regarded as a manifestation of brain processes related to preparation for or in response to discrete events meaningful to the subject. The ongoing electrical activity of the brain, the EEG, is comprised of relatively slow fluctuations, in the range of 0.1 - 100 Hz, with magnitudes of 10 - 100 uV. ERP's are characterized by overlapping spectra with the EEG, but with significantly lower magnitudes of 0.1 - 10 uV. The unfavorable Signal to Noise Ratio (SNR) requires filtering of the raw signals to enable analysis of the time-locked signals. The common method used for this purpose is signal averaging, synchronized to repeated occurrences of a specific event. Averaging-based techniques assume a deterministic signal within the averaged session, and thus signal variability can not be modeled unless a-priori stimulus- or response-based categorization is available; it is the purpose of this paper to provide an alternative working method to enhance conventional averaging techniques, and thus facilitating identification and analysis of variable brain responses. 1.2 COMPETITIVE LEARNING Competitive learning is a well-known branch of the general unsupervised learning theme. The elementary principles of competitive learning are (Rumelhart & Zipser, 1985): (a) start with a set of units that are all the same except for some randomly distributed parameter which makes each of them respond slightly differently to a set of input patterns, (b) limit the strength of each unit, and (c) allow the units to compete in some way for the right to respond to a given subset of inputs. Applying these three principles yields a learning paradigm where individual units learn to specialize on sets of similar patterns and thus become feature detectors. Competitive learning is a mechanism well-suited for regularity detection (H aykin , 1994), where there is a popUlation of input patterns each of which is presented with some probability. The detector is supposed to discover statistically salient features of the input population, without a-priori categorization into which the patterns are to be classified. Thus the detector needs to develop its own featural representation of the population of input patterns capturing its most salient features. 1.3 PROBLEM STATEMENT The complicated, generally unknown relationships between the stimulus and its associated brain response, and the extremely low SNR of the brain responses which are practically masked by the background brain activity, make the choice of a self organizing structure for post-stimulus epoch analysis most appropriate. The competitive network, having the property that its weights converge to the actual embedded signal patterns while inherently averaging out the additive background EEG, is thus an evident choice. A Generic Approach for Identification of Event Related Brain Potentials 2 2.1 903 THE COMPETITIVE NEURAL NETWORK THEORY The common architecture of a competitive learning system appears in Fig. 1. The system consists of a set of hierarchically layered neurons in which each layer is connected via excitatory connections with the following layer. Within a layer, the neurons are divided into sets of inhibitory clusters in which all neurons within a cluster inhibit all other neurons in the cluster, which results in a competition among the neurons to respond to the pattern appearing on the previous layer. Let Wji denote the synaptic weight connecting input node i to neuron j. A neuron learns by shifting synaptic weights from its inactive to active input nodes. If a neuron does not respond to some input pattern, no learning occurs in that neuron. When a single neuron wins the competition, each of its input nodes gives up some proportion of its synaptic weight, which is distributed equally among the active input nodes, fulfilling: 2:i Wji = 1. According to the standard competitive learning rule, for a winning neuron to an input vector Xi, the change llWji is defined by: llWji = 7J(Xi - Wji), where 7J is a learning rate coefficient. The effect of this rule is that the synaptic weights of a winning neuron are shifted towards the input pattern; thus assuming zero-mean additive background EEG, once converged, the network operates as a matched filter bank classifier. 2.2 MATCHED FILTERING From an information theoretic perspective, once the network has converged, our classification problem coincides with the general detection problem of known signals in additive noise. For simplicity, we shall limit the discussion to the binary decision problem of a known signal in additive white Gaussian noise, expandable to the M-ary detection in colored noise (Van Trees, 1968). Adopting the common assumption of EEG and ERP additivity (Gevins, 1984), and distinct signal categories, the competitive NN weights inherently converge to the general signal patterns embedded within the background brain activity; therefore the converged network operates as a matched filter bank. Assuming the simplest binary decision problem, the received signal under one hypothesis consists of a completely known signal, VEs(t), representing the EP, corrupted by an additive zero-mean Gaussian noise w(t) with variance (72; the received signal under the other hypothesis consists of the noise w(t) alone. Thus: = wet), 0 $ t $ T HI: ret) = ../Es(t) + wet), 0$ t $ T For convenience we assume that JoT s2(t)dt = 1, so that E represents the signal energy. Ho: ret) The problem is to observe r(t) over the interval [0, T] and decide whether Ho or Hl is true. It can be shown that the matched filter is the optimal detector, its impulse response being simply the signal reversed in time and shifted: her) = s(T - r) (1) Assuming that there is no a-priori knowledge of the probability of signal presence, the total probability of error depends only on the SNR and is given by (Van Trees, 1968): Pe 1 fOO = Vr.c exp( 21r IJi: V-;;2 ~2 )dz 2 (2) Fig. 2 presents the probability of true detection: (a) as a function of SNR, for minimized error probability, and (b) as a function of the probability of false detection. These D. H. Lange, H. T. Siegelmann, H. Pratt and G. F. Inbar 904 results are applicable to our detection problem assuming approximate Gaussian EEG characteristics (Gersch, 1970), or optimally by using a pre-whitening approach (Lange et. al., 1997). Probability of True Detection with Minimum Error J~ r L ll yer 1 Illhlb," n ry C lu l le rs 1.xIllIlOry ("" "nUn OIl S Layer :! Inhibi to ry CJ u~ lcn. ).K llillory ('Ollonec.u ons ? 0 00 0 0 00 0 0 . o ? oooo ~ 0 .0 0 . 1 }! 20dB ~O.B 0 ~0.6 0 ? tt "0 ~04 ~ 0.2 ? 0.1 INPUT PATffiRN Figure 1: The architecture of a competitive learning structure: learning takes place in hierarchically layered units, presented as filled (active) and empty (inactive) dots . 2.3 SNR In dB Detection performance : SNA "" +2 0 ? ? ,0. 0 , - 10 and -20 dB ~ F,~~~;;~::~====~~~~~~~~;:~ 9 1??lyel I I" pull lnils. : ? : :I ~?~0----~3~ 0 ----~ 20~---~ '0~--~ 0 --~1~ 0 --~2~ 0 --~3~ 0 --~40 0 .2 0 .3 0.4 05 O.B 0 .7 Probability of False Detection 0 ,8 Figure 2: Detection performance. Top: probability of detection as a function of the SNR. Bottom: detection characteristics. NETWORK TRAINING AND CONVERGENCE Our net includes a 300-node input layer and a competitive layer consisting of singlelayered competing neurons. The network weights are initialized with random values and trained with the standard competitive learning rule, applied to the normalized input vectors: z? AWji = 77( ~ - Wji) (3) L.Ji Xi The training is applied to the winning neuron of each epoch, while increasing the bias of the frequently winning neuron to gradually reduce its chance of winning consecutively (eliminating the dead neuron effect (Freeman & Skapura, 1992)). Symmetrically, its bias is reduced with the winnings of other neurons. In order to evaluate the network performance, we explore its convergence by analyzing the learning process via the continuously adapting weights: pj(n) ~ J~!lWl. ; ;~ 1,2, .??,C (4) where C represents the pre-defined number of categories. We define a set of classification confidence coefficients of the converged network: (5) Assuming existence of a null category, in which the measurements include only background noise (EEG), maxj{pj(N)} corresponds to the noise variance. Thus the values of r j, the confidence coefficients, ranging from 0 to 1 (random classification to completely separated categories), indicate the reliability of classification, which breaks down with the fall of SNR. Finally, it should be noted that an explicit statistical evaluation of the network convergence properties can be found in (Lange, 1997). 0 .9 A Generic Approachfor Identification of Event Related Brain Potentials 2.4 905 SIMULATION STUDY A simulation study was carried out to assess the performance of the competitive network classification system. A moving average (MA) process of order 8 (selected according to Akaike's condition applied to ongoing EEG (Gersch, 1970)), driven by a deterministic realization of a Gaussian white noise series, simulated the ongoing background activity x(n). An average of 40 single-trials from a cognitive odd-ball type experiment (to be explained in the Experimental Study), was used as the signal s(n). Then, five 100-trial ensembles were synthesized, to study the classification performance under variable SNR conditions. A sample realization and its constituents, at an SNR of dB, is shown in Fig. 3. The simulation included embedding the signal s(n) in the synthesized background activity x(n) at five SNR levels (-20,-10,0,+10, and +20 dB), and training the network with 750 sweeps (per SNR level). Fig. 4 shows the convergence patterns and classification confidences of the two neurons, where it can be seen that for SNR's lower than -10dB the classification confidence declines sharply. ? .. Templat., "2100 ~d cotalu:lanoe..o ~7 i,o'z . . . ~- -~Il __ ..r..l...:.. ___ _ NOIU de5hed RNllzallon dolted SNR..od8 C 100 ito' com.cMn-o g z ... _--. __ ._----_ .. _---100 200 300 '0' .---"'''''"''"'''''''''''''''"''P''''-="----, SNR __ 20 dB oon""'_006 100 200 300 10-0 100 200 300 i?j):~??--20 - ,0 0 I 20 SNR Figure 3: A sample single realization (dotted) and its constituents (signal solid, noise - dashed). SNR = 0 dB . Figure 4: Convergence patterns and classification confidence values for varying SNR levels. The classification results, tested on 100 input vectors, 50 of each category, for each SNR, are presented in the table below; due to the competitive scheme, Positives and False Negatives as well as Negatives and False Positives are complementary. These empirical results are in agreement with the analytical results presented in the above Matched Filtering sectioll. Table 1: Classification Results I Pos I Neg I FP 3 3.1 I FN EXPERIMENTAL STUDY MOTIVATION An important task in ERP research is to identify effects related to cognitive processes triggered by meaningful versus non-relevant stimuli. A common procedure to study these effects is the classic odd-ball paradigm, where the subject is exposed to a random 906 D. H. Lange, H. T. Siegelmann, H. Pratt and G. F Inbar sequence of stimuli and is instructed to respond only to the task-relevant (Target) ones. Typically, the brain responses are extracted via selective averaging of the recorded data, ensembled according to the types of related stimuli. This method of analysis assumes that the brain responds equally to the members of each type of stimulus; however the validity of this assumption is unknown in this case where cognition itself is being studied. Using our proposed approach, a-priori grouping of the recorded data is not required, thus overcoming the above severe assumption on cognitive brain function. The results of applying our method are described below. 3.2 EXPERIMENTAL PARADIGM Cognitive event-related potential data was acquired during an odd-ball type paradigm from pz referenced to the mid-lower jaw, with a sample frequency of 250 Hz (Lange et. al., 1995). The subject was exposed to repeated visual stimuli, consisting of the digits '3' and '5', appearing on a PC screen. The subject was instructed to press a pushbutton upon the appearance of '5' - the Target stimulus, and ignore the appearances of the digit '3'. With odd-ball type paradigms, the Target stimulus is known to elicit a prominent positive component in the ongoing brain activity, related to the identification of a meaningful stimulus. This component has been labeled P300, indicating its polarity (positive) and timing of appearance (300 ms after stimulus presentation). The parameters of the P 300 component (latency and amplitude) are used by neurophysiologists to assess effects related to the relevance of stimulus and level of attention (Lange et. al., 1995). 3.3 IDENTIFICATION RESULTS The competitive network was trained with 80 input vectors, half of which were Target ERP's and the other half were Non Target. The network converged after approximately 300 iterations (per neuron), yielding a reasonable confidence coefficient of 0.7. A sample of two single-trial post-stimulus sweeps, of the Target and Non-Target averaged ERP templates and of the NN identified signal categories, are presented in Fig. 5. The convergence pattern is shown in Fig. 6. The automatic identification procedure has provided two signal categories, with almost perfect matches to the stimulus-related selective averaged signals. The obtained categorization confirms the usage of averaging methods for this classic experiment, and thus presents an important result in itself. 4 DISCUSSION AND CONCLUSION A generic system for identification and classification of single-trial ERP's was presented. The simulation study demonstrated the powerful capabilities of the competitive neural net in classifying the low amplitude signals embedded within the large background noise. The detection performance declined rapidly for SNR's lower than -10dB, which is in general agreement with the theoretical statistical results, where loss of significance in detection probability is evident for SNR's lower than -20dB. Empirically, high classification performance was maintained with SNR's of down to -10dB, yielding confidences in the order of 0.7 or higher. The experimental study presented an unsupervised identification and classification of the raw data into Target and Non-Target responses, dismissing the requirement of stimulus-related selective data grouping. The presented results indicate that the noisy brain responses may be identified and classified objectively in cases where relevance of 907 A Generic Approachfor Identification of Event Related Brain Potentials the stimuli is unknown or needs to be determined, e.g. in lie-detection scenarios (Lange & Inbar, 1996), and thus open new possibilities in ERP research. J;:s::1 ~F?'S2 -20~ -20L~;:--_-;:-;;o.'-----:, ,:F?tl ,:F=l S NR _ _ 6 dB '0' .~g,~g o 05 Soc I 0 06 Soc , Figure 5: Top row: sample raw Target and Non- Target sweeps. Middle row: Target and Non- Target ERP templates. Bottom row: the NN categorized patterns. 10-40!;-----;:;;,--------; 200:;;----==""';;---=----;;:;;--~ ~ of~. Figure 6: Convergence pattern of the ERP categorization process; convergence is achieved after 300 iterations per neuron. References [1] Freeman J.A. and Skapura D.M. Neural Network6: Algorithm6, Application6, and Programming Technique6: Addison-Wesley Publishing Company, USA, 1992. [2] Gersch W., "Spectral Analysis of EEG's by Autoregressive Decomposition of Time Series," Math. Bi06c., vol. 7, pp. 205-222, 1970. [3] Gevins A.S., "Analysis of the Electromagnetic Signals of the Human Brain: Milestones, Obstacles, and Goals," IEEE Tran6. Biomed. Eng., vol. BME-31, pp. 833-850, 1984. [4] Haykin S. Neural Network6: A Comprehen6ive Foundation. Macmillan College Publishing Company, Inc., USA, 1994. [5] Lange D. H. Modeling and E6timation of Tran6ient, Evoked Brain Potential6. D.Sc. dissertation, Techion - Israel Institute of Technology, 1997. [6] Lange D.H. and Inbar G.F., "Brain Wave Based Polygraphy," Proceeding6 of the IEEE EMBS96 - the 18th Annual International Conference of the IEEE Engineering on Medicine and Biology Society, Amsterdam, October 1996. [7] Lange D.H., Pratt H. and Inbar G.F., "Modeling and Estimation of Single Evoked Brain Potential Components", IEEE. Tran6. Biomed. Eng., vol. BME-44, pp. 791-799, 1997. [8] Lange D.H., Pratt H., and Inbar G.F., "Segmented Matched Filtering of Single Event Related Evoked Potentials," IEEE. Tran6. Biomed. Eng., vol. BME-42, pp. 317-321, 1995. [9] Rumelhart D.E. and Zipser D., "Feature Discovery by Competitive Learning," Cognitive Science, vol. 9, pp. 75-112, 1985. [10] Van Trees H.L. Detection, E6timation, and Modulation Theory: Part 1: John Wiley and Sons, Inc., USA, 1968. 

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Hybrid reinforcement learning and its application to biped robot control Satoshi Yamada, Akira Watanabe, M:ichio Nakashima {yamada, watanabe, naka}~bio.crl.melco.co.jp Advanced Technology R&D Center Mitsubishi Electric Corporation Amagasaki, Hyogo 661-0001, Japan Abstract A learning system composed of linear control modules, reinforcement learning modules and selection modules (a hybrid reinforcement learning system) is proposed for the fast learning of real-world control problems. The selection modules choose one appropriate control module dependent on the state. This hybrid learning system was applied to the control of a stilt-type biped robot. It learned the control on a sloped floor more quickly than the usual reinforcement learning because it did not need to learn the control on a flat floor, where the linear control module can control the robot. When it was trained by a 2-step learning (during the first learning step, the selection module was trained by a training procedure controlled only by the linear controller), it learned the control more quickly. The average number of trials (about 50) is so small that the learning system is applicable to real robot control. 1 Introduction Reinforcement learning has the ability to solve general control problems because it learns behavior through trial-and-error interactions with a dynamic environment. It has been applied to many problems, e.g., pole-balance [1], back-gammon [2], manipulator [3], and biped robot [4]. However, reinforcement learning has rarely been applied to real robot control because it requires too many trials to learn the control even for simple problems. For the fast learning of real-world control problems, we propose a new learning system which is a combination of a known controller and reinforcement learning. It is called the hybrid reinforcement learning system. One example of a known controller is a linear controller obtained by linear approximation. The hybrid learning system S. Yamada, A. Watanabe and M. Nakashima 1072 will learn the control more quickly than usual reinforcement learning because it does not need to learn the control in the state where the known controller can control the object. A stilt-type biped walking robot was used to test the hybrid reinforcement learning system. A real robot walked stably on a flat floor when controlled by a linear controller [5]. Robot motions could be approximated by linear differential equations. In this study, we will describe hybrid reinforcement learning of the control of the biped robot model on a sloped floor, where the linear controller cannot control the robot. 2 Biped Robot a) b) pitch axis Figure 1: Stilt-type biped robot. a) a photograph of a real biped robot, b) a model structure of the biped robot. Ul, U2, U3 denote torques. Figure I-a shows a stilt-type biped robot [5J. It has no knee or ankle, has 1 m legs and weighs 33 kg. It is modeled by 3 rigid bodies as shown in Figure I-b. By assuming that motions around a roll axis and those around a pitch axis are independent, 5-dimensional differential equations in a single supporting phase were obtained. Motions of the real biped robot were simulated by the combination of these equations and conditions at a leg exchange period. If angles are approximately zero, these equations can be approximated by linear equations. The following linear controller is obtained from the linear equations. The biped robot will walk if the angles of the free leg are controlled by a position-derivative (PD) controller whose desired angles are calculated as follows: r{J (J+~+{3 if; - (J ( A = + 2~ -A7) + 6 If (1) where~, {3, 6, and 9 are a desired angle between the body and the leg (7?), a constant to make up a loss caused by a leg exchange (1.3?), a constant corresponding to walking speed, and gravitational acceleration (9.8 ms- 2 ), respectively. The linear controller controlled walking of the real biped robot on a flat floor [5]. However, it failed to control walking on a slope (Figure 2). In this study, the objective of the learning system was to control walking on the sloped floor shown in Figure 2-a. Hybrid Reinforcement Learning for Biped Robot Control a) 1073 lOem] Oem i I 1m b) 45 ?'s , -Angular Velocity - - I 2m 3m - ---------,-,- ----, I fall down iJ -45 .'sO~---------------:' IOcm b Time(s) 1.0 Height of Free Leg's Tip . ______ fall down VVy';.:?!Pft'~',....~'t:, -2cmO ??~ Robo' 10 Time(s) Po".: I = -lm O I 10 Time(s) Figure 2: Biped robot motion on a sloped floor controlled by the linear controller. a) a shape of a floor, b) changes in angular velocity, height of free leg's tip, and robot position 3 Hybrid Reinforcement Learning reinforcement: r(t) state inputs 71,~,iJ linear control module 1-----10( decision of k Figure 3: Hybrid reinforcement learning system. We propose a hybrid reinforcement learning system to learn control quickly, The hybrid reinforcement learning system shown in Figure 3 is composed of a linear control module, a reinforcement learning module, and a selection module. The reinforcement learning module and the selection module select an action and a module dependent on their respective Q-values. This learning system is similar to the modular reinforcement learning system proposed by Tham [6] which was based on hierarchical mixtures of the experts (HME) [7]. In the hybrid learning system , the selection module is trained by Q-Iearning. To combine the reinforcement learning with the linear controller described in (1), the ~u!put of the reinforcement learning module is set to k in the adaptable equation for (, ( = -kiJ + 6. The angle and the angular velocity of the supporting leg at the leg exchange period ('T], iJJi) are used as inputs. The k values are kept constant until the next leg exchange. The reinforcement learning module is trained by "Q-sarsa" learning [8]. Q values are calculated by CMAC neural networks [9], [10]. The Q values for action k (Q c (x, k)) and those for module s selection (Q s (x, s)) are s. Yamada, A Watanabe and M. Nakashima 1074 calculated as follows: L we(k, m, i, t)y(m, i, t) Tn ,i Q.. (x, s) = L w.. (s , m, i, t)y(m, i, t), (2) m,i where we{k,m,i,t) and w.. {s,m,i,t) denote synaptic strengths and y{m,i,t) represents neurons' outputs in CMAC networks at time t . Modules were selected and actions performed according to the ?-greedy policy [8] with ? = O. The temporal difference (TD) error for the reinforcement learning module (fe(t)) is calculated by sel(t) 1 0 = lin + Qe(x(t + l),per(t + 1)) - Qe{x{t),per{t)) sel{t) = rein sel(t + 1) = rein r{t) + Q.. {x(t + 1), sel(t + 1)) - Qe(x(t),per{t)), sel{t) = rein sel{t + 1) = lin r(t) fe(t) = (3) where r{t), per{t), sel(t), lin and rein denote reinforcement signals (r{t) = -1 if the robot falls down, 0 otherwise), performed actions, selected modules, the linear control module and the reinforcement learning module, respectively. TD error (ft (t)) calculated by Q.. (x, s) is considered to be a sum of TD error caused by the reinforcement learning module and that by the selection module. TD error (f .. {t)) used in the selection-module's learning is calculated as follows: f .. {t) = = ft(t) - fe(t) r{t) + ,Q .. {x(t + 1), sel(t + 1)) - Q.. (x{t), sel{t)) - fe(t), (4) where, denotes a discount factor. The reinforcement learning module used replacing eligibility traces {e c (k, m, i, t)) [11]. Synaptic strengths are updated as follows: we(k, m, i, t + 1) w.. (s,m,i ,t + 1) ee(k, m, i, t) = wc{k, m, i, t) + Qefc{t)ee{k, m, i, t)/nt { w.. (s , m, i, t) + Q.. f .. (t)y(m , i , t)/nt s = sel(t) otherwise w.. (s,m,i,t) k=per(t),y(m,i,t)=l 1 { o k ::f: per(t), y(m, i, t) = 1 >.ec(k, m, i, t - 1) otherwise (5) where Qe, Q .. , >. and nt are a learning constant for the reinforcement learning module, that for the selection module, decay rates and the number of tHings, respectively. In this study, the CMAC used 10 tHings. Each of the three dimensions was divided into 12 intervals. The reinforcement learning module had 5 actions (k = 0, A/2, A, 3A/2, 2A). The parameter values were Q .. = 0.2, Q e = 0.4, >. = 0.3, , = 0.9 and 6 = 0.05. Each run consisted of a sequence of trials, where each trial began with robot state of position=O, _5? < () < -2.5?,1 .5? < "I < 3?, cp = ()+~, 'I/J = cp+~, ( = "1+ 2?,9 = cp = "j; = iJ = ( = 0, and ended with a failure signals indicating robot's falling down. Runs were terminated if the number of walking steps of three consecutive trials exceeded 100. All results reported are an average of 50 runs. Hybrid Reinforcement Learning for Biped Robot Control ft''" 1075 80 '" bll ~ 60 C; ~ 40 o o Z 20 '- -?? . .... ???... ----................................... . .. .. ? .. . o~~~~~~~~~~ o jO 100 IjO 200 Trials Figure 4: Learning profiles for control of walking on the sloped floor. (0) hybrid reinforcement learning, (0) 2-step hybrid reinforcement learning, (\7) reinforcement learning and (6) HME-type modular reinforcement learning 4 Results Walking control on the sloped floor (Figure 2-a) was first trained by the usual reinforcement learning. The usual reinforcement learning system needed many trials for successful termination (about 800, see Figure 4(\7)). Because the usual reinforcement learning system must learn the control for each input, it requires many trials. Figure 4(0) also shows the learning curve for the hybrid reinforcement learning. The hybrid system learned the control more quickly than the usual reinforcement learning (about 190 trials). Because it has a higher probability of succeeding on the flat floor, it learned the control quickly. On the other hand, HME-type modular reinforcement learning [6] required many trials to learn the control (Figure 4(6)). 45?'s ,--- - - - - - - - - - -- -------, Angular Vel~ity ~A AAAAAA A hAA Ab AAA AAA AAAAAAAAAA~ I VYVV vvrW{VYvvvVYVVlJm rv YVvVVl () -45?,s ' - - - - - - - - - - - - - - - - - - - ' o Time(s) 20 0: "hfrl}f~iM?f11tfWfNWV\{\/YVVVy1 Height Oem \ . Free Leg's Tip ~ -2cm RObotP~i::[ ~ -1m o Time(s) 2 1? 20 Figure 5: Biped robot motion controlled by the network trained by the 2-step hybrid reinforcement learning. S. Yamada, A. Watanabe and M Nakashima 1076 In order to improve the learning rate, a 2-step learning was examined. The 2-step learning is proposed to separate the selection-module learning from the reinforcement-learning-module learning. In the 2-step hybrid reinforcement learning, the selection module was first trained by a special training procedure in which the robot was controlled only by the linear control module. And then the network was trained by the hybrid reinforcement learning. The 2-step hybrid reinforcement learning learned the control more quickly than the I-step hybrid reinforcement learning (Figure 4(0)). The average number of trials were about 50. The hybrid learning system may be applicable to the real biped robot. Figure 5 shows the biped robot motion controlled by the trained network. On the slope, the free leg's lifting was magnified irregularly (see changes in the height of the free leg's tip of Figure 5) in order to prevent the reduction of an amplitude of walking rhythm. On the upper flat floor, the robot was again controlled stably by the linear control module. a) b) 004 ~ """ bI) .!! ~ EO." "2 00] t: .~ j 15 8 002 0.4 ...t; .5 001 :: 02 o .2 eOc.........................~.....................~...L.....~ o -0.03 -002 -001 0 0.01 0 .02 003 initial synaptic strength values ?003 -002 -0.01 0 001 002 003 initial synaptic strength values Figure 6: Dependence of (a) the learning rate and (b) the selection ratio of the linear control module on the initial synaptic strength values (wa(rein, m, i, 0)). (a) learning rate of (0) the hybrid reinforcement learning, and (0) the 2-step hybrid reinforcement learning. The learning rate is defined as the inverse of the number of trials where the average walking steps exceed 70. (b) the ratio of the linear-controlmodule selection. Circles represent the selection ratio of the linear control module when controlled by the network trained by the hybrid reinforcement learning, rectangles represent that by the 2-step hybrid reinforcement learning. Open symbols represent the selection ratio on the flat floor, closed symbols represent that on the slope. The dependence of learning characteristics on initial synaptic strengths for the reinforcement-learning-module selection (W3 (rein, m, i, 0)) was considered (other initial synaptic strengths were 0). If initial values of ws(rein, m, i, t) (ws(rein, m, i, 0)) are negative, the Q-values for the reinforcement-learning-module selection (Q8(x,rein)) are smaller than Q8(x,lin) and then the linear control module is selected for all states at the beginning of the learning. In the case of the 2-step learning, if Ws (rein, m, i, 0) are given appropriate negative values, the reinforcement learning module is selected only around failure states, where Qa(x, lin) is trained in the first learning step, and the linear control module is selected otherwise at the beginning of the second learning step. Because the reinforcement learning module only requires training around failure states in the above condition, the 2- Hybrid Reinforcement Learning fo r Biped Robot Control 1077 step hybrid system is expected to learn the control quickly. Figure 6-a shows the dependence of the learning rate on the initial synaptic strength values. The 2-step hybrid reinforcement learning had a higher learning rate when Ws (rein, m, i, 0) were appropriate negative values (-0.01 '" -0.005). The trained system selected the linear control module on the flat floor (more than 80%), and selected both modules on the slope (see Figure 6-b), when ws(rein, m, i, 0) were negative. Three trials were required in the first learning step of the 2-step hybrid reinforcement learning. In order to learn the Q-value function around failure states, the learning system requires 3 trials. 5 Conclusion We proposed the hybrid reinforcement learning which learned the biped robot control quickly. The number of trials for successful termination in the 2-step hybrid reinforcement learning was so small that the hybrid system is applicable to the real biped robot. Although the control of real biped robot was not learned in this study, it is expected to be learned quickly by the 2-step hybrid reinforcement learning. The learning system for real robot control will be easily constructed and should be trained quickly by the hybrid reinforcement learning system. References [1] Barto, A. G., Sutton, R. S. and Anderson, C. W.: Neuron like adaptive elements that can solve difficult learning control problems, IEEE Trans. Sys. Man Cybern., Vol. SMC-13, pp. 834-846 (1983). [2] Tesauro, G.: TD-gammon, a self-teaching backgammon program, achieves master-level play, Neural Computation, Vol. 6, pp. 215-219 (1994). [3] Gullapalli, V., Franklin, J. A. and Benbrahim, H.: Acquiring robot skills via reinforcement learning, IEEE Control System, Vol. 14, No.1, pp. 13-24 (1994). [4] Miller, W. T.: Real-time neural network control of a biped walking robot, IEEE Control Systems, Vol. 14, pp. 41-48 (1994). [5] Watanabe, A., Inoue, M. and Yamada, S.: Development of a stilts type biped robot stabilized by inertial sensors (in Japanese), in Proceedings of 14th Annual Conference of RSJ, pp. 195-196 (1996). [6] Tham, C. K.: Reinforcement learning of multiple tasks using a hierarchical CMAC architecture, Robotics and Autonomous Systems, Vol. 15, pp. 247-274 (1995). [7] Jordan, M. I. and Jacobs, R. A.: Hierarchical mixtures of experts and the EM algorithm, Neural Computation, Vol. 6, pp. 181-214 (1994). [8] Sutton, R. S.: Generalization in reinforcement learning: successful examples using sparse coarse coding, Advances in NIPS, Vol. 8, pp. 1038-1044 (1996). [9] Albus, J. S.: A new approach to manipulator control: The cerebellar model articulation controller (CMAC), Transaction on ASME J. Dynamical Systems, Measurement, and Controls, pp. 220-227 (1975). [10] Albus, J. S.: Data storage in the cerebellar articulation controller (CMAC), Transaction on ASME J. Dynamical Systems, Measurement, and Controls, pp. 228-233 (1975). [11] Singh, S. P. and Sutton, R. S.: Reinforcement learning with replacing eligibility traces, Machine Learning, Vol. 22, pp. 123-158 (1996). 

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Incorporating Contextual Information in White Blood Cell Identification Xubo Song* Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125 xubosong@fire.work.caltech.edu Yaser Abu-Mostafa Dept. of Electrical Engineering and Dept. of Computer Science California Institute of Technology Pasadena, CA 91125 Yaser@over. work.caltech.edu Joseph Sill Computation and Neural Systems Program California Institute of Technology Pasadena, CA 91125 joe@busy.work.caltech.edu Harvey Kasdan International Remote Imaging Systems 9162 Eton Ave., Chatsworth, CA 91311 Abstract In this paper we propose a technique to incorporate contextual information into object classification. In the real world there are cases where the identity of an object is ambiguous due to the noise in the measurements based on which the classification should be made. It is helpful to reduce the ambiguity by utilizing extra information referred to as context, which in our case is the identities of the accompanying objects. This technique is applied to white blood cell classification. Comparisons are made against "no context" approach, which demonstrates the superior classification performance achieved by using context. In our particular application, it significantly reduces false alarm rate and thus greatly reduces the cost due to expensive clinical tests. ? Author for correspondence. Incorporating Contextual Information in White Blood Cell Identification 951 1 INTRODUCTION One of the most common assumptions made in the study of machine learning is that the examples are drawn independently from some joint input-output distribution. There are cases, however, where this assumption is not valid. One application where the independence assumption does not hold is the identification of white blood cell images. Abnormal cells are much more likely to appear in bunches than in isolation. Specifically, in a sample of several hundred cells, it is more likely to find either no abnormal cells or many abnormal cells than it is to find just a few. In this paper, we present a framework for pattern classification in situations where the independence assumption is not satisfied. In our case, the identity of an object is dependent of the identities of the accompanying objects, which provides the contextual information. Our method takes into consideration the joint distribution of all the classes, and uses it to adjust the object-by-object classification. In section 2, the framework for incorporating contextual information is presented, and an efficient algorithm is developed. In section 3 we discuss the application area of white blood cell classification, and address the importance of using context for this application. Empirical testing results are shown in Section 4, followed by conclusions in Section 5. 2 INCORPORATING CONTEXTUAL INFORMATION INTO CLASSIFICATION 2.1 THE FRAMEWORK Let Xi be the feature vector of an object, and Ci = C(Xi} be the classification for Xi, i = I, ... N, where N is the total number of objects. Ci E {I, ..., D}, where D is the number of total classes. According to Bayes rule, ( I ) -- pC X p(xlc}p(c} p(x} It follows that the "with context" a posteriori probability of the class labels of all the objects assuming values Cl, C2, "', CN, given all the feature vectors, is It is reasonable to assume that the feature distribution given a class is independent of the feature distributions of other classes, i.e., P(Xb X2, ... , xNlcl, C2, ... , CN} = p(xllcd???p(XNlcN} Then Equation (1) can be rewritten as - p(cllxd",p(CNlxN }P(Xl}???P(XN )P(Cl ' C2, ... , CN) P(Cl}? ..P(CN }P(Xl ' X2, ... , XN} X Song, Y. Abu-Mostafa, 1. Sill and H. Kasdan 952 where p( cilxi) is the "no context" object-by-object Bayesian a posteriori probability, and p( Ci) is the a priori probability of the classes, p( Xi) is the marginal probability of the features, and P(Xl' X2, ... , XN) is the joint distribution of all the feature vectors. Since the features (Xl, X2, ... , XN) are given, p(Xb X2, ... , XN) and p(xd are constant, where (3) The quantity p( Cl, C2, ?.. , CN ), which we call context ratio and through which the context plays its role, captures the dependence among the objects. In the case where all the objects are independent, p( Cl, C2, ..? , CN) equals one - there will be no context. In the dependent case, p( Cl, C2, ..? , CN) will not equal one, and the context has an effect on the classifications. We deal with the application of object classification where it is the count in each class, rather than the particular ordering or numbering of the objects, that matters. As a result, p (Cl , C2, .?. , CN) is only a function of the count in each class. Let N d be the count in class d, and Vd = !ft, d = 1..., D, (4) where Pd is the prior distribution of class d, for d = 1, ... D. 2:f=l Nd Nand L:f=l Vd = l. The decision rule is to choose class labels (Cl' C2, ?.. , CN) = Cl, argmax C2, .. . , CN such that P(Cll C2, .?. , cNIXl, X2, ... , XN) (5) (Cl ,C2 , ... ,CN) When implementing the decision rule, we need to compute and compare DN cases for Equation 5. In the case of white blood cell recognition, D = 14 and N is typically around 600, which makes it virtually impossible to implement. In many cases, additional constraints can be used to reduce computation, as is the case in white blood cell identification, which will be demonstrated in the following section. 3 WmTE BLOOD CELL RECOGNITION Leukocyte analysis is one of the major routine laboratory examinations. The utility of leukocyte classification in clinical diagnosis relates to the fact that in various physiological and pathological conditions the relative percentage composition of the blood leukocytes Incorporating Contextual Infonnation in White Blood Cell Identification 953 changes. An estimate of the percentage of each class present in a blood sample conveys information which is pertinent to the hematological diagnosis. Typical commercial differential WBC counting systems are designed to identify five major mature cell types. But blood samples may also contain other types of cells, i.e. immature cells. These cells occur infrequently in normal specimen, and most commercial systems will simply indicate the presence of these cells because they can't be individually identified by the systems. But it is precisely these cell types that relate to the production rate and maturation of new cells and thus are important indicators of hematological disorders. Our system is designed to differentiate fourteen WBC types which includes the five major mature types: segmented neutrophils, lymphocytes, monocytes, eosinophils, and basophils; and the immature types: bands (unsegmented neutrophils), metamyelocytes, myelocytes, promyelocytes, blasts, and variant lymphocytes; as well as nucleated red blood cells and artifacts. Differential counts are made based on the cell classifications, which further leads to diagnosis or prognosis. The data was provided by IRIS, Inc. Blood specimens are collected at Harbor UCLA Medical Center from local patients, then dyed with Basic Orange 21 metachromatic dye supravital stain. The specimen is then passed through a flow microscopic imaging and image processing instrument, where the blood cell images are captured. Each image contains a single cell with full color. There are typically 600 images from each specimen. The task of the cell recognition system is to categorize the cells based on the images. 3.1 PREPROCESSING AND FEATURE EXTRACTION The size of cell images are automatically tailored according to the size of the cell in the images. Images containing larger cells have bigger sizes than those with small cells. The range varies from 20x20 to 40x40 pixels. The average size is around 25x25. See Figure 3.1. At the preprocessing stage, the images are segmented to set the cell interior apart from the background. Features based on the interior of the cells are extracted from the images. The features include size, shape, color 1 and texture. See Table 1 for the list of features. 2 Figure 1: Example of some of the cell images. 3.2 CELL?BY?CELL CLASSIFICATION The features are fed into a nonlinear feed-forward neural network with 20 inputs, 15 hidden units with sigmoid transfer functions, and 14 sigmoid output units. A cross-entropy error 1 A color image is decomposed into three intensity images - red, green and blue respectively 2The red-blue distribution is the pixel-by-pixel log(red)- log(blue) distribution for pixels in cell interior. The red distribution is the distribution of the red intensity in cell interior. X Song, Y. Abu-Mostafa, 1. Sill and H. Kasdan 954 feature number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 feature description cell area number of pixels on cell edge the 4th quantile of red-blue distribution the 4th quantile of green-red distribution the median of red-blue distribution the median of green-red distribution the median of blue-green distribution the standard deviation of red-blue distribution the standard deviation of green-red distribution the standard deviation of blue-green distribution the 4th quantile of red distribution the 4th quantile of green distribution the 4th quantile of blue distribution the median of red distribution the median of green distribution the median of blue distribution the standard deviation of red distribution the standard deviation of green distribution the standard deviation of blue distribution the standard deviation of the distance from the edge to the mass center function is used in order to give the output a probability interpretation. Denote the input feature vector as x, the network outputs a D dimensional vector ( D = 14 in our case) p = {p(dlx)}, d = 1, ... , D, where p(dlx) is p(dlx) = Prob( a cell belongs to class dl feature x) The decision made at this stage is d(x) = argmax p(dlx) d 3.3 COMBINING CONTEXTUAL INFORMATION The "no-context" cell-by-cell decision is only based on the features presented by a cell, without looking at any other cells. When human experts make decisions, they always look at the whole specimen, taking into consideration the identities of other cells and adjusting the cell-by-cell decision on a single cell according to the company it keeps. On top of the visual perception of the cell patterns, such as shape, color, size, texture, etc., comparisons and associations, either mental or visual, with other cells in the same specimen are made to infer the final decision. A cell is assigned a certain identity if the company it keeps supports that identity. For instance, the difference between lymphocyte and blast can be very subtle sometimes, especially when the cell is large. A large unusual mononuclear cell with the characteristics of both blast and lymphocyte is more likely to be a blast if surrounded by or accompanied by other abnormal cells or abnormal distribution of the cells. 955 Incorporating Contextual Infonnation in White Blood Cell Identification This scenario fits in the framework we described in section 2. The Combining Contextual Information algorithm was used as the post-precessing of the cell-by-cell decisions. 3.4 OBSERVATIONS AND SIMPLIFICATIONS Direct implementation of the proposed algorithm is difficult due to the computational complexity. In the application of WBC identification, simplification is possible. We observed the following: First, we are primarily concerned with one class blast, the presence of which has clinical significance. Secondly, we only confuse blast with another class lymphocyte. In other words, for a potential blast, p(blastlx) ? 0, p(lymphocytelx) ? 0, p(any other classlx) ~ O. Finally, we are fairly certain about the classification of all other classes, i.e. p(a certain c1asslx) ~ 1, p(any other c1asslx) ~ O. Based on the above observations, we can simplify the algorithm, instead of doing an exhaustive search. Let pf = P(Ci = dlxi), i = 1, ... , N. More specifically, let pf = p(blastlxd, pf = p(lymphocytelxi) and pi = p(c1ass * IXi) where * is neither a blast nor a lymphocyte. Suppose there are K potential blasts. Order the over i, such that B B pf, pf, ... , pf 's in a descending manner B Pl ~ P2 ~ ... ~ PK then the probability that there are k blasts is PB(k) = PP ???pfpr+l???pj( PK+1"' piv p(VB = t, VL = v~ + K;/, V3, "" VD) where v~ is the proportion of unambiguous lymphocytes and V3, "" VD are the proportions of the other cell types, We can compute the PB(k)'s recursively, for k:::: 1, "" K-l, and This way we only need to compute K terms to get PB(k)'s . Pick the optimal number of blasts k* that maximizes PB (k), k = 1, "., K, An important step is to calculate p(Vl, "" VD) which can be estimated from the database, 3.5 THE ALGORITHM Step 1 Estimate P(Vl' ,." VD) from the database, for d = 1"", D, Step 2 Compute the object-by-object "no context" a posteriori probability p(cilxi), i 1, "', N, and Ci E {I, ... , D}, Step 3 Compute PB (k) and find k* for k = 1, .'" K, and relabel the cells accordingly, X Song, Y. Abu-Mostafa, 1. Sill and H. Kasdan 956 4 EMPIRICAL TESTING The algorithm has been intensively tested at IRIS, Inc. on the specimens obtained at Harbor UCLA medical center. We compared the performances with or without using contextual information on blood samples from 220 specimens (consisting of 13,200 cells). In about 50% of the cases, a false alarm would have occurred had context not been used. Most cells are correctly classified, but a few are incorrectly labelled as immature cells, which raises a flag for the doctors. Change of the classification of the specimen to abnormal requires expert intervention before the false alarm is eliminated, and it may cause unnecessary worry. When context is applied, the false alarms for most of the specimens were eliminated, and no false negative was introduced. methods no context with context cell classification 88% 89% normality identification 50% 90% I"V I"V false positive 1"V50% ,....10% false negative 0% 0% Table 2: Comparison of with and without using contextual information 5 CONCLUSIONS In this paper we presented a novel framework for incorporating contextual information into object identification, developed an algorithm to implement it efficiently, and applied it to white blood cell recognition. Empirical tests showed that the "with context" approach is significantly superior than the "no context" approach. The technique described could be generalized to a number of domains where contextual information plays an essential role, such a speech recognition, character recognition and other medical diagnosis regimes. Acknowledgments The authors would like to thank the members of Learning Systems Group at Caltech for helpful suggestions and advice: Dr. Amir Atiya, Zehra Cataltepe, Malik Magdon-Ismail, and Alexander Nicholson. References Richard, M.D., & Lippmann, R.P., (1991) Neural network classifiers estimate Bayesian a posteriori probabilities. Neural Computation 3. pp.461-483. Cambridge, MA: MIT Press. Kasdan, H.K., Pelmulder, J.P., Spolter, L., Levitt, G.B., Lincir, M.R., Coward, G.N., Haiby, S. 1., Lives, J., Sun, N.C.J., & Deindoerfer, F.H., (1994) The WhiteIRISTM Leukocyte differential analyzer for rapid high-precision differentials based on images of cytoprobereacted cells. Clinical Chemistry. Vol. 40, No.9, pp.1850-1861. Haralick, R.M., & Shapiro, L.G.,(l992),Computer and Robot Vision, Vol.1 , AddisonWelsley. Aus, H. A., Harms, H., ter Meulen, v., & Gunzer, U. (1987) Statistical evaluation of computer extracted blood cell features for screening population to detect leukemias. In Pierre A. Devijver and Josef Kittler (eds.) Pattern Recognition Theory and Applications, pp. 509518. Springer-Verlag. Kittler, J., (1987) Relaxation labelling. In Pierre A. Devijver and Josef Kittler (eds.) Pattern Recognition Theory and Applications, pp. 99-108. Springer-Verlag. 

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Ensemble and Modular Approaches for Face Detection: a Comparison Raphael Feraud ?and Olivier Bernier t France-Telecom CNET DTLjDLI Technopole Anticipa, 2 avenue Pierre Marzin, 22307 Lannion cedex, FRANCE Abstract A new learning model based on autoassociative neural networks is developped and applied to face detection. To extend the detection ability in orientation and to decrease the number of false alarms, different combinations of networks are tested: ensemble, conditional ensemble and conditional mixture of networks. The use of a conditional mixture of networks allows to obtain state of the art results on different benchmark face databases. 1 A constrained generative model Our purpose is to classify an extracted window x from an image as a face (x E V) or non-face (x EN). The set of all possible windows is E = V uN, with V n N = 0. Since collecting a representative set of non-face examples is impossible, face detection by a statistical model is a difficult task. An autoassociative network, using five layers of neurons, is able to perform a non-linear dimensionnality reduction [Kramer, 1991]. However, its use as an estimator, to classify an extracted window as face or non-face, raises two problems: 1. V', the obtained sub-manifold can contain non-face examples (V C V'), 2. owing to local minima, the obtained solution can be close to the linear solution: the principal components analysis. Our approach is to use counter-examples in order to find a sub-manifold as close as possible to V and to constrain the algorithm to converge to a non-linear solution [Feraud, R. et al., 1997]. Each non-face example is constrained to be reconstructed as its projection on V. The projection P of a point x of the input space E on V, is defined by: ?email: feraud@lannion.cnet.fr t email: bernier@lannion.cnet.fr 473 Ensemble and Modular Approaches for Face Detection: A Comparison ? if x E V, then P{ x) = x, ? if x rJ. V: P{x) = argminYEv{d(x, V)), where During the learning process, the projection by: P(x) ,...., ~ 2:7=1 Vi, where VI, V2, ?.. , V n , in the training set of faces, of v, the nearest d is the Euclidian distance. P of x on V is approximated are the n nearest neighbours, face example of x. The goal of the learning process is to approximate the distance V of an input space element x to the set of faces V: it (x - ?)2, where M is the size of input image x and ? the image reconstructed by the neural network, ? V{x, V) = Ilx - P(x)11 ,...., ? let x E ?, then x E V if and only if V{x, V) S T, with threshold used to adjust the sensitivity of the model. T E IR, where T is a 15 x 20 outputs 50 neurons 35 neurons 15 x 20 inputs Figure 1: The use of two hidden layers and counter-examples in a compression neural network allows to realize a non-linear dimensionality reduction. In the case of non-linear dimensionnality reduction, the reconstruction error is related to the position of a point to the non-linear principal components in the input space. Nevertheless, a point can be near to a principal component and far from the set of faces. With the algorithm proposed, the reconstruction error is related to the distance between a point to the set of faces. As a consequence, if we assume that the learning process is consistent [Vapnik, 1995], our algorithm is able to evaluate the probability that a point belongs to the set of faces. Let y be a binary random variable: y 1 corresponds to a face example and y 0 to a non-face example, we use: = P(y = = llx) = e- (,,_r):l 0'2 ,where (j depends on the threshold T The size of the training windows is 15x20 pixels. The faces are normalized in position and scale. The windows are enhanced by histogram equalization to obtain a relative independence to lighting conditions, smoothed to remove the noise and normalized by the average face, evaluated on the training set. Three face databases are used : after vertical mirroring, B fl is composed of 3600 different faces with orientation between 0 degree and 20 degree, Bf2 is composed of 1600 different faces with orientation between 20 degree and 60 degree and B f3 is the concatenation of Bfl and Bf2, giving a total of 5200 faces. All of the training faces are extracted R. Feraud and O. Bernier 474 from the usenix face database(**), from the test set B of CMU(**), and from 100 images containing faces and complex backgrounds . ? Figure 2: Left to right: the counter-examples successively chosen by the algorithm are increasingly similar to real faces (iteration 1 to 8) . The non-face databases (Bn!! ,B n !2,Bn!3), corresponding to each face database, are collected by an iterative algorithm similar to the one used in [Sung, K . and Poggio, T ., 1994] or in [Rowley, H. et al., 1995] : ? ? ? ? 1) 2) 3) 4) Bn! = 0, T = Tmin , the neural network is trained with B! + B n!, the face detection system is tested on a set of background images, a maximum of 100 su bimages Xi are collected with V ? 5) Bn! = Bn! + {xo, .. . , x n } , T = T + Jl , with Jl ? 6) while T < Tmax go back to step 2. (Xi , V) ~ T , > 0, After vertical mirroring, the size of the obtained set of non-face examples is respectively 1500 for B n!! , 600 for Bn!2 and 2600 for B n !3. Since the non-face set (N) is too large, it is not possible to prove that this algorithm converge in a finite time . Nevertheless, in only 8 iterations, collected counter-examples are close to the set of faces (Figure 2) . Using this algorithm, three elementary face detectors are constructed: the front view face detector trained on Bfl and Bnfl (CGM1), the turned face detector trained on Bf2 and Bn!2 (CGM2) and the general face detector trained on B!3 and Bn!3 (CGM3). To obtain a non-linear dimensionnality reduction, five layers are necessary. However, our experiments show that four layers are sufficient. Consequently, each CGM has four layers (Figure 1) . The first and last layers consist each of 300 neurons, corresponding to the image size 15x20. The first hidden layer has 35 neurons and the second hidden layer 50 neurons. In order to reduce the false alarm rate and to extend the face detection ability in orientation , different combinations of networks are tested. The use of ensemble of networks to reduce false alarm rate was shown by [Rowley, H. et al., 1995]. However, considering that to detect a face in an image, there are two subproblems to solve, detection of front view faces and turned faces, a modular architecture can also be used. 2 Ensemble of CGMs Generalization error of an estimator can be decomposed in two terms: the bias and the variance [Geman , S. et al., 1992] . The bias is reduced with prior knowledge. The use of an ensemble of estimators can reduce the variance when these estimators are independently and identically distributed [Raviv, Y. and Intrator, N., 1996]. Each face detector i produces : Assuming that the three face detectors (CGM1,CGM2,CGM3) are independently and identically distributed (iid), the ouput of the ensemble is: Ensemble and Modular Approaches for Face Detection: A Comparison 3 475 Conditional mixture of CGMs To extend the detection ability in orientation, a conditional mixture of CGMs is tested. The training set is separated in two subsets: front view faces and the corresponding counter-examples (B = 1) and turned faces and the corresponding counter-examples (B = 0). The first subnetwork (CGMl) evaluates the probability of the tested image to be a front view face, knowing the label equals 1 (P(y = llx, B = 1)). The second (CGM2) evaluates the probability of the tested image to be a turned face, knowing the label equals 0 (P(y = llx, B = 0)). A gating network is trained to evaluate P(B llx), supposing that the partition B 1, B 0 can be generalized to every input: = E[Ylx] = = = E[yIB = 1, x]f(x) + E[yIB = 0, x](l- f(x)) Where f(x) is the estimated value of P{B = llx) . This system is different from a mixture of experts introduced by [Jacobs, R. A. et aI., 1991]: each module is trained separately on a subset of the training set and then the gating network learns to combine the outputs. 4 Conditional ensemble of CGMs To reduce the false alarm rate and to detect front view and turned faces, an original combination, using (CGMl,CGM2) and a gate network, is proposed . Four sets are defined: ? F is the front view face set, ? P is the turned face set, with F n p = 0, ? V = F U P is the face set, ? N is the non-face set, with V n N = 0, Our goal is to evaluate P{x E Vlx). Each estimator computes respectively: ? P(x E Fix E FuN, x) (CGMl(x)), ? P(x E Pix E puN, x) (CGM2(x)), Using the Bayes theorem, we obtain: P(x E Fix) P(x E Fix E FUN,x) = P(x E FUNlx)P(x E FuNlx E F,x) Since x E F => x E FuN, then: P(x E Fix) P(x E Flx,x E FuN) = P(x E FUNlx) R. Feraud and o. Bernier 476 ~ ~ P(x E Fix) = P(x E Fix E FUN,x)P(x E FUNlx) P(x E Fix) = P(x E Fix E FUN,x)[P(x E Fix) + P(x E Nix)] In the same way, we have: P(x E Pix) = P(x E Pix E puN, x)[P(x E Pix) + P(x E Nix)] Then: P(x E Vlx) = P(x E Nlx)[P(x E pix E puN, x) + P(x E Fix E FuN, x)] +P(x E Plx)P(x E Pix E puN, x) + P(x E Flx)P(x E Fix E FuN, x) Rewriting the previous equation using the following notation, CGMl(x) for P(x E Fix E FuN, x) and CGMl(x) for P(x E Pix E puN, x), we have: P(x E Vlx) = P(x E Nlx)[CGMl(x) + CGM2(x)] (1) +P(x E Plx)CGM2(x) + P(x E Flx)CGMl(x) (2) Then, we can deduce the behaviour of the conditional ensemble: ? in N, if the output of the gate network is 0.5, as in the case of ensembles, the conditional ensemble reduces the variance of the error (first term of the right side of the equation (1)), ? in V, as in the case of the conditional mixture, the conditional ensemble permits to combine two different tasks (second term of the right side of the equation (2)) : detection of turned faces and detection of front view faces. The gate network f(x) is trained to calculate the probability that the tested image is a face (P(x E Vlx)), using the following cost function: C = ~ ([f(xi)MGCl(x) x.EV 5 + (1- f(xd)]MGC2(x) - Yi)2 + ~ (J(xd - 0.5)2 xiEN Discussion Each 15x20 subimage is extracted and normalized by enhancing, smoothing and substracting the average face, before being processed by the network. The detection threshold T is fixed for all the tested images. To detect a face at different scales, the image is subsampled. The first test allows to evaluate the limits in orientation of the face detectors. The sussex face database(**), containing different faces with ten orientations betwen 0 degree and 90 degrees, is used (Table 1). The general face detector (CGM3) uses the same learning face database than the different mixtures of CGMs. Nevertheless, CGM3 has a smaller orientation range than the conditional mixtures of CGMs, and the conditional ensemble of CGMs. Since the performances of the ensemble of CGMS are low, the corresponding hypothesis (the CGMs are iid) is invalid. Moreover, this test shows that the combination by a gating neural network ofCGMs, Ensemble and Modular Approaches for Face Detection: A Comparison 477 Table l :Results on Sussex face database orientation (degree) CGM1 CGM2 0 10 20 30 40 50 60 70 100.0 0 62 .5 % 50.0 % 12.5 % 0.0 % 0.0 % 0.0 % 0.0 % CGM3 Ensemble (1,2 ,3) Conditional ensemble Conditional trained on different training set , allows to extend the detection ability to both front view and turned faces. The conditional mixture of CGMs obtains results in term of orientation and false alarm rate close to the best CGMs used to contruct it (see Table 1 and Table 2). The second test allows to evaluate the false alarms rate and to compare our results with the best results published so far on the test set A [Rowley, H. et al., 1995] of the CMU (**), containing 42 images of various quality. First, these results show that the model , trained without counter-examples (GM) , overestimates the distribution offaces and its false alarm rate is too important to use it as a face detector. Second, the estimation of the probability distribution of the face performed by one CGM (CGM3) is more precise than the one obtained by [Rowley, H. et al., 1995] with one SWN (see Table 2). The conditional ensemble of CGMs and the conditional mixture of CG Ms obtained a similar detection rate than an ensemble of SWN s [Rowley, H. et al. , 1995], but with a false alarm rate two or three times lower. Since the results of the conditional ensemble of CG Ms and the conditional mixture of CGMs are close on this test , the detection rate versus the number of false alarms is plotted (Figure 3), for different thresholds . The conditional mixture of CGMs curve is above the one for the conditional ensemble of CG Ms. l 00~--~----~----~----~----~--~ ~ ..... - ............. ........__ ........ ,,- 10 3D Cl so Numt.rolt. . .wnw 110 70 10 Figure 3: Detection rate versus number of false alarms on the CMU test set A. In dashed line conditional ensemble and in solid line conditional mixture. The conditionnal mixture of CG Ms is used in an application called LISTEN [Collobert, M. et al. , 1996] : a camera detects, tracks a face and controls a microphone array towards a speaker, in real time. The small size of the subimages (15x20) processed allows to detect a face far from the camera (with an aperture of 60 degrees, the maximum distance to the camera is 6 meters). To detect a face in real time, the number of tested hypothesis is reduced by motion and color analysis . R. Feraud and o. Bernier 478 Table 2:Results on the CMU test set A GM: the model trained without counter-examples, CGM1 : face detector, CGM2: turned face detector, CGM3: general face detector. SWN: shared weight network. (*) Considering that our goal is to detect human faces, non-human faces and rough face drawings have not been taken into account. Model GM CGMI CGM2 CGM3 [Rowley, 1995] (one SWN) Ensemble (CGM1,CGM2,CGM3) Conditional ensemble (CGM1,CGM2,gate) [Rowley, 1995] (three SWNs) Conditional mixture (CGM1,CGM2,gate) Detection rate 84% 77 % 127/164 85 % 139/164 85 % 139/164 84 % 142/169* 74 % 121/164 82 % 134/164 85 % 144/169* 87 % 142/164 False alarms rate 1/1,000 +/- 5 % +/- 5 % 5.43/1,000,000 47/33,700,000 6.3/1,000,000 +/- 5 % 212/33,700,000 1.36/1,000,000 +/- 5 % 46/33,700,000 8.13/1,000,000 +/- 5 % 179/22,000,000 0.71/1,000,000 +/- 5 % 24/33,700,000 0.77/1,000,000 +/- 5 % 26/33,700,000 2.13/1,000,000 +/- 5 % 47/22,000,000 1.15/1,000,000 +/- 5 % 39/33,700,000 +/- 0.1/1,000,00( +/- 0.38/1,000,00 +/- 0.37/1,000,00 +/- 0.41/1,000,00 +/- 0.4/1,000,00( +/- 0.43/1,000,00 +/- 0.38/1,000,00 +/- 0.42/1,000,00 +/- 0.35/1,000,001 (**) usenix face database, sussex face database and CMU test sets can be retrieved at www.cs.rug.nl/peterkr/FACE/face.html. References [Collobert, M. et al., 1996] Collobert, M., Feraud, R, Le Tourneur, G ., Bernier, 0., Viallet, J.E, Mahieux, Y., and Collobert, D. (1996). Listen: a system for locating and tracking individual speaker. In Second International Conference On Automatic Face and Gesture Recognition. [Feraud, R et al., 1997] Feraud, R, Bernier, 0., and Collobert, D. (1997). A constrained generative model applied to face detection. Neural Processing Letters. [Geman, S. et al., 1992] Geman, S., Bienenstock, E., and Doursat, R (1992). networks and the bias-variance dilemma. Neural Computation, 4:1-58. Neural [Jacobs, R A. et al., 1991] Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. (1991). Adaptative mixtures of local experts. Neural Computation, 3:79-87. [Kramer, 1991] Kramer, M. (1991). Nonlinear principal component analysis using autoassociative neural networks. AIChE Journal, 37:233-243. [Raviv, Y. and Intrator, N., 1996] Raviv, Y. and Intrator, N . (1996). Bootstrapping with noise: An effective regularization technique. Connection Science, 8:355-372. [Rowley, H. et al., 1995] Rowley, H., Baluja, S., and Kanade, T. (1995). detection in visual scenes. In Neural Information Processing Systems 8. Human face [Sung, K. and Poggio, T ., 1994] Sung, K. and Poggio, T. (1994). Example-based learning for view-based human face detection. Technical report, M.I.T. [Vapnik, 1995] Vapnik, V. (1995). The Nature of Statistical Learning Theory. SpringerVerlag New York Heidelberg Berlin. 

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Wavelet Models for Video Time-Series Sheng Ma and Chuanyi Ji Department of Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute, Troy, NY 12180 e-mail: shengm@ecse.rpi.edu, chuanyi@ecse.rpi.edu Abstract In this work, we tackle the problem of time-series modeling of video traffic. Different from the existing methods which model the timeseries in the time domain, we model the wavelet coefficients in the wavelet domain. The strength of the wavelet model includes (1) a unified approach to model both the long-range and the short-range dependence in the video traffic simultaneously, (2) a computationally efficient method on developing the model and generating high quality video traffic, and (3) feasibility of performance analysis using the model. 1 Introduction As multi-media (compressed Variable Bit Rate (VBR) video, data and voice) traffic is expected to be the main loading component in future communication networks, accurate modeling of the multi-media traffic is crucial to many important applications such as video-conferencing and video-on-demand. From modeling standpoint, multi-media traffic can be regarded as a time-series, which can in principle be modeled by techniques in time-seres modeling. Modeling such a time-series, however, turns out to be difficult, since it has been found recently that real-time video and Ethernet traffic possesses the complicated temporal behavior which fails to be modeled by conventional methods[3] [4]. One of the significant statistical properties found recently on VBR video traffic is the co-existence of the long-range (LRD) and the short-range (SRD) dependence (see for example [4][6] and references therein). Intuitively, this property results from scene changes, and suggests a complex behavior of video traffic in the time domain[7]. This complex temporal behavior makes accurate modeling of video traffic a challenging task. The goal of this work is to develop a unified and computationally efficient method to model both the long-range and the short-range dependence in real video sources. Ideally, a good traffic model needs to be (a) accurate enough to characterize pertinent statistical properties in the traffic, (b) computationally efficient, and (c) fea- s. Ma and C. Jj 916 sible for the analysis needed for network design. The existing models developed to capture both the long-range and the short-range dependence include Fractional Auto-regressive Integrated Moving Average (FARIMA) models[4]' a model based on Hosking's procedure[6], Transform-Expand-Sample (TES) model[9] and scenebased models[7]. All these methods model both LRD and SRD in the time domain . The scene-based modeling[7] provides a physically interpretable model feasible for analysis but difficult to be made very accurate. TES method is reasonably fast but too complex for the analysis. The rest of the methods suffer from computational complexity too high to be used to generate a large volume of synthesized video traffic. To circumvent these problems, we will model the video traffic in the wavelet domain rather than in the time domain. Motivated by the previous work on wavelet representations of (the LRD alone) Fractional Gaussian Noise (FGN) process (see [2] and references therein), we will show in this paper simple wavelet models can simultaneously capture the short-range and the long-rage dependence through modeling two video traces. Intuitively, this is due to the fact that the (deterministic) similar structure of wavelets provides a natural match to the (statistical) self-similarity of the long-range dependence. Then wavelet coefficients at each time scale is modeled based on simple statistics. Since wavelet transforms and inverse transforms is in the order of O(N) , our approach will be able to attain the lowest computational complexity to generate wavelet models. Furthermore, through our theoretical analysis on the buffer loss rate, we will also demonstrate the feasibility of using wavelet models for theoretical analysis. 1.1 Wavelet Transforms In L2(R) space, discrete wavelets ?j(t)'s are ortho-normal basis which can be represented as ?j(t) = 2- j / 2 ?(2- i t - m), for t E [0,2 K - 1] with K ~ 1 being an integer. ?(t) is the so-called mother wavelet. 1 ~ j ~ K and 0 ~ m ~ 2K - j - 1 represent the time-scale and the time-shift, respectively. Since wavelets are the dilation and shift of a mother wavelet, they possess a deterministic similar structure at different time scales. For simplicity, the mother wavelet in this work is chosen to be the Haar wavelet, where ?(t) is 1 for 0 ~ t < 1/2, -1 for 1/2 ~ t < 1 and 0 otherwise. Let dj's be wavelet coefficients of a discrete-time process x(t) (t E [0,2 K 1]) . Then dj can be obtained through the wavelet transform dj = K L:;=O-l x(t)?j(t). x(t) can be represented through the inverse wavelet transform K 2 1 X( t) =L:j=l L:m=O - dj?j(t) + ?o, where ?o is equal to the average of x(t). K - , 2 2.1 Wavelet Modeling of Video Traffic The Video Sources Two video sources are used to test our wavelet models: (1) "Star Wars" [4]' where each frame is encoded by JPEG-like encoder, and (2) MPEG coded videos at Group of Pictures (GOP) level[7][ll] called "MPEG GOP" in the rest of the paper. The modeling is done at either the frame level or the GOP level. Wavelet Models for Video Time-Series 917 31 & .. , ,.. 32 9 ;30 21 > ? i 21 u 22 20 .. ? AMIA('.0.4.o) ? - .' 34 i 20 ? .+ 31 ? ? . :. + .1 dR{' ) ~.O ~ ? ? ':QOP . :Si90Soutlt . 0 " .. ... .0 .2 ~ . .0 ijs ? ? .. ~ AR1IIA{O,Q.4.o) ? ? . .? .. , :.: . .- . -- +. ..a . ~ ? "0 12 10 -I 14 .. .,\ 8 0 TWoScoIoI Figure 1: Log 2 of Variance of dJ versus Figure 2: Log 2 of Variance of dJ versus the time scale j the time scale j 0 .? - : StarW.,. .. :GOP 0.8 j J 0 ?? 0.2 0 -0.20 2 8 8 10 Lag 12 1. 18 18 Figure 3: The sample auto correlations of d 2.2 20 s. The Variances and Auto-correlation of Wavelet Coefficients As the first step to understand how wavelets capture the LRD and SRD, we plot in Figure (1) the variance of the wavelet coefficients dj's at different time scales for both sources. To understand what the curves mean, we also plot in Figure (2) the variances of wavelet coefficients for three well-known processes: FARIMA(O, 0.4, 0), FARIMA(l, 0.4, 0), and AR(l). FARIMA(O, 0.4,0) is a long-range dependent ptocess with Hurst parameter H = 0.9. AR(l) is a short-range dependent process, and FARIMA(l, 0.4,0) is a mixture of the long-range and the short-range dependent process. As observed, for FARIMA(O, 0.4, 0) process (LRD alone), the variance increases with j exponentially for all j. For AR(l) (SRD alone), the variance increases at an even faster rate than that of FARIMA(O, 0.4, 0) when j is small but saturates when j is large. For FARIMA(l, 0.4,0), the variance shows the mixed properties from both AR(l) and FARIMA(O, 0.4, 0). The variance of the video sources behaves similarly to that of FARIMA(l, 0.4,0), and thus demonstrate the co-existence of the SRD and LRD in the video sources in the wavelet domain. s Figure 3 gives the sample auto-correlation of d in terms of m's. The autocorrelation function of the wavelet coefficients approaches zero very rapidly, and s. Ma and C. Ji 918 . , ~ ~ . ; I m -... -2 0 so ... Quantll._ of Stand.rd Norm ?? Figure 4: Quantile-Quantile of d';' for j = 3. Left: Star Wars. Right: GOP. thus indicates the short-range dependence in the wavelet domain. This suggests that although the autocorrelation of the video traffic is complex in the time-domain, modeling wavelet coefficients may be done using simple statistics within each time scale. Similar auto-correlations have been observed for the other j's. 2.3 Marginal Probability Density Functions Is variance sufficient for modeling wavelet coefficients? Figure (4) plots the Q - Q plots for the wavelet coefficients of the two sources at j 31 . The figure shows that the sample marginal density functions of wavelet coefficients for both the "Star Wars" and the MPEG GOP source at the given time scale have a much heavier tail than that of the normal distribution. Therefore, the variance alone is only sufficient when the marginal density function is normal, and in general a marginal density function should be considered as another pertinent statistical property. = It should be noted that correlation among wavelet coefficients at different time scales is neglected in this work for simplicity. We will show both empirically and theoretically that good performance in terms of sample auto-correlation and sample buffer loss probability can be obtained by a corresponding simple algorithm. More careful treatment can be found in [8]. 2.4 An Algorithm for Generating Wavelet Models The algorithm we derive include three main steps: (a) obtain sample variances of wavelet coefficients at each time scale, (b) generate wavelet coefficients independently from the normal marginal density function using the sample mean and variance 2, and (c) perform a transformation on the wavelet coefficients so that the ISimilar behaviors have been observed at the other time scales. A Q - Q plot is a standard statistical tool to measure the deviation of a marginal density function from a normal density. The Q - Q plots of a process with a normal marginal is a straight line. The deviation from the line indicates the deviation from the normal density. See [4] and references therein for more details. 2The mean of the wavelet coefficients can be shown to be zero for stationary processes. Wavelet Models for Video Time-Series 919 resulting wavelet coefficients have a marginal density function required by the traffic. The obtained wavelet coefficients form a wavelet model from which synthesized video traffic can be generated. The algorithm can be summarized as follows. Let x(t) be the video trace oflength N. Algorithm 1. Obtain wavelet coefficients from x(t) through the wavelet transform. 2. Compute the sample variance Uj of wavelet coefficients at each time scale j. 3. Generate new wavelet coefficients dj's for all j and m independently through Gaussian distributions with variances Uj 's obtained at the previous step. 4. Perform a transformation on the wavelet coefficients so that the marginal density function of wavelet coefficients is consistent with that determined by the video traffic (see [6] for details on the transformation). 5. Do inverse wavelet transform using the wavelet coefficients obtained at the previous step to get the synthesized video traffic in the time domain . The computational complexity of both the wavelet transform (Step 1) and the inverse transform (Step 5) is O(N). So is for Steps 2, 3 and 4. Then O(N) is the computational cost of the algorithm, which is the lowest attainable for traffic models. 2.5 Experimental Results Video traces of length 171, 000 for "Star Wars" and 66369 for "MPEG GOP" are used to obtain wavelet models. FARIMA models with 45 parameters are also obtained using the same data for comparison. The synthesized video traffic from both models are generated and used to obtain sample auto-correlation functions in the time-domain, and to estimate the buffer loss rate. The results3 are given in Figure (6). Wavelet models have shown to outperform the FARIMA model. For the computation time, it takes more than 5-hour CPU time 4 on a SunSPARC 5 workstation to develop the FARIMA model and to generate synthesized video traffic of length 171, 000 5 . It only takes 3 minutes on the same machine for our algorithm to complete the same tasks. 3 Theory It has been demonstrated empirically in the previous section that the wavelet model, which ignores the correlation among wavelet coefficients of a video trace, can match well the sample auto-correlation function and the buffer loss probability. To further evaluate the feasibility of the wavelet model, the buffer overflow probability has been analyzed theoretically in [8]. Our result can be summarized in the following theorem. 3Due to page limit, we only provide plots for JPEG. GOP has similar results and was reported in [8]. 4Computation time includes both parameter estimation and synthesized traffic generation. 5The computational complexity to generate synthesized video traffic of length N is O(N2) for an FARIMA model[5][4]. S. Ma and C. Ii 920 ..- -2 -2.5 OJ 01 -4.5 I I I -5 ~.5 0.2 4~1 0.1 Figure 5: "-": Autocorrelation of "Star Wars"; "- -": ARIMA(25,d,20); " ". Our Algorithm O.li 0.4 0.45 o.s 0.&6 0.8 085 07 0.71 OJ Figure 6: Loss rate attained via simulation. Vertical axis: loglO (Loss Rate); horizontal axis: work load. "-": the single video source; "". FARIMA(25,d,20); "-" Our algorithm. The normalized buffer size: 0.1, 1, 10,30 and 100 from the top down. Theorem Let BN and EN be the buffer sizes at the Nth time slot due to the syn- thesized traffic by the our wavelet model, and by the FGN process, respectively. Let C and B represent the capacity, and the maximum allowable buffer size respectively. Then InPr(BN > B) InPr(EN > B) (C - JL)2(i!:;?(1-H)e~7f-)2H 20- 2 (1- H)2 (1) where ~ < H < 1 is the Hurst parameter. JL and 0- 2 is the mean and the variance of the traffic, respectively. B is assume to be (C - It )2ko, where ko is a positive integer. This result demonstrates that using our simple wavelet model which neglects the correlations among wavelet coefficients, buffer overflow probability obtained is similar to that of the original FGN process as given in[10]. In other words, it shows that the wavelet model for a FG N process can have good modeling performance in terms of the buffer overflow criterion. We would like to point out that the above theorem is held for a FGN process. Further work are needed to account for more general processes. 4 Concl usions In this work, we have described an important application on time-series modeling: modeling video traffic. We have developed a wavelet model for the timeseries. Through analyzing statistical properties of the time-series and comparing the wavelet model with FARIMA models, we show that one of the key factors to successfully model a time-series is to choose an appropriate model which naturally fits the pertinant statistical properties of the time-series. We have shown wavelets are particularly feasible for modeling the self-similar time-series due to the video traffic. Wavelet Models for Video Time-Series 921 We have developed a simple algorithm for the wavelet models, and shown that the models are accurate, computationally efficient and simple enough for analysis. References [1] I. Daubechies, Ten Lectures on Wavelets. Philadelphia: SIAM, 1992. [2] Patrick Flandrin, "Wavelet Analysis and Synthesis of Fractional Brownian Motion", IEEE transactions on Information Theory, vol. 38, No.2, pp.910-917, 1992. [3] W.E Leland, M.S . Taqqu, W. Willinger and D.V. Wilson, "On the Self-Similar Nature of Ethernet Traffic (Extended Version)", IEEE/ACM Transactions on Networking, vo1.2, 1-14, 1994. [4] Mark W. Garrett and Walter Willinger. "Analysis, Modeling and Generation of Self-Similar VBR Video Traffic.", in Proceedings of ACM SIGCOMM'94, London, U.K, Aug., 1994 [5] J .R.M. Hosking, "Modeling Persistence in Hydrological Time Series Using Fractional Differencing", Water Resources Research, 20, pp . 1898-1908, 1984. [6] C. Huang, M. Devetsikiotis, I. Lambadaris and A.R. Kaye, "Modeling and Simulation of Self-Similar Variable Bit Rate Compressed Video: A Unified Approach", in Proceedings of ACM SIGCOMM'95, pp. 114-125. [7] Predrag R. Jelenlnovic, Aurel A. Lazar, and Nemo Semret. The effect of multiple time scales and subexponentiality in mpeg video streams on queuing behavior. IEEE Journal on Selected Area of Communications, 15, to appear in May 1997. [8] S. Ma and C. Ji, "Modeling Video Traffic in Wavelet Domain" , to appear IEEE INFO COM, 1998. [9] B. Melamed, D. Raychaudhuri, B. Sengupta, and J. Zdepski. Tes-based video source modeling for performance evaluation of integrated networks. IEEE Transactions on Communications, 10, 1994. [10] Ilkka Norros, "A storage model with self-similar input," Queuing Systems, vol.16, 387-396, 1994. [11] O. Rose. "Statistical properties of mpeg video traffic and their impact on traffic modeling in atm traffic engineering", Technical Report 101, University of Wurzburg, 1995. , ) 

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Learning Path Distributions using Nonequilibrium Diffusion Networks Paul Mineiro * Javier Movellan pmineiro~cogsci.ucsd.edu movellan~cogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92093-0515 Department of Cognitive Science University of California, San Diego La Jolla, CA 92093-0515 Ruth J. Williams williams~math.ucsd.edu Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 Abstract We propose diffusion networks, a type of recurrent neural network with probabilistic dynamics, as models for learning natural signals that are continuous in time and space. We give a formula for the gradient of the log-likelihood of a path with respect to the drift parameters for a diffusion network. This gradient can be used to optimize diffusion networks in the nonequilibrium regime for a wide variety of problems paralleling techniques which have succeeded in engineering fields such as system identification, state estimation and signal filtering. An aspect of this work which is of particular interest to computational neuroscience and hardware design is that with a suitable choice of activation function, e.g., quasi-linear sigmoidal, the gradient formula is local in space and time. 1 Introduction Many natural signals, like pixel gray-levels, line orientations, object position, velocity and shape parameters, are well described as continuous-time continuous-valued stochastic processes; however, the neural network literature has seldom explored the continuous stochastic case. Since the solutions to many decision theoretic problems of interest are naturally formulated using probability distributions, it is desirable to have a flexible framework for approximating probability distributions on continuous path spaces. Such a framework could prove as useful for problems involving continuous-time continuous-valued processes as conventional hidden Markov models have proven for problems involving discrete-time sequences. Diffusion networks are similar to recurrent neural networks, but have probabilistic dynamics. Instead of a set of ordinary differential equations (ODEs), diffusion networks are described by a set of stochastic differential equations (SDEs). SDEs provide a rich language for expressing stochastic temporal dynamics and have proven ? To whom correspondence should be addressed. Learning Path Distributions Using Nonequilibriwn Diffusion Networks 599 Figure 1: An example where the average of desirable paths yields an undesirable path, namely one that collides with the tree. useful in formulating continuous-time statistical inference problems, resulting in such solutions as the continuous Kalman filter and generalizations of it like the condensation algorithm (Isard & Blake, 1996). A formula is given here for the gradient of the log-likelihood of a path with respect to the drift parameters for a diffusion network. Using this gradient we can potentially optimize the model to approximate an entire probability distribution of continuous paths, not just average paths or equilibrium points. Figure 1 illustrates the importance of this kind of learning by showing a case in which learning average paths would have undesirable results, namely collision with a tree. Experience has shown that learning distributions of paths, not just averages, is crucial for dynamic perceptual tasks in realistic environments, e.g., visual contour tracking (Isard & Blake, 1996). Interestingly, with a suitable choice of activation function, e.g., quasilinear sigmoidal, the gradient formula depends only upon local computations, i.e., no time unfolding or explicit backpropagation of error is needed. The fact that noise localizes the gradient is of potential interest for domains such as theoretical neuroscience, cognitive modeling and hardware design. 2 Diffusion Networks ? Hereafter Cn refers to the space of continuous Rn-valued functions over the time interval [0, TJ, with T E R, T > fixed throughout this discussion. A diffusion network with parameter A E RP is a random process defined via an SD E of the form dX(t) = JL(t, X(t), A)dt + adB(t), X(O) '" Ito (1) 1I, where X is a Cn-valued process that represents the temporal dynamics of the n nodes in the network; JL: [0, T] x R n x RP -+ Rn is a deterministic function called the drift; A E RP is the vector of drift parameters, e.g., synaptic weights, which are to be optimized; B is a Brownian motion process which provides the random driving term for the dynamics; 1I is the initial distribution of the solution; and a E R, a > 0, is a fixed constant called the dispersion coefficient, which determines the strength of the noise term. In this paper we do not address the problem of optimizing the dispersion or the initial distribution of X. For the existence and uniqueness in law of the solution to (1) JL(',', A) must satisfy some conditions. For example, it is sufficient that it is Borel measurable and satisfies a linear growth condition: IJL(t, x, A)I S; K-\(l + Ixl) for some K-\ > and all t E [0, T], x ERn; see ? 600 P. Mineiro, 1. Movellan and R. 1. Williams (Karatzas & Shreve, 1991, page 303) for details. It is typically the case that the n-dimensional diffusion network will be used to model d-dimensional observations with n > d. In this case we divide X into hidden and observable l components, denoted Hand 0 respectively, so that X = (H,O). Note that with a = 0 in equation (1), the model becomes equivalent to a continuoustime deterministic recurrent neural network. Diffusion networks can therefore be thought of as neural networks with "synaptic noise" represented by a Brownian motion process. In addition, diffusion networks have Markovian dynamics, and hidden states if n > d; therefore, they are also continuous-time continuous-state hidden Markov models. As with conventional hidden Markov models, the probability density of an observable state sequence plays an important role in the optimization of diffusion networks. However, because X is a continuous-time process, care must taken in defining a probability density. 2.1 Density of a continuous observable path Let (XA, B>') defined on some filtered probability space (0, F, {Fd, p) be a (weak) solution of (1) with fixed parameter A. Here X>' = (HA,O>') represents the states of the network and is adapted to the filtration {Fd, B>' is an n-dimensional {Fdmartingale Brownian motion and the filtration {Ft } satisfies the usual conditions (Karatzas and Shreve, 1991, page 300). Let Q>' be the unique probability law generated by any weak solution of (1) with fixed parameter A Q>'(A) = p(X>' E A) for all A E F, (2) where F is the Borel sigma algebra generated by the open sets of Cn. Setting n = Cn , nh = Cn- d, and no = Cd with associated Borel a-algebras F, Fh and Fo, respectively, we have n = nh x no, F = Fh ? Fo, and we can define the marginal laws for the hidden and observable components of the network by Q~(Ah) = QA(Ah x Cd) ~ P(H A E A h ) for all Ah E Fh, (3) Q~(Ao) = Q>'(Cn-d Ao) ~ p(O>' E Ao) for all Ao E Fo. (4) For our purposes the appropriate generalization of the notion of a probability density on Rm to the general probability spaces considered here is the Radon-Nikodym derivative with respect to a reference measure that dominates all members of the family {Q>'} >'ERp (Poor, 1994, p.264ff). A suitable reference measure P is the law of the solution to (1) with zero drift (IJ = 0). The measures induced by this reference measure over Fh and Fo are denoted by Ph and Po, respectively. Since in the reference model there are no couplings between any of the nodes in the network, the hidden and observable processes are independent and it follows that P(Ah x Ao) = Ph (Ah)Po(Ao) for all Ah E Fh,Ao E .1'0' (5) The conditions on IJ mentioned above are sufficient to ensure a Radon-Nikodym derivative for each QA with respect to the reference measure. Using Girsanov's Theorem (Karatzas & Shreve, 1991, p.190ff) its form can be shown to be Z'(w) = X ~~ (w) = exp { : ' lT - I'(t,w(t), A) . dw(t) 2~' lT (6) II'(t, w(t), A)I'dt} , wE \1, lIn our treatment we make no distinction between observables which are inputs and those which are outputs. Inputs can be conceptualized as observables under "environmental control," i.e., whose drifts are independent of both A and the hidden and output processes. Learning Path Distributions Using Nonequilibriwn Diffusion Networks 601 where the first integral is an Ito stochastic integral. The random variable Z>. can be interpreted as a likelihood or probability density with respect to the reference mode1 2 . However equation (6) defines the density of Rn-valued paths of the entire network, whereas our real concern is the density of Rd-valued observable paths. Denoting wE 0 as w = (Wh,w o) where Wh E Oh and Wo E 0 0 , note that (7) (8) and therefore the Radon-Nikodym derivative of Q~ with respect to Po, the density of interest, is given by Z;(wo) = ~~: (wo) = EPh[Z>'( ., wo?)' Wo E 0 0 , 2.2 (9) Gradient of the density of an observable path Z; The gradient of with respect to A is an important quantity for iterative optimization of cost functionals corresponding to a variety of problems of interest, e.g., maximum likelihood estimation of diffusion parameters for continuous path density estimation. Formal differentiation 3 of (9) yields \7>.logZ~(wo) = EPh[Z~lo( ? ,wo)\7),logZ'\(.,wo )), (10) >. 6 Z)'(w) Zh lo (w) = Z;(wo) ' (11) where r a Jo \7),logZ)'(w) = -\- T J(t,W(t),A)' dl(w,t), J. ( A) ~ 8J.lk(t, x , A) Jk t,x, 8A ' ' (12) (13) J lew, t) ~ wet) - w(O) -lot J.l(s,w(s), A)ds. (14) Equation (10) states that the gradient of the density of an observable path can be found by clamping the observable nodes to that path and performing an average of Z~lo \7.x log Z)' with respect to Ph, i.e., average with respect to the hidden paths distributed as a scaled Brownian motion. This makes intuitive sense: the output gradient of the log density is a weighted average of the total gradient of the log density, where each hidden path contributes according to its likelihood Z~lo given the output. In practice to evaluate the gradient, equation (10) must be approximated. Here we use Monte Carlo techniques, the efficiency of which can be improved by sampling according to a density which reduces the variance of the integrand. Such a density 2To ease interpretation of (6) consider the simpler case of a one-dimensional Gaussian random variable with mean JL and variance (J"2. The ratio of the density of such a model with respect to an equivalent model with zero mean is exp(-f<JLx - ~JL2). Equation (6) can be viewed as a generalization of this same idea to Brownian motion. 3S ee (Levanony et aL, 1990) for sufficient conditions for the differentiation in equation (10) to be valid. 602 P. Mineiro, J. Movellan and R. 1. Williams is available for models with hidden dynamics which do not explicitly depend upon the observables, i.e., the observable nodes do not send feedback connections to the hidden states. Models which obey this constraint are henceforth denoted factorial. Denoting J-Lh and J-Lo as the hidden and observable components, respectively, of the drift vector, and Bh and Bo as the hidden and observable components, respectively, of the Brownian motion, for a factorial network we have dH(t) = J-Lh(t, H(t), >')dt + adBh(t), (15) (16) dO(t) = J-Lo(t, H(t), O(t), >')dt + adBo(t). The drift for the hidden variables does not depend on the observables, and Girsanov's theorem gives us an explicit formula for the density of the hidden process. iT Z~(Wh) = dQ>' dP: (Wh) = exp { a12 0 J-Lh(t, Wh(t), >.) . dwh(t) 2~2 /,T Il'h(t,wh(t), ,\)I'dt} . - (17) Equations (9) and (10) can then be written in the form Z;(wo) = EQ~ [Zolh(', wo)], (18) 1 >. v\ log Zo>. (wo) = E Q).h [Z;lh("WO) Z;(wo) yo >.log Z (', wo) , where Z;lh(W) Z>'(w) = Z~(Wh) = exp { II 1 a2 iT 0 - (19) J-Lo(t,w(t), >.) . dwo(t) 2~' /,T lI'o(t, w( t), ,\) I' dt } (20) . Note the expectations are now performed using the measure Q~. We can easily generate samples according to Q~ by numerically integrating equation (15), and in practice this leads to more efficient Monte Carlo approximations of the likelihood and gradient. 3 Example: Noisy Sinusoidal Detection This problem is a simple example of using diffusion networks for signal detection. The task was to detect a sinusoid in the presence of additive Gaussian noise. Stimuli were generated according to the following process Y(t,w) = 1A(w).!.sin(47l't) 7l' + B(t,w), (21) where t E [0,1/2]. Here Y is assumed anchored in a probability space (0, F, P) large enough to accommodate the event A which indicates a signal or noise trial. Note that under P, B is a Brownian motion on Cd independent of A. A model was optimized using 100 samples of equation (21) given W E A, i.e., 100 stimuli containing a signal. The model had four hidden units and one observable unit (n = 5, d = 1). The drift of the model was given by J-L(t, x, >.) = () + W . g(x), gj(x) = 1+ 1 e-Zj , j E {I, 2, 3, 4, 5}, (22) Learning Path Distributions Using Nonequilibriwn Diffusion Networks 603 ROC Curve, Sinewave Detection Problem ...?.---A----~----~---?~?? ??-?T-.- ..-. d'=1.89 0.8 ---.---- 1" ....... l" ! 0.6 ,I t 0.4 rf ~ 0.2 o L -_ _ _ _ o ~ __ ~~ 0.2 __ ~ 0.4 ____ 0.6 ~ ____ ~ 0.8 hit rate Figure 2: Receiver operating characteristic (ROC) curve for a diffusion network performing a signal detection task involving noisy sinusoids, Dotted line: Detection performance estimated numerically using 10000 novel stimuli. Solid line: Best fit curve corresponding to d' = 1.89. This value of d' corresponds to performance within 1.5% of the Bayesian limit. where 0 E R5 and W is a 5x5 real-valued connection matrix. In this case ~ = {{Oi}, {Wij }, i,j = 1, ... ,5}. The connections from output to hidden units were set to zero, allowing use of the more efficient techniques for factorial networks described above. The initial distribution for the model was a 8-function at (1, -1, 1, -1,0). The model was numerically simulated with ilt = 0.01, and 100 hidden samples were used to approximate the likelihood and gradient of the log- likelihood, according to equations (18) and (19). The conjugate gradient algorithm was used for training, with the log-likelihood of the data as the cost function. Once training was complete, the parameter estimation was tested using 10000 novel stimuli and the following procedure. Given a new stimuli y we used the model to estimate the likelihood Zo(Y I A) ~ Z~(Y), where ~ is the parameter vector at the end of training. The decision rule employed was D(Y) = {sig.nal nOIse if Zo(~ I A) otherwIse, > b, (23) where b E R is a bias term representing assumptions about the apriori probability of a signal trial. By sweeping across different values of b the receiver-operator characteristic (ROC) curve is generated. This curve shows how the probability of a hit, P(D = signal I A), and the probability of a false alarm, P(D = signal I AC), are related. From this curve the parameter d', a measure of sensitivity independent of apriori assumptions, can be estimated. Figure 2 shows the ROC curve as found by numerical simulation, and the curve obtained by the best fit value d' = 1.89. This value of d' corresponds to a 82.7% correct detection rate for equal prior signal probabilities. The theoretically ideal observer can be derived for this problem, since the profile of the unperturbed signal is known exactly (Poor, 1994, p. 278ff). For this problem the optimal observer achieves d'max = 2, which implies at equal probabilities for signal and noise trials, the Bayesian limit corresponds to a 84.1 % correct detection rate. The detection system based upon the diffusion network is therefore operating close to the Bayesian limit, but was designed using only implicit information, i.e., 100 training examples, about the structure of the signal to be detected, in contrast to the explicit information required to design the optimal Bayesian classifier. 

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Neural Basis of Object-Centered Representations Sophie Deneve and Alexandre Pouget Georgetown Institute for Computational and Cognitive Sciences Georgetown University Washington, DC 20007-2197 sophie, alex@giccs.georgetown.edu Abstract We present a neural model that can perform eye movements to a particular side of an object regardless of the position and orientation of the object in space, a generalization of a task which has been recently used by Olson and Gettner [4] to investigate the neural structure of object-centered representations. Our model uses an intermediate representation in which units have oculocentric receptive fields- just like collicular neurons- whose gain is modulated by the side of the object to which the movement is directed, as well as the orientation of the object. We show that these gain modulations are consistent with Olson and Gettner's single cell recordings in the supplementary eye field. This demonstrates that it is possible to perform an object-centered task without a representation involving an object-centered map, viz., without neurons whose receptive fields are defined in object-centered coordinates. We also show that the same approach can account for object-centered neglect, a situation in which patients with a right parietal lesion neglect the left side of objects regardless of the orientation of the objects. Several authors have argued that tasks such as object recognition [3] and manipulation [4] are easier to perform if the object is represented in object-centered coordinates, a representation in which the subparts of the object are encoded with respect to a frame of reference centered on the object. Compelling evidence for the existence of such representations in the cortex comes from experiments on hemineglect- a neurological syndrome resulting from unilateral lesions of the parietal cortex such that a right lesion, for example, leads patients to ignore stimuli located on the left side of their egocentric space. Recently, Driver et al. (1994) showed that the deficit can also be object-centered. Hence, hemineglect patients can detect a gap in the upper edge of a triangle when this gap is associated with the right side of the object 25 Neural Basis of Object-Centered Representations A. B. Obje~tnt.recl cu?"'S ./ Spotiol cueing I I !1 e ? : 1 ' I ,, !'2 e1 2 C. I Three possible locations: " eI ;.~ , ""~ ; 1 I e I, .1 I I 3 1 [4 I I ! eI! 1! ? ! i 1 l5/ . ? Time Figure 1: A- Driver et al. (1994) experiment demonstrating object-centered neglect. Subjects were asked to detect a gap in the upper part of the middle triangle, while fixating at the cross, when the overall figure is tilted clockwise (top) or counterclockwise (bottom). Patients perform worse for the clockwise condition, when the gap is perceived to be on the left of the overall figure. B- Sequence of screens presented on each trial in Olson and Gettner experiment (1995). 1- Fixation, 2- apparition of a cue indicating where the saccade should go, either in object-centered coordinates (object-centered cueing), or in screen coordinates (spatial cueing), 3- delay period, 4- apparition of the bar in one of three possible locations (dotted lines), and 5- saccade to the cued location. C- Schematic response of an SEF neuron for 4 different conditions. Adapted from [4]. but not when it belongs to the left side (figure I-A). What could be the neural basis of these object-centered representations? The simplest scheme would involve neurons with receptIve fields defined in object-centered coordinates, i.e., the cells respond to a particular side of an object regardless of the position and orientation of the object. A recent experiment by Olson and Gettner (1995) supports this possibility. They recorded the activity of neurons in the supplementary eye field (SEF) while the monkey was performing object-directed saccades. The task consisted of making a saccade to the right or left side of a bar, independently of the position of the bar on the screen and according to the instruction provided by a visual cue. For instance, the cue corresponding to the instruction 'Go to the right side of the bar' was provided by highlighting the right side of a small bar briefly flashed at the beginning of the trial (step 2 in figure I-B). By changing the position of the object on the screen, it is possible to compare the activity of a neuron for movements involving different saccade directions but directed to the same side of the object, and vice-versa, for movements involving the same saccade direction but directed to opposite sides of the object. Olson and Gettner found that many neurons responded more prior to saccades directed to a particular side of the bar, independently of the direction of the saccades. For example, some neurons responded more for an upward right saccade directed to the left side of the bar but not at all for the same upward right saccade directed to the right side of the bar (column 1 and 3, figure I-C). This would suggest that these neurons have bar-centered receptive! fields, i.e., their receptive fields are centered lwe use the term receptive field in a general sense, meaning either receptive or motor s. Deneve andA. Pouget 26 ... "Do E -;:" i~!: ?c o~ ~. ~ mop Figure 2: Schematic structure of the network with activity patterns in response to the horizontal bar shown in the VI map and the command 'Go to the right'. Only one SEF map is active in this case, the one selective to the right edge of the bar (where right is defined in retinal coordinates), object orientation of 0? and the command 'Go to the right'. The letter a, b, c and d indicate which map would be active for the same command but for various orientations of the object, respectively, 0?, 90?, 180?, 270?. The dotted lines on the maps indicate the outline of the bar. Only a few representative connections are shown. on the bar and not on the retina. This would correspond to what we will call an explicit object-centered representation. We argue in this paper that these data are compatible with a different type of representation which is more suitable for the task performed by the monkey. We describe a neural network which can perform a saccade to the right, or left, boundary of an object, regardless of its orientation, position or size- a generalization of the task used by Olson and Gettner. This network uses units with receptive fields defined in oculocentric coordinates, i.e., they are selective for the direction and amplitude of saccades with respect to the fixation point, just like collicular neurons. These tuning curves, however, are also modulated by two types of signals, the orientation of the object, and the command indicating the side of the object to which the saccade should be directed. We show that these response properties are compatible with the Olson and Gettner data and provide predictions for future experiments. We also show that a simulated lesion leads to object-centered neglect as observed by Driver et al. (1994). 1 Network Architecture The network performs a mapping from the image of the bar and the command (indicating the side of the object to which the saccade must be directed) to the appropriate motor command in oculocentric coordinates (the kind of command observed in the frontal eye field, FEF). We use a bar whose left and right sides are defined with respect to a a triangle appearing on the top of the bar (see figure 2). The network is composed of four parts. The first two parts of the network models the field. Neural Basis of Object-Centered Representations 27 lower areas in visual cortex, where visual features are segmented within retinotopic maps. In the first layer, the image is projected on a very simple VI-like map (10 by 10 neurons with activity equal to one if a visual feature appears within their receptive field, and zero otherwise). The second part on the network contains 4 different V2 retinotopic maps, responding respectively to the right, left, top and bottom boundary of the bar. This model of V2 is intended to reproduce the response properties of a subset of cells recently discovered by Zhou et al. (1996). These cells respond to oriented edges, like VI cells, but when the edge belongs to a closed figure, they also show a selectivity for the side on which the figure appears with respect to the edge. For example, a cell might respond to a vertical edge only if this edge is on the right side of the figure but not on the left (where right and left are defined with respect to the viewer, not the object itself). This was observed for any orientation of the edge, but we limit ourselves in this model to horizontal and vertical ones. The third part of the network models the SEF and is divided into 4 groups of 4 maps, each group receiving connections from the corresponding map in V2 (figure 2). Within each group of maps, visual activity is modulated by signals related to the orientation of the object (assumed to be computed in temporal cortex) such that each of the 4 maps respond best for one particular orientation (respectively 0?, 90?, 180? and 270?). For example, a neuron in the second map of the top group responds maximally if: 1- there is an edge in its receptive field and the figure is below, and 2- the object has an orientation of 90? counterclockwise. Note that this situation arises only if the left side of the object appears in the cell's receptive field; it will never occur for the right side. However, the cell is only partially selective to the left side of the object, e.g., it does not respond when the left side is in the retinal receptive field and the orientation of the object is 270? counterclockwise. These collection of responses can be used to generate an object-centered saccade by selecting the maps which are partially selective for the side of the object specified by the command. This is implemented in our network by modulating the SEF maps by signals related to the command. For example, the unit encoding 'go to the right' send a positive weight to any map compatible with the right side of the object while inhibiting the other maps (figure 2). Therefore, the activity, Bfj' of a neuron at position ij on the map k in the SEF is the product of three functIOns: where Vij is the visual receptive field from the V2 map, h((J) is a gaussian function of orientation centered on the cell preferred orientation, (Jk, and gk(C) is the modulation by the command unit. Olson and Gettner also used a condition with spatial cueing, viz., the command was provided by a spatial cue indicating where the saccade should go, as opposed to an object-centered instruction (see figure I-B). We modeled this condition by simply multiplying the activity of neurons coding for this location in all the SEF maps by a fixed constant (10 in the simulations presented here). We also assume that there is no modulation by orientation of the object since this information is irrelevant in this experimental condition. Finally, the fourth part of the network consists of an oculocentric map similar to the one found in the frontal eye field (FEF) or superior colliculus (SC) in which the command for the saccade is generated in oculocentric coordinates. The activity in the output map, {Oij}, is obtained by simply summing the activities of all the S. Deneve and A. Pouget 28 ... ...,-G: ___... , I 06L __ _ ..t.. I I .. ; ..t.. .6J [!1] B. Figure 3: A- Polar plots showing the selectivity for saccade direction of a representative SEF units. The first three plots are for various combinations of command (L: left, R: right, P: spatial cueing) and object orientation. The left plot corresponds to saccades to a single dot. B- Data for the same unit but for a subset of the conditions. The first four columns can be directly compared to the experimental data plotted in figure 1-C from Olson and Gettner. The 5th and 6th columns show responses when the object is inverted. The seventh columns corresponds to spatial cueing. SEF maps. The result is typically a broad two dimensional bell-shaped pattern of activity from which one can read out the horizontal and vertical components, Xs and Ys, of the intended saccade by applying a center-of-mass operator. (1) 2 Results This network is able to generate a saccade to the right or to left of a bar, whatever its position, size and orientation. This architecture is basically like a radial basis function network, i.e., a look-up table with broad tuning curves allowing for interpolation. Consequently, one or two of the SEF maps light up at the appropriate location for any combination of the command and, position and orientation of the bar. Neurons in the SEF maps have the following property: 1- they have an invariant tuning curve for the direction (and amplitude) of saccadic eye movements in oculacentric coordinates, just like neurons in the FEF or the SC, and 2- the gain of this tuning curve is modulated by the orientation of the object as well as the command. Figure 3-A shows how the tuning curve for saccade direction of one particular unit varies as a function of these two variables. In this particular case, the cell responds best to a right-upward saccade directed to the left side of the object when the object is horizontal. Therefore, the SEF units in our model do not have an invariant receptive field in object-centered coordinates but, nevertheless, the gain modulation is sufficient to perform the object-centered saccade. We predict that similar response properties should be found in the SEF, and perhaps parietal cortex, of a monkey trained on a task analogous to the one described here. Since Olson and Gettner (1995) tested only three positions of the bar and held its Neural Basis of Object-Centered Representations 29 orientation constant, we cannot determine from their data whether SEF neurons are gain modulated in the way we just described. However, they found that all the SEF neurons had oculocentric receptive fields when tested on saccades to a single dot (personal communication), an observation which is consistent with our hypothesis (see fourth plot in figure 3-A) while being difficult to reconcile with an explicit object-centered representation. Second, if we sample our data for the conditions used by Olson and Gettner, we find that our units behave like real SEF cells. The first four columns of figure 3-B shows a unit with the same response properties as the cell represented in figure I-C. Figure 3-B also shows the response of the same unit when the object is upside down and when we use a spatial cue. Note that this unit responds to the left side of the bar when the object is upright (1st column), but not when it is rotated by 1800 (5th column), unless we used a spatial cue (7th column). The absence of response for the inverted object is due to the selectivity of the cell to orientation. The cell nevertheless responds in the spatial cueing conditions because we have assumed that orientation does not modulate the activity of the units in this case, since it is irrelevant to the task. Therefore, the gain modulation observed in our units is consistent with available experimental data and makes predictions for future experiments. 3 Simulation of Neglect Our representational scheme can account for neglect if the parietal cortex contains gain modulated cells like the ones we have described and if each cortical hemisphere contains more units selective for the contralateral side of space. This is known to be the case for the retinal input; hence most cells in the left hemisphere have their receptive field on the right hemiretina. We propose that the left hemisphere also over-represents the right side of objects and vice-versa (where right is defined in object-centered coordinates). Recall that the SEF maps in our model are partially selective for the side of objects. A hemispheric preference for the contralateral side of objects could therefore be achieved by having all the maps responding to the left side of objects in the right hemisphere. Clearly, in this case, a right lesion would lead to left object-centered neglect; our network would no longer be able to perform a saccade to the left side of an object. IT we add the retinal gradient and make the previous gradient not quite as binary, then we predict that a left lesion leads to a syndrome in which the network has difficulty with sac cades to the left side of an object but more so if the object is shown in the left hemiretina. Preliminary data from Olson and Gettner (personal communication) are compatible with this prediction. The same model can also account for Driver et al. (1994) experiment depicted in figure I-A. If the hemispheric gradients are as we propose, a right parietal lesion would lead to a situation in which the overall activity associated with the gap, i.e., the summed activity of all the neurons responding to this retinal location, is greater when the object is rotated counterclockwise- the condition in which the gap is perceived as belonging to the right side of the object- than in the clockwise condition. This activity difference, which can be thought as being a difference in the saliency of the upper edge of the triangle, may be sufficient to account for patients' performance. Note that object-centered neglect should be observed only if the orientation of the object is taken into consideration by the SEF units. If the experimental conditions S. Deneve and A. Pouget 30 are such that the orientation of the object can be ignored by the subject -a situation similar to the spatial cueing condition modeled here- we do not expect to observe neglect. This may explain why several groups (such as Farah et al., 1990) have failed to find object-centered neglect even though they used a paradigm similar to Driver et al. (1994). 4 Discussion We have demonstrated how object-centered saccades can be performed using neurons with oculocentric receptive fields, gain modulated by the orientation of the object and the command. The same representational scheme can also account for object-centered neglect without invoking an explicit object-centered representation, i.e., representation in which neurons' receptive fields are defined in object-centered coordinates. The gain modulation by the command is consistent with the single cell data available [4], but the modulation by the orientation of the object is a prediction for future experiments. Whether explicit object-centered representations exist, remains an empirical issue. In some cases, such representations would be computationally inefficient. In the Olson and Gettner experiment, for instance, having a stage in which motor commands are specified in object-centered coordinates does not simplify the task. Encoding the motor command in object-centered coordinates in the intermediate stage of processing requires (i) recoding the sensory input into object-centered coordinates, (ii) decoding the object-centered command into an oculocentric command, which is ultimately what the oculomotor system needs to ,generate the appropriate saccade. Each of these steps are computationally as complex as performing the overall transformation directly as we have done in this paper. Therefore, gain modulation provides a simple algorithm for performing objectcentered saccades. Interestingly, the same basic mechanism underlies spatial representations in other frames of reference, such as head-centered and body-centered. We have shown previously that these responses can be formalized as being basis functions of their sensory and postures inputs, a set of function which is particularly useful for sensory-motor transformations [5]. The same result applies to the SEF neurons considered in this paper, suggesting that basis functions may provide a unified theory of spatial representations in any spatial frame of reference. References [1] J. Driver, G. Baylis, S. Goodrich, and R. Rafal. Axis-based neglect of visual shapes. Neuropsychologia, 32{11}:1353-1365, 1994. [2] M. Farah, J. Brunn, A. Wong, M. Wallace, and P. Carpenter. Frames of reference for allocating attention to space: evidence from the neglect syndrome. Neuropsychologia, 28(4):335-47, 1990. [3] G. Hinton. Mapping part-whole hierarchies into connectionist networks. Artificial Intelligence, 46(1}:47-76, 1990. [4] C. Olson and S. Gettner. Object-centered direction selectivity in the macaque supplementary eye. Science, 269:985-988, 1995. [5] A. Pouget and T. Sejnowski. Spatial transformations in the parietal cortex using basis functions. Journal of Cognitive Neuroscience, 9(2):222-237, 1997. [6] H. Zhou, H. Friedman, and R. von der Heydt. Edge selective cells code for figure-ground in area V2 of monkey visual cortex. In Society For Neuroscience Abstracts, volume 22, page 160.1, 1996. 

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662 A PASSIVE SHARED ELEMENT ANALOG ELECTRICAL COCHLEA Joe Eisenberg Bioeng. Group U.C. Berkeley David Feld Dept. Elect. Eng. 207-30 Cory Hall U.C. Berkeley Berkeley, CA. 94720 Edwin Lewis Dept Elect. Eng. U.C. Berkeley ABSTRACT We present a simplified model of the micromechanics of the human cochlea, realized with electrical elements. Simulation of the model shows that it retains four signal processing features whose importance we argue on the basis of engineering logic and evolutionary evidence. Furthermore, just as the cochlea does, the model achieves massively parallel signal processing in a structurally economic way, by means of shared elements. By extracting what we believe are the five essential features of the cochlea, we hope to design a useful front-end filter to process acoustic images and to obtain a better understanding of the auditory system. INTRODUCTION Results of psychoacoustical and physiological experiments in humans indicate that the auditory system creates acoustic images via massively parallel neural computations. These computations enable the brain to perform voice detection, sound localization, and many other complex taskS. For example, by recording a random signal with a wide range of frequency components, and playing this signal simultaneously through both channels of a stereo headset, one causes the brain to create an acoustical image of a "shsh" sound in the center of the head. Delaying the presentation of just one frequency component in the random signal going to one ear and simultaneously playing the original signal to the other ear, one would still have the image of a "shsh" in the center of the head; however if one mentally searches the acoustical image space carefully, a clear tone can be found far off to one side of the head. The frequency of this tone will be that of the component with the time delay to one ear. Both ears still are receiving wide-band random signals. The isolated tone will not be perceptible from the signal to either ear alone; but with both signals together, the brain has enough data to isolate the delayed tone in an acoustical image. The brain achieves this by massively parallel neural computation. Because the acoustic front-end fIlter for the brain is the cochlea, people have proposed that analogs of the cochlea might serve well as front-end fIlters for man-made processors of acoustical images (Lyon, Mead, 1988). If we were to base a cochlear analog on current biophysical models of this structure, it would be extraordinarily complicated and extremely difficult to realize with hardware. Because of this, we want to start with a cochlear model that incorporates a minimum set of essential ingredients. The ears of lower vertebrates, such as alligators and frogs, provide some clues to help identify those ingredients. These animals presumably have to compute acoustic images similar to ours, A Passive Shared Element Analog Electrical Cochlea but they do not have cochleas. The acoustic front-end filters in the ears of these animals evolved independently and in parallel to the evolution of the cochlea. Nevertheless, those front-end filters share four functional properties with the part of the cochlea which responds to the lower 7 out of 10 octaves of hearing (20 Hz. to 2560 Hz.): 1. 2. 3. 4. They are multichannel ftlters with each channel covering a different part of the frequency spectrum. Each channel is a relatively broad-band frequency filter. Each filter has an extremely steep high-frequency band edge (typically 60 to 200 db/oct). Each filter has nearly linear phase shift as a function of frequency, within its passband. The front-end acoustical filters of lower vertebrates also have at least one structural feature in common with the cochlea: namely, the various filter channels share their dynamic components. This is the fifth property we choose to include. Properties 1 and 3 provide good resolution in frequency; properties 2 and 4 are what filter designers would add to provide good resolution in time. In order to compute acoustical images with the neural networks in our brain, we need both kinds of resolution: time and frequency. Shared elements, a structural feature, has obvious advantages for economy of construction. The fact that evolution has come to these same front-end filter properties repeatedly suggests that these properties have compelling advantages with respect to an animal's survival. We submit that we can realize all of these properties very well with the simplest of the modern cochlear models, namely that of Joseph Zwislocki (1965). This is a transmission line model made entirely of passive elements. Figure 1 - Drawing of the ear with the cochlea represented by an electrical analog. 663 664 Feld, Eisenberg and Lewis A COCHLEAR MODEL In order to illustrate Zwislocki's model, a quick review of the mechanics of the cochlea is useful. Figure 1 depicts the ear with the cochlea represented by an electrical analog. A sound pressure wave enters the outer ear and strikes the ear drum which, in turn, causes the three bones of the middle ear to vibrate. The last bone, known as the stapes, is connected to the oval window (represented in figure 1 by the voltage source at the beginning of the electrical analog), where the acoustic energy enters the cochlea. As the acoustic energy is transferred to the oval window, a fluid-mechanical wave is formed along a structure known as the basilar membrane ([his membrane and the surrounding fluid is represented by the series and shunt circuit elements of figure 1). As the basilar membrane vibrates, the acoustical signal is transduced to neural impulses which travel along the auditory nerve, carrying the data used by the nervous system to compute auditory images. Figure 2 is taken from a paper by Zweig et al. (1976), and depicts an uncoiled cochlea. As the fluid-mechanical wave travels through the cochlea: 1) The wave gradually slows down, and 2) The higher-frequency components of the wave are absorbed, leaving an increasingly narrower band of low-frequency components proceeding on toward the far end of the cochlea. If we were to uncoil and enlarge the basilar membrane it would look like a swim fin (figure 3). If we now were to push on the basilar membrane, it would push back like a spring. It is most compliant at the wide, thin end of the fin. Thus as one moves along the basilar membrane from its basal to apical end, its compliance increases. Zwislocki's transmission-line model was tapered in this same way. Scala vestibuli Helicotrema Figure 2 - Uncoiled cochlea (Zweig. 1976). A Passive Shared Element Analog Electrical Cochlea Basal End Figure 3 - Simplified uncoiled and enlarged drawing of the basilar membrane. Zwislocki's model of the cochlea is a distributed parameter transmission line. Figure 4 shows a lumped electrical analog of the model. The series elements (L 1,,, .Lo) represent the local inertia of the cochlear fluid. The shunt capacitive elements (CI,...Co) represent the local compliance of the basilar membrane. The shunt resistive elements (R 1,...Rn) represent the local viscous resistance of the basilar membrane and associated fluid. The model has one input and a huge 'number of outputs. The input, sound pressure at the oval window, is represented here as a voltage source. The outputs are either the displacements or the velocities of the various regions of the basilar membrane. Figure 4 - Transmission line model of the cochlea represented as an electrical circuit. In the electrical analog, shown in figure 4, we have selected velocities as the outputs (in order to compare our data to real neural tuning curves) and we have represented those velocities as the currents (11 , .. .In ). The original analysis of Zwislocki's tapered transmission line model did not produce the steep high frequency band edges that are observed in real cochleas. This deficiency was a major driving force behind the early development of more complex cochlear models. Recently, it was found that the original analysis placed the Zwislocki model in an inappropriate mode of operation (Lewis, 1984). In this mode,determined by the relative parameter values, the high frequency band edges had very gentle slopes. Solving the partial differential equations for the model with the help of a commonly used simplification (the WKB approximation), one finds a second mode of operation. In this mode, the model produces all five of the properties that we desire, including extraordinarily steep high-frequency band edges. 665 666 Feld, Eisenberg and Lewis RESULTS We were curious to know whether or not the newly-found mode of operation, with its very steep high-frequency band edges, could be found in a finite-element version of the model. If so, we should be able to realize a lumped, analog version of the Zwislocki model for use as a practical front-end filter for acoustical image formation and processing. We decided to implement the finite element model in SPICE. SPICE is a software package that is used for electrical circuit simulation. Our SPICE model showed the following: As long as the model was made up of enough segments, and as long as the elements had appropriate parameter values, the second mode of operation indeed was available. Furthermore, it was the predominant mode of operation when the parameter values of the model were matched to biophysical data for the cochlea. CD :3 L\.l :z: < Q: Q:I "~ ::E ~ u Z ~ ~ < 1-0 ~ ,. CI) 100000 L OG (FREQUENC Y) ]/\ I 1.1 " , " ,,,I U I I ", ",I " , ... FREQUENCY (kHz) Figure 5 - Frequency response of Figure 6 - Inverted neural tuning curves the basilar membrane velocity. from three afferent fibers of a cat cochlea (Kiang and Moxon, 1974). Figure 5 shows the magnitude of the electrical analog's response plotted against frequency on log-log coordinates. The five curves correspond to five different locations along our model. The cutoff frequencies span approximately seven octaves. Further adjustments of the parameters will be needed in order to shift these curves to span the lower seven octaves of human audition. For comparison, figure 6 shows threshold response curves of a cat cochlea from a paper by Kiang and Moxon (1974). These curves are inverted intentionally because Kiang and Moxon plotted stimulus threshold vs. frequency rather than response amplitude vs. frequency. We use these neural tuning curves for comparison because direct observations of cochlear mechanics have been limited to the basal end. Furthermore, in the realm of single frequencies and small signals, Evans has produced compelling evidence that this is a valid comparison (Evans, in press). These three curves are typical of the lower seven octaves of hearing. One obvious discrepancy between Kiang and Moxon's data and our results is that our model does not exhibit the sharp corners occurring at the band edges. The term sharp corner denotes the fact that the transition between the shallow rising edge and steep falling edge of a given curve is abrupt i.e. the corner is not rounded. A Passive Shared Element Analog Electrical Cochlea Figure 7 shows what happens to the response curve at a single location along our model as the number of stages is increased. The curve on the right, in figure 7, was derived with 500 stages and does not change much as we increase the number of stages indefinitely. Thus the curve represents a convergence of the solution of the lumped parameter Zwislocki model to the distributed parameter model. The middle curve was derived with 100 stages and the left-hand curve was derived with 50. In any lumpedelement transmission line, there occurs an artifactual cutoff which occurs roughly at the point where the given input wavelength exceeds the dimensions of the lumped elements. If we do not lump the stages in our model finely enough, we observe this artifactual cutoff as opposed to the true cutoff of Zwislocki's distributed parameter model. This phenomena is clearly observed in the curve derived from 50 stages and may account for the sharper comers in response curves from real cochleas. However, in order to make our finite element model operate in a manner analogous to that of the distributed parameter Zwislocki model we need approximately 500 stages. ~ c ~ ~ ~ ~ "' "' ." ~ 0 0 ~z ~ . ~ ~ 10 10 0 00 0 L OG !FREQUENCY ) IdB} 10000 0 L OG (FREQUENCY) (HZ) Figure 7 - Convergence of cut- Figure 8 - Frequency response of the toff points as the number of branches increase. basilar membrane velocity without the Heliocotrema. A critical element in the Zwislocki model is a terminating resistor, representing the heliocotrema (see Rh in figure 3). The heliocotrema is a small hole at the end of the basilar membrane. Figure 8, shows the effects of removing that resistor. The irregular frequency characteristics are quite different from the experimental data and represent possibly wild excursions of the basilar membrane. Figure 9. shows phase data for the Zwislocki model, which is linear as a function of frequency. Anderson et aI (1971), show similar results in the squirrel monkey. With lumped-element analysis we are able to obtain temporal as well as spectral responses. For a temporal waveform such as an acoustic pulse, the linear relationship between phase and frequency guarantees that those Fourier components which pass through the spectral filter will be reassembled with proper phase relationships at the output of the filter. As it travels down the basilar membrane. the temporal waveform will simply be smoothed, due to loss of its higher-frequency components and delayed, due to the linear phase shift. Figure 10 shows the response of our electrical analog to a 1 msec wide square pulse at the input. The curves represent the time courses of basilar membrane displacement at four equally spaced locations along the cochlea. The curve on the right represents the response at the apical end of the cochlea (the end farthest from the input). The curve on the left represents the response at a point 25 percent of the distance 667 668 Feld, Eisenberg and Lewis input end. The impulse responses of mammalian cochleas and of the auditory filters of lower vertebrates all show a slight ringing, again indicating a deficiency in our model. w ~ d > Apical End 2000 LINEA.R FREOUENCY {HZ' 04 TIfoilE (SECONDS) Figure 9 - Phase response of the Figure 10 - Traveling square wave basilar membrane velocity. pulse along the membrane from the basal to apical end. CONCLUSION Research activity studying the function of higher level brain processing is in its infancy and little is known about how the various features of the cochlea, such as linear phase, sharp band edges, as well as nonlinear features, such f1S two-tone suppression and cubic difference tone excitation, are used by the brain. Therefore, our approach, in developing a cochlear model, is to incorporate only the most essential ingredients. We have incorporated the five properties mentioned in the introduction which provide simplicity of analysis, economy of hardware construction, and the preservation of both temporal and spectral resolution. The inclusion of these properties is also consistent with the fact that they are found in numerous species. We have found that in the correct mode of operation a tapered transmission line model can exhibit these five important cochlear properties. A lumped-element approximation can be used to simulate this model as long as at least 500 stages are used. As observed in figure 7, by decreasing the number of stages below 500, the solution to the lumped-element model no longer adheres to the Zwislocki model. In fact, the output of the coursely lumped model more closely resembles the neural tuning data of the cochlea in that it produces very sharp corners. There is some evidence that indicates the cochlea is constructed of discrete components. Indeed, the hair cells themselves are discretized. If this idea is valid, a model constructed of as little as 50 branches may more accurately represent the cochlear mechanics then the Zwislocki model. Our simple model has some drawbacks in that it does not replicate various properties of the cochlea. For example, it does not span the full ten octaves of human audition, nor does it explain any of the experimentally observed nonlinear aspects seen in the cochlea. However, we take this approach because it provides us with a powerful analysis tool that will enable us to study the behavior of lumped-element cochlear models. This tool will allow us to proceed to the next step; the building of a hardware analog of the cochlea. A Passive Shared Element Analog Electrical Cochlea allow US to proceed to the next step; the building of a hardware analog of the cochlea. RESEARCH DIRECTIONS In and of itself, the tapered shared element travelling wave structure we have chosen is interesting to analyze. In order to get even further insight into how this filter works and to aid in the building of a hardware version of such a filter, we plan to study the placement of the poles and zeroes of the transfer function at each tap along the structure. In a travelling wave transmission line we expect that the transfer function at each tap will have the same denominator. Therefore, it must be the numerators of the transfer functions which will change, i.e. the zeroes will change from tap to tap. It will be of interest to see what role the zeroes play in such a ladder structure. Furthennore, it will be of great interest to us to study what happens to the poles and zeroes of the transfer function at each tap as the number of stag~s is increased (approaching the distributed parameter ftlter), or decreased (apprpaching the lumped-element cutoff version of the filter with sharper corners). We should emphasize that our circuit is bi-directional, i.e. there is loading from the stages before and after each tap, as in the real cochlea. For this reason, we must consider carefully the options for hardware realization of our circuit. We might choose to make a mechanical structure on silicon or some oilier medium, or we could convert our structure into a uni-directional circuit and build it as a digital or analog circuit Using this design we plan to build an acoustic imaging device that will enable us to explore various signal processing tasks. One such task would be to extract acoustic signals from noise. All species need to cope with two types of noise, internal sensor and amplifier noise, and external noise such as that generated by wind. Spectral decomposition is on effective way to deal with internal noise. For example, the amplitudes of the spectral components in the passband of a filter are largely undiminished, whereas the broadband noise, passed by the filter, is proportional to the square root of the bandwidth. External noise reduction can be accomplished by spatial decomposition. When temporal resolution is preserved in signals, spatial decomposition can be achieved by cross correlation of the signals from two ears. Therefore, from these two properties, spectral and temporal resolution, one can construct an acoustic imaging system in which signals buried in a sea of noise can be extracted. Acknowledgments We would like to thank Thuan Nguyen for figure 1, Eva Poinar who helped with the figures, Michael Sneary for valuable discussion, and Bruce Parnas for help with programming. 669 670 Feld, Eisenberg and Lewis References Anderson, DJ., Rose, I.E., Hind, I.E .? Brugge I.F.. Temporal Position of Discharges in Single Auditory Nerve Fibers Within the Cycle of a Sine-Wave Stimulus: Frequency and Intensity Effects, J. Acoust. Soc. A~z.., 49. 1131-1139, 1971. Evans, E.F.? Cochlear Filtering: A View Seen Through the Temporal Discharge Patterns of Single Cochlear Nerve Fibers. A talk given at the 1988 NATO advanced workshop. to be published as (J.P. Wilson, D.T. Kemp, eds.) Mechanics of Hearing. Plenum Press. N.Y. Kiang, N.Y.S .? Moxon, E.C .? Tails of Tuning Curves of Auditory-Nerve Fibers, J. Acoust. Soc. Am., 55, 620-630. 1974. Lewis. E.R., High Frequency Rolloff in a Cochlear Model Without critical-layer resonance. J. Acoust. Soc. Am.? 76 (3) September, 1984. Lyon, R.F., Mead, C.A.? An Analog Electronic Cochlea, IEEE Trans.-ASSP, 36. 11191134. 1988. Zweig. G., Lipes. R.. Pierce. I.R.. The Cochlear Compromise, J. Acoust. Soc. Am., 59, 975-982. 1976. Zwislocki.I., Analysis of Some Auditory Characteristics, in Handbook of Mathematical Psychology, Vol. 3. (Wiley. New York), pp. 1-97, 1965. 

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Two Approaches to Optimal Annealing Todd K. Leen Dept of Compo Sci. & Engineering Oregon Graduate Institute of Science and Technology P.O.Box 91000, Portland, Oregon 97291-1000 tleen@cse.ogi.edu Bernhard Schottky and David Saad Neural Computing Research Group Dept of Compo Sci. & Appl. Math. Aston University Birmingham, B4 7ET, UK schottba{ saadd}@aston.ac.uk Abstract We employ both master equation and order parameter approaches to analyze the asymptotic dynamics of on-line learning with different learning rate annealing schedules. We examine the relations between the results obtained by the two approaches and obtain new results on the optimal decay coefficients and their dependence on the number of hidden nodes in a two layer architecture. 1 Introduction The asymptotic dynamics of stochastic on-line learning and it's dependence on the annealing schedule adopted for the learning coefficients have been studied for some time in the stochastic approximation literature [1, 2] and more recently in the neural network literature [3, 4, 5]. The latter studies are based on examining the KramersMoyal expansion of the master equation for the weight space probability densities. A different approach, based on the deterministic dynamics of macroscopic quantities called order parameters, has been recently presented [6, 7]. This approach enables one to monitor the evolution of the order parameters and the system performance at all times. In this paper we examine the relation between the two approaches and contrast the results obtained for different learning rate annealing schedules in the asymptotic regime. We employ the order parameter approach to examine the dependence of the dynamics on the number of hidden nodes in a multilayer system. In addition, we report some lesser-known results on non-standard annealing schedules T. K. Leen, B. Schottky and D. Saad 302 2 Master Equation Most on-line learning algorithms assume the form Wt+l = Wt + 1]o/tP H(wt,xt) where Wt is the weight at time t, Xt is the training example, and H(w,x) is the weight update. The description of the algorithm's dynamics in terms of weight space probability densities starts from the master equation P(w',t+1)= JdW (8(w'-w-~~H(w,x)))xP(w,t) (1) where ( .. '}x indicates averaging with respect to the measure on x, P(w,t) is the probability density on weights at time t, and 8( ... ) is the Dirac function. One may use the Kramers-Moyal expansion of Eq.(l) to derive a partial differential equation for the weight probability density (here in one dimension for simplicity) {3, 4] at P (w, t) = t r (~~) i (7~ a~ [ (Hi (w, x) >x P (w, t)] t=l . (2) Following {3], we make a small noise expansion for (2) by decomposing the weight trajectory into a deterministic and stochastic pieces or ~ = ( TJo)-"( tP (w-</>(t)) (3) where </>( t) is the deterministic trajectory, and ~ are the fluctuations. Apart from the factor (1]0/ tP)'Y that scales the fluctuations, this is identical to the formulation for constant learning in {3]. The proper value for the unspecified exponent, will emerge from homogeneity requirements. Next, the dependence of the jump moments (Hi (w, x) > on 1]0 is explicated by a Taylor series expansion about the deterministic path </>. T!{e coefficients in this series expansion are denoted a~i) == ai (Hi(w,x))x /awilw=tI> Finally one rewrites (2) in terms of </> and ~ and the expansion of the jump moments, taking care to transform the differential operators in accordance with (3). These transformations leave equations of motion for </> and the density the fluctuations d</> dt = I1(~, t) on (4) = = For stochastic descent H (w, x) -V w E( w, x) and (4) describes the evolution of </> as descent on the average cost. The fluctuation equation (5) requires further manipulation whose form depends on the context. For the usual case of descent in a quadratic minimum (ail) = -G, minus the cost function curvature), we take ''i 1/2 to insure that for any m, terms in the sum are homogeneous in T}o/t P = For constant learning rate (p = 0), rescaling time as t ~ 1]ot allows (5) to be written in a form convenient for perturbative analysis in 1]0 Typically, the limit 1]0 ~ 0 is invoked and only the lowest order terms in 1]0 retained (e.g. [3]). These comprise a diffusion operator, which results in a Gaussian approximation for equilibrium densities. Higher order terms have been successfully used to calculate corrections to the equilibrium moments in powers of 1]0 [8]. Two Approaches to Optimal Annealing 303 Of primary interest here is the case of annealed learning, as required for convergence of the parameter estimates. Again assuming a quadratic bowl and 'Y = 1/2, the first few terms of (5) are Gt II = As t -+ 00 o < p S 1). :t Gd~II) - all) i; Ge(~II) + ~a~O): Gl II + 0 (: r/2. (6) the right hand side of (6) is dominated by the first three terms (since Precisely which terms dominate depends on p. . We will first review the classical case p = 1. Asymptotically 1> -+ w"', a local optimum. The first three leading terms on the right hand side of (6) are all of order lit. For t -+ 00, we discard the remaining terms. From the resulting equation we recover a Gaussian equilibrium distribution for ~, or equivalently for Vt (w - w"') == Vtv where v is called the weight error. The asymptotically normal distribution for Vtv has variance 0'0v from which the asymptotic expected squared weight error can be derived 2 (0) 1 ? E[I v 12] - 0 ' 2 - - 7Jo a 2 1 11m (7) t->oo -,;tv t 27]0 G'" - 1 t == G(w"') is the curvature at the local optimum. Positive O',;t v requires T]o > 11 (2G"'). If this condition is not met the expected where G'" squared weight offset converges as (1It)1-2'1oG', slower than lit [5, for example, and references therein]. The above confirms the classical results [1] on asymptotic normality and convergence rate for 1I t annealing. For the case 0 < p < 1, the second and third terms on the right hand side of (6) will dominate as t -+ 00. Again, we have a Gaussian equilibrium density for ~. Consequently ViP v is asymptotically normal with variance O'~v leading to the expected squared weight error 0' 2 r;-;; ytPv - 1 = tP T]o a~O) 2G .!.. tP (8) Notice that the convergence is slower than lit and that there is no critical value of the learning rate to obtain a sensible equilibrium distribution. (See [9] for earlier results on 11tP annealing.) The generalization error follows the same decay rate as the expected weight offset. In one dimension, the expected squared weight offset is directly related to excess generalization error (the generalization error minus the least generalization error achievable) Eg = G E[v 2 ]. In multiple dimensions, the expected squared weight offset, together with the maximum and minimum eigenvalues of G'" provide upper and lower bounds on the excess generalization error proportional to E[lvI 2 ], with the criticality condition on G'" (for p = 1 )replaced with an analogous condition on its eigenvalues. 3 Order parameters In the Master equation approach, one focuses attention on the weight space distribution P( w, t) and calculates quantities of interested by averaging over this density. An alternative approach is to choose a smaller set of macroscopic variables that are sufficient for describing principal properties of the system such as the generalization error (in contrast to the evolution of the weights w which are microscopic). T. K. Leen, B. Schottky and D. Saad 304 Formally, one can replace the parameter dynamics presented in Eq.(1) by the corresponding equation for macroscopic observables which can be easily derived from the corresponding expressions for w. By choosing an appropriate set of macroscopic variables and invoking the thermodynamic limit (i.e., looking at systems where the number of parameters is infinite), one obtains point distributions for the order parameters, rendering the dynamics deterministic. Several researchers [6, 7] have employed this approach for calculating the tr~ing dynamics of a soft committee machine (SCM) . The SCM maps inputs x E RN to a scalar, through a model p{w,x) 2:~lg{Wi' x). The activation function of the hidden units is g{u) == erf{u/V2) and Wi is the set of input-to-hidden adaptive weights for the i = 1 ... K hidden nodes. The hidden-to-output weights are set to 1. This architecture preserves most of the properties of the learning dynamics and the evolution of the generalization error as a general two-layer network, and the formalism can be easily extended to accommodate adaptive hidden-to-output weights [10]. = Input vectors x are independently drawn with zero mean and unit variance, and the corresponding targets y are generated by deterministic teacher network corrupted by additive Gaussian output noise of zero mean and variance O'~. The teacher network is also a SCM, with input-to-hidden weights wi. The order parameters sufficient to close the dynamics, and to describe the network generalization error are overlaps between various input-to-hidden vectors Wi . Wk == Qik, Wi' W~ _ Rin, and w~? w~ == Tnm . Network performance is measured in terms of the generalization error Eg{W) _ (1/2 [ p(w, x) - Y ]2)~. The generalization error can be expressed in closed form in terms of the order parameters in the thermodynamiclimit (N -+ 00). The dynamics of the latter are also obtained in closed form [7]. These dynamics are coupled nonlinear ordinary differential equations whose solution can only be obtained through numerical integration. However, the asymptotic behavior in the case of annealed learning is amenable to analysis, and this is one of the primary results of the paper. We assume an isotropic teacher Tnm = 8 nm and use this symmetry to reduce the system to a vector of four order parameters uT = (r, q, s, c) related to the overlaps by Rin = 8in (1 + r) + (1- 8in)S and Qik = 8 i k(1 + q) + {1- 8 ik )C. ? With learning rate annealing and limt-+oo u = we describe the dynamics in this vicinity by a linearization of the equations of motion in [7]. The linearization is d dt U where O'~ is the noise variance, b T -4 M = 2 3V31T 4 3 --V3 2 3V3 3 = rJ 1\d U + rJ 2 0' 2b , (9) /I = ~ (0,1/,;3,0,1/2), rJ = rJo/tP, -~(K -lhI(3) 4 3 -(K - 1)V3 3 2 2 2 3 -(K -1)V3 ~ --(K - 1)V3 2 o --(K-2)+3V3(K - 2) J3 +~ and M is -3V3(K - 2) (10) +~ V3 The asymptotic equations of motion (9) were derived by dropping terms of order O(rJlluI12) and higher, and terms of order O{rJ2 u). While the latter are linear in the order parameters, they are dominated by the rJu and rJ20'~b terms in (9) as t -+ 00. Two Approaches to Optimal Annealing 305 This choice of truncations sheds light on the approach to equilibrium that is not implicit in the master equation approach. In the latter, the dominant terms for the asymptotics of (6) were identified by time scale of the coefficients, there was no identification of system observables that signal when the asymptotic regime is entered. For the order parameter approach, the conditions for validity of the asymptotic approximations are cast in terms of system observables 1JU vs rlu VS 1J2 fI~. The solution to (9) is u(t) = -yet, to) Uo + fI~ j3(t, to) b (11) where Uo == u(to) and -y(t, to) = exp {M lot dr 1J(r)} and j3(t, to) = t dr -yet, r) 1J2( r). lto (12) The asymptotic order parameter dynamics allow us to compute the generalization error (to first order in u) E/ K(l K-1 = -:; J3(q- 2r) + -2-(C28) ) . (13) Using the solution of Eq.(l1), the generalization error consists of two pieces: a contribution depending on the actual initial conditions Uo and a contribution due to the second term on the r.h.s. of Eq.(l1), independent of Uo. The former decays more rapidly than the latter, and we ignore it in what follows. Asymptotically, the generalization error is of the form E/ = fI;(CI0l(t) + C202(t)), where Cj are K dependent coefficients, and OJ are eigenmodes that evolve as OJ =- 1J5 1 + (}:j1Jo [!t _ to'it) Ot;;-(O'it)O+I)] . (14) = -~ (~ +2(K -1)) (15) with eigenvalues (Fig. l(a)) (}:1 = -~ (~ -2) and (}:2 The critical learning rate 1J~rit, above which the generalization decays as lit is, for K> - 2, crit 7r (16) 110 = max - (}:1 , - (}:2 41 J3 - 2 . (1 1) = For 1Jo > 1J~rit E/ both modes OJ, i = 1,2 decay as lit, and so C2) 1 = -fI 2 11o2 (Cl 1 + (}:11JO + 1 + (}:21Jo t 1I 2 ( == fIll f 1Jo,K) t1 (17) Minimizing the prefactor f (1Jo, K) in (17) minimizes the asymptotic error. The values 1J~Pt (K) are shown in Fig. 1(b), where the special case of K = 1 (see below) is also included: There is a significant difference between the values for K = 1 and K = 2 and a rather weak dependence on K for K ~ 2. The sensitivity of the generalization error decay factor on the choice of 1Jo is shown in Fig. 1 (c). The influence of the noise strength on the generalization error can be seen directly from (17): the noise variance fI; is just a prefactor scaling the lit decay. Neither the value for the critical nor for the optimal1Jo is influenced by it. T. K. Leen, B. Schottky and D. Saad 306 The calculation above holds for the case K = 1 (where c and s and the mode 01 are absent). In this case - 2 crit(R" -- 1) -__ 2. 110opt(K -_ 1) -1]0 <l:2 _ J311" . - 2 (18) Finally, for the general annealing schedule of the form 1] = 1]oltP with 0 < p < 1 the equations of motion (11) can be investigated, and one again finds 11tP decay. 4 Discussion and summary We employed master equation and order parameter approaches to study the convergence of on-line learning under different annealing schedules. For the lit annealing schedule, the small noise expansion provides a critical value of 1]0 (7) in terms of the curvature, above which Vt v is asymptotically normal, and the generalization decays as lit. The approach is general, but requires knowledge of the first two jump moments in the asymptotic regime for calculating tl~e relevant properties. By restricting the order parameters approach to a symmetric task characterized by a set of isotropic teacher vectors, one can explicitly solve the dynamics in the asymptotic regime for any number of hidden nodes, and provide explicit expressions for the decaying generalization error and for the critical (16) and optimal learning rate prefactors for any number of hidden nodes K. Moreover, one can study the sensitivity of the generalization error decay to the choice of this prefactor. Similar results have been obtained for the critical learning rate prefactors using both methods, and both methods have been used to study general11tP annealing. However, the order parameters approach enables one to gain a complete description of the dynamics and additional insight by restricting the task examined. Finally the order parameters approach expresses the dynamics in terms of ordinary differential equations, rather than partial differential equations; a clear advantage for numerical investigations. The order parameter approach provides a potentially helpful insight on the passage into the asymptotic regime. Unlike the truncation of the small noise expansion, the truncation of the order parameter equations to obtain the asymptotic dynamics is couched in terms of system observables (c.f. the discussion following (10)). That is, one knows exactly which observables must be dominant for the system to be in the asymptotic regime. Equivalently, starting from the full equations, the order parameters approach can tell us when the system is close to the equilibrium distribution. Although we obtained a full description of the asymptotic dynamics, it is still unclear how relevant it is in the larger picture which includes all stages of the training process, as in many cases it takes a prohibitively long time for the system to reach the asymptotic regime. It would be interesting to find a way of extending this framework to gain insight into earlier stages of the learning process. Acknowledgements: DS and BS would like to thank the Leverhulme Trust for their support (F J250JK). TL thanks the International Human Frontier Science Program (SF 473-96), and the NSF (ECS-9704094) for their support. Two Approaches to Optimal Annealing 307 I wr----~~~-~~~~ " ~. -2 ..... "'''. -4 ::v .... . . ... -' II;. "." .... ", .... -8 =~~' I 8 . . . . .. 4 6 8 10 12 14 16 18 4 I (b) 20 18 16 -K=I ...... K=2 - - K=3 .. .. K=5 ,. 14 ,-, ::.:: 0 .,f . 12 I\ . .. 10 ~\ ~ ;;::: 8 .... .. .. ... . . . . . \\. \ 6 \..~'----------- 4 ....... ............. 2 0 2 4 6 (c) 8 ..... ....?. OL-~~--~--'-~--~~~~ 20 K (a) ? .. ....? ... . 2 L -_ _....i.---'--....L.~"'--'-.....&--..L.---'''-'?? ........J 2 .. ? 6 ......... '. 10 12 14 16 . ?...... . 14 -'." ". -6 -12 .. .... f(l1O"'. K) 16 " -10 -T/"'" 0 18 " . 2 3 4 5 6 7 8 9 10 K Figure 1: (a) The dependence of the eigenvalues of M on the the number of hidden units K. Note that the constant eigenvalue al dominates the convergence for K ~ 2_ (b) 1]gpt and the resulting generalization prefactor / (1]gpt ,K) of (17) as a function of K_ (c) The dependence of the generalization error decay prefactor / (1]0, K) on the choice of 1]0. 18 t'lo References [1] V. Fabian. Ann. Math . Statist., 39, 1327 1968. [2] L. Goldstein. Technical Report DRB-306, Dept. of Mathematics, University of Southern California, LA, 1987. [3] T. M. Heskes and B. Kappen, Phys. Rev. A 44, 2718 (1991)_ [4] T. K. Leen and J. E. Moody. In Giles, Hanson, and Cowan, editors, Advances in Neural In/ormation Processing Systems, 5, 451 , San Mateo, CA, 1993. Morgan Kaufmann. [5] T. K. Leen and G. B. Orr. In J.D. Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing Systems 6, 477 ,San Francisco, CA., 1994. Morgan Kaufmann Publishers. [6] M. Biehl and H. Schwarze, J. Phys. A 28, 643 (1995). [7] D. Saad and S_A. Solla Phys. Rev. Lett. 74, 4337 (1995) and Phys. Rev. E 52 4225 (1995). [8] G. B. Orr. Dynamics and Algorithms for Stochastic Search. PhD thesis, Oregon Graduate Institute, October 1996. [9] Naama Barkai. Statistical Mechanics of Learning_ PhD thesis, Hebrew University of Jerusalem, August 1995. [10] P. Riegler and M. Biehl J_ Phys. A 28, L507 (1995). 

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On the infeasibility of training neural networks with small squared errors Van H. Vu Department of Mathematics, Yale University vuha@math.yale.edu Abstract We demonstrate that the problem of training neural networks with small (average) squared error is computationally intractable. Consider a data set of M points (Xi, Yi), i = 1,2, ... , M, where Xi are input vectors from R d , Yi are real outputs (Yi E R). For a network 10 in some class F of neural networks, (11M) L~l (fO(Xi)Yi)2)1/2 - inlfEF(l/ M) "2:f!1 (f(Xi) - YJ2)1/2 is the (avarage) relative error occurs when one tries to fit the data set by 10. We will prove for several classes F of neural networks that achieving a relative error smaller than some fixed positive threshold (independent from the size of the data set) is NP-hard. 1 Introduction Given a data set (Xi, Yi), i = 1,2, ... , M. Xi are input vectors from R d , Yi are real outputs (Yi E R). We call the points (Xi, Yi) data points. The training problem for neural networks is to find a network from some class (usually with fixed number of nodes and layers), which fits the data set with small error. In the following we describe the problem with more details. Let F be a class (set) of neural networks, and a be a metric norm in RM. To each 1 E F, associate an error vector Ef = (1/(Xd - Yil)f;l (EF depends on the data set, of course, though we prefer this notation to avoid difficulty of having too many subindices). The norm of Ej in a shows how well the network 1 fits the data regarding to this particular norm. Furthermore, let eo:,F denote the smallest error achieved by a network in F, namely: eo: F , = min liEf 110: fEF In this context, the training problem we consider here is to find 1 E F such that v.n 372 Vu IIEfila - ea ,F ~ fF, where fF is a positive number given in advance, and does not depend on the size M of the data set. We will call fF relative error. The norm a is chosen by the nature of the training process, the most common norms are: 100 norm: 12 norm: lem). Ilvll oo = maxlvi/ (interpolation problem) IIvl12 = (l/M2::;l v[)1/2, where v = (Vi)t;l (least square error prob- The quantity liEf 1112 is usually referred to as the emperical error of the training process. The first goal of this paper is to show that achieving small emperical error is NP-hard. From now on, we work with 12 norm, if not otherwise specified. A question of great importance is: given the data set, F and fF in advance, could one find an efficient algorithm to solve the training problem formulated above. By efficiency we mean an algorithm terminating in polynomial time (polynomial in the size of the input). This question is closely related to the problem of learning neural networks in polynomial time (see [3]). The input in the algorithm is the data set, by its size we means the number of bits required to write down all (Xi, Yi). Question 1. Given F and fF and a data set. Could one find an efficient algorithm which produces a function f E F such that liEf II < eF + fF Question 1 is very difficult to answer in general. In this paper we will investigate the following important sub-question: Question 2. Can one achieve arbitrary small relative error using polynomial algorithms ? Our purpose is to give a negative answer for Question 2. This question was posed by 1. Jones in his seminar at Yale (1996). The crucial point here is that we are dealing with 12 norm, which is very important from statistical point of view. Our investigation is also inspired by former works done in [2], [6], [7], etc, which show negative results in the 100 norm case. Definition. A positive number f is a threshold of a class F of neural networks if the training problem by networks from F with relative error less than f is NP-hard (i.e., computationally infeasible) . In order to provide a negative answer to Question 2, we are going to show the existence of thresholds (which is independent from the size of the data set) for the following classes of networks. = {flf(x) = (l/n)(2:~=l step (ai x - bi)} ? F~ = {flf(x) = (2:7=1 Cistep (ai x - bd} ? On = {glg(x) = 2:~1 ci<!>i(ai x - bi)} where n is a positive integer, step(x) = 1 if x is positive and zero otherwise, ai and ? Fn are vectors from R d , bi are real numbers, and Ci are positive numbel's. It is clear that the class F~ contains Fn; the reason why we distinguish these two cases is that the proof for Fn is relatively easy to present, while contains the most important ideas. In the third class, the functions 1>i are sigmoid functions which satisfy certain Lipchitzian conditions (for more details see [9]) x Main Theorem (i) The classes F1, F2, F~ and 02 have absolute constant (positive) thresholds On the Infeasibility of Training Neural Networks with Small Squared Errors 373 (ii) For ellery class F n+2, n > 0, there is a threshold of form (n- 3/'2d- 1 /'2. (iii) For every F~+'2' 11 > 0, (iv) For every class 9n+2, n there is a threshold of form (n-3/2d-3/'2 . > 0, there is a threshold of form (n- 5 / 2 d- 1 / 2 . In the last three statements. ( is an absolute positive constant . Here is the key argument of the proof. Assume that there is an algorithm A which solves the training problem in some class (say Fn ) with relative error f. From some (properly chosen) NP-hard problem. we will construct a data set so that if f is sufficiently small, then the solution found by A (given the constructed data set as input) in Fn implies a solution for the original NP-hard problem. This will give a lower bound on f, if we assume that the algorithm A is polynomial. In all proofs the leading parameter is d (the dimension of data inputs). So by polynomial we mean a polynomial with d as variable. All the input (data) sets constructed have polynomial size in d. ""ill The paper is organized as follow. In the next Section, we discuss earlier results concerning the 100 norm. In Section 3, we display the NP-hard results we will use in the reduction. In Section 4, we prove the main Theorem for class F2 and mention the method to handle more general cases. We conclude with some remarks and open questions in Section 5. To end this Section, let us mention one important corollary. The Main Theorem implies that learning F n, F~ and 9n (with respect to 12 norm) is hard. For more about the connection between the complexity of training and learning problems, we refer to [3], [5]. Notation: Through the paper Ud denotes the unit hypercube in Rd. For any number x, Xd denotes the vector (x, X,." x) of length d. In particular, Od denotes the origin of Rd. For any half space H, fI is the complement of H. For any set A, IAI is the number of elements in A. A function y( d) is said to have order of magnitude 0(F(d)), if there are c < C positive constants such that c < y(d)jF(d) < C for all d. 2 Previous works in the loo case The case Q = 100 (interpolation problem) was considered by several authors for many different classes of (usually) 2-layer networks (see [6],[2], [7], [8]). Most of the authors investigate the case when there is a perfect fit, i.e., eleo,F = O. In [2], the authors proved that training 2-layer networks containing 3 step function nodes with zero relative error is NP-hard. Their proof can be extended for networks with more inner nodes and various logistic output nodes. This generalized a former result of Maggido [8] on data set with rational inputs. Combining the techniques used in [2] with analysis arguments, Lee Jones [6] showed that the training problem with relative error 1/10 by networks with two monotone Lipschitzian Sigmoid inner nodes and linear output node, is also NP-hard (NP-complete under certain circumstances). This implies a threshold (in the sense of our definition) (1/10)M- 1/ 2 for the class examined. However, this threshold is rather weak, since it is decreasing in M. This result was also extended for the n inner nodes case [6]. It is also interesting to compare our results with Judd's. In [7] he considered the following problem "Given a network and a set of training examples (a data set), does there exist a set of weights so that the network gives correct output for all training examples ?" He proved that this problem is NP-hard even if the network is V. H. Vu 374 required to produce the correct output for two-third of the traing examples. In fact, it was shown that there is a class of networks and a data sets so that any algorithm will produce poorly on some networks and data sets in the class. However, from this result one could not tell if there is a network which is "hard to train" for all algorithms. Moreover, the number of nodes in the networks grows with the size of the data set. Therefore, in some sense, the result is not independent from the size of the data set. In our proofs we will exploit many techniques provided in these former works. The crucial one is the reduction used by A. Blum and R. Rivest, which involves the NP-hardness of the Hypergraph 2-Coloring problem. 3 Sonle NP hard problems Definition Let B be a CNF formula, where each clause has at most k literals. Let max(B) be the maximum number of clauses which can be satisfied by a truth assignment. The APP MAX k-SAT problem is to find a truth assignment which satisfies (1 - f)max(B) clauses. The following Theorem says that this approximation problem is NP -hard, for some small f. Theorem 3.1.1 Fix k 2: 2. There is fl > 0, such that finding a truth assignment. which satisfies at least (1- fdmax(B) clauses is NP-h ard. The problem is still hard, when every literal in B appears in only few clauses, and every clause contains only few literals. Let B3(5) denote the class of CNFs with at most 3 literals in a clause and every literal appears in at most 5 clauses (see [1]). Theorem 3.1.2 There is t2 > 0 such that finding a truth assignment, which satisfies at least (1 - f)max(B) clauses in a formula B E B3(5) is NP-hard. The optimal thresholds in these theorems can be computed, due to recent results in Thereotical Computer Science. Because of space limitation, we do not go into this matter. Let H = (V, E) be a hypergraph on the set V, and E is the set of edges (collection of subsets of V). Elements of V are called vertices. The degree of a vertex is the number of edges containing the vertex. We could assume that each edge contains at least two vertices. Color the vertices with color Blue or Red. An edge is colorful if it contains vertices of both colors, otherwise we call it monochromatic. Let c( H) be the maximum number of colorful edges one can achieve by a coloring. By a probabilistic argument, it is easy to show that c(H) is at least IEII2 (in a random coloring, an edge will be colorful with probability at least 1/2). Using 3.1.2, we could prove the following theorem (for the proof see [9]) Theorem 3.1.3 There is a constant f3 > 0 such that finding a coloring with at least (1 - t3)c(H) colorful edges is NP-hard. This statement holds even in the case when every but one degree in H is at most 10 4 Proof for :F2 We follow the reduction used in [2]. Consider a hypergraph H(V, E) described Theorem 3.2.1. Let V = {I, 2, . . " d + I}, where with the possible exception of the vertex d + 1, all other vertices have degree at most 10. Every edge will have at least 2 and at most 4 vertices. So the number of edges is at least (d + 1) /4. On the Infeasibility of Training Neural Networks with Small Squared Errors Let Pi be the ith 375 unit vector in R d +l , Pi = (0,0 , . .. ,0,1,0, .. . ,0). Furthermore, Xc = LiE C Pi for every edge C E E. Let S be a coloring with maximum number of colorful edges. In this coloring denote by Al the set of colorful edges and by A2 the set of monochromatic edges. Clearly IAII = e(H). Our data set will be the following (inputs are from R d +l instead of from R d , but it makes no difference) where (Pd+1,1/2)t and (Od+l , l)t means (Pd+1, 1/2) and (Od+l, 1) are repeated t times in the data set, resp. Similarly to [2], consider two vectors a and b in R d +l where a = (al,"" ad+l), ai = -1 if i is Red and ai b = (b l , . .. , bd+l) , bi = -1 if i is Blue and bi = d + 1 otherwise = d + 1 otherwise = It is not difficult to verify that the function fa (1/2)(step (ax + 1/2) + step (bx 1/2)) fits the data perfectly, thus e:F2 = IIEjal1 = O. Suppose f = (1/2) (step (ex - + I) + step (dx - 6? satisfies M MllEjW = 2)f(Xd - Yi)2 < Mc 2 i=l Since if f(X i ) Po = l{i.J(Xd # Yi then U(Xi ) # Ydl < 4Mc 2 = - Yi)2 2: 1/4, the previous inequality implies: p The ratio po/Mis called misclassification ratio, and we will show that this ratio cannot be arbitrary small. In order to avoid unnecessary ceiling and floor symbols, we assume the upper-bound p is an integer. We choose t P so that we can also assume that (Od+l, 1) and (Pd+l, 1/2) are well classified. Let Hl (H2) be the half space consisting of x: ex - 'Y > 0 (dx - 6 > 0). Note that Od E HI n H2 and Pd+l E fI I U fI 2. Now let P l denote the set of i where Pi t/:. HI, and P 2 the set of i such that Pi E Hl n H 2 ? Clearly, if j E P2 , then f(pj) # Yj, hence: IP2 1::; p. Let Q = {C E EIC n P2 # 0}. Note that for each j E P2, the degree of j is at most 10, thus: IQI ::; 10!?:?1 ::; lOp = Let A~ = {Clf(xc) = I}. Since less than p points are misclassified, IA~ .0. A I I < p. Color V by the following rule: (1) if Pi E PI, then i is Red; (2) if Pi E P2 , color i arbitrarily, either Red or Blue; (3) if Pi t/:. P l U P2 , then i is Blue. Now we can finish the proof by the following two claims: Claim 1: Every edge in statement. A~ \Q is colorful. It is left to readers to verify this simple Claim 2: IA~ \QI is close to IAII ? Notice that: IAI \(A~ \Q)I ::; IAI.0.A~ 1+ IQI ::; p + lOp = IIp Observe that the size of the data set is M = d + 2t + lEI, so lEI + d 2: M - 2t = M - 2p. Moreover, lEI 2: (d + 1)/4, so lEI 2: (1/5)(M - 2p). On the other hand, IAII2: (1/2)IEI, all together we obtain; IAII2: (1/10)(M - p), which yields: V. H. Vu 376 = Choose f f4 such that k(f4) ~ f3 (see Theortm 3.1.3). Then for the class ;:2. This completes the proof. Q.E.D. f4 will be a threshold Due to space limitation, we omit the proofs for other classes and refer to [9]. However, let us at least describe (roughly) the general method to handle these cases. The method consists of following steps: ? Extend the data set in the previous proof by a set of (special) points. ? Set the multiplicities of the special points sufficiently high so that those points should be well-classified. ? If we choose the special points properly, the fact that these points are well-classified will determine (roughly) the behavior of all but 2 nodes. In general we will show that all but 2 nodes have little influence on the outputs of non-special data points. ? The problem basically reduces to the case of two nodes. By modifying the previous proof, we could achieve the desired thresholds. 5 Remarks and open problems ? Readers may argue about the existence of (somewhat less natural) data points of high multiplicities. We can avoid using these data points by a combinatorial trick described in [9]. ? The proof in Section 4 could be carried out using Theorem 3.1.2. However, we prefer using the hypergraph coloring terminology (Theorem 3.1.3), which is more convenient and standard. Moreover, Theorem 3.1.3 itself is interesting, and has not been listed among well known "approximation is hard" theorems. ? It remains an open question to determine the right order of magnitude of thresholds for all the classes we considered. (see Section 1). By technical reasons, in the Main theorem, the thresholds for more than two nodes involve the dimension (d). We conjecture that there are dimension-free thresholds. Acknowledgement We wish to thank A. Blum, A. Barron and 1. Lovasz for many useful ideas and discussions. References [1] S. Arora and C. Lund Hardness of approximation, book chapter, preprint [2] A. Blum, R. Rivest Training a 3-node neural network is NP-hard Neutral Networks, Vol 5., p 117-127, 1992 [3] A. Blumer, A. Ehrenfeucht, D. Haussler, M. Warmuth, Learnability and the Vepnik-Chervonenkis Dimension, Journal ofthe Association for computing Machinery, Vol 36, No.4, 929-965, 1989. [4] M. Garey and D. Johnson, Computers and intractability: A guide to the theory of NP-completeness, San Francisco, W.H.Freeman, 1979 On the Infeasibility o/Training Neural Networks with Small Squared Errors 377 [5] D. Haussler, Generalizing the PAC model for neural net and other learning applications (Tech. Rep. UCSC-CRL-89-30). Santa Cruz. CA: University of California 1989. [6] L. J ones, The computational intractability of training sigmoidal neural networks (preprint) [7] J. Judd Neutral Networks and Complexity of learning, MIT Press 1990. [8] N. Meggido, On the complexity of polyhedral separability (Tech. Rep. RJ 5252) IBM Almaden Research Center, San Jose, CA [9] V. H. Vu, On the infeasibility of training neural networks with small squared error. manuscript. 

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Using Helmholtz Machines to analyze multi-channel neuronal recordings Virginia R. de Sa desa@phy.ucsf.edu R. Christopher deC harms decharms@phy.ucsf.edu Michael M. Merzenich merz@phy.ucsf.edu Sloan Center for Theoretical Neurobiology and W. M. Keck Center for Integrative Neuroscience University of California, San Francisco CA 94143 Abstract One of the current challenges to understanding neural information processing in biological systems is to decipher the "code" carried by large populations of neurons acting in parallel. We present an algorithm for automated discovery of stochastic firing patterns in large ensembles of neurons. The algorithm, from the "Helmholtz Machine" family, attempts to predict the observed spike patterns in the data. The model consists of an observable layer which is directly activated by the input spike patterns, and hidden units that are activated through ascending connections from the input layer. The hidden unit activity can be propagated down to the observable layer to create a prediction of the data pattern that produced it. Hidden units are added incrementally and their weights are adjusted to improve the fit between the predictions and data, that is, to increase a bound on the probability of the data given the model. This greedy strategy is not globally optimal but is computationally tractable for large populations of neurons. We show benchmark data on artificially constructed spike trains and promising early results on neurophysiological data collected from our chronic multi-electrode cortical implant. 1 Introduction Understanding neural processing will ultimately require observing the response patterns and interactions of large populations of neurons. While many studies have demonstrated that neurons can show significant pairwise interactions, and that these pairwise interactions can code stimulus information [Gray et aI., 1989, Meister et aI., 1995, deCharms and Merzenich, 1996, Vaadia et al., 1995], there is currently little understanding of how large ensembles of neurons might function together to represent stimuli. This situation has arisen partly out of the historical V R. d. Sa, R. C. deCharms and M. M. Merzenich 132 difficulty of recording from large numbers of neurons simultaneously. Now that this is becoming technically feasible, the remaining analytical challenge is to understand how to decipher the information carried in distributed neuronal responses. Extracting information from the firing patterns in large neuronal populations is difficult largely due to the combinatorial complexity of the problem, and the uncertainty about how information may be encoded. There have been several attempts to look for higher order correlations [Martignon et al., 1997] or decipher the activity from multiple neurons, but existing methods are limited in the type of patterns they can extract assuming absolute reliability of spikes within temporal patterns of small numbers of neurons [Abeles, 1982, Abeles and Gerstein, 1988, Abeles et al., 1993, Schnitzer and Meister,] or considering only rate codes [Gat and Tishby, 1993, Abeles et al., 1995]. Given the large numbers of neurons involved in coding sensory events and the high variability of cortical action potentials, we suspect that meaningful ensemble coding events may be statistically similar from instance to instance while not being identical. Searching for these type of stochastic patterns is a more challenging task. One way to extract the structure in a pattern dataset is to construct a generative model that produces representative data from hidden stochastic variables. Helmholtz machines [Hinton et al., 1995, Dayan et al., 1995] efficiently [Frey et al., 1996] produce generative models of datasets by maximizing a lower bound on the log likelihood of the data. Cascaded Redundancy Reduction [de Sa and Hinton, 1998] is a particularly simple form of Helmholtz machine in which hidden units are incrementally added. As each unit is added, it greedily attempts to best model the data using all the previous units. In this paper we describe how to apply the Cascaded Redundancy Reduction algorithm to the problem of finding patterns in neuronal ensemble data, test the performance of this method on artificial data, and apply the method to example neuronal spike trains. 1.1 Cascaded Redundancy Reduction The simplest form of generative model is to model each observed (or input) unit as a stochastic binary random variable with generative bias bi. This generative input is passed through a transfer function to give a probability of firing. p. = a(b?) = ~ ~ 1 1 + e- bi (1) While this can model the individual firing rates of binary units, it cannot account for correlations in firing between units. Correlations can be modeled by introducing hidden units with generative weights to the correlated observed units. By cascading hidden units as in Figure 1, we can represent higher order correlations. Lower units sum up their total generative input from higher units and their generative bias. Xi = bi + L Sj9j,i (2) j>i Finding the optimal generative weights (9j,i, bi) for a given dataset involves an intractable search through an exponential number of possible states of the hidden units. Helmholtz machines approximate this problem by using forward recognition connections to compute an approximate distribution over hidden states for each data pattern. Cascaded Redundancy Reduction takes this approximation one step further by approximating the distribution by a single state. This makes the search for recognition and generative weights much simpler. Given a data vector, d, considering the state produced by the recognition connections as Sd gives a lower bound on the log Using Helmholtz Machines to Analyze Multi-channel Neuronal Recordings 133 -:.> generative connections ~ recognition connections TO,k T i,k Figure 1: The Cascaded Redundancy Reduction Network. Hidden units are added incrementally to help better model the data. likelihood of the data. Units are added incrementally with the goal of maximizing this lower bound, C, C = 'L[(s% log a(b k ) + (l-s%) log(l-a(bk))+ L d sf log a(xf)+ (I-sf) log(l-a(xf))] i (3) Before a unit is added it is considered as a temporary addition. Once its weights have been learned, it is added to the permanent network only if adding it reduces the cost on an independent validation set from the same data distribution. This is to prevent overtraining. While a unit is considered for addition, all weights other than those to and from the new unit and the generative bias weights are fixed. The learning of the weights to and from the new unit is then a fairly simple optimization problem involving treating the unit as stochastic, and performing gradient descent on the resulting modified lower bound. 2 Method This generic pattern finding algorithm can be applied to multi-unit spike trains by treating time as another spatial dimension as is often done for time series data. The spikes are binned on the order of a few to tens of milliseconds and the algorithm looks for patterns in finite time length windows by sliding a window centered on each spike from a chosen trigger channel. An example extracted window using channel 4 as the trigger channel is shown in Figure 2. Because the number of spikes can be larger than one, the observed units (bins) are modeled as discrete Poisson random variables rather than binary random variables (the hidden units are still kept as binary units). To reflect the constraint that the expected number of spikes cannot be negative but may be larger than one, the transfer function for these observed bins was chosen to be exponential. Thus if Xi is the total summed generative input, Ai, the expected mean number of spikes in bin i, is calculated as eX; and the probability of finding s spikes in that bin is given by s! (4) V.R. d. Sa, R. C. deChanns and M. M. Merzenich 134 Figure 2: The input patterns for the algorithm are windows from the full spatiatemporal firing patterns. The full dataset is windows centered about every spike in the trigger channel. The terms in the lower bound objective function due to the observed bins are modified accordingly. 3 Experimental Results Before applying the algorithm to real neural spike trains we have characterized its properties under controlled conditions. We constructed sample data containing two random patterns across 10 units spanning 100 msec. The patterns were stochastic such that each neuron had a probability of firing in each time bin of the pattern. Sample patterns were drawn from the stochastic pattern templates and embedded in other "noise" spikes. The sample pattern templates are shown in the first column of Figure 3. 300 seconds of independent training, validation and test data were generated. All results are reported on the test data . After training the network, performance was assessed by stepping through the test data and observing the pattern of activation across the hidden units obtained from propagating activity through the forward (recognition) connections and their corresponding generative pattern {Ad obtained from the generative connections from the binary hidden unit pattern. Typically, many of the theoretically possible 2n hidden unit patterns do not occur. Of the ones that do, several may code for the noise background. A crucial issue for interpreting patterns in real neural data is to discover which of the hidden unit activity patterns correspond to actual meaningful patterns. We use a measure that calculates the quality of the match of the observed pattern and the generative pattern it invokes. As the algorithm was not trained on the test data, close matches between the generative pattern and the observed pattern imply real structure that is common to the training and test dataset. With real neural data, this question can also be addressed by correlating the occurrence of patterns to stimuli or behavioural states of the animal. One match measure we have used to pick out temporally modulated structure is the cost of coding the observed units using the hidden unit pattern compared to the cost of using the optimal rate code for that pattern (derived by calculating the firing rate for each channel in the window excluding the trigger bin). Match values were calculated for each hidden unit pattern by averaging the results across all its contributing observed patterns. Typical generative patterns of the added template patterns (in noise) are shown in the second column of Figure 3. The third column in the figure shows example matches from the test set, (Le. patterns that activated the hidden unit pattern corresponding to the generative pattern in column 2). Note that the instances of the patterns are missing some spikes present in the template, and are surrounded by many additional spikes. Using Helmholtz Machines to Analyze Multi-channel Neuronal Recordings Generative Pattern Template 135 Test set Example Pattern 1 Pattern 2 Figure 3: Pattern templates, resulting generative patterns after training (showing the expected number of spikes the algorithm predicts for each bin), and example test set occurrences. The size and shade of the squares represents the probability of activation of that bin (or 0/1 for the actual occurrences), the colorbars go from 0 to 1. We varied both the frequency of the pattern occurrences and that of the added background spikes. Performance as a function of the frequency of the background spikes is shown on the left in Figure 4 for a pattern frequency of .4 Hz. Performance as a function of the pattern frequency for a noise spike frequency of 15Hz is shown on the right of the Figure. False alarm rates were extremely low ranging from 0-4% across all the tested conditions. Also, importantly, when we ran three trials with no added patterns"no patterns were detected by the algorithm. ~! i e- ...., .118, . ~1 0(.88, .95, .97) x(.73, .94, .94) i e- 0.8 "i ~ "iN s:: "2 '" .~ 0.6 co 0 .4 .. 0(0, .73, .82) 0(0, .45, .9) E CD Q. '0 I, .~ " ~ x(.73, .94, .94) 0.8 pattem 1 pattem 2 shifted Pattem 1 0.2 ~I '" 0.6 . 0.4 .~ E CD co Q. '0 .(0, .83, .90) .~ x(O, 0, .51) Q. " ~ 0(0, 0, .87) 0.2 Q. x(O, 0, .32) O L - - L_ _~_ _~-L_ _~_ _~~_ _~ 0 14 17 18 19 16 20 average firing 'ale 01 back ground spikas (Hz) 15 21 0.1 0.15 0.2 0.25 0.3 0.35 0.4 frequency 01 pattem occurrence (Hz) 0.45 0 .5 Figure 4: Graphs showing the effect of adding more background spikes (left) and decreasing the number of pattern occurrences in the dataset (right) on the percentage of patterns correctly detected. The detection of shifted pattern is due to the presence of a second spike in channel 4 in the pattern (hits for this case are only calculated for the times when this spike was present - the others would all be missed). In fact in some cases the presence of the only slightly probable 3rd bin in channel 4 was enough to detect another shifted pattern 1. Means over 3 trials are plotted with the individual trial values given in braces The algorithm was then applied to recordings made from a chronic array of extracellular microelectrodes placed in the primary auditory cortex of one adult marmoset monkey and one adult owl monkey [deC harms and Merzenich, 1998]. On some elec- V.R. d. Sa, R. C. deCharms and M. M. Merzenich 136 Figure 5: Data examples (all but top left) from neural recordings in an awake marmoset monkey that invoke the same generative pattern (top left). The instances are patterns from the test data that activated the same hidden unit activity pattern resulting in the generative pattern in the top left. The data windows were centered around all the spikes in channel 4. The brightest bins in the generative pattern represent an expected number of spikes of 1. 7. In the actual patterns, The darkest and smallest bins represent a bin with 1 spike; each discrete grayscale/size jump represents an additional spike. Each subfigure is indiv!dually normalized to the bin with the most spikes. trodes spikes were isolated from individual neurons; others were derived from small clusters of nearby neurons. Figure 5 shows an example generative pattern (accounting for 2.8% of the test data) that had a high match value together with example occurrences in the test data. The data were responses recorded to vocalizations played to the marmoset monkey, channel 4 was used as the trigger channel and 7 hidden units were added. 4 Discussion We have introduced a procedure for searching for structure in multineuron spike trains, and particularly for searching for statistically reproducible stochastic temporal events among ensembles of neurons. We believe this method has great promise for exploring the important question of ensemble coding in many neuronal systems, a crucial part of the problem of understanding neural information coding. The strengths of this method include the ability to deal with stochastic patterns, the search for any type of reproducible structure including the extraction of patterns of unsuspected nature, and its efficient, greedy, search mechanism that allows it to be applied to large numbers of neurons. Acknowledgements We would like to acknowledge Geoff Hinton for useful suggestions in the early stages of this work, David MacKay for helpful comments on an earlier version of the manuscript, and the Sloan Foundation for financial support. Using Helmholtz Machines to Analyze Multi-channel Neuronal Recordings 137 References [Abeles, 1982] Abeles, M. (1982). Local Cortical Circuits An Electrophysiological Study, volume 6 of Studies of Brain Function. Springer-Verlag. [Abeles et al., 1995] Abeles, M., Bergman, H., Gat, I., Meilijson, I., Seidemann, E., Tishby, N., and Vaadia, E. (1995). Cortical activity flips among quasi-stationary states. Proceedings of the National Academy of Science, 92:8616- 8620. [Abeles et al., 1993] Abeles, M., Bergman, H., Margalit, E., and Vaadia, E. (1993). Spatiotemporal firing patterns in the frontal cortex of behaving monkeys. Journal of Neurophysiology, 70(4):1629-1638. [Abeles and Gerstein, 1988] Abeles, M. and Gerstein, G. L. (1988). Detecting spatiotemporal firing patterns among simultaneously recorded single neurons. Journal of Neurophysiology, 60(3). [Dayan et al., 1995] Dayan, P., Hinton, G. E., Neal, R. M., and Zemel, R. S. (1995). The helmholtz machine. Neural Computation, 7:889-904. [de Sa and Hinton, 1998] de Sa, V. R. and Hinton, G. E. (1998). Cascaded redundancy reduction. to appear in Network{February). [deC harms and Merzenich, 1996] deCharms, R. C. and Merzenich, M. M. (1996). Primary cortical representation of sounds by the coordination of action-potential timing. Nature, 381:610-613. [deCharms and Merzenich, 1998] deCharms, R. C. and Merzenich, M. M. (1998). Characterizing neurons in the primary auditory cortex of the awake primate using reverse correlation. this volume. [Frey et al., 1996] Frey, B. J., Hinton, G. E., and Dayan, P. (1996). Does the wakesleep algorithm produce good density estimators? In Touretzky, D., Mozer, M., and Hasselmo, M., editors, Advances in Neural Information Processing Systems 8, pages 661-667. MIT Press. [Gat and Tishby, 1993] Gat, I. and Tishby, N. (1993). Statistical modeling of cellassemblies activities in associative cortex of behaving monkeys. In Hanson, S., Cowan, J., and Giles, C., editors, Advances in Neural Information Processing Systems 5, pages 945-952. Morgan Kaufmann. [Grayet al., 1989] Gray, C. M., Konig, P., Engel, A. K., and Singer, W. (1989). Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature, 338:334-337. [Hinton et al., 1995] Hinton, G. E., Dayan, P., Frey, B. J., and Neal, R. M. (1995). The wake-sleep algorithm for unsupervised neural networks. Science, 268:1158116l. [Martignon et al., 1997] Martignon, L., Laskey, K., Deco, G., and Vaadia, E. (1997). Learning exact patterns of quasi-synchronization among spiking neurons from data on multi-unit recordings. In Mozer, M., Jordan, M., and Petsche, T., editors, Advances in Neural Information Processing Systems 9, pages 76-82. MIT Press. [Meister et al., 1995] Meister, M., Lagnado, L., and Baylor, D. (1995). Concerted signaling by retinal ganglion cells. Science, 270:95-106. [Schnitzer and Meister,] Schnitzer, M. J. and Meister, M. Information theoretic identification of neural firing patterns from multi-electrode recordings. in preparation. [Vaadia et al., 1995] Vaadia, E., Haalman, I., Abeles, M., Bergman, H., Prut, Y., Slovin, H., and Aertsen, A. (1995). Dynamics of neuronal interactions in monkey cortex in relation to behavioural events. Nature, 373:515-518. 

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An Annealed Self-Organizing Map for Source Channel Coding Matthias Burger, Thore Graepel, and Klaus Obermayer Department of Computer Science Technical University of Berlin FR 2-1, Franklinstr. 28/29, 10587 Berlin, Germany {burger, graepel2, oby} @cs.tu-berlin.de Abstract We derive and analyse robust optimization schemes for noisy vector quantization on the basis of deterministic annealing. Starting from a cost function for central clustering that incorporates distortions from channel noise we develop a soft topographic vector quantization algorithm (STVQ) which is based on the maximum entropy principle and which performs a maximum-likelihood estimate in an expectationmaximization (EM) fashion. Annealing in the temperature parameter f3 leads to phase transitions in the existing code vector representation during the cooling process for which we calculate critical temperatures and modes as a function of eigenvectors and eigenvalues of the covariance matrix of the data and the transition matrix of the channel noise. A whole family of vector quantization algorithms is derived from STVQ, among them a deterministic annealing scheme for Kohonen's self-organizing map (SOM). This algorithm, which we call SSOM, is then applied to vector quantization of image data to be sent via a noisy binary symmetric channel. The algorithm's performance is compared to those of LBG and STVQ. While it is naturally superior to LBG, which does not take into account channel noise, its results compare very well to those of STVQ, which is computationally much more demanding. 1 INTRODUCTION Noisy vector quantization is an important lossy coding scheme for data to be transmitted over noisy communication lines. It is especially suited for speech and image data which in many applieations have to be transmitted under low bandwidth/high noise level conditions. Following the idea of (Farvardin, 1990) and (Luttrell, 1989) of jointly optimizing the codebook and the data representation w.r.t. to a given channel noise we apply a deterministic annealing scheme (Rose, 1990; Buhmann, 1997) to the problem and develop a 431 An Annealed Self-Organizing Map for Source Channel Coding soft topographic vector quantization algorithm (STVQ) (cf. Heskes, 1995; Miller, 1994). From STVQ we can derive a class of vector quantization algorithms, among which we find SSOM, a deterministic annealing variant of Kohonen's self-organizing map (Kohonen, 1995), as an approximation. While the SSOM like the SOM does not minimize any known energy function (Luttre11, 1989) it is computationally less demanding than STVQ. The deterministic annealing scheme enables us to use the neighborhood function of the SOM solely to encode the desired transition probabiliti~;;s of the channel noise and thus opens up new possibilities for the usage of SOMs with arbitrary neighborhood functions. We analyse phase transitions during the annealing and demonstrate the performance of SSOM by applying it to lossy image data compression for transmission via noisy channels. 2 DERIVATION OF A CLASS OF VECTOR QUANTIZERS Vector quantization is a method of encoding data by grouping the data vectors and providing a representative in data space for each group. Given a set X of data vectors Xi E ~, i 1, ... , D, the objective of vector quantization is to find a set W of code vectors Wr 1 r = 0, ... , N- 1, and a set M of binary assignment variables IDir. Lr IDir = 1, Vi, such that a given cost function = (1) r is minimized. Er (Xi, W) denotes the cost of assigning data point Xi to code vector Wr. Following an idea by (Luttrell, 1994) we consider the case that the code labels r form a compressed encoding of the data for the purpose of transmission via a noisy channel (see Figure 1). The distortion caused by the channel noise is modeled by a matrix H of tran1 , Vr, for the noise induced change of assignment of sition probabilities hrs. La hrs a data vector Xi from code vector Wr to code vector W 8 ? After transmission the received index s is decoded using its code vector w 8 ? Averaging the squared Euclidean distance IIxi - w sll2 over a11 possible transitions yields the assignment costs = (2) where the factor 1/2 is introduced for computational convenience. Starting from the cost function E given in Eqs. (1), (2) the Gibbs-distribution P (M, WI X) exp ( -,8 E (M, WI X)) can be obtained via the principle of maximum entropy under the constraint of a given average cost (E). The Lagrangian multiplier ,B is associated with {E) and is interpreted as an inverse temperature that determines the fuzziness of assignments. In order to generalize from the given training set X we calculate the most likely set of code vectors from the probability distribution P (M, WI X) marginalized over all legal sets of assignments M. For a given value of ,B we obtain =! Wr where P(xi E s) = LiXi L 8 hrsP(xi E s) LiLa hraP(xi E s) ' Vr, (3) = (mis). P (Xi E s) = e:x;p (-~ Lthat Jlxi- Wtll 2 ) Lu exp ( -~ Lt hut llxi- Wtll 2 ), (4) is the assignment probability of data vector Xi to code vector Wa. Solving Eqs. (3), (4) by fixed-point iteration comprises an expectation-maximization algorithm, where the E-step, 432 M Burger, T. Graepel and K Obermayer Figure 1: Cartoon of a generic data communication problem. The encoder assigns input vectors Xi to labeled code vectors Wr. Their indices r are the~ transmitted via a noisy channel which is characterized by a set of transition probabilities hrs? The decoder expands the received index s to its code vector W 8 which represents the data vectors assigned to it during encoding. The total error is measured via the squared Euclidean distance between the original data vector Xi and its representative w 5 averaged over all transitions r -t s. Q I Em'OdPr r------------., : Distortion " I Chlllmel Noiie hn : r - s I x, - w, X; II x; - w? 11 2 : 1.------------..1 I 1? w, Decod~r s - w, Eq. (4), determines the assignment probabilities P(xi E s) for all data points Xi and the old code vectors w 8 and theM-step, Eq. (3), determines the new code vectors Wr from the new assignment probabilities P(xi E s). In order to find the global minimum ofE, (3 0 is increased according to an annealing schedule which tracks the solution from the easily solvable convex problem at low f3 to the exact solution of Eqs. (1 ), (2) at infinite j3. In the following we call the solution of Eqs. (3), (4) soft topographic vector quantizer (STVQ). = Eqs. (3), (4) are the starting point for a whole class of vector quantization algorithms (Figure 2). The approximation hrs -t drs applied to Eq. (4) leads to a soft version of Kohonen's self-organzing map (SSOM), if additionally applied to Eq. (3) soft-clustering (SC) (Rose, 1990) is recovered. f3 -t oo leads to the corresponding "hard" versions topographic vector quantisation (TVQ) (Luttrell, 1989), self-organizing map (SOM) (Kohonen, 1995), and LBG. In the following, we will focus on the soft self-organizing map (SSOM). SSOM is computationally less demanding than STVQ, but offers - in contrast to the traditional SOM - a robust deterministic annealing optimization scheme. Hence it is possible to extend the SOM approach to arbitrary non-trivial neighborhood functions hrs as required, e.g. for source channel coding problems for noisy channels. 3 PHASE TRANSITIONS IN THE ANNEALING From (Rose, 1990) it is known that annealing in f3 changes the representation of the data. Code vectors split with increasing f3 and the size of the codebook for a fixed f3 is given by the number of code vectors that have split up to that point. With non-diagonal H, however, permutation symmetry is broken and the "splitting" behavior of the code vectors changes. At infinite temperature every data vector Xi is assigned to every code vector w r with equal probability P 0 (xi E r) = 1/N, where N is the size of the codebook. Hence all code f:s 2:'::i Xi , Vr, of the data. Expanding the vectors are located in the center of mass, w~ r.h.s. of Eq. (3) to first order around the fixed point { w~} and assuming hrs hsr, 'r/ r, s, we obtain the critical value = = (5) An Annealed Self-Organizing Map for Source Channel Coding 433 r hrs-- t5rs STVQ hrs-- t5rs (3--- M- Step (3- 00 L TVQ (3-- .J 00 - - - - .., hrs- t5rs SOM E-Step --- ! 00 r hrs- t5rs ., sc SSOM E-Step --- LBG M- Step L ---- .J Figure 2: Class of vector quantizers derived from STVQ, together with approximations and limits (see text). The "S" in front stands for "soft" to indicate the probabilistic approach. for the inverse temperature, at which the center of mass solution becomes unstable . .\~ax is the largest eigenvalue of the covariance matrix C = ~ Li XiXf of the data and corresponds to their variance .\~ax = IT~ax along the principal axis which is given by the associated eigenvector v:;;ax and along which code vectors split. .\~ax is the largest eigenvalue of a matrix G whose elements are given by grt = Ls hrs (hst- h). The rth component of the corresponding eigenvector v~ax determines for each code vector Wr in which direction along the principal axis it departs from w~ and how it moves relative to the other code vectors. For SSOM a similar result is obtained with Gin Eq. (5) simply being replaced by GssoM, g;~oM hrt - ~. See (Graepel, 1997) for details. = 4 NUMERICAL RESULTS In the following we consider a binary symmetric channel (BSC) with a bit error rate (BER) ?. Assuming that the length of the code indices is n bits, the matrix elements of the transition matrix Hare hrs = (1 _ c)n-dH(r,s) cdH(r,s) l ( 6) where dH (r, s) is the Hamming-distance between the binary representations ofr and s. 4.1 TOY PROBLEM The numerical analysis of the phase transitions described in the previous section was performed on a toy data set consisting of 2000 data vectors drawn from a two-dimensional elongated Gaussian distribution P(x) = (211')- 1 ICI-~ exp(-~xTc- 1 x) with diagonal covariance matrix C = diag(1, 0.04). The size of the codebook was N = 4 corresponding ton = 2 bits. Figure 3 (left) shows the x-coordinates of the positions of the code vectors in data space as functions of the inverse temperature {3. At a critical inverse temperature {3* the code vectors split along the x-axis which is the principal axis of the distribution of data points. In accordance with the eigenvector v~ax = (1, 0, 0, -1) T for the largest eigenvalue .\~ax of the matrix G two code vectors with Hamming distance dH 2 move to opposite positions along the principal axis, and two remain at the center. Note the degeneracy of eigenvalues for matrix (6). Figure 3 (right) shows the critical inverse temperature /3* as a function of the BER for both STVQ (crosses) and SSOM (dots). Results are in very good agreement with the theoretical predictions of Eq. (5) (solid line). The inset displays the average cost (E) = ~ Li Lr P(xi E r) Ls hrs \\xi- Ws\\ 2 as a function of f3 for = M Burger, T. Graepel and K. Obennayer 434 = f 0.08 for STVQ and SSOM. The drop of the average cost occurs at the critical inverse temperature {3"'. 4 3.5 .. X ., . Dl ... 0.5 11 3 I! I; 'rn2.5 .. ?, -0.5 ... -1 0 2 8 3 4 5 0.05 0.1 BER 0.15 0.2 0.25 Figure 3: Phase transitions in the 2 bit "toy" problem. (left) X-coordinate of code vectors for the SSOM case plotted vs. inverse temperature {3, f = 0.08. The splitting of the four code vectors occurs at {3 1.25 which is in very good accordance with the theory. (right) Critical values of {3 for SSOM (dots) and STVQ (crosses), determined via the kink in the 0.08, top line STVQ), which indicates the phase transition. Solid average cost (inset: f lines denote theoretical predictions. Convergence parameter for the fixed-point iteration, giving the upper limit for the difference in successive code vector positions per dimension, was d = 5.0E- 10. = = 4.2 SOURCE CHANNEL CODING FOR IMAGE DATA In order to demonstrate the applicability of STVQ and in particular of SSOM to source channel coding we employed both algorithms to the compression of image data, which were then sent via a noisy channel and decoded after transmission. As a training set we used three 512 x 512 pixel 256 gray-value images from different scenes with blocksize d = 2 x 2. The size of the codebook was chosen to beN = 16 in order to achieve a com2 f3t pression to 1 bpp. We applied an exponential annealing schedule given by f3t+l and determined the start value f3o to be just below the critical {3"' for the first split as given in Eq. (5). Note that with the transition matrix as given in Eq. (6) this optimization corresponds to the embedding of an n = 4 dimensional hypercube in the d = 4 dimensional data space. We tested the resulting codebooks by encoding our test image Lena1 (Figure 5), which had not been used for determining the codebook, simulating the transmission of the indices via a noisy binary symmetric channel with given bit error rate and reconstructing the image using the codebook. = The results are summarized in Figure 4 which shows a plot of the signal-to-noise-ratio (SNR) as a function of the bit-error rate for STVQ (dots), SSOM (vertical crosses), and LBG (oblique crosses). STVQ shows the best performance especially for high BERs, where it is naturally far superior to the LBG-algorithm which does not take into account channel noise. SSOM, however, performs only slightly worse (approx. 1 dB) than STVQ. Considering the fact that SSOM is computationally much less demanding than STVQ 1The Lenna Story can be found at http://www.isr.com/ chuck/lennapgllenna.shtml An Annealed Self-Organizing Map for Source Channel Coding 435 (O(N) for encoding)- due to the omission of the convolution with hrs in Eq. (4)- theresult demonstrates the efficiency of SSOM for source channel coding. Figure 4 also shows the generalization behavior of a SSOM codebook optimized for a BER of 0.05 (rectan0.05 it performs worse than approprigles). Since this codebook was optimized fort: ately trained SSOM codebooks for other values of BER, but still performs better than LBG except for low values of BERs. At low values, SSOMs trained for the noisy case are outperformed by LBG because robustness w.r.t. channel noise is achieved at the expense of an optimal data representation in the noise free case. Figure 5, finally, provides a vis..Ial impression of the performance of the different vector quantizers at a BER of 0.033. While the reconstruction for STVQ is only slightly better than the one for SSOM, both are clearly superior to the reconstruction for LBG. = Figure 4: Comparison between different vector quantizers for image compression, noisy channel (BSC) transmission and reconstruction. The plot shows the signal-to-noise-ratio (SNR), defined as 10 loglo(O'signat/ O'noise). as a function of bit-error rate (BER) for STVQ z and SSOM, each optimized for the given ~ channel noise, for SSOM, optimized for f a BER of 0.05, and for LBG. The training set consisted of three 512 x 512 pixel 256 gray-value images with blocksize 2 x 2. The codebook size was N d 16 corresponding to 1 bpp. The annealing schedule was given by f3t+I = 2 f3t and Lena was used as a test image. Convergence parameter was 1. 0E - 5. = SSQoi5%8ER ?II-? LOO-.. = o 5 SlVQ++- SSQ,I 14 -2 BER CONCLUSION We presented an algorithm for noisy vector quantization which is based on deterministic annealing (STVQ). Phase transitions in the annealing process were analysed and a whole class of vector quantizers could be derived, includings standard algorithms such as LBG and "soft" versions as special cases of STVQ. In particular, a fuzzy version of Kohonen's SOM was introduced, which is computationally more efficient than STVQ and still yields very good results as demonstrated for noisy vector quantization of image data. The deterministic annealing scheme opens up many new possibilities for the usage of SOMs, in particular, when its neighborhood function represents non-trivial neighborhood relations. Acknowledgements This work was supported by TU Berlin (FIP 13/41). We thank H. Bartsch for help and advice with regard to the image processing example. References J. M. Buhmann and T. Hofmann. Robust Vector Quantization by Competitive Learning. Proceedings ofiCASSP'97, Munich, (1997). N. Farvardin. A Study of Vector Quantization/or Noisy Channels. IEEE Transactions on Infonnation Theory, vol. 36, p. 799-809 (1990). 436 M. Burger, T. Graepel and K. Obermayer Original STVQ LBG SNR9.00dB SSOM SNR 4.64 dB SNR 7.80 dB Figure 5: Lena transmitted over a binary symmetric channel with BER of 0.033 encoded and reconstructed using different vector quantization algorithms. ? T. Graepel, M. Burger, and K. Obermayer. Phase Transitions in Stochastic Self-Organizing Maps. Physical Review E, vol. 56, no. 4, p. 3876-3890 (1997). T. Heskes and B. Kappen. Self-Organizing and Nonparametric Regression. Neural Networks- ICANN'95, vol.l,p. 81-86 (1995). Artificial T. Kohonen. Self-Organizing Maps. Springer-Verlag, 1995. S. P. Luttrell. Self-Organisation: A Derivationjromfirst Principles of a Class of Learning Algorithms. Proceedings of IJCNN'89, Washington DC, vol. 2, p. 495-498 (1989). S. P. Luttrell. A Baysian Analysis of Self-Organizing Maps. Neural Computation, vol. 6, p. 767-794 (1994). D. Miller and K. Rose. Combined Source-Channel Vector Quantization Using Deterministic Annealing. IEEE Transactions on Communications, vol. 42, p. 347-356 (1994). K. Rose, E. Gurewitz, and G. C. Fox. Statistical Mechanics and Phase Transitions in Clustering. Physical Review Letters, vol. 65, No.8, p. 945-948 (1990). 

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Regression with Input-dependent Noise: A Gaussian Process Treatment Paul W. Goldberg Department of Computer Science University of Warwick Coventry, CV 4 7AL, UK pvgGdcs.varvick.ac.uk Christopher K.I. Williams Neural Computing Research Group Aston University Birmingham B4 7ET, UK c.k.i.villiamsGaston.ac.uk Christopher M. Bishop Microsoft Research St. George House 1 Guildhall Street Cambridge, CB2 3NH, UK cmbishopOmicrosoft.com Abstract Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance. 1 Background and Motivation A very natural approach to regression problems is to place a prior on the kinds of function that we expect, and then after observing the data to obtain a posterior. The prior can be obtained by placing prior distributions on the weights in a neural 494 P. W Goldberg, C. K. L Williams and C. M. Bishop network, although we would argue that it is perhaps more natural to place priors directly over functions. One tractable way of doing this is to create a Gaussian process prior. This has the advantage that predictions can be made from the posterior using only matrix multiplication for fixed hyperparameters and a global noise level. In contrast, for neural networks (with fixed hyperparameters and a global noise level) it is necessary to use approximations or Markov chain Monte Carlo (MCMC) methods. Rasmussen (1996) has demonstrated that predictions obtained with Gaussian processes are as good as or better than other state-of-the art predictors. In much of the work on regression problems in the statistical and neural networks literatures, it is assumed that there is a global noise level, independent of the input vector x. The book by Bishop (1995) and the papers by Bishop (1994), MacKay (1995) and Bishop and Qazaz (1997) have examined the case of input-dependent noise for parametric models such as neural networks. (Such models are said to heteroscedastic in the statistics literature.) In this paper we develop the treatment of an input-dependent noise model for Gaussian process regression, where the noise is assumed to be Gaussian but its variance depends on x. As the noise level is nonnegative we place a Gaussian process prior on the log noise level. Thus there are two Gaussian processes involved in making predictions: the usual Gaussian process for predicting the function values (the y-process), and another one (the z-process) for predicting the log noise level. Below we present a Markov chain Monte Carlo method for carrying out inference with this model and demonstrate its performance on a test problem. 1.1 Gaussian processes A stochastic process is a collection of random variables {Y(x)lx E X} indexed by a set X. Often X will be a space such as 'R,d for some dimension d, although it could be more general. The stochastic process is specified by giving the probability distribution for every finite subset of variables Y(Xl), ... , Y(Xk) in a consistent manner. A Gaussian process is a stochastic process which can be fully specified by its mean function J.L(x) = E[Y(x)] and its covariance function Cp(x,x') = E[(Y(x)-J.L(x?)(Y(x')-J.L(x'?]; any finite set of points will have a joint multivariate Gaussian distribution. Below we consider Gaussian processes which have J.L(x) == O. This assumes that any known offset or trend in the data has been. removed. A non-zero I' (x ) is easily incorporated into the framework at the expense of extra notational complexity. A covariance junction is used to define a Gaussian process; it is a parametrised function from pairs of x-values to their covariance. The form of the covariance function that we shall use for the prior over functions is given by Cy(x(i),x U? =vyexp (-~ tWYl(x~i) _x~j?2) + J y 8(i,j) (1) 1=1 where vy specifies the overall y-scale and W;:/2 is the length-scale associated with the lth coordinate. Jy is a "jitter" term (as discussed by Neal, 1997), which is added to prevent ill-conditioning of the covariance matrix of the outputs. J y is a typically given a small value, e.g. 10- 6 . For the prediction problem we are given n data points 1) = ((Xl,t1),(X2,t2), Input-dependent Noise: A Gaussian Process Treatment 495 ... , (x n , t n ?), where ti is the observed output value at Xi. The t's are assumed to have been generated from the true y-values by adding independent Gaussian noise whose variance is x-dependent. Let the noise variance at the n data points be r = (r(xl),r(x2), ... ,r(x n )). Given the assumption of a Gaussian process prior over functions, it is a standard result (e.g. Whittle, 1963) that the predictive distribution P(t*lx*) corresponding to a new input x* is t* "'" N(t(X*),0'2(X*)), where i(x*) 0'2(X*) - k~(x*)(Ky + KN)-lt Cy(x*, x*) + r(x*) - k~(x*)(Ky (2) + KN )-lky(x*) (3) where the noise-free covariance matrix K y satisfies [KY] ij = C y (x i, X j ), and ky(x*) = (Cy(x*,xd, ... ,Cy(x*,xn?T, KN = diag(r) and t = (tb ... ,tn)T, and V0'2(X*) gives the "error bars" or confidence interval of the prediction. In this paper we do not specify a functional form for the noise level r(x) but we do place a prior over it. An independent Gaussian process (the z-process) is defined to be the log of the noise level. Its values at the training data points are denoted by z = (zl, . .. ,zn),sothatr = (exp(zl), ... ,exp(zn?. The priorforz has a covariance function CZ(X(i), xU? similar to that given in equation 1, although the parameters vz and the WZI'S can be chosen to be different to those for the y-process. We also add the jitter term J z t5(i,j) to the covariance function for Z, where Jz is given the value 10- 2 ? This value is larger than usual, for technical reasons discussed later. We use a zero-mean process for z which carries a prior assumption that the average noise rate is approximately 1 (being e to the power of components of z). This is suitable for the experiment described in section 3. In general it is easy to add an offset to the z-process to shift the prior noise rate. 2 An input-dependent noise process We discuss, in turn, sampling the noise rates and making predictions with fixed values of the parameters that control both processes, and sampling from the posterior on these parameters. 2.1 Sampling the Noise Rates The predictive distribution for t*, the output at a point x*, is P(t*lt) = f P(t*lt,r(z?P(zlt)dz. Given a z vector, the prediction P(t*lt,r(z? is Gaussian with mean and variance given by equations 2 and 3, but P(zlt) is difficult to handle analytically, so we use a Monte Carlo approximation to the integral. Given a representative sample {Zb ... ' Zk} of log noise rate vectors we can approximate the integral by the sum E j P(t*lt,r(zj?. i We wish to sample from the distribution P(zlt). As this is quite difficult, we sample instead from P(y, zit); a sample for P(zlt) can then be obtained by ignoring the y values. This is a similar approach to that taken by Neal (1997) in the case of Gaussian processes used for classification or robust regression with t-distributed noise. We find that (4) P(y, zit) oc P(tly, r(z?P(y)P(z). We use Gibbs sampling to sample from P(y, zit) by alternately sampling from P(zly, t) and P(ylz, t). Intuitively were are alternating the "fitting" of the curve (or P. W. Goldberg, C. K. 1. Williams and C. M Bishop 496 y-process) with "fitting" the noise level (z-process) . These two steps are discussed in turn . ? Sampling from P(ylt, z) For y we have that P(ylt, z) ex P(tly, r(z?P(y) (5) where P(tly, r(z? = TI n 1 ( (21l'Ti)l/2 exp - (ti - Yi)2 ) 2Ti . (6) Equation (6) can also be written as P(tly,r(z? '" N(t,KN) ' Thus P(ylt,z) is a multivariate Gaussian with mean (Kyl + Ki/)-l K;/t and covariance matrix (Kyl + KN1)-1 which can be sampled by standard methods . ? Sampling from P(zlt,y) For fixed y and t we obtain P(zly, t) ex P(tly, z)P(z). (7) The form of equation 6 means that it is not easy to sample z as a vector. Instead we can sample its components separately, which is a standard Gibbs sampling algorithm. Let Zi denote the ith component of z and let Z-i denote the remaining components. Then (8) P(Zilz-i) is the distribution of Zi conditioned on the values of Z-i' The computation of P(zilz-i) is very similar to that described by equations (2) and (3), except that Cy ( " .) is replaced by C z ( " .) and there is no noise so that T (.) will be identically zero. We sample from P(zilz-i' y, t) using rejection sampling. We first sample from P(zilz-i), and then reject according to the term exp{ -Zi/2 - Hti - Yi)2 exp( -Zi)} (the likelihood of local noise rate Zi), which can be rescaled to have a maximum value of lover Zi. Note that it is not necessary to perform a separate matrix inversion for each i when computing the P(zilz-i) terms; the required matrices can be computed efficiently from the inverse of K z. We find that the average rejection rate is approximately two-thirds, which makes the method we currently use reasonably efficient. Note that it is also possible to incorporate the term exp( -Zi/2) from the likelihood into the mean of the Gaussian P(zilz-i) to reduce the rejection rate. As an alternative approach, it is possible to carry out Gibbs sampling for P(zilz-i' t) without explicitly representing y, using the fact that 10gP(tlz) = -~logIKI? T K-1t + canst, where K = K y + K N . We have implemented this and found similar results to those obtained using sampling of the y's. However, explicitly representing the y-process is useful when adapting the parameters, as described in section 2.3. !t Input-dependent Noise: A Gaussian Process Treatment 2.2 497 Making predictions So far we have explained how to obtain a sample from P(zlt). To make predictions we use P(t*lt) ~ ~ l: P(t*lt, r(zj)). (9) j However, P(t*lt,r(zj)) is not immediately available, as z*, the noise level at x* is unknown. In fact (10) P(z*IZj, t) is simply a Gaussian distribution for z* conditioned on Zj, and is obtained in a similar way to P(zilz-i). As P(t*lz*, t, r(zj)) is a Gaussian distribution as given by equations (2) and (3), P(t*\t, r(z j)) is an infinite mixture of Gaussians with weights P(z*IZj) . Note, however, that each ofthese components has the same mean i(x*) as given by equation (2), but a different variance. We approximate P(t*lt, r(zj)) by taking s = 10 samples of P(z*lzj) and thus obtain a mixture of s Gaussians as the approximating distribution. The approximation for P(t*lt) is then obtained by averaging these s-component mixtures over the k samples Z1> ??? , Zk to obtain an sk-component mixture of Gaussians. 2.3 Adapting the parameters Above we have described how to obtain a sample from the posterior distribution P(z\t) and to use this to make predictions, based on the assumption that the parameters Oy (Le. Vy,Jy,WYl, . .. ,WYd) and Oz (Le. vz,JZ,WZl, ... ,WZd) have been set to the correct values. In practice we are unlikely to know what these settings should be, and so introduce a hierarchical model, as shown in Figure l. This graphical model shows that the joint probability distribution decomposes as P(Oy,OZ, y, z, t) = P(Oy)P(Oz)P(yIOy )P(z\Oz)P(t\y, z). Our goal now becomes to obtain a sample from the posterior P(Oy,Oz,y,zlt), which can be used for making predictions as before. (Again, the y samples are not needed for making predictions, but they will turn out to be useful for sampling Oy .) Sampling from the joint posterior can be achieved by interleaving updates of Oy and Oz with y and Z updates. Gibbs sampling for Oy and Oz is not feasible as these parameters are buried deeply in the K y and K N matrices, so we use the Metropolis algorithm for their updates. As usual, we consider moving from our current state 0 = (Oy,Oz) to a new state 0 using a proposal distribution J(O,O). In practice we take J to be an isotropic Gaussian centered on 0?. Denote the ratio of P(Oy)P(Oz)P(yIOy)P(z\Oz) in states 9 and 0 by r. Then the proposed state 0 is accepted with probability min{r, 1}. It would also be possible to use more sophisticated MCMC algorithms such as the Hybrid Monte Carlo algorithm which uses derivative information, as discussed in Neal (1997). 3 Results We have tested the method on a one-dimensional synthetic problem. 60 data points P. W. Goldberg, C. K. l Williams and C. M Bishop 498 Figure 1: The hierarchical model including parameters. were generated from the function y = 2 sin(271"x) on [0, 1] by adding independent Gaussian noise. This noise has a standard deviation that increases linearly from 0.5 at x 0 to 1.5 at x 1. The function and the training data set are illustrated in Figure 2(a). = = As the parameters are non-negative quantities, we actually compute with their log values. logvy, logvz, logwy and log Wz were given N(O, 1) prior distributions. The jitter values were fixed at J y = 10- 6 and J z = 10- 2 ? The relatively large value for J z assists the convergence of the Gibbs sampling, since it is responsible for most of the variance of the conditional distribution P(Zi/Z-i}. The broadening of this distribution leads to samples whose likelihoods are more variable, allowing the likelihood term (used for rejection) to be more influential. In our simulations, on each iteration we made three Metropolis updates for the parameters, along with sampling from all of the y and Z variables. The Metropolis proposal distribution was an isotropic Gaussian with variance 0.01. We ran for 3000 iterations, and discarded the first one-third of iterations as "burn-in", after which plots of each of the parameters seemed to have settled down. The parameters and Z values were stored every 100 iterations. In Figure 2(b) the average standard deviation of the inferred noise has been plotted, along with with two standard deviation error-bars. Notice how the standard deviation increases from left to right, in close agreement with the data generator. Studying the posterior distributions of the parameters, we find that the ylength scale Ay d;j (wy) -1/2 is well localized around 0.22 ? 0.1, in good agreement with the wavelength of the sinusoidal generator. (For the covariance function in equation 1, the expected number of zero crossings per unit length is 1/7I"Ay.) (WZ)-1/2 is less tightly constrained, which makes sense as it corresponds to a longer wavelength process, and with only a short segment of data available there is still considerable posterior uncertainty. 4 Conclusions We have introduced a natural non-parametric prior on variable noise rates, and given an effective method of sampling the posterior distribution, using a MCMC 499 Input-dependent Noise: A Gaussian Process Treatment .' .. " .' .. -2 o. ?O~~ O I~~ 0 2--0~'~O~'~ O ~'~"--O~7~O~'~ "~ (a) (b) Figure 2: (a) shows the training set (crosses); the solid line depicts the x-dependent mean of the output. (b) The solid curve shows the average standard deviation of the noise process, with two standard deviation error bars plotted as dashed lines. The dotted line indicates the true standard deviation of the data generator. method. When applied to a data set with varying noise, the posterior noise rates obtained are well-matched to the known structure. We are currently experimenting with the method on some more challenging real-world problems. Acknowledgements This work was carried out at Aston University under EPSRC Grant Ref. GR/K 51792 Validation and Verification of Neural Network Systems. References [1] C.M. Bishop (1994). Mixture Density Networks. Technical report NCRG/94/001, Neural Computing Research Group, Aston University, Birmingham, UK. [2] C.M. Bishop (1995). Neural Networks for Pattern Recognition. Oxford University Press. [3] C.M. Bishop and C. Qazaz (1997). Regression with Input-dependent Noise: A Bayesian Treatment. In M. C. Mozer, M. I. Jordan and T. Petsche (Eds) Advances in Neural Information Processing Systems 9 Cambridge MA MIT Press. [4] D. J. C. MacKay (1995). Probabilistic networks: new models and new methods. In F. Fogelman-Soulie and P. Gallinari (Eds), Proceedings ICANN'95 International Conference on Neural Networks, pp. 331-337. Paris, EC2 & Cie. [5] R. Neal (1997). Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification. Technical Report 9702, Department of Statistics, University of Toronto. Available from http://vvv.cs.toronto.edurradford/. [6] C.E. Rasmussen (1996). Evaluation of Gaussian Processes and Other Methods for Nonlinear Regression. PhD thesis, Department of Computer Science, University of Toronto. Available from http://vwv . cs .utoronto. carcarl/ . [7] C.K.I. Williams and C.E. Rasmussen (1996). Gaussian Processes for Regression. In D. S. Touretzky, M. C. Mozer and M. E . Hasselmo Advances in Neural Information Processing Systems 8 pp. 514-520, Cambridge MA MIT Press. [8] P. Whittle (1963). Prediction and regulation by linear least-square methods. English Universities Press. 

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Structure Driven Image Database Retrieval Jeremy S. De Bonet &, Paul Viola Artificial Intelligence Laboratory Learning & Vision Group 545 Technology Square Massachusetts Institute of Technology . Cambridge, MA 02139 EMAIL: jsdCOaLmit. edu & violaCOaLmit. edu HOMEPAGE: http://www.ai . mit. edu/pro j ects/l v Abstract A new algorithm is presented which approximates the perceived visual similarity between images. The images are initially transformed into a feature space which captures visual structure, texture and color using a tree of filters. Similarity is the inverse of the distance in this perceptual feature space. Using this algorithm we have constructed an image database system which can perform example based retrieval on large image databases. Using carefully constructed target sets, which limit variation to only a single visual characteristic, retrieval rates are quantitatively compared to those of standard methods. 1 Introduction Without supplementary information, there exists no way to directly measure the similarity between the content of images. In general, one cannot answer a question of the form: "is image A more like image B or image C?" without defining the criteria by which this comparison is to be made. People perform such tasks by inferring some criterion, based on their visual experience or by complex reasoning about the situations depicted in the images. Humans are very capable database searchers. They can perform simple searches like, "find me images of cars", or more complex or loosely defined searches like, "find me images that depict pride in America". In either case one must examine all, or a large portion, of the database. As the prevalence and size of multimedia databases increases, automated techniques will become critical in the successful retrieval of relevant information. Such techniques must be able to measure the similarity between the visual content of natural images. Structure Driven Image Database Retrieval 867 Many algorithms have been proposed for image database retrieval. For the most part these techniques compute a feature vector from an image which is made up of a handful of image measurements. Visual or semantic distance is then equated with feature distance. Examples include color histograms, texture histograms, shape boundary descriptors, eigenimages, and hybrid schemes (QBIC, ; Niblack et aI., 1993; Virage, ; Kelly, Cannon and Hush, 1995; Pentland, Picard and ScIaroff, 1995; Picard and Kabir, 1993; Santini and Jain, 1996). A query to such a system typically consists of specifying two types of parameters: the target values of each of the measurements; and a set of weights, which determine the relative importance of deviations from the target in each measurement dimension. The features used by these systems each capture some very general property of images. As a result of their lack of specificity however, many images which are actually very different in content generate the same feature responses. In contrast our approach extracts thousands of very specific features. These features measure both local texture and global structure. The feature extraction algorithm computes color, edge orientation, and other local properties at many resolutions. This sort of multi-scale feature analysis is of critical importance in visual recognition and has been used successfully in the context of object recognition (von der Malsburg, 1988; Rao and Ballard, 1995; Viola, 1996) Our system differs from others because it detects not only first order relationships, such as edges or color, but also measures how these first order relationships are related to one another. Thus by finding patterns between image regions with particular local properties, more complex - and therefore more discriminating - features can be extracted. This type of repeated, non-linear, feature detection bears a strong resemblance to the response properties of visual cortex cells (Desimone et al., 1984). While the mechanism for the responses of these cells is not yet clear, this work supports the conclusion that this type of representation is very useful in practical visual processing. 2 Computing the Characteristic Signature The "texture-of-texture" measurements are based on the outputs of a tree of nonlinear filtering operations. Each path through the tree creates a particular filter network, which responds to certain structural organization in the image. Measuring the appropriately weighted difference between the signatures of images in the database and the set of query-images, produces a similarity measure which can be used to rank and sort the images in the database. The computation of the characteristic signature is straightforward. At the highest level of resolution the image is convolved with a set of 25 local linear features including oriented edges and bars. The results of these convolutions are 25 feature response images. These images are then rectified by squaring, which extracts the texture energy in the image, and then downsampled by a factor of two. At this point there are 25 half scale output images which each measure a local textural property of the input image. For example one image is sensitive to vertical edges, and responds strongly to both skyscrapers and picket fences. Convolution, rectification and downsampling is then repeated on each of these 25 half resolution images producing 625 quarter scale "texture-of-texture" images. The second layer will respond strongly to regions where the texture specified in the first layer has a particular spatial arrangement. For example if horizontal alignments of vertical texture are detected, there will be a strong response to a picket fence and little response to a skyscraper. With additional layers additional specificity is 868 J. S. D. Bonet and P. A. Viola achieved; repeating this procedure a third time yields 15,625 meta-texture images at eighth scale. Each of the resulting meta-textures is then summed to compute a single value and provides one element in the characteristic signature. When three channels of color are included there are a total of 46,875 elements in the characteristic signature. Once computed, the signature elements are normalized to reduce the effects of contrast changes. More formally the characteristic signature of an image is given by: Si,j,k,e(I) = L (1) Ei,j,k(Ie) pixels where I is the image, i, j and k index over the different types of linear filters, and Ie are the different color channels of the image. The definition of E is: Ei(I) Ei,j (I) Ei,j,k(I) 2 .j.. [(Fi 0 1)2] 2.j.. [(Fj 0 Ei(I))2] (2) (3) 2.j.. [(Fk 0 E i ,j(I))2] . (4) where F j is the ith filter and 2.j.. is the downsampling operation. 3 U sing Characteristic Signatures To Form Image Queries In the "query by image" paradigm, we describe similarity in terms of the difference between an image and a group of example query images. This is done by comparing the characteristic signature of the image to the mean signature of the query images. The relative importance of each element of the characteristic signature in determining similarity is proportional to the inverse variance of that element across the example-image group: L= - L L L L [S;,j,k,O( Iq) - S;,j,k,o (I.".) i j k e Var[Si,j,k,e(lq )] r (5) where Si,j,k,e(Iq ) and Var [Si,j,k,c(Iq )] are the mean and variance of the characteristic signatures computed over the query set. This is a diagonal approximation of the Mahalanobis distance (Duda and Hart, 1973). It has the effect of normalizing the vector-space defined by the characteristic signatures, so that characteristic elements which are salient within the group of query images contribute more to the overall similarity of an image. In Figure 1 three 2D projections of these 46,875 dimensional characteristic signature space are shown. The data points marked with circles are generated by the 10 images shown at the top of Figure 3. The remaining points are generated by 2900 distract or images. Comparing (a) and (b) we see that in some projections the images cluster tightly, while in others they are distributed. Given a sample of images from the target set we can observe the variation in each possible projection axis. Most of the time the axes shown in (a) will be strongly discounted by the algorithm because these features are not consistent across the query set. Similarly the axes from (b) will receive a large weight because the target images have very consistent values. The axes along which target groups cluster, however, differ from target group to target group. As a result it is not possible to conclude that the axes in (b) are simply better than the axes in (a). In Figure 1 (c) the same projection is shown again this time with a different target set highlighted (with asterisks). Structure Driven Image Database Retrieval 869 9 ~i~~:;:,~?~~~~'?:.~ .' .' ~~~" 9.:'.): t'K. o o (a) (b) (c) Figure 1: In some projections target groups do not cluster (a), and they do in others (b). However, different target groups will not necessarily cluster in the same projections (c). 4 Experiments In the first set of experiments we use a database of 2900 images from 29 Corel Photo CD (collections 1000-2900.) Figure 2 shows the results of typical user query on this system. The top windows in each Figure contain the query-images submitted by the user. The bottom windows show the thirty images found to be most similar; similarity decreases from upper left (most similar) to lower right. Though these examples provide an anecdotal indication that the system is generating similarity measures which roughly conform to human perception, it is difficult to fully characterize the performance of this image retrieval technique. This is a fundamental problem of the domain. Images vary from each other in an astronomical number of ways, and similarity is perceived by human observers based upon complex interactions between recognition, cognition, and assumption. It seems unlikely that an absolute criterion for image similarity can ever be determined, if one truly exists. However using sets of images which we believe are visually similar, we can establish a basis for comparing algorithms. To better measure the performance of the system we added a set of 10 images to the 2900 image database and attempted to retrieve these new images. We compare the performance of the present system to ten other techniques. Though these techniques are not as sophisticated as those used in systems developed by other researchers, they are indicative of the types of methods which are prevalent in the literature. In each experiment we measure the retrieval rates for a set of ten target images which we believe to be visually similar because they consist of images of a single scene. Images in the target set differ due to variation of a single visual characteristic. In some of the target sets the photographic conditions have been changed, either by moving the camera, the objects or the light. In other target sets post photograph image manipulation has been performed. Two example target sets are shown in Figure 3. In each experiment we perform 45 database queries using every possible pair of images from the target set. The retrieval methods compared are: ToT The current textures-of-textures system; RGB-216(or 512)C R,G,B color histograms using 216 (or 512) bins by dividing each color dimension into 6 (or 8) regions. The target histogram is generated by combining the histograms from the two model images; HSV-216(or 512)C same using H,S,V color space; RGB(or HSV)-216(or 870 J. s. D. Figure 2: Sample queries and top 30 responses. Bonet and P. A. Viola Structure Driven Image Database Retrieval 871 , .~t . ? ., 1:: '" ~ . "t- '. :tic ~~ ~~ p .. ~. -' 'i'~,.' .. ,L1 ~ ~< .. J, rf. -" ~ ~ . '\\ ' ";"~. .. .,'" " :". : ... . ' ~' .. . "r;IiB ,~~-o: ~ ., ~,~ Figure 3: Two target sets used in the retrieval experiments. Top: Variation of object location. BOTTOM: Variation of hard shadows. c:: o +=' U Q) 0.5 CD ~ ~ O~--~L-~LL~? Bnghtness Camera Position Contrast Hard Shadows Soft shadows NOise 0.75 Object Location Object Pose - c:: o U Q) 0.5 Q) :3~ o NOise 0.25 NOise 0 .5 - c:: o U Q) 0 .5 - I ~ ROe-l16C RGe- O,, 2C "lGa-2I SNN ~GB - ~l2N.'" . ~S\l-2 16C H'>V-2"i~ I"+S"'~ l 2N" COr-lU"M c .~ ,,' Q) :3~ Occlusion Rotation (360deg) Rotation (60deg) Translation Zoom Figure 4: The percentage of target images ranked above all the distract or images, shown for 15 target sets such as those in Figure 3. The textures-of-textures model presented here achieves perfect performance in 13 of the 15 experiments. 512)NN histograms in which similarity is measured using a nearest neighbor metric; COR-full(and low)res full resolution (and 4x downsampled) image correlation. J. S. D. Bonet and P. A. Viola 872 Rankings for all 10 target images in each of the 45 queries are obtained for each variation. To get a comparative sense of the overall performance of each technique, we show the number of target images retrieved with a Neyman-Pearson criterion of zero, i.e. no false positives Figure 4. The textures-of-textures model substantially outperforms all of the other techniques, achieving perfect performance in 13 experiments. 5 Discussion We have presented a technique for approximating perceived visual similarUsing ity, by measuring the structural content similarity between images. the high dimensional "characteristic signature" space representation, we diA world wide rectly compare database-images to a set of query-images. web interface to system has been created and is available via the URL: http://www.ai.mit.edu/~jsd/Research/lmageDatabase/Demo Experiments indicate that the present system can retrieve images which share visual characteristics with the query-images, from a large non-homogeneous database. Further, it greatly outperforms many of the standard methods which form the basis of other systems. Though the results presented here are encouraging, on real world queries, the retrieved images often contain many false alarms, such as those in Figure 2; however, we believe that with additional analysis performance can be improved. References Desimone, R., Albright, T. D., Gross, C. G., and Bruce, C. (1984) . Stimulus selective properties of inferior temporal neurons in the macaque. Journal of Neuroscience, 4:2051-2062. Duda, R . and Hart, P. (1973) . Pattern Classification and Scene Analysis. John Wiley and Sons. Kelly, M., Cannon, T. M., and Hush, D. R. (1995). Query by image example: the candid approach. SPIE Vol. 2420 Storage and Retrieval for Image and Video Databases III, pages 238-248. Niblack, V., Barber, R ., Equitz, W ., Flickner, M. , Glasman, E. , Petkovic, D., Yanker, P., Faloutsos, C., and Taubin, G. (1993). The qbic project: querying images by content using color, texture, and shape. ISf3TjSPIE 1993 International Symposium on Electronic Imaging: Science f3 Technology, 1908:173-187. Pentland, A., Picard, R. W., and Sclaroff, S. (1995). Photobook: Content-based manipulation of image databases. Technical Report 255, MIT Media Lab. Picard, R . W. and Kabir, T . (1993) . Finding similar patterns in large image databases. ICASSP, V:161-164. QBIC. The ibm qbic project. Web: http://wwwqbic.almaden.ibm.comf. Rao, R. P. N. and Ballard, D. (1995). Object indexing using an iconic sparse distributed memory. Technical Report TR-559, University of Rochester. Santini, S. and Jain, R. (1996) . Gabor space and the development of preattentive similarity. In Proceedings of ICPR 96. International Conference on Pattern Recognition, Vienna. Viola, P. (1996). Complex feature recognition: A bayesian approach for learning to recognize objects. Technical Report 1591, MIT AI Lab. Virage. The virage project. Web: http://www.virage.com/ . von der Malsburg, C. (1988). Pattern recognition by labeled graph matching. Neural Networks, 1:141-148. 

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Factorizing Multivariate Function Classes Juan K. Lin* Department of Physics University of Chicago Chicago, IL 60637 Abstract The mathematical framework for factorizing equivalence classes of multivariate functions is formulated in this paper. Independent component analysis is shown to be a special case of this decomposition. Using only the local geometric structure of a class representative, we derive an analytic solution for the factorization. We demonstrate the factorization solution with numerical experiments and present a preliminary tie to decorrelation. 1 FORMALISM In independent component analysis (ICA), the goal is to find an unknown linear coordinate system where the joint distribution function admits a factorization into the product of one dimensional functions. However, this decomposition is only rarely possible. To formalize the notion of multivariate function factorization, we begin by defining an equivalence relation. Definition. We say that two functions f, 9 : IRn -t IR are equivalent if there exists A,b and c such that: f(x) = cg(Ax+b), where A is a non-singular matrix and c f. O. Thus, the equivalence class of a function consists of aU invertible linear transformations of it. To avoid confusion, equivalence classes will be denoted in upper case, and class representatives in lower case. We now define the product of two equivalence classes. Consider representatives b : IRn -t IR, and c : IRm -t IR of corresponding equivalence classes Band C. Let Xl E IRn , x"2 E IRm , and x = (xl, x"2). From the scalar product of the two functions, define the function a : IRn+m -+ IR by a(x) = b(xI)c(x"2). Let the product of Band C be the equivalence class A with * Current address: E25-201, MIT, Cambridge, MA 02139. Email: jklin@ai.mit.edu 1. K Lin 564 representative a(x). This product is independent of the choice of representatives of Band C, and hence is a well defined operation on equivalence classes. We proceed to define the notion of an irreducible class. Definition. Denote the equivalence class of constants by I. We say that A is irreducible if A = BC implies either B = A, C = I, or B = I, C = A. From the way products of equivalence classes are defined, we know that all equivalence classes of one dimensional functions are irreducible. Our formulation of the factorization of multivariate function classes is now complete. Given a multivariate function, we seek a factorization of the equivalence class of the given representative into a product of irreducibles. Intuitively, in the context of joint distribution functions, the irreducible classes constitute the underlying sources. This factorization generalizes independent component analysis to allow for higher dimensional "vector" sources. Consequently, this decomposition is well-defined for all multivariate function classes. We now present a local geometric approach to accomplishing this factorization. 2 LOCAL GEOMETRIC INFORMATION Given that the joint distribution factorizes into a product in the "source" coordinate system, what information can be extracted locally from the joint distribution in a "mixed" coordinate frame? We assume that the relevant multivariate function is twice differentiable in the re!jion of interest, and denote H f, the Hessian of I, to be the matrix with elements Hij = oiod, where Ok = a~k' Proposition: H' is block diagonal everywhere, oiojllso = 0 for all points and all i ~ k, j > k, il and only il 1 is separable into a sum I(SI,"" sn) g( SI, ... , Sk) + h( Sk+l, ... , sn) for some functions 9 and h. Proof - Sufficiency: Given l(sl, . .. , sn) = g(SI, . .. , Sk) So = + h(sk+1, . .. , Sn), 0 21 _ ~ Oh(Sk+1,"" Sn) _ 0 OSiOSj - OSi OSj everywhere for all i ~ k, j > k. Necessity: From H{n = 0, we can decompose 1 into l(sl, S2,???, sn) = 9(SI, ... , sn-t} for some functions j > k, we find 9 and h. + h(S2"'" sn), Continuing by imposing the constraints H [. = 0 for all J l(sl, S2,"" sn) = 9(S1, ... , Sk) + h(S2"'" sn). Combining with Htj = 0 for all j > k yields I(SI, S2,???, sn) = 9(SI, ... , Sk) + h(S3, ... , sn). Finally, inducting on i, from the constraints Ht. arrive at the desired functional form ~ I(SI,S2"",Sn) = 0 for all i < - k and J' > k , we = g(SI,. ",Sk) + h(Sk+l, ... ,Sn). Factorizing Multivariate Function Classes 565 More explicitly, a twice-differentiable function satisfies the set of coupled partial differential equations represented by the block diagonal structure of H if and only if it admits the corresponding separation of variables decomposition. By letting log p = f, the additive decomposition of f translates to a product decomposition of p. The more general decomposition into an arbitrary number of factors is obtained by iterative application of the above proposition. The special case of independent component analysis corresponds to a strict diagonalization of H. Thus, in the context of smooth joint distribution functions, pairwise conditional independence is necessary and sufficient for statistical independence. To use this information in a transformed "mixture" frame, we must understand how the matrix Hlog p transforms. From the relation between the mixture and source As, we have 8~; Aji 8~j' where we use Eincoordinate systems given by fl stein's convention of summation over repeated indices. From the relation between the joint distributions in the mixture and source frames, Ps(S) = IAIPx (fl), direct differentiation gives 8 2 10gps(S) _ A A 8 2 10gPx(i) = OSi 8s 1 . H .. L et tmg lJ = . 8 2 10gp (8) 8s; 8Sj an d H-lJ.. = ji - = kl 8x j 8 x k 8 2 10gp (x). Z 8x; 8Xj , m . . . matnx notatIOn we have H = AT if A. In other words, H is a second rank (symmetric) covariant tensor. The joint distribution admits a product decomposition in the source frame if and only if H and hence AT if A has the corresponding block diagonal structure. Thus multivariate function class factorization is solved by joint block diagonalization of symmetric matrices, with constraints on A of the form AjiifjkAkl = 0. Because the Hessian is symmetric, its diagonalization involves only (n choose 2) constraints. Consequently, in the independent component analysis case where the joint distribution function admits a factorization into one dimensional functions, if the mixing transformation is orthogonal, the independent component coordinate system will lie along the eigenvector directions of if. Generally however, n(n 1) independent constraints corresponding to information from the Hessian at two points are needed to determine the n arbitrary coordinate directions. 3 NUMERICAL EXPERIMENTS In the simplest attack on the factorization problem, we solve the constraint equations from two points simultaneously. The analytic solution is demonstrated in two dimensions. Without loss of generality, the mixing matrix A is taken to be of the form A=(~ ~) . The constraints from the two points are: ax + b(xy + 1) + cy = 0, and a'x + b'(xy + 1) + e'y = 0, where Hu = a, H21 = H12 = b and H22 = e at the first point, and the primed coefficients denote the values at the second point. Solving the simultaneous quadratic equations, we find x a'e - ae' ? v(ale - ae' )2 - 4(a' b - ab' ) (b'e - be') 2(ab' - a'b) 566 1. K. Lin y a'e - ae' ? v(ale - ae')2 - 4(a'b - ab')(b'e - be') 2(bc' - b'e) The ? double roots is indicative of the (x, y) ~ (l/y, l/x) symmetry in the equations, and together only give two distinct orientation solutions. These independent component orientation solutions are given by 81 = tan-l(l/x) and 82 = tan-ley). 3.1 Natural Audio Sources To demonstrate the analytic factorization solution, we present some proof of concept numerics. Generality is pursued over optimization concerns. First, we perform the standard separation of two linearly mixed natural audio sources. The input dataset consists of 32000 un-ordered datapoints, since no use will be made of the temporal information. The process for obtaining estimates of the Hessian matrix if is as follows. A histogram of the input distribution was first acquired and smoothed by a low-pass Gaussian mask in spatial-frequency space. The elements of if were then obtained via convolution with a discrete approximation of the derivative operator. The width of the Gaussian mask and the support of the derivative operator were chosen to reduce sensitivity to low spatial-frequency uncertainty. It should be noted that the analytic factorization solution makes no assumptions about the mixing transformation, consequently, a blind determination of the smoothing length scale is not possible because of the multiplicative degree of freedom in each source. Because of the need to take the logarithm of p before differentiation, or equivalently to divide by p afterwards, we set a threshold and only extracted information from points where the number of counts was greater than threshold. This is justified from a counting uncertainty perspective, and also from the understanding that regions with vanishing probability measure contain no information. With our sample of 32000 datapoints, we considered only the bin-points with a corresponding bin count greater than 30. From the 394 bin locations that satisfied this constraint, the solutions (8 1 Jh) for all (394 choose 2) = (394?393/2) pairs of the corresponding factorization equations are plotted in Fig. 1. A histogram ofthese solutions are shown in Fig. 2. The two peaks in the solution histogram correspond to orientations that differ from the two actual independent component orientations by 0.008 and 0.013 radians. The signal to mixture ratio of the two outputs generated from the solution are 158 and 49. 3.2 Effect of Noise Because the solution is analytic, uncertainty in the sampling just propagates through to the solution, giving rise to a finite width in the solution's distribution. We investigated the effect of noise and counting uncertainty by performing numerics starting from analytic forms for the source distributions. The joint distribution in the source frame was taken to be: Normalization is irrelevant since a function's decomposition into product form is preserved in scalar multiplication. This is also reflected in the equivalence between Hiogp and Hiog cp for e an arbitrary positive constant. The joint distribution in the mixture frame was obtained from the relation Px(x) = IAI-Ips(S'). To simulate Factorizing Multivariate Function Classes . . 567 ._ ... -'.o r' ~ . . . . ~. n/4 82 ..????? n:I..:-l re/4 ??-Tt/4 Figure 1: Scatterplot of the independent component orientation solutions. All unordered solution pairs ((h, (J2) are plotted. The solutions are taken in the range from -7r /2 to 7r /2. ~~~~r--------'r--------'.--------,--------~--------~--------~ ??_????n/2 ?_??. ?n!4 e n/4 rr/2 Figure 2: Histogram of the orientation solutions plotted in the previous figure. The range is still taken from -7r /2 to 7r /2, with the histogram wrapped around to ease the circular identification. The mixing matrix used was: all = 0.0514, a21 = 0.779, a12 = 0.930, a22 = -0.579, giving independent component orientations at -0.557 and 1.505 radians. Gaussian fit to the centers of the two solution peaks give -0.570 ? 0.066 and 1.513 ? 0.077 radians for the two orientations. sampling, Px (x) was multiplied with the number of samples M, onto which was added Gaussian distributed noise with amplitude given by the (M Px(X))1/2. This reflects the fact that counting uncertainty scales as the square root of the number of counts. The result was rounded to the nearest integer, with all negative count values set to zero. The subsequent processing coincided with that for natural audio sources. From the source distribution equation above, the minimum number of expected counts is M, and the maximum is 9M. The results in Figures 3 and 4 show that, as expected, increasing the number of samplings decreases the widths of the solution peaks. By fitting Gaussians to the two peaks, we find that the uncertainty (peak widths) in the independent component orientations changes from 0.06 to 0.1 radians as the sampling is decreased from for M = 20 to M = 2. So even with few samplings, a relatively accurate determination of the independent component coordinate system can be made. 1. K Lin 568 - IT.!::! e - rr/4 rr/ 2 rr/ 4 Figure 3: Histogram of the independent component orientation solutions for four different samplings. Solutions were generated from 20000 randomly chosen pairs of positions. The curves, from darkest to lightest, correspond to solutions for the noiseless, M = 20,11 and 2 simulations. The noiseless solution histogram curve extends to a height of approximately 15000 counts, and is accurate to the width of the bin. The slight scatter is due to discretization noise. Spikes at (} = 0 and -7r /2 correspond to pairs of positions which contain no information. . . . . . rrJ2 . . . . . . ~4- -?~---_ - -;r-- _ "?_- "? +- - -" _ - -r-- ?""? Y- " ?~'" e - +''---%-- .____4- - . . . . . . -+''.---3;-'' .. - .-3;- ---T... - -Y-".".+ - - - -rr./4 --rr./2 p ..... -T .........;!;- ,,- ?? ;E-.. ...-T ........;!;-??.. ... ;E-.. " .?. .;E-. . .. .;1;-- ???? .;!;- .. ?? ;E- . . . . . . ",..... -r ..".. .;E- . .-1'-" .. '" .... -:E-. os "" . ... -:E-. ..:r ??cr? = = = M Figure 4: The centers and widths of the solution peaks as a function of the minimum expected number of counts M . From the source distribution, the maximum expected number of counts is 9M. Information was only extracted from regions with more than 2M counts. The actual independent component orientation as determined from the mixing matrix A are shown by the two dashed lines. The solutions are very accurate even for small samplings. 4 RELATION TO DECORRELATION Ideally, if a mixed tensor (transforms as J = A-I j A) with the full degrees of freedom can be found which is diagonal if and only if the joint distribution appears in product form, then the independent component coordinate directions will coincide with that of the tensor's eigenvectors. However, the preceding analysis shows that a maximum of n(n -1)/2 constraints contain all the information that exists locally. This, however, provides a nice connection with decorrelation. Starting with the characteristic function of log p(i) , ?(k) = the off diagonal terms of Hlogp are given by J ei k.;E logp(X) di, which can loosely be seen as the second order cross-moments in ?(k). Thus di- Factorizing Multivariate Function Classes 569 agonalization of Hiog P roughly translates into decor relation in ?(k). It should be noted that ?(k) is not a proper distribution function. In fact, it is a complex valued function with ?ek) = ?* (-k). Consequently, the summation in the above equation is not an expectation value, and needs to be interpreted as a superposition of plane waves with specified wavelengths, amplitudes and phases. 5 DISCUSSION The introduced functional decomposition defines a generalization of independent component analysis which is valid for all multivariate functions. A rigorous notion of the decomposition of a multivariate function into a set of lower dimensional factors is presented. With only the assumption of local twice differentiability, we derive an analytic solution for this factorization [1]. A new algorithm is presented, which in contrast to iterative non-local parametric density estimation ICA algorithms [2, 3, 4], performs the decomposition analytically using local geometric information. The analytic nature of this approach allows for a proper treatment of source separation in the presence of uncertainty, while the local nature allows for a local determination of the source coordinate system. This leaves open the possibility of describing a position dependent independent component coordinate system with local linear coordinates patches. The presented class factorization formalism removes the decomposition assumptions needed for independent component analysis, and reinforces the well known fact that sources are recoverable only up to linear transformation. By modifying the equivalence class relation, a rich underlying algebraic structure with both multiplication and addition can be constructed. Also, it is clear that the matrix of second derivatives reveals an even more general combinatorial undirected graphical structure of the multivariate function. These topics, as well as uniqueness issues of the factorization will be addressed elsewhere [5]. The author is grateful to Jack Cowan, David Grier and Robert Wald for many invaluable discussions. References [1] J. K. Lin, Local Independent Component Analysis, Ph. D. thesis, University of Chicago, 1997. [2] A. J. Bell and T. J . Sejnowski, Neural Computation 7, 1129 (1995). [3] S. Amari, A. Cichocki, and H. Yang, in Advances in Neural and Information Processing Systems, 8, edited by D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo (MIT Press, Cambridge, MA, 1996), pp. 757-763. [4] B. A. Pearlmutter and L. Parra, in Advances in Neural and Information Processing Systems, 9, edited by M. C. Mozer, M. I. Jordan, and T. Petsche (MIT Press, Cambridge, MA, 1997), pp. 613-619. [5] J. K. Lin, Graphical Structure of Multivariate Functions, in preparation. 

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An Improved Policy Iteratioll Algorithm for Partially Observable MDPs Eric A. Hansen Computer Science Department University of Massachusetts Amherst, MA 01003 hansen@cs.umass.edu Abstract A new policy iteration algorithm for partially observable Markov decision processes is presented that is simpler and more efficient than an earlier policy iteration algorithm of Sondik (1971,1978). The key simplification is representation of a policy as a finite-state controller. This representation makes policy evaluation straightforward. The paper's contribution is to show that the dynamic-programming update used in the policy improvement step can be interpreted as the transformation of a finite-state controller into an improved finite-state controller. The new algorithm consistently outperforms value iteration as an approach to solving infinite-horizon problems. 1 Introduction A partially observable Markov decision process (POMDP) is a generalization of the standard completely observable Markov decision process that allows imperfect information about the state of the system. First studied as a model of decision-making in operations research, it has recently been used as a framework for decision-theoretic planning and reinforcement learning with hidden state (Monahan, 1982; Cassandra, Kaelbling, & Littman, 1994; Jaakkola, Singh, & Jordan, 1995). Value iteration and policy iteration algorithms for POMDPs were first developed by Sondik and rely on a piecewise linear and convex representation of the value function (Sondik, 1971; Smallwood & Sondik,1973; Sondik, 1978). Sondik's policy iteration algorithm has proved to be impractical, however, because its policy evaluation step is extremely complicated and difficult to implement. As a result, almost all subsequent work on dynamic programming for POMDPs has used value iteration. In this paper, we describe an improved policy iteration algorithm for POMDPs that avoids the difficulties of Sondik's algorithm. We show that these difficulties hinge on the choice of a policy representation and can be avoided by representing a policy as a finite-state E. A. Hansen 1016 controller. This representation makes the policy evaluation step easy to implement and efficient. We show that the policy improvement step can be interpreted in a natural way as the transformation of a finite-state controller into an improved finite-state controller. Although it is not always possible to represent an optimal policy for an infinite-horizon POMDP as a finite-state controller, it is always possible to do so when the optimal value function is piecewise linear and convex. Therefore representation of a poiicy as a finite-state controller is no more limiting than representation of the value function as piecewise linear and convex. In fact, it is the close relationship between representation of a policy as a finite-state controller and representation of a value function as piecewise linear and convex that the new algorithm successfully exploits. The paper is organized as follows. Section 2 briefly reviews the POMDP model and Sondik's policy iteration algorithm. Section 3 describes an improved policy iteration algorithm. Section 4 illustrates the algorithm with a simple example and reports a comparison of its performance to value iteration. The paper concludes with a discussion of the significance of this work. 2 Background Consider a discrete-time POMDP with a finite set of states 5, a finite set of actions A, and a finite set of observations e. Each time period, the system is in some state i E 5, an agent chooses an action a E A for which it receives a reward with expected value ri, the system makes a transition to state j E 5 with probability pij' and the agent observes () E e with probability tje. We assume the performance objective is to maximize expected total discounted reward over an infinite horizon. Although the state of the system cannot be directly observed, the probability that it is in a given state can be calculated. Let 7r denote a vector of state probabilities, called an information state, where 7ri denotes the probability that the system is in state i. If action a is taken in information state 7r and () is observed, the successor information state is determined by revising each state probability using Bayes' theorem: trj = LiEs 7riPijQje/ Li,jES 7riPijQje' Geometrically, each information state 7r is a point in the (151 - I)-dimensional unit simplex, denoted II. It is well-known that an information state 7r is a sufficient statistic that summarizes all information about the history of a POMDP necessary for optimal action selection. Therefore a POMDP can be recast as a completely observable MDP with a continuous state space II and it can be theoretically solved using dynamic programming. The key to practical implementation of a dynamic-programming algorithm is a piecewise-linear and convex representation of the value function. Smallwood and Sondik (1973) show that the dynamic-programming update for POMDPs preserves the piecewise linearity and convexity of the value function. They also show that an optimal value function fot a finite-horizon POMDP is always piecewise linear and convex. For infinite-horizon POMDPs, Sondik (1978) shows that an optimal value function is sometimes piecewise linear and convex and can be aproximated arbitrarily closely by a piecewise linear and convex function otherwise. A piecewise linear and convex value function V can be represented by a finite set of lSI-dimensional vectors, r = {aO,a i , ?.. }, such that V(7r) = maxkLi s7riaf. A dynamic-programming update transforms a value function V representedEfiy a set r of a-vectors into an improved value function V' represented by a set r' of a-vectors. Each possible a-vector in r' corresponds to choice of an action, and for each possible observation, choice of a successor vector in r. Given the combinatorial number of choices that can be made, the maximum n4mber of vectors in r' is IAllfll91. However most of these potential vectors are not needed to define the updated value function and can be pruned. Thus the dynamic-programming update problem is to find a An Improved Policy Iteration Algorithmfor Partially Observable MDPs J017 minimal set of vectors r' that represents V', given a set of vectors r that represents V . Several algorithms for performing this dynamic-programming update have been developed but describing them is beyond the scope of this paper. Any algorithm for performing the dynamic-programming update can be used in the policy improvement step of policy iteration. The algorithm that is presently the fastest is described by (Cassandra, Littman, & Zhang, 1997). For value iteration, it is sufficient to have a representation of the value function because a policy is defined implicitly by the value function, as follows, 8(11") = a(arg mF L 1I"i o f), (1) iES where a(k) denotes the action associated with vector ok. But for policy iteration, a policy must be represented independently of the value function because the policy evaluation step computes the value function of a given policy. Sondik's choice of a policy representation is influenced by Blackwell's proof that for a continuous-space infinite-horizon MDP, there is a stationary, deterministic Markov policy that is optimal (Blackwell, 1965). Based on this result, Sondik restricts policy space to stationary and deterministic Markov policies that map the continuum of information space II into action space A. Because it is important for a policy to have a finite representation, Sondik defines an admissible policy as a mapping from a finite number of polyhedral regions of II to A . Each region is represented by a set of linear inequalities, where each linear inequality corresponds to a boundary of the region. This is Sondik's canonical representation of a policy, but his policy iteration algorithm makes use of two other representations. In the policy evaluation step, he converts a policy from this representation to an equivalent, or approximately equivalent, finitestate controller. Although no method is known for computing the value function of a policy represented as a mapping from II to A, the value function of a finite-state controller can be computed in a straightforward way. In the policy improvement step, Sondik converts a policy represented implicitly by the updated value function and equation (1) back to his canonical representation. The complexity of translating between these different policy representations - especially in the policy evaluation step - makes Sondik's policy iteration algorithm difficult to implement and explains why it is not used in practice. 3 Algorithm We now show that policy iteration for POMDPs can be simplified - both conceptually and computationally - by using a single representation of a policy as a finite-state controller. 3.1 Policy evaluation As Sondik recognized, policy evaluation is straightforward when a policy is represented as a finite-state controller. An o-vector representation of the value function of a finitestate controller is computed by solving the system of linear equations, k _ 0i - a(k) ri + (3'"' a(k) a(k) s(k ,8) L.JPij qj8 OJ , (2) j ,8 where k is an index of a state of the finite-state controller, a(k) is the action associated with machine state k, and s(k,O) is the index of the successor machine state if 0 is observed. This value function is convex as well as piecewise linear because the expected value of an information state is determined by assuming the controller is started in the machine state that optimizes it. E. A. Hansen 1018 1. Specify an initial finite-state controller, <5, and select f. for detecting conver- gence to an f.-optimal policy. 2. Policy evaluation: Calculate a set r of a-vectors that represents the value function for <5 by solving the system of equations given by equation 2. 3. Policy improvement: Perform a dynamic-programming update and use the new set of vectors r' to transform <5 into a new finite-state controller, <5', as follows: (a) For each vector a in r': l. If the action and successor links associated with a duplicate those of a machine state of <5, then keep that machine state unchanged in 8'. ii. Else if a pointwise dominates a vector associated with a machine state of <5, change the action and successor links of that machine state to those used to create a. (If it pointwise dominates the vectors of more than one machine state, they can be combined into a single machine state.) iii. Otherwise add a machine state to <5' that has the same action and successor links used to create a. (b) Prune any machine state for which there is no corresponding vector in r', as long as it is not reachable from a machine state to which a vector in r' does correspond. 4. Termination test. If the Bellman residual is less than or equal to f.(1 - /3)//3, exit with f.-optimal policy. Otherwise set <5 to <5' and go to step 2. Figure 1: Policy iteration algorithm. 3.2 Policy improvement The policy improvement step uses the dynamic-programming update to transform a value function V represented by a set r of a-vectors into an improved value function V' represented by a set r' of a-vectors. We now show that the dynamic-programming update can also be interpreted as the transformation of a finite-state controller 8 into an improved finite-state controller <5'. The transformation is made based on a simple comparison of r' and r. First note that some of the a-vectors in r' are duplicates of a-vectors in r, that is, their action and successor links match (and their vector values are pointwise equal). Any machine state of <5 for which there is a duplicate vector in r' is left unchanged. The vectors in r' that are not duplicates of vectors in r indicate how to change the finite-state controller. If a non-duplicate vector in r' pointwise dominates a vector in r, the machine state that corresponds to the pointwise dominated vector in r is changed so that its action and successor links match those of the dominating vector in r'. If a non-duplicate vector in r' does not pointwise dominate a vector in r, a machine state is added to the finite-state controller with the same action and successor links used to generate the vector. There may be some machine states for which there is no corresponding vector in r' and they can be pruned, but only if they are not reachable from a machine state that corresponds to a vector in r'. This last point is important because it preserves the integrity of the finite-state controller. A policy iteration algorithm that uses these simple transformations to change a finitestate controller in the policy improvement step is summarized in Figure 1. An algorithm that performs this transformation is easy to implement and runs very efficiently because it simply compares the a-vectors in r' to the a-vectors in r and modifies the finite-state controller accordingly. The policy evaluation step is invoked to compute the value function of the transformed finite-state controller. (This is only necessary An Improved Policy Iteration Algorithmfor Partially Observable MDPs 1019 if a machine state has been changed, not if machine states have simply been added.) It is easy to show that the value function of the transformed finite-state controller /j' dominates the value function of the original finite-state controller, /j, and we omit the proof which appears in (Hansen, 1998). Theorem 1 If a finite-state controller is not optimal, policy improvement transforms it into a finite-state controller with a value function that is as good or better for every information state and better for some information state. 3.3 Convergence If a finite-state controller cannot be improved in the policy improvement step (Le., all the vectors in r' are duplicates of vectors in r), it must be optimal because the value function satisfies the optimality equation. However policy iteration does not necessarily converge to an optimal finite-state controller after a finite number of iterations because there is not necessarily an optimal finite-state controller. Therefore we use the same stopping condition used by Sondik to detect t-optimality: a finite-state controller is t-optimal when the Bellman residual is less than or equal to t(l- {3) / {3, where {3 denotes the discount factor. Representation of a policy as a finite-state controller makes the following proof straightforward (Hansen, 1998). Theorem 2 Policy iteration converges to an t-optimal finite-state controller after a finite number of iterations. 4 Example and performance We illustrate the algorithm using the same example used by Sondik: a simple twostate, two-action, two-observation POMDP that models the problem of finding an optimal marketing strategy given imperfect information about consumer preferences (Sondik,1971,1978). The two states of the problem represent consumer preference or lack of preference for the manufacturers brand; let B denote brand preference and ....,B denote lack of brand preference. Although consumer preferences cannot be observed, they can be infered based on observed purchasing behavior; let P denote purchase of the product and let ....,p denote no purchase. There are two marketing alternatives or actions; the company can market a luxury version of the product (L) or a standard version (S). The luxury version is more expensive to market but can bring greater profit. Marketing the luxury version also increases brand preference. However consumers are more likely to purchase the less expensive, standard product. The transition probabilities, observation probabilities, and reward function for this example are shown in Figure 2. The discount factor is 0.9. Both Sondik's policy iteration algorithm and the new policy iteration algorithm converge in three iterations from a starting policy that is equivalent to the finite-state AClions Markel luxury producl (L) Markel slandard producl (S) Transilion probabililies B -B B/O.8/0.2\ -B 0.5 0.5 B -B -BB~ 0.4 o. Observalion probabililies P -p B10.81 0.2\ -B 0.60.4 P Expecled reward -BB?j ?4 -p -BB~ O. 0. -BBbj ?3 Figure 2: Parameters for marketing example of Sondik (1971,1978) . E A. Hansen 1020 . ~; " -~ '''' .. " a=L \ ~ 9,96 : '- 18.86 <,8=-P '~~9~~_:~~~;~<~._ = S " \\ 14.82! \ : ' .1 : \ 18.20 / \ ', __ __ - '~~ P \., ''', ,.;,\ /""---.. ,9=-p,y: " a.= S \ .. ,'" : 14.86 t ____ . . \_: 8.1~ / 8=P (.) (b) (e) (d) (e) Figure 3: (a) shows the initial finite-state controller, (b) uses dashed circles to show the vectors in r' generated in the first policy improvement step and (c) shows the transformed finite-state controller, (d) uses dashed circles to show the vectors in r' generated in the second policy improvement step and (e) shows the transformed finite-state controller after policy evaluation. The optimality of this finite-state controller is detected on the third iteration, which is not shown. Arcs are labeled with one of two possible observations and machine states are labeled with one of two possible actions and a 2-dimensional vector that contains a value for each of the two possible system states. controller shown in Figure 3a. Figure 3 shows how the initial finite-state controller is transformed into an optimal finite-state controller by the new algorithm. In the first iteration, the updated set of vectors r' (indicated by dashed circles in Figure 3b) includes two duplicate vectors and one non-duplicate that results in an added machine state. Figure 3c shows the improved finite-state controller after the first iteration. In the second iteration, each of the three vectors in the updated set of vectors r' (indicated by dashed circles in Figure 3d) pointwise dominates a vector that corresponds to a current machine state. Thus each of these machine states is changed. Figure 4e shows the improved finite-state controller after the second iteration. The optimality of this finite-state controller is detected in the third iteration. This is the only example for which Sondik reports using policy iteration to find an optimal policy. For POMDPs with more than two states, Sondik's algorithm is especially difficult to implement. Sondik reports that his algorithm finds a suboptimal policy for an example described in (Smallwood & Sondik, 1973). No further computational experience with his algorithm has been reported. The new policy iteration algorithm described in this paper easily finds an optimal finite-state controller for the example described in (Smallwood & Sondik, 1973) and has been used to solve many other POMDPs. In fact, it consistently outperforms value iteration. We compared its performance to the performance of value iteration on a suite of ten POMDPs that represent a range of problem sizes for which exact dynamicprogramming updates are currently feasible. (Presently, exact dynamic-prorgramming updates are not feasible for POMDPs with more than about ten or fifteen states, actions, or observations.) Starting from the same point, we measured how soon each algorithm converged to f-optimality for f values of 10.0, 1.0, 0.1 , and 0.01. Policy iteration was consistently faster than value iteration by a factor that ranged from a low of about 10 times faster to a high of over 120 times faster. On average, its rate of convergence was between 40 and 50 times faster than value iteration for this set of examples. The finite-state controllers it found had as many as several hundred machine states, although optimal finite-state controllers were sometimes found with just a few machine states. An Improved Policy Iteration Algorithm for Partially Observable MDPs 5 1021 Discussion We have demonstrated that the dynamic-programming update for POMDPs can be interpreted as the improvement of a finite-state controller. This interpretation can be applied to both value iteration and policy iteration. It provides no computational speedup for value iteration, but for policy iteration it results in substantial speedup by making policy evaluation straightforward and easy to implement. This representation also has the advantage that it makes a policy easier to understand and execute than representation as a mapping from regions of information space to actions. In particular, a policy can be executed without maintaining an information state at run-time. It is well-known that policy iteration converges to f-optimality (or optimality) in fewer iterations than value iteration. For completely observable MDPs, this is not a clear advantage because the policy evaluation step is more computationally expensive than the dynamic-programming update. But for POMDPs, policy evaluation has loworder polynomial complexity compared to the worst-case exponential complexity of the dynamic-programming update (Littman et al., 1995). Therefore, policy iteration appears to have a clearer advantage over value iteration for POMDPs. Preliminary testing bears this out and suggests that policy iteration significantly outperforms value iteration as an approach to solving infinite-horizon POMDPs. Acknowledgements Thanks to Shlomo Zilberstein and especially Michael Littman for helpful discussions. Support for this work was provided in part by the National Science Foundation under grants IRI-9409827 and IRI-9624992. References Blackwell, D. {1965} Discounted dynamic programming. Ann. Math. Stat. 36:226235. Cassandra, A.; Kaelbling, L.P.; Littman, M.L. {1994} Acting optimally in partially observable stochastic domains. In Proc. 13th National Conf. on AI, 1023-1028. Cassandra, A.; Littman, M.L.; & Zhang, N.L. (1997) Incremental pruning: A simple, fast, exact algorithm for partially observable Markov decision processes. In Proc. 13th A nnual Con/. on Uncertainty in AI. Hansen, E.A. (1998). Finite-Memory Control of Partially Observable Systems. PhD thesis, Department of Computer Science, University of Massachusetts at Amherst. Jaakkola, T.; Singh, S.P. ; & Jordan, M.I. (1995) Reinforcement learning algorithm for partially observable Markov decision problems. In NIPS-7. Littman, M.L.; Cassandra, A.R.; & Kaebling, L.P. (1995) Efficient dynamicprogramming updates in partially observable Markov decision processes. Computer Science Technical Report CS-95-19, Brown University. Monahan, G.E. (1982) A survey of partially observable Markov decision processes: Theory, models, and algorithms. Management Science 28:1-16. Smallwood, R.D. & Sondik, E.J. (1973) The optimal control of partially observable Markov processes over a finite horizon. Operations Research 21:1071-1088. Sondik, E.J. (1971) The Optimal Control of Partially Observable Markov Processes. PhD thesis, Department of Electrical Engineering, Stanford University. Sondik, E.J. (1978) The optimal control of partially observable Markov processes over the infinite horizon: Discounted costs. Operations Research 26:282-304. 

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Adaptation in Speech Motor Control John F. Houde* UCSF Keck Center Box 0732 San Francisco, CA 94143 Michael I. Jordan MIT Dept. of Brain and Cognitive Sci. EI0-034D Cambridge, MA 02139 houde~phy.ucsf.edu jordan~psyche.mit.edu Abstract Human subjects are known to adapt their motor behavior to a shift of the visual field brought about by wearing prism glasses over their eyes. We have studied the analog of this effect in speech. Using a device that can feed back transformed speech signals in real time, we exposed subjects to alterations of their own speech feedback. We found that speakers learn to adjust their production of a vowel to compensate for feedback alterations that change the vowel's perceived phonetic identity; moreover, the effect generalizes across consonant contexts and to different vowels. 1 INTRODUCTION For more than a century, it has been know that humans will adapt their reaches to altered visual feedback [8]. One of the most studied examples of this adaptation is prism adaptation, which is seen when a subject reaches to targets while wearing image-shifting prism glasses [2]. Initially, the subject misses the targets, but he soon learns to compensate and reach accurately. This compensation is retained beyond the time that the glasses are worn: when the glasses are removed, the subject's reaches now overshoot targets in the direction that he compensated. This retained compensation is called adaptation, and its generation from exposure to altered sensory feedback is called sensorimotor adaptation (SA). In the study reported here, we investigated whether SA could be observed in a motor task that is quite different from reaching - speech production. Specifically, we examined whether the control of phonetically relevant speech features would respond adaptively to altered auditory feedback. By itself, this is an important theoretical question because various aspects of speech production have already been shown to be sensitive to auditory feedback [5, 1, 4]. Moreover, we were particularly *To whom correspondence should be addressed. 39 Adaptation in Speech Motor Control interested in whether speech SA would also exhibit generalization. If so, speech SA could be used to examine the organization of speech motor control. For example, suppose we observed adaptation of [c) in "get". We could then examine whether we also see adaptation of [c) in "peg". IT so, then producing [c) in the two different words must access a common, adapted representation - evidence for a hierarchical speech production system in which word productions are composed from smaller units such as phonemes. We could also examine whether adapting [c) in "get" causes adaptation of [re) in "gat". IT so, then the production representations of [c) and [re] could not be independent, supporting the idea that vowels are produced by controlling a common set of features. Such theories about the organization of the speech production system have been postulated in phonology and phonetics, but the empirical evidence supporting these theories has generally been observational and hence not entirely conclusive [7,6]. 2 METHODS To study speech SA, we focused on vowel production because the phonetically relevant features of vowel sounds are formant frequencies, which are feasible to alter in real time. 1 To alter the formants of a subject's speech feedback, we built the apparatus shown in Figure 1. The subject wears earphones and a microphone and sits in front of a PC video monitor that presents words to be spoken aloud. The signal from the microphone is sent to a Digital Signal Processing board, which collects a 64ms time interval from which a magnitude spectrum is calculated. From this spectrum, formant frequencies and amplitudes are estimated. To alter the speech, the first three formant frequencies are shifted, and the shifted formants drive a formant synthesizer that creates the output speech sent to the subject's earphones. This analysis-synthesis process was accomplished with only 16ms of feedback delay. To minimize how much the subject directly heard of his own voice via bone conduction, the subject produced only whispered speech, masked with mild noise. altered feedback Altered Fonnants I earphones. ;-------... .... I ?. / pep DSP board in PC I ? PC video monitor microphone .' .'" L....-_~ ..... Fl.F2.F3 Frequency Alteration. '. Fonnants V\IFonnant /\ ?Estimation I L.....---' . . ~. Magnitude Spectrum Figure 1: The apparatus used in the study. For each subject in our experiment, we shifted formants along the path defined by the (F1,F2,F3) frequencies of a subject's productions of the vowels [i) , [t.]' [c], [re], 1 See [3] for detailed discussion of the methods used in this study. J. F. Houde and M. I Jordan 40 and [a].2 Figure 2 shows examples of this shifting process in (Fl,F2) space for the feedback transformations that were used in the study. TOIshift formants along the subject's [i]-[a] path, we extend the path at both ends and we number the endpoints and vowels to make a path position measure that normalizes the distances between vowels. The formants of each speech sound F produced by the subject were then re-represented in terms of path projection - the path posiUon of nearest path point P, and path deviation - the distance D to this point P. Feedback transformations were constructed to alter path projections while preserving path deviations. Two different transformations were used. The +2.0 transformation added 2.0 to path projections: under this transform, if the subject produced speech sound F (a sound near [cD, he heard instead sound F+ (a sound near [aD. The subject could compensate for this transform and hear sound F only by shifting his production of F to F- (a sound near [iD. The -2.0 transformation subtracted 2.0 from path projections: under this transformation, if the subject produced F, he heard F-. Thus, in this case, the subject could compensate by shifting production to F+. F- F+ F+ [ah)O 5 end /)6 end /)6 Fl Fl (a) +2.0 Transformation (b) -2.0 Transformation Figure 2: Feedback transformations used in the study. These feedback transformations were used in an experiment in which a subject was visually prompted to whisper words with a 300ms target duration_ Word promptings occurred in groups of ten called epochs. Within each epoch, the first six words came from a set of training words and the last four came from a set of testing words. The subject heard feedback of his first five word productions in each epoch, while masking noise blocked his hearing for his remaining five word productions in the epoch. Thus, the subject only heard feedback of his production of the first five training words and never heard his productions of the testing words. 2Where possible, we use standard phonetic symbols for vowel sounds: [i] as in "seat", [L] as in "hit", [c] as in "get", [re] as in "hat", and [a] as in "pop". Where font limitations prevent us from using these symbols, we use the alternate notation of [i], [ib], [eh] , rae], and lab], respectively, for the same vowel sounds. Adaptation in Speech Motor Control 41 The experiment lasted 2 hours and consisted of 422 epochs divided over five phases: 1. A 10 minute warmup phase used to acclimate the subject to the experimen- tal setup. 2. A 17 minute baseline phase used to measure formants of the subject's normal vowel productions. 3. A 20 minute ramp phase in which the subject's feedback was increasingly altered up to a maximum value. 4. A 1 hour training phase in which the subject produced words while the feedback was maximally altered. 5. A 17 minute test phase used to measure formants of the subject's postexposure vowel productions while his feedback was maximally altered. By the end of the ramp phase, feedback alteration reached its maximum strength, which was +2.0 for half the subjects and -2.0 for the other subjects. In addition, all subjects were run in a control experiment in which feedback was never altered. The two word sets from which prompted words were selected were both sets of eve words. Training words (in which adaptation was induced) were all bilabials with [c] as the vowel ("pep", "peb", "bep", and "beb"). Testing words (in which generalization of the training word adaptation was measured) were divided into two subsets, each designed to measure a different type of generalization: (1) context generalization words, which had the same vowel [c] as the training words but varied the consonant context ("peg", "gep", and "teg"); (2) vowel target generalization words, which had the same consonant context as the training words but varied the vowel ("pip,", "peep,", "pap" , and "pop"). Eight male MIT students participated in the study. All were native speakers of North American English and all were naive to the purpose of the study. 3 RESULTS To illustrate how we measured compensation and adaptation in the experiments, we first show the results for an individual subject. Figure 3 shows (F1,F2) plots of response of subject OB in both the adaptation experiment (in which he was exposed to the -2.0 feedback transformation) and the control experiment. In each figure, the dotted line is OB's [i]-[a] path. Figure 3(a) shows OB's compensation responses, which were measured from his productions of the training words made when he heard feedback of his whispering. The solid arrow labeled "-2.0 xform" shows how much his mean vowel formants changed (testing phase - baseline phase) after being exposed to the -2.0 feedback transformation. It shows he shifted his production of [c] to something a bit past [re], which corresponds to a p;;tth projection change of slightly more than one vowel interval towards [a]. Thus, since the path projection shift of the transform was -2.0 (2.0 vowel intervals towards liD, the figure shows that OB compensates for over half the action of the transformation. The hollow arrow in Figure 3(a) shows how OB heard his compensation. It shows he heard his actual production shift from [c] towards [a] as a shift from [i] back towards [c]. Figure 3(b) shows how much of OB's compensation was retained when he whispered the training words with feedback blocked by noise .. This retained compensation is called adaptation, and it was measured from path projection changes by the same method used to measure compensation. In the figure, we see OB's adaptation J. F. Houde and M. I Jordan 42 response (the solid "-2.0 xform" arrow) is a path projection shift of slightly less than one vowel interval, so his adaptation is slightly less than half. Thus, the figure shows that OB retains an appreciable amount of his compensation in the absence of feedback. Finally, in both plots of Figure 3, the almost non-existent "control" arrows show that OB exhibited almost no formant change in the control experiment - as we would expect since feedback was never altered in this experiment. 2600 2600 2400 2400 [i) ?. [i) ~. :~ 2200 N 2000 .\\. N 1800 \. control "': ~ ?2.0 dorm C'I "' [ab) ? .. [eb) '-. ~ . 1800 control , \. 2.0 dorm ~ [ae) \ 1600 '. 1400 [ib)", '-' rae) \ 1600 \~ 2000 X [eb;??? ~ ~ X C'I '\\ 2200 \ [ab)? 1400 1200 1200 300 SOO 700 900 Fl (Hz) (a) compensation II 00 300 SOO 700 900 1100 Fl (Hz) (b) adaptation Figure 3: Subject OB compensation and adaptation. The plots in Figure 4 show that there was significant compensation and adaptation across all subjects. In these plots, the vertical scale indicates how much the changes in mean vowel formants (testing phase - baseline phase) in each subject's productions of the training words compensated for the action of the feedback transformation he was exposed to. The filled circles linked by the solid line show compensation (Figure 4(a? and retained compensation, or adaptation (Figure 4(b)) across subjects in the adaptation experiment in which feedback was altered; the open circles linked by the dotted line show the same measures from the control experiment in which feedback was not altered. (The solid and dotted lines facilitate comparison of results across subjects but do not signify any relationship between subjects.) In the control experiment, for each subject, compensation and adaptation were measured with respect to the feedback transformation used in the adaptation experiment. The plots show that there are large variations in compensation and adaptation across subjects, but overall there was significantly more compensation (p < 0.006) and adaptation (p < 0.023) in the adaptation experiments that in the control experiments. Figure 5 shows plots of how much of the adaptation observed in the training words carried over the the testing words. For each testing word shown, a measure of this carryover called mean generalization is plotted, which was calculated as a ratio of adaptations: the adaptation seen in the testing word divided by the adaptation seen in the training words (adaptation values observed in the control experiment were Adaptation in Speech Motor Control 43 1.2 1.2 1.0 1.0 0 .80 0.80 0 .60 0 .60 0.40 ? ; 0.20 \ /rs \" /i 0.00 0.40 ro 0.20 \~_ vs .J: . .~ ah ty -0.20 vs sr 0.00 sr -0.20 ro ab cw -0.40 -0 .40 (a) mean compensation (b) mean adaptatioll Figure 4: Mean compensation and adaptation across all subjects. subtracted out to remove any effects not arising from exposure to altered feedback). Figure 5(a) shows mean generalization for the context generalization words except for "pep" (since "pep" was also a training word) . The plot shows large variance in mean generalization for each of the three words, but overall there was significant (p < 0.040) mean generalization. Thus, there was significant carryover of the adaptation of [c] in the training words to different consonant contexts. Figure 5(b) shows mean generalization for the vowel target generalization words. Not all of these words are shown: unfortunately, we weren't able to accurately estimate the formants of [i] and [a], so "peep" and "pop" were dropped from our generalization analysis. For the remaining two vowel target generalization words, the plot shows large variance in mean generalization for each of the words, but overall there was significant (p < 0.013) mean generalization. Thus, there was significant generalization of the adaptation of [c] to the vowels [t] and [eel. 4 DISCUSSION Several conclusions can be drawn from the experiment described above. First, comparison of the adaptation and control experiment results seen in Figure 4 shows a clear effect of exposure to the altered feedback: this exposure caused compensation responses in most subjects. Furthermore, the adaptation results show that this compensation was retained in absence of acoustic feedback. Next, the context generalization results seen in Figure 5(a) show that some adapted representation of [c] is shared across the training and testing words. These results provide evidence for a hierarchical speech production system in which words are composed from smaller phoneme-like units. Finally, the vowel target generalization results seen in Figure 5(b) show that the production representations of [t], [cl, and [ee] are not independent, suggesting that these vowels are produced by controlling a common set of features. 1. F. Houde and M. I. Jordan 44 2.0 2.0 1.5 1.5 peg .,: Q) .,: 1.0 0 c: <'S Q) ~ gep teg 0 c: <'S 0.50 pap 1.0 Q) Q) ~ 0.50 0.00 0.00 -0.50 ?0.50 (a) context gen. words pip A ~ (b) vowel target gen. words Figure 5: Mean generalization for the testing words, averaged across subjects. Thus, in summary, our study has shown (1) that speech production, like reaching, can be made to exhibit sensorimotor adaptation, and (2) that this adaptation effect exhibits generalization that can be used to make inferences about the structure of the speech production system. Acknowledgments We thank J. Perkell, K. Stevens, R. Held and P. Sabes for helpful discussions. References [1] V. L. Gracco et al., (1994) J. Acoust. Soc. Am. 95:2821 [2] H. V. Helmholtz, (1867) Treatise on physiological optics, Vol. 3 (Eng. Trans. by Optical Soc. of America, Rochester, NY, 1925) [3] J. F. Houde (1997), Sensorimotor Adaptation in Speech Production, Doctoral Dissertation, M. I. T., Cambridge, MA. [4) H. Kawahara (1993) J. Acoust. Soc. Am. 94:1883. [5] B. S. Lee (1950) J. Acoust. Soc. Am. 22:639. [6] W. J. M. Levelt (1989), Speaking: from intention to articulation, MIT Press, Cambridge, MA. [7] A. S. Meyer (1992), Cognition 42:18l. [8) R. B. Welch (1986), Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, J. P. Thomas Eds., John Wiley and Sons, New York. 

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459 A BIFURCATION THEORY APPROACH TO THE PROGRAMMING OF PERIODIC A TTRACTORS IN NETWORK MODELS OF OLFACTORY CORTEX Bill Baird Department of Biophysics U.C. Berkeley ABSTRACT A new learning algorithm for the storage of static and periodic attractors in biologically inspired recurrent analog neural networks is introduced. For a network of n nodes, n static or n/2 periodic attractors may be stored. The algorithm allows programming of the network vector field independent of the patterns to be stored. Stability of patterns, basin geometry, and rates of convergence may be controlled. For orthonormal patterns, the l~grning operation reduces to a kind of periodic outer product rule that allows local, additive, commutative, incremental learning. Standing or traveling wave cycles may be stored to mimic the kind of oscillating spatial patterns that appear in the neural activity of the olfactory bulb and prepyriform cortex during inspiration and suffice, in the bulb, to predict the pattern recognition behavior of rabbits in classical conditioning experiments. These attractors arise, during simulated inspiration, through a multiple Hopf bifurcation, which can act as a critical "decision pOint" for their selection by a very small input pattern. INTRODUCTION This approach allows the construction of biological models and the exploration of engineering or cognitive networks that employ the type of dynamics found in the brain. Patterns of 40 to 80 hz oscillation have been observed in the large scale activity of the olfactory bulb and cortex(Freeman and Baird 86) and even visual neocortex(Freeman 87,Grey and Singer 88), and found to predict the olfactory and visual pattern recognition responses of a trained animal. Here we use analytic methods of bifurcation theory to design algorithms for determining synaptic weights in recurrent network architectures, like those 460 Baird found in olfactory cortex, for associative memory storage of these kinds of dynamic patterns. The "projection algorithm" introduced here employs higher order correlations, and is the most analytically transparent of the algorithms to come from the bifurcation theory approach(Baird 88). Alternative numerical algorithms employing unused capacity or hidden units instead of higher order correlations are discussed in (Baird 89). All of these methods provide solutions to the problem of storing exact analog attractors, static or dynamic, in recurrent neural networks, and allow programming of the ambient vector field independent of the patterns to be stored. The stability of cycles or equilibria, geometry of basins of attraction, rates of convergence to attractors, and the location in parameter space of primary and secondary bifurcations can be programmed in a prototype vector field - the normal form. To store cycles by the projection algorithm, we start with the amplitude equations of a polar coordinate normal form, with coupling coefficients chosen to give stable fixed points on the axes, and transform to Cartesian coordinates. The axes of this system of nonlinear ordinary differential equations are then linearly transformed into desired spatial or spatio-temporal patterns by projecting the system into network coordinates - the standard basis - using the desired vectors as columns of the transformation matrix. This method of network synthesis is roughly the inverse of the usual procedure in bifurcation theory for analysis of a given physical system. Proper choice of normal form couplings will ensure that the axis attractors are the only attractors in the system - there are no "spurious attractors". If symmetric normal form coefficients are chosen, then the normal form becomes a gradient vector field. It is exactly the gradient of an explicit potential function which is therefore a strict Liapunov function for the system. Identical normal form coefficients make the normal form vector field equivariant under permutation of the axes, which forces identical scale and rotation invariant basins of attraction bounded by hyperplanes. Very complex periodic a~tractors may be established by a kind of Fourier synthesis as linear combinations of the simple cycles chosen for a subset of the axes, when those are programmed to be unstable, and a single "mixed mode" in the interior of that subspace is made stable. Proofs and details on vectorfield programming appear in (Baird 89). In the general case, the network resulting from the projection A Bifurcation Theory Approach to Programming algorithm has fourth order correlations, but the use of restrictions on the detail of vector field programming and the types of patterns to be stored result in network architectures requiring only s~cond order correlations. For biological modeling, where possibly the patterns to be stored are sparse and nearly orthogonal, the learning rule for periodic patterns becomes a "periodic" outer product rule which is local, additive, commutative, and incremental. It reduces to the usual Hebb-like rule for static attractors. CYCLES The observed physiological activity may be idealized mathe11 y as a " 1 e " , r Xj e 1(ej + wt) , J? 1 , 2 , ... ,n. S uc h a . ma t 1ca cyc cycle is ~ "periodic attractor" if it is stable. The global amplitude r is just a scaling factor for the pattern ~ , and the global phase w in e 1wt is a periodic scaling that scales x by a factor between ? 1 at frequency w as t varies. The same vector X S or "pattern" of relative amplitudes can appear in space as a standing wave, like that seen in the bulb, if the relative phase eS1 of each compartment (component) is the same, eS 1 eS 1 , or as a traveling wave, like that seen in the ~repyriform cortex. if the relative phase components of ~s form a gradient in space, eS 1+1 - 1/a e\. The traveling wave will "sweep out" the amplitude pattern XS in time, but the root-mean-square amplitude measured in an experiment will be the same ~s, regardless of the phase pattern. For an arbitrary phase vector, t~~se "simple" single frequency cycles can make very complicated looking spatio-temporal patterns. From the mathematical point of view, the relative phase pattern ~ is a degree of freedom in the kind patterns that can be stored. Patterns of uniform amplitude ~ which differed only in the phase locking pattern ~ could be stored as well. +, - To store the kind of patterns seen in bulb, the amplitude vector ~ is assumed to be parsed into equal numbers of excitatory and inhibitory components, where each class of component has identical phase. but there is a phase difference of 60 90 degrees between the classes. The traveling wave in the prepyriform cortex is modeled by introducing an additional phase g~adient into both excitatory and inhibitory classes. PROJECTION ALGORITHM The central result of this paper is most compactly stated as the following: 461 462 Baird THEOREM Any set S, s - 1,2, ... , n/2 , of cycles r S x. s e1(9js + wst) of linearly independent vectors of relative comJonent amplitudes xS E Rn and phases ~s E Sn, with frequencies wS E R and global amplitudes r S E R, may be established in the vector field of the analog fourth order network: by some variant of the projection operation : -1 Tij ... Emn Pim J mn P nj , T ijk1? EPA mn im mn p-1. p- 1 mJ nk p- 1 n1' where the n x n matrix P contains the real and imaginary components [~S cos ~s , ~s sin ~S] of the complex eigenvectors x S e 19s as columns, J is an n x n matrix of complex conjugate eigenvalues in diagonal blocks, Amn is an n x n matrix of 2x2 blocks of repeated coefficients of the normal form equations, and the input b i &(t) is a delta function in time that establishes an initial condition. The vector field of the dynamics of the global amplitudes rs and phases -s is then given exactly by the normal form equations : rs == Us r s In particular, for ask > 0 , and ass/a kS < 1 , for all sand k, the cycles s - 1,2, ... ,n/2 are stable, and have amplitudes rs ;; (u s/a ss )1I2, where us? 1 - "T ? Note that there is a multiple Hopf bifurcation of codimension n/2 at "T = 1. Since there are no approximations here, however, the theorem is not restricted to the neighborhood of this bifurcation, and can be discussed without further reference to bifurcation theory. The normal form equations for drs/dt and d_s/dt determine how r S and _s for pattern s evolve in time in interaction with all the other patterns of the set S. This could be thought of as the process of phase locking of the pattern that finally emerges. The unusual power of this algorithm lies in the ability to precisely specify these ~ linear interactions. In general, determination of the modes of the linearized system alone (li and Hopfield 89) is insufficient to say what the attractors of the nonlinear system will be. A Bifurcation Theory Approach to Programming PROOF The proof of the theorem is instructive since it is a constructive proof, and we can use it to explain the learning algorithm. We proceed by showing first that there are always fixed points on the axes of these amplitude equations, whose stability is given by the coefficients of the nonlinear terms. Then the network above is constructed from these equations by two coordinate transformations. The first is from polar to Cartesian coordinates, and the second is a linear transformation from these canonical "mode" coordinates into the standard basis e 1, e 2, ... , eN' or "network coordinates". This second transformation constitutes the "learning algorithm", because it tra"nSfrirms the simple fixed points of the amplitude equations into the specific spatio-temporal memory patterns desired for the network. Amplitude Fixed Points Because the amplitude equations are independent of the rotation _, the fixed points of the amplitude equations characterize the asymptotic states of the underlying oscillatory modes. The stability of these cycles is therefore given by the stability of the fixed points of the amplitude equations. On each axis r s ' the other components rj are zero, by definition, rj = rj ( uj - Ek a jk r k2 ) ? 0, r s - rs ( Us - ass r s2 ), and for rj ? 0, which leaves rs - 0 There is an equilibrium on each axis s, at r s.(u s /a ss )1I2, as claimed. Now the Jacobian of the amplitude equations at some fixed point r~ has elements J lJ . . - - 2 a lJ .. r~.1 r.....J , J 11 = u.1 - :5 a 11 .. r~.2 - ]7-i ~ a .. r~.2 . 1 lJ J For a fixed point r~s on axis s, J ij ? 0 , since r~i or r~j ? 0, making J a diagonal matrix whose entries are therefore its eigenvalues. Now J l1 ? u1 - a is r~ s2, for i /. s, and J ss ? Us :5 ass r~/. Since r~/ ? us/ass' J ss ? - 2 us' and J ii ? u i - a is (us/ass). This gives aisfass > u1/u s as the condition for negative eigenvalues that assures the stability of r ....s . Choice of aji/a ii ) uj/u i , for all i, j , therefore guarantees stability of all axis fixed points. Coordinate Transformations We now construct the neural network from these well behaved equations by the following transformations, First; polar to Cartesian, (rs'-s) to (v2s-1.v2s) : Using V 2s - 1 '" r s cos -s v 2s = r s sin -s ,and differentiating these 463 464 Baird gives: V 2s - 1 ? r s cos "s by the chain rule. Now substituting and r s sin "s ? v 2s , cos tis ? v 2s - 1/r s ' gives: v 2s - v2s rs + (v 2s- l/ r s ) .. s Entering the expressions of the normal form for rs and tis' gives: and since rs 222 = v 2s- 1 + v 2s n/2 v 2s - 1 - Us v 2s -1 - Ws v 2s + E j [v 2s -1 a sj - v 2s bsj ] (v 2j -/ + v 2/) Similarly, n/2 v 2s - Us v 2s + Ws v 2s-' + E j [v 2s a sj + v 2s - 1 bSj ] (v 2j _/ + v 2/)? Setting the bsj - 0 for simplicity, choosing Us - - T + 1 to get a standard network form, and reindexing i,j-l,2, ... ,n , we get the Cartesian equivalent of the polar normal form equations. n n Here J is a matrix containing 2x2 blocks along the diagonal of the local couplings of the linear terms of each pair of the previous equations v 2s -1 ' v 2s ? with - T separated out of the diagonal terms. The matrix A has 2x2 blocks of identical coefficients a sj of the nonlinear terms from each pair. J 1 - w, w, 1 "- = 1 w2 - w2 1 " ~ a'l a" a" a 1, a 12 a'2 a'2 a'2 a 21 a 21 a 21 a 21 a 22 a 22 a 22 a 22 ., A Bifurcation Theory Approach to Programming Learning Transformation - Linear Term Second; J is the canonical form of a real matrix with complex conjugate eigenvalues, where the conjugate pairs appear in blocks along the diagonal as shown. The Cartesian normal form equations describe the interaction of these linearly uncoupled complex modes due to the coupling of the nonlinear terms. We can interpret the normal form equations as network equations in eigenvector (or "memory") coordinates, given by some diagonalizing transformation P, containing those eigenvectors as its columns, so that J a p- 1 T P. Then it is clear that T may instead be determined by the reverse projection T _ P J p- 1 back into network coordinates, if we start with desired eigenvectors and eigenvalues. We are free to choose as columns in P, the real and imaginary vectors [X S cos 9s , XS sin 9S ] of the cycles ~s e i9s of any linearly independent- set -S of p~tterns to be learned. If we write the matrix expression for the projection in component form, we recover the expression given in the theorem for Tij , Nonlinear Term Projection The nonlinear terms are transformed as well, but the expression cannot be easily written in matrix form. Using the component form of the transformation, substituting into the Cartesian normal form, gives: Xi - (-'T+1) E j Pij (E k P- 1jk x k) + E j Pij Ek J jk (E I P-\I xl) + E j Pij (E k P- 1jk xk) EI Ajl (Em p-\m xm) (En p-\n x n) Rearranging the orders of summation gives, Xi = (-'T+1) Ek (E j Pi j P- 1jk ) x k + EI (E k E j Pij J jk P-\l) xl + En Em Ek ( EI E j -1 -1 Pij P jk AjI P 1m p-1 In ) x k xm xn Finally, performing the bracketed summations and relabeling indices gives us the network of the theorem, xi =- 'T xi + E j T1j Xj + Ejkl Tijkl Xj Xk xl with the expression for the tensor of the nonlinear term, 465 466 Baird T ijk1 - Emn P im Amn P -1 mj P -1 nk P -1 n1 Q.E.D. LEARNING RULE EXTENSIONS This is the core of the mathematical story, and it may be extended in many ways. When the columns of P are orthonormal, then p-1 ? pT, and the formula above for the linear network coupling becomes T = pJpT. Then, for complex eigenvectors, This is now a local, additive, incremental learning rule for synapse ij, and the system can be truly self-organizing because the net can modify itself based on its own activity. Between units of equal phase, or when 9i s = 9j S - 0 for a static pattern, this reduces to the usual Hebb rule. In a similar fashion, the learning rule for the higher order nonlinear terms becomes a multiple periodic outer product rule when the matrix A is chosen to have a simple form. Given our present ignorance of the full biophysics of intracellular processing, it is not entirely impossible that some dimensionality of the higher order weights in the mathematical network coul~ be implemented locally within the cells of a biological network, using the information available on the primary lines given by the linear connections discussed above. When the A matrix is chosen to have uniform entries Aij - c for all its off-diagonal 2 x 2 blocks, and uniform entries Aij - c - d for the diagonal blocks, then, T ijk1 ? This reduces to the multiple outer product The network architecture generated by this learning rule is This reduces to an architecture without higher order correlations in the case that we choose a completely uniform A matrix (A 1j - c , for all i,j). Then + + A Bifurcation Theory Approach to Programming This network has fixed points on the axes of the normal form as always, but the stability condition is not satisfied since the diagonal normal form coefficients are equal, not less, than the remaining A matrix entries. In (Baird 89) we describe how clamped input (inspiration) can break this symmetry and make the nearest stored pattern be the only attractor. All of the above results hold as well for networks with sigmoids, provided their coupling is such that they have a Taylor's expansion which is equal to the above networks up to third order. The results then hold only in the neighborhood of the origin for which the truncated expansion is accurate. The expected performance of such systems has been verified in simulations. Acknowledgements Supported by AFOSR-87-0317. I am very grateful for the support of Walter Freeman and invaluable assistance of Morris Hirsch. References B. Baird. Bifurcation Theory Methods For Programming Static or Periodic Attractors and Their Bifurcations in Dynamic Neural Networks. Proc. IEEE Int. Conf. Neural Networks, San Diego, Ca.,pI-9, July(1988). B. Baird. Bifurcation Theory Approach to Vectorfield Programming for Periodic Attractors. Proc. INNS/IEEE Int. Conf. on Neural Networks. Washington D.C., June(1989). W. J. Freeman & B. Baird. Relation of Olfactory EEG to Behavior: Spatial Analysis. Behavioral Neuroscience (1986). W. J. Freeman & B. W. van Dijk. Spatial Patterns of Visual Cortical EEG During Conditioned Reflex in a Rhesus Monkey. Brain Research, 422, p267(1987). C. M. Grey and W. Singer. Stimulus Specific Neuronal Oscillations in Orientation Columns of Cat Visual Cortex. PNAS. In Press(1988). Z. Li & J.J. Hopfield. Modeling The Olfactory Bulb. Biological Cybernetics. Submitted(1989}. 467 

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A Non-parametric Multi-Scale Statistical Model for Natural Images Jeremy S. De Bonet & Paul Viola Artificial Intelligence Laboratory Learning & Vision Group 545 Technology Square Massachusetts Institute of Technology Cambridge, MA 02139 EMAIL: jsd@ai.mit.edu & viola@ai.mit.edu HOMEPAGE: http://www.ai .mit. edu/projects/lv Abstract The observed distribution of natural images is far from uniform. On the contrary, real images have complex and important structure that can be exploited for image processing, recognition and analysis. There have been many proposed approaches to the principled statistical modeling of images, but each has been limited in either the complexity of the models or the complexity of the images. We present a non-parametric multi-scale statistical model for images that can be used for recognition, image de-noising, and in a "generative mode" to synthesize high quality textures. 1 Introduction In this paper we describe a multi-scale structure of natural images across many it can be used to recognize novel images, tasks is reasonably efficient, requiring no workstation. statistical model which can capture the scales. Once trained on example images, or to generate new images. Each of these more than a few seconds or minutes on a The statistical modeling of images is an endeavor which reaches back to the 60's and 70's (Duda and Hart, 1973). Statistical approaches are alluring because they provide a unified view of learning, classification and generation. To date however, a generic, efficient and unified statistical model for natural images has yet to appear. Nevertheless, many approaches have shown significant competence in specific areas. Perhaps the most influential statistical model for generic images is the Markov random field (MRF) (Geman and Geman, 1984). MRF's define a distribution over 774 J. S. D. Bonet and P. A. Viola images that is based on simple and local interactions between pixels. Though MRF's have been very successfully used for restoration of images, their generative properties are weak. This is due to the inability of the MRF 's to capture long range (low frequency) interactions between pixels. Recently there has been a great deal of interest in hierarchical models such as the Helmholtz machine (Hinton et al., 1995; Dayan et al. , 1995). Though the Helmholtz machine can be trained to discover long range structure, it is not easily applied to natural images. Multi-scale wavelet models have emerged as an effective technique for modeling realistic natural images. These techniques hypothesize that the wavelet transform measures the underlying causes of natural images which are assumed to be statistically independent. The primary evidence for this conjecture is that the coefficients of wavelet transformed images are uncorrelated and low in entropy (hence the success of wavelet compression) . These insights have been used for noise reduction (Donoho and Johnstone, 1993; Simoncelli and Adelson, 1996), and example driven texture synthesis (Heeger and Bergen, 1995). The main drawback of wavelet algorithms is the assumption of complete independence between coefficients. We conjecture that in fact there is strong cross-scale dependence between the wavelet coefficients of an image, which is consistent with observations in (De Bonet, 1997) and (Buccigrossi and Simoncelli, 1997). 2 Multi-scale Statistical Models Multi-scale wavelet techniques assume that images are a linear transform of a collection of statistically independent random variables: 1= W-1C, where I is an image, W- 1 is the inverse wavelet transform, and C = {Ck} is a vector of random variable "causes" which are assumed to be independent. The distribution of each cause Ck is Pk ( . ), and since the Ck'S are independent it follows that: p( C) = k Pk (Ck). Various wavelet transforms have been developed, but all share the same type of multi-scale structure - each row of the wavelet matrix W is a spatially localized filter that is a shifted and scaled version of a single basis function. n The wavelet transform is most efficiently computed as an iterative convolution using a bank of filters . First a "pyramid" of low frequency downsampled images is created: Go = I , G 1 = 2 ..!-(9 ? Go), and Gi+l = 2 ..!-(9 ? Gi ), where 2..!- downsamples an image by a factor of 2 in each dimension, ? is the convolution operation, and 9 is a low pass filter. At each level a series offilter functions are applied: Fj = h (j!) Gj, where the Ii 's are various types of filters. Computation of the Fj's is a linear transformation that can thought of as a single matrix W. With careful selection of 9 and h this matrix can be constructed so that W- 1 = W T (Simoncelli et al., 1992)1. Where convenient we will combine the pixels of the feature images Fj(x, y) into a single cause vector C. The expected distribution of causes, Ck, is a function of the image classes that are being modeled. For example it is possible to attempt to model the space of all natural images. In that case it appears as though the most accurate Pk (.) 's are highly kurtotic which indicates that the Ck ' S are most often zero but in rare cases take on very large values (Donoho and Johnstone, 1993; Simoncelli and Adelson, 1996) . This is in direct contrast to the distribution of Ck'S for white-noise images which is gaussian. The difference in these distributions can be used as the basis of noise reduction algorithms, by reducing the wavelet coefficients which are more lComputation of the inverse wavelet transform is algorithmically similar to the computation of the forward wavelet transform. Non-Parametric Multi-Scale Statistical Image Model 775 likely to be noise than signal. Specific image classes can be modeled using similar methods (Heeger and Bergen, 1995)2. For a given set of input images the empirical distribution of the Ck'S is observed. To generate a novel example of a texture a new set of causes, (}, is sampled from the assumed independent empirical distributions Pk (.). The generated images are computed using the inverse wavelet transform: l' = W- 1 (},. Bergen and Heeger have used this approach to build a probabilistic model of a texture from a ::;ingle example image. To do this they assume that textures are spatially ergodic - that the expected distribution is not a function of position in the image. As a result the pixels in anyone feature image, Fj(x, y), are samples from the same distribution and can be combined 3 . Heeger and Bergen's work is at or near the current state of the art in texture generation. Figure 1 contains some example textures. Notice however, that this technique is much better at generating smooth or noise-like textures than those with well defined structure. Image structures, such as the sharp edges at the border of the tiles in the rightmost texture can not be modeled with their approach. These image features directly contradict the assumption that the wavelet coefficients, or causes, of the image are independent. For many types of natural images the coefficients of the wavelet transform are not independent, for example images which contain long edges. While wavelets are local both in frequency and space, a long edge is not local in frequency nor in space. As a result the wavelet representation of such a feature requires many coefficients. The high frequencies of the edge are captured by many small high frequency wavelets. The long scale is captured by a number of larger low frequency wavelets. A model which assumes these coefficients are independent can never accurately model images which contain these non-local features. Conversely a model which captures the conditional dependencies between coefficients will be much more effective. We chose to approximate the joint distribution of coefficients as a chain, in which coefficients that occur higher in the wavelet pyramid condition the distribution of coefficients at lower levels (Le. low frequencies condition the generation of higher frequencies). For every pixel in an image define the parent vector of that pixel: V.... (x, y) = [ Fo0 (x, y), Fo1 (x, y), . .. ,FoN (x, y), -nO X y 1 X y) l'{(l2"J, l2"J),F 1 (L2"J, L2"J , ... N ,Fl X (l2"J, L2"YJ) , ... o x Y 1 X J Y) x J' L2M Y J)] FM(L2MJ,l2MJ),FM(l2M ,L 2M J , ... ,FMN ( L2M (1) where M is the top level of the pyramid and N is the number of features. Rather than generating each of these coefficients independently, we define a chain across scale. In this chain the generation of the lower levels depend on the higher levels: p(V(x, y)) = p(VM(x, y)) x p(VM- 1 (x, y)IVM(x, y)) x p(VM-2(X, y)!VM-l (x, y), VM(x, y)) x ... x p(Vo(x, y)IVl (x, y), ... , VM-l (x, y), VM(x, y)) (2) 2See (Zhu, Wu and Mumford, 1996) for a related but more formal model. 3Their generation process is slightly more complex than this, involving a iteration designed to match the pixel histogram. The implementation used for generating the images in Figure 1 incorporates this, but we do not discuss it here. 776 1. S. D. Bonet and P. A. Viola Figure 1: Synthesis results for the Heeger and Bergen (1995) model. Top: Input textures. BOTTOM: Synthesis results. This technique is much better at generating fine or noisy textures then it is at generating textures which require co-occurrence of wavelets at multiple scales. Figure 2: Synthesis results using our technique for the input textures shown in Figure 1 (Top). where Yt(x , y) is the a subset of the elements of Vex , y) computed from C/. Usually we will assume ergodicity, i.e. that p(V(x, y)) is independent of x and y. The generative process starts from the top of the pyramid, choosing values for the VM (x, y) at all points. Once these are generated the values at the next level, VM -1 (x , y) , are generated. The process continues until all of the wavelet coefficients are generated. Finally the image is computed using the inverse wavelet transform. It is important to note that this probabilistic model is not made up of a collection of independent chains, one for each Vex , y). Parent vectors for neighboring pixels have substantial overlap as coefficients in the higher pyramid levels (which are Non-Parametric Multi-Scale Statistical Image Model 777 lower resolution) are shared by neighboring pixels at lower pyramid levels. Thus, the generation of nearby pixels will be strongly dependent. In a related approach a similar arrangement of generative chains has been termed a Markov tree (Basseville et al., 1992). 2.1 Estimating the Conditional Distributions The additional descriptive power of our generative model does not come without cost. The conditional distributions that appear in (2) must be estimated from observations. We choose to do this directly from the data in a non-parametric fashion. Given a sample of parent vectors {8(x, y)} from an example image we estimate the conditional distribution as a ratio of Parzen window density estimators: p( ~( I x,y )1v,M ( /+1 x,y )) _ p(ViM(x,y)) - '" Lx',y' R(V;M(X,y), 8r(x', y')) .... M '" "'M ....M p(V/+1(x,y)) Lx',y' R(V/+l (x, y), S/+1 (x',y')) (3) where Vik(x,y) is a subset of the parent vector V(x,y) that contains information from level I to level k, and R(?) is a function of two vectors that returns maximal values when the vectors are similar and smaller values when the vectors are dissimilar. We have explored various R(?) functions. In the results presented the R( .) function returns a fixed constant 1/ z if all of the coefficients of the vectors are within some threshold () and zero otherwise. Given this simple definition for R(?) sampling from p(Vz(x,Y)IV;~1(X,y)) is very straightforward: find all x', y' such that R(8#1 (x', y'), 8#1 (x, y)) = 1/ z and pick from among them to set Vz(x,y) = SI(X',y'). 3 Experiments We have applied this approach to the problems of texture generation, texture recognition, target recognition, and signal de-noising. In each case our results are competitive with the best published approaches. In Figure 2 we show the results of our technique on the textures from Figure 1. For these textures we are better able to model features which are caused by a conjunction of wavelets. This is especially striking in the rightmost texture where the geometrical tiling is almost, but not quite, preserved. In our model, knowledge of the joint distribution provides constraints which are critical in the overall perceived appearance of the synthesized texture. Using this same model, we can measure the textural similarity between a known and novel image. We do this by measuring the likelihood of generating the parent vectors in the novel image under the chain model of the known image. On "easy" data sets, such as the the MeasTex Brodatz texture test suite, performance is slightly higher than other techniques, our approach achieved 100% correct classification compared to 97% achieved by a gaussian MRF approach (Chellappa and Chatterjee, 1985). The MeasTex lattice test suite is slightly more difficult because each texture is actually a composition of textures containing different spatial frequencies. Our approach achieved 97% while the best alternate method, in this case Gabor Convolution Energy method (Fogel and Sagi, 1989) achieved 89%. Gaussian MRF's explicitly assume that the texture is a unimodal distribution and as a result achieve only 79% correct recognition. We also measured performance on a set of 20 types of natural texture and compared the classification power of this model to that of human observers (humans discriminate textures extremely accurately.) On this 1. S. D. Bonet and P. A. Viola 778 Original Denoise Shrinkage Shrinkage Residual Noised Denoise Ours Our Residual Figure 3: (Original) the original image; (Noised) the image corrupted with white gaussian noise (SNR 8.9 dB); (Denoise Shrinkage) the results of de-noising using wavelet shrinkage or coring (Donoho and Johnstone, 1993; Simoncelli and Adelson, 1996) (SNR 9.8 dB); (Shrinkage Residual) the residual error between the shrinkage de-noised result and the original - notice that the error contains a great deal of interpretable structure; (Denoise Ours) our de-noising approach (SNR 13.2 dB); and (Our Residual) the residual error - these errors are much less structured. test, humans achieved 86% accuracy, our approach achieved an accuracy of 81%, and GMRF's achieved 68%. A strong probabilistic model for images can be used to perform a variety of image processing tasks including de-noising and sharpening. De-noising of an observed image i can be performed by Monte Carlo averaging: draw a number of sample images according to the prior density P(I), compute the likelihood of the noise for each image P(v = (1) - 1), and then find the weighted average over these images. The weighted average is the estimated mean over all possible ways that the image might have been generated given the observation. Image de-noising frequently relies on generic image models which simply enforce image smoothness. These priors either leave a lot of residual noise or remove much of the original image. In contrast, we construct a probability density model from the noisy image itself. In effect we assume that the image is redundant, containing many examples of the same visual structures, as if it were a texture. The value of this approach is directly related to the redundancy in the image. If the redundancy in the image is very low, then the parent structures will be everywhere different, and the only resampled images with significant likelihood will be the original image. But if there is some redundancy in the image - that might arise from a regular texture or smoothly varying patch - the resampling will freely average across these similar regions. This will have the effect of reducing noise in these images. In Figure 3 we show results of this de-noising approach. Non-Parametric Multi-Scale Statistical Image Model 4 779 Conclusions We have presented a statistical model of texture which can be trained using example images. The form of the model is a conditional chain across scale on a pyramid of wavelet coefficients. The cross scale condtional distributions are estimated non-parametrically. This is important because many of the observed conditional distributions are complex and contain multiple modes. We believe that there are two main weaknesses of the current approach: i) the tree on which the distributions are defined are fixed and non-overlapping; and ii) the conditional distributions are estimated from a small number of samples. We hope to address these limitations in future work. Acknowledgments In this research, Jeremy De Bonet is supported by the DOD Multidisciplinary Research Program of the University Research Initiative, and Paul Viola by Office of Naval Research Grant No. N00014-96-1-0311. References Basseville, M., Benveniste, A., Chou, K. C., Golden, S. A., Nikoukhah, R., and Will sky, A. S. (1992). Modeling and estimation of multiresolution stochastic processes. IEEE Transactions on Information Theory, 38(2) :766-784. Buccigrossi, R. W. and Simoncelli, E. P. (1997). Progressive wavelet image coding based on a conditional probability model. In Proceedings ICASSP-97, Munich, Germany. Chellappa, R . and Chatterjee, S. (1985). Classification of textures using gaussian markov random fields. In Proceedings of the International Joint Conference on Acoustics, Speech and Signal Processing, volume 33, pages 959-963. Dayan, P., Hinton, G., Neal, R., and Zemel, R. (1995). The helmholtz machine. Neural Computation, 7:1022-1037. De Bonet, J. S. (1997). Multiresolution sampling procedure for analysis and synthesis of texture images. In Computer Graphics. ACM SIGGRAPH. Donoho, D. L. and Johnstone, 1. M. (1993). Adaptation to unknown smoothness via wavelet shrinkage. Technical report, Stanford University, Department of Statistics. Also Tech. Report 425. Duda, R. and Hart, P. (1973). Pattern Classification and Scene Analysis. John Wiley and Sons. Fogel, I. and Sagi, D. (1989). Gabor filters as texture discriminator. Biological Cybernetics, 61: 103- 113. Geman, S. and Geman, D. (1984). Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721-741. Heeger, D. J. and Bergen, J. R. (1995). Pyramid-based texture analysis/synthesis. In Computer Graphics Proceedings, pages 229-238. Hinton, G., Dayan, P., Frey, B., and Neal, R. (1995). The "wake-sleep" algorithm for unsupervised neural networks. Science, 268:1158-116l. Simoncelli, E. P. and Adelson, E. H. (1996). Noise removal via bayesian wavelet coring. In IEEE Third Int'l Conf on Image Processing, Laussanne Switzerland. IEEE. Simoncelli, E. P., Freeman, W. T., Adelson, E. H., and Heeger, D. J. (1992). Shiftable multiscale transforms. IEEE Transactions on Information Theory, 38(2):587-607. Zhu, S. C., Wu, Y., and Mumford, D. (1996). Filters random fields and maximum entropy(frame): To a unified theory for texture modeling. To appear in Int'l Journal of Computer Vision. 

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Radial Basis Functions: a Bayesian treatment David Barber* Bernhard Schottky Neural Computing Research Group Department of Applied Mathematics and Computer Science Aston University, Birmingham B4 7ET, U.K. http://www.ncrg.aston.ac.uk/ {D.Barber,B.Schottky}~aston.ac.uk Abstract Bayesian methods have been successfully applied to regression and classification problems in multi-layer perceptrons. We present a novel application of Bayesian techniques to Radial Basis Function networks by developing a Gaussian approximation to the posterior distribution which, for fixed basis function widths, is analytic in the parameters. The setting of regularization constants by crossvalidation is wasteful as only a single optimal parameter estimate is retained. We treat this issue by assigning prior distributions to these constants, which are then adapted in light of the data under a simple re-estimation formula. 1 Introduction Radial Basis Function networks are popular regression and classification tools[lO]. For fixed basis function centers, RBFs are linear in their parameters and can therefore be trained with simple one shot linear algebra techniques[lO]. The use of unsupervised techniques to fix the basis function centers is, however, not generally optimal since setting the basis function centers using density estimation on the input data alone takes no account of the target values associated with that data. Ideally, therefore, we should include the target values in the training procedure[7, 3, 9]. Unfortunately, allowing centers to adapt to the training targets leads to the RBF being a nonlinear function of its parameters, and training becomes more problematic. Most methods that perform supervised training of RBF parameters minimize the ?Present address: SNN, University of Nijmegen, Geert Grooteplein 21, Nijmegen, The Netherlands. http://wwv.mbfys.kun.nl/snn/ email: davidb<llmbfys.kun.nl Radial Basis Functions: A Bayesian Treatment 403 training error, or penalized training error in the case of regularized networks[7, 3, 9]. The setting of the associated regularization constants is often achieved by computationally expensive approaches such as cross-validation which search through a set of regularization constants chosen a priori. Furthermore, much of the information contained in such computation is discarded in favour of keeping only a single regularization constant. A single set of RBF parameters is subsequently found by minimizing the penalized training error with the determined regularization constant. In this work, we assign prior distributions over these regularization constants, both for the hidden to output weights and the basis function centers. Together with a noise model, this defines an ideal Bayesian procedure in which the beliefs expressed in the distribution of regularization constants are combined with the information in the data to yield a posterior distribution of network parameters[6]. The beauty of this approach is that none of the information is discarded, in contrast to cross-validation type procedures. Bayesian techniques applied to such non-linear, non-parametric models, however, can also be computationally extremely expensive, as predictions require averaging over the high-dimensional posterior parameter distribution. One approach is to use Markov chain Monte Carlo techniques to draw samples from the posterior[8]. A simpler approach is the Laplace approximation which fits a Gaussian distribution with mean set to a mode of the posterior, and covariance set to the inverse Hessian evaluated at that mode. This can be viewed as a local posterior approximation, as the form of the posterior away from the mode does not affect the Gaussian fit. A third approach, called ensemble learning, also fits a Gaussian, but is based on a less local fit criterion, the Kullback-Leibler divergence[4, 5]. As shown in [1], this method can be applied successfully to multi-layer perceptrons, whereby the KL divergence is an almost analytic quantity in the adaptable parameters. For fixed basis function widths, the KL divergence for RBF networks is completely analytic in the adaptable parameters, leading to a relatively fast optimization procedure. 2 Bayesian Radial Basis Function Networks For an N dimensional input vector x, we consider RBFs that compute the linear combination of K Gaussian basis functions, K f(x, m) = L WI exp {-Atllx - ctll2} (1) 1=1 where we denote collectivel~ the ce,nters Cl ..? cKl and wei~hts w = w~ ... Wk. by the parameter vector m = [c I , ... ,CK,Wl, ... ,WK]. We consIder the basIS functIOn widths AI, ... Ak to be fixed although, in principle, they can also be adapted by a similar technique to the one presented below. The data set that we wish to regress is a set of P input-output pairs D = {xl', yJL, JL = 1 ... Pl. Assuming that the target outputs y have been corrupted with additive Gaussian noise of variance (3-I, the likelihood of the data is l (2) p(Dlm,{3) = exp (-{3ED) jZD, where the training error is defined, 1 ED = p '2 L (f(x JL , m) - yJL)2 (3) 1'=1 To discourage overfitting, we choose a prior regularizing distribution for m p(mla) = exp (-Em(m)) jZp lIn the following, ZD, Zp and ZF are normalising constants (4) D. Barber and B. Schottky 404 where we take Em(m) = ~mTAm for a matrix A of hyperparameters. More complicated regularization terms, such as those that penalize centers that move away from specified points are easily incorporated in our formalism . For expositional clarity, we deal here with only the simple case of a diagonal regularizer matrix A = aI. The conditional distribution p(mID, a,.8) is then given by p(mID,a,.8) = exp(-.8ED(m) - Em(m?/ZF We choose to model the hyperparameters a and .8 by Gamma distributions, p(a) ex: aa-le-o:/b p(.8) ex: a c - 1 e-/3/d , (5) (6) where a, b, c, d are chosen constants. This completely specifies the joint posterior, p(m,a,.BID) = p(mID,a,.8)p(a)p(.B) . (7) A Bayesian prediction for a new test point x is then given by the posterior average (f(x, m?)p(In,o:,/3I D). If the centers are fixed, p(wID, a,.8) is Gaussian and computing the posterior average is trivial. However, with adaptive centers,the posterior distribution is typically highly complex and computing this average is difficult 2 . We describe below approaches that approximate the posterior by a simpler distribution which can then be used to find the Bayesian predictions and error bars analytically. 3 Approximating the posterior 3.1 Laplace's method Laplace's method is an approximation to the Bayesian procedure that fits a Gaussian to the mode mo of p(m, ID,a,.8) by extremizing the exponent in (5) ~llmW + .8ED(m) T = (8) with respect to m. The mean of the approximating distribution is then set to the mode roo, and the covariance is taken to be the inverse Hessian around roo; this is then used to approximately compute the posterior average. This is a local method as no account is taken for the fit of the Gaussian away from the mode. 3.2 Kullback-Leibler method The Kullback-Leibler divergence between the posterior p(m, a, .8ID) and an approximating distribution q(m, a,.8) is defined by KL[q] = - J q(m,a,.B) In ( p(m,a,.8ID)) q(m,a,.8) . (9) K L[q] is zero only if p and q are identical, and is greater than zero otherwise. Since in (5) ZF is unknown, we can compute the KL divergence only up to an additive constant, L[q] = KL[q] - InZF. We seek then a posterior approximation of the form q(m, a,.8) = Q(m)R(a)S(B) where Q(m) is Gaussian and the distributions R and S are determined by minimization of the functional L[q][5]. We first consider optimizing L with respect to the mean m and covariance C of the Gaussian distribution Q(m) ex: exp {-~(m - m)TC-l(m - m)}. Omitting all constant terms and integrating out a and .8, the Q(m) dependency in Lis, L[Q(m)] = - J Q(m) [-i3ED(m) - ~Q:llmW -In Q(m)] dm + const. (10) 2The fixed and adaptive center Bayesian approaches are contrasted more fully in [2]. Radial Basis Functions: A Bayesian Treatment 405 where Q =J E= J aR(a)da, f3S(f3)df3 (11) are the mean values of the hyperparameters. For Gaussian basis functions, the remaining integration in (10) over Q(m) can be evaluated analytically, giving 3 L[Q(m)] = ~Q {tr(C) + IImW} + E(ED(m?)Q - ~ In(det C) + const. (12) where (ED(m))Q p ( = "21 ~ (y~)2 - K 2y~ ~ sj + J;1K) s~1 (13) The analytical formulae for sj = = S~, (wlexp{-'\jllxl'-CIW})Q (14) (WkWj exp{ -,\kllxtL - Ck 112} exp{ -'\l\lx~ - cdI 2})Q (15) are straightforward to compute, requiring only Gaussian integration[2]. The values for C and m can then be found by optimizing (12). We now turn to the functional optimisation of (9) with respect to R. Integrating out m and f3 leaves, up to a constant, L[R] = JR(a){a["~W + tr~C) +t] + [K(~+l) +a-l]lna+lnR(a)}da (16) As the first two terms in (16) constitute the log of a Gamma distribution (6), the functional (16) is optimized by choosing a Gamma distribution for a, (17) with K(N + 1) ! = IIml12 s 2 The same procedure for S(f3) yields r= 2 + a, + !tr(C) + ! 2 a=rs. b' S(f3) ex: f3 U - 1 e-/3/v with P = -2 1 = (19) 1 = (ED(m?)Q + -d' -:e uv, (20) v where the averaged training error is given by (13). The optimization of the approximating distribution Q(m)R(a)S(f3) can then be performed using an iterative procedure in which we first optimize (12) with respect to m and C for fixed a, 73, and then update a and 7J according to the re-estimation formulae (18,20). u + c, (18) After this iterative procedure has converged, we have an approximating distribution of parameters, both for the hidden to output weights and center positions (figure l(a?). The actual predictions are then given by the posterior average over this distribution of networks. The model averaging effect inherent in the Bayesian procedure produces a final function potentially much more complex than that achievable by a single network. A significant advantage of our procedure over the Laplace procedure is that we can lower bound model the likelihood Inp(Dlmodel) ~ -(L+lnZD+lnZp). Hence, decreasing L increases p(Dlmodel). We can use this bound to rank different models, leading to principled Bayesian model selection. 3( ... }Q denotes J Q(m) ... dm D. Barber and B. Schottky 406 Center Fluctuations Widths Figure 1: Regressing a surface from 40 noisy training examples. (a) The KL approximate Bayesian treatment fits 6 basis functions to the data. The posterior distribution for the parameters gives rise to a posterior weighted average of a distribution of the 6 Gaussians. We plot here the posterior standard deviation of the centers (center fluctuations) and the mean centers. The widths were fixed a priori using Maximum Likelihood. (b) Fixing a basis function on each training point with fixed widths. The hidden-output weights were determined by cross-validation of the penalised training error. 4 Relation to non-Bayesian treatments One non-Bayesian approach to training RBFs is to minimze the training error (3) plus a regularizing term of the form (8) for fixed centers[7, 3, 9]. In figure l(b) we fix a center on each training input. For fixed hyperparameters a and (3, the optimal hidden-to-output weights can then be found by minimizing (8). To set the hyperparameters, we iterate this procedure using cross-validation. This results in a single estimate for the parameters mo which is then used for predictions f(x, mo). In figure(l), both the Bayesian adaptive center and the fixed center methods have similar performance in terms of test error on this problem. However, the parsimonious representiation of the data by the Bayesian adaptive center method may be advantageous if interpreting the data is important. In principle, in the Bayesian approach, there is no need to carry out a crossvalidation type procedure for the regularization parameters a, {3. After deciding on a particular Bayesian model with suitable hyperprior constants (here a, b, c, d), our procedure will combine these beliefs about the regularity of the RBF with the dataset in a principled manner, returning a-posteriori probabilities for the values of the regularization constants. Error bars on the predictions are easily calculated as the posterior distribution quantifies our uncertainty in the parameter estimates. One way of viewing the connection between the CV and Bayesian approaches, is to identify the a-priori choice of CV regularization coefficients ai that one wishes to examine as a uniform prior over the set {ad. The posterior regularizer distribution is then a delta peak centred at that a* with minimal CV error. This delta peak represents a loss of information regarding the performance of all the other networks trained with ai :la*. In contrast, in our Bayesian approach we assign a continuous prior distribution on a, which is updated according to the evidence in the data. Any loss of information then occurs in approximating the resulting posterior distribution. Radial Basis Functions: A Bayesian Treatment (a) Minimum KL 407 (b) Laplace (c) Regularized (non Bayesian) 0.8 0.8 0.8 0.& 0.& 0.8 0.' 0.' 0.' 0.2 " , , ' 0.2 ", " -0.& \ 0.2 , , . \, . ~ .& " \ '., : ~.& Figure 2: Minimal KL Gaussian fit, Laplace Gaussian, and a non-Bayesian procedure on regressing with 6 Gaussian basis functions. The training points are labelled by crosses and the target function 9 is given by the solid lines. For both (a) and (b), the mean prediction is given by the dashed lines, and standard errors are given by the dots. (a) Approximate Bayesian solution based on Kullback-Leibler divergence. The regularization constant Q and inverse noise level f3 are adapted as described in the text. (b) Laplace method based on equation (8). Both Q and f3 are set to the mean of the hyperparameter distributions (6). The mean prediction is given by averaging over the locally approximated posterior. Note that the error bars are somewhat large, suggesting that the local posterior mass has been underestimated. (c) The broken line is the Laplace solution without averaging over the posterior, showing much greater variation than the averaged prediction in (b). The dashed line corresponds to fixing the basis function centers at each data point, and estimating the regularization constants a by cross-validation. 5 Demonstration We apply the above outlined Bayesian framework to a simple one-dimensional regression problem. The function to be learned is given by (21) and is plotted in figure(2). The training patterns are sampled uniformly between [-4,4] and the output is corrupted with additive Gaussian noise of variance a 2 = 0.005. The number of basis function is K = 6, giving a reasonably flexible model for this problem. In figure(2), we compare the Bayesian approaches (a),(b) to non-Bayesian approaches(c). In this demonstration, the basis function widths were chosen by penalised training error minimization and fixed throughout all experiments. For the Bayesian procedures, we chose hyperprior constants, a = 2, b = 1/4, c = 4, d = 50, corresponding to mean values it = 0.5 and E= 200. In (c), we plot a more conventional approach using cross-validation to set the regularization constant. A useful feature of the Bayesian approaches lies in the principled theory for the error bars. In (c), although we know the test error for each regularization constant in the set of constants we choose to examine, we do not know any principled procedure for using these values for error bar assessment. 408 6 D. Barber and B. Schottky Conclusions We have incorporated Radial Basis FUnctions within a Bayesian framework, arguing that the selection of regularization constants by non-Bayesian methods such as cross-validation is wasteful of the information contained in our prior beliefs and the data set. Our framework encompasses flexible priors such as hard assigning a basis function center to each data point or penalizing centers that wander far from pre-assigned points. We have developed an approximation to the ideal Bayesian procedure by fitting a Gaussian distribution to the posterior based on minimizing the Kullback-Leibler divergence. This is an objectively better and more controlled approximation to the Bayesian procedure than the Laplace method. FUrthermore, the KL divergence is an analytic quantity for fixed basis function widths. This framework also includes the automatic adaptation of regularization constants under the influence of data and provides a rigorous lower bound on the likelihood of the model. Acknowledgements We would like to thank Chris Bishop and Chris Williams for useful discussions. BS thanks the Leverhulme Trust for support (F /250/K). References [1] D. Barber and C. M. Bishop. On computing the KL divergence for Bayesian Neural Networks. Technical report, Neural Computing Research Group, Aston University, Birmingham, 1998. See also D. Barber and C. M. Bishop These proceedings. [2] D. Barber and B. Schottky. Bayesian Radial Basis Functions. Technical report, Neural Computing Research Group, Aston University, Birmingham, 1998. [3] C. M. Bishop. Improving the Generalization Properties of Radial Basis Function Networks. Neural Computation, 4(3):579-588, 1991. [4] G. E. Hinton and D. van Camp. Keeping neural networks simple by minimizing the description length of the weights. In Proceedings of the Seventh Annual ACM Workshop on Computational Learning Theory (COLT '93), 1993. [5J D. J. C. MacKay. 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, Nijmegan, Netherlands, 14-15 September 1995, pages 191-198. Springer. [6] D. J. C. MacKay. Bayesian Interpolation. Neural Computation, 4(3):415-447, 1992. [7] J. Moody and C. J . Darken. Fast Learning in Networks of Locally-Tuned Processing Units. Neural Computation, 1:281-294, 1989. [8] Neal, R. M. Bayesian Learning for Neural Networks. Springer, New York, 1996. Lecture Notes in Statistics 118. [9J M. J . L. Orr. Regularization in the Selection of Radial Basis Function Centers. Neural Computation, 7(3) :606-623, 1995. [10] M. J . L. Orr. Introduction to Radial Basis Function Networks. Technical report, Centre for Cognitive Science, Univeristy of Edinburgh, Edinburgh, ER8 9LW, U.K., 1996. 

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Asymptotic Theory for Regularization: One-Dimensional Linear Case Petri Koistinen Rolf Nevanlinna Institute, P.O. Box 4, FIN-00014 University of Helsinki, Finland. Email: PetrLKoistinen@rnLhelsinkLfi Abstract The generalization ability of a neural network can sometimes be improved dramatically by regularization. To analyze the improvement one needs more refined results than the asymptotic distribution of the weight vector. Here we study the simple case of one-dimensional linear regression under quadratic regularization, i.e., ridge regression. We study the random design, misspecified case, where we derive expansions for the optimal regularization parameter and the ensuing improvement. It is possible to construct examples where it is best to use no regularization. 1 INTRODUCTION Suppose that we have available training data (Xl, Yd, .. 0' (Xn' Y n ) consisting of pairs of vectors, and we try to predict Yi on the basis of Xi with a neural network with weight vector w. One popular way of selecting w is by the criterion 1 (1) n -n L ?(Xi' Yi, w) + >..Q(w) = min!, I where the loss ?(x,y,w) is, e.g., the squared error Ily - g(x,w)11 2 , the function g(., w) is the input/output function of the neural network, the penalty Q(w) is a real function which takes on small values when the mapping g(o, w) is smooth and high values when it changes rapidly, and the regularization parameter >.. is a nonnegative scalar (which might depend on the training sample). We refer to the setup (1) as (training with) regularization, and to the same setup with the choice >.. = 0 as training without regularization. Regularization has been found to be very effective for improving the generalization ability of a neural network especially when the sample size n is of the same order of magnitude as the dimensionality of the parameter vector w, see, e.g., the textbooks (Bishop, 1995; Ripley, 1996). Asymptotic Theory for Regularization: One-Dimensional Linear Case 295 In this paper we deal with asymptotics in the case where the architecture of the network is fixed but the sample size grows . To fix ideas, let us assume that the training data is part of an Li.d. (independent, identically distributed) sequence (X,Y);(Xl'Yl),(X2'Y2)"" of pairs of random vectors, i.e., for each i the pair (Xi, Yi) has the same distribution as the pair (X, Y) and the collection of pairs is independent (X and Y can be dependent) . Then we can define the (prediction) risk of a network with weights w as the expected value (2) r(w) := IE:f(X, Y, w). Let us denote the minimizer of (1) by Wn (.),) , and a minimizer of the risk r by w*. The quantity r(w n (>.)) is the average prediction error for data independent of the training sample. This quantity r(w n (>.)) is a random variable which describes the generalization performance of the network: it is bounded below by r( w*) and the more concentrated it is about r(w*), the better the performance . We will quantify this concentration by a single number, the expected value IE:r(w n (>.)) . We are interested in quantifying the gain (if any) in generalization for training with versus training without regularization defined by (3) When regularization helps, this is positive. However, relatively little can be said about the quantity (3) without specifying in detail how the regularization parameter is determined. We show in the next section that provided>' converges to zero sufficiently quickly (at the rate op(n- 1 / 2 )), then IE: r(wn(O)) and IE: r(w n (>.)) are equal to leading order. It turns out, that the optimal regularization parameter resides in this asymptotic regime. For this reason, delicate analysis is required in order to get an asymptotic approximation for (3). In this article we derive the needed asymptotic expansions only for the simplest possible case: one-dimensional linear regression where the regularization parameter is chosen independently of the training sample. 2 REGULARIZATION IN LINEAR REGRESSION We now specialize the setup (1) to the case of linear regression and a quadratic smoothness penalty, i.e. , we take f(x,y,w) = [y-x T wJ2 and Q(w) = wTRw, where now y is scalar, x and w are vectors, and R is a symmetric, positive definite matrix. It is well known (and easy to show) that then the minimizer of (1) is (4) wn (>') = [ 1 -1 n ~ ~ XiX! + >'R ] 1 n ~ ~ XiYi. This is called the generalized ridge regression estimator, see, e.g., (Titterington, 1985); ridge regression corresponds to the choice R = I, see (Hoerl and Kennard, 1988) for a survey. Notice that (generalized) ridge regression is usually studied in the fixed design case, where Xi:s are nonrandom. Further, it is usually assumed that the model is correctly specified, i.e., that there exists a parameter such that Y i = Xr w* + ?i , and such that the distribution of the noise term ?i does not depend on Xi. In contrast, we study the random design, misspecified case. Assuming that IE: IIXI1 2 < 00 and that IE: [XXT] is invertible, the minimizer of the risk (2) and the risk itself can be written as (5) (6) w* = A-lIE: [XY], r(w) = r(w*) + (w - with A:=IE:[XXT] w*f A(w - w*). P. Koistinen 296 If Zn is a sequence of random variables, then the notation Zn = open-a) means that n a Zn converges to zero in probability as n -+ 00 . For this notation and the mathematical tools needed for the following proposition see, e.g., (Serfiing, 1980, Ch. 1) or (Brockwell and Davis, 1987, Ch. 6). Proposition 1 Suppose that IE: y4 < 00, IE: IIXII 4 < 00 and that A = IE: [X XTj is invertible. If,\ = op(n- I / 2), then both y'n(wn(O) -w*) and y'n(w n ('\) - w*) converge in distribution to N (0, C), a normal distribution with mean zero and covariance matrix C. The previous proposition also generalizes to the nonlinear case (under more complicated conditions). Given this proposition, it follows (under certain additional conditions) by Taylor expansion that both IE:r(w n ('\)) - r(w*) and IEr(wn(O)) - r(w*) admit the expansion f31 n -} + o( n -}) with the same constant f3I. Hence, in the regime ,\ = op(n-I/2) we need to consider higher order expansions in order to compare the performance of wn (,\) and wn(O). 3 ONE-DIMENSIONAL LINEAR REGRESSION We now specialize the setting of the previous section to the case where x is scalar. Also, from now on, we only consider the case where the regularization parameter for given sample size n is deterministic; especially ,\ is not allowed to depend on the training sample. This is necessary, since coefficients in the following type of asymptotic expansions depend on the details of how the regularization parameter is determined. The deterministic case is the easiest one to analyze. We develop asymptotic expansions for the criterion (7) where now the regularization parameter k is deterministic and nonnegative. The expansions we get turn out to be valid uniformly for k ~ O. We then develop asymptotic formulas for the minimizer of I n , and also for In(O) - inf I n . The last quantity can be interpreted as the average improvement in generalization performance gained by optimal level of regularization, when the regularization constant is allowed to depend on n but not on the training sample. From now on we take Q(w) = w 2 and assume that A = IEX2 = 1 (which could be arranged by a linear change of variables). Referring back to formulas in the previous section, we see that (8) r(wn(k)) - r(w*) = ern - kw*)2/(Un + 1 + k)2 =: h(Un , Vn, k), whence In(k) = IE:h(Un , Vn , k), where we have introduced the function h (used heavily in what follows) as well as the arithmetic means Un and Vn _ (9) 1 n L Vi, n Vn:= - with I (10) Vi := XiYi - w* xl For convenience, also define U := X2 - 1 and V := Xy - w* X2 . Notice that U; UI, U2 , ? .. are zero mean Li.d. random variables, and that V; Vi, V2 ,. " satisfy the same conditions. Hence Un and Vn converge to zero, and this leads to the idea of using the Taylor expansion of h(u, v, k) about the point (u, v) = (0,0) in order to get an expansion for In(k). 297 Asymptotic Theory for Regularization: One-Dimensional Linear Case To outline the ideas, let Tj(u,v,k) be the degree j Taylor polynomial of (u,v) f-7 h(u, v, k) about (0,0), i.e., Tj(u, v, k) is a polynomial in u and v whose coefficients are functions of k and whose degree with respect to u and v is j. Then IETj(Un,Vn,k) depends on n and moments of U and V. By deriving an upper bound for the quantity IE Ih(Un , Vn , k) - Tj(Un , Vn , k)1 we get an upper bound for the error committed in approximating In(k) by IE Tj(Un , Vn , k). It turns out that for odd degrees j the error is of the same order of magnitude in n as for degree j - 1. Therefore we only consider even degrees j. It also turns out that the error bounds are uniform in k ~ 0 whenever j ~ 2. To proceed, we need to introduce assumptions. Assumption 1 IE IXlr < 00 and IE IYls < 00 for high enough rand s. Assumption 2 Either (a) for some constant j3 > 0 almost surely X has a density which is bounded in some neighborhood of zero. IXI ;::: j3 or (b) Assumption 1 guarantees the existence of high enough moments; the values r = 20 and s = 8 are sufficient for the following proofs. E.g., if the pair (X, Y) has a normal distribution or a distribution with compact support, then moments of all orders exist and hence in this case assumption 1 would be satisfied. Without some condition such as assumption 2, In(O) might fail to be meaningful or finite. The following technical result is stated without proof. Proposition 2 Let p > 0 and let 0 < IE X 2 < 00. If assumption 2 holds, then where the expectation on the left is finite (a) for n assumption 2 (a), respectively 2 (b) holds. ~ 1 (b) for n > 2p provided that Proposition 3 Let assumptions 1 and 2 hold. Then there exist constants no and M such that In(k) = JET2(Un , Vn , k) + R(n, k) where _ _ (w*)2k 2 W*kIEUV] -1 [IEV2 (w*)2k2JEU 2 IET2(Un , Vn , k) = (1+k)2 +n (1+k)2 +3 (1+k)4 +4 (1+k)3 IR(n, k) I :s; Mn- 3/ 2(k + 1)-1, "In;::: no, k ;::: o. PROOF SKETCH The formula for IE T 2 (Un , Vn. k) follows easily by integrating the degree two Taylor polynomial term by term. To get the upper bound for R(n, k), consider the residual where we have omitted four similar terms. Using the bound P. Koistinen 298 the Ll triangle inequality, and the Cauchy-Schwartz inequality, we get r IR(n, k)1 = IJE [h(Un , Vn , k) - T2(Un , Vn , k)]1 ., (k+ {2(k W' {Ii: [(~ ~Xl)-'] + 1)3[JE (lUn I2IVnI 4 )]l/2 + 4(w*)2k 2(k + 1)[18 IUn I6 ]l/2 ... } By proposition 2, here 18 [(~ 2:~ X[)-4] = 0(1). Next we use the following fact, cf. (Serfiing, 1980, Lemma B, p. 68). Fact 1 Let {Zd be i.i.d. with 18 [Zd = 0 and with 18 IZI/v < 00 for some v Then v ~ 2. Applying the Cauchy-Schwartz inequality and this fact, we get, e.g., that [18 (IUn I2 IVnI 4 )]l/2 ~ [(18 IUn I4 )1/2(E IVnI 8)1/2p/2 = 0(n- 3 / 2). o Going through all the terms carefully, we see that the bound holds. Proposition 4 Let assumptions 1 and 2 hold, assume that w* :j; 0, and set al := (18 V2 - 2w*E [UVD/(w*)2. If al > 0, then there exists a constant ni such that for all n ~ nl the function k ~ ET2(Un , Vn,k) has a unique minimum on [0,(0) at the point k~ admitting the expanszon k~ = aIn- 1 + 0(n- 2); further, In(O) - inf{Jn(k) : k ~ O} = In(O) - In(aln- 1 ) = ar(w*)2n- 2 If a ~ + 0(n- 5 / 2). 0, then The proof is based on perturbation expansio!1 c~nsidering lin a small parameter. By the previous proposition, Sn(k) := ET2 (Un , Vn , k) is the sum of (w*)2k 2/(1 + k)2 and a term whose supremum over k ~ ko > -1 goes to zero as n ~ 00. Here the first term has a unique minimum on (-1,00) at k = O. Differentiating Sn we get PROOF SKETCH S~(k) = [2(w*)2k(k + 1)2 + n- 1 p2(k)]/(k + 1)5, where P2(k) is a second degree polynomial in k. The numerator polynomial has three roots, one of which converges to zero as n ~ 00. A regular perturbation expansion for this root, k~ = aln- I + a2n-2 + ... , yields the stated formula for al. This point is a minimum for all sufficiently large n; further, it is greater than zero for all sufficiently large n if and only if al > O. The estimate for J n (0) - inf {J n (k) : k ~ O} in the case al > 0 follows by noticing that In(O) - In(k) = 18 [h(Un , Vn , 0) - h(Un , Vn , k)), where we now use a third degree Taylor expansion about (u, v, k) = (0,0,0) h(u,v,O) - h(u,v,k) = 2w* kv - (w*)2k2 - 4w*kuv + 2(w*?k 2u + 2kv 2 - 4w*k 2v + 2(W*)2k 3 + r(u, v, k). Asymptotic Theory for Regularization: One-Dimensional Linear Case 299 0.2 0.18 0.16 0.14 0.12 0.1 __ __ __ __ __ __ __ 0.05 0.1 0.15 0 .2 0.25 0.3 0.35 0.4 ~~ o ~ ~ ~ ~ ~ ~ L-~ __ 0.45 ~ 0.5 Figure 1: Illustration of the asymptotic approximations in the situation of equation (11) . Horizontal axis kj vertical axis .In(l?) and its asymptotic appr~ximations. Legend: markers In(k); solid line IE T 2 (Un , Vn , k)j dashed line IET4 (Un , Vn , k). Usin~ t~e techniques of the previous proposition, it can be shown that IE Ir(Un , Vn , k~)1 = O(n- S/ 2 ). Integrating the Taylor polynomial and using this estimate gives In(O) - In(aI/n) = af(w*)2n- 2 + O(n- S / 2 ). Finally, by the mean value theorem, In(O) -inf{ In(k) : k ~ O} = In(O) -In(aI/n) + ! = In(O) - (In(O) - In(k)]lk=8(k~ -aI/n) In(aI/n) + O(n-1)O(n-2) where () lies between k~ and aI/n, and where we have used the fact that the indicated derivative evaluated at () is of order O(n- 1 ), as can be shown with moderate effort. 0 Remark In the preceding we assumed that A = IEX 2 equals 1. If this is not the case, then the formula for a1 has to be divided by A; again, if a1 > 0, then k~ = a1n-1 + O(n- 2 ) . If the model is correctly specified in the sense that Y = w* X + E, where E is independent of X and IE E = 0, then V = X E and IE [UV] = O. Hence we have a1 = IE [E 2]j(w*)2, and this is strictly positive expect in the degenerate case where E = 0 with probability one. This means that here regularization helps provided the regularization parameter is chosen around the value aI/n and n is large enough. See Figure 1 for an illustration in the case (11) X "'" N(O, 1) , Y = w* X +f, f "'" N(O, 1), w* = 1, where E and X are independent. In(k) is estimated on the basis of 1000 repetitions of the task for n = 8. In addition to IE T2(Un , Vn , k) the function IE T4 (Un , lin, k) is also plotted. The latter can be shown to give In(k) correctly up to order O(n- s/ 2 (k+ 1)-3). Notice that although IE T2 (Un , Vn , k) does not give that good an approximation for In(k), its minimizer is near the minimizer of In(k), and both of these minimizers lie near the point al/n = 0.125 as predicted by the theory. In the situation (11) it can actually be shown by lengthy calculations that the minimizer of In(k) is exactly al/n for each sample size n ~ 1. It is possible to construct cases where X "'" Uniform (a, b), Y = cjX + d+ Z, a1 < O. For instance, take 1 1 b=-(3Vs-l) 2' 4 c= -5,d= 8, a=- P. Koistinen 300 and Z '" N (0, a 2 ) with Z and X independent and 0 :::; a < 1.1. In such a case regularization using a positive regularization parameter only makes matters worse; using a properly chosen negative regularization parameter would, however, help in this particular case. This would, however, amount to rewarding rapidly changing functions. In the case (11) regularization using a negative value for the regularization parameter would be catastrophic. 4 DISCUSSION We have obtained asymptotic approximations for the optimal regularization parameter in (1) and the amount of improvement (3) in the simple case of one-dimensional linear regression when the regularization parameter is chosen independently of the training sample. It turned out that the optimal regularization parameter is, to leading order, given by Qln- 1 and the resulting improvement is of order O(n- 2 ). We have also seen that if Ql < 0 then regularization only makes matters worse. Also (Larsen and Hansen, 1994) have obtained asymptotic results for the optimal regularization parameter in (1). They consider the case of a nonlinear network; however, they assume that the neural network model is correctly specified. The generalization of the present results to the nonlinear, misspecified case might be possible using, e.g., techniques from (Bhattacharya and Ghosh, 1978). Generalization to the case where the regularization parameter is chosen on the basis of the sample (say, by cross validation) would be desirable. Acknowledgements This paper was prepared while the author was visiting the Department for Statistics and Probability Theory at the Vienna University of Technology with financial support from the Academy of Finland. I thank F. Leisch for useful discussions. References Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. The Annals of Statistics, 6(2):434-45l. Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press. Brockwell, P. J. and Davis, R. A. (1987). Time Series: Theory and Methods. Springer series in statistics. Springer-Verlag. Hoerl, A. E. and Kennard, R. W. (1988). Ridge regression. In Kotz, S., Johnson, N. L., and Read, C. B., editors, Encyclopedia of Statistical Sciences. John Wiley & Sons, Inc. Larsen, J. and Hansen, L. K. (1994). Generalization performance of regularized neural network models. In Vlontos, J., Whang, J.-N., and Wilson, E., editors, Proc. of the 4th IEEE Workshop on Neural Networks for Signal Processing, pages 42-51. IEEE Press. Ripley, B. D. (1996). Pattern Recognition and Neural Networks. Cambridge University Press. Serfiing, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, Inc. Titterington, D. M. (1985). Common structure of smoothing techniques in statistics. International Statistical Review, 53:141-170. 

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Graph Matching with Hierarchical Discrete Relaxation Richard C. Wilson and Edwin R. Hancock Department of Computer Science, University of York York, YOl 5DD, UK. Abstract Our aim in this paper is to develop a Bayesian framework for matching hierarchical relational models. The goal is to make discrete label assignments so as to optimise a global cost function that draws information concerning the consistency of match from different levels of the hierarchy. Our Bayesian development naturally distinguishes between intra-level and inter-level constraints. This allows the impact of reassigning a match to be assessed not only at its own (or peer) level ofrepresentation, but also upon its parents and children in the hierarchy. 1 Introd uction Hierarchical graphical structures are of critical importance in the interpretation of sensory or perceptual data. For instance, following the influential work of Marr [6] there has been sustained efforts at effectively organising and processing hierarchical information in vision systems. There are a plethora of concrete examples which include pyramidal hierarchies [3] that are concerned with multi-resolution information processing and conceptual hierarchies [4] which are concerned with processing at different levels of abstraction. Key to the development of techniques for hierarchical information processing is the desire to exploit not only the intra-level constraints applying at the individual levels of representation but also inter-level constraints operating between different levels of the hierarchy. If used effectively these interlevel constraints can be brought to bear on the interpretation of uncertain image entities in such a way as to improve the fidelity of interpretation achieved by single level means. Viewed as an additional information source, inter-level constraints can be used to resolve ambiguities that would persist if single-level constraints alone were used. 690 R. C. Wilson and E. R. Hancock In the connectionist literature graphical structures have been widely used to represent probabilistic causation in hierarchical systems [5, 9]. Although this literature has provided a powerful battery of techniques, they have proved to be of limited use in practical sensory processing systems. The main reason for this is that the underpinning independence assumptions and the resulting restrictions on graph topology are rarely realised in practice. In particular there are severe technical problems in dealing with structures that contain loops or are not tree-like. One way to overcome this difficulty is to edit intractable structures to produce tractable ones [8]. Our aim in this paper is to extend this discrete relaxation framework to hierarchical graphical structures. We develop a label-error process to model the violation of both inter-level and intra-level constraints. These two sets of constraints have distinct probability distributions. Because we are concerned with directly comparing the topology graphical structures rather than propagating causation, the resulting framework is not restricted by the topology of the hierarchy. In ,particular, we illustrate the effectiveness of the method on amoral graphs used to represent scene-structure in an image interpretation problem. This is a heterogeneous structure [2, 4] in which different label types and different classes of constraint operate at different levels of abstraction. This is to be contrasted with the more familiar pyramidal hierarchy which is effectively homogeneous [1, 3]. Since we are dealing with discrete entities inter-level information communication is via a symbolic interpretation of the objects under consideration. 2 Hierarchical Consistency The hierarchy consists of a number of levels, each containing objects which are fully described by their children at the level below. Formally each level is described by an attributed relational graph G I = (Vi, EI, Xl), Vi E L, with L being the index-set of levels in the hierarchy; the indices t and b are used to denote the top and bottom levels of the hierarchy respectively. According to our notation for level i of the hierarchy, Vi is the set of nodes, EI is the set of intra-level edges and Xl = {~~, Vu E Vi} is a set of unary attributes residing on the nodes. The children or descendents which form the representation of an element j at a lower level are denoted by V j . In other words, if U l - I is in Vj then there is a link in the hierarchy between element j at level i and element u at level i-I. According to our assumptions, the elements of Vj are drawn exclusively from Vi-I. The goal of performing relaxation operations is to find the match between scene graph G 1 and model graph G 2 ? At each individual level of the hierarchy this match is represented by a mapping function p, Vi E L, where II: Vi -t Vi. The development of a hierarchical consistency measure proceeds along a similar line to the Single-level work of Wilson and Hancock [10]. The quantity of interest is the MAP estimate for the mapping function I given the available unary attributes, i.e. I = argmaxj P(jt, Vi E LIX I , Vi E L). We factorize the measurement information over the set of nodes by application of Bayes rule under the assumption of measurement independence on the nodes. As a result P(/, Vi E L/X l , Vi E L) = (Xl p, ~i E L) {II II p(X~ll(u))}P(fI, Vi E L) (1) IELuEVI The critical modelling ingredient in developing a discrete relaxation scheme from the above MAP criterion is the joint prior for the mapping function, i.e. p(fl, Vi E L) 691 Graph Matching with Hierarchical Discrete Relaxation Parents Children A Possible mappings or children: 1,2,3 -- B c A,B,C C,B,A Figure 1: Example constrained children mappings which represents the influence of structural information on the matching process. The joint measurement density, p(XI, 'VI E L), on the other hand is a fixed property of the hierarchy and can be eliminated from further consideration. Raw perceptual information resides on the lowest level of the hierarchy. Our task is to propagate this information upwards through the hierarchy. To commence our development, we assume that individual levels are conditionally dependent only on the immediately adjacent descendant and ancestor levels. This assumption allows the factorisation of the joint probability in a manner analogous to a Markov chain [3]. Since we wish to draw information from the bottom upwards, the factorisation commences from the highest level of labelling. The expression for the joint probability of the hierarchical labelling is p(fl, 'VI E L) = p(fb) II P(fl+Ill) (2) IEL,I#t We can now focus our attention on the conditional probabilities P(fI+1lfl). These quantities express the probability of a labelling at the level I + 1 given the previously defined labelling at the descendant level l. We develop tractable expressions for these probabilities by decomposing the hierarchical graph into convenient structural units. Here we build on the idea of decomposing Single-level graphs into supercliques that was successfully exploited in our previous work [10]. Super-cliques are the sets of nodes connected to a centre-object by intra-level edges. However, in the hierarchical case the relational units are more complex since we must also consider the graph-structure conveyed by inter-level edges. We follow the philosophy adopted in the single-level case [10] by averaging the superclique probabilities to estimate the conditional matching probabilities P(fI+1lfl). If r~ C fl denotes the current match of the super-clique centred on the object j E ~l then we write P(f'lfl-I) = ~l L p(r~lfl-l) I I jEV' (3) In order to model this probability, we require a dictionary of constraint relations for the corresponding graph sub-units (super-cliques) from the model graph G 2 ? The allowed mappings between the model graph and the data graph which preserve the topology of the graph structure at a particular level of representation are referred 692 R. C. Wilson and E R. Hancock to as "structure preserving mappings" or SPM's. It is important to note that we need only explore those mappings which are topologically identical to the superclique centred on object j and therefore the possible mappings of the child nodes are heavily constrained by the mappings of their parents (Figure 1). We denote the set of SPM's by P. Since the set P is effectively the state-space of legal matching, we can apply the Bayes theorem to compute the conditional super-clique probability in the following manner p(r~I/I-l) = 2: p(r~15,/I-l)P(5Ii-l) (4) SEP According to this expression, there are two distinct components to our model. The first involves the comparison between our mapped realisation of the super-clique from graph G 1 , i.e. with the selected unit from graph G 2 and the mapping from level 1- 1. Here we take the view that once we have hypothesised a particular mapping 5 from P, the mapping P-l provides us with no further information, i.e. p(r~ 15, /1-1) = p(r~ 15). The matched super-clique r~ is conditionally independent given a mapping from the set of SPM's and we may write the first term as p(r~15). In other words, this first conditional probability models only intra-level constraints. The second term is the significant one in evaluating the impact inter-level constraints on the labelling at the previous level. In this term the probability of the hypothesised mapping 5 is conditioned according to the match of the child levell. q, All that remains now is to evaluate the conditional probabilities. Under the assumption of memoryless matching errors, the first term may be factorised over the marginal probabilities for the assigned matches lIon the individual nodes of the matched super-clique given their counterparts Si belonging to the structure preserving mapping 5. In other words, q p(r;15) = II P('~lsi) (5) 1'! Ef~ In order to proceed we need to specify a probability distribution for the different matching possibilities. There are three cases. Firstly, the match Ii may be to a dummy-node d inserted into to raise it to the same size as 5 so as to facilitate comparison. This process effectively models structural errors in the data-graph. The second and third cases, relate to whether the match is correct or in error. Assuming that dummy node insertions may be made with probability Ps and that matching errors occur with probability Pe , then we can write down the following distribution rule q if Ii = d or 1'f Ii1 = Si otherwise Si = d (6) The second term in Equation (5) is more subtle; it represents the conditional probability of the SPM 5 given a previously determined labelling at the level below. However, the mapping contains labels only from the current levell, not labels from level I - 1. We can reconcile this difference by noting that selection of a particular mapping at level I limits the number of consistent mappings allowed topologically at the level below. In other words if one node is mapped to another at level I, Graph Matching with Hierarchical Discrete Relaxation 693 the consistent interpretation is that the children of the nodes must match to each other. Provided that a set of mappings is available for the child-nodes, then this constraint can be used to model P(SljI-1). The required child-node mappings are referred to as "Hierarchy Preserving Mappings" or HPM's . It is these hierarchical mappings that lift the requirements for moralization in our matching scheme, since they effectively encode potentially incestuous vertical relations. We will denote the set of HPM's for the descendants of the SPM S as Qs and a member of this set as Q = {qi, 'Vi E Vj}. Using this model the conditional probability P(SIfI-l) is given by p(SIfI-1) L = P(SIQ,/I-1)P(Qll- 1 ) (7) QEQs Following our modelling of the intra-level probabilities, in this inter-level case assume that S is conditionally independent of 11- 1 given Q, i.e. P(SIQ, / 1- 1) = P(SIQ)? Traditionally, dictionary based hierarchical schemes have operated by using a labelling determined at a preceding level to prune the dictionary set by elimination of vertically inconsistent items [4]. This approach can easily be incorporated into our scheme by setting P(QI/I-l) equal to unity for consistent items and to zero for those which are inconsistent. However we propose a different approach; by adopting the same kind of label distribution used in Equation 6 we can grade the SPM's according to their consistency with the match at level 1 - 1, i.e. jI-l. The model is developed by factorising over the child nodes qi E Q in the following manner P(Qll- 1 ) = II P(qih,!-1) (8) qiEQ The conditional probabilities are assigned by a re-application of the distribution rule given in Equation (6), i.e. if dummy node match 'f 1-1 1 qi = Ii otherwise (9) For the conditional probability of the SPM given the HPM Q, we adopt a simple uniform model under the assumption that all legitimate mappings are equivalent, i.e. P(SIQ) = P(S) = I~I' The various simplifications can be assembled along the lines outlined in [10] to develop a discrete update rule for matching the two hierarchical structures. The MAP update decision depends only on the label configurations residing on levels 1 - 1, 1 and 1 + 1 together with the measurements residing on levell. Specifically, the level 1 matching configuration satisfies the condition II = argm!F{ f II p(~~ljt(j)) }P(fI-lli )P(PI1 + l 1 1) (10) jEV! Here consistency of match between levels land 1 - 1 of the hierarchy is gauged by R. C. Wilson and E. R. Hancock 694 the quantity PUl-llf l ) = :1 L L 1 L + ks~(rL S))] K(rD Qs exp [-(keH(r LS) K(r!-l) exp [- (keH (r~-l ,Q) 1 iEVI SEP + ks ~(r~-l, Q)) }11) QEQs In the above expression H (r j, S) is the "Hamming distance" which counts the number of label conflicts between the assigned match rj and the structure preserving mapping S. This quantity measures the consistency of the matched labels. The number of dummy nodes inserted into r j by the mapping S is denoted by ~ (r j, S). This second quantity measures the structural compatibility of the two hierarchical graphs. The exponential constants ke = In (l-PeMI-P,) and ks = In 1Ft, are related to the probabilities of structural errors and mis-assignment errors. Finally, K(rj) = (1- P e ){1- Pe)lrjl is a normalisation constant. Finally, it is worth pointing out that the discrete relaxation scheme of Equation (10) can be applied at any level in the hierarchy. In other words the process can be operated in top-down or bottom-up modes if required. 3 Matching SAR Data In our experimental evaluation of the discrete relaxation scheme we will focus on the matching of perceptual groupings of line-segments in radar images. Here the model graph is elicited from a digital map for the same area as the radar image. The line tokens extracted from the radar data correspond to hedges in the landscape. These are mapped as quadrilateral field boundaries in the cartographic model. To support this application, we develop a hierarchical matching scheme based on linesegments and corner groupings. The method used to extract these features from the radar images is explained in detail in [10]. Straight line segments extracted from intensity ridges are organised into corner groupings. The intra-level graph is a constrained Delaunay triangulation of the line-segments. Inter-level relations represent the subsumption of the bottom-level line segments into corners. The raw image data used in this study is shown in Figure 2a. The extracted linesegments are shown in Figure 2c. The map used for matching is shown in Figure 2b. The experimental matching study is based on 95 linear segments in the SAR data and 30 segments contained in the map. However only 23 of the SAR segments have feasible matches within the map representation. Figure 2c shows the matches obtained by non-hierarchical means. The lines are coded as follows; the black lines are correct matches while the grey lines are matching errors. With the same coding scheme Figure 2d. shows the result obtained using the hierarchical method outlined in this paper. Comparing Figures 2c and 2d it is clear that the hierarchical method has been effective at grouping significant line structure and excluding clutter. To give some idea of relative performance merit, in the case of the non-hierarchical method, 20 of the 23 matchable segments are correctly identified with 75 incorrect matches. Application of the hierarchical method gives 19 correct matches, only 17 residual clutter segments with 59 nodes correctly labelled as clutter. 4 Concl usions We have developed graph matching technique which is tailored to hierarchical relational descriptions. The key element is this development is to quantify the match- Graph Matching with Hierarchical Discrete Relaxation , a) b) \ " ~\ 695 - " - "\ ' /\{-\'\(" '\ c) \ d) Figure 2: Graph editing: a) Original image, b) Digital map, c) Non hierarchical match, d) Hierarchical match. ing consistency using the concept of hierarchy preserving mappings between two graphs. Central to the development of this novel technique is the idea of computing the probability of a particular node match by drawing on the topologically allowed mappings of the child nodes in the hierarchy. Results on image data with lines and corners as graph nodes reveal that the technique is capable of matching perceptual groupings under moderate levels of corruption. References [1] F. Cohen and D. Cooper. Simple Parallel Hierarchical and Relaxation Algorithms for Segmenting Non-Causal Markovian Random Fields. IEEE PAMI, 9, 1987, pp.195219. [2] L. Davis and T. Henderson. Hierarchical Constraint Processes for Shape Analysis. IEEE PAMI, 3, 1981, pp.265-277. [3] B. Gidas. A Renormalization Group Approach to Image Processing Problems. IEEE PAMI, 11, 1989, pp.164-180. [4] T . Henderson. Discrete Relaxation Techniques . Oxford University Press, 1990. [5] D.J. Spiegelhalter and S.L. Lauritzen, Sequential updating of conditional probabilities on directed Graphical structures, Networks, 1990, 20, pp.579-605. [6] D. Marr, Vision. W.H. Freeman and Co., San Francisco. [7] J. Pearl, Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, 1988. [8] M. Meila and M. Jordan, Optimal triangulation with continuous cost functions, Advances in Neural Information Processing Systems 9, to appear 1997. [9] P.Smyth, D. Heckerman, M.1. Jordan, Probabilistic independence networks for hidden Markov probability models, Neural Computation, 9, 1997, pp. 227-269. [10] R.C . Wilson and E . R. Hancock, Structural Matching by Discrete Relaxation. IEEE PAMI, 19, 1997, pp .634- 648. IEEE PAMI, June 1997. 

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Asymptotic Theory for Regularization: One-Dimensional Linear Case Petri Koistinen Rolf Nevanlinna Institute, P.O. Box 4, FIN-00014 University of Helsinki, Finland. Email: PetrLKoistinen@rnLhelsinkLfi Abstract The generalization ability of a neural network can sometimes be improved dramatically by regularization. To analyze the improvement one needs more refined results than the asymptotic distribution of the weight vector. Here we study the simple case of one-dimensional linear regression under quadratic regularization, i.e., ridge regression. We study the random design, misspecified case, where we derive expansions for the optimal regularization parameter and the ensuing improvement. It is possible to construct examples where it is best to use no regularization. 1 INTRODUCTION Suppose that we have available training data (Xl, Yd, .. 0' (Xn' Y n ) consisting of pairs of vectors, and we try to predict Yi on the basis of Xi with a neural network with weight vector w. One popular way of selecting w is by the criterion 1 (1) n -n L ?(Xi' Yi, w) + >..Q(w) = min!, I where the loss ?(x,y,w) is, e.g., the squared error Ily - g(x,w)11 2 , the function g(., w) is the input/output function of the neural network, the penalty Q(w) is a real function which takes on small values when the mapping g(o, w) is smooth and high values when it changes rapidly, and the regularization parameter >.. is a nonnegative scalar (which might depend on the training sample). We refer to the setup (1) as (training with) regularization, and to the same setup with the choice >.. = 0 as training without regularization. Regularization has been found to be very effective for improving the generalization ability of a neural network especially when the sample size n is of the same order of magnitude as the dimensionality of the parameter vector w, see, e.g., the textbooks (Bishop, 1995; Ripley, 1996). Asymptotic Theory for Regularization: One-Dimensional Linear Case 295 In this paper we deal with asymptotics in the case where the architecture of the network is fixed but the sample size grows . To fix ideas, let us assume that the training data is part of an Li.d. (independent, identically distributed) sequence (X,Y);(Xl'Yl),(X2'Y2)"" of pairs of random vectors, i.e., for each i the pair (Xi, Yi) has the same distribution as the pair (X, Y) and the collection of pairs is independent (X and Y can be dependent) . Then we can define the (prediction) risk of a network with weights w as the expected value (2) r(w) := IE:f(X, Y, w). Let us denote the minimizer of (1) by Wn (.),) , and a minimizer of the risk r by w*. The quantity r(w n (>.)) is the average prediction error for data independent of the training sample. This quantity r(w n (>.)) is a random variable which describes the generalization performance of the network: it is bounded below by r( w*) and the more concentrated it is about r(w*), the better the performance . We will quantify this concentration by a single number, the expected value IE:r(w n (>.)) . We are interested in quantifying the gain (if any) in generalization for training with versus training without regularization defined by (3) When regularization helps, this is positive. However, relatively little can be said about the quantity (3) without specifying in detail how the regularization parameter is determined. We show in the next section that provided>' converges to zero sufficiently quickly (at the rate op(n- 1 / 2 )), then IE: r(wn(O)) and IE: r(w n (>.)) are equal to leading order. It turns out, that the optimal regularization parameter resides in this asymptotic regime. For this reason, delicate analysis is required in order to get an asymptotic approximation for (3). In this article we derive the needed asymptotic expansions only for the simplest possible case: one-dimensional linear regression where the regularization parameter is chosen independently of the training sample. 2 REGULARIZATION IN LINEAR REGRESSION We now specialize the setup (1) to the case of linear regression and a quadratic smoothness penalty, i.e. , we take f(x,y,w) = [y-x T wJ2 and Q(w) = wTRw, where now y is scalar, x and w are vectors, and R is a symmetric, positive definite matrix. It is well known (and easy to show) that then the minimizer of (1) is (4) wn (>') = [ 1 -1 n ~ ~ XiX! + >'R ] 1 n ~ ~ XiYi. This is called the generalized ridge regression estimator, see, e.g., (Titterington, 1985); ridge regression corresponds to the choice R = I, see (Hoerl and Kennard, 1988) for a survey. Notice that (generalized) ridge regression is usually studied in the fixed design case, where Xi:s are nonrandom. Further, it is usually assumed that the model is correctly specified, i.e., that there exists a parameter such that Y i = Xr w* + ?i , and such that the distribution of the noise term ?i does not depend on Xi. In contrast, we study the random design, misspecified case. Assuming that IE: IIXI1 2 < 00 and that IE: [XXT] is invertible, the minimizer of the risk (2) and the risk itself can be written as (5) (6) w* = A-lIE: [XY], r(w) = r(w*) + (w - with A:=IE:[XXT] w*f A(w - w*). P. Koistinen 296 If Zn is a sequence of random variables, then the notation Zn = open-a) means that n a Zn converges to zero in probability as n -+ 00 . For this notation and the mathematical tools needed for the following proposition see, e.g., (Serfiing, 1980, Ch. 1) or (Brockwell and Davis, 1987, Ch. 6). Proposition 1 Suppose that IE: y4 < 00, IE: IIXII 4 < 00 and that A = IE: [X XTj is invertible. If,\ = op(n- I / 2), then both y'n(wn(O) -w*) and y'n(w n ('\) - w*) converge in distribution to N (0, C), a normal distribution with mean zero and covariance matrix C. The previous proposition also generalizes to the nonlinear case (under more complicated conditions). Given this proposition, it follows (under certain additional conditions) by Taylor expansion that both IE:r(w n ('\)) - r(w*) and IEr(wn(O)) - r(w*) admit the expansion f31 n -} + o( n -}) with the same constant f3I. Hence, in the regime ,\ = op(n-I/2) we need to consider higher order expansions in order to compare the performance of wn (,\) and wn(O). 3 ONE-DIMENSIONAL LINEAR REGRESSION We now specialize the setting of the previous section to the case where x is scalar. Also, from now on, we only consider the case where the regularization parameter for given sample size n is deterministic; especially ,\ is not allowed to depend on the training sample. This is necessary, since coefficients in the following type of asymptotic expansions depend on the details of how the regularization parameter is determined. The deterministic case is the easiest one to analyze. We develop asymptotic expansions for the criterion (7) where now the regularization parameter k is deterministic and nonnegative. The expansions we get turn out to be valid uniformly for k ~ O. We then develop asymptotic formulas for the minimizer of I n , and also for In(O) - inf I n . The last quantity can be interpreted as the average improvement in generalization performance gained by optimal level of regularization, when the regularization constant is allowed to depend on n but not on the training sample. From now on we take Q(w) = w 2 and assume that A = IEX2 = 1 (which could be arranged by a linear change of variables). Referring back to formulas in the previous section, we see that (8) r(wn(k)) - r(w*) = ern - kw*)2/(Un + 1 + k)2 =: h(Un , Vn, k), whence In(k) = IE:h(Un , Vn , k), where we have introduced the function h (used heavily in what follows) as well as the arithmetic means Un and Vn _ (9) 1 n L Vi, n Vn:= - with I (10) Vi := XiYi - w* xl For convenience, also define U := X2 - 1 and V := Xy - w* X2 . Notice that U; UI, U2 , ? .. are zero mean Li.d. random variables, and that V; Vi, V2 ,. " satisfy the same conditions. Hence Un and Vn converge to zero, and this leads to the idea of using the Taylor expansion of h(u, v, k) about the point (u, v) = (0,0) in order to get an expansion for In(k). 297 Asymptotic Theory for Regularization: One-Dimensional Linear Case To outline the ideas, let Tj(u,v,k) be the degree j Taylor polynomial of (u,v) f-7 h(u, v, k) about (0,0), i.e., Tj(u, v, k) is a polynomial in u and v whose coefficients are functions of k and whose degree with respect to u and v is j. Then IETj(Un,Vn,k) depends on n and moments of U and V. By deriving an upper bound for the quantity IE Ih(Un , Vn , k) - Tj(Un , Vn , k)1 we get an upper bound for the error committed in approximating In(k) by IE Tj(Un , Vn , k). It turns out that for odd degrees j the error is of the same order of magnitude in n as for degree j - 1. Therefore we only consider even degrees j. It also turns out that the error bounds are uniform in k ~ 0 whenever j ~ 2. To proceed, we need to introduce assumptions. Assumption 1 IE IXlr < 00 and IE IYls < 00 for high enough rand s. Assumption 2 Either (a) for some constant j3 > 0 almost surely X has a density which is bounded in some neighborhood of zero. IXI ;::: j3 or (b) Assumption 1 guarantees the existence of high enough moments; the values r = 20 and s = 8 are sufficient for the following proofs. E.g., if the pair (X, Y) has a normal distribution or a distribution with compact support, then moments of all orders exist and hence in this case assumption 1 would be satisfied. Without some condition such as assumption 2, In(O) might fail to be meaningful or finite. The following technical result is stated without proof. Proposition 2 Let p > 0 and let 0 < IE X 2 < 00. If assumption 2 holds, then where the expectation on the left is finite (a) for n assumption 2 (a), respectively 2 (b) holds. ~ 1 (b) for n > 2p provided that Proposition 3 Let assumptions 1 and 2 hold. Then there exist constants no and M such that In(k) = JET2(Un , Vn , k) + R(n, k) where _ _ (w*)2k 2 W*kIEUV] -1 [IEV2 (w*)2k2JEU 2 IET2(Un , Vn , k) = (1+k)2 +n (1+k)2 +3 (1+k)4 +4 (1+k)3 IR(n, k) I :s; Mn- 3/ 2(k + 1)-1, "In;::: no, k ;::: o. PROOF SKETCH The formula for IE T 2 (Un , Vn. k) follows easily by integrating the degree two Taylor polynomial term by term. To get the upper bound for R(n, k), consider the residual where we have omitted four similar terms. Using the bound P. Koistinen 298 the Ll triangle inequality, and the Cauchy-Schwartz inequality, we get r IR(n, k)1 = IJE [h(Un , Vn , k) - T2(Un , Vn , k)]1 ., (k+ {2(k W' {Ii: [(~ ~Xl)-'] + 1)3[JE (lUn I2IVnI 4 )]l/2 + 4(w*)2k 2(k + 1)[18 IUn I6 ]l/2 ... } By proposition 2, here 18 [(~ 2:~ X[)-4] = 0(1). Next we use the following fact, cf. (Serfiing, 1980, Lemma B, p. 68). Fact 1 Let {Zd be i.i.d. with 18 [Zd = 0 and with 18 IZI/v < 00 for some v Then v ~ 2. Applying the Cauchy-Schwartz inequality and this fact, we get, e.g., that [18 (IUn I2 IVnI 4 )]l/2 ~ [(18 IUn I4 )1/2(E IVnI 8)1/2p/2 = 0(n- 3 / 2). o Going through all the terms carefully, we see that the bound holds. Proposition 4 Let assumptions 1 and 2 hold, assume that w* :j; 0, and set al := (18 V2 - 2w*E [UVD/(w*)2. If al > 0, then there exists a constant ni such that for all n ~ nl the function k ~ ET2(Un , Vn,k) has a unique minimum on [0,(0) at the point k~ admitting the expanszon k~ = aIn- 1 + 0(n- 2); further, In(O) - inf{Jn(k) : k ~ O} = In(O) - In(aln- 1 ) = ar(w*)2n- 2 If a ~ + 0(n- 5 / 2). 0, then The proof is based on perturbation expansio!1 c~nsidering lin a small parameter. By the previous proposition, Sn(k) := ET2 (Un , Vn , k) is the sum of (w*)2k 2/(1 + k)2 and a term whose supremum over k ~ ko > -1 goes to zero as n ~ 00. Here the first term has a unique minimum on (-1,00) at k = O. Differentiating Sn we get PROOF SKETCH S~(k) = [2(w*)2k(k + 1)2 + n- 1 p2(k)]/(k + 1)5, where P2(k) is a second degree polynomial in k. The numerator polynomial has three roots, one of which converges to zero as n ~ 00. A regular perturbation expansion for this root, k~ = aln- I + a2n-2 + ... , yields the stated formula for al. This point is a minimum for all sufficiently large n; further, it is greater than zero for all sufficiently large n if and only if al > O. The estimate for J n (0) - inf {J n (k) : k ~ O} in the case al > 0 follows by noticing that In(O) - In(k) = 18 [h(Un , Vn , 0) - h(Un , Vn , k)), where we now use a third degree Taylor expansion about (u, v, k) = (0,0,0) h(u,v,O) - h(u,v,k) = 2w* kv - (w*)2k2 - 4w*kuv + 2(w*?k 2u + 2kv 2 - 4w*k 2v + 2(W*)2k 3 + r(u, v, k). Asymptotic Theory for Regularization: One-Dimensional Linear Case 299 0.2 0.18 0.16 0.14 0.12 0.1 __ __ __ __ __ __ __ 0.05 0.1 0.15 0 .2 0.25 0.3 0.35 0.4 ~~ o ~ ~ ~ ~ ~ ~ L-~ __ 0.45 ~ 0.5 Figure 1: Illustration of the asymptotic approximations in the situation of equation (11) . Horizontal axis kj vertical axis .In(l?) and its asymptotic appr~ximations. Legend: markers In(k); solid line IE T 2 (Un , Vn , k)j dashed line IET4 (Un , Vn , k). Usin~ t~e techniques of the previous proposition, it can be shown that IE Ir(Un , Vn , k~)1 = O(n- S/ 2 ). Integrating the Taylor polynomial and using this estimate gives In(O) - In(aI/n) = af(w*)2n- 2 + O(n- S / 2 ). Finally, by the mean value theorem, In(O) -inf{ In(k) : k ~ O} = In(O) -In(aI/n) + ! = In(O) - (In(O) - In(k)]lk=8(k~ -aI/n) In(aI/n) + O(n-1)O(n-2) where () lies between k~ and aI/n, and where we have used the fact that the indicated derivative evaluated at () is of order O(n- 1 ), as can be shown with moderate effort. 0 Remark In the preceding we assumed that A = IEX 2 equals 1. If this is not the case, then the formula for a1 has to be divided by A; again, if a1 > 0, then k~ = a1n-1 + O(n- 2 ) . If the model is correctly specified in the sense that Y = w* X + E, where E is independent of X and IE E = 0, then V = X E and IE [UV] = O. Hence we have a1 = IE [E 2]j(w*)2, and this is strictly positive expect in the degenerate case where E = 0 with probability one. This means that here regularization helps provided the regularization parameter is chosen around the value aI/n and n is large enough. See Figure 1 for an illustration in the case (11) X "'" N(O, 1) , Y = w* X +f, f "'" N(O, 1), w* = 1, where E and X are independent. In(k) is estimated on the basis of 1000 repetitions of the task for n = 8. In addition to IE T2(Un , Vn , k) the function IE T4 (Un , lin, k) is also plotted. The latter can be shown to give In(k) correctly up to order O(n- s/ 2 (k+ 1)-3). Notice that although IE T2 (Un , Vn , k) does not give that good an approximation for In(k), its minimizer is near the minimizer of In(k), and both of these minimizers lie near the point al/n = 0.125 as predicted by the theory. In the situation (11) it can actually be shown by lengthy calculations that the minimizer of In(k) is exactly al/n for each sample size n ~ 1. It is possible to construct cases where X "'" Uniform (a, b), Y = cjX + d+ Z, a1 < O. For instance, take 1 1 b=-(3Vs-l) 2' 4 c= -5,d= 8, a=- P. Koistinen 300 and Z '" N (0, a 2 ) with Z and X independent and 0 :::; a < 1.1. In such a case regularization using a positive regularization parameter only makes matters worse; using a properly chosen negative regularization parameter would, however, help in this particular case. This would, however, amount to rewarding rapidly changing functions. In the case (11) regularization using a negative value for the regularization parameter would be catastrophic. 4 DISCUSSION We have obtained asymptotic approximations for the optimal regularization parameter in (1) and the amount of improvement (3) in the simple case of one-dimensional linear regression when the regularization parameter is chosen independently of the training sample. It turned out that the optimal regularization parameter is, to leading order, given by Qln- 1 and the resulting improvement is of order O(n- 2 ). We have also seen that if Ql < 0 then regularization only makes matters worse. Also (Larsen and Hansen, 1994) have obtained asymptotic results for the optimal regularization parameter in (1). They consider the case of a nonlinear network; however, they assume that the neural network model is correctly specified. The generalization of the present results to the nonlinear, misspecified case might be possible using, e.g., techniques from (Bhattacharya and Ghosh, 1978). Generalization to the case where the regularization parameter is chosen on the basis of the sample (say, by cross validation) would be desirable. Acknowledgements This paper was prepared while the author was visiting the Department for Statistics and Probability Theory at the Vienna University of Technology with financial support from the Academy of Finland. I thank F. Leisch for useful discussions. References Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. The Annals of Statistics, 6(2):434-45l. Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press. Brockwell, P. J. and Davis, R. A. (1987). Time Series: Theory and Methods. Springer series in statistics. Springer-Verlag. Hoerl, A. E. and Kennard, R. W. (1988). Ridge regression. In Kotz, S., Johnson, N. L., and Read, C. B., editors, Encyclopedia of Statistical Sciences. John Wiley & Sons, Inc. Larsen, J. and Hansen, L. K. (1994). Generalization performance of regularized neural network models. In Vlontos, J., Whang, J.-N., and Wilson, E., editors, Proc. of the 4th IEEE Workshop on Neural Networks for Signal Processing, pages 42-51. IEEE Press. Ripley, B. D. (1996). Pattern Recognition and Neural Networks. Cambridge University Press. Serfiing, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, Inc. Titterington, D. M. (1985). Common structure of smoothing techniques in statistics. International Statistical Review, 53:141-170. A General Purpose Image Processing Chip: Orientation Detection Ralph Etienne-Cummings and Donghui Cai Department of Electrical Engineering Southern Illinois University Carbondale, IL 6290 1-6603 Abstract A 80 x 78 pixel general purpose vision chip for spatial focal plane processing is presented. The size and configuration of the processing receptive field are programmable. The chip's architecture allows the photoreceptor cells to be small and densely packed by performing all computation on the read-out, away from the array. In addition to the raw intensity image, the chip outputs four processed images in parallel. Also presented is an application of the chip to line segment orientation detection, as found in the retinal receptive fields of toads. 1 INTRODUCTION The front-end of the biological vision system is the retina, which is a layered structure responsible for image acquisition and pre-processing. The early processing is used to extract spatiotemporal information which helps perception and survival. This is accomplished with cells having feature detecting receptive fields, such as the edge detecting center-surround spatial receptive fields of the primate and cat bipolar cells [Spillmann, 1990]. In toads, the receptive fields of the retinal cells are even more specialized for survival by detecting ''prey'' and "predator" (from size and orientation filters) at this very early stage [Spi11mann, 1990]. The receptive of the retinal cells performs a convolution with the incident image in parallel and continuous time. This has inspired many engineers to develop retinomorphic vision systems which also imitate these parallel processing capabilities [Mead, 1989; Camp, 1994]. While this approach is ideal for fast early processing, it is not space efficient. That is, in realizing the receptive field within each pixel, considerable die area is required to implement the convolution kernel. In addition, should programmability be required, the complexity of each pixel increases drastically. The space constraints are eliminated if the processing is performed serially during read-out. The benefits of this approach are 1) each pixel can be as small as possible to allow high resolution imaging, 2) a single processor unit is used for the entire retina thus reducing mis-match problems, 3) programmability can be obtained with no impact on the density of imaging array, and R. Etienne-Cummings and D. Cai 874 4) compact general purpose focal plane visual processing is realizable. The space constrains are then transfonned into temporal restrictions since the scanning clock speed and response time of the processing circuits must scale with the size of the array. Dividing the array into sub-arrays which are scanned in parallel can help this problem. Clearly this approach departs from the architecture of its biological counterpart, however, this method capitalizes on the main advantage of silicon which is its speed. This is an example of mixed signal neuromorphic engineering, where biological ideas are mapped onto silicon not using direct imitation (which has been the preferred approach in the past) but rather by realizing their essence with the best silicon architecture and computational circuits. This paper presents a general purpose vision chip for spatial focal plane processing. Its architecture allows the photoreceptor cells to be small and densely packed by performing all computation on the read-out, away from the array. Performing computation during read-out is ideal for silicon implementation since no additional temporal over-head is required, provided that the processing circuits are fast enough. The chip uses a single convolution kernel, per parallel sub-array, and the scanning bit pattern to realize various receptive fields. This is different from other focal plane image processors which am usually restricted to hardwired convolution kernels, such as oriented 20 Gabor filters [Camp, 1994]. In addition to the raw intensity image, the chip outputs four processed versions per sub-array. Also presented is an application of the chip to line segment orientation detection, as found in the retinal receptive fields of toads [Spillmann, 1990]. 2 THE GENERAL PURPOSE IMAGE PROCESSING CHIP 2.1 System Overview This chip has an 80 row by 78 column photocell array partitioned into four independent sub-arrays, which are scanned and output in parallel, (see figure I). Each block is 40 row by 39 column, and has its own convolution kernel and output circuit. The scanning circuit includes three parts: virtual ground, control signal generator (CSG), and scanning output transformer. Each block has its own virtual ground and scanning output transformer in both x direction (horizontal) and y direction (vertical). The control signal generator is shared among blocks. 2.2 Hardware Implementation The photocell is composed of phototransistor, photo current amplifier, and output control. The phototransistor performance light transduction, while the amplifier magnifies the photocurrent by three orders of magnitude. The output control provides multiple copies of the amplified photocun-ent which is subsequently used for focal plane image processing. The phototransistor is a parasitic PNP transistor in an Nwell CMOS process. The current amplifier uses a pair of diode connected pmosfets to obtain a logarithmic relationship between light intensity and output current. This circuit also amplifies the photocurrent from nanoamperes to microamperes. The photocell sends three copies of the output currents into three independent buses. The connections from the photocell to the buses are controlled by pass transistors, as shown in Fig. 2. The three current outputs allow the image to be processed using mUltiple receptive field organization (convolution kernels), while the raw image is also output. The row (column) buses provides currents for extracting horizontally (vertically) oriented image features, while the original bus provides the logarithmically compressed intensity image. The scanning circuit addresses the photocell array by selecting groups of cells at one time. Since the output of the cells are currents, virtual ground circuits are used on each bus to mask the> I pF capacitance of the buses. The CSG, implemented with shift registers A General Purpose Image Processing Chip: Orientation Detection " " " " t J, -=_"ODat~tl'g<I"O'''''' I 1: ~!; ~~.+ I :': :~ I ~'!"' I 1" ,, " :;"" o;::!' ,, : 1 ...... . - 875 J9~ .,. . .-- I ~~".... i ! I ! ! f W V lloc',I: .... , ..-. ~ m ~ 1 1 V ... . ......,. ..,. 1I..... ,2orf. _ e69r>0 . . .o-Y......... 1IIocIf,.?""._ lIIIoc~' l: Drl, ...... ..... . . . " . "'!J" rty ..... , 1Id,..,.Mpwy A r 1 f 1 -f L 1 I 1 : >< ~;;:-i :':'< :1 - -t.". ciii-. - - { ~ I ,I , 1 ! f " :, L I I f = -t *",1,,",,1, ,',,1,><,.,,', ~I.. _ .... " " " " ' 1 ~ I 1 ? , I_ t ...f~;;;i [!J ..?..?.? ?? jir;i c.... oA""; - - w d"' ~ ;;; " " I ~~ - ~ ~ Figure 1: Block diagram of the chip. produces signals which select photocells and control the scanning output transformer. The scanning output transformer converts currents from all row buses into Ipe? and Icenx' and converts currents from all row buses into lpery and Iccny. This transformation is required to implement the various convolution kernels discussed later. The output transformer circuits are controlled by a central CSG and a peripheral CSG. These two generators have identical structures but different initial values. It consists of an n-bit shift register in x direction (horizontally) and an m-bit shift register in y direction (vertically). A feedback circuit is used to restore the scanning pattern into the x shift register after each row scan is completed. This is repeated until all the row in each block are scanned. The control signals from the peripheral and central CSGs select all the cells covered by a 2D convolution mask (receptive field). The selected cells send Ixy to the original bus, Ixp to the row bus, and Iyp to the column bus. The function of the scanning output transformer is to identify which rows (columns) are considered as the center (Icenx or Ircny) or periphery (Ire rx or Ipcry) of the convolution kernel, respectively. Figure 3 shows how a 3x3 convolution kernel can be constructed. Figure 4 shows how the output transformer works for a 3x3 mask. Only row bus transformation is shown in this example, but the same mechanism applies to the column bus as well. The photocell array is m row by n column, and the size is 3x3. The XC (x center) and YC (y center) come from the central CSG; while XP (x peripheral) and YP (y peripheral) come from the peripheral CSG. After loading the CSG, the initial values of XP and YP are both 00011...1. The initial values of XC and YC are both 10 111.. .1. This identifies the central cell as location (2, 2). The currents from the central row (column) are summed to form Iren ? and leeny, while all the peripheral cells are summed to form Iperx and lpery. This is achieved by activating the switches labeled XC, YC, XP and YP in figure 2. XPj (YP,) {i= I, 2, ... , n} controls whether the output current of one cell R. Etienne-Cummings and D. Cai 876 YC~ IO----+----yp ,.b---~p XC---<::A. Jyp hp Original Bus--<E,..-I'----+_--+_ _ Column Bus_~_--,_-+_ _ Row (1 ,1) (1 ,2) (2,1) (2,2) (3,1) (3,2) lori (3,:l BUS_~_ _ _-----4_ _ Figure 2: Connections between a photocell and the current buses. Figure 3: field . Constructing a 3x3 receptive goes to the row (column) bus. Since XP j (YP) is connected to the gate of a pmos switch, a 0 in XP j (YPj) it turns on . YC j (XC j ) {i=l, 2, ... , n} controls whether a row (column) bus connects to Icenx bus in the same way. On the other hand, the connection from a row (column) bus to Ipcrx bus is controlled by an nmos and a pmos switch. The connection is made if and only if YC, (XCi)' an nmos switch, is 1 and YP i (XPi), a pmos switches, is O. The intensity image is obtained directly when XCi and YC j are both O. Hence, lori = 1(2,2), Icenx = lrow2= 1(2,1) + 1(2,2) + 1(2,3) and Iperx = lrowl + lrow3 = 1(1,1) + 1(1,2) + 1(1,3) + 1(3,1) + 1(3,2) + 1(3,3). The convolution kernel can be programmed to perform many image processing tasks by loading the scanning circuit with the appropriate bit pattern. This is illustrated by configuring the chip to perform image smoothing and edge extraction (x edge, y edge, and 20 edge), which are all computed simultaneously on read-out. It receives five inputs (lori' Iccn ,' lperx, Iceny, Ipcry) from the scanning circuit and produces five outputs (lori' ledge.> ledgey' Ismllllth,ledge2d). The kernel (receptive field) size is programmable from 3x3, 5x5, 7x7, 9x9 and 11 x 11 . Fig. 5 shows the 3x3 masks for this processing. Repeating the above steps for 5x5, 7x7, 9x9, and II x 11 masks, we can get similar results. p.. u >- >- c:s:l c:s:l c:s:l ..... -- YPI VCI Yl'2 YC2 c:s:l ---- YP3 YC3 YPIoI YCN Figure 4: Scanning output transformer for an m row by n column photo cell array. A General Purpose Image Processing Chip: Orientation Detection 877 1 I 1 -I -I -I -1 2 -I 0 -1 0 1 1 1 2 2 2 -I 2 -I -I 4 -I 1 1 1 -I -I -1 -I 2 -I 0 -I 0 (a) smooth (b) edge_x (c) edge-y (d) edge_2D Figure 5: 3x3 convolution masks for various image processing. In general, the convolution results under different mask sizes can be expressed as follows: I~mooth=Icen. +Ire... Iedge.=Kld *Icen. -Ipc", Iedgey=Kld *Iceny-Ipcry Iedge2D=K2d *I"ri-Icen.-Iceny Where Kid and K2d are the programmable coefficients (from 2-6 and 6-14, respectively) for ID edge extraction and 2D edge extraction, respectively. By varying the locations of the O's in the scanning circuits, different types of receptive fields (convolution kernels) can be realized. 2.3 Results The chip contains 65K transistors in a footprint of 4.6 mm x 4.7 mm. There are 80 x 78 photocells in the chip, each of which is 45.6 11m x 45 !lm and a fill factor of 15%. The convolution kernel occupies 690.6 !lm x 102.6 11m. The power consumption of the chip for a 3x3 (1\ x 11) receptive field, indoor light, and 5V power supply is < 2 m W (8 m W). To capitalize on the programmability of this chip, an ND card in a Pentium 133MHz PC is used to load the scanning circuit and to collect data. The card, which has a maximum analog throughput of 100KHz limits the frame rate of the chip to 12 frames per second. At this rate, five processed versions of the image is collected and displayed. The scanning and processing circuits can operate at 10 MHz (6250 fps), however, the phototransistors have much slower dynamics. Temporal smoothing (smear) can be observed on the scope when the frame rate exceeds 100 fps. The chip displays a logarithmic relationship between light intensity and output current (unprocessed imaged) from 0.1 lux (100 nA) to 6000 lux (10 IlA). The fixed pattern noise, defined as standard-deviation/mean, decreases abruptly from 25% in the dark to 2% at room light (800 lux). This behavior is expected since the variation of individual pixel current is large compared to the mean output when the mean is small. The logarithmic response of the photocell results in high sensitivity at low light, thus increasing the mean value sharply. Little variation is observed between chips. The contrast sensitivity of the edge detection masks is also measured for the 3x3 and 5x5 receptive fields. Here contrast is defined as (1m .. - Imin)/(Im .. + I min ) and sensitivity is given as a percentage of the maximum output. The measurements are performed for normal room and bright lighting conditions. Since the two conditions corresponded to the saturated part of the logarithmic transfer function of the photocells, then a linear relationship between output response and contrast is expected. Figure 6 shows contrast sensitivity plot. Figure 7 shows examples of chip's outputs. The top two images are the raw and smoothed (5x5) images. The bottom two are the 1D edge_x (left) and 2D edge (right) images. The pixels with positive values have been thresholded to white. The vertical black line in the image is not visible in the edge_x image, but can be clearly seen in the edge_2D image. 878 R. Etienne-Cummings and D. Cai 80 >< ...~o 60 ~ :; 40 & ::I o ..... ??5,5 Bri!,hl -e-5xS Normal 20 ... ? ?? 3,3 Bri!,hl _ _ 3,3 Nonnal Contrast [%] Figure 6: Contrast sensitivity function of the x edge detection mask. Figure 7: (Clockwise) Raw image. 5x5 smoothed image. edge_2D and edge_x. 3 APPLICATION: ORIENTATION DETECTION 3.1 Algorithm Overview This vision chip can be elegantly used to measure the orientation of line segments which fall across the receptive field of each pixel. The output of the 10 Laplacian operators, edge_x and edge_y, shown in figure 5, can be used to detennine the orientation of edge segments. Consider a continuous line through the origin, represented by a delta function in 20 space by IX y-xtan()). If the origin is the center of the receptive field. the response ofthe edge_x kernel can be computed by evaluating the convolution equation (1). where W(x) = u(x+m)-u(x-m) is the x window over which smoothing is performed, 2m+ J is the width of the window and 2n+ J is the number of coefficients realizing the discrete Laplacian operator. In our case, n =m. Evaluating this equation and substituting the origin for the pixel location yields equation (2), which indicates that the output of the 10 edge_x (edge-y) detectors have a discretized linear relationship to orientation from on to 45" (45? to 90?). At 0", the second term in equation (2) is zero. As e increase, more terms are subtracted until all tenns are subtracted at 45?. Above 45 0 (below 45?), the edge_x (edge-y) detectors output zero since equal numbers of positive and negative coefficients are summed. Provided that contrast can be nonnalized. the output of the detectors can be used to extract the orientation of the line. Clearly these responses are even about the x- and y-axis. respectively. Hence, a second pair of edge detectors. oriented at 45", is required to uniquely extract the angle of the line segment. 10 - - 3 7 0 Lux 8 ~ ..=. 0 - ,0..,. 9- -260 Lux - -. - - (R!) Lux -?? .. -? - (25 Lu x b~ 6 :; s:::I 0 o: -o._~ 4 "I ~ ? ? . ? ?.? ? 2 ,,b , 'b . _. . '0, ' . .. . . o o 20 CI' ...' -. ... ... . ...~ .~ ....:- 40 . ~ . ': ? ? . , , ; : : . :-:-' . - - ?? 60 T -0- - _ _ _ 80 Angle ['-'] Figure 8: Measured orientation transfer function of edge_x detectors. A General Purpose Image Processing Chip: Orientation Detection. 879 n 0edge_Ax,y) = [2nW(x ?m)8(y)- ...EW(x ?m)8(y?i)]*8(y-xtane) (I) ;=} n? ~ I . -I Oedge AO.O) = 2n-[ ~(W(--)+ W(--)] ;=} tane fane 3.2 (2) Results Figure 8 shows the measured output of the edge_x detectors for various lighting conditions as a line is rotated. The average positive outputs are plotted. As expected, the output is maximum for bright ambients when the line is horizontal. As the line is rotated, the output current decreases linearly and levels off at approximately 45". On the other hand, the edge_y (not shown) begins its linear increase at 45" and maximizes at 90?. After normalizing for brightness. the four curves are very similar (not shown). To further demonstrate orientation detection with this chip, a character consisting of a circle and some straight lines is presented. The intensity image of the character is shown in figure 9(a). Figures 9(b) and 9(c) show the outputs of the edge_x and edge-y detectors, respectively. Since a 7x7 receptive field is used in this experiment, some outer pixels of each block are lost. The orientation selectivity of the 1D edge detectors are clearly visible in the figures, where edge_x highlights horizontal edges and edge_y vertical edges. Figure 9(d) shows the reported angles. A program is written which takes the two I D edge images, finds the location of the edges from the edge_2D image, the intensity at the edges (positive lobe) and then computes the angle of the edge segment. In figure 9(d), the black background is chosen for locations where no edges are detected, white is used for 0? and gray for 90?. (a) (b) (c) (d) Figure 9: Orientation detection using ID Laplacian Operators. 4 CONCLUSION A 80x78 pixel general purpose vision chip for spatial focal plane processing has been presented. The size and configuration of the processing receptive field are programmable. In addition to the raw intensity image, the chip outputs four processed images in parallel. The chip has been successfully used for compact line segment orientation detection, which can be used in character recognition. The programmability and relatively low power consumption makes it ideal for many visual processing tasks. References Camp W. and J. Van cler Spiegel, "A Silicon VLSI Optical Sensor for Pattern Recognition, " Sensors and Actuators A, Vol. 43, No. 1-3, pp. 188-195, 1994. Mead C. and M. Ismail (Eds.), Analog VLSI Implementation of Neural Networks, Kluwer Academic Press, Newell, MA, 1989. Spi11mann L. and J. Werner (Eds.), Visual Perception: The Neurophysiological Foundations, Academic Press, San Diego, CA, 1990. 

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On the Separation of Signals from Neighboring Cells in Tetrode Recordings Maneesh Sahani, John S. Pezaris and Richard A. Andersen maneesh@caltech.edu, pz@caltech.edu, andersen@vis.caltech.edu Computation and Neural Systems California Institute of Technology 216-76 Caltech, Pasadena, CA 91125 USA Abstract We discuss a solution to the problem of separating waveforms produced by multiple cells in an extracellular neural recording. We take an explicitly probabilistic approach, using latent-variable models of varying sophistication to describe the distribution of waveforms produced by a single cell. The models range from a single Gaussian distribution of waveforms for each cell to a mixture of hidden Markov models. We stress the overall statistical structure of the approach, allowing the details of the generative model chosen to depend on the specific neural preparation. 1 INTRODUCTION Much of our empirical understanding of the systems-level functioning of the brain has come from a procedure called extracellular recording. The electrophysiologist inserts an insulated electrode with exposed tip into the extracellular space near one or more neuron cell bodies. Transient currents due to action potentials across nearby cell membranes are then recorded as deflections in potential, spikes, at the electrode tip. At an arbitrary location in gray matter, an extracellular probe is likely to see pertubations due to firing in many nearby cells, each cell exhibiting a distinct waveform due to the differences in current path between the cells and the electrode tip. Commonly, the electrode is maneuvered until all the recorded deflections have almost the same shape; the spikes are then all presumed to have arisen from a single isolated cell. This process of cell isolation is time-consuming, and it permits recording from only one cell at a time. IT differences in spike waveform can be exploited to sort recorded events by cell, the experimental cost of extracellular recording can be reduced, and data on interactions between simultaneously recorded cells can be obtained. Separation of Signals from Neighboring Cells 223 Many ad hoc solutions to spike sorting have been proposed and implemented, but thus far an explicit statistical foundation, with its accompanying benefits, has mostly been lacking. Lewicki (1994) is the exception to this rule and provides a wellfounded probabilistic approach, but uses assumptions (such as isotropic Gaussian variability) that are not well supported in many data sets (see Fee et al (1996)). A first step in the construction of a solution to the spike-sorting problem is the specification of a model by which the data are taken to be generated. The model has to be powerful enough to account for most of the variability observed in the data, while being simple enough to allow tractable and robust inference. In this paper we will discuss a number of models, of varying sophistication, that fall into a general framework. We will focus on the assumptions and inferential components that are common to these models and consider the specific models only briefly. In particular, we will state the inference algorithms for each model without derivation or proof; the derivations, as well as measures of performance, will appear elsewhere. 2 DATA COLLECTION The algorithms that appear in this paper are likely to be of general applicability. They have been developed, however, with reference to data collected from the parietal cortex of adult rhesus macaques using tetrodes (Pezaris et a11997). The tetrode is a bundle of four individually insulated 13/lm-diameter wires twisted together and cut so that the exposed ends lie close together. The potential on each wire is amplified (custom electronics), low-pass filtered (9-pole Bessel filter, Ie = 6.4 kHz) to prevent aliasing, and digitized (fs between 12.8 and 20 kHz) (filters and AjD converter from Thcker Davis Technologies). This data stream is recorded to digital media; subsequent operations are currently performed off-line. In preparation for inference, candidate events (where at least one cell fired) are identified in the data stream. The signal is digitally high-pass filtered (fe = 0.05Is) and the root-mean-square (RMS) amplitude on each channel is calculated. This value is an upper bound on the noise power, and approaches the actual value when the firing rates of resolvable cells are low. Epochs where the signal rises above three times the RMS amplitude for two consecutive signals are taken to be spike events. The signal is upsampled in the region of each such threshold crossing, and the time of the maximal subsequent peak across all channels is determined to within onetenth of a sample. A short section is then extracted at the original Is such that this peak time falls at a fixed position in the extracted segment. One such waveform is extracted for each threshold crossing. 3 GENERATIVE FRAMEWORK Our basic model is as follows. The recorded potential trace V(t) is the sum of influences that are due to resolvable foreground cells (which have a relatively large e~ect) and a background noise process. We write (1) Here, c~ is an indicator variable that takes the value 1 if the mth cell fires at time and 0 otherwise. If cell m fires at T it adds a deflection of shape S::n (t - T) to the recorded potential. The effect of all background neural sources, and any electrical noise, is gathered into a single term 7](t). For a multichannel probe, such as a tetrode, all of V(t), 7](t) and S::n(t) are vector-valued. Note that we have indexed T M. Sahani, 1. S. pezaris and R. A Andersen 224 8) @) ! ! Figure 1: Schematic graph of the general framework. the spike shapes from the mth cell by time; this allows us to model changes in the spike waveform due to intrinsic biophysical processes (such as sodium inactivation during a burst of spikes) as separate to the additive background process. We will discuss models where the choice of S::n is purely stochastic, as well as models in which both the probability of firing and the shape of the action potential depend on the recent history of the cell. It will be useful to rewrite (1) in terms of the event waveforms described in section 2. At times r when no foreground cell fires all the c~ are zero. We index the remaining times (when at least one cell fired) by i and write c!n for c~ at ri (similarly for S:n) to obtain (2) This basic model is sketched, for the case of two cells, in figure 1. Circles represent stochastic variables and squares deterministic functions, while arrows indicate conditional or functional dependence. We have not drawn nodes for 0Tl and O. The representation chosen is similar to, and motivated by, a directed acyclic graph (DAG) model of the generative distribution. For clarity, we have not drawn edges that represent dependencies across time steps; the measurement V(t) depends on many nearby values of S::n and c~, and f/(t) may be autocorrelated. We will continue to omit these edges, even when we later show connections in time between c~ and S::n . 4 INFERENCE We have two statistical objectives. The first is model selection, which includes the choice of the number of cells in the foreground . The second is inference: finding good estimates for the c~ given the measured V(t) . We will have little to say on the subject of model selection in this paper, besides making the observation that standard techniques such as cross-validation, penalized likelihood or approximation of the marginal likelihood (or "evidence") are all plausible approaches. We will instead focus on the inference of the spike times. Rather than calculating the marginalized posterior for the c~ we will find the distribution conditioned on the most probable values of the other variables. This is a common approximation to the true posterior (compare Lewicki (1994)). A simple property of the data allows us to estimate the most probable values of Separation of Signals from Neighboring Cells 225 the parameters in stages; times at which at least one foreground cell fires can be identified by a threshold, as described in section 2. We can then estimate the noise parameters 8Tf by looking at segments of the signal with no foreground spikes, the waveform distribution and firing time parameters 8 from the collection of spike events, and finally the spike times c:-n and the waveforms by a filtering process applied to the complete data V(t) given these model parameters. S:n 4.1 NOISE We study the noise distribution as follows. We extract lms segments from a bandpassed recording sampled at 16 kHz from a four-channel electrode, avoiding the foreground spikes identified as in section 2. Each segment is thus a 64-dimensional object. We find the principal components of the ensemble of such vectors, and construct histograms of the projections of the vectors in these directions. A few of these histograms are shown on a log-scale in figure 2 (points), as well as a zero-mean Gaussian fit to the distribution projected along the same axes (lines). It is clear that the Gaussian is a reasonable description, although a slight excess in kurtosis is visible in the higher principal components. pc 1 2l102~. c: ::l 8 0 10 pc 24 pc 12 pc 36 pc 48 ., 10-2 -100 0 IlV 100 -50 o 50 -20 o 20 -20 o 20 -20 o 20 Figure 2: Distribution of background noise. The noise parameters are now seen to be the covariance of the noise, ETf (we represent it as a covariance matrix taken over the length of a spike). In general, we can fit an autoregressive process description to the background and apply a filter that will whiten the noise. This will prove to be quite useful during the filtering stages. 4.2 WAVEFORM PARAMETERS We can make some general remarks about the process of inferring the parameters of the models for S~ and c:-n. Specific models and their inference algorithms will appear in section 5. The models will, in general, be fit to the collection of segments extracted and aligned as described in section 2. At other times they have no influence on the waveform recorded. We will represent these segments by Vi, implying a connection to the firing events ri used in (2). It should be borne in mind that the threshold-based trigger scheme will not exactly identify all of the true ri correctly. We will assume that each segment represents a single Sm, that is, that no two cells fire at times close enough for their spike waveforms to overlap. This is an unreasonable assumption; we can shore it up partially by eliminating from our collection of Vi segments that appear heuristically to contain overlaps (for example, double-peaked waveforms). Ultimately, however, we will need to make our inference procedure robust enough that the parameters describing the model are well estimated despite the errors in the data. M. Sahani, 1. S. Pezaris and R. A Andersen 226 Figure 3: The mixture model for Vi. The advantage to making this assumption is that the overall model for the distribution of the Vi becomes a mixture: a single control variable c i sets exactly one of the c~ to 1. Vi is then drawn from the distribution of waveforms for the selected cell, convolved with the noise. This is a formal statement of the "clustering" approach to spike-sorting. Mixture models such as these are easy to fit using the ExpectationMaximization (EM) algorithm (Dempster et al1977). We will also consider models with additional latent state variables, which are used to describe the distributions of the Sm and Cm, where again EM will be of considerable utility. The measured ensemble Vi will be incorrect on a number of counts. The threshold may make either false positive or false negative errors in selecting ri, and some of the identified Vi will represent overlaps. We can use heuristics to minimize such errors, but need to account for any remaining outliers in our models. We do so by introducing additional mixture components. Segments of noise that are incorrectly identified as foreground events are handled by an explicit zero mixture component whose variability is entirely due to the background noise. Overlaps are handled by providing very broad low-probability components spanning large areas in waveform space; clusters of overlap waveforms are likely to be diffuse and sparse. The mixture model is sketched in figure 3. In the basic model the variables are chosen independently for each cross-threshold event. The dynamic models discussed below will introduce dependencies in time. 4.3 SPIKE TIMES In our final stage of inference, we make estimates of the c~ given the V(t) and the most-probable parameters fit in the previous two stages. This is exactly the signal detection problem of identifying pulses (perhaps with random or else adapting parameters) in Gaussian noise of known covariance. Solutions to this are well known (McDonough and Whalen 1995) and easily adapted to the problem at hand (Sahani et al1998). Separation of Signals from Neighboring Cells 5 227 SPECIFIC MODELS Finally, we describe examples of models that may be used within this framework. As stated before, in this brief catalog we summarize the motivation for each, and state without derivation or proof the algorithms for inference. The details of these algorithms, as well as tests of performance, will appear elsewhere. 5.1 CONSTANT WAVEFORM The simplest model is one in which we take the waveform of the mth cell to remain unchanged and the firing probability of each cell to be constant. In this case we drop the index r or i on the waveform shape and just write Sm (t - r i ) . We write Pm for the probability that a given event is due to the mth cell firing. The mixture model is then a mixture of multivariate Gaussian distributions, each with covariance E r" mean Sm and mixture fraction Pm. The EM algorithm for such a mixture is well known (Nowlan 1990). Given parameters 8(n) = {S~), p~)} from the nth iteration, we find the expected values of the e~ (called the responsibilities), ri = ?[e i I {Vi} m m 8(n)] , = (n) N(Vi. S(n) E ) pm , m , 1) '"'p~n) N(Vi. S~n) E )' L.Jm in (3) 'm'1) and then reestimate the parameters from the data weighted by the responsibilities. P(n+l) = L:r~ _i_ _ . 5.2 (4) N' m REFRACTORY FIRING A simple modification to this scheme can be used to account for the refractory period between spikes from the same cell (Sahani et al1998). The model is similar to the Gaussian mixture above, except that the choice of mixture component is no longer independent for each waveform. If the waveforms arrive within a refractory period they cannot have come from the same cell. This leads to the altered responsibilities: IT i i rm sm = Zi j ;(i,j) (5) refractory where Z is a normalizing constant. The M step here is identical to (4), with the responsibilities s~ replacing the r~. 5.3 STATIC MIXTURE As we have suggested above, the waveform of the mth cell is not, in fact, unchanged each time the cell fires. Variability in excess of the additive background noise is introduced by changes in the biophysical properties of the cell (due to recent firing patterns, or external modulators) as well as by background activity that may be correlated with foreground events. We can attempt to model this variability as giving rise to a discrete set of distinct waveforms, which are then convolved with the previously measured noise covariance to obtain the distribution of measurements. In effect, we are tiling an irregularly shaped distribution with a mixture of Gaussians M. Sahani, J S. pezaris and R. A Andersen 228 of fixed shape, E1J" We obtain a hierarchical mixture distribution in which each component corresponding to a cell is itself is a mixture of Gaussians. Given a particular hierarchical arrangement the parameters can be fit exactly as above. While this approach seems attractive, it suffers from the flaw that model selection is not well defined. In particular, the hierarchical mixture is equivalent in terms of likelihood and parameters to a single-layer, flat, mixture. To avoid this problem we may introduce a prior requiring that the Gaussian components from a single cell overlap, or otherwise lie close together. It is, however, difficult to avoid excessive sensitivity to such a prior. 5.4 DYNAMICAL MIXTURE An alternative approach is to replace the independent transitions between the components of the mixture distribution of a single cell with a dynamical process that reflects the manner in which both firing probability and waveform shape depend on the recent history of the cell. In this view we may construct a mixture of hidden Markov models (HMMs), one for each cell. Our earlier mixture assumption now means that the models must be coupled so that on anyone time step at most one makes a transition to a state corresponding to firing. This structure may be thought of as a special case of the factorial HMM discussed by Gharamani and Jordan (1997). The general model is known to be intractable ..In this special case, however, the standard forward-backward procedure for a single HMM can be modified to operate on reponsibiIity-weighted data, where the reponsibilities are themselves calculated during the forward phase. This is empirically found to provide an effective Estep. The M step is then straightforward. Acknowledgements This work has benefited considerably from important discussions with both Bill Bialek and Sam Roweis. John Hopfield has provided invaluable advice and mentoring to MS. We thank Jennifer Linden and Philip Sabes for useful comments on an earlier version of the manuscript. Funding for various components of the work has been provided by the Keck Foundation, the Sloan Center for Theoretical Neuroscience at Caltech, the Center for Neuromorphic Systems Engineering at Caltech, and the National Institutes of Health. References Dempster, A. P., N. M. Laird, and D. B. Rubin (1977). J. Royal Stat. Soc. B 39, 1-38. Fee, M. S., P. P. Mitra, and D. Kleinfeld (1996). J. Neurophys. 76(3),3823-3833. Gharamani, Z. and M. I. Jordan (1997). Machine Learning 29, 245-275. Lewicki, M. S. (1994). Neural Compo 6(5), 1005-1030. McDonough, R. N. and A. D. Whalen (1995). Detection of Signals in Noise (2nd ed.). San Diego: Academic Press. Nowlan, S. J. (1990). In D. S. Touretzky (Ed.), Advances in Neural Information Processing Systems 2, San Mateo, CA. Morgan Kaufmann. Pezaris, J. S., M. Sahani, and R. A. Andersen (1997). In J. M. Bower (Ed.), Computational Neuroscience: Trends in Research, 1997. Sahani, M., J. S. Pezaris, and R. A. Andersen (1998). In J. M. Bower (Ed.), Computational Neuroscience: Trends in Research, 1998. 

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Prior Knowledge in Support Vector Kernels Bernhard Scholkopf*t, Patrice Simard t , Alex Smola t, & Vladimir Vapnikt * Max-Planck-Institut fur biologische Kybernetik, Tiibingen, Gennany t GMD FIRST, Rudower Chaussee 5, 12489 Berlin, Gennany t AT&T Research, 100 Schulz Drive, Red Bank, NJ, USA bS@first.gmd.de Abstract We explore methods for incorporating prior knowledge about a problem at hand in Support Vector learning machines. We show that both invariances under group transfonnations and prior knowledge about locality in images can be incorporated by constructing appropriate kernel functions. 1 INTRODUCTION When we are trying to extract regularities from data, we often have additional knowledge about functions that we estimate. For instance, in image classification tasks, there exist transfonnations which leave class membership invariant (e.g. local translations); moreover, it is usually the case that images have a local structure in that not all correlations between image regions carry equal amounts of infonnation. The present study investigates the question how to make use of these two sources of knowledge by designing appropriate Support Vector (SV) kernel functions. We start by giving a brief introduction to SV machines (Vapnik & Chervonenkis, 1979; Vapnik, 1995) (Sec. 2). Regarding prior knowledge about invariances, we present a method to design kernel functions for invariant classification hyperplanes (Sec. 3). The method is applicable to invariances under the action of differentiable local 1-parameter groups of local transfonnations, e.g. translational invariance in pattern recognition. In Sec. 4, we describe kernels which take into account image locality by using localized receptive fields. Sec. 5 presents experimental results on both types of kernels, followed by a discussion (Sec. 6). 2 OPTIMAL MARGIN HYPERPLANES For linear hyperplane decision functions f(x) = sgn((w? x) + b), the VC-dimension can be controlled by controlling the nonn of the weight vector w. Given training data (xl,yd, ... ,(Xl,Yl), Xi E RN,Yi E {?1}, a separating hyperplane which generalizes 641 Prior Knowledge in Support Vector Kernels well can be found by minimizing ~llwl12 subject to Yi' ((Xi' w) + b) ~ 1 for i = 1, ... , f, (I) the latter being the conditions for separating the training data with a margin. Nonseparable cases are dealt with by introducing slack variables (Cortes & Vapnik 1995), but we shall omit this modification to simplify the exposition. All of the following also applies for the nonseparable case. To solve the above convex optimization problem, one introduces a Lagrangian with multipliers Qi and derives the dual form of the optimization problem: maximize i I i L i=l Qi - '2 L QiYiQkYk(Xi . Xk) subject to Qi ~ 0, i,k=l i L QiYi = 0. (2) i=l It turns out that the solution vector has an expansion in terms of training examples, W = L:=l QiYiXi, where only those Qi corresponding to constraints (1) which are met can become nonzero; the respective examples Xi are called Support Vectors. Substituting this expansion for W yields the decision function (t f(x) = sgn o,y,(x. Xi) + b) . (3) It can be shown that minimizing (2) corresponds to minimizing an upper bound on the VC dimension of separating hyperplanes, or, equivalently, to maximizing the separation margin between the two classes. In the next section, we shall depart from this and modify the dot product used such that the minimization of (2) corresponds to enforcing transformation invariance, while at the same time the constraints (1) still hold. 3 INVARIANT HYPERPLANES Invariance by a self-consistency argument. We face the following problem: to express the condition of invariance of the decision function, we already need to know its coefficients which are found only during the optimization, which in turn should already take into account the desired invariances. As a way out of this circle, we use the following ansatz: consider decision functions f = (sgn 0 g), where g is defined as i g(Xj) := L QiYi(Bxj . BXi) + b, (4) i=l with a matrix B to be determined below. This follows Vapnik (1995), who suggested to incorporate invariances by modifying the dot product used. Any nonsingular B defines a dot product, which can equivalently be written as (Xj . AXi), with a positive definite matrix A = BTB. Clearly, invariance of g under local transformations of all Xj is a sufficient condition for the local invariance of f, which is what we are aiming for. Strictly speaking, however, invariance of g is not necessary at points which are not Support Vectors, since these lie in a region where (sgn 0 g) is constant - however, before training, it is hard to predict which examples will turn out to become SVs. In the Virtual SV method (Scholkopf, Burges, & Vapnik, 1996), a first run of the standard SV algorithm is carried out to obtain an initial SV set; similar heuristics could be applied in the present case. Local invariance of g for each pattern Xj under transformations of a differentiable local I-parameter group of local transformations Lt, ~ It=o g(LtXj) = 0, ut (5) B. Schllikopf, P. Simard, A. 1 Smola and V. Vapnik 642 can be approximately enforced by minimizing the regularizer 1 (88tlt=og(.c Xj) )2 eL j=1 i (6) t Note that the sum may run over labelled as well as unlabelled data, so in principle one could also require the decision function to be invariant with respect to transformations of elements of a test set. Moreover, we could use different transformations for different patterns. For (4), the local invariance term (5) becomes using the chain rule. Here, 81(B.coxj . BXi) denotes the gradient of (x? y) with respect to x, evaluated at the point (x . y) = (B.coxj . BXi). Substituting (7) into (6), using the facts that.co = I and 81(x, y) YT, yields the regularizer = 1 (i i 8)2 i (};iYi(};kYk(Bxi' BCBT BXk) - ' " ' " OWi(Bxi)T B8 t 1t=O .ctXj e~ ~ j=1 ~=1 where 1i C:=-'" e~ j=1 = '" ~ (8) i,k=1 (8-I )(8-I 8t t=O .ctx? 8t J t=O .ctx?J )T (9) We now choose B such that (8) reduces to the standard SV target function IlwW in the form 2::=1 (};iYiXi into it (cf. the quadratic term of obtained by substituting the expansion w (2?, utilizing the dot product chosen in (4), i.e. such that (BXi . BCB T BXk) = (f3Xi . BXk). Assuming that the Xi span the whole space, this condition becomes BT BC B B = B T B, or, by requiring B to be nonsingular, i.e. that no information get lost during the preprocessing, BCB T = I. This can be satisfied by a preprocessing (whitening) matrix = B =C-t (10) (modulo a unitary matrix, which we disregard), the nonnegative square root of the inverse of the nonnegative matrix C defined in (9). In practice, we use a matrix C>. o< A ~ := (1 - A)C + AI, (11 ) 1, instead of C. As C is nonnegative, C>. is invertible. For A = 1, we recover the standard SV optimal hyperplane algorithm, other values of A determine the trade-off between invariance and model complexity control. It can be shown that using C>. corresponds to using an objective function 4>(w) = (1 - A) 2:i(W' ttlt=0.c t Xi)2 + Allw112. By choosing the preprocessing matrix B according to (10), we have obtained a formulation of the problem where the standard SV quadratic optimization technique does in effect minimize the tangent regularizer (6): the maximum of (2), using the modified dot product as in (4), coincides with the minimum of (6) subject to the separation conditions Yi . g(Xi) 2: I, where 9 is defined as in (4). Note that preprocessing with B does not affect classification speed: since (Bxj . BXi) = (Xj . BT BXi), we can precompute BT BXi for all SVs Xi and thus obtain a machine (with modified SVs) which is as fast as a standard SV machine (cf. (4?. Relationship to Principal Component Analysis (PCA). Let us now provide some interpretation of (10) and (9). The tangent vectors ? tt It=o.ctxj have zero mean, thus C is a Prior Knowledge in Support Vector Kernels 643 sample estimate of the covariance matrix of the random vector s . %t It=OLtX, s E {?1} being a random sign. Based on this observation, we call C (9) the Tangent Covariance Matrix of the data set {Xi: i = 1, . .. ,f} with respect to the transformations Lt. Being positive definite,1 C can be diagonalized, C = SDS T , with an orthogonal matrix S consisting of C's Eigenvectors and a diagonal matrix D containing the corresponding positive Eigenvalues. Then we can compute B = C-! = SD-! ST, where D- ~ is the diagonal matrix obtained from D by taking the inverse square roots of the diagonal elements. Since the dot product is invariant under orthogonal transformations, we may drop the leading S and (4) becomes l g(Xj) = 2:>~iYi(D-t ST Xj . D-~ sT Xi) + b. ( 12) i=l A given pattern X is thus first transformed by projecting it onto the Eigenvectors of the tangent covariance matrix C, which are the rows of ST. The resulting feature vector is then rescaled by dividing by the square roots of C's Eigenvalues. 2 In other words, the directions of main variance of the random vector %t It=OLtX are scaled back, thus more emphasis is put on features which are less variant under Lt. For example, in image analysis, if the Lt represent translations, more emphasis is put on the relative proportions of ink in the image rather than the positions of lines. The peA interpretation of our preprocessing matrix suggests the possibility to regularize and reduce dimensionality by discarding part of the features, as it is common usage when doing peA. In the present work, the ideas described in this section have only been tested in the linear case. More generally, SV machines use a nonlinear kernel function which can be shown to compute a dot product in a high-dimensional space F nonlinearly related to input space via some map <P, i.e. k(x, y) = (<fl(x) . <fl(y)). In that case, the above analysis leads to a tangent covariance matrix C in P, and it can be shown that (12) can be evaluated in terms of the kernel function (Scholkopf, 1997). To this end, one diagonalizes C using techniques of kernel peA (Scholkopf, Smola, & Muller, 1996). 4 KERNELS USING LOCAL CORRELATIONS By using a kernel k(x,y) = (x? y)d, one implicitly constructs a decision boundary in the space of all possible products of d pixels. This may not be desirable, since in natural images, correlations over short distances are much more reliable as features than long-range correlations are. To take this into account, we define a kernel k~l ,d2 as follows (cf. Fig. 1): 1. compute a third image z, defined as the pixel-wise product of x and y 2. sample Z with pyramidal receptive fields of diameter p, centered at ~ 11 locations (i,j), to obtain the values Zij 3. raise each Zij to the power d 1 , to take into account local correlations within the range of the pyramid 4. sum ztJ over the whole image, and raise the result to the power d 2 to allow for longe-range correlations of order d2 lIt is understood that we use C>. if C is not definite (cf. (11)). Alternatively, we can below use the pseudoinverse. 2 As an aside, note that our goal to build invariant SV machines has thus serendipitously provided us with an approach for an open problem in SV learning, namely the one of scaling: in SV machines, there has so far been no way of automatically assigning different weight to different directions in input space - in a trained SV machine, the weights of the first layer (the SV s) form a subset of the training set. Choosing these Support Vectors from the training set only gives rather limited possibilities for appropriately dealing with different scales in different directions of input space. 644 B. SchOlkopf, P. Simard, A. 1. Smola and V. Vapnik ~d C.'.) j I (. )d l Figure I: Kernel utilizing local correlations in images, corresponding to a dot product in a polynomial space which is spanned mainly by local correlations between pixels (see text). The resulting kernel will be of order d 1 ? d 2 , however, it will not contain all possible correlations of d 1 . d 2 pixels. 5 EXPERIMENTAL RESULTS In the experiments, we used a subset of the MNIST data base of handwritten characters (Bottou et aI., 1994), consisting of 5000 training examples and 10000 test examples at a resolution of 20x20 pixels, with entries in [-1, 1]. Using a linear SV machine (i.e. a separating hyperplane), we obtain a test error rate of 9.8% (training 10 binary classifiers, and using the maximum value of 9 (cf. (4? for lO-class classification); by using a polynomial kernel of degree 4, this drops to 4.0%. In all of the following experiments, we used degree 4 kernels of various types. The number 4 was chosen as it can be written as a product of two integers, thus we could compare results to a kernel k~l , d 2 with d 1 = d2 = 2. For the considered classification task, results for higher polynomial degrees are very similar. In a series of experiments with a homogeneous polynomial kernel k(x, y) = (x? y)4, using preprocessing with Gaussian smoothing kernels of standard deviation 0.1, 0.2, ... ,1.0, we obtained error rates which gradually increased from 4.0% to 4.3%; thus no improvement of this performance was possible by a simple smoothing operation. Applying the Virtual SV method (retraining the SV machine on translated SVs; Scholkopf, Burges, & Vapnik,1996) to this problem results in an improved error rate of 2.8%. For training on the full 60000 pattern set, the Virtual SV performance is 0.8% (Scholkopf, 1997). Invariant hyperplanes. Table 1 reports results obtained by preprocessing all patterns with B (cf. (10?, choosing different values of ..\ (cf. (11?. In the experiments, the patterns were first rescaled to have entries in [0,1], then B was computed, using horizontal and vertical translations, and preprocessing was carried out; finally, the resulting patterns were scaled back again. This was done to ensure that patterns and derivatives lie in comparable regions of RN (note that if the pattern background level is a constant -1, then its derivative is 0). The results show that even though (9) was derived for the linear case, it can lead to improvements in the nonlinear case (here, for a degree 4 polynomial), too. Dimensionality reduction. The above [0, 1] scaling operation is affine rather than linear, hence the argument leading to (12) does not hold for this case. We thus only report results on dimensionality reduction for the case where the data is kept in [0, 1] scaling from the very Prior Knowledge in Support Vector Kernels 645 Table I: Classification error rates for modifying the kernel k(x, y) = (X?y)4 with the invariI ant hyperplane preprocessing matrix B).. = C~ 'i ; cf. (10) and (11). Enforcing invariance with 0.1 < A < 1 leads to improvements over the original performance (A = 1). A error rate in % 0.1 4.2 0.2 3.8 0.4 3.6 0.6 3.8 Table 2: Dropping directions corresponding to smaIl Eigenvalues of C (cf. (12)) leads to substantial improvements. AIl results given are for the case A = 0.4 (cf. Table 1); degree 4 homogeneous polynomial kernel. principal components discarded error rate in % beginning on. Dropping principal components which are less important leads to substantial improvements (Table 2); cf. the explanation foIlowing (12). The results in Table 2 are somewhat distorted by the fact that the polynomial kernel is not translation invariant, and performs poorly on the [0, 1] data, which becomes evident in the case where none of the principal components are discarded. Better results have been obtained using translation invariant kernels, e.g. Gaussian REFs (Scholkopf, 1997). Kernels using local correlations. To exploit locality in images, we used a pyramidal receptive field kernel k;l,d 2 with diameter p = 9 (cf. Sec. 4). For d 1 = d2 = 2, we obtained an improved error rate of 3.1%, another degree 4 kernel with only local correlations (d l = 4, d2 = 1) led to 3.4%. Albeit significantly better than the 4.0% for the degree 4 homogeneous polynomial (the error rates on the 10000 element test set have an accuracy of about 0.1%, cf. Bottouet aI., 1994), this is still worse than the Virtual SV resultof2.8%. As the two methods, however, exploit different types of prior knowledge, it could be expected that combining them leads to still better performance; and indeed, this yielded the best performance of all (2.0%). For the purpose of benchmarking, we also ran our system on the US postal service database of 7291 +2007 handwritten digits at a resolution of 16 x 16. In that case, we obtained the foIlowing test error rates: SV with degree 4 polynomial kernel 4.2%, Virtual SV (same kernel) 3.5%, SV with k~,2 3.6%, Virtual SV with k~,2 3.0%. The latter compares favourably to almost all known results on that data base, and is second only to a memory-based tangentdistance nearest neighbour classifier at 2.6% (Simard, LeCun, & Denker, 1993). 6 DISCUSSION With its rather general class of admissible kernel functions, the SV algorithm provides ample possibilities for constructing task-specific kernels. We have considered an image classification task and used two forms of domain knowledge: first, pattern classes were required to be locally translationaIly invariant, and second, local correlations in the images were assumed to be more reliable than long-range correlations. The second requirement can be seen as a more general form of prior knowledge - it can be thought of as arising partiaIly from the fact that patterns possess a whole variety of transformations; in object recognition, for instance, we have object rotations and deformations. Typically, these transformations are continuous, which implies that local relationships in an image are fairly stable, whereas global relationships are less reliable. We have incorporated both types of domain knowledge into the SV algorithm by constructing appropriate kernel functions, leading to substantial improvements on the considered pattern recognition tasks. Our method for constructing kernels for transformation invariant SV machines, put forward to deal with the first type of domain knowledge, so far has 646 B. SchOlkopf, P. Simard, A. 1. Smola and V. Vapnik only been applied in the linear case, which partially explains why it only led to moderate improvements (also, we so far only used translational invariance). It is applicable for differentiable transformations - other types, e.g. for mirror symmetry, have to be dealt with using other techniques, e.g. Virtual SVs (Scholkopf, Burges, & Vapnik, 1996). Its main advantages compared to the latter technique is that it does not slow down testing speed, and that using more invariances leaves training time almost unchanged. The proposed kernels respecting locality in images led to large improvements; they are applicable not only in image classification but in all cases where the relative importance of subsets of products features can be specified appropriately. They do, however, slow down both training and testing by a constant factor which depends on the specific kernel used. Both described techniques should be directly applicable to other kernel-based methods as SV regression (Vapnik, 1995) and kernel PCA (Scholkopf, Smola, & Muller, 1996). Future work will include the nonlinear case (cf. our remarks in Sec. 3), the incorporation of invariances other than translation, and the construction of kernels incorporating local feature extractors (e.g. edge detectors) different from the pyramids described in Sec. 4. Acknowledgements. We thank Chris Burges and Uon Bottou for parts of the code and for helpful discussions, and Tony Bell for his remarks. References B. E. Boser, I .M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory, pages 144-152, PittSburgh, PA, 1992. ACM Press. L. Bottou, C. Cortes, J. S. Denker, H. Drucker, I. Guyon, L. D. Jackel, Y. LeCun, U. A. Muller, E. Sackinger, P. Simard, and V. Vapnik. Comparison of classifier methods: a case study in handwritten digit recognition. In Proceedings of the J2th International Conference on Pattern Recognition and Neural Networks, Jerusalem, pages 77 - 87. IEEE Computer Society Press, 1994. C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273 - 297, 1995. B. Scholkopf. Support Vector Learning. R. Oldenbourg Verlag, Munich, 1997. ISBN 3-486-24632-1. B. Scholkopf, C. Burges, and V. Vapnik. Incorporating invariances in support vector learning machines. In C. von der Malsburg, W. von Seelen, J. C. Vorbriiggen, and B. Sendhoff, editors, Artificial Neural Networks-ICANN'96, pages 47 - 52, Berlin, 1996a. Springer Lecture Notes in Computer Science, Vol. 1112. B. Scholkopf, A. Smola, and K.-R. MulIer. Nonlinear component analysis as a kernel eigenvalue problem. Technical Report 44, Max-Planck-Institut fUr biologische Kybernetik, 1996b. in press (Neural Computation). P. Simard, Y. LeCun, and J. Denker. Efficient pattern recognition using a new transformation distance. In S. 1. Hanson, J. D. Cowan, and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 50-58, San Mateo, CA, 1993. Morgan Kaufmann. P. Simard, B. Victorri, Y. LeCun, and 1. Denker. Tangent prop - a formalism for specifying selected invariances in an adaptive network. In J. E. Moody, S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 895903, San Mateo, CA, 1992. Morgan Kaufmann. V. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York, 1995. V. Vapnik and A. Chervonenkis. Theory of Pattern Recognition [in Russian}. Nauka, Moscow, 1974. (German Translation: W. Wapnik & A. Tscherwonenkis, Theorie der Zeichenerkennung, Akademie-Verlag, Berlin, 1979). 

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