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Transformation Invariant Autoassociation with Application to Handwritten Character Recognition Holger Schwenk Maurice Milgram PARC Universite Pierre et Marie Curie tour 66-56, boite 164 4, place Jussieu, 75252 Paris cedex 05, France. e-mail: schwenk@robo.jussieu.fr Abstract When training neural networks by the classical backpropagation algorithm the whole problem to learn must be expressed by a set of inputs and desired outputs. However, we often have high-level knowledge about the learning problem. In optical character recognition (OCR), for instance, we know that the classification should be invariant under a set of transformations like rotation or translation. We propose a new modular classification system based on several autoassociative multilayer perceptrons which allows the efficient incorporation of such knowledge. Results are reported on the NIST database of upper case handwritten letters and compared to other approaches to the invariance problem. 1 INCORPORATION OF EXPLICIT KNOWLEDGE The aim of supervised learning is to learn a mapping between the input and the output space from a set of example pairs (input, desired output). The classical implementation in the domain of neural networks is the backpropagation algorithm. If this learning set is sufficiently representative of the underlying data distributions, one hopes that after learning, the system is able to generalize correctly to other inputs of the same distribution. 992 Holger Schwenk, Maurice Milgram It would be better to have more powerful techniques to incorporate knowledge into the learning process than the choice of a set of examples. The use of additional knowledge is often limited to the feature extraction module. Besides simple operations like (size) normalization, we can find more sophisticated approaches like zernike moments in the domain of optical character recognition (OCR). In this paper we will not investigate this possibility, all discussed classifiers work directly on almost non preprocessed data (pixels). In the context of OCR interest focuses on invariance of the classifier under a number of given transformations (translation, rotation, ... ) of the data to classify. In general a neural network could extract those properties of a large enough learning set, but it is very hard to learn and will probably take a lot of time. In the last years two main approaches for this invariance problem have been proposed: tangent-prop and tangent-distance. An indirect incorporation can be achieved by boosting (Drucker, Schapire and Simard, 1993). In this paper we briefly discuss these approaches and will present a new classification system which allows the efficient incorporation of transformation invariances. 1.1 TANGENT PROPAGATION The principle of tangent-prop is to specify besides desired outputs also desired changes jJJ. of the output vector when transforming the net input x by the transformations tJJ. (Simard, Victorri, LeCun and Denker, 1992). For this, let us define a transformation of pattern p as t(p, a) : P --t P where P is the space of all patterns and a a parameter. Such transformations are in general highly nonlinear operations in the pixel space P and their analytical expressions are seldom known. It is therefore favorable to use a first order approximation: t(p, a) :::::: p + atp with - at(p, a) tp _ aa I (1) a=O tp is called the tangent vector. This definition can be generalized to c transformations: t(p, Q) :::::: p + a] tp] + ... + a c tpc = P + TpQ where Tp is a n x c matrix, each column corresponding to a tangent vector. (2) Let us define R (x) the function calculated by the network. The desired behavior of the net outputs can be obtained by adding a regularization term Er to the objective function: E, = ~ k -aR(t;~.a)tJ '" ~ k -a~~x) t~ I ' (3) ti is the tangent vector for transformation tJJ. of the input vector x and a R (x) / ax is the gradient of the network with respect to the inputs. Transformation invariance of the outputs is obtained by setting jJJ. = 0, so we want that aR (x) / ax is orthogonal to ti . Tangent-prop improved the learning time and the generalization on small databases, but its applicability to highly constraint networks (many shared weights) trained on large databases remains unknown. 1.2 TANGENT DISTANCE Another class of classifiers are memory based learning methods which rely on distance metrics. The incorporation of knowledge in such classifiers can be done by a distance Transformation Invariant Autoassociation 993 measure which is (locally) invariant under a set of specified transformations. (Simard, LeCun and Denker, 1993) define tangent distance as the minimal distance between the two hyperplanes spanned up by the tangent vectors Tp in point P and Tq in point q: Dpq(p, q) = ~i!J (p + Tpii 01,/3 q - TqiJ) 2 = (p + Tpii* _ q _ TqiJ*) 2 (4) The optimality condition is that the partial derivatives 8Dpq /8ii* and 8D pq /8iJ* should be zero. The values ii* and iJ* minimizing (4) can be computed by solving these two linear systems numerically. (Simard, LeCun and Denker, 1993) obtained very good results on handwritten digits and letters using tangent distance with a I-nearest-neighborclassifier (I-nn) . A big problem of every nn-classifier, however, is that it uses no compilation of the data and it needs therefore numerous reference vectors resulting in long classification time and high memory usage. Like reported in (Simard, 1994) and (Sperdutti and Stork, 1995) important improvements are possible, but often a trade-off between speed and memory usage must be made. 2 ARCHITECTURE OF THE CLASSIFIER The main idea of our approach is to use an autoassociative multilayer perceptron with a low dimensional hidden layer for each class to recognize. These networks, called diabolo network in the following, are trained only with examples of the corresponding class. This can be seen as supervised learning without counter-examples. Each network learns a hidden layer representation which preserves optimally the information of the examples of one class. These learned networks can now be used like discriminant functions: the reconstruction error is in general much lower for examples of the learned class than for the other ones. In order to build a classifier we use a decision module which interprets the distances between the reconstructed output vectors and the presented example. In our studies we have used until now a simple minimum operator which associates the class of the net with the smallest distance (Fig. 1). The figure illustrates also typical classification behavior, here when presenting a "D" out of the test set. One can see clearly that the distance of the network "D" is much lower than for the two other ones. The character is therefore correctly classified. It is also interesting to analyze the outputs of the two networks with the next nearest distances: the network "0" tries to output a more round character and the network "B" wants to add a horizontal bar in the middle. The basic classification architecture can be adapted in two ways to the problem to be solved. One on hand we can imagine different architectures for each diabolo network, e.g. several encoding/decoding layers which allow nonlinear dimension reduction. It is even possible to use shared weights realizing local feature detectors (see (Schwenk and Milgram, 1994) for more details). One the other hand we can change the underlying distance measure, as long as the derivatives with respect to the weights can be calculated. This offers a powerful and efficient mechanism to introduce explicit knowledge into the learning algorithm of a neural network. In the discussed case, the recognition of characters represented as pixel images, we can use a 994 Holger Schwenk, Maurice Milgram score B 8.07 I--t-~ A U. scoreD --~ 4.49 score 0 8.54 character to classify input vector diabolo networks output vectors distance measures decision module Figure 1: Basic Architecture of a Diabolo Classifier transformation invariant distance measure between the net output 0 and the desired output d (that is of course identical with the net input). The networks do not need to learn each example separately any more, but they can use the set of specified transformations in order to find a common non linear model of each class. The advantage of this approach, besides a better expected generalization behavior of course, is a very low additional complexity. In comparison to the original k-nn approach, and supposedly any possible optimization, we need to calculate only one distance measure for each class to recognize, regardless of the number of learning data. We used two different versions of tangent distance with increasing complexity: 1. single sided tangent distance: Dd(d,o) =mjn~ (d+Td a - o f =~ ( d+Td a * -of (5) This is the minimal distance between the hyperplane spanned up by the tangent vectors Td in input vector d and the untransformed output vector o. 2. double sided tangent distance: Ddo(d,o) = mi!?! (d Ci,/3 2 + Tda - 0 *g - ToiJ) 2 (6) The convolution of the net output with a Gaussian 9 is necessary for the computation of the tangent vectors To (the net input d is convolved during preprocessing). Figure 2 shows a graphical comparison of Euclidean distance with the two tangent distances. Transformation Invariant Autoassociation 995 d : desired output td : tangent vector in d o : net output to : tangent vector in 0 D : Euclidean distance D d : single sided tangent distance (only d is transformed) Ddo : double sided tangent distance (both points are transformed) V' D d : gradient of D d Figure 2: Comparison of Euclidean Distance with the Different Tangent Distances The major advantage of the single sided version is that we can now calculate easily the optimal multipliers &* and therefore the whole distance (the double sided version demands expensive matrix multiplications and the numerical solution of a system of linear equations). The optimality condition 8Dd(d, 0)/8&* ~ OT gives: (7) The tangent vectors Td and the matrix Ti} = (TITd)-1 can be precomputed and stored in memory. Note that it is the same for all diabolo networks, regardless of their class. 2.1 LEARNING ALGORITHM When using a tangent distance with an autoencoder we must calculate its derivatives with respect to the weights, i.e. after application of the chain rule with respect to the output vector o. In the case of the single sided tangent distance we get: (8) The resulting learning algorithm is therefore barely more complicated than with standard Euclidean error. Furthermore it has a pleasant graphical interpretation: the net output doesn't approach directly the desired output any more, but it takes the shortest way towards the tangent hyperplane (see also fig. 2). The derivation of the double sided tangent distance with respect to the net output is more complicated. In particular we must derivate the convolution of the net output with a Gaussian as well as the tangent vectors To. These equations will be published elsewhere. Training of the whole system is stopped when the error on the cross validation set reaches a minimum. Using stochastic gradient descent convergence is typically achieved after some ten iterations. 996 3 Holger Schwenk, Maurice Milgram APPLICATION TO CHARACTER RECOGNITION In 1992 the National Institute of Standards and Technology provided a Database of handwritten digits and letters, known under the name NIST Special-Database 3. This database contains about 45000 upper case segmented characters which we have divided into learning and cross-validation set (60%) and test set (40%) respectively. We only applied a very simple preprocessing: the binary characters were centered and sizenormalized (the aspect-ratio was kept). The net input is 16 x 16 pixels with real-values. 3.1 EXPERIMENTAL RESULTS All the following results were obtained by fully connected diabolo networks with one low dimensional hidden layer, and a set of eight transformations (x- and y-translation, rotation, scaling, axis-deformation, diagonal-deformation, x- and y-thickness). Figure 3 illustrates how the networks use the transformations. Euclidean distance: 11 .1 , , input output Euclidean distance: 20.0 , , input output optimally ttansformed input output optimally ttansformed input output '" tangent distance: 0.61 tangent distance: 0.94 ., Figure 3: Reconstruction Examples (test set). The left side of each screen dump depicts the input character and the right side the one reconstructed by the network. In the middle, finally, one can see the optimally transformed patterns as calculated when evaluating the double sided tangent distance, i.e. transformed by a* and jJ* respectively. Although the "I.:' in the first example has an unusual short horizontal line, the network reconstructs a normally sized character. It is clearly visible how the input transformation lengthens and the output transformation shortens this line in order to get a small tangent distance. The right side shows a very difficult classification problem: a heavily deformed "T". Nevertheless we get a small tangent distance, so that the character is correctly classified. In summary we note a big difference between the Euclidean and the tangent distances, this is a good indicator that the networks really use the transformations. The performances on the whole test set of about 18 000 characters are summarized in figure 4. For comparison we give also the results of a one nearest neighbor classifier on the same test set. The incorporation of knowledge improved in both cases dramatically the performances. The diabolo classifier, for instance, achieves an error rate of 4.7 % with simple Euclidean distance which goes down to 3.7 % with the single sided and to only 2.6 % with the double sided tangent distance. In order to get the same results with the I-nn approach, the whole set of 27000 reference vectors had to be used. It's worth to note the results with less references: when using only 18 000 reference vectors the error rates increased to 3.7% for the single sided and to 2.8% for the double sided version respectively. Transformation Invariant Autoassociation % 997 5.3 4.7 r;; 5- - 11 F 4.0 3.7 c Euclidean - one sided - two sided 2.5 I-nn Diabolo LeNet (27 000 refs) Figure 4: Raw Error Rate with NIST Upper Case Letters (test set) In practical applications we are not only interested in low error rates, but we need also low computationally costs. An important factor is the recognition speed. The overall processing time of a diabolo classifier using the full tangent distance corresponds to the calculation of about 7 000 Euclidean and to less than 50 tangent distances. This should be less than for any algorithm of the k-nn family. If we assume the precalculation of all the tangent vectors and other expensive matrix multiplications, we could evaluate about 80 tangent distances, but the price would be exploding memory requirements. A diabolo classifier, on the other hand, needs only few memory: the storage of the weights corresponds to about 60 reference vectors per class. On a HP 715/50 workstation we obtained a recognition rate of 7.5 chis with the single sided and of more than 2.5 chis with the double sided tangent distance. We have also a method to combine both by rejection, resulting in up to 4 chis at the same low error rates (corresponds to the calculation of 32 double sided tangent distances). The table contains also the results of a large multilayer perceptron with extensive use of shared weights, known as LeNet. (Drucker, Schapire and Simard, 1993) give an error rate of 4.0% when used alone and of 2.4% for an ensemble of three such networks trained by boosting. The networks were trained on a basic set of 10 000 examples, the cross validation and test set consisted of 2 000 and 3 000 examples respectively (Drucker, personal communication). Due to the different number of examples, the results are perhaps not exactly comparable, but we can deduce nevertheless that the state of the art on this database seems to be around 2.5 %. 4 DISCUSSION We have proposed a new classification architecture that allows the efficient incorporation of knowledge into the learning algorithm. The system is easy to train and only one structural parameter must be chosen by the supervisor: the size of the hidden layer. It achieved state of the art recognition rates on the NIST database of handwritten upper case letters at a very low computational complexity. Furthermore we can say that a hardware implementation seems to be promising. Fully connected networks with only two layers are easy to put into standardized hardware chips. We could even propagate all diabolo networks in parallel. Speedups of several orders of magnitude should therefore be possible. 998 Holger Schwenk, Maurice Milgram On this year NIPS conference several authors presented related approaches. A comparable classification architecture was proposed by (Hinton, Revow and Dayan, 1995). Instead of one non-linear global model per class, several local linear models were used by performing separately principal component analysis (PCA) on subsets of each class. Since diabolo networks with one hidden layer and linear activation functions perform PCA, this architecture can be interpreted as an hierarchical diabolo classifier with linear nets and Euclidean distance. Such an hierarchisation could also be done with our classifier, i.e. with tangent distance and sigmoidal units, and might improve the results even further. (Hastie, Simard and Sackinger, 1995) developed an iterative algorithm that learns optimal reference vectors in the sense of tangent distance. An extension allows also to learn typical invariant transformations, i.e. tangent vectors, of each class. These two algorithms allowed to reduce drastically the number of reference vectors, but the results of the original approach couldn't be achieved no longer. Acknowledgements The first author is supported by the German Academic Exchange Service under grant HSP II 516.006.512.3. The simulations were performed with the Aspirin/ MIGRAINES neural network simulator developed by the MITRE Corporation. References H. Drucker, R. Schapire, and P. Simard (1993), "Boosting performance in neural networks," Int. Journal of Pattern Recognition and Artificial Intelligence, vol. 7, no. 4, pp. 705-719. T. Hastie, P. Simard and E. Sackinger (1995), "Learning prototype models for tangent distance," in NIPS 7 (G. Tesauro, D. Touretzky, and T. Leen, eds.), Morgan Kaufmann. G. Hinton, M. Revow, and P. Dayan (1995), "Recognizing handwritten digits using mixtures of linear models," in NIPS 7 (G. Tesauro, D. Touretzky, and T. Leen, eds.), Morgan Kaufmann. H. Schwenk and M. Milgram (1994), "Structured diabolo-networks for handwritten character recognition," in International Conference on Artificial Neural Networks, pp. 985-988, Springer-Verlag. P. Simard, B. Victorri, Y. LeCun, and 1. Denker (1992), "Tangent prop - a formalism for specifying selected invariances in an adaptive network," in NIPS 4 (1. Moody, S. Hanson, and R. Lippmann, eds.), pp. 895-903, Morgan Kaufmann. P. Simard, Y. LeCun, and J. Denker (1993), "Efficient pattern recognition using a new transformation distance," in NIPS 5 (S. Hanson, J. Cowan, and C. Giles, eds.), pp. 50-58, Morgan Kaufmann. P. Simard (1994), "Efficient Computation of complex distance measures using hierarchical filtering," in NIPS 6 (J.D. Cowan, G. Tesauro, and J. Alspector, eds.), pp. 50-58, Morgan Kaufmann. A. Sperdutti and D.G. Stork (1995), ''A rapid graph-based method for arbitrary transformation invariant pattern classification," in NIPS 7 (G. Tesauro, D. Touretzky, and T. Leen, eds.), Morgan Kaufmann. 

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Recognizing Handwritten Digits Using Mixtures of Linear Models Geoffrey E Hinton Michael Revow Peter Dayan Deparbnent of Computer Science, University of Toronto Toronto, Ontario, Canada M5S lA4 Abstract We construct a mixture of locally linear generative models of a collection of pixel-based images of digits, and use them for recognition. Different models of a given digit are used to capture different styles of writing, and new images are classified by evaluating their log-likelihoods under each model. We use an EM-based algorithm in which the M-step is computationally straightforward principal components analysis (PCA). Incorporating tangent-plane information [12] about expected local deformations only requires adding tangent vectors into the sample covariance matrices for the PCA, and it demonstrably improves performance. 1 Introduction The usual way of using a neural network for digit recognition is to train it to output one of the ten classes. When the training data is limited to N examples equally distributed among the classes, there are only N log2 10 bits of constraint in the class labels so the number of free parameters that can be allowed in a discriminative neural net model is severely limited. An alternative approach, motivated by density estimation, is to train a separate autoencoder network on examples of each digit class and to recognise digits by seeing which autoencoder network gives the best reconstruction of the data. Because the output of each autoencoder has the same dimensionality as the image, each training example provides many more bits of constraint on the parameters. In the example we describe, 7000 training images are sufficient to fit 384, 000 parameters and the training procedure is fast enough to do the fitting overnight on an R4400-based machine. Auto-encoders can be viewed in terms of minimum description length descriptions of data in which the input-hidden weights produce a code for a particular case and 1016 Geoffrey E. Hinton, Michael Revow, Peter Dayan the hidden-output weights embody a generative model which turns this code back into a close approximation of the original example [14,7]. Code costs (under some prior) and reconstruction error (squared error assuming an isotropic Gaussian misfit model) sum to give the overall code length which can be viewed as a lower bound on the log probability density that the autoencoder assigns to the image. This properly places the emphasis on designing appropriate generative models for the data which will generalise well by assigning high log probabilities to new patterns from the same classes. We apply this idea to recognising handwritten digits from grey-level pixel images using linear auto-encoders. Linear hidden units for autoencoders are barely worse than non-linear ones when squared reconstruction error is used [I], but have the great computational advantage during training that input-hidden and hiddenoutput weights can be derived from principal components analysis (PCA) of the training data. In effect a PCA encoder approximates the entire N dimensional distribution of the data with a lower dimensional"Gaussian pancake" [13], choosing, for optimal data compression, to retain just a few of the PCs. One could build a single PCA model for each digit - however the many different styles of writing suggest that more than one Gaussian pancake should be used, by dividing the population of a single digit class into a number of sub-classes and approximating each class by its own model. A similar idea for data compression was used by [9] where vector quantization was used to define sub-classes and PCA was performed within each sub-class (see also [2]). We used an iterative method based on the Expectation Maximisation (EM) algorithm [4] to fit mixtures of linear models. The reductio of the local linear approach would have just one training pattern in each model. This approach would amount to a nearest neighbour method for recognition using a Euclidean metric for error, a technique which is known to be infelicitous. One reason for this poor performance is that characters do not only differ in writing styles, but also can undergo simple transformations such as translations and rotations. These give rise to somewhat non-linear changes as measured in pixel space. Nearest neighbour methods were dramatically improved methods by defining a metric in which locally linearised versions of these transformations cost nothing [12]. In their tangent distance method, each training or test point is represented by a h-dimensionallinear subspace, where each of these dimensions corresponds to a linear version of one of the transformations, and distances are measured between subspaces rather than points. The local linear autoencoder method can be seen just like this - variations along one of the h principal component directions are free, while variations along the remaining principal directions cost. However, rather than storing and testing each training pattern, the local models summarise the regularities over a number of patterns. This reduces storage and recognition time, and allows the directions of free variation to be averaged over numbers of patterns and also to be determined by the data rather than being pre-specified. A priori knowledge that particular transformations are important can be incorporated using a version of the tangent-prop procedure [12], which is equivalent in this case to adding in slightly transformed versions of the patterns. Also reconstruction error could be assessed either at a test pattern, or, more like tangent distance, between its transformation subspace and the models' principal components subs paces. Recognizing Handwritten Digits Using Mixtures of Linear Models 1017 ,a ,, , Figure 1: Didactic example of tangent information and local linear models. See text for details. Figure 1 illustrates the idea. Imagine that the four points 1-4 portray in image space different examples of the same digit, subject to some smooth transformation. As in tangent distance, one could represent this curve using the points and their local tangents (thick lines). However one might do better splitting it into three local linear models rather than four - model 'a' Gust a line in this simple case) averages the upper part of the curvp more effectively than the combination of the two tangents at '1' and '2'. However, given just the points, one might c~nstruct model 'b' for '3' and '4', which would be unfortunate. Incorporating information about the tangents as well would encourage the separation of these segments. Care should be taken in generalising this picture to high dimensional spaces. The next section develops the theory behind variants of these systems (which is very similar to that in [5, 10]), and section 3 discusses how they perform. 2 Theory Linear auto-encoders embody a model in which variations from the mean of a population along certain directions are cheaper than along others, as measured by the log-unlikelihood of examples. Creating such a generative model is straightforward. Principal component analysis (PCA) is performed on the training data and the leading h principal components are retained, defining an h-dimensional subspace onto which the n-dimensional inputs are projected. We choose h using cross-validation, although a minimum description length criterion could also be used. We ignore the effect of different variances along the different principal components and use a model in which the code length for example i (the negative log-likelihood) is proportional to the reconstruction error - the squared Euclidean distance between the output of the autoencoder and the pattern itself. Rather than having just one autoencoder for each digit, there is a whole collection, and therefore we use EM to assign examples to n sub-classes, just as in clustering using a mixture of Gaussians generative model. During the E-step, the responsibility for each pattern is assigned amongst the sub-classes, and in the M-step PCA is performed, altering the parameters of a sub-class appropriately to minimise the 1018 Geoffrey E. Hinton, Michael Revow, Peter Dayan reconstruction cost of the data for which it is responsible. Formally, the algorithm is: 1. Choose initial autoencoder assignments for each example in the training set (typically using a K-means clustering algorithm). 2. Perform PCA separately for each autoencoder; 3. Reassign patterns to the autoencoder that reconstructs them the best; 4. Stop if no patterns have changed sub-class, otherwise return to step 2. There is a 'soft' version of the algorithm in which the responsibility of autoencoder q for example i is calculated as ri.q = e-IIE,q 112 /Za 2 /(L.r e-IIE'TII2 /Z( 2 ) where Ei.r is the reconstruction error. For this, in step 2, the examples are weighted for the PCA by the responsibilities, and convergence is assessed by examining the change in the log-likelihood of the data at each iteration. The soft version requires a choice of O'z, the assumed variance in the directions orthogonal to the pancake. The algorithm generates a set of local linear models for each digit. Given a test pattern we evaluate the code length (the log-likelihood) against all the models for all the digits. We use a hard method for classification - determining the identity of the pattern only by the model which reconstructs it best. The absolute quality of the best reconstruction and the relative qualities of slightly sub-optimal reconstructions are available to reject ambiguous cases. For a given linear model, not counting the code cost implies that deformations of the images along the principal components for the sub-class are free. This is like the metric used by [11] except that they explicitly specified the directions in which deformations should be free, rather than learning them from the data. We wished to incorporate information about the preferred directions without losing the summarisation capacity of the local models, and therefore turned to the tangent prop algorithm [12]. Tangent prop takes into account information about how the output of the system should vary locally with particular distortions of the input by penalising the system for having incorrect derivatives in the relevant directions. In our case, the overall classification process is highly non-linear, making the application of tangent-prop to it computationally unattractive, but it is easy to add a tangent constraint to the reconstruction step because it is linear. Imagine requiring the system f(p) = A.p to reconstruct x + .M and x -7-. t as well as x, for an input example x, and distortion t (the tangent vector), and where 7-. is a weight. Their contribution to the error is proportional to Ix - A.xl z + 7-. z lt - A.tI Z , where the second term is equivalent to the error term that tangent-prop would add. Incorporating this into the PCA is as simple as adding a weighted version of tt T to the covariance matrix - the tangent vectors are never added to the means of the sub-classes. 3 Results We have evaluated the performance of the system on data from the CEDAR CDROM 1 database containing handwritten digits lifted from mail pieces passing through a United States Post Office [8]. We divided the br training set of binary Recognizing Handwritten Digits Using Mixtures of Linear Models 1019 Table 1: Classification errors on the validation test when different weightings are used for the tangent vectors during clustering and recognition. No rejections were allowed. segmented digits into 7,000 training examples, 2,000 validation examples and 2,000 "internal test" examples. All digits were equally represented in these sets. The binary images in the database are all of different sizes, so they were scaled onto a 16 x 16 grid and then smoothed with a Gaussian filter. The validation set was used to investigate different choices for the Gaussian filter variances, the numbers of sub-classes per digit and principal components per model, and the different weightings on the tangent vectors. Clearly this is a large parameter space and we were only able to perform a very coarse search. In the results reported here all digits have the same number of sub-classes (10) and the number of principal components per model was picked so as to explain 95% of the training set variance assigned to that model. Once a reasonable set of parameter settings had been decided upon, we used all 11,000 images to train a final version which was tested on the official bs set (2711 images). There are two major steps to the algorithm; defining sub-classes within each digit and reconstructing for recognition. We found that the tangent vectors should be more heavily weighted in the sub-class clustering step than during ultimate recognition. Figure 2 shows the means of the 10 sub-classes for the digit two where the clustering has been done (a) without or (b) with tangent vectors. It is clear that the clusters defined in (b) capture different styles of 2s in a way that those in (a) do not - they are more diverse and less noisy. They also perform better. The raw error rate (no rejections allowed) on the validation set with different amounts of tangent vector weightings are shown in Table 1. The results on the official test set (2711 examples) are shown in Table 2. 4 Discussion and Conclusions A mixture of local linear models is an effective way to capture the underlying styles of handwritten digits. The first few principal components (less than 20 on 256-dimensional data) extract a significant proportion of the variance in the images within each sub-class. The resulting models classify surprisingly well without requiring large amounts of either storage or computation during recognition. Further, it is computationally easy and demonstrably useful to incorporate tangent information requiring reconstruction to be good for small transformations of the sample patterns in a handful of pre-specified directions. Adding tangent information is exactly equivalent to replacing each digit by a Gaussian cloud of digits perturbed along the tangent plane and is much more efficient than adding extra Geoffrey E. Hinton, Michael Revow, Peter Dayan 1020 a) b) Figure 2: Cluster means for images of twos. (a) Without tangent vectors. (b) Cluster means using translation, rotation and scaling tangent vectors. stochastically perturbed examples. The weightings applied to reconstruction of the tangent vectors are equivalent to the variances of the cloud. There is an interesting relationship between mixtures of linear models and mixtures of Gaussians. Fitting a mixture of full-covariance Gaussians is typically infeasible when the dimensionality of the input image space, n, is high and the number of training examples is limited. Each Gaussian requires n parameters to define its mean and n(n + 1)/2 more parameters to define its covariance matrix. One way to reduce the number of parameters is to use a diagonal covariance matrix but this is a very poor approximation for images because neighbouring pixel intensities are very highly correlated. An alternative way to simplify the covariance matrix is to flatten the Gaussian into a pancake that has h dimensions within the pancake and n - h dimensions orthogonal to the pancake. Within this orthogonal subspace we assume an isotropic Gaussian distribution (ie a diagonal covariance matrix with equal, small variances along the diagonal), so we eliminate (n - h)(n - h + 1)/2 degrees of freedom which is nearly all the degrees of freedom of the full covariance matrix if h is small compared with n. Not counting the mean, this leaves h(2n - h + 1) /2 degrees of freedom which is exactly the number of parameters required to define the first h principal components. l Thus we can view PCA as a way of fiercely constraining a full covariance Gaussian but nevertheless leaving it free to model lThe h th principal component only requires n - h orthogonal to all previous components. + 1 parameters because it must be Recognizing Handwritten Digits Using Mixtures of Linear Models ? ? ..' I???. ?'? ?1??. ?. ?' '1 . . . , ~it:I; ..f.... :~l ~ .??? . ..... ..... " ,,~:I!:::';;.?; .I'" .....?. ........ _._....?? :d...: .1.: .......... f!~~, -. :rL .................. ....o.? 1021 ~ "'lii!;~" ?. . ? I;':~?~; .,. ... roo r. ? ........ " . .. ........ a. .-. ..... ., . ...... .o... :H:fiiiiP.~!? ,.. ..... .. i;;::~?: ..::P.'a .......... ?. .. ' ..?? . .u .. .?. . Li. a:!:? .:t-ir:~:,. .!si??. ~ .....??.. ....'ii: ?~HiiiH~ a&,. :-!'**!Hr:~:' Figure 3: Reconstruction of a 2 (left column) using 2 (upper) and 0 (lower) models. Linear 0 e Linear Models + Tangent Dist Tan ent Distance Table 2: Classification errors on the official test test when no rejections are allowed. Memory requirements are indicated in terms of the number of 16 x 16 images that need to be stored. important correlations. We have investigated a number of extensions to the basic scheme. As only one of these yielded improved results on the validation set, we will only briefly review them. If instead of assuming an isotropic Gaussian within the pancake we use an ellipsoidal subspace, then we can can take into account the different variances along each of the h principal directions. This is akin to incorporating a code cost [14]. Similarly, the squared reconstruction error used in the basic scheme also assumes an isotropic distribution. Again a diagonal covariance matrix may be substituted. We surmise that this was not successful because we had insufficient training data to estimate the variances along some dimensions reliably; for example some of the edge pixels were never turned on in the training data for some models. The Euclidean metric for the reconstruction error is convenient as it authorises the use of a powerful methods such as principal components; however as pointed out in the introduction it is deficient for situations like character recognition. We tried a scheme in which the models were trained as described above, but tangent distance [11] was used during testing. This method yielded marginally improved results on both the validation and test sets (Table 2). More adventurous tangent options, including using them in the clustering phase, were explored by [5, 10]. PCA models are not ideal as generative models of the data because they say nothing about how to generate values of components in the directions orthogonal to the pancake. To ameliorate this, the generative model may be formulated as f(p) = A.p + ?, where the components of ? are independent and the factors p have some prior covariance matrix. The generative weights of this autoencoder can 1022 Geoffrey E. Hinton, Michael Revow, Peter Dayan be obtained using the technique of maximum likelihood factor analysis (which is closely related to PCA) and the resulting architecture and hierarchical variants of it can be formulated as real valued versions of the Helmholtz machine [3,6]. The cost of coding the factors relative to their prior is implicitly included in this formulation, as is the possibility that different input pixels are subject to different amounts of noise. Unlike PCA, factor analysis privileges the particular input coordinates (rather than being invariant to rotations of the input covariance matrix). References [1] Bourlard, H & Kamp, Y (1988). Auto-association by Multilayer Perceptrons and Singular Value Decomposition. BioI. Cybernetics 59,291-294. [2] Bregler, C & Omohundro, SM (1995). Non-linear image interpolation using surface learning. This volume. [3] Dayan, P, Hinton, GE, Neal, RM & Zemel, RS (1995). The Helmholtz machine. Neural Computation, in press. [4] Dempster, AP, Laird, NM & Rubin, DB (1976). Maximum likelihood from incomplete data via the EM algorithm. Proceedings of the Royal Statistical Society, 1-38. [5] Hastie, T, Simard, P & Sackinger, E (1995). Learning prototype models for tangent distance. This volume. [6] Hinton, GE, Dayan, P, Frey, BJ, Neal, RM (1995). The wake-sleep algorithm for unsupervised neural networks. Submitted for publication. [7] Hinton, GE & Zemel, RS (1994). Autoencoders, minimum deSCription length and Helmholtz free energy. In JD Cowan, G Tesauro & J Alspector, editors, Advances in Neural Information Processing Systems 6. San Mateo, CA: Morgan Kaufmann. [8] Hull, JJ (1994). A database for handwritten text recognition research. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16, 550-554. [9] Kambhatla, N & Leen, TK (1994). Fast non-linear dimension reduction. In JD Cowan, G Tesauro & J Alspector, editors, Advances in Neural Information Processing Systems 6. San Mateo, CA: Morgan Kaufmann. [10] Schwenk, H & Milgram, M (1995). Transformation invariant autoassociation with application to handwritten character recognition. This volume. [11] Simard, P, Le Cun, Y & and Denker, J (1993). Efficient pattern recognition using a new transformation distance. In SJ Hanson, JD Cowan & CL Giles, editors, Advances in Neural Information Processing Systems 5, 50-58. San Mateo, CA: Morgan Kaufmann. [12] Simard, P, Victorri, B, LeCun, Y & Denker, J (1992). Tangent Prop - A formalism for specifying selected invariances in an adaptive network. In JE Moody, SJ Hanson & RP Lippmann, editors, Advances in Neural Information Processing Systems 4. San Mateo, CA: Morgan Kaufmann. [13] Williams, CKI, Zemel, RS & Mozer, MC (1993). Unsupervised learning of object models. In AAAI FalI1993 Symposium on Machine Learning in Computer Vision, 20-24. [14] Zemel, RS (1993). A Minimum Description Length Framework for Unsupervised Learning. PhD Dissertation, Computer Science, University of Toronto, Canada. IThis research was funded by the Ontario Information Technology Research Centre and NSERC. We thank Patrice Simard, Chris Williams, Rob Tibshirani and Yann Le Cun for helpful discussions. Geoffrey Hinton is the Noranda Fellow of the Canadian Institute for Advanced Research. 

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An Auditory Localization and Coordinate Transform Chip Timothy K. Horiuchi timmer@cns.caltech.edu Computation and Neural Systems Program California Institute of Technology Pasadena, CA 91125 Abstract The localization and orientation to various novel or interesting events in the environment is a critical sensorimotor ability in all animals, predator or prey. In mammals, the superior colliculus (SC) plays a major role in this behavior, the deeper layers exhibiting topographically mapped responses to visual, auditory, and somatosensory stimuli. Sensory information arriving from different modalities should then be represented in the same coordinate frame. Auditory cues, in particular, are thought to be computed in head-based coordinates which must then be transformed to retinal coordinates. In this paper, an analog VLSI implementation for auditory localization in the azimuthal plane is described which extends the architecture proposed for the barn owl to a primate eye movement system where further transformation is required. This transformation is intended to model the projection in primates from auditory cortical areas to the deeper layers of the primate superior colliculus. This system is interfaced with an analog VLSI-based saccadic eye movement system also being constructed in our laboratory. Introduction Auditory localization has been studied in many animals, particularly the barn owl. Most birds have a resolution of only 10 to 20 degrees, but owls are able to orient 788 Timothy Horiuchi to sound with an accuracy of 1 to 2 degrees which is comparable with humans. One important cue for localizing sounds is the relative time of arrival of a sound to two spatially separated ears. A neural architecture first described by Jeffress (1948) for measuring this time difference has been shown to exist in the barn owl auditory localization system (Konishi 1986) . An analog VLSI implementation of the barn owl system constructed by Lazzaro (1990) is extended here to include a transformation from head coordinates to retinal coordinates. In comparison to the barn owl, the neurophysiology of auditory localization in cats and primates is not as well understood and a clear map of auditory space does not appear to be present in the inferior colliculus as it is in the owl. It has been suggested that cortical auditory regions may provide the head-based map of auditory space (Groh and Sparks 1992). In primates, where much of the oculomotor system is based in retinotopic coordinates , head-based information must ultimately be transformed in order to be used. While other models of coordinate transformation have been proposed for visual information (e.g. Zipser and Andersen 1988, Krommenhoek et al. 1993) and for auditory information (Groh and Sparks 1992), the model of coordinate transformation used in this system is a switching network which shifts the entire projection of the head-based map of auditory space onto a retinotopic "colliculus" circuit. This particular model is similar to a basis function approach where intermediate units have compact receptive fields in an eye-position / head-based azimuth space and the output units sum the outputs of a subset of these units. The auditory localization system described here provides acoustic target information to an analog VLSI-based saccadic eye movement system (Horiuchi, Bishofberger, and Koch 1994) being developed in our laboratory for multimodal operation. Bandpass _::::::10-,.,.... . . . Filtering @3.2kHz Thresholded Zero-crossings electret Localization condenser microphones and Bandpass Filtering @3.2kHz Thresholded Zero-crossings Coordinate ~--Eye Position Transform Chip Sound Source Location (in retinal coordinates) Figure 1: Block diagram of the auditory localization system. The analog front end consists of external discrete analog electronics. 789 An Auditory Localization and Coordinate Transform Chip M? Snop II 45 degrees R of Center .' .. .- i f 0 ?1 .. . " ~+-----~-----+----~~----+-----~----~ o M? Snop II CerRr POIIition i I ?1 ~+-----~-----+------~----+-----~----~ o Finga' Snop 0145 degrees LofCCDtcr I f .. ' : : ?1 ~ : :~ .. . .. ". ~ ~ : :~ ~~----~-----+----~~----+-----~----~ o Figure 2: Filtered signals of the left and right microphones from 3 different angles. 790 Timothy Horiuchi The Localization System The analog front-end of the system (see figure 1) consists of three basic components, the microphones, the filter stage, and the thresholded, zero-crossing stage. Two microphones are placed with their centers about 2 inches apart. For any given time difference in arrival of acoustic stimuli, there are many possible locations from which the sound could have originated. These points describe a hyperbola with the two microphones as the two foci. If the sound source is distant enough, we can estimate the angle since the hyperbola approaches an asymptote. The current system operates on a single frequency and the inter-microphone distance has been chosen to be just under one wavelength apart at the filter frequency. The filter frequency chosen was 3.2 kHz because the author's finger snap, used extensively during development contained a large component at that frequency. The next step in the computation consists of triggering a digital pulse at the moment of zerocrossing if the acoustic signal is large enough. Thn::sb.olded Zct<Hlrossing Detection Pulses 6 7.5 5 5.0 25 4 '-- '-- ~ .e } '-- '-- '-- '-- 0.0 ?25 ~ ?5.0 ~ 2 ] 0 .? .!! 1\ !< it '-- 3 !..... '" ..... I i J'L ",... VV....., I!' II flA~ V V '" V ?1 V V ?75 ?10.0 'R II ~ ?125 ?2 -15.0 ? 175 ?3 0 2 3 4 5 6 1imo (aeconda) (10") Figure 3: Example of output pulses from the external circuitry. Zero phase is chosen to be the positive-slope zero-crossing. Top: Digital pulses are generated at the time of zero phase for signals whose derivative is larger than a preset threshold. Bottom: 3.2 kHz Bandpass filtered signal for a finger snap. Phase Detection and Coordinate Transform in Analog VLSI The analog VLSI component of the system consists of two axon delay lines (Mead 1988) which propagate the left and right microphone pulse signals in opposing directions in order to compute the cross correlation (see Fig 4.) The location of the peak in this correlation technique represents the relative phase of the two signals. An Auditory Localization and Coordinate Transform Chip 791 This technique is described in more detail and with more biological justification by Lazzaro (1990). The current implementation contains 15 axon circuits in each delay line. This is shown in figure 4. At each position in the correlation delay line are logical AND circuits which output a logic one when there are two active axon units at that location. Since these units only turn on for specific time delays, they define auditory "receptive fields". The output of this subsystem are 15 digital lines which are passed on to the coordinate transform. Left Channel JlllL Right Channel Jllll Figure 4: Diagram of the double axon delay line which accepts digital spikes on the inputs and propagates them across the array. Whenever two spikes meet, a pulse is generated on the output AND units. The position of the AND circuit which gets activated indicates the relative time of arrival of the left and right inputs. NOTE: the actual circuit contains 15 axon units. A Head-based Auditory Units left ear-....IIIUI=llI1l~..... - right ear f3 'S ::> s:: .g ? .-4 (I) A ~ ~ ~ eye position Retinotopic Auditory Units Figure 5: For the one-dimensional case described in this project, the appropriate transform from head to retinal coordinates is a rotation which subtracts the eye position. The eye position information on the chip is represented as a voltage which activates one of the eye position units. The spatial pattern of activation from the auditory units is then "steered" to the output stage with the appropriate shift. (See figure 5). This 792 Timothy Horiuchi is similar to a shift scheme proposed by Pitts and McCulloch (1947) for obtaining pitch invariance for chord recognition. The eye position units are constructed from an array of "bump" circuits (Delbriick 1993) which compare the eye position voltage with its local voltage reference. The two dimensional array of intermediate units take the digital signal from the auditory units and switch the "bump" currents onto the output lines. The output current lines drive the inputs of a centroid circuit. The current implementation of the shift can be viewed as a basis function approach where a population of intermediate units respond to limited "ball-like" regions in the two-dimensional space of horizontal eye position and sound source azimuth (head-coordinates). The output units then sum the outputs of only those intermediate units which represent the same retinal location. It should be noted that this coordinate transformation is closely related to the "dendrite model" proposed for the projection of cortical auditory information to the deep SC by Groh and Sparks (1992). The final output stage converts this spatial array of current carrying lines into a single output voltage which represents the centroid of the stimulus in retinal coordinates. This centroid circuit (DeWeerth 1991) is intended to represent the primate SC where a similar computation is believed to occur. Results and Conclusions Figure 6 shows three plots of the chip's output voltage as a function of the interpulse time interval. Figure 7 shows three plots of the full system's output voltage for different eye position voltages. The output is roughly linear with azimuth and linear with eye position voltage. In operation, the system input consists of a sound entering the two microphones and the output consists of an analog voltage representing the position of the sound source and a digital signal indicating that the analog data is valid. The auditory localization system described here is currently in use with an analog VLSI-based model of the primate saccadic system to expand its operation into the auditory domain (Horiuchi, Bishofberger, & Koch 1994). In addition to the effort of our laboratory to model and understand biological computing structures in real-time systems, we are exploring the use of these low power integrated sensors in portable applications such as mobile robotics. Analog VLSI provides a compact and efficient implementation for many neuromorphic computing architectures which can potentially be used to provide, small, fast, low power sensors for a wide variety of applications. Acknowledgements The author would like to acknowledge Prof. Christof Koch for his academic support and use of laboratory facilities for this project, Brooks Bishofberger for his assistance in constructing some of the discrete electronics and Prof. Carver Mead for running the CNS184 course under which this chip was fabricated. The author is supported by an AASERT grant from the Office of Naval Research. 793 An Auditory Localization and Coordinate Transform Chip Output vs. Arrival Time Difference (3 eye positions) 3.1 3.0 2.9 6' ..; old 2.8 ..j N d N 2.7 II 0 ? 2.6 0 1 0 2.5 > <;; g 2.4 2.3 2.2 2.1 -600 -200 o 200 400 600 R to L delay (microseconda) Figure 6: Chip output vs. input pulse timing: The chip was driven with a signal generator and the output voltage was plotted for three different eye position voltages. Due to the discretized nature of the axon, there are only 15 axon locations at which pulses can meet. This creates the staircase response. References T. Delbriick (1993) Investigations of Analog VLSI Visual Transduction and Motion Processing, Ph.D. Thesis, California Institute of Technology J. Groh and D. Sparks (1992) 2 Models for Transforming Auditory Signals from Head-Centered to Eye-Centered Coordinates Bioi. Cybern. 67(4) 291-302. T. Horiuchi, B. Bishofberger, & C. Koch, (1994) An Analog VLSI-based Saccadic System, In (ed.), Advances in Neural Information Processing Systems 6 San Mateo, CA: Morgan Kaufman L. A. Jeffress (1948) A Place Theory of Sound Localization J. Comp_ Physiol. Psychol. 41: 35-39. M. Konishi (1986) Centrally Synthesized Maps of Sensory Space. TINS April, pp. 163-168. K. P. Krommenhoek, A. J. Van Opstal, C. C. A. M. Gielen, J. A. ,M. Van Gisbergen. (1993) Remapping of Neural Activity in the Motor Colliculus: A Neural Network Study. Vision Research 33(9):1287-1298. Timothy Horiuchi 794 J. Lazzaro. (1990) Silicon Models of Early Audition, Ph.D. Thesis, California Institute of Technology C. Mead, (1988) Analog VLSI and Neural Systems Menlo Park: Addison-Wesley W. Pitts and W. S. McCulloch, (1947) How we know universals: the perception of auditory and visual forms. Bulletin of Mathematical Biophysics 9:127-147. D. Zipser and R. A. Andersen (1988) A back-propagation programmed network that simulates response properties of a subset of posterior parietal neurons. Nature 331:679-684. Localization Output vs. Sound Source Azimuth 3.6 3.4 3.2 3.0 II -e,. on ..; 28 'Ij N 'Ij II t S -e 26 24 22 ~ 20 1.8 1.6 1.4 ?20 0 Figure 7: Performance of the full system on continuous input (sinusoidal) delivered by a speaker from different angles. Note that 90 degrees denotes the center position. The three plots are the outputs for three different settings of the eye position input voltage. 

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An Input Output HMM Architecture Yoshua Bengio* Dept. Informatique et Recherche Operationnelle Universite de Montreal, Qc H3C-3J7 bengioyOIRO.UMontreal.CA Paolo Frasconi Dipartimento di Sistemi e Informatica Universita di Firenze (Italy) paoloOmcculloch.ing.unifi.it Abstract We introduce a recurrent architecture having a modular structure and we formulate a training procedure based on the EM algorithm. The resulting model has similarities to hidden Markov models, but supports recurrent networks processing style and allows to exploit the supervised learning paradigm while using maximum likelihood estimation. 1 INTRODUCTION Learning problems involving sequentially structured data cannot be effectively dealt with static models such as feedforward networks. Recurrent networks allow to model complex dynamical systems and can store and retrieve contextual information in a flexible way. Up until the present time, research efforts of supervised learning for recurrent networks have almost exclusively focused on error minimization by gradient descent methods. Although effective for learning short term memories, practical difficulties have been reported in training recurrent neural networks to perform tasks in which the temporal contingencies present in the input/output sequences span long intervals (Bengio et al., 1994; Mozer, 1992). Previous work on alternative training algorithms (Bengio et al., 1994) could suggest that the root of the problem lies in the essentially discrete nature of the process of storing information for an indefinite amount of time. Thus, a potential solution is to propagate, backward in time, targets in a discrete state space rather than differential error information. Extending previous work (Bengio & Frasconi, 1994a), in this paper we propose a statistical approach to target propagation, based on the EM algorithm. We consider a parametric dynamical system with discrete states and we introduce a modular architecture, with subnetworks associated to discrete states. The architecture can be interpreted as a statistical model and can be trained by the EM or generalized EM (GEM) algorithms (Dempster et al., 1977), considering the internal state trajectories as missing data. In this way learning is decoupled into ? also, AT&T Bell Labs, Holmdel, N J 07733 428 Yoshua Bengio, Paolo Frasconi a temporal credit assignment subproblem and a static learning subproblem that consists of fitting parameters to the next-state and output mappings defined by the estimated trajectories . In order to iteratively tune parameters with the EM or GEM algorithms, the system propagates forward and backward a discrete distribution over the n states, resulting in a procedure similar to the Baum-Welch algorithm used to train standard hidden Markov models (HMMs) (Levinson et al., 1983) . HMMs however adjust their parameters using unsupervised learning, whereas we use EM in a supervised fashion . Furthermore, the model presented here could be called Input/Output HMM, or IOHMM , because it can be used to learn to map input sequences to output sequences (unlike standard HMMs, which learn the output sequence distribution) . This model can also be seen as a recurrent version of the Mixture of Experts architecture (Jacobs et al. , 1991) , related to the model already proposed in (Cacciatore and Nowlan, 1994). Experiments on artificial tasks (Bengio & Frasconi , 1994a) have shown that EM recurrent learning can deal with long term dependencies more effectively than backpropa~ation through time and other alternative algorithms. However, the model used in (Bengio & Frasconi, 1994a) has very limited representational capabilities and can only map an input sequence to a final discrete state. In the present paper we describe an extended architecture that allows to fully exploit both input and output portions of the data, as required by the supervised learning paradigm. In this way, general sequence processing tasks, such as production, classification, or prediction, can be dealt with. 2 THE PROPOSED ARCHITECTURE We consider a discrete state dynamical system based on the following state space x - f(x U ) description: t t-l, t (1) Yt = 9(Xt, Ut) where Ut E R m is the input vector at time t, Yt E R r is the output vector, and Xt E {I , 2, .. . , n} is a discrete state. These equations define a generalized Mealy finite state machine, in which inputs and outputs may take on continuous values . In this paper , we consider a probabilistic version of these dynamics, where the current inputs and the current state distribution are used to estimate the state distribution and the output distribution for the next time step. Admissible state transitions will be specified by a directed graph 9 whose vertices correspond to the model 's states and the set of successors for state j is Sj . The system defined by equations (1) can be modeled by the recurrent architecture depicted in Figure l(a) . The architecture is composed by a set of state n etworks N j, j = 1 .. . n and a set of output networks OJ, j = 1 . .. n. Each one of the state and output networks is uniquely associated to one of the states,and all networks share the same input Ut . Each state network M has the task of predicting the next state distribution , based on the current input and given that Xt-l = j . Similarly, each output network OJ predicts the output of the system, given the current state and input. All the subnetworks are assumed to be static and they are defined by means of smooth mappings Nj (Ut ;9j ) and OJ (Ut; 1Jj), where 9j and 1Jj are vectors of adjustable parameters (e .g., connection weights). The ranges of the functions N j 0 may be constrained in order to account for the underlying transition graph 9 . Each output 'Pij ,t of the state subnetwork Nj (at time t) is associated to one of the successors i of state j . Thus the last layer of M has as many units as the cardinality of Sj. For convenience of notation, we suppose that 'Pij,t are defined for each i, j = 1, ... , n and we impose the condition 'Pij ,t = 0 for each i not belonging to S j . The softmax function is used in the last layer: 'Pij ,t = e a,j ,t ILlEsj ea l j,t, j = 1, ... , n , i E Sj where aij, t are intermediate variables that can be thought of as the An Input Output HMM Architecture cu ... nt Input EIYt ... Xt-l current ??pectod output, given PIlat Input Mquenc. 11 t 429 current atilt. dlatrtbutton Ct= Pl' t I Ul ) lull ... ... Xt Xt+l 1 1 1 Yt-l Yt Yt+l Xt-l X( Xt+l '-'t \.Yt-l \Yt I I Ut-l HMM \.Yt+l I Ut .. .... IOHMM Ut+l (b) (a) Figure 1: (a): The proposed IOHMM architecture. (b): Bottom: Bayesian network expressing conditional dependencies for an IOHMM; top: Bayesian network for a standard HMM 't activations of the output units of subnetwork N j . In this way L:7=1 'Pij ,t = 1 Tij,t. The vector E R n represents the internal state of the model and it is computed as a linear combination of the outputs of the state networks, gated by the previously n computed internal state: 't = L ( j ,t-IIPj,t (2) j=1 where IPj,t = ['PIj,t, ... , 'Pnj,t]'. Output networks compete to predict the global output of the system 1Jt E R r : n 1Jt = L (jt1Jjt (3) j=1 where 1Jjt E R r is the output of subnetwork OJ. At this level , we do not need to further specify the internal architecture of the state and output subnetworks. Depending on the task, the designer may decide whether to include hidden layers and what activation rule to use for the hidden units. This connectionist architecture can be also interpreted as a probability model. Let us assume a multinomial distribution for the state variable Xt and let us consider the main variable of the temporal recurrence (2). If we initialize the vector to positive numbers summing to 1, it can be interpreted as a vector of initial state probabilities. In general, we obtain relation (it = P(Xt = i I having denoted with ui the subsequence of inputs from time 1 to t, inclusively. Equation (2) then has the following probabilistic interpretation: 't, un, '0 n P(Xt = i lui) = L P(Xt = i I Xt-I = j, ut}P(Xt-1 = j lui-I) (4) j=l i.e., the subnetworks N j compute transition probabilities conditioned on the input sequence Ut: P( Xt = ~. I Xt-l = ),.Ut ) (5) 'Pij,t = As in neural networks trained to minimize the output squared error, the output 1Jt of this architecture can be interpreted as an expected "position parameter" for the probability distribution of the output Yt. However, in addition to being conditional on an input Ut, this expectation is also conditional on the state Xt, i.e. Yoshua Bengio, Paolo Frasconi 430 = E[Yt I Xt,Ut] . The actual form of the output density, denoted !Y(Yt ; 7]t), will be chosen according to the task . For example a multinomial distribution is suitable for sequence classification, or for symbolic mutually exclusive outputs. Instead, a Gaussian distribution is adequate for producing continuous outputs. In the first case we use a softmax function at the output of subnetworks OJ; in the second case we use linear output units for the subnetworks OJ. 7]t In order to reduce the amount of computation, we introduce an independency model among the variables involved in the probabilistic interpretation of the architecture. We shall use a Bayesian network to characterize the probabilistic dependencies among these variables. Specifically, we suppose that the directed acyclic graph 9 depicted at the bottom of Figure 1b is a Bayesian network for the dependency One of the most evident consequences model associated to the variables u of this independency model is that only the previous state and the current input are relevant to determine the next-state. This one-step memory property is analogue to the Markov assumption in hidden Markov models (HMM). In fact, the Bayesian network for HMMs can be obtained by simply removing the Ut nodes and arcs from them (see top of Figure Ib) . I ,xI, YI. 3 A SUPERVISED LEARNING ALGORITHM The learning algorithm for the proposed architecture is derived from the maximum likelihood principle. The training data are a set of P pairs of input/ output sequences (of length Tp): 1) = {(uip(p),Yip(p));p = 1 .. . P}. Let ?J denote the vector of parameters obtained by collecting all the parameters (Jj and iJi of the architecture. The likelihood function is then given by p L(?J; 1)) = II p(Yip(p) I uip(p); ?J). (6) p=l The output values (used here as targets) may also be specified intermittently. For example, in sequence classification tasks , one may only be interested in the output YT at the end of each sequence. The modification of the likelihood to account for intermittent targets is straightforward. According to the maximum likelihood principle, the optimal parameters are obtained by maximizing (6). In order to apply EM to our case we begin by noting that the state variables Xt are not observed. Knowledge of the model's state trajectories would allow one to decompose the temporal learning problem into 2n static learning subproblems. Indeed , if Xt were known, the probabilities (it would be either 0 or 1 and it would be possible to train each subnetwork separately, without taking into account any temporal dependency. This observation allows to link EM learning to the target propagation approach discussed in the introduction . Note that if we used a Viterbi-like approximation (i .e., considering only the most likely path) , we would indeed have 2n static learning problems at each epoch. In order to we derive the learning equations, let us define the complete data as 1)c = {(uiP(p),yiP(p),xiP(p));p = 1 ... P}. The corresponding complete data l%-likelihood is T T T Ic(?J;1)c) = """ L...IOgP(YIP(P),ZlP(P) I u1P(p); ?J). (7) p=l Since lc( ?J; 1)c) depends on the hidden state variables it cannot be maximized directly. The MLE optimization is then solved by introducing the auxiliary function Q(?J; 0) and iterating the following two,steps for k = 1, 2 ... :, Estimation: Compute Q(?J ; ?J) = E[lc(?J; 1)c) r 1), ?J] (8) Maximization: Update the parameters as 0 t- arg max?J Q( ?J; 0) An Input Output HMM Architecture 431 The expectation of (7) can be expressed as p Q(0 ; 0) Tp N N = L: L: L: (it!og P(Yt I Xt = i , Ut i 0) + L: hij,tlog<Pij ,t p=1 t=1 i=1 (9) j=1 I where hij ,t = EIZitzj,t-l uf, yf; 0J, denoting Zit for an indicator variable = 1 if Xt = i and 0 otherwise. The hat in ( it and hij ,t means that these variables are computed using the "old" parameters 0 . In order to compute hij ,t we introduce the forward probabilities Qit = P(YL Xt = i ; uD and the backward probabilities f3it = p(yf I Xt = i , that are updated as follows: un, f3it = fY(Yt;l1it) Lt <Pti(Ut+df3l,t+l (10) Qit = fY(Yt; l1it) Lt <pa(ut} Qt ,t-l . (11) - f3it Qj ,t-l<Pij (ut) " wi QiT Each iteration of the EM algorithm requires to maximize Q(0 ; 0). We first consider a simplified case, in which the inputs are quantized (i .e., belonging to a finite alphabet {0"1, "" O"K}) and the subnetworks behave like lookup tables addressed by the input symbols O"t, i.e. we interpret each parameter as Wi' k = P(Xt = i I Xt-l = j , O"t = k). For simplicity, we restrict the analysis to classification tasks and we suppose that targets are specified as desired final states for each sequence. Furthermore, no output subnetworks are used in this particular application of the algorithm. In this case we obtain the reestimation formulae: h Wijk = ij ,t - " "p" (12) . . {j,t(j,t-l w i ESj wp=1 wt :Ut=k , + T ? "'T ' In general, however, if the subnetworks have hidden sigmoidal units, or use a softmax function to constrain their outputs to sum to one, the maximum of Q cannot be found analytically. In these cases we can resort to a GEM algorithm, that simply produces an increase in Q, for example by gradient ascent. In this case, the derivatives of Q with respect to the parameters can be easily computed as follows. Let Ojlt be a generic weight in the state subnetwork N j . From equation (9): 8Q(0;0) 80jk = L:L:L:hij,t_l_8<pij ,t p t i <Pij,t 80jk (13) where the partial derivatives &:e~;t can be computed using backpropagation . Similarly, denoting with t'Jik a generic weight of the output subnetwork Oi, we have: 8Q( 0; 0) '" '" '" . 8 87]a t 8t'J ? = L..JL..JL..J(i , t~logfY(Yy;l1it) 8t'J .' p t t 7],t,t ,k ,k (14) where ~;;:~t are also computed using backpropagation. Intuitively, the parameters are updated as if the estimation step of EM had provided targets for the outputs of the 2n subnetworks , for each time t . Although GEM algorithms are also guaranteed to find a local maximum of the likelihood, their convergence may be significantly slower compared to EM. In several experiments we noticed that convergence can be accelerated with stochastic gradient ascent . 432 4 Yoshua Bengio, Paolo Frasconi COMPARISONS It appears natural to find similarities between the recurrent architecture described so far and standard HMMs (Levinson et al., 1983). The architecture proposed in this paper differs from standard HMMs in two respects: computing style and learning. With IOHMMs, sequences are processed similarly to recurrent networks, e.g., an input sequence can be synchronously transformed into an output sequence. This computing style is real-time and predictions of the outputs are available as the input sequence is being processed. This architecture thus allows one to implement all three fundamental sequence processing tasks: production, prediction, and classification. Finally, transition probabilities in standard HMMs are fixed, i.e. states form a homogeneous Markov chain. In IOHMMs, transition probabilities are conditional on the input and thus depend on time, resulting in an inhomogeneous Markov chain. Consequently, the dynamics of the system (specified by the transition probabilities) are not fixed but are adapted in time depending on the input sequence. The other fundamental difference is in the learning procedure. While interesting for their capabilities of modeling sequential phenomena, a major weakness of standard HMMs is their poor discrimination power due to unsupervised learning. An approach that has been found useful to improve discrimination in HMMs is based on maximum mutual information (MMI) training . It has been pointed out that supervised learning and discriminant learning criteria like MMI are actually strictly related (Bridle, 1989). Although the parameter adjusting procedure we have defined is based on MLE, is used as desired output in response to the input resulting in discriminant supervised learning. Finally, it is worth mentioning that a number of hybrid approaches have been proposed to integrate connectionist approaches into the HMM frame\'Vork. For example in (Bengio et al. , 1992) the observations used by the HMM are generated by a feedforward neural network. In (Bourlard and Wellekens, 1990) a feedforward network is used to estimate state probabilities, conditional to the acoustic sequence. A common feature of these algorithms and the one proposed in this paper is that neural networks are used to extract temporally local information whereas a Markovian system integrates long-term constraints. yf uf , We can also establish a link between IOHMMs and adaptive mixtures of experts (ME) (Jacobs et al., 1991). Recently, Cacciatore & Nowlan (1994) have proposed a recurrent extension to the ME architecture, called mixture of controllers (MC), in which the gating network has feedback connections, thus allowing to take temporal context into account . Our IOHMM architecture can be interpreted as a special case of the MC architecture, in which the set of state subnetworks play the role of a gating network having a modular structure and second order connections. 5 REGULAR GRAMMAR INFERENCE In this section we describe an application of our architecture to the problem of grammatical inference. In this task the learner is presented a set of labeled strings and is requested to infer a set of rules that define a formal language. It can be considered as a prototype for more complex language processing problems. However, even in the "simplest" case, i.e. regular grammars, the task can be proved to be NP-complete (Angluin and Smith, 1983) . We report experimental results on a set of regular grammars introduced by Tomita (1982) and afterwards used by other researchers to measure the accuracy of inference methods based on recurrent networks (Giles et al. , 1992; Pollack, 1991; Watrous and Kuhn , 1992) . We used a scalar output with supervision on the final output YT that was modeled as a Bernoulli variable fy (YT ; 7]T) = 7]~T (1 - 7] ) l-YT, with YT = 0 if the string is rejected and YT = 1 if it is accepted . In tbis application we did not apply An Input Output HMM Architecture 433 Table 1: Summary of experimental results on the seven Tomita's grammars. Grammar n* 1 2 3 4 5 6 7 2 8 7 4 4 3 3 Sizes FSA min 2 3 5 4 4 3 5 Convergence .600 .800 .150 .100 .100 .350 .450 Average 1.000 .965 .867 1.000 1.000 1.000 .856 Accuracies Worst Best 1.000 1.000 .834 1.000 .775 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .815 1.000 W&K Best 1.000 1.000 .783 .609 .668 .462 .557 external inputs to the output networks . This corresponds to modeling a Moore finite state machine. Given the absence of prior knowledge about plausible state paths, we used an ergodic transition graph (i.e., fully connected).In the experiments we measured convergence and generalization performance using different sizes for the recurrent architecture. For each setting we ran 20 trials with different seeds for the initial weights. We considered a trial successful if the trained network was able to correctly label all the training strings. The model size was chosen using a cross-validation criterion based on performance on 20 randomly generated strings of length T ::; 12. For comparison, in Table 1 we also report for each grammar the number of states of the minimal recognizing FSA (Tomita, 1982). We tested the trained networks on a corpus of 213 - 1 binary strings of length T ::; 12. The final results are summarized in Table 1. The column "Convergence" reports the fraction of trials that succeeded to separate the training set. The next three columns report averages and order statistics (worst and best trial) of the fraction of correctly classified strings, measured on the successful trials. For each grammar these results refer to the model size n* selected by cross-validation. Generalization was always perfect on grammars 1,4,5 and 6. For each grammar, the best trial also attained perfect generalization. These results compare very favorably to those obtained with second-order networks trained by gradient descent, when using the learning sets proposed by Tomita. For comparison , in the last column of Table 1 we reproduce the results reported by Watrous & Kuhn (1992) in the best of five trials. In most of the successful trials the model learned an actual FSA behavior with transition probabilities asymptotically converging either to 0 or to 1. This renders trivial the extraction of the corresponding FSA . Indeed, for grammars 1,4,5, and 6, we found that the trained networks behave exactly like the minimal recognizing FSA . A potential training problem is the presence of local maxima in the likelihood function. For example, the number of converged trials for grammars 3, 4, and 5 is quite small and the difficulty of discovering the optimal solution might become a serious restriction for tasks involving a large number of states. In other experiments (Bengio & Frasconi , 1994a) , we noticed that restricting the connectivity of the transition graph can significantly help to remove problems of convergence. Of course, this approach can be effectively exploited only if some prior knowledge about the state space is available. For example, applications of HMMs to speech recognition always rely on structured topologies. 6 CONCLUSIONS There are still a number of open questions . In particular, the effectiveness of the model on tasks involving large or very large state spaces needs to be carefully evaluated. In (Bengio & Frasconi 1994b) we show that learning long term dependencies in these models becomes more difficult as we increase the connectivity of the state 434 Yoshua Bengio, Paolo Frasconi transition graph. However, because transition probabilities of IOHMMs change at each t, they deal better with this problem of long-term dependencies than standard HMMs. Another interesting aspect to be investigated is the capability of the model to successfully perform tasks of sequence production or prediction. For example, interesting tasks that could also be approached are those related to time series modeling and motor control learning. References Angluin, D. and Smith, C. (1983). Inductive inference: Theory and methods. Computing Surveys, 15(3):237-269. Bengio, Y. and Frasconi, P. (1994a) . Credit assignment through time: Alternatives to backpropagation. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6. Morgan Kaufmann. Bengio, Y. and Frasconi, P. (1994b). An EM Approach to Learning Sequential Behavior. Tech. Rep. RT-DSI/11-94, University of Florence. Bengio, Y., De Mori, R., Flammia, G., and Kompe, R. (1992). Global optimization of a neural network-hidden markov model hybrid. IEEE Transactions on Neural Networks, 3(2):252-259. Bengio, Y., Simard, P., and Frasconi, P. (1994) . Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Networks, 5(2). Bourlard, H. and Wellekens, C. (1990). Links between hidden markov models and multilayer perceptrons. IEEE Trans. Pattern An. Mach. Intell., 12:1167-1178. Bridle, J. S. (1989). Training stochastic model recognition algorithms as networks can lead to maximum mutual information estimation of parameters. In D.S.Touretzky, ed., NIPS2, pages 211-217. Morgan Kaufmann. Cacciatore, T. W. and Nowlan, S. J. (1994). Mixtures of controllers for jump linear and non-linear plants. In Cowan, J. et. al., editors, Advances in Neural Information Processing Systems 6, San Mateo, CA. Morgan Kaufmann. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum-likelihood from incomplete data via the EM algorithm. J. Royal Stat. Soc. B,39:1-38. Giles, C. L., Miller, C. B ., Chen, D., Sun, G. Z., Chen, H. H., and Lee, Y. C. (1992). Learning and extracting finite state automata with second-order recurrent neural networks. Neural Computation, 4(3):393-405. Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. (1991). Adaptive mixture of local experts. Neural Computation, 3:79-87. Levinson, S. E., Rabiner, L. R., and Sondhi, M. M. (1983). An introduction to the application of the theory of probabilistic functIons of a markov process to automatic speech recognition. Bell System Technical Journal, 64(4):1035-1074. Mozer, M. C. (1992). The induction of multiscale temporal structure. In Moody, J. et. al., eds, NIPS 4 pages 275-282. Morgan Kaufmann. Pollack, J. B. (1991) . The induction of dynamical recognizers. Machine Learning, 7(2):196-227. Tomita, M. (1982). Dynamic construction of finite-state automata from examples using hill-climbing. Proc. 4th Cog. Science Con!, pp. 105-108, Ann Arbor MI. Watrous, R. 1. and Kuhn, G. M. (1992). Induction of finite-state languages using second-order recurrent networks. Neural Computation, 4(3):406-414 . 

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Learning from queries for maximum information gain in imperfectly learnable problems Peter Sollich David Saad Department of Physics, University of Edinburgh Edinburgh EH9 3JZ, U.K. P.Sollich~ed.ac.uk. D.Saad~ed.ac.uk Abstract In supervised learning, learning from queries rather than from random examples can improve generalization performance significantly. We study the performance of query learning for problems where the student cannot learn the teacher perfectly, which occur frequently in practice. As a prototypical scenario of this kind, we consider a linear perceptron student learning a binary perceptron teacher. Two kinds of queries for maximum information gain, i.e., minimum entropy, are investigated: Minimum student space entropy (MSSE) queries, which are appropriate if the teacher space is unknown, and minimum teacher space entropy (MTSE) queries, which can be used if the teacher space is assumed to be known, but a student of a simpler form has deliberately been chosen. We find that for MSSE queries, the structure of the student space determines the efficacy of query learning, whereas MTSE queries lead to a higher generalization error than random examples, due to a lack of feedback about the progress of the student in the way queries are selected. 1 INTRODUCTION In systems that learn from examples, the traditional approach has been to study generalization from random examples, where each example is an input-output pair 288 Peter Sollich, David Saad with the input chosen randomly from some fixed distribution and the corresponding output provided by a teacher that one is trying to approximate. However, random examples contain less and less new information as learning proceeds. Therefore, generalization performance can be improved by learning from queries, i. e., by choosing the input of each new training example such that it will be, together with its expected output, in some sense 'maximally useful'. The most widely used measure of 'usefulness' is the information gain, i.e ., the decrease in entropy of the post-training probability distributions in the parameter space of the student or the teacher. We shall call the resulting queries 'minimum (student or teacher space) entropy (MSSE/MTSE) queries'; their effect on generalization performance has recently been investigated for perfectly learnable problems, where student and teacher space are identical (Seung et al., 1992, Freund et al., 1993, Sollich, 1994), and was found to depend qualitatively on the structure of the teacher. For a linear perceptron, for example, one obtains a relative reduction in generalization error compared to learning from random examples which becomes insignificant as the number of training examples, p, tends to infinity. For a perceptron with binary output, on the other hand, minimum entropy queries result in a generalization error which decays exponentially as p increases, a marked improvement over the much slower algebraic decay with p in the case of random examples. In practical situations, one almost always encounters imperfectly learnable problems, where the student can only approximate the teacher, but not learn it perfectly. Imperfectly learnable problems can arise for two reasons: Firstly, the teacher space (i.e., the space of models generating the data) might be unknown. Because the teacher space entropy is then also unknown, MSSE (and not MTSE) queries have to be used for query learning. Secondly, the teacher space may be known, but a student of a simpler structure might have deliberately been chosen to facilitate or speed up training, for example. In this case, MTSE queries could be employed as an alternative to MSSE queries. The motivation for doing this would be strongest if, as in the learning scenario that we consider below, it is known from analyses of perfectly learnable problems that the structure of the teacher space allows more significant improvements in generalization performance from query learning than the structure of the student space. With the above motivation in mind, we investigate in this paper the performance of both MSSE and MTSE queries for a prototypical imperfectly learnable problem, in which a linear perceptron student is trained on data generated by a binary perceptron teacher. Both student and teacher are specified by an N-dimensional weight vector with real components, and we will consider the thermodynamic limit N -+ 00, p -+ 00,0:' = piN = const. In Section 2 below we calculate the generalization error for learning from random examples. In Sections 3 and 4 we compare the result to MSSE and MTSE queries. Throughout, we only outline the necessary calculations; for details, we refer the reader to a forthcoming publication. We conclude in Section 5 with a summary and brief discussion of our results. 2 LEARNING FROM RANDOM EXAMPLES We denote students and teachers by N (for 'Neural network') and V (for 'element of the Version space', see Section 4), respectively, and their corresponding weight Learning from Queries for Maximum Information Gain 289 vectors by W Nand w v . For an input vector x, the outputs of a given student and teacher are YN = XTWN' Yv = sgn( xTwv) . 7N 7N Assuming that inputs are drawn from a uniform distribution over the hypersphere x 2 = N, and taking as our error measure the standard squared output difference (YN - Yv)2, the generalization error, i. e., the average error between student Nand teacher V when tested on random test inputs, is given by ! ?g(N, V) 1[ = '2 QN (2) 1/2] , R + 1- 2..ftTv;: (1) where we have set R = ];WrrWv,QN = ];w~,Qv = ];w~. As our training algorithm we take stochastic gradient descent on the trammg error E t , which for a training set e(p) = {( x~ ,Y~ Yv( x~)), J.l 1 ... p} is Et ! 2:~(Y~ - YN(X~))2. A weight decay term !AW~ is added for regularization, i.e., to prevent overfitting. Stochastic gradient descent on the resulting energy function E = E t + !AW~ yields a Gibbs post-training distribution of students, p(wNle(p)) ex exp(-E/T), where the training temperature T measures the amount of stochasticity in the training algorithm. For the linear perceptron students considered here, this distribution is Gaussian, with covariance matrix TM,N-l, where (IN denotes the N x N identity matrix) = = MN = AlN = + ~ 2:~=1 x~(x~f? Since the length ofthe teacher weight vector Wv does not affect the teacher outputs, we assume a spherical prior on teacher space, P(w v ) ex 6(w~-N), for which Qv = 1. Restricting attention to the limit of zero training temperature, it is straightforward to calculate from eq . (1) the average generalization error obtained by training on random examples ?g - ?g,min =;:1 [ AoptG + A(Aopt - A) OG] OA ' (2) with the function G = (]; tr M,N-1) P( {xl'}) given by (Krogh and Hertz, 1992) G= 2\ [1 - a - A+ )(1- a - A)2 + 4A] . (3) In eq. (2) we have explicitly subtracted the minimum achievable generalization error, = !(1-2/11'), which is nonzero since a linear perceptron cannot approximate a binary percept ron perfectly. At finite a, the generalization error is minimized when the weight decay is set to its optimal value A = Aopt 11'/2 - 1. Note that since both G and OG/OA tend to zero as a --+ 00, the generalization error for random examples approaches the minimum achievable generalization error in this limit. ?g,min = 3 MINIMUM STUDENT SPACE ENTROPY QUERIES We now calculate the generalization performance resulting from MSSE queries. For the training algorithm introduced in the last section, the student space entropy (normalized by N) is given by 290 Peter Sollich, David Saad ""T""----r----r---------------, 3.0 - - A== 0.01 ---- A == 0.1 -------- A == 1 2.5 2.0 1.5 -------------------::~-::-=.:--~~.J .. _-----------1.0-+--.::..::......-.-----,----.-----r-----I 3 o 4 5 2 a Figure 1: Relative improvement weight decay A = 0.01,0 .1,1. K, in generalization error due to MSSE queries, for 1 S'" = - 2N lndet M N , where we have omitted an unimportant constant which depends on the training temperature only. This entropy is minimized by choosing each new query along the direction corresponding to the minimal eigenvalue of the existing MN (Sollich, 1994). The expression for the resulting average generalization error is given by eq. (2) with G replaced by its analogue for MSSE queries (Sollich, 1994) ~a 1- ~a GQ +----:- A + [a] + 1 A + [a] , where [a] is the greatest integer less than or equal to a and ~o' = a-raj. We define the improvement factor K, as the ratio of the generalization error (with the minimum achievable generalization error subtracted as in eq. (2)) for random examples to that for MSSE queries. Figure 1 shows K(O') for several values of the weight decay A. Comparing with existing results (Sollich, 1994), we find that K, is exactly the same as if our linear student were trying to approximate a linear teacher with additive noise of variance Aopt on the outputs. For large a, one can show (Sollich, 1994) that K, = 1 + I/O' + O(I/a 2 ) and hence the relative reduction in generalization error due to querying tends to zero as a - I - 00. We investigate in the next section whether it is possible to improve generalization performance more significantly by using MTSE queries. 4 MINIMUM TEACHER SPACE ENTROPY QUERIES We now consider the generalization performance achieved by MTSE queries. We remind readers that such queries could be used if the teacher space is known, but a student of a simpler functional form has deliberately chosen. The aim in using MTSE rather than MSSE queries would be to exploit the structure of the teacher space if this is known (for perfectly learnable problems) to make query learning very efficient compared to random examples. For the case of noise free training data under consideration, the posterior probability distribution in teacher space given a certain training set is proportional to the prior 291 Learning from Queries for Maximum Information Gain distribution on the version space (the set of all teachers that could have produced the training set without error) and zero everywhere else. From this the (normalized) teacher space entropy can be derived to be, up to an additive constant, 1 Sv = N In V(p), where the version space volume V(p) is given by (8(z) = 1 for z V(p) = > 0 and 0 otherwise) JdwvP(w v) n~=l 8(JNy/Jw~x/J). It can easily be verified that this entropy is minimized l by choosing queries x which (bisect' the existing version space, i. e., for which the hyperplane perpendicular to x splits the version space into two equal halves (Seung et al., 1992, Freund et al., 1993). Such queries lead to an exponentially shrinking version space, V(p) = 2- P , and hence a linear decrease of the entropy, Sv = -a In 2. We consider instead queries which achieve qualitatively the same effect, but permit a much simpler analysis of the resulting student performance. They are similar to those studied in the context of a learnable problem by Watkin and Rau (1992), and are defined as follows . The (p + 1)th query is obtained by first picking a random teacher vector wp from the version space defined by the existing p training examples, and then picking the new training input Xp+l from the distribution of random inputs but under the constraint that x;+1wp = O. For the calculation of the student performance, i. e., the average generalization error, achieved by the approximate MTSE queries described above, we use an approximation based on the following observation. As the number of training examples, p, increases, the teacher vectors wp from the version space will align themselves with the true teacher w~; their components along the direction of w~ will increase, whereas their components perpendicular to we will decrease, varying widely across the N - 1 dimensional hyperplane perpendicular to we . Following Watkin and Rau (1992), we therefore assume that the only significant effect of choosing queries xp+1 with X;+lWP = 0 is on the distribution of the component ofxp+l along we . Writing this component as x~+1 = x;+1 w~/lwel, its probability distribution can readily be shown to be P(x~+1) ex: exp (-~(x~+1lsp)2) , (4) where sp is the sine of the angle between w p and we. For finite N, the value of sp is dependent on the p previous training examples that define the existing version space and on the teacher vector wp sampled randomly from this version space. In the thermodynamic limit, however, the variations of sp become vanishingly small and we can thus replace sp by its average value, which is a function of palone. In the thermodynamic limit, this average value becomes a continuous function of a = pIN , the number of training examples per weight, which we denote simply by sea) . The calculation can then be split into two parts: First, the function sea) is obtained from a calculation of the teacher space entropy using the replica method, generalizing the results of Gyorgi and Tishby (1990) . The average generalization 1 More precisely, what is minimized is the value of the entropy after a new training example (x, y) is added, averaged over the distribution of the unknown new training output y given the new training input x and the existing training set; see Sollich (1994). 292 Peter Sollich, David Saad 0 ................?, .......,.,.....,. -1 --':''''''::-.::, -............ ----------- -2 ..... ' -3 . ...?.....??...... -4 -5 0 2 3 4 5 a 6 7 8 9 10 Figure 2: MTSE queries: Teacher space entropy, Sv (with value for random examples plotted for comparison), and In s, the log of the sine of the angle between the true teacher and a random teacher from the version space. error can then be calculated by using an extension of the response function method described in (Sollich, 1994b) or by another replica calculation (now in student space) as in (Dunmur and Wallace, 1993). Figure 2 shows the effects of (approximate) MTSE queries in teacher space. For large a values, the teacher space entropy decreases linearly with a, with gradient c : : : : 0.44, whereas the entropy for random examples , also shown for comparison , decreases much more slowly (asymptotically like -In a, see (Gyorgi and Tishby, 1990)). The linear a-dependence of the entropy for queries corresponds to an average reduction of the version space volume with each new training example by a factor of exp( -c) : : : : 0.64, which is reasonably close to the factor ~ for proper bisection of the version space. This justifies our choice of analysing approximate MTSE queries rather than true MTSE queries, since the former achieve qualitatively the same results as the latter. Before discussing the student performance achieved by (approximate) MTSE queries, we note from figure 2 that In s( a) decreases linearly with a for large a, with the same gradient as the teacher space entropy. Hence s( a) ex: exp( -ca) for large a, and MTSE queries force the average teacher from the version space to approach the true teacher exponentially quickly. It can easily be shown that if we were learning with a binary perceptIOn student, i. e., if the problem were perfectly learnable, then this would result in an exponentially decaying generalization error, ?g ex: exp( -ca). MTSE queries would thus lead to a marked improvement in generalization performance over random examples (for which ?g ex: l/a, see (Gyorgi and Tishby, 1990)). It is this significant benefit (in teacher space) of query learning that provides the motivation for using MTSE queries in imperfectly learnable problems such as the one considered here. The results plotted in Figure 3 for the average generalization error achieved by the linear perceptron student show, however, that MTSE queries do not have the desired effect. Far from translating the benefits in teacher space into improvements in generalization performance for the linear student, they actually lead to a deterioration of generalization performance, i. e., a larger generalization error than that Learning from Queries for Maximum Information Gain - - >. = 0.01 ---- >. = 0.1 -------- >. = 1 0.6 0.5 fg ...\ - , " 0.4 , '..". ,, '......., .. 0.3 -------- - ---- --------......-=---=--=---=--d-- -~ 0.2 o 293 2 3 4 5 a 6 7 8 9 10 Figure 3: Generalization error for MTSE queries (higher curves of each pair) and random examples (lower curves), for weight decay>. = 0.01,0.1,1. The curves for random examples (which are virtually indistinguishable from one another already at a = 10) converge to the minimum achievable generalization error fg,min (dotted line) as a -+ 00. obtained for random examples. Worse still, they 'mislead' the student to such an extent that the minimum achievable generalization error is not reached even for an infinite number of training examples, a -+ 00. How does this happen? It can be verified that the angle between the student and teacher weight vectors tends to zero for a -+ 00 as expected, while QN, the normalized squared length of the student weight vector, approaches QN(a -+ 00) S = -2 ( ---= 11' >. + s2 )2 , (5) where s = Jooo da s( a), s2 = Jooo da s2 (a). Unless the weight decay parameter >. happens to be equal to s - s2, this is different from the optimal asymptotic value, which is 2/11'. This is the reason why in general the linear student does not reach the minimum possible generalization error even as a -+ 00. The approach of QN to its non-optimal asymptotic value can cause an increase in the generalization error for large a and a corresponding minimum of the generalization error at some finite a, as can be seen in the plots for>. = 0.01 and 0.1 in Figure 3. For>. = 0, eq. (5) has the following intuitive interpretation: As a increases, the version space shrinks around the true teacher w~, and hence MTSE queries become 'more and more orthogonal' to w~ . As a consequence, the distribution of training inputs along the direction of is narrowed down progressively (compare eq. (4)). Trying to find a best fit to the teacher's binary output function over this narrower range of inputs, the linear student learns a function which is steeper than the best fit over the range of random inputs (which would give minimum generalization error). This corresponds to a suboptimally large length of the student weight vector in agreement with eq. (5): QN(a -+ 00) > 2/11' for>. = a because s2 < s. we Summarizing the results of this section, we have found that although MTSE queries are very beneficial in teacher space, they are entirely misleading for the linear student, to the extent that the student does not learn to approximate the teacher optimally even for an infinite number of training examples. 294 5 Peter Sollich, David Saad SUMMARY AND DISCUSSION We have found in our study of an imperfectly learnable problem with a linear student and a binary teacher that queries for minimum student and teacher space entropy, respectively, have very different effects on generalization performance. Minimum student space entropy (MSSE) queries essentially have the same effect as for a linear student learning a noisy linear teacher, apart from a nonzero minimum value of the generalization error due to the unlearnability of the problem . Hence the structure of the student space is the dominating influence on the efficacy of query learning. Minimum teacher space entropy queries (MTSE), on the other hand, perform worse than random examples, leading to a higher generalization error even for an infinite number of training examples. With the benefit of hindsight, we note that this makes intuitive sense since the teacher space entropy, according to which MTSE queries are selected, contains no feedback about the progress of the student in learning the required generalization task, and thus MTSE queries cannot be guaranteed to have a positive effect. Our results, then, are a mixture of good and bad news for query learning for maximum information gain in imperfectly learnable problems: The bad news is that MTSE queries, due to a lack of feedback information about student progress, are not enough to translate significant benefits in teacher space into similar improvements of student performance and may in fact yield worse performance than random examples. The good news is that for MSSE queries, we have found evidence that the structure of the student space is the key factor in determining the efficacy of query learning. If this result holds more generally, then statements about the benefits of query learning can be made on the basis of how one is trying to learn only, independently of what one is trying to learn-a result of great practical significance. References A P Dunmur and D J Wallace (1993). Learning and generalization in a linear perceptron stochastically trained with noisy data. J. Phys. A, 26:5767-5779. Y Freund, H S Seung, E Shamjr, and N Tishby (1993). Information, prediction, and query by committee. In S J Hanson, J D Cowan, and C Lee Giles, editors, NIPS 5, pages 483-490, San Mateo, CA, Morgan Kaufmann. G Gyorgi and N Tishby (1990). Statistical theory of learning a rule. In W Theumann and R Koberle, editors, Neu.ral Networks and Spin Glasses, pages 3-36. Singapore , World Scientific. A Krogh and J A Hertz (1992). Generalization in a linear perceptron in the presence of noise. J. Phys. A,25:1135-1147. P Sollich (1994). Query construction , entropy, and generalization in neural network models. Phys. Rev. E,49:4637-4651. P Sollich (1994b) . Finite-size effects in learning and generalization in linear perceptrons. J. Phys. A, 27:7771-7784. H S Seung, M Opper, and H Sompolinsky (1992) . Query by committee. In Proceedings of COLT '92, pages 287-294, New York, ACM . T L H Watkin and A Rau (1992). Selecting examples for perceptrons. J . Phys. A, 25:113-121. 

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Boltzmann Chains and Hidden Markov Models Lawrence K. Saul and Michael I. Jordan lksaulOpsyche.mit.edu, jordanOpsyche.mit.edu Center for Biological and Computational Learning Massachusetts Institute of Technology 79 Amherst Street, E10-243 Cambridge, MA 02139 Abstract We propose a statistical mechanical framework for the modeling of discrete time series. Maximum likelihood estimation is done via Boltzmann learning in one-dimensional networks with tied weights. We call these networks Boltzmann chains and show that they contain hidden Markov models (HMMs) as a special case. Our framework also motivates new architectures that address particular shortcomings of HMMs. We look at two such architectures: parallel chains that model feature sets with disparate time scales, and looped networks that model long-term dependencies between hidden states. For these networks, we show how to implement the Boltzmann learning rule exactly, in polynomial time, without resort to simulated or mean-field annealing. The necessary computations are done by exact decimation procedures from statistical mechanics. 1 INTRODUCTION AND SUMMARY Statistical models of discrete time series have a wide range of applications, most notably to problems in speech recognition (Juang & Rabiner, 1991) and molecular biology (Baldi, Chauvin, Hunkapiller, & McClure, 1992). A common problem in these fields is to find a probabilistic model, and a set of model parameters, that 436 Lawrence K. Saul, Michael I. Jordan account for sequences of observed data. Hidden Markov models (HMMs) have been particularly successful at modeling discrete time series. One reason for this is the powerful learning rule (Baum) 1972?) a special case of the Expectation- Maximization (EM) procedure for maximum likelihood estimation (Dempster) Laird) & Rubin) 1977). In this work) we develop a statistical mechanical framework for the modeling of discrete time series. The framework enables us to relate HMMs to a large family of exactly solvable models in statistical mechanics. The connection to statistical mechanics was first noticed by Sourlas (1989?) who studied spin glass models of error-correcting codes. We view the estimation procedure for HMMs as a special (and particularly tractable) case of the Boltzmann learning rule (Ackley) Hinton) & Sejnowski) 1985; Byrne) 1992). The rest of this paper is organized as follows . In Section 2) we review the modeling problem for discrete time series and establish the connection between HMMs and Boltzmann machines. In Section 3) we show how to quickly determine whether or not a particular Boltzmann machine is tractable) and if so) how to efficiently compute the correlations in the Boltzmann learning rule. Finally) in Section 4) we look at two architectures that address particular weaknesses of HMMs: the modelling of disparate time scales and long-term dependencies. 2 MODELING DISCRETE TIME SERIES A discrete time series is a sequence of symbols {jdr=l in which each symbol belongs to a finite countable set) i.e. jl E {1) 2) .. . ) m}. Given one long sequence) or perhaps many shorter ones) the modeling task is to characterize the probability distribution from which the time series are generated. 2.1 HIDDEN MARKOV MODELS A first-order Hidden Markov Model (HMM) is characterized by a set of n hidden states) an alphabet of m symbols) a transmission matrix ajj') an emission matrix bjj ) and a prior distribution 7I'j over the initial hidden state. The sequence of states {idr=l and symbols {jdr=l is modeled to occur with probability (1) The modeling problem is to find the parameter values (ajj' , bij ) 7I'j) that maximize the likelihood of observed sequences of training data. We will elaborate on the learning rule in section 2.3) but first let us make the connection to a well-known family of stochastic neural networks , namely Boltzmann machines. 2.2 BOLTZMANN MACHINES Consider a Boltzmann machine with m-state visible units) n-state hidden units) tied weights) and the linear architecture shown in Figure 1. This example represents the simplest possible Boltzmann "chain))) one that is essentially equivalent to a firstorder HMM unfolded in time (MacKay) 1994). The transition weights Aii' connect adjacent hidden units) while the emission weights Bj j connect each hidden unit to Boltzmann Chains and Hidden Markov Models 437 1J Bij ViS~ble un~ts Bij hidden units ? ? ? A i i , Au' Figure 1: Boltzmann chain with n-state hidden units, m-state visible units, transition weights Aiil, emission weights Bij, and boundary weights IIi. its visible counterpart. In addition, boundary weights IIi model an extra bias on the first hidden unit. Each configuration of units represents a state of energy L-1 1t[{il' jd] = -Ilil - L L Ailil+t - l=l 2: Bitio (2) l=l where {idf=l ({jl }f=l) is the sequence of states over the hidden (visible) units. The probability to find the network in a particular configuration is given by ' '}) = Ze 1 -{31-l , P({ Zl,)l where f3 (3) = I/T is the inverse temperature, and the partition function L Z = e-fJ'H. (4) {idd is the sum over states that normalizes the Boltzmann distribution, eq. (3). Comparing this to the HMM distribution, eq. (1), it is clear that any first-order HMM can be represented by the Boltzmann chain of figure 1, provided we take 1 IIi = TIn 7ri? (5) Later, in Section 4, we will consider more complicated chains whose architectures address particular shortcomings of HMMs. For now, however, let us continue to develop the example of figure 1, making explicit the connection to HMMs. Aii' 2.3 = TIn aij/, Bij = TIn bij , LEARNING RULES In the framework of Boltzmann learning (Williams & Hinton, 1990), the data for our problem consist of sequences of states over the visible units; the goal is to find the weights (Ail, B ij , IIi) that maximize the likelihood of the observed data. The likelihood of a sequence {jd is given by the ratio . P({il,jd) P({Jd) = P({idl{jl}) e-{3'H./Z Zc = e-{3'H./Zc = Z' (6) 1 Note, however, that the reverse statement-that for any set of parameters, this Boltzmann chain can be represented as an HMM-is not true. The weights in the Boltzmann chain represent arbitrary energies between ?oo, whereas the HMM parameters represent probabilities that are constrained to obey sum rules, such as Lil aiil = 1. The Boltzmann chain of figure 1 therefore has slightly more degrees of freedom than a first-order HMM. An interpretation of these extra degrees of freedom is given by MacKay (1994). 438 Lawrence K. Saul, Michael I. Jordan where Zc is the clamped partition function Zc = L e-/3 1i . (7) {it} Note that the sum in Zc is only over the hidden states in the network, while the visible states are clamped to the observed values bt}. The Boltzmann learning rule adjusts the weights of the network by gradient-ascent on the log-likelihood. For the example of figure 1, this leads to weight updates L-l ~Aii' = 7J/3 L [(6iil6ilil+Jc - (8) (6iil6ilil+l)] ; l=1 L ~Bij 7J/3 L [(6iil6jjl)C - (9) (6i il 6jjl)] , l=1 ~ni 7J/3 [(6ii1 )c - (6ii1 )] , (10) where 6ij stands for the Kronecker delta function, 7J is a learning rate, and (-) and (-) c denote expectations over the free and clamped Boltzmann distributions. The Boltzmann learning rule may also be derived as an Expectation-Maximization (EM) algorithm. The EM procedure is an alternating two-step method for maximum likelihood estimation in probability models with hidden and observed variables. For Boltzmann machines in general, neither the E-step nor the M-step can be done exactly; one must estimate the necessary statistics by Monte Carlo simulation (Ackley et al., 1985) or mean-field theory (Peterson & Anderson, 1987). In certain special cases (e.g. trees and chains) , however, the necessary statistics can be computed to perform an exact E-step (as shown below). While the Mstep in these Boltzmann machines cannot be done exactly, the weight updates can be approximated by gradient descent. This leads to learning rules in the form of eqs. (8-10). HMMs may be viewed as a special case of Boltzmann chains for which both the E-step and the M-step are analytically tractable. In this case, the maximization in the M-step is performed subject to the constraints 2:i e/3 Il ? = 1, 2:il e/3 A ;;1 = 1, and 2:j e/3 B ;i = 1. These constraints imply Z = 1 and lead to closed-form equations for the weight updates in HMMs. 3 EXACT METHODS FOR BOLTZMANN LEARNING The key technique to compute partition functions and correlations in Boltzmann chains is known as decimation. The idea behind decimation 2 is the following. Consider three units connected in series, as shown in Figure 2a. Though not directly connected, the end units have an effective interaction that is mediated by the middle one. In fact, the two weights in series exert the same influence as a single effective weight, given by (11) jl 2A related method, the transfer matrix, is described by Stolarz (1994). Boltzmann Chains and Hidden Markov Models 439 A~~)' 1.1. + A~~). 11. (b) (a) (c) Figure 2: Decimation, pruning, and joining in Boltzmann machines. Replacing the weights in this way amounts to integrating out, or decimating, the degree offreedom represented by the middle unit. An analogous rule may be derived for the situation shown in Figure 2b. Summing over the degrees of freedom of the dangling unit generates an effective bias on its parent, given by ef3 B ? = L:: ef3B ? j ? (12) j We call this the pruning rule. Another type of equivalence is shown in Figure 2c. The two weights in parallel have the same effect as the sum total weight Ajjl = A~P + A~i) . (13) We call this the joining rule. It holds trivially for biases as well as weights. The rules for decimating, pruning, and joining have simple analogs in other types of networks (e.g. the law for combining resistors in electric circuits), and the strategy for exploiting them is a familiar one. Starting with a complicated network, we iterate the rules until we have a simple network whose properties are easily computed. A network is tractable for Boltzmann learning if it can be reduced to any pair of connected units. In this case, we may use the rules to compute all the correlations required for Boltzmann learning. Clearly, the rules do not make all networks tractable; certain networks (e.g. trees and chains), however, lend themselves naturally to these types of operations. 4 DESIGNER NETS The rules in section 3 can be used to quickly assess whether or not a network is tractable for Boltzmann learning. Conversely, they can be used to design networks that are computationally tractable. This section looks at two networks designed to address particular shortcomings of HMMs. 4.1 PARALLEL CHAINS AND DISPARATE TIME SCALES An important problem in speech recognition (Juang et al., 1991) is how to "combine feature sets with fundamentally different time scales." Spectral parameters, such 440 Lawrence K. Saul, Michael I. Jordan fast features coupled hidden units slow features Figure 3: Coupled parallel chains for features with different time scales. as the cepstrum and delta-cepstrum, vary on a time scale of 10 msec; on the other hand, prosodic parameters, such as the signal energy and pitch, vary on a time scale of 100 msec. A model that takes into account this disparity should avoid two things. The first is redundancy-in particular, the rather lame solution of oversampling the nonspectral features. The second is overfitting. How might this arise? Suppose we have trained two separate HMMs on sequences of spectral and prosodic features, knowing that the different features "may not warrant a single, unified Markov chain" (Juang et al., 1991). To exploit the correlation between feature sets, we must now couple the two HMMs. A naive solution is to form the Cartesian product of their hidden state spaces and resume training. Unfortunately, this results in an explosion in the number of parameters that must be fit from the training data. The likely consequences are overfitting and poor generalization. Figure 3 shows a network for modeling feature sets with disparate time scales-in this case, a 2: 1 disparity. Two parallel Boltzmann chains are coupled by weights that connect their hidden units. Like the transition and emission weights within each chain, the coupling weights are tied across the length of the network. Note that coupling the time scales in this way introduces far fewer parameters than forming the Cartesian product of the hidden state spaces. Moreover, the network is tractable by the rules of section 3. Suppose, for example, that we wish to compute the correlation between two neighboring hidden units in the middle of the network. This is done by first pruning all the visible units, then repeatedly decimating hidden units from both ends of the network. Figure 4 shows typical results on a simple benchmark problem, with data generated by an artificially constructed HMM. We tested the parallel chains model on 10 training sets, with varying levels of built-in correlation between features. A twostep method was used to train the parallel chains. First, we set the coupling weights to zero and trained each chain by a separate Baum-Welch procedure. Then, after learning in this phase was complete, we lifted the zero constraints and resumed training with the full Boltzmann learning rule. The percent gain in this second phase was directly related to the degree of correlation built into the training data, suggesting that the coupling weights were indeed capturing the correlation between feature sets. We also compared the performance of this Boltzmann machine versus that of a simple Cartesian-product HMM trained by an additional Baum-Welch procedure. While in both cases the second phase of learning led to reduced training error, the Cartesian product HMMs were decidedly more prone to overfitting. Boltzmann Chains and Hidden Markov Models I ?1500 441 I-J"anv::luu::a:cnu:nnuI I II JOU"'XJ:o:x:o:a:J::) /tfA='=u"na"m.....,m"l 1-1rainl"O ! - - croaa-vaJdation ~ 20 '" '" 10 -1700 200 eoo 400 eoo 0.2 epoch (a) 0.6 0.' feature colT8latkwl 0.8 (b) Figure 4: (a) Log-likelihood versus epoch for parallel chains with 4-state hidden units, 6-state visible units, and 100 hidden-visible unit pairs (per chain) . The second jump in log-likelihood occurred at the onset of Boltzmann learning (see text). (b) Percent gain in log-likelihood versus built-in correlation between feature sets. 4.2 LOOPS AND LONG-TERM DEPENDENCIES Another shortcoming of first-order HMMs is that they cannot exhibit long-term dependencies between the hidden states (Juang et aL , 1991). Higher-order and duration-based HMMs have been used in this regard with varying degrees of success. The rules of section 3 suggest another approach-namely, designing tractable networks with limited long-range connectivity. As an example, Figure 5a shows a Boltzmann chain with an internal loop and a long-range connection between the first and last hidden units. These extra features could be used to enforce known periodicities in the time series. Though tractable for Boltzmann learning, the loops in this network do not fit naturally into the framework of HMMs . Figure 5b shows learning curves for a toy problem, with data generated by another looped network. Carefully chosen loops and long-range connections provide additional flexibility in the design of probabilistic models for time series. Can networks with these extra features capture the long-term dependencies exhibited by real data? This remains an important issue for future research . Acknowledgements We thank G. Hinton, D. MacKay, P. Stolorz, and C. Williams for useful discussions. This work was funded by ATR Human Information Processing Laboratories , Siemens Corporate Research, and NSF grant CDA-9404932 . References D. H. Ackley, G . E. Hinton, and T . J. Sejnowski. (1985) A Learning Algorithm for Boltzmann Machines. Cog. Sci. 9: 147- 160. P. Baldi, Y. Chauvin , T . Hunkapiller, and M. A. McClure. (1992) Proc. Nat . Acad. Sci. (USA) 91: 1059-1063. Lawrence K. Saul, Michael I. Jordan 442 I-tralnin ~ g crosa..validation I 12 1. ?700 o (a) 8 epoch 10 (b) Figure 5: (a) Looped network. (b) Log-likelihood versus epoch for a looped network with 4-state hidden units, 6-state visible units, and 100 hidden-visible unit pairs . L. Baum. (1972) An Inequality and Associated Maximization Technique in Statistical Estimation of Probabilistic Functions of Markov Processes, Inequalities 3:1-8. Byrne, W. (1992) Alternating Minimization and Boltzmann Machine Learning. IEEE Trans. Neural Networks 3:612-620. A. P. Dempster, N. M. Laird, and D. B. Rubin . (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm . J. Roy. Statist. Soc. B, 39 :1-38 . C. Itzykson and J . Drouffe. (1991) Statistical Field Theory, Cambridge: Cambridge University Press. B. H. Juang and L. R. Rabiner. (1991) Hidden Markov Models for Speech Recognition, Technometrics 33: 251-272. D. J. MacKay. (1994) Equivalence of Boltzmann Chains and Hidden Markov Models, submitted to Neural Compo C. Peterson and J. R. Anderson. (1987) A Mean Field Theory Learning Algorithm for Neural Networks, Complex Systems 1:995-1019. 1. Saul and M. Jordan . (1994) Learning in Boltzmann Trees. Neural Comp o 6 : 1174-1184. N. Sourlas. (1989) Spin Glass Models as Error Correcting Codes. Nature 339: 693-695 . P. Stolorz. (1994) Links Between Dynamic Programming and Statistical Physics for Heterogeneous Systems, JPL/Caltech preprint . C. Williams and G. E. Hinton. (1990) Mean Field Networks That Learn To Discriminate Temporally Distorted Strings. Proc. Connectionist Models Summer School: 18-22. 

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An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems Robert L. Tokar2 Brian D.McVey 2 'Center for Nonlinear Studies, 2Applied Theoretical Physics Division Los Alamos National Laboratory, Los Alamos, NM, 87545 Abstract In this study, an integrated neural network control architecture for nonlinear dynamic systems is presented. Most of the recent emphasis in the neural network control field has no error feedback as the control input, which rises the lack of adaptation problem. The integrated architecture in this paper combines feed forward control and error feedback adaptive control using neural networks. The paper reveals the different internal functionality of these two kinds of neural network controllers for certain input styles, e.g., state feedback and error feedback. With error feedback, neural network controllers learn the slopes or the gains with respect to the error feedback, producing an error driven adaptive control systems. The results demonstrate that the two kinds of control scheme can be combined to realize their individual advantages. Testing with disturbances added to the plant shows good tracking and adaptation with the integrated neural control architecture. 1 INTRODUCTION Neural networks are used for control systems because of their capability to approximate nonlinear system dynamics. Most neural network control architectures originate from work presented by Narendra[I), Psaltis[2) and Lightbody[3) . In these architectures, an identification neural network is trained to function as a model for the plant. Based on the neural network identification model, a neural network controller is trained by backpropagating the error through the identification network. After training, the identification network is replaced by the real plant. As is illustrated in Figure 1, the controller receives external inputs as well as plant state feedback inputs. Training procedures are employed such that the networks approximate feed forward control surfaces that are functions of external inputs and state feedbacks of the plant (or the identification network during training). It is worth noting that in this architecture, the error between the plant output and the desired output of the reference model is not fed back to the controller, after the training phase. In other words, this error information is ignored when the neural network applies its control. It is well known in control theory that the error feedback plays a significant role in adaptation. Therefore, when model uncertainty or noise/disturbances are present, a feed forward neural network controller with only state feedback will not adaptively update the control signal. On line training for the neural controller has been proposed to obtaip adaptive ability[I)[3). However, the stability for the on line training of the neural network controller is unresolved[1][4] . In this study, an additional nonlinear recurrent network is combined with the feed forward neural network controller to form an adaptive controller. This added neural network uses feedback error between the reference model output and the plant output as an input In addition, the system's external Liu Ke, Robert L. Tokar, Brian D. McVey 1032 inputs and the plant states are also input to the feedback network. This architecture is used in the control community, but not with neural network components. The approach differs from a conventional error feedback controller, such as a gain scheduled PID controller, in that the neural network error feedback controller implements a continuous nonlinear gain scheduled hypersurface, and after training, adaptive model reference control for nonlinear dynamic systems is achieved without further parameter computation. The approach is tested on well-known nonlinear control problems in the neural network literature, and good results are obtained. 2 NEURAL NETWORK CONTROL In this section, several different neural network control architectures are presented. In these structures, identification neural networks, viewed as accurate models for real plants, are used. 2.1 NEURAL NETWORK FEED FORWARD CONTROL The neural network controllers are trained by backpropagation of errors through a well trained neural identification network. In this architecture, the state variable yet) of the system is sent back to the neural network, and the external input x(t) also is input to the network. With these inputs, the neural network estabJishes a feed forward mapping from the external input x(t) to the control signal u(t). This control mapping is expressed as a function of the external input x(t) and the plant state yet) : u(t)==j(x(t), yet?~ (1) where x(t)=[x(t), x(t-l), .. J, andy(t)=[y(t), y(t-l), ..Y. This neural network control architecture is denoted in this study as feed forward neural control even though it includes state feedback. Neural control with error feedback is denoted as feedback neural control. x(t) -..:...r-----~Ref. x(t) -~----~Ref. Modelf-------, Modell-----.., e(t+ 1) Control NN y(t+ 1) Figure I Neural Network Control Architecture. ID NN represents the identification network. Ref. Model means reference model, and NN means neural network. u(t) ,-------, f----+-.. y(t+ 1) Figure 2 Neural Network Feedback Control Architecture During the training phases, based on the assumption that the neural identification network provides a model for the plant, the gradient information needed for error backpropagation is obtained by calculating the Jacobian of the identification network. The following equation describes this process for the control architecture shown in Figure I. If the cost function is defined as E, then the gradient of the cost function with respect to weight w of the neural controller is An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems a E a E au (a E au aE Ja Yt-l w + a u a Yt-l +a Yt-l --a;- a: = a; a 1033 (2) where u is tbe control signal and YI-1 is tbe plant feedback state. After tbe training stage, tbe neural network supplies a control law. Because neural networks have the ability to approximate any arbitrary nonlinear functions[5], a feed forward neural network can build a nonlinear controller, which is crucial to tbe use of tbe neural network in control engineering. Also, since all tbe parameters of the neural network identification model and tbe neural network controller are obtained from learning through samples, matbematically untraceable features of tbe plant can be extracted from tbe samples and imbedded into tbe control system. However, because tbe feed forward controller has no error feedback, tbe controller can not adapt to tbe disturbances occurring in tbe plant or tbe reference model. This problem is of substantial importance in tbe context of adaptive control. In tbe next subsection, error feedback between tbe reference models and tbe plant outputs is introduced into neural network controllers for adaptation. 2.2 NEURAL ADAPTIVE CONTROL WITH ERROR FEEDBACK It is known that feedback errors from the system are important for adaptation. Due to the flexibility of the neural network architecture, the error between the reference model and the plant can be sent back to the controller as an extra input. In such an architecture, neural networks become nonlinear gain scheduled controllers with smooth continuous gains. Figure 2 shows the architecture for the feedback neural control. With tbis architecture, tbe neural network control surface is not tbe fixed mapping from tbe x(t) to u(t) for each state y(t), but instead it learns tbe slope or tbe gain referring to tbe feedback error e(t) for control. This gain is a continuous nonlinear function of tbe external input x(t) and tbe state feedback yet). Figure 3 shows tbe recurrent network architecture of tbe feedback neural controller. The output node needs to be recurrent because tbe output witbout tbe recurrent link from tbe neural controller is only a correction to tbe old control signal, and tbe new control signal should be tbe combination of old control signal and tbe correction. The otber nodes of tbe network can be feed forward or recurrent. If we denote tbe weight for tbe output node's recurrent link as w., tben tbe output from tbe recurrent link is w.u(t-l). The following equation describes the feedback network. u(t) = wbu(t-I )+j(X(t), y(t), e(t? (3) where j(.) is a nonlinear function established by tbe network for which tbe recurrent link output is not included and e(t)=[e(t), e(t-I), ... f To compare tbe control gain expression with conventional control theory, consider tbe Taylor series expansion of tbe network forward mappingj(.), equation (3) becomes u(t) = w.u(t-l) +!'(x(t). yet?~ e(t)+ j"(x(t), yet?~ e 2(t)+... where f'(x(t), y(t?=[ i1j(x(t), y(t), e(t?/ae(t), aJ!:x(t), y(t), e(t?Ii1e(t-I), ... ]. ignored and gO representsf'O, we get u(t) =wbu(t-I)+ g(x(t), yet?~ e(t) (4) If high order terms are (5) 1034 Liu Ke, Robert L. Tokar, Brian D. McVey which is a gain scheduled controller and the gain is the function of external input x(/) and the plant state It is clear that when w.=l.O, g(.) is a constant vector and e(/)=[e(t), e(t-l), e(t-2)]T, the feedback neural network controller degenerates to a discrete PID controUer. Because the neural network can approximate arbitrary nonlinear functions through learning, the neural network feedback controller can generate a nonlinear continuous gain hypersurface. y(/). Ref. Model Figure 3 Feedback Neural Network Controller Figure 4 Integrated NN Control Architeture. In the training process, error backpropagating through the identification network is used. The process is similar to the training of a feed forward neural controller, but the resulting control surface is completely different due to the different inputs. After training, the neural network is able to provide a nonlinear control law, that is, the desired model following response can be obtained with fixed controller parameters for nonlinear dynamic systems. Traditionally, the control of the nonlinear plant is derived from continuous computing of the controller gains. This feedback controller is error driven . As long as an error exists, the control signal is updated according to the error and the gain. This kind of neural controller is an adaptive controller in principle. 2.3 INTEGRATED NEURAL NETWORK CONTROLLER The characteristics of feed forward and error feedback neural control networks are described in the previous subsections. In this section. the two controllers are combined. Figure 4 shows the architecture. In this architecture, we include both feed forward and feedback neural network controllers. The control signal is the combination from these two networks' outputs. In the training stage, it is our experience that the feed forward network should be trained first. The feedback network is not included while training the feed forward network. After training the feed forward controller, the error feedback network is trained with the feed forward network, but the feed forward networks' weights are unchanged. Backpropagating the error through the identification network is applied for the training of both networks. When training the feedback control network, the feed forward calculation is =ujt)+u/b(t). (6) y(t+ 1) = P(x(t), y(t), u(t?, (7) u(t) where uj/) is the output from the feed forward controller network and u,..(t) is the output from the feedback controller network, P(.) is the identification mapping. 1035 An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems 3 CONTROL ON EXAMPLE PROBLEMS In this section, the control architecture described above is applied to a well-known problem from the literature[I). The plants and the reference model of the sample problems are described by difference equations y(t + 1) plant: = yet) 2 + (u(t) -1.O)u(t)(u(t) + 1.0) 1.0+ Y (t) (II) y(t + 1) = 0.6y(t) + u(t) reference model: (12) This is a nonlinear time varying dynamic system with no analytical inverse. 3.1 FEED FORWARD CONTROL A feed forward neural network is trained to control the system to follow the reference model. The plant state yet) and external inputx(t) are fed to the controller. During the training, the x(t) is randomly generated. After training, the controller generates a control signal u(t) such that the plant can follow the reference model output. Figure 5 shows the testing result of the reference model output and the controlled plant output. The input function is x(t)=sin(21ttf25)+sin(21tt/1O). The controller network architecture is (2, 20, 1). 4 2 OJ 0 c Q) 2 2 ? (j 10- OJ ~ '" 1J ::J 0 '" 0 c ... 1 ~ c 0 0 () 0 u -2 20 40 60 80 ~0 -1 ...2 100 Figure 5 Tracking Result From the Feed Forward NN. Output of reference (solid line) and plant (dash line). Figure 6 Feed Forward Control Surface The output surface of the controller network is shown in Figure 6. By examining the controller output surface, we can see that the neural network builds a feed forward mapping from x(t) to u(t). This feed forward mapping is also a function of the plant state yet). Under each state, the neural network controller accepts input x(t) to produce control signal u(t) such that the plant follows the reference model reasonably well. In Figure 6, the x axis is the external input x(t) and the y axis is the plant feedback output yet) . The z axis represents the control surface. The feed forward controller laCks the ability to adapt to plant uncertainty, noise or changes in the reference model. As an example, we apply this feed forward controller to the disturbed plant with a bias 0.5 added to the original plant. The tracking result is shown in Figure 7. With this slight bias, the plant does not follow the reference model. Clearly, the feed forward controller has no adaptive ability to this model bias. 1036 Liu Ke, Robert L. Tokar, Brian D. McVey 3.2 FEEDBACK CONTROL FtrSt, we compare the neural network feedback controller with fixed gain PID controllers. For many nonlinear systems, the fixed gain PID controllers will give poor tracking and continuous adaptation of the controller parameters is needed. The neural network approach offers an alternative control approach for nonlinear systems. Through the training, control gains, imbedded in the neural network, are established as a continuous function of system external inputs x(t) and plant states yet). The sample problem in the above section is now employed to describe how the neural network creates a nonlinear control gain surface with error feedback and additional inputs. First, we show one simple case of neural adaptive feedback controller. This controller can only adapt to the system nonlinearity with a fixed linear input pattern. The reason to show this simple adaptation case first is that its control gain surface can be illustrated graphically. Figure 8 illustrates, for the system in equations (11) and (12) that a fixed gain PI controller fails to track the reference model, for even one fixed linear input pattern x(t)=0.2t-2.5, because the plant nonlinearity. Figure 9 illustrates the result from a recurrent neural network with feedback error e(t) and x(t) as inputs. The neural network is trained by backpropagation error through the identification network. Compared to the flXed gain PI controller, the neural network improves the tracking ability significantly. ., .,.c ., 4 >. 0 6 '!! -.::I C 0 >. , OJ 0 <J C ~ 2 .2 ~ I I 3 0 >. '0 c -3 0 -2 >. 20 40 60 80 100 t Figure 7 Tracking Result for Shifted Plant, plant output (dash line) and reference output (solid line). o 5 10 15 20 25 30 35 t Figure 8 Reference Model Output (solid line) and PID Controlled Plant Output (dashed line) The control surface of the updating output fl.) is shown in Figure 10, which is the output from the neural network controller without recurrent link (see equation (3?. We plot the surface of the updating output from the controller with respect to input x(t) and error feed back input e(t). The gain of the controller is equivalent to the updating output from the network when error=l.O. As shown in the figure, the gain in the neighborhood about x(t)=O changes largely according to the direction of changes in the plant in the corresponding region. The updating surface for a PID controller is a plane. The neural network implements a nonlinear continuous control gain surface. For a more complicated case, we addx(t-I) as another input to the neural network as well as e(t-l), and train by error backpropagation through the identification network. These two inputs, x(t) and x(t-I) add difference information to the network. The network can adapt to not only different operating regions indicated by x(t), but also different input patterns. Figure 11 shows the tracking results with two different input patterns. In Figure II (a), input pattern is x(t)=4.0sin(tI4.0). In Figure 11 (b) input pattern is x(t)=sin(21t1!25)+sin(21t111O) . An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems 1037 6 Q) u c ~ 2 ~ 3 0 , , >. -0. c 0 >. I I -3 -6 o 5 10 15 20 25 30 35 t Figure 9 Reference Model Output (solid line) and Neural Network Controled Output (dashed line) Figure 10 Feedback Neural Controller Updating Surface .,u OJ u c ~ ~ ~ c ~ OJ l' 0 -2 "c:0 - 4 >. -6 -8 -10 >. >. " c 5 4 3 2 1 0 -1 -2 0 -3 >. - 4 -5 -6 20 0 40 60 80 10C (a) 0 20 40 60 80 100 (b) Figure 11 Output of the Reference Model (solid line) and the Plant (dash line) 3.3 INTEGRA TED NEURAL CONTROLLER As shown in the above section, when only error feedback neural controller is used, the control result is not very accurate. Now we combine feed forward and feedback control to realize good tracking and adaptation. Figure 12 shows the control result from the integrated controller when the plant is shifted O.S. Compared to only feed forward control (Figure 7), the integrated controller has much better adaptation to the shifted plant. When the plant changes, adding an extra feed back controller can avoid on-line training of feed forward network which may induce potential instability, and the adaptation is achieved. The output from the feedback network controller is driven by the error between the reference model and the plant. 4 DISCUSSIONS We have emphasized in the above sections that a feed forward controller with only state feedback does not adapt when model uncertainties or noise/disturbance are present. The presence of a feed back controller can make the on line training of the feed forward network unnecessary, thus avoiding potential instability. The main reason for the instability of on-line training is the incompleteness of sample sets, which is referred to as a lack of persistent excitation in control theory[6]. First, it leads to an inaccurate identification network. Training with this network can result in an unstable controller. Second, it makes the training of controller away from global representation . With an error feedback adaptive network, the output from the feedback network controller is driven by the error between the reference model and the plant. In the simplest case when all the activity functions are linear and only the feedback errors are inputs, this kind of neural network is equivalent to a PID controller. However, Liu Ke, Robert L Tokar, Brian D. McVey 1038 beyond the scope of PID controllers, the neural networks are capable to approximating nonlinear time variant control gain surfaces corresponding to different operating regions. Also, unlike a PID controller, the coefficients for the neural adaptive controller are obtained through a training procedure. 4 Q) u c 1:' Q) 0; ... >, "0 C 0 >, o 20 40 60 80 100 Figure 12 Integrated Network Controller Tracking Result for Shifted Plant. Plant Output (dash line) and Reference Output (solid line). The error feedback network behaves as a gain scheduling controller. It has rise time, overshoot consideration and delay problem. Feed forward control can compensate for these problems to some degree. For example, the feed forward network can perform a nonlinear mapping with designed time delay. Therefore with the feed forward network, the delay problem maybe overcame significantly. Also the feed forward controller can help to reduce rise time compare to use only feedback controller. With the feed forward network, the feedback network controller can have much smaller gains compared to using a feedback network alone. This increases the noise rejection ability. Also this reduces the overshoot as well as settle time. The neural network control architecture offers an alternative to the conventional approach. It gives a generic model for the broadest class of systems considered in control theory . However this model needs to be configured depending on the details of the control problem. With different inputs, the neural network controllers establish different internal hyperstates. When plant states are fed back to the network, a feed forward mapping is established as a function of the plant states by the neural network controller. When the errors between the reference model and the plant are used as the error feedback inputs to a dynamic neural network controller, the network functions as an associative memory nonlinear gain scheduled controller. The above two kinds of neural controller can be combined and complemented to achieve accurate tracking and adaptation. References [1] Kumpati S. Narendra and Kannan Parthasarathy. "Gradient Methods for the Optimization of DynamiCal Systems Containing Neural Networks," IEEE Trans. Neural Networks. vol. 2. pp252-262 Mar. 1991 [2] Psaltis. D .? Sideris. A. and Yamamura. A., "Neural controllers." Proc. of 1st International Conference on Neural Networks. Vol. 4. pp551-558. San Diego. CA. 1987 [3) G. lightbOdy. Q. H. Wu and G. W. Irwin. "Control applications for feed forward networks." Chapter 4. Neural Networks for Control and Systems. Edited by K.warwich, G. W. Irwin and K. J. Hunt 1992 [4) R. Abikowski and P. 1. Gawthrop. "A survey of neural networks for control" Chapter 3. NeUral Networks for Control and Systems. ISBN 0-86341-279-3. Edited by K.warwich. G. W. Irwin and K. 1. Hunt 1992 [5] John Hertz. Anders Krogh and Richard G. Palmer. "Introduction to the Theory of Neural Computation." [6J Thomas Miller. RiChard S. Sutton and Paul 1. Werbos. "Neural Networks for Control" 

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Capacity and Information Efficiency of a Brain-like Associative Net Bruce Graham and David Willshaw Centre for Cognitive Science, University of Edinburgh 2 Buccleuch Place, Edinburgh, EH8 9LW, UK Email: bruce@cns.ed.ac.uk&david@cns.ed.ac.uk Abstract We have determined the capacity and information efficiency of an associative net configured in a brain-like way with partial connectivity and noisy input cues. Recall theory was used to calculate the capacity when pattern recall is achieved using a winners-takeall strategy. Transforming the dendritic sum according to input activity and unit usage can greatly increase the capacity of the associative net under these conditions. For moderately sparse patterns, maximum information efficiency is achieved with very low connectivity levels (~ 10%). This corresponds to the level of connectivity commonly seen in the brain and invites speculation that the brain is connected in the most information efficient way. 1 INTRODUCTION Standard network associative memories become more plausible as models of associative memory in the brain if they incorporate (1) partial connectivity, (2) sparse activity and (3) recall from noisy cues. In this paper we consider the capacity of a binary associative net (Willshaw, Buneman, & Longuet-Higgins, 1969; Willshaw, 1971; Buckingham, 1991) containing these features. While the associative net is a very simple model of associative memory, its behaviour as a storage device is not trivial and yet it is tractable to theoretical analysis. We are able to calculate 514 Bruce Graham, David Willshaw the capacity of the net in different configurations and with different pattern recall strategies. Here we consider the capacity as a function of connectivity level when winners-take-all recall is used. The associative net is an heteroassociative memory in which pairs of binary patterns are stored by altering the connection weights between input and output units via a Hebbian learning rule. After pattern storage, an output pattern is recalled by presenting a previously stored input pattern on the input units. Which output units become active during recall is determined by applying a threshold of activation to measurements that each output unit makes of the input cue pattern. The most commonly used measurement is the weighted sum of the inputs, or dendritic sum. Amongst the simpler thresholding strategies is the winners-take-all (WTA) approach, which chooses the required number of output units with the highest dendritic sums to be active. This works well when the net is fully connected (each input unit is connected to every output unit), and input cues are noise-free. However, recall performance deteriorates rapidly if the net is partially connected (each input unit is connected to only some of the output units) and cues are noisy. Marr (1971) recognised that when an associative net is only partially connected, another useful measurement for threshold setting is the total input activity (sum of the inputs, regardless of the connection weights). The ratio of the dendritic sum to the input activity can be a better discriminator of which output units should be active than the dendritic sum alone. Buckingham and Willshaw (1993) showed that differences in unit usage (the number of patterns in which an output unit is active during storage) causes variations in the dendritic sums that makes accurate recall difficult when the input cues are noisy. They incorporated both input activity and unit usage measurements into a recall strategy that minimised the number of errors in the output pattern by setting the activity threshold on a unit by unit basis. This is a rather more complex threshold setting strategy than a simple winners-take-all. We have previously demonstrated via computer simulations (Graham & Wills haw , 1994) that the WTA threshold strategy can achieve the same recall performance as this minimisation approach if the dendritic sums are transformed by certain functions of the input activity and unit usage before a threshold is applied. Here we calculate the capacity of the associative net when WTA recall is used with three different functions of the dendritic sums: (1) pure dendritic sums, (2) modified by input activity and (3) modified by input activity and unit usage. The results show that up to four times the capacity can be obtained by transforming the dendritic sums by a function of both input activity and unit usage. This increase in capacity was obtained without a loss of information efficiency. For the moderately sparse patterns used, WTA recall is most information efficient at low levels of connectivity (~ 10%), as is the minimisation approach to threshold setting (Buckingham, 1991). This connectivity range is similar to that commonly seen in the brain. Capacity and Infonnation Efficiency of a Brain-Like Associative Net 2 515 NOTATION AND OPERATION The associative net consists of N B binary output units each connected to a proportion Z of the N A binary input units. Pairs of binary patterns are stored in the net. Input and output patterns contain MA and MB active units, respectively (activity level a = M / N ? 1). All connection weights start at zero. On presentation to the net of a pattern pair for storage, the connection weight between an active input unit and an active output unit is set to 1. During recall an input cue pattern is presented on the input units. The input cue is a noisy version of a previously stored input pattern in which a fraction, s, of the MA active units do not come from the stored pattern. A thresholding strategy is applied to the output units to determine which of them should be active. Those that should be active in response to the input cue will be called high units, and those that should be inactive will be called low units. We consider winners-take-all (WTA) thresholding strategies that choose to be active the MB output units with the highest values of three functions of the dendritic sum, d, the input activity, a, and the unit usage, r. These functions are listed in Table 1. The normalised strategy deals with partial connectivity. The transformed strategy reduces variations in the dendritic sums due to differences in unit usage. This function minimises the variance of the low unit dendritic sums with respect to the unit usage (Graham & Willshaw, 1994). Table 1: WTA Strategies WTA Strategy Basic Normalised Transformed 3 FUnction d d' = d/a d* = 1 - (1 - d/a)l/r RECALL THEORY The capacity of the associative net is defined to be the number of pattern pairs that can be stored before there is one bit in error in a recalled output pattern. This cannot be calculated analytically for the net configuration under study. However, it can be determined numerically for the WTA recall strategy by calculating the recall response for different numbers of stored patterns, R, until the minimum value of R is found for which a recall error occurs. The WTA recall response can be calculated theoretically using expressions for the distributions of the dendritic sums of low and high output units. The probability that the dendritic sum of a low or high output unit should have a particular value x is, respectively (Buckingham & Willshaw, 1993; Buckingham, 1991) P(d, = x) = t. (~ ) "'8(1- "B)R-, ( "!,A ) (Zp[rJ)"(I- Zp[rJ)MA-" (I) 516 Bruce Graham. David Willshaw P(dh = x) = ~ ( ~ ) a;'(l-aB)R-" ( ~A ) (ZI'[r +1])"(1- ZI'[r+1])MA-" (2) where p[r] and /L[r] are the probabilities that an arbitrarily selected active input is on a connection with weight 1. For a low unit, p[r] = 1- (1- OA)r. For a high unit a good approximation for /L is /L[r + 1] ~ g + sp[r] = 1 - s(l - OAY where g and s are the probabilities that a particular active input in the cue pattern is genuine (belongs to the stored pattern) or spurious, respectively (g + s = 1) (Buckingham & Willshaw, 1993). The basic WTA response is calculated using these distributions by finding the threshold, T, that gives (3) The number of false positive and false negative errors of the response is given by (4) The actual distributions of the normalised dendritic sums are the distributions of dja. For the purposes of calculating the normalised WTA response, it is possible to use the basic distributions for the situation where every unit has the mean input activity, am = MAZ. In this case the low and high unit distributions are approximately P(til P(d'h = x) = t. (~ ) ,,;'(1- "B)R-" ( "; ) (p[r])"(l- p[r])"m-" = x) = ~ ( ~ ) aB(1- "B )R-" ( (5) a; ) (I'[r + 1])" (l-l'[r +1])"m -" (6) Due to the nonlinear transformation used, it is not possible to calculate the transformed distributions as simple sums of binomials, so the following approach is used to generate the transformed WTA response. For a given transformed threshold, T*, and for each possible value of unit usage, r, an equivalent normalised threshold is calculated via T'[r] = am (l - (1 - T*Y) (7) The transformed cumulative probabilities can then be calculated from the normalised distributions: P(dj :2: TO) = P( di. :2: T') t. (~ ) aB(1 - "B )R-" P( til :2: T'[r]) = ~ ( ~ ) ,,;'(1 - "B )R-" P(d' :2: T'[r + 1]) (8) (9) The normalised and transformed WTA responses are calculated in the same manner as the basic response, using the appropriate probability distributions. Capacity and Information Efficiency of a Brain-Like Associative Net (a) 0% noise 5000 g ~ ?u --B - - N, T (b) 40% noise ... ..................... ~ .-~ ...-., 3000 --B ?-N 2500 ---T ./" 4000 517 2000 3000 1500 ca Co ca 2000 U 1000 1000 500 O~---...L 20 40 60 80 Connectivity (%) 100 0 ___--'---'-.....L-....o...-..I.-..........J 20 40 60 80 100 Connectivity (%) Figure 1: Capacity Versus Connectivity 4 RESULTS Extensive simulations were previously carried out of WTA recall from a large associative net with the following specifications (Graham & Willshaw, 1994): NA = 48000, MA = 1440, NB = 6144, MB = 180. Agreement between the simulations and the theoretical recall described above is extremely good, indicating that the approximations used in the theory are valid. Here we use the theoretical recall to calculate capacity results for this large associative net that are not easily obtained via simulations. All the results shown have been generated using the theory described in the previous section. Figure 1 shows the capacity as a function of connectivity for the different WTA strategies when there is no noise in the input cue, or 40% noise in the cue (legend: B = basic WTA, N = normalised WTA, T = transformed WTA; for clarity, individual data points are omitted). With no noise in the cue the normalised and transformed methods perform identically, so only the normalised results are shown. Figure l(a) highlights the effectiveness of normalising the dendritic sums against input activity when the net is partially connected. Figure 1(b) shows the effect of noise on capacity. The capacity of each recall strategy at a given connectivity level is much reduced compared to the noise-free case. However, for connectivities greater than 10% the capacity of the transformed WTA is now much greater than either the normalised or basic WTA. The relative capacities of the different strategies are shown in Figure 2 (legend: NIB = ratio of normalised to basic capacity, T I B = ratio of transformed to basic, TIN = ratio of transformed to normalised). In the noise-free case (Figure 2(a)), at low levels of connectivity the relative capacity is distorted because the basic capacity 518 Bruce Graham, David Willshaw drops to near zero, so that even low normalised capacities are relatively very large. For most connectivity levels (10-90%) the normalised WTA provides 2-4 times the capacity of the basic WTA. In the noisy case (Figure 2(b)), the normalised capacity is only up to 1.5 times the basic capacity over this range of connectivities. The transformed WTA, however, provides 3 to nearly 4 times the basic capacity and 2.5 to nearly 3 times the normalised capacity for connectivities greater than 10%. The capacities can be interpreted in information theoretic terms by considering the information efficiency of the net. This is the ratio of the amount of information that can be retrieved from the net to the number of bits of storage available and is given by "'0 = Ro10jZNANB' where Ro is the capacity, 10 is the amount of information contained in an output pattern and Z NANB is the number of weights, or bits of storage required (Willshaw et al., 1969; Buckingham & Willshaw, 1992). Information efficiency as a function of connectivity is shown in Figure 3. There is a distinct peak in information efficiency for each of the recall strategies at some low level of connectivity. The peak information efficiencies and the efficiencies at full connectivity are summarised in Table 2. The greatest contrast between full and partial connectivity is seen with the normalised WTA and noise-free cues. At 1% connectivity the normalised WTA is nearly 14 times more efficient than at full connectivity. In absolute terms, however, the normalised capacity is only 694 at 1% connectivity, compared with 5122 at full connectivity. The peak efficiency of 53% obtained by the normalised WTA is approaching the theoretically approximate maximum of 69% for a fully connected net (Willshaw et al., 1969). 5 DISCUSSION Previous simulations (Graham & Willshaw, 1994) have shown that, when the input cues are noisy, the recall performance of the winners-take-all thresholding strategy applied to the partially connected associative net is greatly improved if the dendritic sums of the output units are transformed by functions of input activity and unit usage. We have confirmed and extended these results here by calculating the theoretical capacity of the associative net as a function of connectivity. For the moderately sparse patterns used, all of the recall strategies are most information efficient at very low levels of connectivity (~ 10%). However, the optimum connectivity level is dependent on the pattern coding rate. Extending the analysis of Willshaw et al. (1969) to a partially connected net using normalised WTA recall yields that maximum information efficiency is obtained when ZMA = log2(NB). So for input coding rates higher than log2(NB), a partially connected net is most information efficient. For the input coding rate used here, this relationship gives an optimum connectivity level of 0.87%, very close to the 1% obtained from the recall theory. Comparing the peak efficiencies across the different strategies for the noisy cue case, the normalised WTA is about twice as efficient as the basic WTA, while the transformed WTA is three times as efficient. This comparison does not include the Capacity and Information Efficiency of a Brain-Like Associative Net (a) 0% noise (b) 40% noise 12 4 - - NIB, TIB 10 .~ 0 III a. I ~ ;; , 3 ( 8 III Q) ~------ 6 4 a: 0 40 60 20 80 Connectivity (%) 0 , I I \1 ' ...... - .............. 1;?- ,, - -___ I, - - NIB II 1 -_ ' --- T/B I TIN \ 100 ..... , , ,..- ...... -------- -~ }{' ... 2 ., 2 "" I III u 519 0 20 40 60 80 Connectivity (%) 0 100 Figure 2: Relative Capacity Versus Connectivity (a) 0% noise (b) 40% noise 60 - 6 ~ 50 g Q) 40 ffic: 30 .(3 ia 20 E .... 0 1: 10 o 4 \ .Q ~ r-o 3 2 ----"::7 20 40 60 80 Connectivity (%) ,,,' 5 --B ._._- N, T >. ", 100 0 --B --N ---T '\ '\ '\ '\ , f.l .\ I '" " ., ;'\ ., ' ...'-- " ............ - ... --------- L-'--'........- - - L _ - - L _ - - - L . - - ' - - - I 0 20 40 60 80 Connectivity (%) Figure 3: Information Efficiency Versus Connectivity Table 2: Information Efficiency WTA Strategy Basic Normalised Transformed at Peak (%) 6.1 53.3 53.3 TJo 0% Noise Z at TJo at Peak Z=1 (%) (%) 4 3.9 1 3.9 1 3.9 at Peak T]o 40% Noise Z at TJo at Peak Z=1 (%) (%) (%) 2.0 3.6 5.7 7 5 10 0.8 0.8 2.3 100 520 Bruce Graham, David Willshaw cost of storing input activity and unit usage information. If one bit of storage per connection is required for the input activity, and another bit for the unit usage, then the information efficiency of the normalised WTA is halved, and the information efficiency of the transformed WTA is reduced by two thirds. This results in all the strategies having about the same peak efficiency. However, the absolute capacities of the different strategies at their peak efficiencies are 183,237 and 741 for the basic, normalised and transformed WTA, respectively. So, at the same level of efficiency, the transformed WTA delivers four times the capacity of the basic WTA. In conclusion, numerical calculations of the capacity of the associative net show that it is most information efficient at a very low level of connectivity when moderately sparse patterns are stored. Including input activity and unit usage information into the recall calculations results in a four-fold increase in storage capacity without loss of efficiency. Acknowledgements To the Medical Research Council for financial support under Programme Grant PG 9119632 References Buckingham, J., & Willshaw, D. (1992). Performance characteristics of the associative net. Network, 8, 407-414. Buckingham, J., & Willshaw, D. (1993). On setting unit thresholds in an incompletely connected associative net. Network, 4, 441-459. Buckingham, J. (1991). Delicate nets, faint recollections: a study of partially connected associative network memories. Ph.D. thesis, University of Edinburgh. Graham, B., & Willshaw, D. (1994). Improving recall from an associative memory. Bioi. Cybem., in press. Marr, D. (1971). Simple memory: a theory for archicortex. Phil. Trans. Roy. Soc. Lond. B, 262, 23-81. Shepherd, G. (Ed.). (1990). The Synaptic Organization of the Brain (Third edition). Oxford University Press, New York, Oxford. Willshaw, D. (1971). Models of distributed associative memory. Ph.D. thesis, University of Edinburgh. Willshaw, D., Buneman, 0., & Longuet-Higgins, H. (1969). Non-holographic associative memory. Nature, 222, 960-962. 

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Adaptive Elastic Input Field for Recognition Improvement Minoru Asogawa C&C Research Laboratories, NEe Miyamae, Miyazaki, Kawasaki Kanagawa 213 Japan asogawa~csl.cl.nec.co.jp Abstract For machines to perform classification tasks, such as speech and character recognition, appropriately handling deformed patterns is a key to achieving high performance. The authors presents a new type of classification system, an Adaptive Input Field Neural Network (AIFNN), which includes a simple pre-trained neural network and an elastic input field attached to an input layer. By using an iterative method, AIFNN can determine an optimal affine translation for an elastic input field to compensate for the original deformations. The convergence of the AIFNN algorithm is shown. AIFNN is applied for handwritten numerals recognition. Consequently, 10.83% of originally misclassified patterns are correctly categorized and total performance is improved, without modifying the neural network. 1 Introduction For machines to accomplish classification tasks, such as speech and character recognition, appropriately handling deformed patterns is a key to achieving high performance [Simard 92] [Simard 93] [Hinton 92] [Barnard 91]. The number of reasonable deformations of patterns is enormous, since they can be either linear translations (an affine translation or a time shifting) or non-linear deformations (a set of combinations of partial translations), or both. Although a simple neural network (e.g. a 3-layered neural network) is able to adapt Minoru Asogawa 1102 ~'JU""'Y'CeUs j-th Input Cell /!/"....-I,ncul Field .;_-;)oun~e Image Figure 1: AIFNN ne~~----~----~~-------------- ~----~--------------------~s Position Figure 2: Delta Force non-linear deformations and to discriminate noises, it is still necessary to have additional methods or data to appropriately process deformations. This paper presents a new type of classification system, an Adaptive Input Field Neural Network (AIFNN), which includes a simple pre-trained neural network and an elastic input field attached to an input layer . The neural network is applied to non-linear deformation compensations and the elastic input field to linear deformations . The AIFNN algorithm can determine an optimal affine translation for compensating for the original patterns' deformations, which are misclassified by the pre-trained neural network. As the result, those misclassified patterns are correctly classified and the final classification performance is improved, compared to that for the original neural network, without modifying the neural network. 2 Adaptive Input Field Neural Network (AIFNN) AIFNN includes a pre-trained neural network and an elastic input field attached to an input layer (Fig. 1) . The elastic input field contains receptors sampling input patterns at each location . Each receptor connects to a cell in the input layer. Each receptor links to its adjacent receptors with an elastic constraint and can move over Adaptive Elastic Input Field for Recognition Improvement 1103 the input pattern independently, as long as its relative elastic constraint is satisfied. The affine translation of the whole receptor (e.g. a shift, rotation, scale and slant translation) satisfies an elastic constraint, since a constraint violation is induced by the receptors' relative locations. 1 Partial deformations are also allowed with a little constraint violation . This feature of the elastic constraint is similar to that of the Elastic Net method [Durbin 87], which can solve NP-hard problems. Although this elastic net method is directly applicable to the template matching method, the performance is highly dependent on the template selection. Therefore, an elaborated feature space for non-linear deformations is mandatory [Hinton 92]. AIFNN utilizes something like an elastic net constraint, but does not require any prominent templates. The AIFNN algorithm is a repeated sequence of a bottom-up process (calculating a guess and comparing with the presumption) and a top-down process (modifying receptor's location to decrease the error and to satisfy the input field constraints). For applying AIFNN as a classifier, a parallel search is performed; all categories are chosen as presumption categories and the AIFNN algorithm is executed. After hundreds of repetitions, an L score is calculated, which is the sum of the error and the constraint violation in the elastic input field. A category which produces the lowest L score is chosen as a plausible category. In Section 3, it is proved that all receptors will settle to an equilibrium state. In the following sections, details about the bottom-up and top-down processes are described. Bottom-Up Process: When a novel pattern is presented, each receptor samples activation corresponding to a pattern intensity at each location. Each receptor activation is directly transmitted to a corresponding neural network input cell. Those input values are forwarded through a pre-trained neural network and an output guess is obtained. This guess is compared to the presumption category, and the negative of this error is defined as the presumption certainty. 2 For example, using the mean squared error criterion, the error ED is defined as follows; ED = ~ I)dk - Ok?, (1) k where Ok is the output value, and dk is the desired value determined by the presumption category. The presumption certainty is defined as _ED. Top-Down Process: To minimize the error and to maximize the presumption certainty, each receptor modifies the activation by moving its location over the input pattern. The new location for each receptor is determined by two elements; a direction which yields less error and a direction which satisfies the input field elastic constraint. The former element is called a Delta Force, since it relates to a delta value of an input layer cell. The latter element is named an Address Force. Each receptor moves to 1 In previous papers, [Asogawa 90] and [Asogawa 91], a shift and rotation translation was taken into account. In those models, a scale and slant translation violated the elastic constraint. 2 Although another category coding schema is also possible, for simplicity, it is presumed that each output cell corresponds to one certain category. 1104 Minoru Asogawa a new location, which is determined by a sum of those two forces. The sum force is called the Combined Force. In the next two sections, details about these forces are described. Delta Force: The Delta Force, which reduces ED by altering receptors' locations, is determined by two elements: a partial derivative for the input value to the error, and a local pattern gradient at each receptor location (Fig. 1). To decrease the error ED, the value divergence for the j-th cell is computed as, !lnet ? } D {)E D D == -a - = a 6? , onetj } (2) where aD is small positive number and 6j is a delta value for the j-th input cell and computed by the back-propagation [Yamada 91]. !lnetj and a local pattern g~adient \1,pj are utilized to calculate a Delta Force !lsf; a scalar value of !lsf is given as, D !lnetj l!lsj I = 1\1,pj I . Force !lsf is chosen as The direction of the Delta Consequently, !lsf is given as, D _ !lnetj \1,pj _ being parallel to that of \1,pj . 6j D !lSj - 1\1,pjll\1,pjl - (3) 1\1,pjI2 \1,pj . a (4) To avoid !lsf becoming infinity, when 1\1,pi I is almost equal to 0, a small constant c( is added to the denominator; therefore, !lsf is defined as, = t) D _ !lSj - a D 6j . 1\1,pj 12 + c\1,p} . (5) Address Force: If each receptor is moved iteratively following only the Delta Force, the error becomes its minimum. However, receptors may not satisfy the input field constraint and induce a large constraint violation EA. Here, EA is defined by a distance between a receptor's lattice S and a lattice which is derived by an affine translation from the original lattice. Therefore, EA is defined as follows; E 1 A N 2"d(S, S) = 2"1", L.,.-IlsiN - sill 2 i 1 0 (6) "2d(T(S ; t), S), where d(?,?) is a distance measure for two receptor's lattices. S is a current receptor lattice . SN is the receptor lattice given by the affine translation T (.) with parameters t and SO. SO is the original receptor lattice. Therefore, as long as the receptor's lattice can be driven by some affine translation, there is no constraint violation. The affine parameters t are estimated so as to minimize EA; {)EA -{)- = 0 ti for i = 1, ? ??,6. (7) Adaptive Elastic Input Field for Recognition Improvement 1105 Since EA is quadratic with respect to ti, computing ti is moderate. The Address Force for the j-th receptor ~st is defined as the partial derivative to EA with respect to the receptor's location Sj; A_ ~Sj = -(): Aa;:-' aEA (8) 1 where (}:A is a small positive constant. Combined Force: Here, all receptors are moved by a Combined Force a sum of the Delta Force ~sD and the Address Force ~sA. ~s, which is After one hundred iterations, all receptors are moved to the location which produces the minimum output error and the minimum constraint violation. Final states are evaluated with a new measurement score, which is the sum of the error and the constraint violation i.e. + L L= ED EA. EA; ED This L score is utilized to choose the correct category in a parallel search. In a parallel search, each category is temporarily chosen as a presumption and converged L scores are calculated. Those scores are compared and the category yielding the smallest L score is chosen as the correct category. This method fully exploits the features of AIFNN, but it requires a large amount of computation, which can fortunately be processed totally in parallel. In the following section, convergence of the AIFNN is shown. 3 Convergence Convergence is shown by proving that the L is a Lyapunov function. When the L is a Lyapunov function, all receptors converge to some locations after iterations. The necessary and sufficient conditions for a Lyapunov function are (1) L has a lower bound and (2) L monotonically decreases by applying the Combined Forces. (12 Lower Bound: E is the squared error at the output layer. Therefore, ED ~ O. EA is the constraint violation, which is defined with a distance between two lattices. Therefore, EA ~ O. Since the L is a sum of ED and EA, the existence of a lower bound for the L is proved. 0 (2) Monotonically Decrease: The derivative of the L is calculated to show that the L decreases monotonically. dL d d dt a:t+dT ED EA I:. { aE aS , D i A ~} + I: {aE dt . aS ~} dt , i ~{(a:: + aa~:) ~~i}, , (9) d Si IS . t he Comb'me d Force an d gIven . h were dt as, d s I? = __ dt ~sD + ~SA. (10) Minoru Asogawa lJ06 When a source image is smooth and 1'V?d is smaller than c, the following approximation is satisfied; (11) By using Eq. (11), the Delta Force is approximated as follows; 6.sD = aD 'V ?i 6i ~ _aD OED. 1\7?d 2 + c 'V ?i OSi (12) By using Eqs. (8) and (12), and by letting aD = a A , the L derivative is computed as follows; dL dt ...... OED _a A ' " ( Li OEA)2 as? + -as? ~ O. I (13) I With Eq. (13), it is proved that L decreases monotonically.D 4 Experiments and Results Hand-written numerals recognition is chosen as one of the applications of AIFNN, since performance improvement is shown by compensating for deformations [Simard 92] [Simard 93] [Hinton 92] . The numeral inputs are bi-Ievel images of 32x40. They are blurred with a 5x5 Gaussian kernel and resampled to 14x 18 pixel gray level images. To calculate an intensity and a local gradient between grids, bi-linear Lagrange interpolation is utilized. A neural network is 3 layered. The numbers of cells for the input layer, the hidden layer and the output layer are 252, 20 and 10, respectively. To obtain a simpler weight configuration, two techniques are utilized; a constant weight decay [Ishikawa 89] and a small constant addition to output function derivatives [Fahlman 88]. Training is repeated for 180 epochs with 2500 numerals, and tested with another 2500. Since image edges are almost blank, about 2400 connections between the input layer and the hidden layer are equal to 0; therefore, the number of parameters is reduced to 2870. In this experiment, a simple decision method is used; the maximum output cell is chosen as a guess and patterns are rejected when the error of the guess is greater than a threshold value . Naturally, a low threshold yields a low misclassification rate, but also yields a high rejection rate [Martin 92]. With the maximum threshold, the rates of rejection, correct classification and misclassification are 0.00%(0 patterns), 95.20%(2380 patterns) and 4.80%(120 patterns), respectively. For the 2500 numerals learning data, these rates are 0.00%(0 patterns), 99.40%(2485 patterns) and 0.60%(15 patterns). When a threshold is 0.001, the rates of rejection, correct classification and misclassification are 43.52%(1088 patterns), 56.40%(1410 patterns) and 0.08%(2 patterns), respectively. AIFNN is applied to these 1088 rejected patterns. and classifies 997 patterns correctly. Therefore, total performances for rejection, correct classification and misclassification become 0.00%(0 patterns), 95.72%(2393 patterns) and 4.28%(107 patterns), respectively. As the classification performance is improved; the number of Adaptive Elastic Input Field for Recognition Improvement 1107 misclassified patterns reduces from 120 to 107 without modifying the neural network. 10.83% of the originally misclassified patterns are correctly categorized. Fig. 3 shows an input field after one hundred iterations. ..... ....... ...... ...... ?????? . ????? ? 6 ? .. " ? III! ? 0.111 ? It . . . . ?atOl" ?? ?????? ????? ??? ...... '?????? .........?iiI". .....?. ? A .... )0 ,. , I:: I 18 ? ??? 6 it . " . .y............ Ilttput Activation I... ....... . ... 1. . 0123456789 In the figure on the left, receptors are located at each grid point in a gray lattice. The circle diameter corresponds to the pattern intensity at each receptor's location. The bottom right figure indicates the source image, and the top right figure indicates the neural network input. This image was initially misclassified as 3 instead of 8. After iteration with presumption as 8, category 8 gets the highest activation and the receptor's lattice is rotated to compensate for the initial deformation. Figure 3: Input Field After Adaptation 5 Discussion It is shown that the AIFNN can improve the classification performance for the original neural network, without modifications. This performance improvement is caused by an optimal affine translations estimation for rejected patterns. Although an affine translation is discussed in this paper, the algorithm is applicable to any deformation mechanism; such as a gain and offset equalization and 3D perspective deformation. 1108 Minoru Asogawa The requirement for a neural network in AIFNN is the capability of calculating partial derivatives for an input layer, so a layered neural network is utilized in this paper. Since partial derivative can be computed by numerical approximation, practically any neural network is applicable for AIFNN. Moreover, any differentiable error criterion is applicable; such as, a KL information and a likelihood. To reduce computation, a sequential searching is also possible; a presumption is chosen as the most plausible category, e.g. the smallest error category. If the L score falls behind a threshold, this presumption is regarded as correct. If it's not, another plausible category is chosen as a presumption and tested [Asogawa 91] . References [As ogawa 90] M. Asogawa, "Adaptive Input Field Neural Network - that can recognize rotated and/or shifted character -", Proceedings of IJCNN '90 at San Diego, vol. 3. pp. 733-738. June 1990. [As ogawa 91] M. Asogawa, "Adaptive Input Field Neural Network", Proceedings of IJCNN '91 at Singapore, vol. 1. pp. 83-88. November 1991. [Barnard 91] E. Barnard et aI., "Invariance and Neural Nets" , IEEE trans. on Neural Networks, vol. 2. no. 5, pp . 498-508. 1992. [Durbin 87] R. Durbin et al., "An analogue approach to the traveling salesman problem using an elastic net method", Nature, vol. 326. pp. 689-691. 1987. [Fahlman 88] S. Fahlman, "An empirical study of learning speed in backpropagation networks", CMU-CS-88-162, 1988. [Hinton 92] G .E. Hinton et al., "Adaptive Elastic Models for Hand-Printed Character Recognition", Advances in Neural Information Processing Systems, vol. 4. pp. 512-519. 1992. [Ishikawa 89] M. Ishikawa, "A structural learning algorithm with forgetting of link weights" , Proceedings of IJCNN '89 at Washington DC., vol. 2, pp. 626, 1989. [Martin 92] G. L. Martin et al., "Recognizing Overlapping Hand-Printed Characters by Centered-Object Integrated Segmentation and Recognition", Advances in Neural Information Processing Systems, vol. 4. pp. 504-511. 1992. [Simard 92] P. Simard et al., "Tangent Prop - A Formalism for Specifying Selected Invariances in an Adaptive Network", Advances in Neural Information Processing Systems, vol. 4. pp. 895-903. 1992. [Simard 93] P . Simard et al., "Efficient Pattern Recognition Using a New Transformation Distance" , Advances in Neural Information Processing Systems, vol. 5. pp. 50-58. 1993. [Yamada 91] K . Yamada, "Learning of category boundaries based on inverse recall by multilayer neural network", Proceedings of IJCNN '91 at Seattle, pp. 7-12 vol.2 1991. 

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289 NEURAL ANALOG DIFFUSION-ENHANCEMENT LAYER AND SPATIO-TEMPORAL GROUPING IN EARLY VISION Allen M. Waxman?,t, Michael Seibert?,t,RobertCunninghamt and I ian Wu? ? Laboratory for Sensory Robotics Boston University Boston, MA 02215 t Machine Intelligence Group MIT Lincoln Laboratory Lexington, MA 02173 ABSTRACT A new class of neural network aimed at early visual processing is described; we call it a Neural Analog Diffusion-Enhancement Layer or "NADEL." The network consists of two levels which are coupled through feedfoward and shunted feedback connections. The lower level is a two-dimensional diffusion map which accepts visual features as input, and spreads activity over larger scales as a function of time. The upper layer is periodically fed the activity from the diffusion layer and locates local maxima in it (an extreme form of contrast enhancement) using a network of local comparators. These local maxima are fed back to the diffusion layer using an on-center/off-surround shunting anatomy. The maxima are also available as output of the network. The network dynamics serves to cluster features on multiple scales as a function of time, and can be used in a variety of early visual processing tasks such as: extraction of comers and high curvature points along edge contours, line end detection, gap filling in contours, generation of fixation points, perceptual grouping on multiple scales, correspondence and path impletion in long-range apparent motion, and building 2-D shape representations that are invariant to location, orientation, scale, and small deformation on the visual field. INTRODUCTION Computer vision is often divided into two main stages, "early vision" and "late vision", which correspond to image processing and knowledge-based recognition/interpretation, respectively. Image processing for early vision involves algorithms for feature enhancement and extraction (e.g. edges and comers), feature grouping (i.e., perceptual We acknowledge support from the Machine Intelligence Group of MIT Lincoln Laboratory. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government 290 Waxman, Seibert, Cunningham and Wu organization), and the extraction of physical properties for object surfaces that comprise a scene (e.g. reflectance, depth, surface slopes and curvatures, discontinuities). The computer vision literature is characterized by a plethora of algorithms to achieve many of these computations, though they are hardly robust in performance. Biological neural network processing does, of course, achieve all of these early vision tasks, as evidenced by psychological studies of the preattentive phase of human visual processing. Often, such studies provide motivation for new algorithms in computer vision. In contrast to this algorithmic approach, computational neural network processing tries to glean organizational and functional insights from the biological realizations, in order to emulate their information processing capabilities. This is desirable. mainly because of the adaptive and real-time nature of the neural network architecture. Here, we shall demonstrate that a single neural architecture based on dynamical diffusionenhancement networks can realize a large variety of early vision tasks that deal mainly with perceptual grouping. The ability to group image features on multiple scales as a function of time, follows from "attractive forces" that emerge from the network dynamics. We have already implemented the NADEL (in 16-bit arithmetic) on a videorate parallel computer, the PIPE [Kent et al .? 1985], as well as a SUN-3 workstation. THE NADEL The Neural Analog Diffusion-Enhancement Layer was recently introduced by Seibert & Waxman [1989]. and is illustrated in Figure 1; it consists primarily of two levels which are coupled via feedforward and shunted feedback connections. Low-level features extracted from the imagery provide input to the lower level (a 2-D map) which spreads input activity over larger scales as time progresses via diffusion, allowing for passive decay of activity. The diffused activity is periodically sampled and passed upward to a contrast-enhancing level (another 2-D map) which locates local maxima in the terrain of diffuse activity. However, this forward pathway is masked by receptive fields which pass only regions of activity with positive Gaussian curvature and negative mean curvature; that is these receptive fields play the role of inhibitory dendro-dendritic modulatory gates. This masking fascilitates the local maxima detection in the upper level. These local maxima detected by the upper level are fed back to the lower diffusion level using a shunting dynamics with on-center/off-surround anatomy (cf. [Grossberg, 1973] on the importance of shunting for automatic gain control, and the role of center/surround anatomies in competitive networks). The local maxima are also available as outputs of the network, and take on different interpretations as a function of the input. A number of examples of spatio-temporal grouping will be illustrated in the next section. The primary result of diffusion-enhancement network dynamics is to create a longrange attractive force between isolated featural inputs. This force manifests itself by shifting the local maxima of activity toward one another. leading to a featural grouping over mUltiple scales as a function of time. This is shown in Figure 1, where two featural inputs spread their initial excitations over time. The individual activities superpose. with the tail of one gaussian crossing the maximum of the other gaussian at an angle. This Neural Analog Diffusion-Enhancement Layer biases the superposition of activities, adding more activity to one side of a maximum than another, causing a shift in the local maxima toward one another. Eventually, the local maxima merge into a single maximum at the centroid of the individual inputs. If we keep track of the local maxima as diffusion progresses (by connecting the output of the enhancement layer to another layer which stores activity in short term memory), then the two initial inputs will become connected by a line. In Figure 1 we also illustrate the grouping of five features in two clusters, a configuration possessing two spatial scales. After little diffusion the local maxima are located where the initial inputs were. Further diffusion causes each cluster to form a single local maximum at the cluster centroid. Eventually, both clusters merge into a single hump of activity with one maximum at the centroid of the five initial inputs. Thus, multi scale grouping over time emerges. The examples of Figure 1 use only diffusion without any feedback, yet they illustrate the importance of localizing the local maxima through a kind of contrast-enhancement on another layer. The local maxima of activity serve as "place tokens" representing grouped features at a particular scale. The feedback pathway re-activates the diffusion layer, thereby allowing the grouping process to proceed to still larger scales, even across featureless areas of imagery. The dynamical evolution of activity in the NADEL can be modeled using a modified diffusion equation [Seibert & Waxman, 1989]. However, in our simulations of the NADEL we don't actually solve this differential equation directly. Instead, each iteration of the NADEL consists of a spreading of activity using gaussian convolution, allowing for passive decay, then sampling the diffusion layer, masking out areas which are not positive Gaussian curvature and negative mean curvature activity surfaces, detecting one local maximum in each of these convex areas, and feeding this back to the diffusion layer with a shunted on-center/off-surround excitation at the local maxima. In the biological system, diffusion can be accomplished via a recurrent network of cells with offcenter/on-surround lateral connectivity, or more directly using electrotonic coupling across gap junctions as in the horizontal cell layer of the retina [Dowling, 1987]. Curvature masking of the activity surface can be accomplished using oriented offcenter/on-surround receptive fields that modulate the connections between the two primary layers of the NADEL. SPATIO-TEMPORAL GROUPING We give several examples of grouping phenomena in early vision, utilizing the NADEL. In all cases its parameters correspond to gaussian spreading with 0'=3 and passive decay of 1% per iteration, and on-center/off-surround feedback with 0'+=11'12 and 0'_=1. Grouping of Two Points: The simple case of two instantaneous point stimuli input simultaneously to the NADEL is summarized in Figure 2. We plot the time (N network iterations) it takes to merge the two inputs, as a function of their initial separation (S pixels). For S:S;6 the points merge in one iteration; for S>24 activity equilibrates and shifting of local maxima never begins. 291 292 Waxman, Seibert, Cunningham and Wu Grouping on Multiple Scales: Figure 3 illusttates the hierarchy of groupings generated by a square outline (31 pixels on a side) with gaps (9 pixels wide). Comer and line-end features are first enhanced using complementary center-surround receptive fields (modeled as a rectified response to a difference-of-gaussians), and located at the local maxima of activity. These features are shown superimposed on the shape in 3a; they serve as input to the NADEL. Figure 3b shows the loci of local maxima determined up to the second stable grouping, superimposed over the shape. Boundary completion fills the gaps in the square. In Figure 3c we show the loci of local maxima on the image plane, after the final grouping has occured (N=I00 iterations). The trajectory of local maxima through space-time (x,y,t) is shown in Figure 3d after the fourth grouping. It reveals a hierarchical organization similar to the "scale-space diagrams" of Witkin [1983]. It can be seen from Figure 3d that successive groupings form stable entities in that the place tokens remain stationary for several iterations of the NADEL. It isn't until activity has diffused farther out to the next representative scale that these local maxima start moving once again, and eventually merge. This relates stable perceptual groupings to place tokens (i.e., local maxima of activity) that are not in motion on the diffusion layer. The motion of place tokens can be measured in the same fashion as feature point motion across the visual field. Real-time receptive fields for measuring the motion of image edge and point features have recently been developed by Waxman et ale [1988]. Grouping of Time-Varying Inputs: The simplest example in this case corresponds to the grouping of two lights that are flashed at different locations at different times. When the time interval between flashes (Stimulus Onset Asynchrony SOA) is set appropriately, one perceives a smooth motion _or "path impletion" between the stimuli. This percept of "long-range apparent motion" is the cornerstone of the Gestalt Psychology moverment, and has remained unexplained for one-hundred years now [Kolers, 1972]. We have applied the NADEL to a variety of classical problems in apparent motion including the "split motion" percept and the mUlti-point Ternus configuration [Waxman et al., 1989]. Here we consider only the case of motion between two stimuli, where we interpret the locus of local maxima as the impleted path in apparent motion. However, the direction of perceived motion is not determined by the grouping process itself; only the path. We make the additional assumption that grouping generates a motion percept only if the second stimulus begins to shift immediately upon input to the NADEL. We suggest that the motion percept occurs only after path impletion is complete. That is, while grouping is active, its outputs are suppressed from our perception (a form of "ttansient-onsustained inhibition" analogous to saccadic suppression). By varying the separation between the two stimuli, and the time (SOA) between their inputs, we can plot regimes for which the NADEL predicts apparent motion. This is shown in Figure 4, which compares favorably with the psychophysical results summarized in Figure 3.2 of [Kolers, 1972]. We find regimes in which complete paths are formed ("smooth motion"), partial paths are formed ("jumpy motion"), and no immediate shifting occurs ("no motion"). The maximum allowable SOA between stimuli (upper curves) is determined by the passive decay rate. Increasing this decay from 1% to 3% will decrease the maximum SOA by a Neural Analog Diffusion-Enhancement Layer factor of five. The minimum allowable SOA (lower curves) increases with increasing separation, since it takes longer for activity from the first stimulus to influence a more distant second stimulus. The linearity of the lower boundary has been interpreted by [Waxman et aI.? 1989] as suggestive of Korte's "third law" [Kolers, 1972], when taken in combination with a logarithmic transformation of the visual field [Schwartz, 1980]. Attentional Cues and Invariant Representations: Place tokens which emerge as stable groupings over time can also provide attentional cues to a vision system. They would typically drive saccadic eye motions during scene inspection, with the relative activities of these maxima and their order of emergence determining the sequence of rapid eye motions. Such eye motions are known to play a key role in human visual perception [Yarbus, 1967]; they are influenced by both bottom-up perceptual cues as well as topdown expectations. The neuromorphic vision system developed by Seibert & Waxman [1989], shown in Figure 5, utilizes the NADEL to drive "eye motions", and thereby achieve translational invariance in 2-D object learning and recognition. This is followed by a log-polar transform (which emulates the geniculo-cortical connections [Schwartz, 1980]) and another NADEL to achieve rotation and scale in variance as well. Further coding of the transformed feature points by overlapping receptive fields provides invariance to small deformation. Pattern learning and recognition is then achieved using an Adaptive Resonance Theory (ART-2) network [Carpenter & Grossberg, 1987]. REFERENCES G. Carpenter & S. Grossberg (1987). ART-2: Self-organization of stable category recognition codes for analog input patterns. Applied Optics 26, pp. 4919-4930. I.E. Dowling (1987). The RETINA: An approachable part or the brain. Cambridge, MA: Harvard University Press. S. Grossberg (1973). Contour enhancement, short term memory, and constancies in reverberating neural networks. Studies in Applied Mathematics 52. pp.217-257. E.W. Kent, M.O. Shneier & R. Lumia (1985). PIPE: Pipelined Image Processing Engine. Journal of Parallel and Distributed Computing 2, pp. 50-78. P.A. Kolers (1972). Aspects or Motion Perception. New York: Pergamon Press. E.L. Schwartz (1980). Computational anatomy and functional architecture of striate cortex: A spatial mapping approach to perceptual coding. Vision Research 20, pp. 645669. M. Seibert & A.M. Waxman (1989). Spreading activation layers, visual saccades and invariant representations for neural pattern recognition systems. N~ural Networks 2, pp. 9-27. 293 294 Waxman, Seibert, Cunningham and Wu A.M. Waxman, J. Wu & F. Bergholm (1988). Convected activation profiles and the measurement of visual motion. Proceeds. 1988 IEEE Conference on Computer Vision and Pattern Recognition. Ann Arbor, MI, pp. 717-723. A.M. Waxman. J. Wu & M. Seibert (1989). Computing visual motion in the short and the long: From receptive fields to neural networks. Proceeds. IEEE 1989 Workshop on Visual Motion. Irvine, CA. A.P. Witkin (1983). Scale space filtering. Proceeds. of the International Joint Conference on Artificial Intelligence. Karlsruhe, pp. 1019-1021. AL. Yarbus (1967). Eye Movements and Vision. New York: Plenum Press. ON?CENTER! o o o OFF?SURROUND OFF?CENTER! a.l-SURAOUNQ SHUNTNG FEE08AO< ON-CENTER! OFF?SURAOUND Figure 1 ? (left) The NADEL takes featural input and diffuses it over a 2-D map as a function of time. Local maxima of activity are detected by the upper layer and fed back to the diffusion layer using an on-center/off-surround shunting anatomy. (right. top) Features spread their activities which superpose to generate an attractive force, causing distant features to group. (right, bottom) Grouping progresses in time over multiple scales. with small clusters emerging before extended clusters. Clusters are represented by their local ~axima of activity, which serve as place tokens. 1ii~Ill'J!I!IIIl!lll!l1llllllm= !,~ ~l" lmilllillUml:l ? 1_? 1000 600 .00 I; 200 /, ~ I so ~ ~~ .. I~~ Ii' ~,I ~ tt~i'><'! k;'~ "~~~"'r..;.'(!S.1-:'~-' ~,~o.~_, ? ~7'~~':;' ~~~~ . ... (b) (a) I ~o / .x ~ ? ':'.~ ~ ~ / 60 .9 ~ 1 ~h t~5~ ~~!:Ji. .J\llt! :??'!"Id I I lOa <! 1~~'l F~ .' ;h~ ' 800 z ~ e. ~ e. / / 20 I / 10 I I 6 I t2 / ~ y / ~ .--- .... ,-_ ..... r!3. g Separalion ~ 5 (pIXelS) g. (e) Cd) I g Figure 2 _ The time (network iterations N) to merge two points input simultaneously to the NADEL, as a function of their initial separation (S pixels). ("t- ~ Figure 3 - Perceptual grouping of a square outline with gaps. ~ ~ ~ '-0 c:.n ~ co (J') E_pectat Ions Spatial relations 200 80 g ,/ 40 Smooth Motion <II ::s e.::s ./ ./' c: .2 ~ a> ,;+ No Motion 60 ~ ./ I?. ./ 20 a ./' -l: ,/ 0 ~ iii .s 10 8 C C/) 6 Jumpy Motion ~~ ./ , / ./ ~ ./ ./ ./ 4 / / / 2 / VIA NADEL / 1 J ....g-~ 100 FEATURE MAP 6 Separation S (pixels) Figure 4 - Apparent motion between two flashed lights: Stimulus Onset Asynchrony SOA (network iterations N) vs. Separation S (pixels). Solid curves indicate boundaries between which introduction of the second light yields immediate shifting of local maxima; dashed curves (above solid curves) indicate when final merge occurs yielding the impleted path. EYE MOTIONS Figure 5 - The neuromorphic vision system for invariant learning and recognition of 2-D object~ utilizes three NADEL networks. 

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Sample Size Requirements For Feedforward Neural Networks Michael J. Turmon Cornell Univ. Electrical Engineering Ithaca, NY 14853 mjt@ee.comell.edu Terrence L. Fine Cornell Univ. Electrical Engineering Ithaca, NY 14853 tlfine@ee.comell.edu Abstract We estimate the number of training samples required to ensure that the performance of a neural network on its training data matches that obtained when fresh data is applied to the network. Existing estimates are higher by orders of magnitude than practice indicates. This work seeks to narrow the gap between theory and practice by transforming the problem into determining the distribution of the supremum of a random field in the space of weight vectors, which in turn is attacked by application of a recent technique called the Poisson clumping heuristic. 1 INTRODUCTION AND KNOWN RESULTS We investigate the tradeofi"s among network complexity, training set size, and statistical performance of feedforward neural networks so as to allow a reasoned choice of network architecture in the face of limited training data. Nets are functions 7](x; w), parameterized by their weight vector w E W ~ Rd , which take as input points x E Rk. For classifiers, network output is restricted to {a, 1} while for forecasting it may be any real number. The architecture of all nets under consideration is N, whose complexity may be gauged by its Vapnik-Chervonenkis (VC) dimension v, the size of the largest set of inputs the architecture can classify in any desired way ('shatter'). Nets 7] EN are chosen on the basis of a training set T {(Xi, YiHr=l. These n samples are i.i.d. according to an unknown probability law P. Performance of a network is measured by the mean-squared error E(w) E(7](x; w) - y)2 (1) = P(7](x;w);/; y) (for classifiers) (2) = 328 Michael Turman, Terrence L. Fine and a good (perhaps not unique) net in the architecture is WO = argmiIlwew ?(w). To select a net using the training set we employ the empirical error 1 n VT(W) = - I)11(Xi; w) - Yi)2 (3) n i=l sustained by 11(?; w) on the training set T. A good choice for a classifier is then w? = argmiIlwew VT(W). In these terms, the issue raised in the first sentence ofthe section can be restated as, "How large must n be in order to ensure ?(w?)-?(WO) $ i with high probability?" For purposes of analysis we can avoid dealing directly with the stochastically chosen network w? by noting ?(w?) - ?(WO) $ IVT(W?) - ?(w?)1 + IVT(WO) - ?(wo)1 $ 2 sup IVT(W) - ?(w)1 wEW A bound on the last quantity is also useful in its own right. The best-known result is in (Vapnik, 1982), introduced to the neural network community by (Baum & Haussler, 1989): (2n)V ~ P( sup IVT(W) - ?(w)1 ~ i) $ 6-,-e- n ( /2 (4) wEW v. This remarkable bound not only involves no unknown constant factors, but holds independent of the data distribution P . Analysis shows that sample sizes of about nc = (4V/i 2) log 3/i (5) are enough to force the bound below unity, after which it drops exponentially to zero. Taking i = .1, v = 50 yields nc = 68000, which disagrees by orders of magnitude with the experience of practitioners who train such simple networks. More recently, Talagrand (1994) has obtained the bound K2ni2)v ~ P( sup IVT(W) - ?(w)1 ~ i) $ Kl ( e- 2n ( , (6) wew V yielding a sufficient condition of order V/i 2, but the values of Kl and K2 are inaccessible so the result is of no practical use. Formulations with finer resolution near ?(w) = 0 are used. Vapnik (1982) bounds P(suPwew IVT(W) - ?(w)I/?(w)1/2 ~ i)-note ?(w)1/2 ~ Var(vT(w?1/2 when ?(w) ~ O-while Blumer et al. (1989) and Anthony and Biggs (1992) work with P(suPWEW IVT(W) - ?(w)ll{o}(VT(W? ~ i). The latter obtain the sufficient condition (7) nc = (5.8v/i) log 12/i for nets, if any, having VT( w) = o. If one is guaranteed to do reasonably well on the training set, a smaller order of dependence results. Results (Turmon & Fine, 1993) for perceptrons and P a Gaussian mixture imply that at least v/280i 2 samples are needed to force ?(w?) - ?(WO) < 2i with high probability. (Here w? is the best linear discriminant with weights estimated from the data.) Combining with Talagrand's result, we see that the general (not assuming small VT(W? functional dependence is V/i 2. Sample Size Requirements for Feedforward Neural Networks 2 329 APPLYING THE POISSON CLUMPING HEURISTIC We adopt a new approach to the problem. For the moderately large values of n we anticipate, the central limit theorem informs us that Vn[lIT(W) - E(w)] has nearly the distribution of a zero-mean Gaussian random variable. It is therefore reasonable l to suppose that P( sup IlIT(W) - E(w)1 ~ f) ~ P( sup IZ(w)1 ~ fJ1i) ~ 2P( sup Z(w) ~ fVn) wEW wEW wEW where Z( w) is a Gaussian process with mean zero and covariance R(w, v) = EZ(w)Z(v) = Cov(y -1J(x; w?2, (y -1J(x; V?2) The problem about extrema of the original empirical process is equivalent to one about extrema of a corresponding Gaussian process. The Poisson clumping heuristic (PCR), introduced in the remarkable (Aldous, 1989), provides a general tool for estimating such exceedance probabilities. Consider the excursions above level b(= fVn ~ 1) by a stochastic process Z(w). At left below, the set {w : Z( w) ~ b} is seen as a group of "clumps" scattered in weight space W. The PCR says that, provided Z has no long-range dependence and the level b is large, the centers of the clumps fall according to the points of a Poisson process on W, and the clump shapes are independent. The vertical arrows (below right) illustrate two clump centers (points of the Poisson process); the clumps are the bars centered about the arrows. w w In fact, with PheW) = P(Z(w) ~ b), Ch(W) the size of a clump located at w, and Ah (w) the rate of occurrence of clump centers, the fundamental equation is (8) The number of clumps in W is a Poisson random variable Nh with parameter The probability of a clump is P(Nb > 0) = 1- exp( - fwAh( w) dW) =::: Ah(W) dw where the approximation holds because our goal is to operate in a regime where this probability is near zero. Letting ~(b) = P(N(0, 1) > b) and (T2(w) = R(w, w), we have PheW) = ~(b/(T(w?. The fundamental equation becomes 1, Ah( w) dw. fw P( sup Z(w) wEW ~ b) ~ r ~(b/(T(w? ECh(W) Jw dw (9) It remains only to find the mean clump size ECh( w) in terms of the network architecture and the statistics of (x, y). lSee ch. 7 of (Pollard, 1984) for treatment of some technical details in this limit. 330 3 Michael Tunnon, Terrence L. Fine POISSON CLUMPING FOR SMOOTH PROCESSES Assume Z(w) has two mean-square derivatives in w. (If the network activation functions have two derivatives in w, for example, Z( w) will have two almost sure derivatives.) Z then has a parabolic approximation about some Wo via its gradient G = 'VZ(w) and Hessian matrix H = 'V'VZ(w) at woo Provided Zo ~ b, that is that there is a clump at Wo, simple computations reveal Cb( wo) ~ Kd (2(Zo - b) - cP'H- 1 G)d/2 IHI I / 2 where Kd is the volume of the unit ball in Rd and I? 1is the determinant. clump size is the expectation of this conditioned on Z(wo) ~ (10) The mean b. The same argument used to show that Z(w) is approximately normal shows that G and H are approximately normal too. In fact, E[HIZ(wo) z (F 2( Wo )A(wo) -EZ(wo)H = -'Vw'VwR(wo, w)lw=wo = z] A(wo) so that, since b (and hence z) is large, the second term in the numerator of (10) may be neglected. The expectation is then easily computed, resulting in Lemma 1 (Smooth process clump size) Let the network activation functions be twice continuously differentiable, and let b ? (F( w). Then ECb(W) ~ (21r)d/21 ~~~) 1- 112 ((F(:?) d Substituting into (9) yields P( sup Z(w) ~ b) ~ (21r)-~ 1A(w) 11/2 (_b_) d-~_b~/2q~(W) dw, (11) (F(w) where use of the asymptotic expansion ~(z) ~ (zv'21r)-l exp( _Z2 /2) is justified since ('v'w)b ? (F( w) is necessary to have the individual P( Z( w) ~ b) low-let alone the supremum. To go farther, we need information about the variance (F2 (w) of (y - 11( x; w?2. In general this must come from the problem at hand, but suppose wEW ( iw (F2(w) for example the process has a unique variance maximum 0'2 at w. Then, since the level b is large, we can use Laplace's method to approximate the d-dimensional integral. Laplace's method finds asymptotic expansions for integrals fw g(w) exp( - f(w)2 /2) dw when few) is C2 with a unique positive minimum at Wo in the interior of W ~ Rd , and g( w) is positive and continuous. Suppose I( wo) ? 1 so that the exponential factor is decreasing much faster than the slowly varying g. Expanding f to second order about Wo, substituting into the exponential, and performing the integral shows that iw g( w) exp( - f(w)2 /2) dw ~ (21r)d/2If( wo)KI- 1/ 2g( wo) exp( - f( wo)2 /2) Sample Size Requirements for Feedforward Neural Networks 331 where K = V'V'f(w)lwo, the Hessian of f. See (Wong, 1989) for a proof. Applying this to (11) and using the asymptotic expansion for ~ in reverse yields Theorem 1 Let the network activation functions be twice continuously differentiable. Let the variance have a unique maximum u at w in the interior of Wand the level b ~ u. Then the peH estimate of exceedance probability is given by IA(w)1 1/ 2 _ P(:~fv Z(w) ~ b) ~ IA(w) _ r(w)1 1/ 2 ~(b/u) (12) where r(w) = V'wV'tlR(w,v)lw=tI=w. Furthermore, A- r is positive-definite at w; it is -1/2 the Hessian of cr 2 (w). The leading constant thus strictly e:cceeds unity. The above probability is just P(Z(w) ~ b) multiplied by a factor accounting for the other networks in the supremum. Letting b = f...;n reveals nc = u2 10g(IA(w)I/IA(w) - r(w)!) f. 2 (13) samples force P(supw IlIT(W) - &(w)\ ~ {) below unity. If the variance maximum is not unique but occurs over a d-dimensional set within W, the sample size estimate becomes proportional to u2d/{2. With d playing the role of VC dimension v, this is similar to Vapnik's bound although we retain dependence on P and N. The above probability is determined by behavior near the maximum-variance point, which for example in classification is where &(w) = 1/2. Such nets are uninteresting as classifiers, and certainly it is undesirable for them to dominate the entire probability. This problem is avoided by replacing Z(w) with Z(w)/cr(w), which additionally allows a finer resolution where &(w) nears zero. Indeed, for classification, if n is such that with high probability sup weW IlIT(W) - &(w)1 IlIT(W) - &(w)1 = sup <{ , cr(w) wew J&(W)(I- &(w? (14) then lIT(W?) = 0 ::} &(w?) < {2(1 + (2)-1 ~ {2 <t:: {. Near lIT(W?) = 0, condition (14)/ is much more powerful than the corresponding unnormalized one. Sample size estimates using this setup give results having a functional form similar to (7). 4 ANOTHER MEANS OF COMPUTING CLUMP SIZE Conditional on there being a clump center at w, the upper bound Cb(W) ~ Db(W) == iw l[o,oo)(Z(w') - b) dw' (15) is evidently valid: the volume of the clump at w is no larger than the total volume of all clumps. (The right hand side is indeed a function of w because we condition on occurrence of a clump center at w.) The bound is an overestimate when the number Nb of clumps exceeds one, but recall that we are in a regime where b (equivalently n) is large enough so that P( Nb > 1)/ P( Nb = 1) ~ fw ).b (w) dw <t:: 1. Thus error in (15) due to this source is negligible. To compute its mean, we approximate EDb(W) = iw P(Z(~f) ~ blw a clump center)dw' 332 Michael Turmon, Terrence L. Fine (16) The point is that occurrence of a clump center at Wo is a smaller class of events than merely Z( wo) ~ b: the latter can arise from a clump center at a nearby w E W capturing woo Since Z(w) and Z(w') are jointly normal, abbreviate u = u(w), u' = u(w'), p = p(w,w') = R(w,w')/(uu'), and let (=(w,w') (U I U,)I;r;!; 1- p2 (1- p)/(1 + p?)1/2 (17) (constant variance case) (18) Evaluating the conditional probabilities of (16) presents no problem, and we obtain Lemma 2 (Clump size estimate) For b ~ u the mean clump size is ECb(W) ~ EDb(W) ~ /w cf>?blu)() dw' (19) Remark 1. This integral will be used in (9) to find P(s~pZ(w) > b) ~ ( cf>(blu) Jw fw~?blu)() dw,dw (20) Since b is large, the main contribution to the outer integral occurs for w near a variance maximum, i.e. for u' I u ~ 1. If the variance is constant then all w E W contribute. In either case ( is nonnegative. By lemma 1 we expect (19) to be, as a function of b, of the form (const ulb)P for, say, p = d. In particular, we do not anticipate the exponentially small clump sizes resulting if (Vw')( w, w') ~ M ~ O. Therefore ( should approach zero over some range of w', which happens only when p ~ 1, that is, for w' near w. The behavior of pew, w') for w' ~ w is the key to finding the clump size. Remark 2. There is a simple interpretation of the clump size; it represents the volume of w' E W for which Z(w') is highly correlated with Z(w) . The exceedance probability is a sum of the point exceedance probabilities (the numerator of (20?, each weighted according to how many other points are correlated with it. In effect, the space W is partitioned into regions that tend to "have exceedances together," with a large clump size ECb( w) indicating a large region. The overall probability can be viewed as a sum over all these regions of the corresponding point exceedance probability. This has a similarity to the Vapnik argument which lumps networks together according to their nV Iv! possible actions on n items in the training set. In this sense the mean clump size is a fundamental quantity expressing the ability of an architecture to generalize. 5 EMPIRICAL ESTIMATES OF CLUMP SIZE The clump size estimate of lemma 2 is useful in its own right if one has information about the covariance of Z. Other known techniques of finding ECb( w) exploit special features of the process at hand (e.g. smoothness or similarity to other wellstudied processes); the above expression is valid for any covariance structure. In Sample Size Requirements for Feedf01ward Neural Networks 333 this section we show how one may estimate the clump size using the training set, and thus obtain probability approximations in the absence of analytical information about the unknown P and the potentially complex network architecture N. Here is a practical way to approximate the integral giving EDb{W). For'Y a set of significant w' S-y{W) = {w' E W: (w,w') $ 'Y} then monotonicity of ~ yields EDb{W) ~ < 1 define V-y{W) = vol{S-y(w)) (21) Is .., ~((b/(1X) dw' ~ V-y(W) ~((b/uh) . This apparently crude lower bound for ~ is accurate enough near the origin to give satisfactory results in the cases we have studied. For example, we can characterize the covariance R( w, w') of the smooth process oflemma 1 and thus find its ( function. The bound above is then easily calculated and differs by only small constant factors from the clump size in the lemma. The lower bound for EDb(W) yields the upper bound 1 ~(b/(1-) P(s~p Z(w) ~ b) $ w V-y(w) ~?b/uh) dw (22) We call V-y(w) the correlation volume, as it represents those weight vectors w' whose errors Z(w') are highly correlated with Z(w); one simple way to estimate the correlation volume is as follows . Select a weight w' and using the training set compute (Yl - 17( Xl; w))2, ... , (Yn - 17( Xn; w)? & (Yl - 17( Xl; w'))2 , ... , (Yn - 17( Xn; w'))2 . It is then easy to estimate u 2, u,2, and p, and finally (w , w'), which is compared to the chosen 'Y to decide if w' E S-y (w) . The difficulty is that for large d, S-y (w) is far smaller than any approximatelyenclosing set. Simple Monte Carlo sampling and even importance sampling methods fail to estimate the volume of such high-dimensional convex bodies because so few hits occur in probing the space (Lovasz, 1991). The simplest way to concentrate the search is to let w' = w except in one coordinate and probe along each coordinate axis. The correlation volume is approximated as the product of the one-dimensional measurements. Simulation studies of the above approach have been performed for a perceptron architecture in input uniform over [-1, l]d. The integral (22) is computed by Monte Carlo sampling, and based ona training set of size lOOd, V-y (w) is computed at each point via the above method. The result is that an estimated sample size of 5.4d/f 2 is enough to ensure (14) with high probability. For nets, if any, having VT(W) = 0, sample sizes larger than 5.4d/f will ensure reliable generalization, which compares favorably with (7) . 6 SUMMARY AND CONCLUSIONS To find realistic estimates of sample size we transform the original problem into one of finding the distribution of the supremum of a derived Gaussian random field, which is defined over the weight space of the network architecture. The latter problem is amenable to solution via the Poisson clumping heuristic. In terms of the PCH the question becomes one of estimating the mean clump size, that 334 Michael Turman, Terrence L. Fine is, the typical volume of an excursion above a given level by the random field. In the "smooth" case we directly find the clump volume and obtain estimates of sample size that are (correctly) of order v/?2. The leading constant, while explicit, depends on properties of the architecture and the data-which has the advantage of being tailored to the given problem but the potential disadvantage of our having to compute them. We also obtain a useful estimate for the clump size of a general process in terms of the correlation volume V-y(w). For normalized error, (22) becomes approximately p (sup lIr(w) - ?(w) weW u(w) > ?) ~ E - [vol(W)] e-(1--y2)nf 2 /2 V-y(w) where the expectation is taken with respect to a uniform distribution on W. The probability of reliable generalization is roughly given by an exponentially decreasing factor (the exceedance probability for a single point) times a number representing degrees of freedom. The latter is the mean size of an equivalence class of "similarlyacting" networks. The parallel with the Vapnik approach, in which a worst-case exceedance probability is multiplied by a growth function bounding the number of classes of networks in N that can act differently on n pieces of data, is striking. In this fashion the correlation volume is an analog of the VC dimension, but one that depends on the interaction of the data and the architecture. Lastly, we have proposed practical methods of estimating the correlation volume empirically from the training data. Initial simulation studies based on a perceptron with input uniform on a region in Rd show that these approximations can indeed yield informative estimates of sample complexity. References Aldous, D. 1989. Probability Approximations via the Poisson Clumping Heuristic. Springer. Anthony, M., & Biggs, N. 1992. Computational Learning Theory. Cambridge Univ. Baum, E., & Haussler, D. 1989. What size net gives valid generalization? Pages 81-90 of' Touretzky, D. S. (ed), NIPS 1. Blumer, A., Ehrenfeucht, A., Haussler, D., & Warmuth, M. K. 1989. Learnability and the Vapnik-Chervonenkis dimension. Jour. Assoc. Compo Mach., 36,929-965. LovMz, L. 1991. Geometric Algorithms and Algorithmic Geometry. In: Proc. Internat. Congr. Mathematicians. The Math. Soc. of Japan. Pollard, D. 1984. Convergence of Stochastic Processes. Springer. Talagrand, M. 1994. Sharper bounds for Gaussian and empirical processes. Ann. Probab., 22, 28-76. Turmon, M. J., & Fine, T. L. 1993. Sample Size Requirements of Feedforward Neural Network Classifiers. In: IEEE 1993 Intern. Sympos. Inform. Theory. Vapnik, V. 1982. Estimation of Dependences Based on Empirical Data. Springer. Wong, R. 1989. Asymptotic Approximations of Integrals. Academic. 

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Combining Estimators Using Non-Constant Weighting Functions Volker Tresp*and Michiaki Taniguchi Siemens AG, Central Research Otto-Hahn-Ring 6 81730 Miinchen, Germany Abstract This paper discusses the linearly weighted combination of estimators in which the weighting functions are dependent on the input . We show that the weighting functions can be derived either by evaluating the input dependent variance of each estimator or by estimating how likely it is that a given estimator has seen data in the region of the input space close to the input pattern. The latter solution is closely related to the mixture of experts approach and we show how learning rules for the mixture of experts can be derived from the theory about learning with missing features. The presented approaches are modular since the weighting functions can easily be modified (no retraining) if more estimators are added. Furthermore, it is easy to incorporate estimators which were not derived from data such as expert systems or algorithms. 1 Introduction Instead of modeling the global dependency between input x E ~D and output y E ~ using a single estimator, it is often very useful to decompose a complex mapping -'\.t the time of the research for this paper, a visiting researcher at the Center for Biological and Computational Learning, MIT. Volker.Tresp@zfe.siemens.de 420 Volker Tresp, Michiaki Taniguchi into simpler mappings in the form l (1) M n(x) = L hi(X) i=l The weighting functions hi(X) act as soft switches for the modules N Ni(X). In the mixture of experts (Jacobs et al., 1991) the decomposition is learned in an unsupervised manner driven by the training data and the main goal is a system which learns quickly. In other cases, the individual modules are trained individually and then combined using Equation 1. We can distinguish two motivations: first, in the work on averaging estimators (Perrone, 1993, Meir , 1994, Breiman, 1992) the modules are trained using identical data and the weighting functions are constant and, in the simplest case, all equal to one. The goal is to achieve improved estimates by averaging the errors of the individual modules. Second, a decomposition as described in Equation 1 might represent some "natural" decomposition of the problem leading to more efficient representation and training (Hampshire and Waibel, 1989). A good example is a decomposition into analysis and action. hi(x) might be the probability of disease i given the symptoms x, the latter consisting of a few dozen variables. The amount of medication the patient should take given disease i on the other hand - represented by the output of module N Ni (x) - might only depend on a few inputs such as weight, gender and age.2 Similarly, we might consider hie x) as the IF-part of the rule, evaluating the weight of the rule given x, and as N Ni(X) the conclusion or action which should be taken under rule i (compare Tresp, Hollatz and Ahmad, 1993). Equation 1 might also be the basis for biological models considering for example the role of neural modulators in the brain. Nowlan and Sejnowsky (1994) recently presented a biologically motivated filter selection model for visual motion in which modules provide estimates of the direction and amount of motion and weighting functions select the most reliable module. In this paper we describe novel ways of designing the weighting functions. Intuitively, the weighting functions should represent the competence or the certainty of a module, given the available information x. One possible measure is related to the number of training data that a module has seen in the neighborhood of x. Therefore, P(xli), which is an estimate of the distribution of the input data which were used to train module i is an obvious candidate as weighting function. Alternatively, the certainty a module assigns to its own prediction, represented by the inverse of the variance 1/ var( N Ni ( x? is a plausible candidate for a weighting function. Both approaches seem to be the flip-sides of the same coin, and indeed, we can show that both approaches are extremes of a unified approach. IThe hat stands for an estimates value. 2Note, that we include the case that the weighting functions and the modules might explicitly only depend on different subsets of x. 421 Combining Estimators Using Non-Constant Weighting Functions (e) 1 0.8 I, 0.6 I / / \ I \ \ \ 0 .4 0.2 I \ (d) (b) 0.5r-r-----T"I I ' I,:' I: j ,l t' (f) 0.5.----------" o :1 ~ -1 -1 -1 -1 o -1 o 1 o -1 1 Figure 1: (a): Two data sets (1:*, 2:0) and the underlying function (continuous). (b) The approximations of the two neural networks trained on the data sets (continuous: 1, dashed: 2). Note, that the approximation of a network is only reliable in the regions of the input space in which it has "seen" data. (c) The weighting functions for variance-based weighting. (d) The approximation using variance-based weighting (continuous). The approximation is excellent, except to the very right. (e) The weighting functions for density-based weighting (Gaussian mixtures approximation). (f) The approximation using density-based weighting (continuous). In particular to the right, the extrapolation is better than in (d). 2 Variance-based Weighting Here, we assume that the different modules N Ni(x) were trained with different data sets {(xLY~)}:~l but that they model identical input-output relationships (see Figure 1 a,b ). To give a concrete example, this would correspond to the case that we trained two handwritten digit classifiers using different data sets and we want to use both for classifying new data. If the errors of the individual modules are uncorrelated and unbiased, 3 the combined estimator is also unbiased and has the smallest variance if we select the weighting functions inversely proportional the the variance of the modules 1 (2) hi(X) = var(NNi(x)) This can be shown using var(2:::'l gi(x)N Ni(X)) = 2:::'1 gl(x)var(N Ni(X)) and 1. Intuitively, using Lagrange mUltiplier to enforce the constraint that 2:i gi(X) = 3The errors are un correlated since the modules were trained with different data; correlation and bias are discussed in Section 8.1. 422 Volker Tresp, Michiaki Taniguchi Equation 2 says that a module which is uncertain about its own prediction should also obtain a smaller weight. We estimate the variance from the training data as var ? '" 8N Ni(xl H-:- 18N Ni(X) * x "" 8w * 8w' (NN .( Hi is the Hessian, which can be approximated as Tibshirani, 1994) 3 ?(12 is the output-noise variance, Density-based Weighting In particular if the different modules were trained with data sets from different regions of the input space, it might be a reasonable assumption that the different modules represent different input-output relationships. In terms of our example, this corresponds to the problem, that we have two handwritten digit classifiers, one trained with American data and one with European data. If the classifiers are used in an international setting, confusions are possible, since, for example, an American seven might be confused with a European one. Formally, we introduce an additional variable which is equal to zero if the writer is American and is equal to one if the writer is European. During recall, we don't know the state of that variable and we are formally faced with the problem of estimation with missing inputs. From previous work (Ahmad and Tresp, 1993) we know that we have to integrate over the unknown input weighted by the conditional probability of the unknown input given the known variables. In this case, this translates into Equation 1, where the weighting function is In our example, P(ilx) would estimate the probability that the writer is American or European given the data. Depending on the problem P(ilx) might be estimated in different ways. If x represents continuous variables, we use a mixture of Gaussians model (3) where G(x; cij, Eij) is our notation for a normal density centered at cij and with covariance Eij. Note that we have obtained a mixture of experts network with P( ilx) as gating network. A novel feature of our approach is that we maintain an estimate of the input data distribution (Equation 3), which is not modeled in the original mixture of experts network. This is advantageous if we have training data which are not assigned 423 Combining Estimators Using Non-Constant Weighting Functions to a module (in the mixture of experts, no data are assigned) which corresponds to training with missing inputs (the missing input is the missing assignment), for which the solution is known (Tresp et ai., 1994). If we use Gaussian mixtures to approximate P(xli), we can use generalized EM learning rules for adaptation. The adaptation of the parameters in the "gating network" which models P(xli) is therefore somewhat simpler than in the original mixture of experts learning rules (see Section 8.2). 4 Unified Approach In reality, the modules will often represent different mappings, but these mappings are not completely independent. Let's assume that we have an excellent American handwritten digit classifier but our European handwritten digit classifier is still very poor, since we only had few training data. We might want to take into account the results of the American classifier, even if we know that the writer was European. Mathematically, we can introduce a coupling between the modules. Let's assume that the prediction ofthe i-th module NNi(X) = li(x)+{i is a noisy version of the true underlying relationship li(x) and that {i is independent Gaussian noise with variance var(N Ni(X?. Furthermore, we assume that the true underlying functions are coupled through a prior distribution (for simplicity we only assume two modules) 1 P(h(x), h(x)) ex: exp (--2-(I1(x) - 12(x ) )2 ). vare We obtain as best estimates A 1 h(x) = K(x) [(var(N N 2 (x)) A 1 + vare) N Nt(x) + var(N Nl(x)) h(x) = K(x) [var(N N2(x? N Nl(X) + (var(N Nt (x? where K(x) + vare) N N2(X)] N N2(X)] = var(N Nl (x? + var(N N2 (x)) + vare. = We use density-based weighting to combine the two estimates: y(x) P(llx)it(x)+ P(2Ix)i2(X). Note, that if vare -- 00 (no coupling) we obtain the density-based solution and for vare -- 0 (the mappings are forced to be identical) we obtain the variance-based solution. A generalization to more complex couplings can be found in Section 8.2.1. 5 Experiments We tested our approaches using the Boston housing data set (13 inputs, one continuous output). The training data set consisted of 170 samples which were divided into 20 groups using k-means clustering. The clusters were then divided randomly into two groups and two multi-layer perceptrons (MLP) were trained using those two 424 Volker Tresp. Michiaki Taniguchi data sets. Table 1 shows that the performances of the individual networks are pretty bad which indicates that both networks have only acquired local knowledge with only limited extrapolation capability. Variance-based weighting gives considerably better performance, although density-based weighting and the unified approach are both slightly better. Considering the assumptions, variance-based weighting should be superior since the underlying mappings are identical. One problem might be that we assumed that the modules are unbiased which might not be true in regions were a given module has seen no data. Table 1: Generalization errors N N2 0.6948 6 1.188 I variance-based I density-based I unified I I 0.4821 I 0.4472 I 0.4235 I Error-based Weighting In most learning tasks only one data set is given and the task is to obtain optimal predictions. Perrone (1994) has shown that simply averaging the estimates of a small number (i. e. 10) of neural network estimators trained on the same training data set often gives better performance than the best estimator out of this ensemble. Alternatively, bootstrap samples of the original data set can be used for training (Breimann, personal communication). Instead of averaging, we propose that Equation 1, where might give superior results (error-based weighting). Res(N Ni(X)) stands for an estimate of the input dependent residual squared error at x. As a simple approximation, Res(N Ni(X)) can be estimated by training a neural network with the residual squared errors of N Ni. Error-based weighting should be superior to simple averaging in particular if the estimators in the pool have different complexity. A more complex system would obtain larger weights in regions where the mapping is complex, since an estimator which is locally too simple has a large residual error, whereas in regions, where the mapping is simple, both estimators have sufficient complexity, but the simpler one has less variance. In our experiments we only tried networks with the same complexity. Preliminary results indicate that variance-based weighting and error-based weighting are sometimes superior to simple averaging. The main reason seems to be that the local overfitting of a network is reflected in a large variance near that location in input space. The overfitting estimator therefore obtains a small weight in that region (compare the overfitting of network 1 in Figure Ib near x 0 and the small weight of network 1 close to x 0 in Figure lc). = = 425 Combining Estimators Using Non-Constant Weighting Functions 7 Conclusions We have presented modular ways for combining estimators. The weighting functions of each module can be determined independently of the other modules such that additional modules can be added without retraining of the previous system. This can be a useful feature in the context of the problem of catastrophic forgetting: additional data can be used to train an additional module and the knowledge in the remaining modules is preserved. Also note that estimators which are not derived from data can be easily included if it is possible to estimate the input dependent certainty or competence of that estimator. Acknowledgements: Valuable discussions with David Cohn, Michael Duff and Cesare Alippi are greatfully acknowledged. The first author would like to thank the Center for Biological and Computational Learning (MIT) for providing and excellent research environment during the summer of 1994. 8 8.1 Appendix Variance-based Weighting: Correlated Errors and Bias We maintain that Li gi(X) = 1. In general (Le. the modules have seen the same data, or partially the same data), we cannot assume that the errors in the individual modules are independent. Let the M x M matrix O( x) be the covariance between the predictions of the modules N Ni(X). With h(x) (h 1(x) .... hM(X)T, the optimal weighting vector becomes = h(x) = 0-1(X) n(x) U = u' 0-1(X) U where u is the M-dimensional unit vector. If the individual modules are biased (biasi(x) = ED(N Ni(X)) - EYI.r(ylx)),~ we form the M x M matrix B(x), with Bii(x) = biasi(x)biasj(x), and the minimum variance solution is found for h(x) = (O(x) 8.2 + B(X))-1 u n(x) = u' (O(x) + B(X))-1 u. Density-based Weighting: GEM-learning Let's assume a training pattern (Xlo' YIo) which is not associated with a particular module. If wi is a parameter in network N Ni the error gradient becomes 8errorlo __ (Ylo -NNo , ( Xlo ))p"(OIS Xlo, Ylo )8NNi(XIo) 8 8 w' w' ? - ? ? This equation can be derived from the solution to the problem of training with missing features (here: the true i is unknown, see Tresp, Ahmad and Neuneier, 1994). This corresponds also to the M-step in a generalized EM algorithm, where the E-step calculates " . P(SIXIo, YIo) P(Ylolxlo , i)P(xloli)P(i) = '" p o" L..Ji (Ylolxlo, s)P(xlols)P(s) 0 " . " ? P(Ylolxlo, s) = G(ylo; N Ni(XIo) , 2 (1 ). ~ E stands for the expected value; the expectation ED is taken with respect to all data sets of the same size. 426 Volker Tresp, Michiaki Taniguchi using the current parameters. The M-step in the "gating network" P(xli) is particularly simple using the well known EM-rules for Gaussian mixtures. Note, that P(module = i, mixture component: ilxk' Yk) needs to be calculated. 8.2.1 Unified Approach: Correlated Errors and General Coupling Let's form the vectors N N(x) = (N N 1(x), ... N NM(X))T and /(x) = (!t(x), ... , /M(x)f. In a more general case, the prior coupling between the underlying functions is described by P(f(x)) = G(f(x);g(x), ~g(x)) = where g{x) (g1{X), ... ,gM{x)f. Furthermore, in a more general case, the estimates are not independent, P{N N{x)l/{x)) = G{N N(x); I(x), ~N{X)). The minimum variance solution is now = The equations in Section 4 are special cases with M = 2, g{x) = 0, ~;1(x) l/varcc x (1, -1)(1, -If, ~N(X) = 1 (var{N N 1{x)), var(N N2{X)))T (1 is the 2D-unit matrix). References Ahmad, S. and Tresp, V. (1993) . Some Solutions to the Missing Feature Problem in Vision. In S. J. Hanson, J. D. Cowan and C. L. Giles, (Eds.), Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann. Breiman, L. (1992). Stacked Regression. Dept. of Statistics, Berkeley, TR No. 367. Hampshire, J. and Waibel, A. (1989). The meta-pi network: Building Distributed Knowledge Representations for Robust Pattern Recognition. TR CMU-CS-89-166, CMU, PA. Jacobs, R. A., Jordan, M. 1., Nowlan, S. J. and Hinton, J. E. (1991). Adaptive Mixtures of Local Experts. Neuml Computation, Vol. 3, pp. 79-87. Meir, R. (1994). Bias, Variance and the Combination of Estimators: The Case of Linear Least Squares. TR: Dept. of Electrical Engineering, Technion, Haifa. Nowlan, S. J and Sejnowski, T. J. (1994). Filter Selection Model for Motion Segmentation and Velocity Integration. J. Opt. Soc. Am. A, Vol. 11, No. 12, pp. 1-24. Perrone, M. P. (1993). Improving Regression Estimates: Averaging Methods for Variance Reduction with Extensions to General Convex Measure Optimization. PhD thesis. Brown University. Tibshirani, R. (1994). A Comparison of Some Error Estimates for Neural Network Models. TR Department of Statistics, University of Toronto. Tresp, V., Ahmad, S. and Neuneier, R. (1994). Training Neural Networks with Deficient Data. In: Cowan, J. D., Tesauro, G., and Alspector, J., eds., Advances in Neural Information Processing Systems 6, San Mateo, CA, Morgan Kaufman. Tresp, V., Hollatz J. and Ahmad, S. (1993). Network Structuring and Training Using Rule-based Knowledge. In S. J. Hanson, J. D. Cowan and C. L. Giles, (Eds.), Advances in Neural Information Processing Systems 5, San Mateo, CA: Morgan Kaufmann. 

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A model of the hippocampus combining selforganization and associative memory function. Michael E. Hasselmo, Eric Schnell Joshua Berke and Edi Barkai Dept. of Psychology, Harvard University 33 Kirkland St., Cambridge, MA 02138 hasselmo@katla.harvard.edu Abstract A model of the hippocampus is presented which forms rapid self-organized representations of input arriving via the perforant path, performs recall of previous associations in region CA3, and performs comparison of this recall with afferent input in region CA 1. This comparison drives feedback regulation of cholinergic modulation to set appropriate dynamics for learning of new representations in region CA3 and CA 1. The network responds to novel patterns with increased cholinergic modulation, allowing storage of new self-organized representations, but responds to familiar patterns with a decrease in acetylcholine, allowing recall based on previous representations. This requires selectivity of the cholinergic suppression of synaptic transmission in stratum radiatum of regions CA3 and CAl, which has been demonstrated experimentally. 1 INTRODUCTION A number of models of hippocampal function have been developed (Burgess et aI., 1994; Myers and Gluck, 1994; Touretzky et al., 1994), but remarkably few simulations have addressed hippocampal function within the constraints provided by physiological and anatomical data. Theories of the function of specific subregions of the hippocampal formation often do not address physiological mechanisms for changing dynamics between learning of novel stimuli and recall of familiar stimuli. For example, the afferent input to the hippocampus has been proposed to form orthogonal representations of entorhinal activity (Marr, 1971; McNaughton and Morris, 1987; Eichenbaum and Buckingham, 1990), but simulations have not addressed the problem of when these representations 78 Michael E. Hasselmo. Eric Schnell. Joshua Berke. Edi Barkai should remain stable, and when they should be altered. In addition, models of autoassociative memory function in region CA3 (Marr, 1971; McNaughton and Morris, 1987; Levy, 1989; Eichenbaum and Buckingham, 1990) and heteroassociative memory function at the Schaffer collaterals projecting from region CA3 to CAl (Levy, 1989; McNaughton, 1991) require very different activation dynamics during learning versus recall. Acetylcholine may set appropriate dynamics for storing new information in the cortex (Hasselmo et aI., 1992, 1993; Hasselmo, 1993, 1994; Hasselmo and Bower, 1993). Acetylcholine has been shown to selectively suppress synaptic transmission at intrinsic but not afferent fiber synapses (Hasselmo and Bower, 1992), to suppress the neuronal adaptation of cortical pyramidal cells (Hasselmo et aI., 1994; Barkai and Hasselmo, 1994), and to enhance long-term potentiation of synaptic potentials (Hasselmo, 1994b). Models show that suppression of synaptic transmission during learning prevents recall of previously stored information from interfering with the storage of new information (Hasselmo et al., 1992, 1993; Hasselmo, 1993, 1994a), while cholinergic enhancement of synaptic modification enhances the rate of learning (Hasselmo, 1994b). Feedback regulation of cholinergic modulation may set the appropriate level of cholinergic modulation dependent upon the novelty or familiarity of a particular input pattern. We have explored possible mechanisms for the feedback regulation of cholinergic modulation in simulations of region CAl (Hasselmo and Schnell, 1994) and region CA3. Here we show that self-regulated learning and recall of self-organized representations can be obtained in a network simulation of the hippocampal formation. This model utilizes selective cholinergic suppression of synaptic transmission in stratum radiatum of region CA3, which has been demonstrated in brain slice preparations of the hippocampus. 2 METHODS 2.1. SIMPLIFIED REPRESENTA nON OF HIPPOCAMPAL NEURONS. In place of the sigmoid input-output functions used in many models, this model uses a simple representation in which the output of a neuron is not explicitly constrained, but the total network activity is regulated by feedback from inhibitory interneurons and adaptation due to intracellular calcium concentration. Separate variables represent pyramidal cell membrane potential a, intracellular calcium concentration c, and the membrane potential of inhibitory interneurons h: l1a i = Ai -l1 ai - J..l.C + L Wijg(aj - e) - Hikg(h k - e h) j I1c?I = 'Vg(a .i? I I1hk e )- Qc C = IWkjg(aj-eo)-l1hk- IHk/g(h/-e) j where A = afferent input, " / =passive decay of membrane potential, Il =strength of cal- A Model of Hippocampus 79 cium-dependent potassium current (proportional to intracellular calcium), Wij = excitatory recurrent synapses (longitudinal association path tenninating in stratum radiatum), gO is a threshold linear function proportional to the amount by which membrane potential exceeds an output threshold 00 or threshold for calcium current Oc' 'Y = strength of voltagedependent calcium current, n = diffusion constant of calcium, Wki = excitatory synapses inhibitory interneurons, Hilc = inhibitory synapses from interneurons to pyramidal cells, Hk}= inhibitory synapses between interneurons. This representation gives neurons adaptation characteristics similar to those observed with intracellular recording (Barkai and Hasselmo, 1994), including a prominent afterhyperpolarization potential (see Figure 1). An B .... .. ~ ....J 'N~JJL C lO \..--14 Figure 1. Comparison of pyramidal cell model with experimental data. In Figure I, A shows the membrane potential of a modeled pyramidal cell in response to simulated current injection. Output of this model is a continuous variable proportional to how much membrane potential exceeds threshold. This is analogous to the reciprocal of interspike interval in real neuronal recordings. Note that the model displays adaptation during current injection and afterhyperpolarization afterwards, due to the calcium-dependent potassium current. B shows the intracellularly recorded membrane potential in a pirifonn cortex pyramidal cell, demonstrating adaptation of firing frequency due to activation of calcium-dependent potassium current. The firing rate falls off in a manner similar to the smooth decrease in firing rate in the simplified representation. C shows an intracellular recording illustrating long-tenn afterhyperpolarization caused by calcium influx induced by spiking of the neuron during current injection. 2.2. NETWORK CONNECTIVITY A schematic representation of the network simulation of the hippocampal fonnation is shown in Figure 2. The anatomy of the hippocampal fonnation is summarized on the left in A, and the function of these different subregions in the model is shown on the right in B. Each of the subregions in the model contained a population of excitatory neurons with a single inhibitory interneuron mediating feedback inhibition and keeping excitatory activity bounded. Thus, the local activation dynamics in each region follow the equations presented above. The connectivity of the network is further summarized in Figure 3 in the Results section. A learning rule of the Hebbian type was utilized at all synaptic connections, with the exception of the mossy fibers from the dentate gyrus to region CA3, and the connections to and from the medial septum. Self-organization of perforant path synapses was obtained through decay of synapses with only pre or post-synaptic activity, and growth of synapses with combined activity. Associative memory function at synapses 80 Michael E. Hasse/mo, Eric Schnell, Joshua Berke, Edi Barkai arising from region CA3 was obtained through synaptic modification during cholinergic suppression of synaptic transmission. B Entorhinal cortex """""""""" ?? ? ??? : Self-organized : representation : r-----..L.\ ? ~ Comparison .L-______~~.-----~~~--~~ Feedback regulation of cholinergic modulation Regulation of learning dynamics Figure 2. Schematic representation of hippocampal circuitry and the corresponding function of connections in the model. 2.3. CHOLINERGIC MODULAnON The total output from region CAl determined the level of cholinergic modulation within both region CA3 and CAl, with increased output causing decreased modulation. This is consistent with experimental evidence suggesting that activity in region CAl and region CA3 can inhibit activity in the medial septum, and thereby downregulate cholinergic modulation. This effect was obtained in the model by excitatory connections from region CAl to an inhibitory interneuron in the medial septum, which suppressed the activity of a cholinergic neuron providing modulation to the full network. When levels of cholinergic modulation were high, there was strong suppression of synaptic transmission at the excitatory recurrent synapses in CA3 and the Schaffer collaterals projecting from region CA3 to CAL This prevented the spread of activity due to previous learning from interfering with self-organization. When levels of cholinergic modulation were decreased, the strength of synaptic transmission was increased, allowing associative recall to dominate. Cholinergic modulation also increased the rate of synaptic modification and depolarized neurons. 2.4. TESTS OF SELF-REGULATED LEARNING AND RECALL Simulations of the full hippocampal network evaluated the response to the sequential presentation of a series of highly overlapping activity patterns in the entorhinal cortex. Recall was tested with interspersed presentation of degraded versions of previously presented activity patterns. For effective recall, the pattern of activity in entorhinal cortex layer IV evoked by degraded patterns matched the pattern evoked by the full learned version of these patterns. The function of the full network is illustrated in Figure 3. In simulations A Model of Hippocampus 81 focused on region CA3, activity patterns were induced sequentially in region CA3, representing afferent input from the entorhinal cortex. Different levels of external activation of the cholinergic neuron resulted in different levels of learning of new overlapping patterns. These results are illustrated in Figure 4. 2.5. BRAIN SLICE EXPERIMENTS The effects in the simulations of region CA3 depended upon the cholinergic suppression of synaptic transmission in stratum radiatum of this region The cholinergic suppression of glutamatergic synaptic transmission in region CA3 was tested in brain slice preparations by analysis of the influence of the cholinergic agonist carbachol on the size of field potentials elicited by stimulation of stratum radiatum. These experiments used techniques similar to previously published work in region CAl (Hasselmo and Schnell, 1994). 3 RESULTS In the full hippocampal simulation, input of an unfamiliar pattern to entorhinal cortex layer II resulted in high levels of acetylcholine. This allowed rapid self-organization of the perforant path input to the dentate gyrus and region CAl. Cholinergic suppression of synaptic transmission in region CAl prevented recall from interfering with self-organization. Instead, recurrent collaterals in region CA3 stored an autoassociative representation of the input from the dentate gyrus to region CA3, and connections from CA3 to CA 1 stored associations between the pattern of activity in CA3 and the associated self-organized representation in region CAl. identity " self-org "matrix ~ ,auto-" ,,~> M assoc u ..... ? ? 'at) u u c >. --I.~ ? c ? c ? :.a.?:i Self-org ~ iden~ity .9 hetero.9 hetero8 matrix ~ assoc ~ assoc C! ~ ~ ? ~ 111111 I "T I' , If ,r 2 I I I II n I I ld II r I" I 'I"'"'I II , , , Q)2dl' r II' I I ' l l I ~ 311111 I' I I I n " j 4 3d 4d ld 2d n,n II II II 'f I I II I n I II I I , , II , I 1I 'I I 'I .. , II f'l't? U i I I Itt ~ I.Ll r" U I " H , I I U I I 1 II r ~' ,, 1 , '( ( U. " I I I 1'1 I I I I I n I"'" , 'I II I II 1 I ' ,I I l 11 I J1 't III I' ?" 'I . . I I , I I II I II ? .1 l " I II I III I I W. Jl 1 lU Neuron # Figure 3. Activity in each subregion of the full network simulation of the hippocampal formation during presentation of a sequence of activity patterns in entorhinal cortex. 82 Michael E. Hasselmo, Eric Schnell, Joshua Berke, Edi Barkai In Figure 3. width of the lines represents the activity of each neuron at a particular time step. As seen here. the network forms a self-organized representation of each new pattern consisting of active neurons in the dentate gyrus and region CAL At the same time. an association is formed between the self-organized representation in region CAl and the same afferent input pattern presented to entorhinal cortex layer IV. Four overlapping patterns (1-4) are presented sequentially. each of which results in learning of a separate selforganized representation in the dentate gyrus and region CAl. with an association formed between this representation and the full input pattern in entorhinal cortex. The recall characteristics of the network are apparent when degraded versions of the afferent input patterns are presented in the sequence (ld-4d). This degraded afferent input weakly activates the same representations previously formed in the dentate gyrus. Recurrent excitation in region CA3 enhances this activity. giving robust recall of the full version of this pattern. This activity then reaches CA 1. where it causes strong activation if it matches the pattern of afferent input from the entorhinal cortex. Strong activation in region CAl decreases cholinergic modulation. preventing formation of a new representation and allowing recall to dominate. Strong activation of the representation stored in region CAl then activates the full representation of the pattern in entorhinal cortex layer IV. Thus. the network can accurately recall each of many highly overlapping patterns. The effect of cholinergic modulation on the level of learning or recall can be seen more clearly in a simulation of auto-associative memory function in region CA3 as shown in Figure 4. Each box shows the response of the network to sequential presentation of full and degraded versions of two highly overlapping input patterns. The width of the black traces represents the activity of each of 10 CA3 pyramidal cells during each simulation step. In the top row. level of cholinergic modulation (ACh) is plotted. In A. external activation of the cholinergic neuron is absent. so there is no cholinergic suppression of synaptic transmission. In this case. the first pattern is learned and recalled properly. but subsequent presentation of a second overlapping pattern results only in recall of the previously learned pattern. In B. with greater cholinergic suppression. recall is suppressed sufficiently to allow learning of a combination of the two input patterns. Finally. in C. strong cholinergic suppression prevents recall. allowing learning of the new overlapping pattern to dominate over the previously stored pattern. A Stored patterns ??? ?? ?? ? ? -.gN.g N - ACh Inhib Q\ .- .--. ACh input = 0.0 ..... ...... ,11",,11. ..... B ACh input =0.15 C __ ACh input =0.3 ...11'_..,.......,.. ...... 111 ... 111...... 11 . . . ."'... 11 ? ?111 ??, ' I ? ? " '? ? _ _ '111" .';:: ~ ~ ?'. _"11.. ? .'.? . .,~ . ",. .u. . . . .... , 0 . . . . . . . . . . 111. . . .... . N Figure 4. Increased cholinergic suppression of synaptic transmission in region CA3 causes greater learning of new aspects of afferent input patterns. A Model of Hippocampus 83 Extracellular recording in brain slice preparations of hippocampal region CA3 have demonstrated that perfusion of the cholinergic agonist carbachol strongly suppresses synaptic potentials recorded in stratum radiatum, as shown in Figure 5. In contrast, suppression of synaptic transmission at the afferent fiber synapses arising from entorhinal cortex is much weaker. At a concentration of 20J..tM, carbachol suppressed synaptic potentials in stratum radiatum on average by 54.4% (n=5). Synaptic potentials elicited in stratum lacunosum were more weakly suppressed, with an average suppression of28%. Control Carbachol (20JlM) Wash Figure 5. Cholinergic suppression of synaptic transmission in stratum radiatum of CA3. 4 DISCUSSION In this model of the hippocampus, self-organization at perforant path synapses forms compressed representations of specific patterns of cortical activity associated with events in the environment. Feedback regulation of cholinergic modulation sets appropriate dynamics for learning in response to novel stimuli, allowing predominance of self-organization, and appropriate dynamics for recall in response to familiar stimuli, allowing predominance of associative memory function. This combination of self-organization and associative memory function may also occur in neocortical structures. The selective cholinergic suppression of feedback and intrinsic synapses has been proposed to allow self-organization of feedforward synapses while feedback synapses mediate storage of associations between higher level representations and activity in primary cortical areas (Hasselmo, 1994b). This previous proposal could provide a physiological justification for a similar mechanism utilized in recent models (Dayan et al., 1995). Detailed modeling of cholinergic effects in the hippocampus provides a theoretical framework for linking the considerable behavioral evidence for a role of acetylcholine in memory function (Hagan and Morris, 1989) to the neurophysiological evidence for the effects of acetylcholine within cortical structures (Hasselmo and Bower, 1992; 1993; Hasselmo, 1994a, 1994b). Acknowledgements This work supported by a pilot grant from the Massachusetts Alzheimer's Disease Research Center and by an NIMH FIRST award MH52732-01. References Barkai E, Hasselmo ME (1994) Modulation of the input/output function of rat piriform cortex pyramidal cells. J. Neurophysiol. 72: 644-658. 84 Michael E. Hasselmo, Eric Schnell, Joshua Berke, Edi Barkai Barkai E, Bergman RE, Horwitz G, Hasselmo ME (1994) Modulation of associative memory function in a biophysical simulation of rat pirifonn cortex. J. Neurophysiol. 72:659677. Burgess N, Reece M, O'Keefe J (1994) A model of hippocampal function. Neural Networks 7: 1065-1081. Dayan P, Hinton GE, Neal RM and Zemel RS (1995) The Helmholtz machine. Neural computation in press. Eichenbaum, H. and Buckingham, J. (1990) Studies on hippocampal processing: experiment, theory and model. In: Learning and computational neuroscience: foundations of adaptive networks, M. Gabriel and J. Moore, eds., Cambridge, MA: MIT Press. Hagan, JJ and Morris, RGM (1989) The cholinergic hypothesis of memory: A review of animal experiments. In Psychopharmacology of the Aging Nervous System, L.L. Iversen, S.D. Iversen and S.H. Snyder, eds. New York: Plenum Press, p. 237-324. Hasselmo, M.E. (1993) Acetylcholine and learning in a cortical associative memory. Neural Compo 5: 22-34. Hasselmo ME (1994a) Runaway synaptic modification in models of cortex: Implications for Alzheimer's disease. Neural Networks 7: 13-40. Hasselmo ME (1994b) Neuromodulation and cortical function. Behav. Brain Res. in press Hasselmo ME, Anderson, BP and Bower, JM (1992) Cholinergic modulation of cortical associative memory function. J. Neurophysiol. 67(5): 1230-1246. Hasselmo ME, Bower JM (1992) Cholinergic suppression specific to intrinsic not afferent fiber synapses in rat pirifonn (olfactory) cortex. J. Neurophysiol. 67(5): 1222-1229. Hasselmo ME, Bower JM (1993) Acetylcholine and memory. Trends Neurosci 16:218222. Hasselmo ME, Barkai E, Horwitz G, Bergman RE (1993) Modulation of neuronal adaptation and cortical associative memory function. In: Computation and Neural Systems II (Eeckman F, Bower JM, ed). Norwell, MA: Kluwer Academic Publishers. Hasselmo ME, Schnell E (1994) Laminar selectivity of the cholinergic suppression of synaptic transmission in rat hippocampal region CAl: Computational modeling and brain slice physiology. J. Neurosci. 14: 3898-3914. Levy WB (1989) A computational approach to hippocampal function. In: Computational models of learning in simple neural systems (Hawkins RD, Bower GH, ed), pp. 243-305. Orlando, FL: Academic Press. Myers CE and Gluck M (1994) Context, conditioning and hippocampal rerepresentation in animal learning. Behav. Neurosci. 108: 835-847. Marr 0 (1971) Simple memory: A theory for archicortex. Phil. Trans. Roy. Soc. B B262:23-81 McNaughton BL (1991) Associative pattern completion in hippocampal circuits: New evidence and new questions. Brain Res. Rev. 16:193-220. McNaughton BL, Morris RGM (1987) Hippocampal synaptic enhancement and infonnation storage within a distributed memory system. Trends Neurosci. 10:408-415. Touretzky OS, Wan HS and Redish AD (1994) Neural representation of space in rats and robots. In Zurada JM and Marks RJ (eds) Computational Intelligence: Imitating life. IEEE Press. 

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Catastrophic Interference in Human Motor Learning Tom Brashers-Krug, Reza Shadmehr t , and Emanuel Todorov Dept. of Brain and Cognitive Sciences, M. I. T., Cambridge, MA 02139 tCurrently at Dept. of Biomedical Eng., Johns Hopkins Univ., Baltimore, MD 21205 Email: tbk@ai.mit.edu, reza@bme.jhu.edu, emo@aLmit.edu Abstract Biological sensorimotor systems are not static maps that transform input (sensory information) into output (motor behavior). Evidence from many lines of research suggests that their representations are plastic, experience-dependent entities. While this plasticity is essential for flexible behavior, it presents the nervous system with difficult organizational challenges. If the sensorimotor system adapts itself to perform well under one set of circumstances, will it then perform poorly when placed in an environment with different demands (negative transfer)? Will a later experience-dependent change undo the benefits of previous learning (catastrophic interference)? We explore the first question in a separate paper in this volume (Shadmehr et al. 1995). Here we present psychophysical and computational results that explore the question of catastrophic interference in the context of a dynamic motor learning task. Under some conditions, subjects show evidence of catastrophic interference. Under other conditions, however, subjects appear to be immune to its effects. These results suggest that motor learning can undergo a process of consolidation. Modular neural networks are well suited for the demands of learning multiple input/output mappings. By incorporating the notion of fast- and slow-changing connections into a modular architecture, we were able to account for the psychophysical results. 20 1 Tom Brashers-Krug, Reza Shadmelzr, Emanuel Todorov Introduction Interacting physically with the world changes the dynamics of one's limbs. For example, when holding a heavy load, a different pattern of muscular activity is needed to move one's arm along a particular path than when not holding a load. Previous work in our laboratory has shown that humans learn a novel dynamic task by forming an internal model of the new inverse dynamics of thier limbs. (Shadmehr and Mussa-Ivaldi 1994, Shadmehr et aI, 1995). Preliminary evidence suggests that subjects can retain one of these internal models over time (BrashersKrug, et al. 1994). Humans are required, however, to move their limbs effectively under a large number of dynamic conditions. Are people able to learn and store an inverse dynamic model appropriate for each condition, or do they form such models from scratch as they need them? In particular, can learning a new inverse dynamic model overwrite or displace a previous model? We will present evidence that certain conditions must be met before a subject is able to retain more than one inverse dynamic model in a given experimental context. These conditions can be modeled as leading to a process of consolidation, whereby learning is transfered from vulnerable, low-capacity storage to a long-term, high-capacity storage. 2 Experimental Protocol We have developed a motor learning paradigm that allows us to alter the dynamics of a subject's arm and so to monitor a subject's ability to learn dynamic tasks. A subject moves the handle on the free end of a two-link planar robot arm-called a manipulandum-to guide a cursor to a series of targets displayed on a computer screen (fig la). The position and velocity of the handle of the manipulandum are recorded at ten-millisecond intervals and are used to deliver state-dependent forces to the subject's hand. In order to test a subject's ability to learn a novel dynamic task, we present the subject with a viscous force field as slhe moves from one target to the next (fig Ib). Initially, such forces perturb the subject's movements, causing them to depart from the smooth, straight-line trajectories of the baseline condition (i .e., the condition before the viscous forces were presented) (figs lc,ld). The extent of learning is measured as the degree to which a subject's movements in the force field over time come to resemble that subject's baseline movements. We have shown in previous work that subjects adapt to the imposed force fileds by forming a predictive model of the velocity-dependent forces, and that subjects use this inverse dynamic model to control their arms in what appears to be a feedforward manner (Shadmehr and Mussa-Ivaldi 1994). 3 3.1 Psychophysical Findings Catastrophic Interference Here, we employed this paradigm to explore the consequences of learning two different dynamic tasks in a row. In an initial series of experiments, we allowed twelve subjects to learn to move the manipulandum in a first force field (Field A) for approximately 5 minutes. Immediately after this first set of movements, we presented the subjects with an anti-correlated force field (Field B). For example, if we pre- Catastrophic Interference in Human Motor Learning <Y'/'/ . C 21 \ D Figure 1: A: The experimental setup. B: An example of a viscous field, plotted in velocity space (mm/sec). The arrows indicate the direction and magnitude of the forces exerted by the manipulandum on the subject's hand at each location in velocity space. C: One subject's trajectories before forces were introduced. Targets are indicated by the open circles. D: Trajectories immediately after the force field in (B) was presented. sen ted the counter-clockwise curl field depicted in fig. 1B as Field A, we would next present a clockwise curl field as Field B. Half the subjects learned the clockwise curl field first and the counter-clockwise field second; the other halflearned the two fields in the reverse order. (The first field will be referred to as Field A and the second field as Field B, whichever field was learned first.) The subjects' mean performance in Field B was worse (p< 0.0001, paired t-test) than in Field A. This phenomenon has been called negative transfer in the psychophysical literature. Negative transfer in this motor learning paradigm is explored more fully in a separate paper in this volume (Shadmehr et aI, 1995). In that paper, we suggested that this negative transfer could result from the fact that the same neural elements are learning both tasks. We predicted that, if this is the case, learning to move in Field B would interfere with a subject's ability to retain an inverse dynamic model of Field A. Learning to move in Field B would, in effect, cause subjects to "unlearn" Field A, resulting in catastrophic interference. In order to test this prediction, we compared the improvement in performance from one day to the next of two groups of subjects, with twelve subjects in each group. The subjects in the control group learned to move in one force field on Day One and were then tested on Day Two in the same force field. The subjects in the experimental group learned two separate force fields in a row on Day One and were then tested on Day Two in the first force field they learned. The experimental group retained significantly less of their learning (p< 0.01, paired t-test) from Day One to Day Two than the control group (figs 2a,2b) . In other words, learning the second force field resulted in catastrophic interference. (The question of whether this represents a storage or a retrieval phenomenon is beyond the scope of this paper.) 22 3.2 Tom Brashers-Krug, Reza Shadmehr, Emanuel Todorov Consolidation Having found evidence for catastrophic interference, we wanted to know whether there were circumstances under which dynamic motor learning was immune to being functionally erased by subsequent learning. We therefore tested two further groups of subjects. We allowed these subjects to practice longer in one field before they learned the second field. We also allowed 24 hours to pass between the time subjects first learned one field and when they learned the second field . The subjects in the experimental group (n = 10) practiced in one force field for approximately 15 minutes on Day One. They returned on Day Two and practiced in the same force field for five more minutes. They were then allowed to practice in a second force field for 15 minutes. By the end of this fifteen minutes, they were performing in the second field at a level comparable to the level they acheived in the first force field. We had the subjects return on Day Three, when they were tested for their retention of the first field they learned. We compared the difference in performance on Day Two and Day Three of this experimental group with that of a control group (n = 9) who followed the same protocol in all respects except that they never learned a second force field . In this way we could determine whether learning the second field resulted in a decrement in performance for the experimental group when compared with the control group. Under these conditions, we found no difference in the retention of learning between the experimental and control groups (fig 2c, 2d) . That is, learning the second field under these conditions no longer resulted in catastrophic interference. What subjects had learned about the first field had become resistant to such interference. It had become consolidated. We can not tell from these experiments whether such consolidation is the result of the increased practice in the first field, or whether it is the result of the 24 hours that elapsed between when the first field was first learned and when the second field was learned. There is evidence that increased practice in a motor task can engage different neural circuits than those that are active during initial learning (Jenkins, et al 1994). The shift to "practiced" circuits may represent the neural substrate of consolidation. There is also evidence that time can be an important factor in the consolidation of skill learning. (Karni and Sagi 1993) In the next section, we present a model of our results that assumes that time is the key variable in the consolidation of motor learning. The model could also be applied to a practice-based model of consolidation with minor modifications. 4 Computational Modeling of the Experimental Results In order to model the results presented above we need a network that learns to compute an appropriate control signal Y given the current state and the desired next state X of the plant. More precisely, it needs to compute a mapping from joint angles 0, joint velocities iJ, and desired joint accelerations jj to torques. Several approaches for solving this problem have been proposed. One way to learn a mapping from X into Y is to use direct inverse modeling: apply a control signal, measure the next state of the plant, and use the current state, new state, and control signal as a training pair for the controller. This approach is not suitable for explaining non- 23 Catastrophic Interference ill Human Motor Learning 0.95 0.95 0.85 0.75 ~ "," 0.75 A 0.65 0 50 0.85 100 0 150 0.95 0.85 0.85 0.75 0 50 100 150 B 0.65 0.95 C ~ 50 100 150 ~ D 0.75 0 50 100 150 Figure 2: Plots of average learning curves. The correlation of trajectories in the force field to baseline trajectories (before the force field was applied) is plotted as a function of movement number. A: Learning curves for the first experimental group. Dark curve: learning curve on Day One in Field A . Light curve: learning curve in Field A on Day Two. This group learned Field B immediately after Field A on Day One (learning curve for Field B not shown). Note minimal improvement from Day One to Day Two. B: Learning curves for the first control group. Dark curve: learning curve in Field A on Day One. Light curve: learning curve in Field A on Day Two. Control group never learned Field B. Note significant improvement from Day One to Day Two. C: Second experimental group. Dark line: learning curve in Field A on Day Two immediately before learning Field B (Field B curve not shown). Light line: learning curve in Field A on Day Three. D: Second control group. Dark and light lines: learning curves in FIeld A on Day Two and Three, respectively. Note the similarity of the curves in C and D. This indicates that learning Field B did not significantly affect the experimental group's retention of Field A. All curves in C and D are significantly higher than curves for the initial learning of Field A on Day One. convex mappings, however. The learning situation we must model is non-convex: we change the dynamic environment the controller operates in by presenting force fileds, and so there will be different Y values corresponding to any X value. A different approach that solves the non convexity problem is distal supervised learning (Jordan and Rumelhart 1992): produce control signals, observe the new state of the plant, and use that information to train a forward model that maps actions into states; then learn a controller, using the forward model to backpropagate error from observable cartesian coordinates to the unknown control space. Distal supervised learning solves the non convexity problem by learning one correct value of Y for each X. But that can not explain the consolidation of learning - when the force field changes back to something already learned, our controller should rapidly recover its performance in that old field, meaning that it should retain information about all Y s that a particular X can map into. An architecture that seems to have most of the desirable properties discussed above is the Mixture of Experts (ME) model (Jordan and Jacobs 1994): several experts learn a mapping from X to Y . A separate gating network selects an expert which is most likely to be correct at each moment. Such a model has been used previously (Jacobs and Jordan 1993) to learn a feedforward controller for a 2-joint planar arm operating under different loads. In their model however they assumed that the 24 Tom Brashers-Krug, Reza Shadmelzr, Emanuel Todorov identity of the load is known to the controller. The subjects in our study were not given any explicit cues about the identity of the fields they were learning. The mixture of experts cannot be used directly here because it decides which expert to select based on a soft partitioning of the input space, and in our experiment any force field is active over any portion of the input space at different moments in time. Here we propose an extension to the ME architecture that is able to deal with mappings overlapping in X space. Another aspect of the results that is difficult to model using standard computational architectures is memory consolidation. To account for this effect we introduce two different learning rates (Alverez and Squire 1994). Some connections in the network change faster, as a result of which they can serve as short-term memory. We also introduce an off-line training phase (possibly corresponding to sleep) in which random inputs are generated, the part of the network containing the fast connections is used to produce a target output, and the resulting input-output pair is used to train the slowly changing connections. During the offline phase the faster changing connections are fixed, after that they are randomized. 4.1 Modified Mixture of Experts The ME model assumes that Y is generated from X by one of N processes (Wl, ... , WN) and therefore the likelihood function is: L(8IXt, Yt) = P(Ytlxt, 8) = I: gi t P(YtIWi, Xt, 8) i g/ = P(WiIXt, 8), where 8 represents the parameters of the model and gi is the prior probability. We want to use the posterior probability P(Wi IXt, Yt) instead, because the processes (different force fields) are separable in XY space, but not in X space. If we want to implement an on-line controller such a term is not available, because at time t, Yt is still unknown (the task of the controller is to produce it). We could approximate P(WiIXt, Yt) with P(WdXt-l. Yt-!) , because dynamic conditions do not change very often. Now the gating network (which computes P(Wdxt) in ME) is going to select expert i based on the previous XY pair. This approach would obviously lead to a single large error at the moment when the force field changes, but so will any model using only Xt to compute Yt. In fact such an error seems to be consistent with our psychophysics data. Thus the learning rule is: ~8i = lIih/(Yt - J.'i(Xt, 8i)) gI.t -_ h.t-1 I h .t I h: = P(W~I I ~ ~ 8) = XII y" gi t P(YtIWi , Xt, 8) '" L..tj g,.tp(Yt IW. "Xt, 8)' where is the posterior probability and J.'i is the output of exper i. J.'i is a linear function of the inputs. In our model we used 4 experts. In order to model the process of consolidation, we gave one expert a learning rate that was 10 times higher than the learning rate of the other 3 experts. Catastrophic Interference in Human Motor Learning 25 4.2 The Model We simulated the dynamics of a 2-joint planar arm similar to the one used in our previous work (Shadmehr and Mussa-Ivaldi, 1994). The torque applied to the arm at every point in time is the sum of the outputs of a fixed controller, a PO controller, and an adaptive controller with the architecture described above. The fixed controller maps (), iJ (current state), and jj (desired next state) into a torque T. The mapping is exact when no external forces are applied. The desired trajectories are minimum-jerk trajectories (Flash and Hogan 1985) sampled at 100Hz. The desired trajectories are 10 cm long and last 0.5 seconds. The PO controller is used to compensate for the troques produced by the force field while the adaptive controller has not yet learned to do that. The adaptive part of the controller consists of a mixture of 4 linear experts (whose initial output is small) and a modified gating network described above. The system operates as follows: (), iJ, jj are sent to the fixed controller, which outputs a torque Tl; the PO controller outputs a torque T2 based on the current deviation from the desired joint position and velocity; 8 terms describing the current state of the arm (and chosen to linearize the mapping to be learned) are sent to the mixture model, which outputs a torque T3; Te = Tl + T2 - T3 is applied to the plant as a control signal; the actual torque T = Te + TJ is computed. The mixture model is trained to produce the torque TJ resulting from the force field. In other words, the adaptive part of the controller learns to compensate for the force field exerted by the environment. The parameters of the mixture model are updated after every movement, so a training pair (Xt, Yt) is actually a batch of 50 points. The input, Xt, consists of the 8 terms describing the current state and the desired next state; the output, Yt, is the torque vector that the force field produces.The compensatory torques for a complete movement are computed before the movement starts. The only processing done during the movement is the computations necessary for the PO controller. 4.3 Results 4.3.1 Negative Transfer When the network was given two successive, incompatable mappings to learn (this corresponds to learning to move in two opposite force fields), the resulting performance very much resembled that of our human subjects. The performance in the second mapping was much poorer than that in the first mapping. The fast-learning expert changed its weights to learn both tasks. Since the two tasks involved anticorrelated maps, the fast expert's weights after learning the first mapping were very inappropriate for the second task, leading to the observed negative transfer. 4.3.2 Catastrophic Interference When the network was trained on two successive, opposite force fields, with no consolidation occurring between the two training sessions, the learning in second training session overwrote the learning that occurred during the first training session (fig 3A). Since the expert with the fast-changing weights attempted to learn both mappings, this catastrophic interference is not unexpected. 4.3.3 Consolidation When the network was allowed to undergo consolidation between learning the first and the second force field, the network no longer suffered from catastrophic inter- 26 Tom Braslzers-Krug, Reza Slzadmelzr, Emanuel Todorov A --:=" D.65~_ _ _ _~_ _ _ o 10 20 Figure 3: A: Learning curves for the ME architecture. Dark line: curve when first learning Field A. light line: curve when given Field A a second time, after learning FIeld B (no consolidation allowed between learning Field A and Field B). Note lack of retention of Field A. B: Learning curves for the same architecture in Field A before and after learning Field B. Consolidations was allowed between learning Field A and Field B. ference (fig 3B). The learning that had initially resided in the fast-learning expert was transfered to one of the slower-learning networks. Thus, when the expert with the fast-changing connections learned the second mapping, the original learning was no longer destroyed. In addition, when the network was allowed to consolidate the second force field, a different slow-learning expert stored the second mapping. In this way, the network stored multiple maps in long-term memory. 5 Conclusions We have presented psychophysical evidence for catastrophic interference. We have also shown results that suggest that motor learning can undergo a process of consolidation. By adding distinct fast- and slow-changing weights to a mixture of experts architecture, we were able to account for these psychophysical findings. We plan to investigate further the neural correlates of consolidation using both brain imaging in humans and electrophysiological studies in primates. References P. Alverez and L. Squire. (1994) Memory consolidation and the medial temporal love: a simple network model, PNAS91:15:7041-7045. T. Brashers-Krug, et al. (1994) Temporal aspects of human motor learning, Soc. Neurosci. Abstract in press. T. Flash and N. Hogan. (1985) The coordination of arm movements: an experimentally confirmed mathematical model. J. Neurosci. 5:1688-1703. R. Jacobs and M. Jordan. (1993) Learning piecewise control strategies in a modular neural network architecture. IEEE Trans. on Systems, Man and Cyber. 23:2: 337-345. I. Jenkins, et al. (1994) Motor sequence learning: a study with positron emission tomography, J. Neurosci.14:3775-3790. M. Jordan and R. Jacobs. (1994) Hierarchical mixture of experts and the EM algorithm, Neural Computation 6:2: 181-214. M. Jordan and D. Rumelhart. (1992) Forward models: supervised learning with a distal teacher Cognitive Sci. 16:307-354 A. Karni and D. Sagi. (1993) Nature 365:250. R. Shadmehr and F . Mussa-Ivaldi. (1994) Adaptive representation of dynamics during learning of a motor task, J. Neurosci.14:5:3208-3224. R. Shadmehr, T. Brashers-Krug, F. Mussa-Ivaldi, (1995) Interference in learning internal models of inverse dynamics in humans, Adv Neural Inform Proc Systvol 7 , in press 

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Model of a Biological Neuron as a Temporal Neural Network Sean D. Murphy and Edward W. Kairiss Interdepartmental Neuroscience Program, Department of Psychology, and The Center for Theoretical and Applied Neuroscience, Yale University, Box 208205, New Haven, CT 06520 Abstract A biological neuron can be viewed as a device that maps a multidimensional temporal event signal (dendritic postsynaptic activations) into a unidimensional temporal event signal (action potentials). We have designed a network, the Spatio-Temporal Event Mapping (STEM) architecture, which can learn to perform this mapping for arbitrary biophysical models of neurons. Such a network appropriately trained, called a STEM cell, can be used in place of a conventional compartmental model in simulations where only the transfer function is important, such as network simulations. The STEM cell offers advantages over compartmental models in terms of computational efficiency, analytical tractabili1ty, and as a framework for VLSI implementations of biological neurons. 1 INTRODUCTION Discovery of the mechanisms by which the mammalian cerebral cortex processes and stores information is the greatest remaining challenge in the brain sciences. Numerous modeling studies have attempted to describe cortical information processing in frameworks as varied as holography, statistical physics, mass action, and nonlinear dynamics. Yet, despite these theoretical studies and extensive experimental efforts, the functional architecture of the cortex and its implementation by cortical neurons are largely a mystery. Our view is that the most promising approach involves the study of computational models with the following key properties: (1) Networks consist of large (> 103 ) numbers of neurons; (2) neurons are connected by modifiable synapses; and (3) the neurons themselves possess biologically-realistic dynamics. Property (1) arises from extensive experimental observations that information processing and storage is distributed over many neurons. Cortical networks are also characterized by sparse connectivity: the probability that any two local cortical neurons are synaptically connected is typically less than 0.1. These and other observations suggest that key features of cortical dynamics may not be apparent unless large, sparsely-connected networks are studied. Property (2) is suggested by the accumulated evidence that (a) memory formation is subserved by use-dependent synaptic modification, and (b) Hebbian synaptic plasticity is present in many areas of the brain thought to be important for memory. It is also well known that artificial networks composed of elements that are connected by Hebb-like synapses have powerful computational properties. 86 Sean D. Murphy, Edward W. Kairiss Property (3) is based on the assumption that biological neurons are computationally more complex than. for example. the processing elements that compose artificial (connectionist) neural networks. Although it has been difficult to infer the computational function of cortical neurons directly from experimental data, models of neurons that explicitly incorporate biophysical components (e.g. neuronal geometry, channel kinetics) suggest a complex, highly non-linear dynamical transfer function. Since the "testability" of a model depends on the ability to make predictions in terms of empirically measurable single-neuron firing behavior, a biologically-realistic nodal element is necessary in the network model. Biological network models with the above properties (e.g. Wilson & Bower, 1992; Traub and Wong, 1992) have been handicapped by the computationally expensive single-neuron representation. These "compartmental" models incorporate the neuron's morphology and membrane biophysics as a large (102 _104) set of coupled, non-linear differential equations. The resulting system is often stiff and requires higher-order numerical methods and small time-steps for accurate solution. Although the result is a realistic approximation of neuronal dynamics, the computational burden precludes exhaustive study of large networks for functionality such as learning and memory. The present study is an effort to develop a computationally efficient representation of a single neuron that does not compromise the biological dynamical behavior. We take the position that the "dynamical transfer function" of a neuron is essential to its computational abstraction, but that the underlying molecular implementation need not be explicitly represented unless it is a target of analysis. We propose that a biological neuron can be viewed as a device that performs a mapping from multidimensional spatio-temporal (synaptic) events to unidimensional temporal events (action potentials). This computational abstraction will be called a Spatio-Temporal Event Mapping (STEM) cell. We describe the architecture of the neural net that implements the neural transfer function, and the training procedure required to develop realistic dynamics. Finally, we discuss our preliminary analyses of the performance of the model when compared with the full biophysical representation. 2 STEM ARCHITECTURE The architecture of the STEM cell is similar to that found in neural nets for temporal sequence processing (e.g. review by Mozer, in press). In general, these networks have 2 components: (1) a short-term memory mechanism that acts as a preprocessor for (2) a nonlinear feedforward network. For example, de Vries & Principe (1992) describe the utility of the gamma net, a real-time neural net for temporal processing, in time series prediction. The preprocessor in the gamma net is the gamma memory structure, implemented as a network of adaptive dispersive elements (de Vries & Principe, 1991). The preprocessor in our model (the "tau layer", described below) is somewhat simpler, and is inspired by the temporal dynamics of membrane conductances found in biological neurons. The STEM architecture (diagrammed in Figure 1) works by building up a vectorial representation of the state of the neuron as it continuously receives incoming synaptic activations, and then labeling that vector space as either "FIRE" or "DON'T FIRE". This is accomplished with the use of four major components: (1) TAU LAYER: a layer of nodes that continuously maps incoming synaptic activations into a finite-dimensional vector space (2) FEEDBACK TAU NODE: a node that maintains a vectorial representation of the past activity of the cell itself (3) MLP: a multilayer perceptron that functions as a nonlinear spatial mapping network that performs the "FIRE" / "NO-FIRE" labeling on the tau layer output (4) OUTPUT FILTER: this adds a refractory period and threshold to the MLP output that contrains the format of the output to be discrete-time events. Model of Biological Neuron as a Temporal Neural Network 87 input (spike train at each synapse) " V Ispatio-temporalto spatial projection layer !_ u l nonlinear spatial mapping network Q - I spike train output I output processor : Figure 1: Information Flow in The STEM Cell The tau layer (Fig. 2) consists of N + 1 tau nodes, where N is the number of synapses on the cell, and the extra node is used for feedback. Each tau node consists of M tau units. Each tau unit within a single tau node receives an identical input signal. Each tau unit within a tau node calculates a second-order rise-and-decay function with unique time constants. The tau units within a tau node translate arbitrary temporal events into a vector form, with each tau-unit corresponding to a different vector component. Taken as a whole, all of the tau unit outputs of the tau node layer comprise a high-dimensional vector that represents the overall state of the neuron. Functionally, the tau layer approximates a oneto-one mapping between the spatio-temporal input and the tau-unit vector space. The output of each tau unit in the tau layer is fed into the input layer of a multilayer perceptron (MLP) which, as will be explained in the next section, has been trained to label the tau-layer vector as either FIRE or NO-FIRE. The output of the MLP is then fed into an output filter with a refractory period and threshold. The STEM architecture is illustrated in Fig. 3. (A) (afferents) (feedback from action potentials) synapse # 1,* ~ 3~?" N~ I ~+TAU_NODE,a~r (all outputs go to MLP) p~esynaptic mput (B) TAU-UNIT DYNAMICS (C) d ' , xiI -xl = a)-dt I 'to I _d T,; , TiI .\ = xl-dt I I 't. I ai(t) I TAU NODE II TAU UNITS x~(~ tM....r"-T~(t~ .~~ Tlll_ _ Figure 2: Tau Layer Schematic. (A) the tau layer has an afferent section, with N tau nodes, and a single-node feedback section. (B) Each tau node contains M tau units, and therefore has 1 input and M outputs (C) Each of the M tau units in a tau node has a rise and decay function with different constants. The equations are given for the ith tau unit of the jth tau node. a is input activity, x an internal state variable, and T the output. Sean D. Murphy, Edward W. Kairiss 88 single tau node from output with B internal tau units afferent inputs (axons) & / IaU node layer sN B units in 1st layer from feedback hidden layer of MLP NxM units in axon subset of 1st layer of multilayer perceptron single output unit of MLP Schematic of output filter processin~ MLP x ',))f\",AU M -n-;f- output filter I I I = full forward connections between layers output axon Figure 3: STEM Architecture. Afferent activity enters the tau layer, where it is converted into a vectorial representation of past spaiotemporal activity. The MLP maps this vector into a FlRE/NO-FIRE output unit, the continuous value of which is converted to a discrete event signal by the refractory period and threshold of the output filter. 3 STEM TRAINING There are six phases to training the STEM cell: (1) Biology: anatomical and physiological data are collected on the cell to be modeled. (2) Compartmental Model: a compartmental model of the cell is designed, typically with a simulation environment such as GENESIS. As much biological detail as possible is incorporated into the model. (3) Transfer Function Trials: many random input sequences are generated for the compartmental model. The firing response of the model is recorded for each input sequence. (4) Samplin~ assi~nments; In the next step, sampling will need to be done on the affect of the input sequences on the STEM tay layer. The timing of the sampling is calculated by separating the response of the compartmental model on each trial into regions where no spikes occur, and regions surrounding spikes. High-rate sampling times are determined for spike regions, and lower rate times are determined for quiet regions. (5) Tau layer trials: the identical input sequences applied to the compartmental model in step #3 are applied to an isolated tau layer of the STEM cell. The spike events from the compartmental model are used as input for the feedback node. For each input sequence, the tay layer is sampled at the times calculated in step #4, and the vector is labeled as FIRE or NO-FIRE (0 or 1). (6) MLP training: conjugate-gradient and line-search methods are used to train the multilayer perceptron using the results of step #5 as training vectors. Model of Biological Neuron as a Temporal Neural Network 89 Training is continued until a minimal performance level is reached, as determined by comparing the response of the STEM cell to the original compartmental model on novel input sequences. 4 RESULTS The STEM cell has been initially evaluated using Roger Traub's (1991) compartmental model of a hippocampal CAl cell, implemented in GENESIS by Dave Beeman. This is a relatively simple model structurally, with 19 compartments connected in a linear segment, with the soma in the middle. Dynamically, however, it is one of the most accurate and sophisticated models published, with on the order of 100 voltage- and Ca++ sensitive membrane channel mechanisms. 94 synapses were placed on the model. Each synapse recevied a random spike train with average frequency 10Hz during training. A diagram of the model and the locations of synaptic input is given in Fig. 4. Inputs going to a single compartment were treated as members of a common synapse, so there were a total of 13 tau nodes, with 5 tau units per node, for a total of 65 tau units, plus 5 additional units from the feedback tau node. These fed into 70 units in the input layer of the MLP. Two STEM cells were trained, one on a passive shell of the CAl cell, and the other with all of the membrane channels included. Both used 70 units in the hidden layer 1200m 880m 6mlfllll_m 14m mwmiil 8m received 8 synaptic inputs Fig. 4 Structure of Traub's CAl cell compartmental model -42. -+4. -44'., -48. -150. -!5Z. -64. -116. -ee. 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 20 2.5 STEM cell 10.0<H'"----t------+-----t----+-----+-------,. B. 4. 2. Figure 5: CAl Passive Cell Output. The somatic voltage for the passive compartmental model and the corresponding output-filtered spike events are given in the upper graph. The lower graph shows the repsonse of the STEM cell to the same input. The upper trace of the lower graph is the output filter response, the lower trace is the raw output of the MLP. Horizontal axes in seconds. Vertical axis: top, m V, and bottom, arbitrary units. 90 Sean D. Murphy, Edward W. Kairiss compartmental model 10, 00-0, 00- -10, 00- -20,,00-30,.00-40,.00- -eo,,00- r--- -60,,00- 0.5 1.0 1.5 r--2.0 r-- I':: 2.5 STEM cell 10,i'lO"!'"----t-----+-----t-----t------t-------r 8, 0.5 1.0 1.5 2.0 2.5 Figure 6: Response of Active CAl Cell, The upper graph is the somatic voltage of the CA 1 cell in response to a random input The lower graph is the response of the STEM cell to the same input. The upper trace of the STEM graph is the output filter response, and the lower trace is the raw MLP output.Horizontal axes in seconds, Vertical axis: top, m V, and bottom, arbitrary units, of the MLP, Comparisons of the compartmental model vs. STEM are shown in Fig. 5 and , for the passive and active models, respectively, Fig. 6. For the active cell model, the STEM cell was approximately 10 times faster, and used 10% of the memory as the compartmental model. 5 DISCUSSION The STEM cell is a general spatio-temporal node architecture and is similar to many context networks that have been previously developed. Its role as a single node in a large meta-network is unexplored, albeit interesting because of its capacity to mimic the transfer functions of biological neurons. Between the complexity range of connectionist networks to biological networks, there may be a multitude of useful computational schemes for solving different types of problems. The STEM cell is an attempt to efficiently capture some elements of biological neural function in a form that can be scaled along this range. Characterization of the computational function of the neuron is a topic of considerable interest and debate. STEM cells may be useful as a rough measure of how complex the transfer function of a given biophysical model is. For example, it might be able to answer the question: Which instantiates a more sophisticated nonlinear spatio-temporal map, a single-compartment cell with complex somatic Ca++ dynamics, or a cell with active Na+ channels in a complex dendritic tree? STEM architecture may also be interesting for theoretical and applied ANN research as a connectionist representation of a biological neuron. The expanding body of work on focused network architectures (Mozer, 1989; Stometta, 1988) may be an avenue towards the formalization of biological neural transfer functions. Because a VLSI implementation Model of Biological Neuron as a Temporal Neural Network 91 of a STEM cell could be reprogrammed on the fly to assume the transfer function of any pre-trained biophysically-modeled cell, a VLSI STEM network chip is a more versatile approach to artifical implementations of biological neurons than reconstructing compartmental models in VLSI. Our future plans include using networks with delay lines and hebbian learning rules, both of which the STEM architecture is directly suited for, to investigate the capacity for STEM networks to perform real-time dynamic pattern tracking. The present implementation of the STEM cell is by no means an optimal one. We are experimenting with alternative components for the MLP, such as modular recurrent networks. Acknowledgements This work was supported by the Yale Center for Theoretical and Applied Neuroscience. References de Vries, B. and Principe, J.C. (1991) A theory for neural networks with time delays. In R.P. Lippmann, J. Moody, & D.S. Touretzlcy (Eds.), Advances in Neural Information Processing Systems 3 (pp. 162-168). San Mateo, CA: Morgan Kaufmann. de Vries, B. and Principe, J.C. (1992) The gamma model - A new neural net model for temporal processing. Neural Networks, 5, 565-576. Mozer, M.C. (1989). A focused back-propagation algorithm for temporal pattern recognition, Complex Systems, 3, 349-381. Mozer, M.C. (in press) Neural net architectures for temporal sequence processing. In A. Weigend & N. Gershenfeld (Eds.), Predicting the Future and Understanding the Past. Redwood City, CA: Addison-Wesley. Stornetta, W.S., Hogg, T., & Huberman, B.A. (1988). A dynamical approach to temporal pattern processing. In Anderson D.Z. (Ed.), Neural Information Processing Systems, 750759. Traub, R.D. and Wong, R. K. S. (1991). Neuronal Networks of the Hippocampus. Cambridge: Cambridge University Press. Wilson, M. and Bower, J.M. (1992) Cortical oscillations and temporal interactions in a computer simulation of piriform cortex. J. Neurophysiol. 67:981-95. 

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Dynamic Cell Structures Jorg Bruske and Gerald Sommer Department of Cognitive Systems Christian Albrechts University at Kiel 24105 Kiel- Germany Abstract Dynamic Cell Structures (DCS) represent a family of artificial neural architectures suited both for unsupervised and supervised learning. They belong to the recently [Martinetz94] introduced class of Topology Representing Networks (TRN) which build perlectly topology preserving feature maps. DCS empI'oy a modified Kohonen learning rule in conjunction with competitive Hebbian learning. The Kohonen type learning rule serves to adjust the synaptic weight vectors while Hebbian learning establishes a dynamic lateral connection structure between the units reflecting the topology of the feature manifold. In case of supervised learning, i.e. function approximation, each neural unit implements a Radial Basis Function, and an additional layer of linear output units adjusts according to a delta-rule. DCS is the first RBF-based approximation scheme attempting to concurrently learn and utilize a perfectly topology preserving map for improved performance. Simulations on a selection of CMU-Benchmarks indicate that the DCS idea applied to the Growing Cell Structure algorithm [Fritzke93] leads to an efficient and elegant algorithm that can beat conventional models on similar tasks. 1 Introduction The quest for smallest topology preserving maps motivated the introduction of growing feature maps like Fritzke's Growing Cell Structures (GCS). In GCS, see [Fritzke93] for details, one starts with a k-dimensional simplex of N = k+ 1 neural units and (k + 1) . kl2 lateral connections (edges). Growing of the network is performed such that after insertion 498 Jorg Bruske, Gerald Sommer of a new unit the network consists solely of k dimensional simplices again. Thus, like Kohonen's SOM, GCS can only learn a perfectly topology preserving feature mapl if k meets the actual dimension of the feature manifold. Assuming that the lateral connections do reflect the actual topology the connections serve to define a neighborhood for a Kohonen like adaptation of the synaptic vectors Wj and guide the insertion of new units. Insertion happens incrementally and does not necessitate a retraining of the network. The principle is to insert new neurons in such a way that the expected value of a certain local error measure, which Fritzke ca11s the resource, becomes equal for all neurons. For instance, the number of times a neuron wins the competition, the sum of distances to stimuli for which the neuron wins or the sum of errors in the neuron's output can all serve as a resource and dramatically change the behavior of GCS. Using different error measures and guiding insertion by the lateral connections contributes much to the success of GCS. The principle of DCS is to avoid any restriction of the topology of the network (lateral connection scheme between the neural units) but to concurrently learn and utilize a perfectly topology preserving map. This is achieved by adapting the lateral connection structure according to a competitive Hebbian learning rule2 : max{Yj'YpCij(t)} : CIj(t+ 1) = { 0 aCjj (t) Yj'Yj~Yk'YI V'(1S,k,IS,N) : Cjj(t) <9 fIfI otherwise, till : (1) where a, 0 < a < 1 is a forgettin?. constant, 9, 0 < 9 < 1 serves as a threshold for deleting lateral connections, and Yj = R (j/v - Wj//) is the activation of the i-th unit with Wj as the centre of its receptive field on presentatIOn of stimulus v. R(.) can be any positive continuously monotonically decreasing function. For batch learning with a training set T of fixed size 111 , a = 17JJ9 is a good choice. Since the isomorphic representation of the topology of the feature manifold M in the lateral connection structure is central to performance, in many situations a DCS algorithm may be the right choice. These situations are characterized by missing a priori knowledge of the topology of the feature manifold M or a topology of M which cannot be readily mapped to the existing models. Of course, if such a priori knowledge is available then models like GCS or Kohonen's SOM allowing to incorporate such knowledge have an advantage, especially if training data are sparse. Note that DCS algorithms can also aid in cluster analysis: In a perfectly topology preserving map clusters which are bounded by regions of P(v) =0 can be identified simply by a connected component analysis. However, without prior knowledge about the feature manifold M it is in principal impossible to check for perfect topology preservation of S. Noise in the input data may render perfect topology learning even more difficult. So what can perfect topology learning be used for? The answer is simply that for every s~t S of reference vectors perfect topology learning yields maximum topology preservation with respect to this set. And connected components with respect to the lateral connection structure C may well serve as an initialization for postprocessing by hierarchical cluster algorithms. 1. We use the term "perfectly topology preserving feature map" in accordance with its rigorous definition in [Martinetz93]. 2. In his very recent and recommendable article [Martinetz94] the term Topology Representing Network (TRN) is coined for any network employing competitive Hebbian learning for topology learning. 3. if topology preservation is measured by the topographic function as defined in [Villmann94]. Dynamic Cell Structures 499 The first neural algorithm attempting to learn perfectly topology preserving feature maps is the Neural Gas algorithm ofT. Martinetz [Martinetz92]. However, unlike DCS the Neu? ral Gas does not further exploit this information: In every step the Neural Gas computes the k nearest neighbors to a given stimulus and, in the supervised case, employs a11 of them for function approximation. DCS avoids this computational burden by utilizing the lateral connection structure (topology) learned so far, and it restricts interpolation between activated units to the submanifold of the current stimulus. Applying the principle ofDCS to Fritzke's GCS yields our DCS?GCS algorithm. This algorithm sticks very closely to the basic structure of its ancestor GCS except the predefined k-dimensional simplex connection structure being replaced by perfect topology learning. Besides the conceptual advantage of perfect topology learning, DCS?GCS does decrease overhead (Fritzke has to handle quite sophisticated data structures in order to maintain the k-dimensional simplex structure after insertion! deletion of units) and can be readily implemented on any serial computer. 2 Unsupervised DCS-GCS The unsupervised DCS-GCS algorithm starts with initializing the network (graph) to two neural units (vertices) n, and n2. Their weight vectors wI' WI (centres of receptive fields) are set to points v I' v2 E M which are drawn from M according to P(v). They are connected by a lateral connection of weight C l2 = C21 = I. Note that lateral connections in DCS are always bidirectional and have symmetric weights. Now the algorithm enters its outer loop which is repeated until some stopping criterium is fulfi11ed. This stopping criterium could for instance be a test whether the quantization error has already dropped below a predefined accuracy. The inner loop is repeated for A. times. In off-line learning A. can be set to the number examples in the training set T. In this case, the inner loop just represents an epoch of training. Within the inner loop, the algorithm first draws an input stimulus v E M from M according to P(v) and then proceeds to calculate the two neural units which weight vectors are first and second closest to v. In the next step, the lateral connections between the neural units are modified according to eq. (1), the competitive Hebbian learning rule. As has already been mentioned, in off-line learning it is a good idea to set a = va . Now the weight vectors Kohonen like fashion: Wj of the best matching unit and its neighbors are adjusted in a .1.wbmu = where the ?8 neighborhood (v - wbmu> and .1.Wj N h U) of a = ?Nh (v unit wj ) , j is (2) defined by NhU) = {il(Cjj*O,l~i~N)} . The inner loop ends with updating the resource value of the best matching unit. The resource of a neuron is a local error measure attached to each neural unit. As has been pointed out, one can choose alternative update functions corresponding to different error measures. For our ~xperim~nts (section 2.1 and sectpn 3.1) we used the accumulated squared distance to the stImulus, I.e . .1.'tbmu = wbmull . The outer loop now proceeds by adding a new neural unit r to the network. This unit is located in-between the unit I with largest resource value and its neighbor n with second largest resource value: 4 Ilv - 500 Jorg Bruske, Gerald Sommer The exact location of its centre of receptive field wr is calculated according to the ratio of the resource values 'tl , 'tn' and the resource values of units n and I are redistributed among r, n and I: Wr = wI + yew n - wI) , 't r = ~'tn + ~'tr' 't l = 't[- ~'tr and't n = 't n - ~'tn? (4) This gives an estimate of the resource values if the new unit had been in the network right from the start. Finally the lateral connections are changed, Cr=C[ rn =0, r r =l,C rn =Crn =IandCr=C n (5) connecting unit r to unit I and disconnecting n and I. This heuristic guided by the lateral connection structure and the resource values promises insertion of new units at good initial positions. It is responsible for the better performance of DCS-GCS and GCS compared to algorithms which do not exploit the neighborhood relation between existing units. ? The outer loop closes by decrementing the resource values of all units, 't i (t + 1) = ~'ti (t) , 1:::; i:::; N, where < ~ < 1 is a constant. This last step just avoids is the natural choice. overflow of the resource variables. For off-line learning, ~ = 2.1 ? Unsupervised DCS simulation results Let us first turn to our simulation on artificial data. The training set T contains 2000 examples randomly drawn from a feature manifold M consisting of three squares, two of them connected by a line. The development of our unsupervised DCS-GCS network is depicted in Figure 1, with the initial situation of only two units shown in the upper left. Examples are represented by small dots, the centres of receptive fields by small circles and the lateral connections by lines connecting the circles. From left to right the network is examined after 0, 9 and 31 epochs of training (i.e. after insertion of 2, 11 and 33 neural units). After 31 epochs the network has built a perfectly topology preserving map of M, the lateral connection structure nicely reflecting the shape of M: Where Mis 2-dimensional the lateral connection structure is 2-dimensional, and it is I-dimensional where M is I-dimensional. Note, that a connected component analysis could recognize that the upper right square is separated from the rest of M. The accumulated squared distance to stimuli served as the resource. Th.e quantization error Eq = ! L Ilv - wbmu (v) 112 dropped from 100% (3 units) to 3% (33 umts). n VE T The second simulation deals with the two-spirals benchmark. Data were obtained by running the program "two-spirals" (provided by eMU) with parameters 5 (density) and 6.5 (spiral radius) resulting in a training set T of 962 examples. The data represent two distinct spirals in the x-y-plane. Unsupervised DCS-GCS at work is shown in Figure 2, after insertion of 80, 154 and, finally, 196 units. With 196 units a perfectly topology preserving map of M has emerged, and the two spirals are clearly separated. Note that the algorithm has learned the separation in a totally unsupervised manner, i.e. not using the labels of the data 4. Fritzke inserts new units at a slightly different location, using not the neighbor with second largest resource but the most distant neighbor. Dynamic Cell Structures 501 Figure 1: Unsupervised DCS-GCS on artificial data points (which are provided by CMU for supervised learning). Again, the accumulated squared distance to stimuli served as the resource. -;..--- ------,--- -- - . , '-::---.:-:r~~ ~.:~:.~.';{';';'; ~.:.;.! ~~~! ~I Figure 2: Unsupervised learning of two spirals 3 Supervised DCS-GCS In supervised DCS-GCS examples consist not only of an input vector v but also include an additional teaching output vector u. The supervised algorithm actually does work very similar to its unsupervised version except ? when a neural unit nj is inserted an output vector OJ will be attached to it with OJ = u. ? the output y of the network is calculated as a weighted sum of the best matching unit's output vector 0hmu and the output vectors of its neighbors OJ' i E Nh (bmu) , Y = (~ ~jE {bmuuNh(hmu)} a.o.) , I I (6) 502 Jorg Brnske, Gerald Sommer = I/(crllv-wiI12+ 1) is the activation of neuron i on stimulus v, cr, cr> (1, representing the size of the receptive fields. In our simulations, the size of where a? receptive fields have been equal for all units. ? adaption of output vectors by the delta-rule: A simple delta-rule is employed to adjust the output vectors of the best matching unit and its neighbors. Most important, the approximation (classification) error can be used for resource updating. This leads to insertion of new units in regions where the approximation error is worst, thus promising to outperform dynamic algorithms which do not employ such a criterion for insertion. In our simulations we used the accumulated squared distance of calculated and teaching output, ~tbmu = Ily - u11 2 . 3.1 Supervised DCS-GCS simulation results We applied our supervised DCS-GCS algorithm to three CMU benchmarks, the supervised two-spiral problem, the speaker independent vowel recognition problem and the sonar mine! rock separation problem. 5 The t~o vE ~ spirals benchmark contains 194 examples, each consisting of an input vector and a binary label indicating to which spiral the point belongs. The spirals can not be linearly separated. The task is to train the examples until the learning system can produce the correct output for all of them and to record the time. The decision regions learned by supervised DCS-GCS are depicted in Figure 3 after 110 and 135 epochs of training, where the classification error on the training set has dropped to 0%. Black indicates assignment to the fist, white assignment to the second spiral. The network and the examples are overlaid. Figure 3: Supervised learning of two spirals Results reported by others are 20000 epochs of Backprop for a MLP by Lang and Witbrok [Lang89], 10000 epochs of Cross Entropy Backprop and 1700 epochs of Cascade-Correlation by Fahlman and Lebiere [Fahlman90] and 180 epochs of GCS training by Fritzke [Fritzke93]. 5. For details of simulation, parameters and additional statistics for all of the reported experiments the reader is refered to [Bruske94] which is also available viaftp.infomuztik.uni-kiel.de in directory publkiellpublicationsffechnicalReportslPs.ZI as 1994tr03.ps.Z Dynamic Cell Structures 503 The data for the speaker independent recognition of 11 vowels comprises a training set of 582 examples and a test set of 462 examples, see [Robinson89]. We obtained 65% correctly classified test samples with only 108 neural units in the DCS? GCS network. This is superior to conventional models (including single and multi layer perceptron, Kanerva Model, Radial Basis Functions, Gaussian Node Network, Square Node Network and Nearest Neighbor) for which figures well below 57% have been reported by Robinson. It also qualitatively compares to GCS Gumps above the 60% margin), for which Fritzke reports best classification results of 61 %(158 units) up to 67% (154 units) for a 3-dim GCS. On the other hand, our best DCS?GCS used much fewer units. Note that DCS?GCS did not rely on a pre-specified connection structure (but learned it!). Our last simulation concerns a data set used by Gorman and Sejnowski in their study of classification of sonar data, [Gorman88]. The training and the test set contain 104 examples each. Gorman and Sejnowski report their best results of 90.4% correctly classified test examples for a standard BP network with 12 hidden units and 82.7% for a nearest neighbor classifier. Supervised DCS?GCS reached a peak classification rate of 95% after only 88 epochs of training. 4 Conclusion We have introduced the idea ofRBF networks which concurrently learn and utilize perfectly topology preserving feature maps for adaptation and interpolation. This family of ANNs, which we termed Dynamic Cell Structures, offers conceptual advantage compared to classical Kohonen type SOMs since the emerging lateral connection structure maximally preserves topology. We have discussed the DCS-GCS algorithm as an instance of DCS. Compared to its ancestor GCS of Fritzke, this algorithm elegantly avoids computational overhead for handling sophisticated data structures. If connection updates (eq.(I)) are restrJcted to the best matching unit and its neighbors, DCS has linear (serial) time complexity and thus may also be considersd as an improvement of Martinetz's Neural Gas idea7 . Space complexity of DCS is 0 (N) in general and can be shown to become linear if the feature manifold M is two dimensional. The simulations on CMU-Benchmarks indicate that DCS indeed has practical relevance for classification and approximation. Thus encouraged, we look forward to apply DCS at various sites in our active computer vision project, including image compression by dynamic vector quantization, sensorimotor maps for the oculomotor system and hand-eye coordination, cartography and associative memories. A recent application can be found in [Bruske95] where a DCS network attempts to learn a continous approximation of the Q-function in a reinforcement learning problem. 6. Here we refer to the serial time a DeS algorithm needs to process a single stimulus (including response calculation and adaptation). 7. The serial time complexity of the Neural Gas is Q (N) ,approaching 0 (NlogN) for k ~ N , k the number of nearest neighbors. 504 larg Bruske, Gerald Sommer References [Bruske94] J. Bruske and G. Sommer. Dynamic Cell Structures: Radial Basis Function Networks with Perfect Topology Preservation. Inst. f. Inf. u. Prakt. Math. CAU zu Kiel. Technical Report 9403. [Bruske9S] J. Bruske. I. Ahms and G. Sommer. Heuristic Q-Learning. submitted to ECML 95. [Fahlman90] S.E. Fahlman. C.Lebiere. The Cascade-Correlation Learning Architecture. Advances in Neural Information processing systems 2, Morgan Kaufman, San Mateo, pp.524-534. [Fahlman93] S.E. Fahlman, CMU Benchmark Collection for Neural Net Learning Algorithms, Carnegie Mellon Univ., School of Computer Science, machine-readable data repository. Pittsburgh. [Fritzke92] B. Fritzke. Growing Cell Structures - a self organizing network in k dimensions, Artificial Neural Networks 2. I.Aleksander & J .Taylor eds .? North-Holland, Amsterdam, 1992. [Fritzke93] B. Fritzke, Growing Cell Structures - a self organizing network for unsupervised and supervised training, ICSI Berkeley, Technical Report, tr-93-026. [Gorman88] R.B. Gorman and TJ. Sejnowski, Analysis of Hidden Units in a Layered Network Trained to Classify Sonar Targets, Neural Networks, VoU. pp. 75-89 [Lang89] K.J. Lang & M.J. Witbrock, Learning to tell two spirals apart, Proc. of the 1988 Connectionist Models Summer School. Morgan Kaufmann, pp.52-59. [Martinetz92] Thomas Martinetz, Selbstorganisierende neuronale Netzwerke zur Bewegungssteuerung, Dissertation. DIFKI-Verlag, 1992. [Martinetz93] Thomas Martinetz, Competitive Hebbian Learning Rule Forms Perfectly Topology Preserving Maps, Proc. of the ICANN 93. p.426-438, 1993. [Martinetz94] Thomas Martinetz and Klaus Schulten, Topology Representing Networks, Neural Networks. No.7, Vol. 3. pp. 505-522, 1994. [Moody89] J.Moody, c.J. Darken. Fast Learning in Networks of Locally-Tuned Processing Units, Neural Computation VoLl Num.2, Summer 1989. [Robinson89] AJ. Robinson, Dynamic Error Propagation Networks, Cambridge Univ., Ph.D. thesis, Cambridge. [Villmann94] T. Villmann and R. Der and T. Martinetz, A Novel Approach to Measure the Topology Preservation of Feature Maps, Proc. of the ICANN 94, 1994. 

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On-line Learning of Dichotomies N. Barkai H. S. Seung AT&T Bell Laboratories Murray Hill, NJ 07974 seungCphysics.att.com Racah Institute of Physics The Hebrew University Jerusalem, Israel 91904 naamaCfiz.huji.ac.il H. Sompolinsky Racah Institute of Physics The Hebrew University Jerusalem, Israel 91904 and AT&T Bell Laboratories haimCfiz.huji.ac.il Abstract The performance of on-line algorithms for learning dichotomies is studied. In on-line learning, the number of examples P is equivalent to the learning time, since each example is presented only once. The learning curve, or generalization error as a function of P, depends on the schedule at which the learning rate is lowered. For a target that is a perceptron rule, the learning curve of the perceptron algorithm can decrease as fast as p- 1 , if the schedule is optimized. If the target is not realizable by a perceptron, the perceptron algorithm does not generally converge to the solution with lowest generalization error. For the case of unrealizability due to a simple output noise, we propose a new on-line algorithm for a perceptron yielding a learning curve that can approach the optimal generalization error as fast as p-l/2. We then generalize the perceptron algorithm to any class of thresholded smooth functions learning a target from that class. For "well-behaved" input distributions, if this algorithm converges to the optimal solution, its learning curve can decrease as fast as p-l. 1 Introduction Much work on the theory of learning from examples has focused on batch learning, in which the learner is given all examples simultaneously, or is allowed to cycle through them repeatedly. In many situations, it is more natural to consider on-line learning paradigms, in which at each time step a new example is chosen. The examples are never recycled, and the learner is not allowed to simply store them (see e.g, Heskes, 1991; Hansen, 1993; Radons, 1993). Stochastic approximation theory (Kushner, 1978) provides a framework for understanding of the local convergence properties of on-line learning of smooth functions. This paper addresses the problem of on-line learning of dichotomies, for which no similarly complete theory yet exists. 304 N. Barkai, H. S. Seung, H. Sompolinsky We begin with on-line learning of perceptron rules. Since its introduction in the early 60's, the perceptron algorithm has been used as a simple model of learning a binary classification rule. The algorithm has been proven to converge in finite time and to yield a half plane separating any set of linearly separable examples. The perceptron algorithm, however, is not efficient in the sense of distribution-free PAC learning (Valiant, 1984), for one can construct input distributions that require an arbitrarily long convergence time. In a recent paper (Baum, 1990) Baum proved that the perceptron algorithm applied in an on-line mode, converges as p-I/3 when learning a half space under a uniform input distribution, where P is the number of presented examples drawn at random. For on-line learning P is also the number of time steps. Baum also generalized his result to any "non-malicious" distribution. Kabashima has found the same power law for learning a two-layer parity machine with non-overlapping inputs, using an on-line least action algorithm (Kabashima, 1994). If efficiency is measured only by the number of examples used (disregarding time), these particular on-line algorithms are much worse than batch algorithms. Any batch algorithm which is able to correctly classify a given set of P examples will converge as p- I (Vapnik, 1982; Amari, 1992; Seung, 1992) . In this paper, we construct on-line algorithms that can actually achieve the same power law as batch algorithms, demonstrating that the results of Baum and Kabashima do not reflect a fundamental limitation of on-line learning. In Section 3, we study on-line algorithms for perceptron learning of a target rule that is not realizable by a perceptron. Here it is nontrivial to construct an algorithm that even converges to the optimal one, let alone to optimize the rate of convergence. For the special case of a target rule that is a percept ron corrupted by output noise this can be done. In Section 4, our results are generalized to dichotomies generated by thresholding smooth functions. In Section 5 we summarize the results. 2 On-line learning of a perceptron rule We consider a half space rule generated by a normalized teacher perceptron Wo ERN, Wo' Wo = 1 such that any vector 5 E RN is given a label uo(5) = sgn(Wo ?5). We study the case of a Gaussian input distribution centered at zero with a unit variance in each direction in space: P(5) = 1 s , /2 II _eN ;=1 2 v'21r (1) Averages over this input distribution will be written with angle brackets (). A student percept ron W is trained by an on-line perceptron algorithm. At each time step, an input 5 E RN is drawn at random, according to distribution Eq. (1) and the student's output u(5) = sgn(W . 5) is calculated. The student is then updated according to the perceptron rule: W' = W + ~f(5;W)uo(5)5 (2) and is then normalized so that W . W = 1 at all times. The factor f(5; W) denotes the error of the student perceptron on the input 5: f = 1 if u(5)uo(5) = 1, and 0 otherwise. The learning rate 1/ is the magnitude of change of the weights at each time step. It is scaled by N to ensure that the change in the overlap R = W? Wo is of order lIN. Thus, a change of 0(1) occurs only after presentation of P = O(N) examples. The performance of the student is measured by the generalization error, defined as the probability of disagreement between the student and the teacher on an arbitrary input fg = (f(5; W?). In the present case, fg is cos- I R (3) fg = - - 7 r - ' Although for simplicity we analyze below the performance of the perceptron rule (2) only for large N, our results apply to finite N as well. Multiplying Eq. (2) by Wo after incorporation of the normalization operation and averaging with respect to the input distribution (1), yields the following differential equation for R(a) where a = PIN, (4) On-line Learning of Dichotomies Here terms of order J17/N 305 have been neglected. The evolution of the overlap R, and thus of the generalization error, depends on the schedule at which the learning rate 17 decreases. We consider two cases, a constant 17 and a time-dependent 17. Constant learning rate: When 17 is held fixed, Eq. (4) has a stable fixed point at R < 1, and hence eg converges to an '7-dependent nonzero value eoo ('7). For '7 ?: 1, 1- Roo ('7) oc '7 2 and eg oc VI - R is therefore proportional to '7, (5) eoo ('7) = '7/..f2i3 . The convergence to this value is exponential in 0, eg(o) - eoo ('7) '" exp(-'7o/$). Time-dependent learning rate: Convergence to eg = 0 can be achieved if '7 decreases slowly enough with o. We study the limiting behaviour of the system for '7 which is decreasing with time as '7 = (7]0$) o-z. z > 1. In this case the rate is reduced too fast before a sufficient number of examples have been seen. This results in R which does not converge to 1 but instead to a smaller value that depends on its initial value. z < 1. The system follows the change in '7 adiabatically. Hence, to first order in Thus, eg converges to zero with an asymptotic rate eg(o) '" o-z. z = 1. 0- 1 , eg(o) = eoo ('7(o?. The behaviour of the system depends on the prefactor '70: (1 '7~)1 eg '70> 1 71"'70- 1 0 y'logo 7]0 = 1 a A 7]0 0'1<> (6) <1 where A depends on the initial condition. Thus the optimal asymptotic change of '7 is 2..;21i/ 0, in which case the error will behave asymptotically as eg(o) '" 1.27/0. This is not far from the batch asymptotic (Seung, 1992) eg(o) '" 0.625/0. We have confirmed these results by numerical simulation of the algorithm Eq. (2). Figure 1 presents the results of the optimalleaming schedule, i.e., '7 = 2..;21i/o. The numerical results are in excellent agreement with the prediction eg(o) = 1.27/0 for the asymptotic behavior. Finally, we note that our analysis of the time-dependent case is similar to that of Kabashima and Shinomoto for a different on-line learning problem (Kabashima, 1993). 3 On-line learning of a percept ron with output noise In the case discussed above, the task can be fully realized by a perceptron, i.e., there is a perceptron W such that eg = O. In more realistic situations a percept ron will only provide an approximation of the target function, so that the minimal value of eg is greater than zero. These cases are called unrealizable tasks. A drawback of the above on-line algorithm is that, for a general unrealizable task, it does not converge to the optimal perceptron, i.e., it does not approach the minimum of ego To illustrate this fact we consider a perceptron rule corrupted by output noise. The label of an input S is O"o(S), where O"o(S) = sgn(Wo . S) with probability 1 - p, and - sgn(Wo . S) with probability p. We assume 0 $ p $ 1/2. For reasons which will become clear later, the input distribution is taken as a Gaussian centered at U P(S) = n..;21i N _1_ e -(S,-u.)'/2 (7) ;=1 In this case eg is given by eg =p+ (l- q -00 DYH(~)+l?O DYH(~?). 1- R 1- R (8) _q where qo = U . W 0 denotes the overlap between the center of the distribution and the teacher perceptron, and q = U . W is the overlap between the center of the distribution and W. The integrals in Eq. (8) are 306 N. Barkai, H. S. Seung, H. Sompolinsky with respect to a Gaussian measure Dy = exp(-y2/2)/.J21i and H(x) = J.,oo Dy. Note that the optimal perceptron is the teacher W = Wo i.e., R = 1, q = 110, which yields the minimal error fmin = p. First, we consider training with the normalized perceptron rule (2). In this case, we obtain differential equations for two variables: R and q. Solving these equations we find that in general, W converges to a vector with a direction which is in the plane of Wo and U and is does not point in the direction of Wo even in the limit of,., -+ O. Here we present the result for the limit of,., -+ 0 and small noise level, i.e., p ?: 1. In this case, we obtain for f oo ('" = 0) foo(O) = P + p (1 - 2H(qo?~ + O(p2) 1+(u2-q~) (9) where u = lUI is the magnitude of the center of the input distribution. For p = 0, the only solution is R = 1 and q = qo, in agreement with the previous results. For p > 0 the optimal solution is retrieved only in the following special cases: (i) the input distribution is isotropic, i.e., qo = u = OJ (ii) when U is parallel to W o, i.e., u = qoj and (iii) when U is orthogonal to W o, i.e., qo = O. This holds also for large value of p . In these special cases, the symmetry of the input distribution relative to the teacher vector, guarantees that the deviations from W = Wo incurred by the inputs that come with the wrong label cancel each other on average. According to Eq. (9), for other directions of U, fg is above the optimal value. Note that the additional term in fg is of the same order of magnitude (O(P? as the minimal error. In the following we suggest a modified on-line algorithm for learning a perceptron rule with output noise. The student weights are changed according to W' = W + ~f(S; W)ITo(S)(S - T(S? (10) followed by a normalization of W. This algorithm differs from the perceptron algorithm in that the change in W is not proportional to the present input, but to a shifted vector. The shifting vector T(S), is determined by the requirement that the teacher Wo will be a fixed point of the algorithm in the limit of,., -+ O. This is equivalent to the condition (11) (fo(S)lTo(S)(S - T(S?) = 0 where foeS) is the error function for S when W = Wo o This condition does not determine T uniquely. A simple choice is one for which T is independent of S. This leads to T = (sgn(Wo' S)S) = (IToS) (sgn(Wo . S? (ITo) (12) where we used the fact that for any S, fo(S)lTo(S) equals - sgn(Wo . S) with probability p, and zero with probability (1 - p). This uniform shift is possible only when (O'o) ~ 0, namely when the average frequencies of +1 and -1 labels are not equal. If this is not the case, one has to choose nonuniform forms of T(S) . Note that in general T has to be learned so that Eq. (10) has to be supplemented by appropriate equations for changing T. In the case of Eq. (12), one can easily learn separately the numerator and denominator by running averages of O'oS and 0'0, respectively. We have studied analytically the above algorithm for the case of the Gaussian input distribution Eq. (7), in the limit of large N. The shifting vector is given by T = U + Wo ~ exp( -q~/2) V1T 1 - 2H(qo) The differential equations for the overlaps R and q in the neighborhood of the point R = 1 and q daR da doq da (13) = qo are, (14) where oR = 1 - R and oq = qo - q. In the limit,., -+ 0, R = 1 and q = qo is indeed a stable fixed point of the algorithm, so that the student converges to the optimal perceptron W o, and hence the feneralization error converges to its minimal value fmin = p. Since, unlike Eq. (4), the coefficient of the,., term in Eq. On-line Learning of Dichotomies 307 (14) is constant, c5Roo(1]) <X 1], for small fixed 1], and not to 1]2. Thus, in this case, the generalization error approaches, in the limit a -t 00, the value exp(-qa/ 4) t:oo (1]) :;;: p +.,fiiP (271"3)1/4 (15) For a time-dependent 1], the convergence to the optimal weights depends on the choice of 1](n), as in the case of the noiseless perceptron rule. For 1] :;;: (1]0 exp( q5 12) ) a-' , with z ~ 1, the error converges to p. For z < 1, to first order in 1/a, t:g(a) :;;: t: oo (1](a)) , yielding J1i72 t:g(a) - p ~ a-?/ 2 ? (16) When z :;;: 1, the rate of convergence depends on the value of 1]0. t:g and logarithmic corrections to a- 1 / 2 for ( ) a a-l /2 1] > 1 p ~ { a-'1oj:~, ~ < 1 - 1]0 :;;: 1. Thus, the optimal rate of convergence is t: (a) _ p 9 which is achieved for 1]0 (17) ~ V~ f2P (18) = 2. We have tested successfully this algorithm by simulations of learning a perceptron rule with output noise with several input distributions, including the Gaussian, of Eq. (7). Figure 2 presents the generalization error as a function of a for the Gaussian distribution, with p = 0.2 , and we have chosen 1]0 = 2. The error converges to the optimal value 0.2 as a- I / 2 in agreement with the theory. For comparison the result of the usual perceptron algorithm is also presented. This algorithm converges to t:g ~ 0.32, clearly larger than the optimal value. 4 On-line learning of thresholded smooth functions Our results for the realizable perceptron can be extended to a more general class of dichotomies, namely thresholded smooth functions. They are defined as dichotomies of the form (19) O"(S; W) = sgn(f(Sj W)) where 1 is a differentiable function of a set of parameters, denoted by W, and S is the input vector. We consider here the case of a realizable task, where the examples are given with labels 0"0 corresponding to a target machine W 0 which is in the W space. For this task we propose the following generalization of the perceptron rule (2) (20) W' = W + 1]t:(Sj W)O"o(S)V I(S; W) where V denotes a gradient w.r.t. W. Then, as we argue below, the vector Wo is a stable fixed point in the limit of 1] -t O. Furthermore, for constant small 1] the residual error scales as t:oo <X 1]. For 1] ~ a-' , z < 1, t:g(a) ~ t:oo(1](a)) '""' a-?. To show this, let us consider for simplicity the one-dimensional case, w' = w + 1]g(w, s), where 81 g(w,s) = 9(-I(w,s)l(wo,s))sgn(f(wo,s))8w . (21) This equation can be converted into a Markov equation for the probability distribution, P(w,n) (Van Kampen, 1981) P(w,n + 1):;;: J dw'W(w'lw)P(w' , n) (22) where W(wlw') =< c5(w' - w -1]g(w, s)) > is the transition rate from w to w'. In the limit of small fixed 1], the equilibrium distribution, Poo, can be shown to have the following scaling form, 1 Poo(w;1]):;;: -F(c5wl1]) (23) 1] 308 N. Barkai, H. S. Seung, H. Sompolinsky where ow = w - Wo and F(x) obeys the following difference equation LF(x) == L 9?(f~ + O"x)J~)I(f~ + O"x)IF(x + o"/~) -lxIF(x) =0 (24) C1=?1 where J6 is the value of the gradient aJ(wo, s)law at the decision boundary of J(wo, s), namely at the point s obeying J(wo,s) = O. Note that since we are interested in normalizable solutions of Eq. (24), F(x) has to vanish for for all x > 1161. This result is valid provided the input distribution is smooth and nonvanishing near the decision boundary. Furthermore, aJ law at Wo may not vanish on the decision boundary. Under the same conditions, it can be shown that the error is homogeneous in ow with degree 1, hence it should scale linearly with "I, i.e., 1':00 oc "I. It should be noted that, unlike other on-line learning problems (Heskes, 1991; Hansen, 1993; Radons, 1993), the equilibrium distribution is our case is not Gaussian. For a time-dependent "I of the form "I = TJon- z , z < I, pew, n) at long times is of the form (25) where F is the stationary distribution, given by Eq. (24) and the coefficient of the correction, G, solves the inhomogeneous equation dF ? (26) zx dx + zF(x) = TJoLG(x) where the linear operator L is defined in Eq. (24). Thus, to leading order in inverse time, the system follows adiabatically the finite-TJ stationary distribution, yielding I':g(n) which vanishes asymptotically as I':g(n) oc TJ(n) ~ n- z . The optimal schedule is obtained for z = 1. In this case, P(w,n) = "1-1 (n)F(ow/TJ(n)) where F(x) solves the homogeneous equation dF zx dx ? + zF(x) = TJoLF(x) For sufficiently large "10, this equation has a solution, implying that (27) I':g oc n- 1 Similarly, the results of Section 3 can also be extended to the case of thresholded- smooth functions with a probability p of an error due to isotropic output noise. In this case, the optimal choice is again "I oc n- I yielding I':g -p Rl ,;ri. It should be noted that for this case, the probability distribution for small "I does reduce to a Gaussian distribution in owl,;ri. Using a multidimensional Markov equation, it is straightforward to extend these results to higher dimensions. The small "I limit yields equations similar to Eqs. (24-26), that involve integration over the decision boundary of J(W, S). 5 Summary and Discussion We have found that the perceptron rule (2) with normalization can lead to a variety of learning curves, depending on the schedule at which the learning rate is decreased. The optimal schedule leads to an inverse power law learning curve, I':g ~ 0- 1 . Baum's results (Baum, 1990) of a non-normalized perceptron with a constant learning rate can be viewed as a special case of the above analysis. In the non-normalized perceptron algorithm, the magnitude of the student's weights grow with 0 as IWI ~ 0 1 / 3 . The time evolution of the overlap R, and thus of the generalization error is governed by the effective learning rate TJe/f = T//IWlleading via Eq. (6) to the result I':g ~ 0- 1 / 3 . Similar results apply to the two-layer parity machine studied in (Kabashima, 1994). Our analysis, leading to the equations of motion (4) and (14), was based on the limit of large N and P, such that 0 = PIN remains finite. We would like to stress however, that this limit is only necessary in deriving the full form of the learning curve, i.e., R(o) for all o. On the other hand, our results for the large P asymptote of the learning curve for smailT/ are valid for finite N as well, as implied by the general treatment of the previous section. Unrealizable percept ron rules present a more complicated problem. We have presented here a modified perceptron algorithm that converges to the optimal solution in the special case of an isotropic output noise. On-line Learning of Dichotomies 309 In this case, the convergence to the optimal error is as 01- 1 / 2 . This is the same power law as obtained in the standard sample complexity upper bounds (Vapnik, 1982) and in the approximate replica symmetric calculations (Seung, 1992) for batch learning of unrealizable rules. It should be stressed however, that the success of the modified algorithm in the case of an output noise depends on the fact that the errors made by the optimal solution are uncorrelated with the input. Thus, finding an on-line algorithm that can cope with other types of unrealizability remains an important problem. The learning algorithms for the perceptron rule, without and with output noise, can be generalized to learning thresholded smooth functions, assuming certain reasonable properties of the input distribution are present, as shown in Section 4. The dependence of the learning curve on the learning rate schedule remains roughly the same as in the percept ron case. This implies that on-line learning of realizable dichotomies, with possible output noise, can achieve the same power laws in the number of examples that is typical of batch learning of such rules. Furthermore, the on-line formulation possesses the theoretical virtues of addressing time as well as sample complexity, so that the same power laws imply the polynomial relationship between the time and the achieved error level. The above conclusions assume that the equilibrium state at small learning rates is unique, which in general is not the case. The issue of overcoming local minima in on-line learning is a difficult problem (Heskes, 1992) Finally, the theoretical results for on-line learning has the important advantage of not requiring the use of the often problematic replica formalism. Acknowledgements We are grateful for helpful discussions with Y. Freund, M. Kearns, R. Schapire, and E. Shamir, and thank Y. Kabashima for bringing his paper to our attention. HS is partially supported by the Fund for Basic Research of the Israeli Academy of Arts and Sciences. References S. Amari, N. Fujita, and S. Shinomoto. Four types of learning curves. Neural Comput., 4:605-618, 1992. E. B. Baum. The perceptron algorithm is fast for nonmalicious distributions. Neural Comput., 2:248-260, 1990. H. J. Kushner and D. S. Clark. Stochastic approximation methods for constrained and unconstrained systems. Springer, Berlin, 1978. L. K. Hansen, R. Pathria, and P. Salamon. Stochastic dynamics of supervised learning. J. Phys., A26:63-71, 1993. T . Heskes and B. Kappen. Learning processes in neural networks. Phys. Rev., A44:2718-2762, 1991. T. Heskes, E. T. P. Slijpen, and B. Kappen. Learning in neural networks with local minima. Phys. Rev., A46:5221-5231, 1992. Y. Kabashima. Perfect loss of generalization due to noise in k = 2 parity machines. J. Phys., A27:1917-1927, 1994. Y. Kabashima and S. Shinomoto. Incremental learning with and without queries in binary choice problems. In Proc. of IJCNN, 1993. G. Radons. On stochastic dynamics of supervised learning. J. Phys., A26:3455-3461, 1993. H. S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of learning from examples. Phys. Rev., A45:6056-6091, 1992. L. G. Valiant. A theory of the learnable. Commun. ACM, 27:1134-1142,1984. N. G. Van Kampen. Stochastic processes in physics and chemistry. North holland 1981. V. N. Vapnik. Estimation of Dependences based on Empirical Data. Springer-Verlag, New York, 1982. N. Barkai, H. S. Seung, H. Sompolinsky 310 0.03 0.025 0.02 0.015 f9 0.01 0.005 0 0.005 0 0.01 0.015 0.02 1/a Figure 1: Asymptotic performance of a realizable perceptron. Simulation results for 110 :;;: 2 and N :;;: 50 (solid curve) are compared with the theoretical prediction f g :;;: 1.271a (dashed curve). 0.35 0.3 0.25 0.2 fg - .' P ..... .... 0.15 o ? ? 'c"..' .. .. .. ..'' . ' .. . ' '.:.... ~ 0.1 0.05 0 0 0.1 0.2 0.3 lIfo 0.4 0.5 0.6 0.7 Figure 2: Simulation results for on-line learning of a perceptron with output noise. Here 1Jo :;;: 2, P :;;: 0.2, U = 4, and qo :;;: -1.95. The regular percept ron learning (dashed curve) is compared with the modified algorithm (solid curve). The dashed line shows the theoretical prediction Eq. (18) N :;;: 250, 

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New Algorithms for 2D and 3D Point Matching: Pose Estimation and Correspondence Steven Gold l , Chien Ping LuI, Anand Rangarajan l , Suguna Pappu l and Eric Mjolsness 2 Department of Computer Science Yale University New Haven, CT 06520-8285 Abstract A fundamental open problem in computer vision-determining pose and correspondence between two sets of points in spaceis solved with a novel, robust and easily implementable algorithm . The technique works on noisy point sets that may be of unequal sizes and may differ by non-rigid transformations. A 2D variation calculates the pose between point sets related by an affine transformation-translation, rotation, scale and shear. A 3D to 3D variation calculates translation and rotation . An objective describing the problem is derived from Mean field theory. The objective is minimized with clocked (EM-like) dynamics. Experiments with both handwritten and synthetic data provide empirical evidence for the method. 1 Introduction Matching the representations of two images has long been the focus of much research in Computer Vision, forming an essential component of many machine-based ob1 E-mail address of authors: lastname-firstname@cs.yale.edu 2Department of Computer Science and Engineering, University of California at San Diego (UCSD), La Jolla, CA 92093-0114. E-mail: emj@cs.ucsd.edu 958 Steven Gold, Chien Ping Lu, Anand Rangarajan, Suguna Pappu, Eric Mjolsness ject recognition systems. Critical to most matching techniques is the determination of correspondence between spatially localized features within each image. This has traditionally been considered a hard problem - especially when the issues of noise, missing or spurious data, and non-rigid transformations are tackled [Grimson, 1990] . Many approaches have been tried, with tree-pruning techniques and generalized Hough transforms being the most common. We introduce anew, robust and easily implementable algorithm to find such poses and correspondences. The algorithm can determine non-rigid transformations between noisy 2D or 3D spatially located unlabeled feature sets despite missing or spurious features. It is derived by minimizing an objective function describing the problem with a combination of optimization techniques, incorporating Mean Field theory, slack variables, iterative projective scaling, and clocked (EM-like) dynamics. 2 2D with Affine Transformations 2.1 Formulating the Objective Our first algorithm calculates the pose between noisy, 2D point sets of unequal size related by an affine transformation - translation, rotation, scale and shear. Given two sets of points {Xj} and {Yk}, one can minimize the following objective to find the affine transformation and permutation which best maps Y onto X : J E2D(m, t, A) = K L L mjkllXj - J t - AYkll2 + g(A) - j=l k=l K aLL mjk j=l k=l with constraints: Vj Ef=l mjk ~ 1 , Vk Ef=l mjk ~ 1 , Vjk mjk ~ 0 and g(A) = 1a2 + /'i,b 2 + AC2 A is decomposed into scale, rotation, vertical shear and oblique shear as follows: where, s(a) = ( eOa o ) e- b ,Sh 2 (c) cosh(c) = ( sm . h( c) sinh(c) ) cosh(c) R(8) is the standard 2x2 rotation matrix. g(A) serves to regularize the affine transformation - bounding the scale and shear components. m is a fuzzy correspondence matrix which matches points in one image with corresponding points in the other image. The constraints on m ensure that each point in each image corresponds to at most one point in the other image. However, partial matches are allowed, in which case the sum of these partial matches may add up to no more than one. The inequality constraint on m permits a null match or multiple partial matches. The a term biases the objective towards matches. The decomposition of A in the above is not required, since A could be left as a 2x2 matrix and solved for directly in the algorithm that follows . The decomposition just provides for more precise regularization, i.e., specification of the likely kinds oftransformations. Also Sh2(C) could New Algorithms for 2D and 3D Point Matching 959 be replaced by another rotation matrix, using the singular value decomposition of A. We transform the inequality constraints into equality constraints by introducing slack variables, a standard technique from linear programming; K L Vj K+1 mj k ::; 1 -+ Vj k=l L mj k = 1 k=l and likewise for the column constraints. An extra row and column are added to the matrix m to hold the slack variables. Following the treatment in [Peterson and Soderberg, 1989; Yuille and Kosowsky, 1994] we employ Lagrange multipliers and an x log x barrier function to enforce the constraints with the following objective: J E2D(m, t, A) K J =L L mjkllXj j=l k=l 1 J+1K+1 +~ L: J L mik(logmjk -1) i=l k=l t - AYk ll 2 K+1 + g(A) K K ct L L mjk j=l k=l J+l + LJlj(L mjk -1) + LlIk(L mjk -1) i=l k=l k=l (1) j=l In this objective we are looking for a saddle point. (1) is minimized with respect to m, t, and A which are the correspondence matrix, translation, and affine transform, and is maximized with respect to Jl and 1I, the Lagrange multipliers that enforce the row and column constraints for m. m is fuzzy, with the degree of fuzziness dependent upon f3. 2.2 The Algorithm The algorithm to minimize the above objective proceeds in two phases. In phase one, while {t, A} are held fixed, m is initialized with a coordinate descent step, described below, and then iteratively normalized across its rows and columns until the procedure converges (iterative projective scaling). This phase is analogous to a softmax update, except that instead of enforcing a one-way, winner-take-all (maximum) constraint, a two-way, assignment constraint is being enforced. Therefore we describe this phase as a softassign. In phase two {t, A} are updated using coordinate descent. Then f3 is increased and the loop repeats. Let E2D be the above objective (1) without the terms that enforce the constraints (i.e. the x log x barrier function and the Lagrange parameters). In phase one (softassign) m is updated via coordinate descent: mjk E 2D = exp (-f3-a -) amjk Then m is iteratively normalized across j and k until Ef=l Ef=l Llmiajk < i : Using coordinate descent the {t, A} are updated in phase two. If a term of {A} cannot be computed analytically (because of its regularization), Newton's method 960 Steven Gold, Chien Ping Lu, Anand Rangarajan, Suguna Pappu, Eric Mjolsness is used to compute the root of the function . So if a is a term of {t, A} then in phase two we update a such that 8!~D = O. Finally f3 is increased and the loop repeats. By setting the partial derivatives of E2D to zero and initializing the Lagrange parameters to zero, the algorithm for phase one may be derived . Beginning with a small f3 allows minimization over a fuzzy correspondence matrix m, for which a global minimum is easier to find. Raising f3 drives the m's closer to 0 or 1, as the algorithm approaches a saddle point. 3 3D with Rotation and Translation The second algorithm solves the 3D-3D pose estimation problem with unknown correspondence. Given two sets of 3D points {Xj} and {Yk} find the rotation R, translation T, and correspondence m that minimize K J E3D(m ,T,R) J K = LLmjkllRXj +T-YkI12-aLLmjk j=l k=1 j=l k=1 with the same constraint on the fuzzy correspondence matrix m as in 2D affine matching. Note that there is no regularization term for the T - R parameters. This algorithm also works in two phases. In the first, m is updated by a soft assign as was described for 2D affine matching. In the second phase, m is fixed, and the problem becomes a 3D to 3D pose estimation problem formulated as a weighted least squares problem. The rotation and translation are represented by a dual number quaternion (r, s) which corresponds to a screw coordinate transform [Walker et al. , 1991] . The rotation can be written as R(r) = W(r)tQ(r) and the translation as W(r)ts . Using these representations, the objective function becomes J K E3D = L L mjkllW(r)tQ(r)xj j=1 k=1 + W(r)t s - Ykl1 2 where Xj = (Xj, 0)1 and Yk = (Yk,O)t are the quaternion representations of Xj and Yk, respectively. Using the properties that Q(a)b = W(b)a and Q(a)tQ(a) = W(a)tW(a) = (ata)J, the objective function can be rewritten as (2) where J K - L L mjkQ(Yk)tW(Xj) j=1k=1 1 J K 2LLmjkI j=1k=1 J C3 = K L L mjk(W(Xj) - Q(Yk)). j=1k=1 New Algorithms for 2D and 3D Point Matching 961 With this new representation, all the information, including the current fuzzy estimate of the correspondence m are absorbed into the three 4-by-4 matrices Cl, C2 , C3 in (2), which can be minimized in closed-form [Walker et al., 1991]. 4 Experimental Results In this section we provide experimental results for both the 20 and 30 matching problems. As an application of the 20 matching algorithm, we present results in the context of handwritten character recognition. 4.1 Handwritten Character Data The data were generated using an X-windows tool which enables us to draw an image with the mouse on a writing pad on the screen. The contours of the images are discretized and are expressed as a set of points in the plane. In the experiments below, we generate 70 points per character on average. The inputs to the point matching algorithm are the x-y coordinates generated by the drawing program. No other pre-processing is done. The output is a correspondence matrix and a pose. In Figures 1 and 2, we show the correspondences found between several images drawn in this fashion.To make the actual point matches easier to see, we have drawn the correspondences only for every other model point. ~ :..... -_........ ..........? _ Figure 1: Correspondence of digits In one experiment, we drew examples of individual digits, one as a model digit and then many different variations of it. In Figure 1, it can be seen that the Steven Gold, Chien Ping Lu, Anand Rangarajan, Suguna Pappu, Eric Mjolsness 962 ..~ ? ?? ?? ?\. I ? ? ???? ?? ?? ????? ? ? .."..: .... ... .. ..?? ..,. ..,.. ? ?? ? ? ' ?? J ? ?? ? Figure 2: Correspondence: "a" found in "cat", "0" found in "song" correspondences are good for a large variation from the model digit. For example, the correspondence is invariant to scale. Also, the correspondence is good between distorted digits, as in 3 and 6, or between different forms of a digit as in 4, 3, and 2. In another experiment (Figure 2), individual letters are correctly identified within words. Here, no pre-processing to segment the cursive word into letters is done. The correspondence returned by the point matching algorithm by itself can be good enough for identification. Even similar letters may be differentiated, for example the "a" in "cat" is correctly identified even though the "e" has a similar shape and the "0" is correctly identified in "song" , despite the similarity of the "s". 4.2 Randomly generated point sets: 2D In the second set of experiments, randomly generated dot patterns were used. In each trial a model is created by randomly generating with a uniform distribution, 50 points on a grid of unit area. Independent Gaussian noise N(O, 0-) is added to each of the points creating a jittered image. Then a fraction, Pd, of points are deleted, and a fraction, P6, of spurious points are added, randomly on the unit square. Finally, a randomly generated transformation is applied to the set to generate a new image. The objective then is to recover the transformation and correspondence between the transformed image and the original point set. The transformations we have considered are A -+ (Translation, rotation, scale) and the full affine transformation, A -+ (Translation, rotation, scale, vertical shear, oblique shear) The transformation parameters, {tz, t y , (J, a, b, c} are bounded in the following way: -0.5 < t z , ty < 0.5, -27 0 < (J < 27 0 , 0.5 ~ ea ~ 2 where a is the scale parameter, and 0.7 ~ eb,ee ~ 1/0.7 where b,c are the parameters for the two shears. Each of the parameters is chosen independently and uniformly from the possible ranges. ocrt.a.1 e ? .,ima"te ? We use the error measure ea = 31 a w'id~h", 1where ea IS the error measure for parameter a and width a is the range of permissible values for a . Dividing by width a is preferable to dividing by aaetual, which incorrectly weights small aaetual values. The reported error (y axes of Figure 3) is the average error over all the parameters. New Algorithms for 2D and 3D Point Matching 963 The time to recover the correspondence and transformation for a problem instance of 50 points is about 50 seconds on a Silicon Graphics workstation with a R4400 processor. By varying parameters such as the annealing rate or stopping criterion, this can be reduced to about 20 seconds with some degradation in accuracy. For each trial combinations of u E {0.01, 0.02, ... , 0.08} and Pd E {O%, 10%,30%, 50%} and P6 E {O%, 10%} were used. Results are reported separately for transformations A and A. For each combination of (U,Pd,P6) 500 test instances were generated. Each data point in Figures 3.a and 3.b represents the average error measure for these 500 experiments. The noise and/or deletion-addition factor increases the error measure monotonically. As expected, the transformation A has better results than the affine transformation A. Translation, rotation, scale Affine 0.6r---~----~----~----~ ~ JE JE ~ ~ e JE JE IE JE JE ? + + 0.2 ~ w ? ? ? ~ oS Q) E 0.4 eas v 0.6r---~----~----~----~ o o ~ + + + + g ~ 0.02 ~ 0.04 + + )( )( ~ ~ 0.06 oS Q) E 0.4 eas ~ co .. 0.2 ? JE JE + + + + ~ w o o 0.08 Standard deviation of noise IE + + ~ e ? QS()()( 0.02 0.04 0.06 0.08 Standard deviation of noise Figure 3: 2D Results for Synthetic Data x: Pd = 0.0,P6 = 0.0, + : Pd = 0.3,P6 = 0.1, 4.3 0: Pd = O.l,p& = 0.1 * : Pd = 0.5,p& = 0.1 Randomly generated point sets: 3D A test instance for 3D point matching involves generating a random 3D point set as a model image, and then generating a test image by applying a random transformation, adding noise and then randomly deleting points. 20 points are generated uniformly within an unit cube. The parameters for the transformation are generated as follows: The three rotation angles for R are selected from a uniform distribution U[20, 70]. Translation parameters T~, Ty, Tz are selected from a uniform distribution U[2.5,7.5]. Gaussian noise N(O, u) is added to the points. The objective then is to recover the three translation and three rotation parameters and to find the correspondence between this and the original point set. The results are summarized in Figure 4. 964 Steven Gold, Chien Ping Lu, Anand Rangarajan, Suguna Pappu, Eric Mjolsness 30 ... 0.8 X ... 0 t:: 20 W C 0 X 0 '.0:; ~ 10 a: ~ X X 0 0 ~ ~ + -+ 0 0 0.2 ...... WO.6 X 0 ~ + 0.4 0 X X 0 ?.0:; J2 en c as ~ 0 0 + 00 0.6 0 0 0 0.4 .= 0.2 X X c 0 X ~ ~ + ~ + 0.2 ~ + + 0.4 ~ + 0.6 Standard deviation of noise Standard deviation of noise Figure 4: 3D Results for Synthetic Data x: Pd = 0.0,P8 = 0.0, 0.2,P8 = 0.2, + : Pd = 5 Pd = 0.1,P8 = 0.1 *: Pd = 0.3, ps = 0.3 0 : Conclusion We have developed an algorithm for solving 2D and 3D correspondence problems. The algorithm handles significant noise, missing or spurious features , and nonrigid transformations. Moreover it works with point feature data alone; inclusion of other types of feature information could improve its accuracy and speed . This approach may also be extended to solve multi-level problems. Additionally, the affine transform might be modified to include higher order transformations. It may also be used as a distance measure in learning [Gold et al.,1994] . Acknowledgements This work has been supported by AFOSR grant F49620-92-J-0465, ONR/DARPA grant N00014-92-J-4048 , and the Yale Center for Theoretical and Applied Neuroscience (CTAN) . Jing Yan developed the handwriting interface. References S. Gold , E. Mjolsness and A. Rangarajan. (1994) Clustering with a domain-specific distance measure. In J.D. Cowan et al., (eds.), NIPS 6. Morgan Kaufmann. E. Grimson, (1990) Object Recognition by Computer, Cambridge, MA : MIT Press C. Peterson and B. Soderberg. (1989) A new method for mapping optimization problems onto neural networks, Int. Journ . of Neural Sys., 1(1) :3:22. M. W. Walker, L. Shoo and R. Volz . (1991) Estimating 3-D location parameters using dual number quaternions, CVGIP: Image Understanding 54(3) :358-367. A. L. Yuille and J. J . Kosowsky. (1994) . Statistical physics algorithms that converge. Neural Computation, 6:341-356. 

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Pattern Playback in the '90s Malcolm Slaney Interval Research Corporation 180 l-C Page Mill Road, Palo Alto, CA 94304 malcolm@interval.com Abstract Deciding the appropriate representation to use for modeling human auditory processing is a critical issue in auditory science. While engineers have successfully performed many single-speaker tasks with LPC and spectrogram methods, more difficult problems will need a richer representation. This paper describes a powerful auditory representation known as the correlogram and shows how this non-linear representation can be converted back into sound, with no loss of perceptually important information. The correlogram is interesting because it is a neurophysiologically plausible representation of sound. This paper shows improved methods for spectrogram inversion (conventional pattern playback), inversion of a cochlear model, and inversion of the correlogram representation. 1 INTRODUCTIONl My interest in auditory models and perceptual displays [2] is motivated by the problem of sound understanding, especially the separation of speech from noisy backgrounds and interfering speakers. The correlogram and related representations are a pattern space within which sounds can be "understood" and "separated" [3][4]. I am therefore interested in resynthesizing sounds from these representations as a way to test and evaluate sound separation algorithms, and as a way to apply sound separation to problems such as speech enhancement. The conversion of sound to a correlogram involves the intermediate representation of a cochleagram, as shown in Figure 1, so cochlear-model inversion is addressed as one piece of the overall problem. 1. Much of this work was performed by Malcolm Slaney, Daniel Naar and Richard F. Lyon while all three were employed at Apple Computer. The mathematical details of this work were presented at the 1994ICASSP[I]. 828 Malcolm Slaney Waveform Cochleagram j 1r- *T'T~" "I-~!"'T!-~ -'-o I-'1 ::::S:ec::ti:n:: h ------------~ Correlogram . . t--'~~'--t.:lfga..~l!!!Jnme ------------~ Figure 1. Three stages in low-level auditory perception are shown here. Sound waves are converted into a detailed representation with broad spectral bands, known as cochleagrams. The correlogram then summarizes the periodicities in the cochleagram with short-time autocorrelation. The result is a perceptual movie synchronized to the acoustic signal. The two inversion problems addressed in this work are indicated with arrows from right to left There are three factors which can be used to judge the quality of an auditory model: psychoacoustic comparisons, neurophysiological plausibility, and does it represent the perceptually relevant information? First, the correlogram has been shown to simply and accurately predict human pitch perception [5]. The neurophysiological basis for the correlogram has not been found, but there are neural circuits performing the same calculation in the mustached bat's echolocation system [6]. Finally, from an information representation point of view, does the correlogram preserve the salient information? The results of this paper show that no information has been lost. Since the psychoacoustic, neurophysiological, and information representation measures are all positive, perhaps the correlogram is the basis of most auditory processing. The inversion techniques described here are important because they allow us to readily evaluate the results of sound separation models that "zero out" unwanted portions of the signal in the correlogram domain. This work extends the convex projection approach of Irino [7] and Yang [8] by considering a different cochlear model, and by including the correlogram inversion. The convex projection approach is well suited to "filling in" missing information. While this paper only describes the process for one particular auditory model, the techniques are equally useful for other models. This paper describes three aspects of the problem: cochleagram inversion, conversion of the correlogram into spectrograms, and spectrogram inversion. A number of reconstruction options are explored in this paper. Some are fast, while other techniques use time-consuming iterations to produce reconstructions perceptually equivalent to the original sound. Fast versions of these algorithms could allow us to separate a speaker's voice from the background noise in real time. 2 COCHLEAGRAM INVERSION Figure 2 shows a block diagram of the cochlear model [9] that is used in this work. The basis of the model is a bank of filters, implemented as a cascade of low-pass filters, that splits the input signal into broad spectral bands. The output from each filter in the bank is called a channel. The energy in each channel is detected and used to adjust the channel gain, implementing a simple model of auditory sensitivity adaptation, or automatic gain control (AGe). The half-wave rectifier (HWR) detection nonlinearity provides a waveform for each channel that roughly represents the instantaneous neural firing rate at each position along the cochlea. E ~---+ > ~ Cochlear Filterbank Detector (HWR) Adaptation or AGC Figure 2. Three stages of the simple cochlear model used in this paper are shown above. Pattern Playback in the '90s 829 The cochleagram is converted back into sound by reversing the three steps shown in Figure 2. First the AGe is divided out, then the negative portions of each cochlear channel are recovered by using the fact that each channel is spectrally limited. Finally, the cochlear filters are inverted by running the filters backwards, and then correcting the resulting spectral slope. The AGe stage in this cochlear model is controlled by its own output. It is a combination of a multiplicative gain and a simple first-order filter to track the history of the output signal. Since the controlling signal is directly available, the AGe can be inverted by tracking the output history and then dividing instead of multiplying. The performance of this algorithm is described by Naar [10] and will not be addressed here. It is worth noting that AGe inversion becomes more difficult as the level of the input signal is raised, resulting in more compression in the forward path. The next stage in the inversion process can be done in one of two ways. After AGC inversion, both the positive values of the signal and the spectral extant of the signal are known. Projections onto convex sets [11], in this case defined by the positive values of the detector output and the spectral extant of the cochlear filters, can be used to find the original signal. This is shown in the left half of Figure 3. Alternatively, the spectral projection filter can be combined with the next stage of processing to make the algorithm more efficient. The increased efficiency is due to better match between the spectral projection and the cochlear filterbank, and due to the simplified computations within each iteration. This is shown in the right half of Figure 3. The result is an algorithm that produces nearly perfect results with no iterations at all. Temporal Projection Filter Bank Inversion Spectral Projection ~ ~ ~ Q) E AGC -go Inversion Other --110.. Inversion ChannelsL..._ _..... ~Qi 0> CI)~ ~ 8 o Figure 3. There are two ways to use convex projections to recover the information lost by the detectors. The conventional approach is shown on the left. The right figure shows a more efficient approach where the spectral projection has been combined with the filterbank inversion Finally, the multiple outputs from the cochlear filterbank are converted back into a single waveform by correcting the phase and summing all channels. In the ideal case, each cochlear channel contains a unique portion of the spectral energy, but with a bit of phase delay and amplitude change. For example, if we run the signal through the same filter the spectral content does not change much but both the phase delay and amplitude change will be doubled. More interestingly, if we run the signal through the filter backwards, the forward and backward phase changes cancel out. After this phase correction, we can sum all channels and get back the Original waveform, with a bit of spectral coloration. The spectral coloration or tilt can be fixed with a simple filter. A more efficient approach to correct the spectral tilt is to scale each channel by an appropriate weight before summing, as shown in Figure 4. The result is a perfect reconstruction, over those frequencies where the cochlear filters are non-zero. Figure 5 shows results from the cochleagram inversion procedure. An impulse is shown on the left, before and after 10 iterations of the HWR inversion (using the algorithm on the right half of Figure 3). With no iterations the result is nearly perfect, except for a bit of noise near the center. The overall curvature of the baseline is due to the fact that informa- 830 ~ Malcolm Slaney c:UI ~'S .... 0- .s5 iIO Time Reversed Filter for Inversion IIR or FIR Filter to Correct Tilt E -0 .... ~ ;:'0> 0> CJ)~ ~ c:,a as;:, ?,e. 0>;:, itO Time Reversed Filter for Inversion E -0 .... c:.g ;:'0> 0> CJ)~ Figure 4. Two approaches are shown here to invert the filterbank. The left diagram shows the normal approach, the right figure shows a more efficient approach where the spectral-tilt filter is converted to a simple multiplication. tion near DC has been lost as it travels through the auditory system and there is no way to recover it with the information that we have. A more interesting example is shown on the right. Here the word "tap" 1 has been reconstructed, with and without the AGC inversion. With the AGe inversion the result is nearly identical to the original. The auditory system is very sensitive to onsets and quickly adapts to steady state sounds like vowels. It is interesting to compare this to the reconstruction withoutAGC inversion. Without the AGC, the result is similar to what the ear hears, the onsets are more prominent and the vowels are deemphasized. This is shown in the right half of Figure 5. Impulse inversion with no iterations Impulse iteration with 10 iterations "Tap" reconstruction "Tap" Reconstruction with AGC Inversion without AGC Inversion Figure 5. The cochlear reconstructions of an impulse and the word "tap" are shown here. The first and second reconstructions show an impulse reconstruction with and without iterations. The third and fourth waveforms are the word "tap" with and without the AGe inversion. 3 CORRELOGRAM INVERSION The correlogram is an efficient way to capture the short-time periodicities in the auditory signal. Many mechanical measurements of the cochlea have shown that the response is highly non-linear. As the signal level changes there are large variations in the bandwidth and center frequency of the cochlear response. With these kinds of changes, it is difficult to imagine a system that can make sense of the spectral profile. This is especially true for decisions like pitch determination and sound separation. But through all these changes in the cochlear filters, the timing information in the signal is preserved. The spectral profile, as measured by the cochlea, might change, but the rate of glottal pulses is preserved. Thus I believe the auditory system is based on a representation of sound that makes short-time periodicities apparent. One such representation is the correlogram. The correlogram measures the temporal correlation within each channel, either using FFfs which are most efficient in computer implementations, or neural delay lines much like those found in the binaural system of the owl. 1. The syllable "tap", samples 14000 through 17000 of the "trainldr5/fcdfll sxl06/sx106.adc" utterance on the TIMIT Speech Database, is used in all voiced examples in this paper. 831 Pattern Playback in the '90s The process of inverting the correlogram is simplified by noting that each autocorrelation is related by the Fourier transform to a power spectrum. By combining many power spectrums into a picture, the result is a spectrogram. This process is shown in Figure 6. In this way, a separate spectrogram is created for each channel. There are known techniques for converting a spectrogram, which has amplitude information but no phase information, back into the original waveform. The process of converting from a spectrogram back into a waveform is described in Section 4. The correlogram inversion process consists of inverting many spectrograms to form an estimate of a cochleagram. The cochleagram is inverted using the techniques described in Section 2. Frame 42 of Correlogram Frame 43 of Correlogram IFFTto get of spectrogram One line of Cochleagram Spectrogram Inversion ., Time Time Figure 6. Correlogram inversion is possible by noting that each row of the correlogram contains the same information as a spectrogram of the same row of cochleagram output. By converting the correlogram into many spectrograms, the spectrogram inversion techniques described in Section 4 can be used. The lower horizontal stripe in the spectrogram is due to the narrow passband of the cochlear channel. Half-wave rectification of the cochlear filter output causes the upper horizontal stripes. One important improvement to the basic method is possible due to the special characteristics of the correlogram. The essence of the spectrogram inversion problem is to recover the phase information that has been thrown away. This is an iterative procedure and would be costly ifit had to be performed on each channel. Fortunately, there is quite a bit of overlap between cochlear channels. Thus the phase recovered from one channel can be used to initialize the spectrogram inversion for the next channel. A difficulty with spectrogram inversion is that the absolute phase is lost. By using the phase from one channel to initialize the next, a more consistent set of cochlear channel outputs is recovered. 4 SPECTROGRAM INVERSION While spectrograms are not an accurate model of human perception, an implementation of a correlogram includes the calculation of many spectrograms. Mathematically, an autocorrelation calculation is similar to a spectrogram or short-time power spectrum. One column of a conventional spectrogram is related to an autocorrelation of a portion of the original waveform ~y a Fourier transform (see Figure 6). Unfortunately, the final representation of both spectrograms and autocorrelations is missing the phase information. The main task of a spectrogram inversion algorithm is to recover a consistent estimate ofthe missing phase. This process is not magical, it can only recover a signal that has the same magnitude spectrum as the original spectrogram. But the consistency constraint on the time evolution of the signal power spectrum also constrains the time evolution of the spectral phase. Malcolm Slaney 832 The basic procedure in spectrogram inversion [12] consists of iterating between the time and the frequency domains. Starting from the frequency domain, the magnitude but not the phase is known. As an initial guess, any phase value can be used. The individual power spectra are inverse Fourier transformed and then summed to arrive at a single waveform. If the original spectrogram used overlapping windows of data, the information from adjacent windows either constructively or destructively interferes to estimate a waveform. A spectrogram of this new data is calculated, and the phase is now retained. We know the original magnitude was correct. Thus we can estimate a better spectrogram by combining the original magnitude information with the new phase information. It can be shown that each iteration will reduce the error. Figure 7 shows an outline of steps that can be used to improve the consistency of phase estimates during the first iteration. As each portion of the waveform is added to the estimated signal, it is possible to add a linear phase so that each waveform lines up with the proceedings segments. The algorithm described in the paragraph above assumes an initial phase of zero. A more likely phase guess is to choose a phase that is consistent with the existing data. The result with no iterations is a waveform that is often closer to the original than that calculated assuming zero initial phase and ten iterations. The total computational cost is minimized by combining these improvements with the initial phase estimates from adjacent channels of the correlogram. Thus when inverting the first channel of the correlogram, a cross-correlation is used to pick the initial phase and a few more iterations insure a consistent result. After the first channel, the phase of the proceeding channel is used to initialize the spectrogram inversion and only a few iterations are necessary to fine tune the waveform. Reconstructions from segments 1 through N o Rotated Segment N+ 1 400 hi\;\; I o Cross Correlation SegmentN+1 New Reconstruction b;&/\; I o 400 400 300 Maximize fit by chOOSing peak o Figure 7. A procedure for adjusting the phase of new segments when inverting a spectrogram is shown above. As each new segment (bottom left) is converted from a power spectrum into a waveform, a linear phase is added to maximize the fit with the existing segments (top left.) The amount of rotation is determined by a cross correlation (middle). Adding the new segment with the proper rotation (top right) produces the new waveform (bottom right.) 833 Pattern Playback in the '90s 5 PUTTING IT TOGETHER This paper has described two steps to convert a correlogram into a sound. These steps are detailed below: I) For each row of the correlogram: a) Convert the autocorrelation data into power spectrum (Section 3). b) Use spectrogram inversion (Section 4) to convert the spectrograms into an estimate of cochlear channel output. c) Assemble the results of spectrogram inversion into an estimate of the cochleagram. Invert the cochleagram using the techniques described in Section 2. 2) This process is diagrammed in Figures I and 6. 6 RESULTS Figure 8 shows the results of the complete reconstruction process for a 200Hz impulse train and the word "tap." In both cases, no iterations were performed for either the spectrogram or filterbank inversion. More iterations reduce the spectral error, but do not make the graphs look better or change the perceptual quality much. It is worth noting that the "tap" reconstruction from a correlogram looks similar to the cochleagram reconstruction without the AGC (see Figure 5.) Reducing the level of the input signal, thus reducing the amount of compression performed by the AGC, results in a correlogram reconstruction similar to the original waveform. 0.02 r---------. 0.05f"'"'"!~----"""" 0.01 o o -0.01 -0.02"'--~~~~~ 50 100 150 200 -0.05 " ' - - - - - - -.... o 1000 2000 3000 Figure 8. Reconstructions from the correlogram representation of an impulse train and the word "tap" are shown above. Reducing the input signal level, thus minimizing the effect of errors when inverting the AGe, produces results identical to the original "tap." It is important to note that the algorithms described in this paper are designed to minimize the error in the mean-square sense. This is a convenient mathematical definition, but it doesn't always correlate with human perception. A trivial example of this is possible by comparing a waveform and a copy of the waveform delayed by lOms. Using the meansquared error, the numerical error is very high yet the two waveforms are perceptually equivalent. Despite this, the results of these algorithms based on mean-square error do sound good. 7 CONCLUSIONS This paper has described several techniques that allow several stages of an auditory model to be converted back into sound. By converting each row of the correlogram into a spectrogram, the spectrogram inversion techniques of Section 4 can be used. The special characteristics of a correlogram described in Section 3 are used to make the calculation more efficient. Finally, the cochlear filterbank can be inverted to recover the original waveform. The results are waveforms, perceptually identical to the original waveforms. 834 Malcolm Slaney These techniques will be especially useful as part of a sound separation system. I do not believe that our auditory system resynthesizes partial waveforms from the auditory scene. Yet, all research systems generate separated sounds so that we can more easily perceive their success. More work is still needed to fine-tune these algorithm and to investigate the ability to reconstruct sounds from partial correlograms. Acknowledgments I am grateful for the inspiration provided by Frank Cooper's work in the early 1950's on pattern playback[13][14]. His work demonstrated that it was possible to convert a spectrogram, painted onto clear plastic, into sound. This work in this paper was performed with Daniel Naar and Richard F. Lyon. We are grateful for the help we have received from Richard Duda (San Jose State), Shihab Shamma (U. of Maryland), Jim Boyles (The MathWorks) and Michele Covell (Interval Research). References [1] Malcolm Slaney, D. Naar, R. F. Lyon, "Auditory model inversion for sound separation," Proc. of IEEE ICASSP, Volume II, pp. 77-80, 1994. [2] M. Slaney and R. F. Lyon, "On the importance of time-A temporal representation of sound," in Visual Representations of Speech Signals, eds. M. Cooke, S. Beet, and M. Crawford, J. Wiley and Sons, Sussex, England, 1993. [3] R. F. Lyon, "A computational model of binaural localization and separation," Proc. of IEEE ICASSP, 1148-1151, 1983. [4] M. Weintraub, "The GRASP sound separation system," Proc. of IEEE ICASSP, pp. 18A.6.1-18A.6.4, 1984. [5] D. Hennes, "Pitch analysis," in Visual Representations of Speech Signals, eds. M. Cooke, S. Beet, and M. Crawford, J. Wiley and Sons, Sussex, England, 1993. [6] N. Suga, "Cortical computational maps for auditory imaging," Neural Networks, 3, 321, 1990. [7] T. lrino, H. Kawahara, "Signal reconstruction from modified auditory wavelet transfonn," IEEE Trans. on Signal Processing, 41,3549-3554, Dec. 1993. [8] x. Yang, K. Wang, and S. Sharnma, "Auditory representations of acoustic signals," IEEE Trans. on Information Theory, 38, 824-839, 1992. [9] R. F. Lyon, "A computational model of filtering, detection, and compression in the cochlea," Proc. of the IEEE ICASSP, 1282-1285,1982. [10] D. Naar, "Sound resynthesis from a correlogram," San Jose State University, Department of Electrical Engineering, Technical Report #3, May 1993. [11] R. W. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Sys., vol. 22, 735, 1975. [12] D. Griffin and J. Lim, "Signal estimation from modified short-time Fourier transfonn," IEEE Trans. on Acoustics, Speech, and Signal Processing, 32, 236-242, 1984. [13] F. S. Cooper, "Some Instrumental Aids to Research on Speech," Report on the Fourth Annual Round Table Meeting on Linguistics and Language Teaching, Georgetown University Press, 46-53, 1953. [14] F. S. Cooper, "Acoustics in human communications: Evolving ideas about the nature of speech," J. Acoust. Soc. Am., 68(1),18-21, July 1980. 

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Learning in large linear perceptrons and why the thermodynamic limit is relevant to the real world Peter Sollich Department of Physics, University of Edinburgh Edinburgh EH9 3JZ, U.K. P.Sollich~ed.ac.uk Abstract We present a new method for obtaining the response function 9 and its average G from which most of the properties of learning and generalization in linear perceptrons can be derived. We first rederive the known results for the 'thermodynamic limit' of infinite perceptron size N and show explicitly that 9 is self-averaging in this limit. We then discuss extensions of our method to more general learning scenarios with anisotropic teacher space priors, input distributions, and weight decay terms. Finally, we use our method to calculate the finite N corrections of order 1/ N to G and discuss the corresponding finite size effects on generalization and learning dynamics. An important spin-off is the observation that results obtained in the thermodynamic limit are often directly relevant to systems of fairly modest, 'real-world' sizes. 1 INTRODUCTION One of the main areas of research within the Neural Networks community is the issue of learning and generalization. Starting from a set of training examples (normally assumed to be input-output pairs) generated by some unknown 'teacher' rule V, one wants to find, using a suitable learning or training algorithm, a student N (read 'Neural Network') which generalizes from the training set, i.e., predicts the outputs corresponding to inputs not contained in the training set as accurately as possible. 208 Peter Sollich If the inputs are N-dimensional vectors x E nN and the outputs are scalars yEn, then one of the simplest functional forms that can be assumed for the student .N is the linear perceptron, which is parametrized in terms of a weight vector W N E nN and implements the linear input-output mapping YN(X) = 7Nw~x. (1) A commonly used learning algorithm for the linear perceptron is gradient descent on the training error, i.e., the error that the student .N makes on the training set . Using the standard squared output deviation error measure, the training error for a given set ofp training examples {(xlJ , ylJ),J-l = 1 . . . p} is E t = L:IJ ~(ylJ-YN(XIJ))2 = ~ L:IJ (ylJ - w~xlJ / VN)2. To prevent the student from fitting noise in the training data, a quadratic weight decay term ~AW~ is normally added to the training error, with the value of the weight decay parameter A determining how strongly large weight vectors are penalized. Gradient descent is thus performed on the function E E t + ~AW~, and the corresponding learning dynamics is, in a continuous time approximation , dw N / dt = -"V wE. As discussed in detail by Krogh and Hertz (1992), this results in an exponential approach of W N to its asymptotic value , with decay constants given by the eigenvalues of the matrix M N , defined by (1 denotes the N x N identity matrix) = MN = Al + A, A = ~ L:IJ xlJ(xlJ)T. To examine what generalization performance is achieved by the above learning algorithm, one has to make an assumption about the functional form of the teacher . The simplest such assumption is that the problem is learnable, i. e., that the teacher , like the student, is a linear perceptron. A teacher V is then specified by a weight vector Wv and maps a given input x to the output Yv(x) = w~ x/VN. We assume that the test inputs for which the student is asked to predict the corresponding outputs are drawn from an isotropic Gaussian distribution , P(x) <X exp( _~x2). The generalization error, i. e., the average error that a student .N makes on a random input when compared to teacher V, is given by fg = ~((YN(X) - yv(x))2)p(x) = 2~(WN - w v )2. (2) Inserting the learning dynamics W N = W N(t), the generalization acquires a time dependence, which in its exact form depends on the specific training set , teacher, and initial value of the student weight vector, WN(t = 0). We shall confine our attention to the average of this time-dependent generalization error over all possible training sets and teachers; to avoid clutter , we write this average simply as fg(t). We assume that the inputs x lJ in the training set are chosen independently and randomly from the same distribution as the test inputs, and that the corresponding training outputs are the teacher outputs corrupted by additive noise, ylJ = yv(xlJ) + 171J, where the 171J have zero mean and variance rr2. If we further assume an isotropic Gaussian prior on the teacher weight vectors, P(w v ) <X exp( -~w~), then the average generalization error for t -> 00 is (Krogh and Hertz, 1992) fg(t -> 1 [ rr 2 C + A(rr2 - A) OC] 00) = 2 OA ' (3) where G is the average of the so-called response function over the training inputs: C = (Q) P( {x"}), Q -- N1 t r 1 MN? (4) Learning in Large Linear Perceptrons 209 The time dependence of the average generalization error for finite but large t is an exponential approach to the asymptotic value (3) with decay constant oX + amin, where am in is the lowest eigenvalue occurring in the average eigenvalue spectrum of the input correlation matrix A (Krogh and Hertz, 1992). This average eigenvalue spectrum, which we denote by p(a), can be calculated from the average response function according to (Krogh, 1992) p(a) = !11' f-O+ lim ImGI.>.=_a-if, (5) where we have assumed p( a) to be normalized, fda p( a) = 1. Eqs. (3,5) show that the key quantity determining learning and generalization in the linear percept ron is the average response function G defined in (4). This function has previously been calculated in the 'thermodynamic limit', N -- 00 at a = piN = const., using a diagrammatic expansion (Hertz et al., 1989) and the replica method (Opper, 1989, Kinzel and Opper, 1991). In Section 2, we present what we believe to be a much simpler method for calculating G , based only on simple matrix identities. We also show explicitly that 9 is self-averaging in the thermodynamic limit, which means that the fluctuations of 9 around its average G become vanishingly small as N -> 00 . This implies, for example , that the generalization error is also self-averaging. In Section 3 we extend the method to more general cases such as anisotropic teacher space priors and input distributions, and general quadratic penalty terms. Finite size effects are considered in Section 4, where we calculate the O(IIN) corrections to G, ?g(t -- (0) and p(a) . We discuss the resulting effects on generalization and learning dynamics and derive explicit conditions on the perceptron size N for results obtained in the thermodynamic limit to be valid. We conclude in Section 5 with a brief summary and discussion of our results. 2 THE BASIC METHOD Our method for calculating the average response function G is based on a recursion relation relating the values of the (unaveraged) response function 9 for p and p + 1 training examples. Assume that we are given a set of p training examples with corresponding matrix M N . By adding a new training example with input x, we obtain the matrix Mt = MN + ~ xx T . It is straightforward to show that the inverse of Mt can be expressed as 1 M-1 TM-1 (MN+) - 1 -_ MN-1 - IV N XX 1N 1 + N1 X TM-N x . (One way of proving this identi~ is to multiply both sides by Mt .and exploit ~he fact that MtM~',l = 1 + kxx M;~} . ) TaklOg the trace, we obtalO the follow 109 recursion relation for g: 1 ..!.x?-2x (6) 9 (p + 1) = 9 (p) - N 1 N 1 T ~ -1 . +NX N X Now denote Zi = ~xTMNix (i = 1,2). With x drawn randomly from the assumed input distribution P(x) ex: exp( - ~x2), the Zi can readily be shown to be random 210 Peter Sollich variables with means and (co-)variances (A A) 2 M-i-j ~Zi~Zj = N1 tr N ? Combining this with the fact that tr M:vk ~ N>.-k = O(N), we have that the (Zi) = 1 M-i N tr N' fluctuations LlZi of the into (6), we obtain Zi around their average values are 0(1/.JN); inserting this 9(p + 1) l I t M- 2 9(p) _ N r N N 1 + ~trM:vl + 0(N-3/2) 9(p) + ~ a9(p) 1 + 0(N-3/2). (7) N ()>. 1 + 9(p) Starting from 9(0) 1/>., we can apply this recursion p aN times to obtain 3 9(p) up to terms which add up to at most 0(pN- /2) = 0(1/.JN). This shows that 9 is self-averaging in the thermodynamic limit: whatever the training set, the value of 9 will always be the same up to fluctuations of 0(1/.JN). In fact, we shall show in Section 4 that the fluctuations of 9 are only 0(1/ N). This means that the 0(N- 3 / 2 ) fluctuations from each iteration of (7) are only weakly correlated, so that they add up like independent random variables to give a total fluctuation for 9(p) of O?p/ N 3 )1/2) 0(1/ N). = = = We have seen that, in the thermodynamic limit, 9 is identical to its average G because its fluctuations are vanishingly small. To calculate the value of G in the thermodynamic limit as a function of a and >., we insert the relation 9(p+1)-9(p) = -ka9(a)/aa + 0(1/ N 2 ) into eq. (7) (with 9 replaced by G) and neglect all finite N corrections. This yields the partial differential equation aG aG 1 aa - a>. 1 + G = 0, (8) which can readily be solved using the method of characteristic curves (see, e.g., John, 1978). Using the initial condition Glo:o = 1/>. gives a/(l + G) = l/G - >., which leads to the well-known result (see, e.g., Hertz et al., 1989) ;>. (1 - a - >. + J(l - a - >.)2 + 4>') . (9) In the complex >. plane, G has a pole at >. = 0 and a branch cut arising from G= the root; according to eq. (5), these singularities determine the average eigenvalue spectrum pea) of A, with the result (Krogh, 1992) pea) = (1- a)8(1 - a)6(a) 1 J(a+ + -2 1ra - a)(a - a_), (10) where 8(x) is the Heaviside step function, 8(x) = 1 for x > 0 and 0 otherwise. The root in eq. (10) only contributes when its argument is non-negative, i.e., for a (1 ? fo)2. between the 'spectral limits' a_ and a+, which have the values a? = 3 EXTENSIONS TO MORE GENERAL LEARNING SCENARIOS We now discuss some extensions of our method to more general learning scenarios. First, consider the case of an anisotropic teacher space prior, P(w v ) ex: Learning in Large Linear Perceptrons 211 exp(-!w~:E~lWV)' with symmetric positive definite :E v . This leaves the defini- tion of the response function unchanged; eq. (3), however, has to be replaced by 1/2{q 2G + A[q2 - A(~tr :Ev)]oGloA}. fg(t -- 00) = As a second extension, assume that the inputs are drawn from an anisotropic distribution, P(x) oc exp(-~xT:E-1x). It can then be shown that the asymptotic value of the average generalization error is still given by eq. (3) if the response function is redefined to be 9 ~ tr :EM~l. This modified response function can be calculated as follows: First we rewrite 9 as ~tr (A:E- 1 + A)-I, where A = ~ L:JJ(xJJ)Tx JJ is the correlation matrix of the transformed input examples xl' = :E- 1/ 2x JJ . Since the x JJ are distributed according to P(x JJ ) oc exp( _~(xJJ)2), the problem is thus reduced to finding the response function 9 = ~ tr (L + A)-1 for isotropically distributed inputs and L A:E- 1. The recursion relations between 9(p + 1) and 9(p) derived in the previous section remain valid, and result, in the thermodynamic limit, in a ?i!f~rential ~~uat.ion for the aver are response function ~ analo.gous. t? eq. (8). The 1mt1al cond1tlon 1S now Gla=o = N tr L -1, and one obtams an 1mphc1t equatiOn for = = G, 1 ( G = N tr L a + 1+G 1 )-1 , (11) where in the case of an anisotropic input distribution considered here, L = A:E- 1. If :E has a particularly simple form, then the dependence of G on a and A can be obtained analytically, but in general eq. (11) has to solved numerically. Finally, one can also investigate the effect of a general quadratic weight decay term, !W;:AWN' in the energy function E. The expression for the average generalization error becomes more cumbersome than eq. (3) in this case, but the result can still be expressed in terms of the average response function G (9) = (~tr (A + A)-l), which can be obtained as the solution of eq. (11) for L = A . = 4 FINITE N CORRECTIONS So far, we have focussed attention on the thermodynamic limit of perceptrons of infinite size N. The results are clearly only approximately valid for real, finite systems, and it is therefore interesting to investigate corrections for finite N. This we do in the present section by calculating the O(IIN) corrections to G and pea). For details of the calculations and results of computer simulations which support our theoretical analysis, we refer the reader to (Sollich, 1994). = First note that, for A 0, the exact result for the average response function is GI.~=o = (a - 1 - IIN)-l for a > 1 + liN (see, e.g., Eaton, 1983), which clearly admits a series expansion in powers of liN. We assume that a similar expansion also exists for nonzero A, and write (12) Go is the value of G in the thermodynamic limit as given by eq. (9). For finite N, the fluctuations ~9 = 9-G of9 around its average value G become relevant; for A = 0, the variance of these fluctuations is known to have a power series expansion in II N, and again we assume a similar expansion for finite A, ((~9)2) = ~2 IN + 0(11 N2), 212 Peter Sollich where the first term is 0(1/ N) and not 0(1) because, as discussed in Section 2, the fluctuations ofg for large N are no greater than 0(1/v'N). To calculate G I and A 2 , one starts again from the recursion relation (6), now expanding everything up to second order in the fluctuation quantities AZi and Ag. Averaging over the training inputs and collecting orders of l/N yields after some straightforward algebra the known eq. (8) for Go and two linear partial differential equations for G I and A 2 , the latter obtained by squaring both sides of eq . (6) . Solving these , one obtains G _ G6(1 - AGO) I (1 + AG6)2 (13) = In the limit A -- 0, G 1 l/(a - 1)2 consistent with the exact result for G quoted above; likewise, the result A 2 == 0 agrees with the exact series expansion of the variance of the fluctuations of 9 for A = 0, which begins with an 0(1/ N 2 ) term (see, e.g., Barber et ai., 1994). (a) (b) 0.5 1.5 - - A=O.OOl 0.4 -------. A =0.1 1.0 0.3 0.5 0.2 0.0 0.1 -0.5 0.0 0.0 0.5 1.0 a 1.5 2.0 0.5 0.0 1.0 a 1.5 2.0 Figure 1: Average generalization error: Result for N -- 00 , (g,O, and coefficient of O(l/N) correction, (g , l. (a) Noise free teacher, (72 O. (b) Noisy teacher, (72 0 .5. Curves are labeled by the value of the weight decay parameter A. = = From the l/N expansion (12) of G we obtain , using eq. (3), a corresponding expansion of the asymptotic value of the average generalization error, which we write as (g(t -- 00) (g ,O + (g , d N + 0(1/ N2). It follows that the thermodynamic limit result for the average generalization error , (g ,O, is a good approximation to the true result for finite N as long as N ~ Nc = /fg ,I/(g ,ol. In Figure 1, we plot (g,O and (g,l for several values of A and (j 2 . It can be seen that the relative size of the first order correction l(g,I/(g,ol and hence the critical system size Nc for validity of the thermodynamic limit result is largest when A is small. Exploiting this fact, Nc can be bounded by 1/(1 - a) for a < 1 and (3a + l)/[a(a - l)J for a > l. It follows, for example , that the critical system size Nc is smaller than 5 as long as a < 0.8 or a > l.72 , for all A and (j2. This bound on Nc can be tightened for non-zero A; for A > 2, for example , one has Nc < (2A - l)/(A + 1)2 < 1/3. We have thus shown explicitly that thermodynamic limit calculations of learning and generalization behaviour can be relevant for fairly small , ' real-world' systems of size N of the order of a few tens or hundreds. This is in contrast to the widespread suspicion = Learning in Large Linear Perceptrons 213 among non-physicists that the methods of statistical physics give valid results only for huge system sizes of the order of N :::::: 10 23 . (b) (a) po PI I (1 - 0-)0(1 - 0-) 4 I 4 a a a __- - - - _ Figure 2: Schematic plot of the average eigenvalue spectrum p( a) of the input correlation matrix A. (a) Result for N -+ 00, po(a). (b) O(l/N) correction, Pl(a). Arrows indicate 6-peaks and are labeled by the corresponding heights. We now consider the 0(1/ N) correction to the average eigenvalue spectrum of the input correlation matrix A. Setting p(a) po(a) + pda)/ N + 0(1/ N 2 ), po(a) is the N -+ 00 result given by eq. (10), and from eq . (13) one derives = Figure 2 shows sketches of Po (a) and PI (a). Note that fda PI (a) = 0 as expected since the normalization of p( a) is independent of N. Furthermore, there is no O(l/N) correction to the 6-peak in po(a) at a = 0, since this peak arises from the N - p zero eigenvalues of A for c.r = p/ N < 1 and therefore has a height of 1- c.r for any finite N. The 6-peaks in PI (a) at the spectral limits a+ and a_ are an artefact of the truncated l/N expansion: p(a) is determined by the singularities of G as a function of A, and the location of these singularities is only obtained correctly by resumming the full l/N expansion. The 6-peaks in pl(a) can be interpreted as 'precursors' of a broadening of the eigenvalue spectrum of A to values which, when the whole 1/ N series is resummed, will lie outside the N -+ 00 spectral range [a_ , a+]. The negative term in PI (a) represents the corresponding 'flattening' of the eigenvalue spectrum between a_ and a+ . We can thus conclude that the average eigenvalue spectrum of A for finite N will be broader than for N -+ 00, which means in particular that the learning dynamics will be slowed down since the smallest eigenvalue amin of A will be smaller than a_. From our result for PI (a) we can also deduce when the N -+ 00 result po(a) is valid for finite N; the condition turns out to be N ~ a/[(a+ - a)(a - a_)]. Consistent with our discussion of the broadening of the eigenvalue spectrum of A, N has to be larger for a near the spectral limits a_, a+ if po(a) is to be a good approximation to the finite N average eigenvalue spectrum of A. 214 5 Peter Sollich SUMMARY AND DISCUSSION We have presented a new method, based on simple matrix identities, for calculating the response function 9 and its average G which determine most of the properties of learning and generalization in linear perceptrons. In the thermodynamic limit, N --+ 00, we have recovered the known result for G and have shown explicitly that 9 is self-averaging. Extensions of our method to more general learning scenarios have also been discussed. Finally, we have obtained the 0(1/ N) corrections to G and the corresponding corrections to the average generalization error, and shown explicitly that the results obtained in the thermodynamic limit can be valid for fairly small, 'real-world' system sizes N . We have also calculated the 0(1/N) correction to the average eigenvalue spectrum of the input correlation matrix A and interpreted it in terms of a broadening of the spectrum for finite N, which will cause a slowing down of the learning dynamics. We remark that the O( 1/N) corrections that we have obtained can also be used in different contexts, for example for calculations of test error fluctuations and optimal test set size (Barber et al., 1994). Another application is in an analysis of the evidence procedure in Bayesian inference for finite N, where optimal values of 'hyperparameters' like the weight decay parameter A are determined on the basis of the training data (G Marion , in preparation). We hope, therefore, that our results will pave the way for a systematic investigation of finite size effects in learning and generalization. References D Barber, D Saad, and P Sollich (1994). Finite size effects and optimal test set size in linear perceptrons. Submitted to J. Phys. A. M LEaton (1983). Multivariate Statistics - A Vector Space Approach. Wiley, New York. J A Hertz, A Krogh, and G I Thorbergsson (1989) . Phase transitions in simple learning. J. Phys. A, 22:2133-2150. F John (1978). Partial Differential Equations. Springer, New York, 3rd ed. W Kinzel and M Opper (1991). Dynamics of learning. In E Domany, J L van Hemmen, and K Schulten, editors, Models of Neural Networks, pages 149-171. Springer, Berlin. A Krogh (1992). Learning with noise in a linear perceptron. J. Phys. A, 25:11191133. A Krogh and J A Hertz (1992). Generalization in a linear percept ron in the presence of noise. J. Phys. A, 25:1135-1147. M Opper (1989). Learning in neural networks: Solvable dynamics. Europhysics Letters, 8:389-392. P Sollich (1994). Finite-size effects in learning and generalization in linear perceptrons. J. Phys. A, 27:7771-7784. 

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553 SPREADING ACTIVATION OVER DISTRIBUTED MICROFEATURES James Hendler * Depart.ment, of Computer Science University of Maryland College Park, MD 20742 ABSTRACT One att?empt at explaining human inferencing is that of spreading activat,ion, particularly in the st.ructured connectionist paradigm. This has resulted in t.he building of systems with semantically nameable nodes which perform inferencing by examining t.he pat,t.erns of activation spread. In this paper we demonst.rate t.hat simple structured network infert'ncing can be p(>rformed by passing art.iva.t.ion over the weights learned by a distributed algarit,hm. Thus , an account, is provided which explains a wellbehaved rela t ionship bet.ween structured and distri butt'd conn('ct.ionist. a.pproachrs. INTRODUCTION A primar~? difference brtween t,he nPllral net.works of 20 years ago and t.he ("urrent genera Lion of connect,ionist models is t.he addit.ion of mechanisms whic h permit t.he s),st,em to create all internal represent,ation. These subsymbolic, semantica.lly unnameable, feat.urrs which a.re induced by connectionist. learning algorithms havr been discussed as bt,ing of import. bot,h in structured and distribut.ed ronnl"ctionist nrtworks (cf. Feldman and Balla.rd , 1982; Rumelhart and McClelland, 198(j). The fact that network learning algorit.hms can creatr these rm?cro!eal'ure,s is not, however. enough in itself t.o aC('Qunt for how rognition works. Most. of what, we call int.elligent thought. dt'rives from being able t,o rea:son about. t.he relatioll:::> I)t'tween object.s, to hypothesize about event.s a.nd things, etc. If we are to do cognit.ive modeling we must. complet.e the story by rxplainiJlg how networks can rea,-;oll in tht> wa.y that, humans (or other int,elligrnt beings) do. aile attempt at (-'xplaining such rea.<;oning is that of spreading activat.ion ill the structured cOllllect,ionist. and marker- passing (cf. Charniak, ]983; Hendler, 1987) ? The aut.hor is also affiliatf' ci wit.h thp Instit.ut.e for Acivanced C'omputpr Studies a.nd t.he Systems Research Center a.t the Universit.y of Ma.ryland. Funding for this work was provided in part by Offi ce of Naval Resf'arch Grallt N00014-88--K - 0,560 . 554 Hendler approaches. In t,hese syst,em::i semantically nanwable nodes permit an energy spread; and reasoning about tilt' world is accounted for by looking at, either stahh' configurations of tJw activation (the st.ruet.ured connectionist. approach) or at. t,he paths fOllnd by examining int.ersect,ions among t.he nodes (the markerpassing te('llnique) . In this paper we will demonst.rate t.hat. simple Rt,ruct.lJrednetwork- like infE'rencing ('an be performed by pa.ssing act.ivat.ion over t,he wt'ightR lea.rned by a distribut.ed algorithm. Thus, an account. is provided which explains a well-behaved relation~hip bet.ween st.ruct,ured and dist.ributed connt'ct.ionist a pproa.ches . THE SPREADING ACTIVATION MODEL In this paper we will demonstrate that local connect.ionist.- Iike net.works can be built by spreading activat.ion ovel' t.he microfeaturcs learned b.\' a dist.ribut.ed network. To show t.his, we st,art wit.h a simple example which c1f'Jl1onst.rat.es the activation spreading mechanism used. ThE' part.icular net.work We will use ill this example is a 6- 3-8 three-layer net.work t rained by t.he hack-propagation learning algorithm. The training set used is shown in table 1. The \w?ights bet.ween the out.put node::; and hidden units which are learned by the Iletwork (after h'arning t.o tlw !)O% level for a typical rlln) are shown in figure 1 . TABLE 1. Training Set. for Examph' 1. Inpll t Pattern Output. Pattern 000000 0000 J 1 001100 OOll}1 110000 10000000 01000000 00100000 00010000 00001000 00000100 00000010 00000001 lIOO]] ] I I 100 111111 Spreading Activation over Distributed Microfeatures Weights n1 n2 n3 n4 n5 n6 n7 nB h1 h2 h3 -4.98 -6.99 -6.11 -6.37 4.36 4.38 0.89 3.88 4.40 -4.99 3.49 -4.68 3.73 -5.97 1.07 -6.95 -2.82 -2.23 0.30 2.53 -5.09 -3.67 3.32 1.88 Figure 1. Weights L~arnecl by Back Propagation To underst.and how t,he act,ivat.ioll spreads, let. us examine what. occurs when activation is started at node nl wit.h a weight of 1. This activation strength is divided by the outbranching of the nodf' and then mult.iplied by t.he weight of each link to t.he hidden units. Thus a.ctiva.tion flows from nl t.o hi with a strength of 1/ 3 1" Weighl(tl1 ,hl}. A similar computat.ion is made t.o each of t.he other hidden units. This act.ivat.ion now spreads to each of the ot.her Ollt.pUt. nodes in turn. Thus, n2 would gain act,ivat.ion of Acti','ation(hl) I Wet"ght(n2,hl)/ 8 -1Activat-ion(h2) 1" lYe'ight(n2)d)/ 8 + Ad1'vation(h3) 1" Weight(n2,h3)/ 8 or .80 from rli . Table 2 shows a graph of t.he act.ivat,ioll spread between the output units. The table, which is symmetric, can t.hus be read as showing t.he out,put at each of the other units when an activation st.rength of 1 is placed at t,he named node . Looking at. the table we see t,hat. t.il(' hight:'st, activat.ion OCClJrs among nodes which share t.he most feat,ures of tJlf' input (i .e. ~anlf' value and posit.ion) while t.he lowest is se('n among tho::;t:' pat.tt'fIls shariug t.he fewest feat.ures. 555 556 Hendler However, as well as having this property, t.able 2 can be seen as providing a matrix which specifies t.he weights bet.ween t.he out.put nodes if viewed as a st.ruct,ured net.work. That. is. ,,1 is conned?ed to rtf! by a strength of + .80, to nB by a st.rength of + 1.0:3, etc . Thus. by I\~ing this t.echniquE' distribut.ed r{'presentations can be turned into connectivity weight.s for st.ructured lIet.works. When non-? ort.hogonal wpights are lIsed. t he same act.ivat ion--spreading a.lgorithm produces a struct.ured network whi('h can be used for more complex inferenciug than can the dist.ributed net work alone . We demonst.rat.e t.his b)' a simple . and aga.in contrin?d. example. This example is mot.ivat.ed by Gary Cot.trell's st.ructured JI10dd for word sense disambiguation (CoUrt'll, Hl85). Cottrell, using weights clerived by hand, demonstrat.ed that. a struct.ured connect.ionis!. net.work could di:';t inguish both word- sense a.nd case- slot assignm('nts for amhiguous lexical items. Pre~enkd wit.h the sent{'nce "John threw the fight." f?he syst.em would ndivate a node for one meaning of "throw/' presented wit.h "John t.hrew t.he ball" it would come up wit.h another. The nodes of Cottrell's network included worels (John. Threw , etc .). word RenseR (Johnl, Propp\. etc .) :lnd cClI'P- sloLs (TAGT (lIgt'llt of tl1(-' throw) . PAGT (ag('nt of t,he Propel). etc.). TABLE 2. Adivat.jon Spn>ad in 6--:3?8 Network. ,,1 nl n2 n.'J n4 '115 116 n7 uB ? .80 1.03 .17 .38 -1.57 - .38 - 2.3 II::! .80 oj ? 1.03 1.02 -.14 .97 - .63 -2 .03 - .03 -l.f!7 1.02 2.60 -1.57 .31 -.79 ? 114 ?//5 .17 2.6 .97 ? -2.42 -.38 -.09 .52 .38 -1.57 - .0:3 - 2.42 ? .64 -.;38 - .77 uB -1.57 .:31 - 2.03 -.:38 .04 ? - .f) 2.1 ?1 Tl7 - .:38 -.79 - .0:3 -.Of! -.;38 -.6 ? .09 u8 - 2.3 .14 --1.97 .52 -.77 2 .14 .09 ? To duplicate Gary's network via training, we presented a 3-layer backprop net... work wit.h a training set in which distribut.ed pat.t.erns, H'ry loosely corresponding to a "dictionary" of word (-,Ileodings l were associa.ted wit,h a vector representing each of the individual noell's which would he represented in Cottrt-ll's system, but. wit.h no struct.ure . Thus, each elemPIlt. in t.he training set. is }- Whieh in any realisti r sy stem would sornp. day be rpplaeed by aduaJ signal pro.;pssing outp"t.S Of Othpf rpprespnt.alinn s of ai'liial worn pfonuneiat.ioll forms. Spreading Activation over Distributed Microfeatures a 16 bit vretor (represent.ing a four word srnt.en('e. ea('h word as a. 4 bit. pattrrn), associated wit.h allot.he)? 16 bit. ve('tor repr<'spnt.ing t.he nodes Bobl Johnl propE'l t.hrfW IJght I ball J jlJgt pob) t agt. to bj bob John t.hrfw U1E' fight ball For t.his example, the system was t.rnineci on th(? E'1I('odings of t.he four srnt.encrs John t.hrew the ball John threw the fight. Bob t.1Hfw t be hall Bob t.hrfw Lllf 6ght wit.h the output set. high foJ' tho:-;e objects a.ppropriatf?\y a.<.;so('iaj,rd . (IS shown in Tabh? :3. TABLE ;~. the second vedor whi('h \v('/,e Tr<lining S(?t fa)' Example Z. Output. Patt.el'n Input. Pattern 011 0 01 J 0 100/ J 00 J /II 0001 0101 OOOJ 0101 OOOJ 010J 000 1 0101 OOlO J010 00/0 10 10 JOO 11000 1J J 01 I 10 lOlOOlJI00101101 0/0/100011011110 01 JOOJ 1100011101 Upon complet.ion of t.he kaming , t.he aet.ivat.ioJl sprrading algorithm was used to derive a table of COlll1f'ctivity weights betw('ell the out.put. unit.s as shown in table 4. These weight.s W(')'e then t.ransferred into a local COllnect ionist simulat.or and a very simple act.ivation spn'adiJlg modrl was llsed to examine t.he r esult?s . \Vhpn Wt' run t.he simulator. u~ing tht' aeti\'atioIl spn?ading oY('r leaJ'lwd wpights. exactly t.he f('sult.s prodllced by Cott.reJl's n{'twork lIr{' seen . Thus : Act.ivation from tht, nodt's corresponding to john. tbroll', tllf-. alld fif/hl cam,e a posit.ive activation at. the node for ('Throw" and a negative a('t.ivat.ion at t.he node for "Propr!." while A('tivat.ion from joh" throlt' the ball sprt'ad positiniy to " Prope'" ,lIlel not. t.o 'I t IIrow . ,. Furt.her, ot?her effrct.s which cHP also predict.ed by C'ou.rdl's model lire seeJl: Act.iYat.ioJl at. TAGT and TOB.! spreads posit.i\'t' activation to TIr.,.o/l' and not t.o Propel. and Activation at PAG?' and POB) causes a spread t.o Propel but. not to Thro'W. 557 558 Hendler TABLE 4. Connectivity Weights for Example 2. *** -?0.12 - 0.12 *** 0.01 -OJ)} -0 .01 0.01 -0.0] 0.01 0,01 -0,01 O.OJ -0.01 0 .01 ?-0.01 -0.01 0.01 -0.01 0.01 0 .12 -0.12 - 0.12 0.12 -0.0:3 0.03 ??0.03 0 .0:3 ??0.0] 0.01 0.00 O.OJ 0.01 -0.01 - 0.01 0.01 *u -0.04 ?-0.04 *** -0.04 0.04 0,04 -0.04 0.04 -0.04 0.05 -0.05 -0,05 0.05 -0.04 0.04 0 .01 -0.01 ?-0,01 0 .01 -0.02 0 .02 ??0.02 0 ,02 -?0.04 0.04 0.04 -0.04 -0.01 0.01 0.01 - 0.01 -0.04 0 .04 0.04 -0 .04 u* -0.04 -0,04 *** -0,05 O.O!) -0,05 0.05 0.05 -0.05 0.04 -0.04 - 0.01 0.01 0.01 -?0.01 0.02 -?0.02 0.02 -0 .02 0.04 - 0.04 ?-0.04 0.04 0.01 -0.01 0 .04 - 0.04 -0.0,5 0.05 *** O.OS -0.05 -0,05 0.01 -0.00 -?0.02 ?-0.02 -?0.04 0.0,5 0.01 - 0.01 -0.01 -0.01 0.01 0.01 0.05 -0.05 - 0.04 -0.05 0.05 0.04 -0,05 0.05 0,04 0,05 -0.0.5 -? 0.04 0.05 -0 .05 -0.05 u* -?0.05 -O.O!) - 0.05 *** 0.05 -0.05 0.05 u* 0 .01 -0.01 -0 .01 -0.01 0 .01 O.OJ -0.02 0.02 0.02 -0.02 0 .0;3 0.02 - 0.04 O.O<J D.O?' 0,05 O.O!) O.OG 0.12 - 0 .12 -?0.12 0 .12 0.01 - 0.01 ?_?0.01 0.01 -0,01 0 ,01 0.01 - 0.01 0.01 ??0.00 0.01 -O.OJ -0.01 O.()] -0.01 0.01 *** ?-O.l:.? ??0.12 *** ?-0.0:3 0.0:3 0.0:3 0.0:) ? 0.01 O.OJ O.OJ ?0.00 ???0.0:3 0.0:3 - 0 .02 0.02 0.02 - 0 .02 -0.02 -0'()2 0 .02 0.02 ?-0.0:3 O.O:~ *** 0.20 0.02 0 .02 -0.0:3 -0.01 0 .00 O.O~ 0.01 ?-0.01 - 0.02 - 0.04 0.04 0.02 0.04 - 0 .04 0.02 0.04 - 0.04 -0.02 - 0.04 0.04 -0.02 - 0 .04 0.05 -0,02 ??0.04 0.05 0.0:3 0 .04 - 0.05 0.02 0.04 -0.05 -0.0:3 ?-0.01 0.0] 0.0:3 0 .01 ??0.00 0 .20 0.02 -0.02 * ** 0.02 - 0.02 0.02 *** . ~.O .? 0.02 ?0.04 *** We believe that results like this one ma.y argue t.hat. :-;tl'uetllrt'd IId?works are int.pgrally linked to dist.ributed networks ill t,hat. distribllt.<?d Ilf'twork le:ullillg t.echuiqut's may provide a fundamental ba~i:-; for ('xplaining the ('ogllit.in' df'VelOPment. of structured networks . III addition, Wt ' see that :,-imple illf{'J'f'llt ial rt'a~on? ing ('an be produ('ed using purdy cOIlJledioni~t. mockl:,. CONCLUDING REMARKS Wt' have at.t.empt.ed t.o ;;;hO\.... that a model u;;;ing an activation spn'ading va.rinnt ean be used t.o take learned connect.ionist models aJld lwrform SOllle limit ed forms of inferencing upon t.helll. Furt.her, WP have argllt"d t.hat t.his tf'chniqlll' Illay pro"idr a comput.ational modd in which strllctured network;;; C;}l) Iw leal'llPd and t.hat st,ruct.ured net.work::; provide the infeI't'ncing (?apnbilities missing in purely dist.rihut.ed models. However. befort' w(' can t.ruly furtht'r thi~ claim. ~ignit1('ant work remain;;; to be done . We must ext.end alld ('xplort' such models. particularly t'xamining whether t.hese t?ypes of t.echniqups ('an be extt'llded t.o handle tlw complt'xitr t?hat can be found in real?- world prol)JpIll~ and sl'rioll~ ('ogllitin models . In part ieula.r we are beginlling an examinat.ion of two cru('ial isslIP::;: First., will t.he technique described above work for realist.ic problem?;? In particular, can t.hl' infen?ncing be designed t.o impact on t.1l(' ]'Pwgnition by t 11(> di.stribut.ed net.work? If so, one ('ould Sf'f', for l'xample, a speech recognit.ion program coupled to a system like Cottrell's natural languagt' sy:.;\.eJll, providing a handle for a t.ext. underst.anding system. Similarly such a t.e('hnique might. allow the int.('gration of top - down and bott.OIll'-IIP pro('('ssing for vision a.nd ot.ht'r such signal Pl'ocf'ssing tasks. Spreading Activation over Distributed Microfeatures Secondly. we wish to see jf more complex spreading activation models could be hooked to this t.ype of model. Could networks such as t.hose proposed by Sha.st.ri (1985), Diedt'rich (HI8!?). and Polla.ck and 'Valt.z (1982) which provide complex inferencing bllt requirt' more ,structure than simply wt'ight.s bet.ween unit.s, be ab.stract.t'cI out. of the learned weight.s? Two partieular areas current.ly being pur~llf'd by t.JH' author. for t'xampk foclls on act.i\'e inhibit.ioll modeb for dt't.ermining whetlwr port.jon~ of the nt'twork C(ln lw ~lIpPJ'f'ssed t.o provide mort' complex inferencing and t.he learning of structures given temporally ordered information. References Charniak, E. Passing markt'rs: A t.heory of contt'xtual influence comprehension Cognitive S6ence, 7(3), 198:~. 171-190. lJl language Cottrell, G.W. A Connection-ist Approach to Word - Sense DiMlmb1'gllation, Doet.oral Disst'rta.t.ion, Comput.t'r Scienc(' Dt'pt.. l fllivl'rsit.y of Hochesler, 1985. Diederich, J. Parallelverarbeil/l'flg in 1IetzlL'erk-ba.~ie1'fen Systemen PhD Disst'rtation, Dt'pt. of Linguistics, llni\'ersity of Rielt~f('ld. J985. Feldman, J.A. and Ballard , D.H. (J 982). Connect.ionist models a.nd t.heir propt:'rt.ies. Cognit1've Science. 6. 205-?254- JlItegrating Maf'l.?er -pali8illg aod Problem Solving: A .~p,.e(/dirtg artivatiou approach to 1'ml)ro'l'cd choice ill ploflJlillg Lawrence Erlbaulll As~ociate~, N .J., Novt'mbt'J', Hlg7. Hendler, J.A. Pollack, J.B. and 'Yalt.7., D.L Nat?u!'al Language Processing lIsiug spr?'ading act.ivat.ion and lateral inhibit.ion Proceedings of the FO'llrth intema.i1'o'l1al Conference of Ihe Cognitive S'o'eure S'ociety, 1982, .50-58. Rurnelhart . D.E. and McClelland. J.L. ((,ds.) Parallel D';strib'llfed Computing Cambridge . ,Ma.: MJT Prel-is. Shastri, L. El'ideul1'al RWMuing ?in SefJ/(llItic /\.retll'{)rk.~: A formal theor:1J ((tid //8 parallel implementatioTl Doct.oral DiS8t'l't at.jon, Computt'r Scit'nce Depart ment, l Jnivefsit.y of Roc-hester, Sept .. J 985. 559 

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Interior Point Implementations of Alternating Minimization Training Michael Lemmon Dept . of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 lemmon@maddog.ee.nd.edu Peter T. Szymanski Dept. of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 pszymans@maddog.ee.nd.edu Abstract This paper presents an alternating minimization (AM) algorithm used in the training of radial basis function and linear regressor networks. The algorithm is a modification of a small-step interior point method used in solving primal linear programs. The algorithm has a convergence rate of O( fo,L) iterations where n is a measure of the network size and L is a measure of the resulting solution's accuracy. Two results are presented that specify how aggressively the two steps of the AM may be pursued to ensure convergence of each step of the alternating minimization. 1 Introduction In recent years, considerable research has investigated the use of alternating minimization (AM) techniques in the supervised training of radial basis function networks. AM techniques were first introduced in soft-competitive learning algorithms[l]. This training procedure was later shown to be closely related to Expectation-Maximization algorithms used by the statistical estimation community[2]. Alternating minimizations search for optimal network weights by breaking the search into two distinct minimization problems. A given network performance functional is extremalized first with respect to one set of network weights and then with respect to the remaining weights. These learning procedures have found applications in the training of local expert systems [3], and in Boltzmann machine training [4]. More recently, convergence rates have been derived by viewing the AM Michael Lemmon. Peter T. Szymanski 570 method as a variable metric algorithm [5]. This paper examines AM as a perturbed linear programming (LP) problem. Recent advances in the application of barrier function methods to LP problems have resulted in the development of "path following" or "interior point" (IP) algorithms [6]. These algorithms are characterized by a fast convergence rate that scales in a sublinear manner with problem size. This paper shows how a small-step IP algorithm for solving a primal LP problem can be modified into an AM training procedure .. The principal results of the paper are bounds on how aggressively the legs of the alternating minimization can be pursued so that the AM algorithm maintains the sublinear convergence rate characteristic of its LP counterpart and so that both legs converge to an optimal solution. 2 Problem Statement Consider a function approximation problem where a stochastic approximator learns a mapping f : IRN -+ IR. The approximator computes a predicted output, Y E IR, given the input z E IRN. The prediction is computed using a finite set of M regressors. The mth regressor is characterized by the pair (Om,w m) E IRN X IRN (m 1, ... , M). The output of the mth regressor, Ym E IR, in response to an input, z E IRN is given by the linear function = Ym =O~z. (1) = The mth regressor (m 1, ... , M) has an associated radial basis function (RBF) with parameter vector Wm E IRN. The mth RBF weights the contribution of the mth output in computing Y and is defined as a normal probability density function Q(mlz) = km exp(-u- 2 1I wm - zW) (2) where k m is a normalizing constant. The set of all weights or gating probabilities is denoted by Q. The parameterization of the mth regressor is em (O~ ,w~)T E IR2N (m = 1, ... , M) and the parameterization of the set of M linear regressors is = e -- (eT I " " , eT)T M . (3) The preceding stochastic approximator can be viewed as a neural network. The network consists of M + 1 neurons. M of the neurons are agent neurons while the other neuron is referred to as a gating neuron. The mth a~ent neuron is parameterized by Om, the first element of the pair em = (O?'n, w?'n) (m = 1, ... , M). The agent neurons receive as input, the vector z E IRN. The output of the mth agent neuron in response to an input z is Ym O?'nz (m 1, ... , M). The gating neuron is parameterized by the conditional gating probabilities, Q. The gating probabilities are defined by the set of vectors, Ii; = {WI, ... ,WM }. The gating neuron receives the agent neurons' outputs and the vector z as its input. The gating neuron computes the network's output, Y, as a hard (4) or soft (5) choice = Y = Ym; m = arg A Y= = max m=I .... ,M Q(mlz) L:~-I Q(mlz)Ym M L:m=1 Q(mlz) (4) (5) Interior Point Implementations of Alternating Minimization Training 571 The network will be said to be "optimal" with respect to a training set T = {( Zi, Yi) : Yi = f(zd, i = 1, ... , R} if a mean square error criterion is minimized. Define the square output error of the mth agent neuron to be em(zi) (Yi - O~Zi)2 and the square weighting or classifier error of the mth RBF to be Cm(Zi) Ilw m - Zi 112. Let the combined square approximation error of the m,th neuron be dm (Zi) = Kee m (Zi) + KcCm(Zi) and let the average square approximation error of the network be = M d(Q, e, T) = L = R LP(zi)Q(mlzddm(zi)' (6) m=l i=l Minimizing (6) corresponds to minimizing both the output error in the M linear regressors and the classifier error in assigning inputs to the M regressors.: Since T is a discrete set and the Q are gating probabilities, the minimization of d(Q, e, T) is constrained so that Q is a valid conditional probability mass function over the training set, T. Network training can therefore be viewed as a constrained optimization problem. In particular, this optimization problem can be expressed in a form very similar to conventional LP problems. The following notational conventions are adopted to highlight this connection. Let x E lR,MR be the gating neuron's weight vector where (7) = Let em (O~,w~)T E lR,2N denote the parameter vectors associated with the regressor and define the cost vector conditioned on e = (er, ... , et)T as c(e) = (p(zt}d1(zt), ... ,p(zR)d1(ZR),p(Z2)d 2(Z2), ... ,p(zi)dm(Zi), ...)T mth (8) With this notation, the network training problem can be stated as follows, mmlmlze cT(e)x with respect to x, e subject to Ax = b, x 2:: 0 where b = (1, ... ,1? E lR,R, A Xi 2:: 0 for i = 1, ... , MR. = [IRxR? .. IRxR] (9) E IR RXMR , and x 2:: 0 implies One approach for solving this problem is to break up the optimization into two steps. The first step involves minimizing the above cost functional with respect to x assuming a fixed e. This is the Q-update of the algorithm. The second leg of the algorithm minimizes the functional with respect to e assuming fixed x. This leg is called the e-update. Because the proposed optimization alternates between two different subsets of weights, this training procedure is often referred to as alternating minimization. Note that the Q-update is an LP problem while the e-update is a quadratic programming problem. Consequently, the AM training procedure can be viewed as a perturbed LP problem. 3 Proposed Training Algorithm The preceding section noted that network training can be viewed as a perturbed LP problem. This observation is significant for there exist very efficient LP solvers 572 Michael Lemmon, Peter T. Szymanski based on barrier function methods used in non-linear optimization. Recent advances in path following or interior point (IP) methods have developed LP solvers which exhibit convergence rates which scale in a sublinear way with problem size [6]. This section introduces a modification of a small-step primal IP algorithm that can be used for neural network training. The proposed modification is later shown to preserve the computational efficiency enjoyed by its LP counterpart. To see how such a modification might arise, we first need to examine path following LP solvers. Consider the following LP problem. cTx mmlmlze with respect to x E IRn subject to Ax = b, x 2: 0 (10) This problem can be solved by solving a sequence of augmented optimization problems arising from the primal parameterization of the LP problem. mmlmlze with respect to subject to O'O:)cTx(k) - 2:i logx~k) x(k) E IRn Ax(k) = b, x(k) 2: 0 (11) where a(k) 2: 0 (k = 1,? .. ,1<) is a finite length, monotone increasing sequence of real numbers. x*(a(k? denotes the solution for the kth optimization problem in the sequence and is referred to as a central point. The locus of all points, x* (O'( k? where a(k) 2: 0 is called the central path. The augmented problem takes the original LP cost function and adds a logarithmic barrier which keeps the central point away from the boundaries of the feasible set. As a increases, the effect of the barrier is decreased, thereby allowing the kth central point to approach the LP problem's solution in a controlled manner. Path following (IP) methods solve the LP problem by approximately solving the sequence of augmented problems shown in (11). The parameter sequence, 0'(0),0'(1), ... , a(K), is chosen to be a monotone increasing sequence so that the central points, x*(a(k?, of each augmented optimization approach the LP solution in a monotone manner. It has been shown that for specific choices of the a sequence, that the sequence of approximate central points will converge to an f-neighborhood of the LP solution after a finite number of iterations. For primal IP algorithms, the required condition is that successive approximations of the central points lie within the region of quadratic convergence for a scaling steepest descent (SSD) algorithm [6]. In particular, it has been shown that if the kth approximating solution is sufficiently close to the kth central point and if O'(k+ 1 ) O'(k)(l + v/..{ii) where v ~ 0.1 controls the distance between successive central points, then the "closeness" to the (k + lyt central point is guaranteed and the resulting algorithm will converge in O( foL) iterations where L = n + p+ 1 specifies the size of the LP problem and p is the total number of bits used to represent the data in A , b, and c. If the algorithm takes small steps, then it is guaranteed to converge efficiently. = The preceding discussion reveals that a key component to a path following method's computational efficiency lies in controlling the iteration so that successive central points lie within the SSD algorithm's region of quadratic convergence. If we are to successfully extend such methods to (9), then this "closeness" of successive solutions must be preserved by the 6-update of the algorithm. Due to the quadratic nature Interior Point Implementations of Alternating Minimization Training 573 of the e-update, this minimization can be done exactly using a single NewtonRaphson iteration. Let e* denote e-update's minimizer. If we update e to e*, it is quite possible that the cost vector, c(e), will be rotated in such a way that the current solution, x(k), no longer lies in the region of quadratic convergence. Therefore, if we are to preserve the IP method's computational efficiency it will be necessary to be less "aggressive" in the e-update. In particular, this paper proposes the following convex combination as the e-update e~+l) = (1 - "Y(k?e~) + "Y(k)e~+l),* (12) where e~) is the mth parameter vector at time k and 0 < "Y(k) of the update. This will ensure convergence of the Q-update. < 1 controls the size Convergence of the AM algorithm also requires convergence of the e-update. For the e-update to converge, "Y(k) in (12) must go to unity as k increases. Convergence of "Y(k) to unity requires that the sequence Ile~+l),* -e~)11 be monotone decreasing. As the e-update minimizer, e(k+l),*, depends upon the current weights, Q(k)(mlz), large changes to Q can prevent the sequence from being monotone decreasing. Thus, it is necessary to also be less "aggressive" in the Q-update. An appropriate bound on v is the proposed solution to guarantee convergence of the e-update. Algorithm 1 (Proposed Training Algorithm) Initialize k = O. Choose X~k). e~). and O'(k) for i= 1,???,(MR). and for m= 1,???,M. repeat O'(k+l) = O'(k)(1 + v/Vn). where v ~ 0.1. Q-update: = X(k) for i = 0, ... , P - 1 Xo Xi+l X(k+l) = ScalingSteepestD.escent(x~k+l), O'(k+l), e(k?) = xp e-update: For m = 1, ... , M e~+l) = (1 - r(k?)e~) + r(k)e~+l),* k=k+l until(.6. < c) 4 Theoretical Results This section provides bounds on the parameter, ,(k), controlling the AM algorithm's e-update so that successive x(k) vector solutions lie within the SSD algorithm's region of quadratic convergence and on v controlling the Q-update so that successive central points are not too far apart, thus allowing convergence of the e-update. Theorem 1 Let e~) and e~ ),* be the current and minimizing parameter vecc(e(k? and c(k),* c(e(k),*). Let 6(x, a, e) tors at time k. Let c(k) = = = 574 Michael Lemmon. Peter T. Szymanski IIPAXX (O'c(8) - x-I) II be the step size of the SSD update where PA = 1AT (AAT)-l A and X = diag(xl, ... , x n ) . Assume that 6(x CHl ), O'CHl), 8 Ck ) = 61 < 0.5 and let e~+1) = (1 - ')'Ck)e~) + ')'Ck)e~+l), ?. If ')'Ck) is chosen as Ck) < 62 - 61 - n(I + 1I/y'n)llcCHl),. - C<k)11 where 61 < 62 = 0.5 , then 6(x(Hl), O'(Hl), 8 CH1 ) ~ 62 = 0.5. (13) ')' Proof: The proof must show that the choice of ')'(k) maintains the nearness ofx(k+l) to the central path after the 8-update. Let h(x,0' , 8) = PAxX(O'c(8)-x-l), hI = h(x CH1 ), O'CHl), 8 Ck ) and h2 = h(x(Hl) , O'(Hl), 8(Hl). Using the triangle inequality produces IIh211 ~ IIh2 - hIli + Ilhlll? Ilhlll = 61 by assumption, so Ilh 211::; 1100(Hl)PAXX(c(Hl) - c(k)11 + 61. Using the convexity of the cost vectors produces (c(Hl) - c(k) ::; (1 - ')'(k)c(k) ,),(k )c Ck +1) ,. - c(k) resulting in IIh211 ~ II00CHl),),Ck)PAXX(cCHl),. - c(k)11 Using the fact that IIPAXXII ~ ~ IIh211 ~ ~ + + 61 . = n/O'(k) (~ is the duality gap), ,),Ck)O'(Hl)IIPAXXllllc(Hl),. - c(k)11 + 61. ')'(k)n(I + 1I/y'n)llc(Hl),. - cCk)11 + 61. Plugging in the value of ')'(k) from (13) and simplifying produces the desired result IIh211 ~ 62 ~ 0.5, guaranteeing that x(Hl) remains close to x(Hl),. after the 8update. 0 Theorem 1 shows that the non-linear optimization can be embedded within the steps of the path following algorithm without it taking solutions too far from successive central points. The following two results, found in [7], provide a bound on II to guarantee convergence of the 8-update. The bound on II forces successive central points to be close and ensures convergence of the 8-update. Proposition 1 Let B = LzpzQ(mlz)zzT , E = Lzpz 6Q(mlz)zzT , w = Lz pzQ(mlz)y(z)z, and 6w = Lz p z 6Q(mlz)y(z)z, where Q(mlz) = Q(k)(mlz) and 6Q(mlz) = Q(Hl)(mlz) - Q(Ck)mlz). Assume that ly(z)1 ~ Y, llzll ::; (, and that B is of full rank for all valid Q 'So Finally, let J1.max = sUPQ IIB- II. Then, 118~+1),. - 8~)'?11 ~ 2(112 where I( = J1.maxY( (1 + Cr!'ru ) and r + 211)K (14) = IIB-lIIIIEII < 1. Theorem 2 Assume that Ilzll ~ (, ly(z)1 ~ Y, 118m ll ~ 8 max , and that B = Lz pzQ(mlz)zzT is of full rank for all valid Q's and that liB-III IIEII = r < 1. If II < min {O.I, -1 + ..jI7"I-+-r-;-/(=2"'7>(2:-J1.-m-ax""") , -1 + JI + ')'min f e/(2K)} (15) Interior Point Implementations of Alternating Minimization Training 575 = where 1\ = ILmax Y?l + (2 ILmax/(l- r?, 1min (6 2 - 6d/(n(1 + 0.1/ y'n)llc(l),? c(O)11) and f0 is the largest lIer!+1),? - er!)11 such that 1(.1:) = 1, then the e-update will converge with Iler!+1),? - er!)II-+ 0 and 1(.1:) -+ 1 as k increases. The preceding results guarantee the convergence of the component minimizations separately. Convergence of the total algorithm relies on the simultaneous convergence of both steps. This is currently being addressed using contraction mapping concepts and stability results from nonlinear stability analysis [8]. The convergence rate of the algorithm is established using the LP problem's duality gap. The duality gap is the difference between the current solutions for the primal and dual formulations of the LP problem. Path following algorithms allow the duality gap to be expressed as follows A( L..). (k? _ n a - + 0.5y'n a(k) (16) . and thus provide a convenient stopping criterion for the algorithm . Note that a(O) /{3 k where {3 ~ (1+11/ y'n). This implies that ~(k) 13k ~(O) ~ {3k2L. If k is chosen so that {3k2L :::; 2- L , then ~(k) :::; 2- L which implies that k ~ 2L/ log(l/{3). Inserting our choice of {3 one finds that k ~ (2y'nL/II)+2L. The preceding argument establishes that the .proposed convergence rate of O( y'nL) iterations. In other words, the procedure's training time scales in a sublinear manner with network size. = a(k) = 5 Simulation Example Simulations were run on a time series prediction task to test the proposed algorithm. The training set is T = {(Zi,Yi) : Yi = y(iT),Zi = (Yi-l,Yi-2, ... ,Yi-Nf E lR,N} for i = 0,1, ... ,100, N = 4, and T = 0.04 where the time series is defined as y(t) = sin(1I"t) - sin(211"t) + sin(311"t) - sin(1I"t/2) (17) The results describe the convergence of the algorithm. These experiments consisted of 100 randomly chosen samples with N 4 and a number of agent neurons ranging from M 4 to 20. This corresponds to an LP problem dimension of 404 to 2020. The stopping criteria for the tests was to run until the solution n was within f = 10- 3 of a local minimum. The number of iterations and floating point operations (FLOPS) for the AM algorithm to converge are shown in Figures l(a) and l(b) with AM results denoted by "0" and the theoretical rates by a solid line . The algorithm exhibits approximately O( y'nL) iterations to converge as predicted. The computational cost, however is O(n 2 L) FLOPS which is better than the predicted O( n 3 .5 L). The difference is due to the use of sparse matrix techniques which reduce the number of computations. The resulting AM algorithm then has the complexity of a matrix multiplication instead of a matrix inversion. The use of the algorithm resulted in networks having mean square errors on the order of 10- 3 . = 6 = = Discussion This paper has presented an AM algorithm which can be proven to converge in O( foL) iterations. The work has established a means by which IP methods can be 576 Michael Lemmon, Peter T. Szymanski 104r---------~--________~ G) ~ 1015 ,....-_ _ _ _ _ _ _ _- - - - - - - - - - - , o ~ > ~G) 8 o c::: 0- C ~ 01 1010 c::: :; ... Z -... E .0 ::J E .2 LL '0 o G) Zl~~--------~----------~ 102 103 LP Problem Size (n) (a) 104 ~105 ~--------~---------~ 102 103 10? LP Problem Size (n) (b) Figure 1: Convergence rates as a function of n applied to NN training in a way which preserves the computational efficiency of IP solvers. The AM algorithm can be used to solve off-line problems such as codebook generation and parameter identification in colony control applications. The method is currently being used to solve hybrid control problems of the type in [9]. Areas of future research concern the study of large-step IP methods and extensions of AM training to other EM algorithms. References [1] S. Nowlan, "Maximum likelihood competitive learning," in Advances in Neural Information Processing Systems 2, pp. 574-582, San Mateo, California: Morgan Kaufmann Publishers, Inc., 1990. [2] M. Jordan and R. Jacobs, "Hierarchical mixtures of experts and the EM algorithm," Tech. Rep. 9301, MIT Computational Cognitive Science, Apr. 1993. [3] R. Jacobs, M. Jordan, S. Nowlan, and G. Hinton, "Adaptive mixtures oflocal experts," Neural Computation, vol. 3, pp. 79-87, 1991. [4] W. Byrne, "Alternating minimization and Boltzmann machine learning," IEEE Transactions on Neural Networks , vol. 3, pp. 612-620, July 1992. [5] M. Jordan and 1. Xu, "Convergence results for the EM approach to mixtures of experts architectures," Tech. Rep. 9303, MIT Computational Cognitive Science, Sept. 1993. [6] C. Gonzaga, "Path-following methods for linear programming," SIAM Review, vol. 34, pp. 167-224, June 1992. [7] P. Szymanski and M. Lemmon, "A modified interior point method for supervisory controller design," in Proceedings of the 99rd IEEE Conference on Decision and Control, pp. 1381-1386, Dec. 1994. [8] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, New Jersey: PrenticeHall, Inc., 1993. [9] M. Lemmon, J. Stiver, and P. Antsaklis, "Event identification and intelligent hybrid control," in Hybrid Systems (R. L. Grossman, A. Nerode, A. P. Ravn, and H. Rischel, eds.), vol. 736 of Lecture Notes in Computer Science, pp. 265-296, Springer-Verlag, 1993. 

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Reinforcement Learning with Soft State Aggregation Satinder P. Singh singh@psyche.mit.edu Tommi Jaakkola tommi@psyche.mit.edu Michael I. Jordan jordan@psyche.mit.edu Dept. of Brain & Cognitive Sciences (E-lO) M.I.T. Cambridge, MA 02139 Abstract It is widely accepted that the use of more compact representations than lookup tables is crucial to scaling reinforcement learning (RL) algorithms to real-world problems. Unfortunately almost all of the theory of reinforcement learning assumes lookup table representations. In this paper we address the pressing issue of combining function approximation and RL, and present 1) a function approximator based on a simple extension to state aggregation (a commonly used form of compact representation), namely soft state aggregation, 2) a theory of convergence for RL with arbitrary, but fixed, soft state aggregation, 3) a novel intuitive understanding of the effect of state aggregation on online RL, and 4) a new heuristic adaptive state aggregation algorithm that finds improved compact representations by exploiting the non-discrete nature of soft state aggregation. Preliminary empirical results are also presented. 1 INTRODUCTION The strong theory of convergence available for reinforcement learning algorithms (e.g., Dayan & Sejnowski, 1994; Watkins & Dayan, 1992; Jaakkola, Jordan & Singh, 1994; Tsitsiklis, 1994) makes them attractive as a basis for building learning control architectures to solve a wide variety of search, planning, and control problems. Unfortunately, almost all of the convergence results assume lookup table representa- 362 Satinder Singh, Tommi Jaakkola, Michael I. Jordan tions for value functions (see Sutton, 1988; Dayan, 1992; Bradtke, 1993; and Vanroy & Tsitsiklis, personal communication; for exceptions). It is widely accepted that the use of more compact representations than lookup tables is crucial to scaling RL algorithms to real-world problems. In this paper we address the pressing issue of combining function approximation and RL, and present 1) a function approximator based on a simple extension to state aggregation (a commonly used form of compact representation, e.g., Moore, 1991), namely soft state aggregation, 2) a theory of convergence for RL with arbitrary, but fixed, soft state aggregation, 3) a novel intuitive understanding of the effect of state aggregation on online RL, and 4) a new heuristic adaptive state aggregation algorithm that finds improved compact representations by exploiting the non-discrete nature of soft state aggregations. Preliminary empirical results are also presented. Problem Definition and Notation: We consider the problem of solving large Markovian decision processes (MDPs) using RL algorithms and compact function approximation. We use the following notation: 8 for state space, A for action space, pa(s, s') for transition probability, Ra(s) for payoff, and 'Y for discount factor. The objective is to maximize the expected, infinite horizon, discounted sum of payoffs. 1.1 FUNCTION APPROXIMATION: SOFT STATE CLUSTERS In this section we describe a new function approximator (FA) for RL. In section 3 we will analyze it theoretically and present convergence results. The FA maps the state space 8 into M > 0 aggregates or clusters from cluster space X. Typically, M < < 181. We allow soft clustering, where each state s belongs to cluster x with probability P(xls), called the clustering probabilities. This allows each state s to belong to several clusters. An interesting special case is that of the usual state aggregation where each state belongs only to one cluster. The theoretical model is that the agent can observe the underlying state but can only update a value function for the clusters. The value of a cluster generalizes to all states in proportion to the clustering probabilities. Throughout we use the symbols x and y to represent individual clusters and the symbols sand s' to represent individual states. 2 A GENERAL CONVERGENCE THEOREM An online RL algorithm essentially sees a sequence of quadruples, < St, at, StH, rt >, representing a transition from current state St to next state StH on current action at with an associated payoff rt. We will first prove a general convergence theorem for Q-learning (Watkins & Dayan, 1992) applied to a sequence of quadruples that mayor may not be generated by a Markov process (Bertsekas, 1987). This is required because the RL problem at the level of the clusters may be non-Markovian. Conceptually, the sequence of quadruples can be thought of as being produced by some process that is allowed to modify the sequence of quadruples produced by a Markov process, e.g., by mapping states to clusters. In Section 3 we will specialize the following theorem to provide specific results for our function approximator. Consider any stochastic process that generates a sequence of random quadruples, \]! = {< Xi, ai, Yi, ri > h, where Xi, Yi E Y, ai E A, and ri is a bounded real number. Note that Xi+l does not have to be equal to Yi. Let IYI and IAI be finite, and define Reinforcement Learning with Soft State Aggregation 363 indicator variables Xi ( x, a, Y) = { and Wi =< x,a,y,. > (for any r) ?1 owhen th erwlse, . when Wi =< x, a, .,. otherwise. > (for any y, and any r) Define ,a ,( ) -_ 2:t-iXk(x,a,y) PIJx,y , , 2:1=i n(x, a) and Theorem 1: If V( > 0, 3ME < 00, such that for all i ~ 0, for all x, y E Y, and for all a E A, the following conditions characterize the infinite sequence W: with probability 1 - (, IPi~i+M.(X,y)-pa(x,y)1 < ( IRi,i+MJX) - Ra(x)1 < (, and (1) where for all x, a, and y, with probability one Pg,oo(x, y) = pa(x, y), and Rg,oo(x) = Ra(x). Then, online Q-learning applied to such a sequence will converge with probability one to the solution of the following system of equations: Vx E Y, and Va E A, Q(x, a) = Ra(x) + 'Y " pa(x, y) maxQ(y, a') L.J a'EA (2) yEY Proof: Consider the semi-batch version of Q-learning that collects the changes to the value function for M steps before making the change. By assumption, for any (, making ME large enough will ensure that with probability 1 - (, the sample (x, y) and Ri,i+M quantities for the ith batch, Pti+M ,?.) < (i) ? (x) are within ( of the asymptotic quantities. In Appendix A we prove that the semi-batch version of Qlearning outlined above converges to the solution of Equation 2 with probability one. The semi-batch proof can be extended to online Q-learning by using the analysis developed in Theorem 3 of Jaakkola et al. (1994). In brief, it can be shown that the difference caused by the online updating vanishes in the limit thereby forcing semi-batch Q-learning and online Q-learning to be equal asymptotically. The use of the analysis in Theorem 3 from Jaakkola et al. (1994) requires that the learning rate parameters CY are such that maxlEM.(Ic)al alex) () _ 1 uniformly w.p.l.; ME(k) is x the kth batch of size ME' If CYt(x) is non-increasing in addition to satisfying the conventional Q-learning conditions, then it will also meet the above requirement . o Theorem 1 provides the most general convergence result available for Q-learning (and TD(O?; it shows that for an arbitrary quadruple sequence satisfying the ergodicity conditions given in Equations 1, Q-learning will converge to the solution of the MDP constructed with the limiting probabilities (Po,oo) and payoffs (Ro,oo). Theorem 1 combines and generalizes the results on hard state aggregation and value iteration presented in Vanroy & Tsitsiklis (personal communication), and on partially observable MDPs in Singh et al. (1994). 364 3 Satinder Singh, Tommi Jaakkola, Michael I. Jordan RL AND SOFT STATE AGGREGATION In this section we apply Theorem 1 to provide convergence results for two cases: 1) using Q-Iearning and our FA to solve MDPs, and 2) using Sutton's (1988) TD(O) and our FA to determine the value function for a fixed policy. As is usual in online RL, we continue to assume that the transition probabilities and the payoff function of the MDP are unknown to the learning agent. Furthermore, being online such algorithms cannot sample states in arbitrary order. In this section, the clustering probabilities P(xls) are assumed to be fixed. Case 1: Q-learning and Fixed Soft State Aggregation Because of function approximation, the domain of the learned Q-value function is constrained to be X x A (X is cluster space). This section develops a "Bellman equation" (e.g., Bertsekas, 1987) for Q-Iearning at the level of the cluster space. We assume that the agent follows a stationary stochastic policy 7r that assigns to each state a non-zero probability of executing every action in every state. Furthermore, we assume that the Markov chain under policy 7r is ergodic. Such a policy 7r is a persistently exciting policy. Under the above conditions p'II"(slx) = :, ;C;,:;p$"C$')' where for all s, p'll"(s) is the steady-state probability of being in state s. Corollary 1: Q-Iearning with soft state aggregation applied to an MDP while following a persistently exciting policy 7r will converge with probability one to the solution of the following system of equations: Vex, a) E (X x A), E$' and pa(s, y) = pa(s, s')P(yls'). The Q-value function for the state space can then be constructed via Q(s, a) = Ex P(xls)Q(x, a) for all (s, a). Proof: It can be shown that the sequence of quadruples produced by following policy 7r and independently mapping the current state s to a cluster x with probability P(xls) satisfies the conditions of Theorem 1. Also, it can be shown that Note that the Q-values found by clustering are dependent on the sampling policy 7r, unlike the lookup table case. Case 2: TD(O) and Fixed Soft State Aggregation We present separate results for TD(O) because it forms the basis for policy-iterationlike methods for solving Markov control problems (e.g., Barto, Sutton & Anderson, 1983) - a fact that we will use in the next section to derive adaptive state aggregation methods. As before, because of function approximation, the domain of the learned value function is constrained to be the cluster space X. Corollary 2: TD(O) with soft state aggregation applied to an MDP while following a policy 7r will converge with probability one to the solution of the following system Reinforcement Learning with Soft State Aggregation 365 of equations: Vx EX, V(x) (4) where again as in Q-Iearning the value function for the state space can be constructed via V(s) = Lx P(xls)V(x) for all s. Proof: Corollary 1 implies Corollary 2 because TD(O) is a special case of Q-Iearning for MDPs with a single (possibly randomized) action in each state. Equation 4 provides a "Bellman equation" for TD(O) at the level of the cluster space. 0 4 ADAPTIVE STATE AGGREGATION In previous sections we restricted attention to a function approximator that had a fixed compact representation. How might one adapt the compact representation online in order to get better approximations of value functions? This section presents a novel heuristic adaptive algorithm that improves the compact representation by finding good clustering probabilities given an a priori fixed number of clusters. Note that for arbitrary clustering, while Corollaries 1 and 2 show that RL will find solutions with zero Bellman error in cluster space, the associated Bellman error in the state space will not be zero in general. Good clustering is therefore naturally defined in terms of reducing the Bellman error for the states of the MDP. Let the clustering probabilities be parametrized as follows P(xls; 0) = L:~~:';~/ . ), where o( x , s) is the weight between state s and cluster x. Then the Bellman error at .state s given parameter (a matrix) is, ? J(sIO) V( slO) - [R' (s) + r ~ P' (8, s')V( S'IO)] ~ P(xls; O)V(xIO) - [w(s) + r ~ P'(s, s') ~ P(xls'; O)V(XIO)] Adaptive State Aggregation (ASA) Algorithm: Step 1: Compute V(xIO) for all x E X using the TD(O) algorithm. 2 (9) Step 2.. Let !:l.0 -- -a oJ89 . Go to step 1. where Step 2 tries to minimize the Bellman error for the states by holding the cluster values fixed to those computed in Step 1. We have 8J 2 (sI0) 80(y, s) 2J(sI0) [P(yls; 0)(1 - ,p7r(s, s?(V(yIO) - V(sIO?]. The Bellman error J(sIO) cannot be computed directly because the transition probabilities P( s, s') are unknown. However, it can be estimated by averaging the sample 366 Sa tinder Singh, Tommi Jaakkola, Michael I. Jordan Bellman error. P(yls; B) is known, and (1 - ,P 7r (s, s)) is always positive, and independent of y, and can therefore be absorbed into the step-size c?. The quantities V(yIB) and V(sIB) are available at the end of Step 1. In practice, Step 1 is only carried out partially before Step 2 is implemented. Partial evaluation works well because the changes in the clustering probabilities at Step 2 are small, and because the final V(xIB) at the previous Step 1 is used to initialize the computation of V(xIB) at the next Step 1. ------ 2 Clusters --------- 4 Clusters to Clusters - - 20 Clusters 4 .. ,' . .. ",' " ....... - -.. . " . ... " .... .' . . " . Iterations of ASA Figure 1: Adaptive State Clustering. See text for explanation. Figure 1 presents preliminary empirical results for the ASA algorithm. It plots the squared Bellman error summed over the state space as a function of the number of iterations of the ASA algorithm with constant step-size c?. It shows error curves for 2, 4, 10 and 20 clusters averaged over ten runs of randomly constructed 20 state Markov chains. Figure 4 shows that ASA is able to adapt the clustering probabilities to reduce the Bellman error in state space, and as expected the more clusters the smaller the asymptotic Bellman error. In future work we plan to test the policy iteration version of the adaptive soft aggregation algorithm on Markov control problems. 5 SUMMARY AND FUTURE WORK Doing RL on aggregated states is potentially very advantageous because the value of each cluster generalizes across all states in proportion to the clustering probabilities. The same generalization is also potentially perilous because it can interfere with the contraction-based convergence of RL algorithms (see Vee, 1992; for a discussion). This paper resolves this debate for the case of soft state aggregation by defining a set of Bellman Equations (3 and 4) for the control and policy evaluation problems in the non-Markovian cluster space, and by proving that Q-Iearning and TD(O) solve them respectively with probability one. Theorem 1 presents a general convergence result that was applied to state aggregation in this paper, but is also a generalization of the results on hidden state presented in Singh et al. (1994), and may be applicable Reinforcement Learning with Soft State Aggregation 367 to other novel problems. It supports the intuitive picture that if a non-Markovian sequence of state transitions and payoffs is ergodic in the sense of Equation 1, then RL algorithms will converge w.p.l. to the solution of an MDP constructed with the limiting transition probabilities and payoffs. We also presented a new algorithm, ASA, for adapting compact representations, that takes advantage of the soft state aggregation proposed here to do gradient descent in clustering probability space to minimize squared Bellman error in the state space. We demonstrated on simple examples that ASA is able to adapt the clustering probabilities to dramatically reduce the Bellman error in state space. In future work we plan to extend the convergence theory presented here to discretizations of continuous state MDPs, and to further test the ASA algorithms. A Convergence of semi-batch Q-Iearning (Theorem 1) Consider a semi-batch algorithm that collects the changes to the Q-value function for M steps before making the change to the Q-value function. Let kM R%(x) = L kM L Mk(X, a) = riXi(x, a); i=(k-l)M Xi(X, a) i=(k-l)M and kM Mk(x,a,y) L = Xi(x,a,y) i=(k-l)M Then the Q-value of (x, a) after the Qk+l(X, a) = kth batch is given by: (1- Mk(X, a)ak(x, a))Qk(x, a) Rk(x) +Mk(x, a) ak(x, a) [ M kx,a ( ) Let Q be Mk(X, a, y) +, ~ ~ M ( kx,a ) m;pcQk(Y, a) I ] a yEY the solution to Equation 2. Define, Fk(x, a) = Rk(x) M ( ) kx,a ~ +, ~ yEY Mk(x, a, y) ,M ( ) m~xQk(Y, a) - Q(x, a), kx,a a then, if Vk(X) = maxa Qk(X, a) and Vex) = maxa Q(x, a), Fk(x, a) = ~ Mk(X, a, y) ) (Vk(y) - V(y)] kx,a y ,~ M ( [(Mk(x,a,y) +, ~ ~ Mk(X,a) a ( Rk(x) a + (Mkx,a ( ) - Ro,oo(x)) ))-()] - PO,oo x,y V y , y The quantity Fk(x, a) can be bounded by IIFk(x, a) II :s; ,llVk - VII + II(~~~~L - Rg,oo(x))11 +,11 Ly(~tX~~~y - Po,oo(x, y))V(y)11 :s; ,llVk - VII + Cfr, a M? th e Iarger 0 f IMk(x,a) R&(x) - R 0,00 ()I I ~ (Mk(x,a,y) na ( x, y ))1 . By h were fk 1S x , an d 'l..Jy Mk(x,a) .ro,oo 368 Sa tinder Singh, Tommi Jaakkola, Michael I. Jordan assumption for any (. > 0, 3Mf < 00 such that (.~< < (. with probability 1 - c The variance of Fk(X, a) can also be shown to be bounded because the variance of the sample probabilities is bounded (everything else is similar to standard Q-learning for MDPs). Therefore by Theorem 1 of Jaakkola et al. (1994), f<2r any (. > 0, with probability (1 - f), Qk(X, a) - Qoo(X, a), where IQoo(x, a) - Q(x, a)1 ~ Cc Therefore, semi-batch Q-learning converges with probability one. 0 Acknowledgements This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, and by a grant from Siemens Corporation. Michael I. Jordan is a NSF Presidential Young Investigator. References A. G. Barto, R. S. Sutton, & C. W. Anderson. (1983) Neuronlike elements that can solve difficult learning control problems. IEEE SMC, 13:835-846. D. P. Bertsekas. (1987) Dynamic Programming: Deterministic and Stochastic Models, Prentice-Hall. S. J. Bradtke. (1993) Reinforcement learning applied to linear quadratic regulation. In Advances in Neural Information Processing Systems 5, pages 295-302. P. Dayan. (1992) The convergence of TD(A) for general A. Machine Learning, 8(3/4):341-362. P. Dayan & T.J. Sejnowski. (1994) TD(A) converges with probability 1. Machine Learning, 13(3). T. Jaakkola, M. I. Jordan, & S. P. Singh. (1994) On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6(6):11851201. A. W. Moore. (1991) Variable resolution dynamic programming: Efficiently learning action maps in multivariate real-valued state-spaces. In M aching Learning: Proceedings of the Eighth International Workshop, pages 333- 337. S. P. Singh, T. Jaakkola, & M. I. Jordan. (1994) Learning without state-estimation in partially observable markovian decision processes. In Machine Learning: Proceedings of the Eleventh International Conference, pages 284-292. R. S. Sutton. (1988) Learning to predict by the methods of temporal differences. Machine Learning, 3:9-44. J. Tsitsiklis. (1994) Asynchronous stochastic approximation and Q-learning. Machine Learning, 16(3):185-202. B. Vanroy & J. Tsitsiklis. (personal communication) C. J. C. H. Watkins & P. Dayan. (1992) Q-learning. Machine Learning, 8(3/4):279292. R. C. Yee. (1992) Abstraction in control learning. Technical Report COINS Technical Report 92-16, Department of Computer and Information Science, University of Massachusetts, Amherst, MA 01003. A dissertation proposal. 

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Coarse-to-Fine Image Search Using Neural Networks Clay D. Spence, John C. Pearson, and Jim Bergen Nationallnfonnation Display Laboratory P.O. Box 8619 Princeton, NJ 08543-8619 cspence@sarnoff.com John_Pearson@maca.sarnoff.com jbergen@sarnoff.com Abstract The efficiency of image search can be greatly improved by using a coarse-to-fine search strategy with a multi-resolution image representation. However, if the resolution is so low that the objects have few distinguishing features, search becomes difficult. We show that the performance of search at such low resolutions can be improved by using context information, i.e., objects visible at low-resolution which are not the objects of interest but are associated with them. The networks can be given explicit context information as inputs, or they can learn to detect the context objects, in which case the user does not have to be aware of their existence. We also use Integrated Feature Pyramids, which represent high-frequency information at low resolutions. The use of multiresolution search techniques allows us to combine information about the appearance of the objects on many scales in an efficient way. A natural fOlm of exemplar selection also arises from these techniques. We illustrate these ideas by training hierarchical systems of neural networks to find clusters of buildings in aerial photographs of farmland. 1 INTRODUCTION Coarse-to-fine image search is a general technique for improving the efficiency of any search algorithm. (Burt, 1988a and b; Burt et al., 1989). One begins by searching a lowresolution (coarse-scale) version of the image, typically obtained by constructing an image 982 Clay D. Spence, John C. Pearson, Jim Bergen pyramid (Burt and Adelson, 1983). Since this version is smaller than the original, there are far fewer pixels to be processed and the search is correspondingly faster. To improve the certainty of detection and refine the location estimates, the search process is repeated at higher resolution (finer scale), but only in those regions-of-interest (ROls) which were identified at lower resolution as likely to contain one of the objects to be found, thus greatly reducing the actual area to be searched. This process is repeated at successively higher resolutions until sufficient certainty and precision is achieved or the original image has been searched. 1 These pyramid techniques scale well with image size and can quickly process large images, but the relatively simple, hand-tuned pattern recognition components that are typically used limit their accuracy. Neural networks provide a complementary set of techniques, since they can learn complex patterns, but they don't scale well with image size. We have developed a Hybrid PyramidlNeural Network (HPNN) system that combines the two techniques so as to leverage their strengths and compensate for their weaknesses. A novel benefit of combining pyramids and neural networks is the potential for automatically learning to use context information, by which we mean any visible characteristics of the object's surroundings which tend to distinguish it from other objects. Examples of context information which might be useful in searching aerial photographs for man-made objects are the proximity of roads and terrain type. Pyramids provide the ability to detect context in a low-resolution image representation, which is necessary for efficient processing of large regions of the image. Neural networks provide the ability to discover context of which we may be unaware, and to learn the relevance of the context to detection, the dependence of detection probability on distance from a context object, etc. Context can help to narrow the search region at low resolutions and improve the detection performance since it is, by definition, relevant information. Of course, the usefulness of the idea will be different for each problem. Context can be explicitly provided, e.g., in the form of a road map. As mentioned above, a neural net may also learn to exploit some context which is not explicitly provided, but must be inferred from the image data. If there is such context and the network's architecture and input features are capable of exploiting it, it should learn to do so even if we use ordinary training methods that do not explicitly take context into account. We currently implement these functions in a hierarchical sequence of ordinary feed-forward networks with sigmoidal units, one network at each resolution in an image pyramid (Fig. 1). Each network receives some features extracted from a window in the image, and its output is interpreted as the probability of finding one of the searched-for objects in the center of the window. The network is scanned over all ROIs in this manner. (At the lowest resolution there is one ROI, which is the entire image.) The networks are trained one at a time, starting at the lowest resolution. Context is made available to networks at higher resolutions by giving them information from the lower-resolution networks as additional inputs, which we will call context inputs. These could be I Several authors have investigated multi-scale processing of one-dimensional signals with neural networks, e.g., Mozer (1994) and Burr and Miyata (1993) studied music composition. Burr and Miyata use sub-sampling as in a pyramid decomposition. Images differ somewhat from music in that there are primitive features at all scales (limited by the sampling rate and image size), whereas the average frequency spectrum of music over a long time span doesn't seem likely to be meaningful. The original paper on pyramid image representation is (Burt and Adelson, 1983). Coarse-To-Fine Image Search Using Neural Networks 983 Target High-Resolution Features Probability Context Information Control of Search Region Features Figure 1. A Hierarchical Search System that Exploits Context. taken from either the hidden units or output units of the lower-resolution networks. By training the networks in sequence starting at low resolution, we can choose to train the higher-resolution networks in the ROls selected by the lower-resolution networks, thus providing a simple means of exemplar selection. This coarse-to-fine training is often useful, since many problems have relatively few positive examples per image, but because of the size of the image there are an enormous number of more-or-Iess redundant negative examples. 2 ANEXAMPLEPROBLEM To demonstrate these ideas we applied them to the problem of finding clusters of buildings in aerial photographs of a rural area. Since buildings are almost always near a road, roads are context objects associated with buildings. We approached the problem in two ways to demonstrate different capabilities. First, we trained systems of networks with explicitl~ provided context in the form of road maps, in order to demonstrate the possible benefits. Second, we trained systems without explicitly-provided context, in order to show that the context could be learned. 2.1 EXPLICIT CONTEXT For comparison purposes, we trained one network to search a high-resolution image directly, with no explicit context inputs and no inputs from other networks. To demonstrate the effect of learned context on search we trained a second system with no explicit 2 It would be surprising if this extra infonnation didn't help, since we know it is relevant. For many applications digitized maps will be available, so demonstrating the possible perfonnance benefit is still worth-while. 984 Clay D. Spence, John C. Pearson, Jim Bergen context provided to the nets, but each received inputs from all of the networks at lower resolutions. These inputs were simply the outputs of those lower-resolution networks. To demonstrate the benefit of explicitly provided context, we trained a third system with both context inputs from lower-resolution networks and explicit context in the form of a road map of the area, which was registered with the image. 2.1.1 Features To preserve some distinguishing features at a given low resolution, we extracted simple features at various resolutions and represented them at the lower resolution. The low-resolution representations were constructed by reducing the feature images with the usual blur-and-sub-sample procedure used to construct a Gaussian pyramid (Burt and Adelson, 1983). The features should not be too computationally expensive to extract, otherwise the efficiency benefit of coarse-to-fine search would be canceled. The features used as inputs to the neural nets in the building-search systems were simple measures of the spatial image energy in several frequency bands. We constructed these feature images by building the Laplacian pyramid of the image3 and then taking the absolute values of the pixels in each image in the pyramid. We then constructed a Gaussian pyramid of each of these images, to provide versions of them at different resolutions. A neural net searching a given resolution received input from each energy image derived from the same resolution and all higher resolutions. 4 Binary road-map images were constructed from the digitized aerial photographs. They were reduced in resolution by first perfonning a binary blur of the image and then subsampling it by two in each dimension. In the binary blur procedure each pixel is set to one if any of its nearest neighbors were road pixels before blurring. This is repeated to get road maps at each resolution. To give the networks a rough measure of distance to a road, we gave the nets inputs from linearly-blurred versions of the road maps, which were made by expanding even lower-resolution versions of the road map with the same linear expand operation used in constructing the Laplacian pyramid. These blurred road maps are therefore not binary. The networks at the fifth and third pyramid levels received inputs from the road-maps at their resolution and from the two lower resolutions, while the network at the first pyramid level received input from the road map at its resolution and the next lower resolution. 2.1.2 Training the Networks To estimate the probability of finding a building cluster at or around a given location in the image, each network received a single pixel from the same location in all of its input images. This should be adequate for search, since a single pixel in a feature image contains information about an extended region of the original. With the features we used, it also makes the system invariant to rotations, so the networks do not have to learn this invariance Also for simplicity, we did not train nets at all resolutions, but only at the fifth, third and first pyramid levels, i.e., on images which were one-thirty-second, one-eighth, 3 The Laplacian pyramid is usually constructed by expanding the lower-resolution levels of a Gaussian pyramid and subtracting each from the next-higher-resolution level. This gives a set of images which are band-passed with one-octave spacing between the bands. 4 There are many examples of more sophisticated features, e.g., Lane, et aI., 1992, Ballard and Wixson, 1993, and Greenspan, et aI., 1994. Simple features were adequate for demonstrating the ideas of this paper. Coarse-To-Fine Image Search Using Neural Networks No Road Maps or context 985 100 ,- - Context ~ ..... .iii o -- -:' . ~ I 90 r ,, , 85 I Q) ... :::l t- ~ - - 95 .~ a.. r ,'- 1- 80~~~~~~~~~~~~~~~~ a 5 15 10 % Folse Positives 20 Figure 2. ROC Curves for the Three Network Search Systems. and one-half the width and height of the original image. A typical building cluster has a linear extent of about 30 pixels in the original images, so in the fifth Gaussian pyramid level they have few distinguishing features. The networks were the usual feed-forward nets with sum-and-sigmoid units and one hidden layer. They were trained using the crossentropy error measure, with a desired output of one in a hand-chosen polygonal region about each building cluster. The standard quadratic weight-decay term was added to this to get the objective function, and only one regularization constant was used. This was adjusted to give lowest error on a test set. We often found that the weights to two or more hidden units would become identical and/or very small as the regularization term was increased, and in these cases we pruned the extraneous units and began the search for the optimal regularization parameter again. We usually ended up with very small networks, with from two to five hidden units. Mter training a net at low-resolution, we expanded the image of that net's output in order to use it as a context input for the networks at higher resolution. 2.1.3 Performance To compare the performance of the three systems, we chose a threshold for the networks' outputs, and considered a building cluster to be detected by a network if the network's output was above the threshold at any pixel within the cluster. The number of detected clusters divided by the total number is the true-positive rate. The false-positive rate is more difficult to define, since we had in mind an application in which the detection system would draw a user's attention to objects which are likely to be of interest. The procedure we used attempts to measure the amount of effort a human would expend in searching regions with false detections by the network. See (Spence, et al., 1994) for details. The performance figures presented here were measured on a validation set, i.e., an image on which the network was not trained and which was not used to set the regularization parameter. The results presented in Tables 1 and 2 are for a single threshold, chosen for each network so that the true-positive rate was 90%. Figure 2 compares the ROC curves of the three systems, i.e. the parametric curves of the true and false-positive rates as the threshold on the network's output is varied. From Table 1, the features we used would seem adequate for search at very low resolution, although the performance could be better. Clay D. Spence, lo/m C. Pearson, Jim Bergen 986 Table 1: False-Positive Rates vs. Resolution. These are results for the system with both road-map and context inputs. PYRAMID LEVEL FALSE-POSITIVE RATE 5 16% 3 4.6% 1 3.6% Table 2: False-Positive Rates of the Search Systems at 90% True-Positive Rate. NETWORK SYSTEM FALSE-POSITIVE RATE No Context or Road-Map Inputs 7.6% Context Inputs, No Road Maps 5.8% Context Inputs and Road Maps 3.6% Table 2 and Figure 2 clearly show the benefits of using the road map and context inputs, although the statistics are somewhat poor at the higher true-positive rates because of the small number of building clusters which are being missed 2.2 LEARNING CONTEXT Two things were changed to demonstrate that the context could be learned. First, the unoriented spatial energy features are not well suited for distinguishing between roads and other objects with a size similar to a road's width, so we used oriented energies instead. These were the oriented energies derived using steerable filters (Freeman and Adelson, 1991) at four orientations. To force orientation imariance, we sorted the four oriented energies at each pixel for each frequency band. These sorted oriented energy images were then reduced in size as appropriate for the resolution being searched by a network. In this case, we extracted energies only from the first, third, and fifth pyramid levels. The second change is the use of hidden unit outputs for context inputs, instead of the output unit's outputs. The output units estimate the probability of finding a building cluster. Although this may reflect information about roads, it is very indirect information about the roads. It is more likely to carry some information about the coarse-scale appearance of the potential building clusters. The hidden unit outputs should contain a richer description of the image at a coarser scale. 2.2.1 Performance The networks were trained in the same way as the networks described in Section 2.1. We trained three networks to search levels five, three, and one. For comparison purposes, we also trained a single-network to search in level one, with the same input features that were used for all of the networks of the hierarchical search system. These include, for example, three versions of each of the oriented energy images from level one (the second highest frequency band). Two of these versions were reduced in size to levels three and five, and then re-expanded to level one, so that they are simply blurred versions of the original energy images. This gives the network a direct source of information on the coarse-scale Coarse-To-Fine Image Search Using Neural Networks 987 100 Non-hierarchical ID 95 .~ Hierarchical .-- ????????? --.. "iii o a. 90 CIl ~ .= ~ 85 80~~~~~~__~~~ a 10 15 5 % False Positives 20 Figure 3. Performance of Hierarchical and Non-hierarchical Algorithms that Learn Context. Figure 4. Image and Context Inputs to Highest-Resolution Network. appearance of the image. although it is an expensive representation in terms of computation and memory. and the network has more inputs so that it takes longer to train. The ROC curves for the two detection algorithms are shown in Figure 3. Their performance is about the same, suggesting that the coarser-scale networks in the hierarchical system are not performing computations that are useful for the finest-scale network, rather they simply pass on information about the coarse-scale appearance. In this example. the advantage of the hierarchical system is in the efficiency of both the training of the algorithm and its use after training. Figure 4 shows the level-one image and the outputs of several hidden units from the level-three network, expanded to the same size. These outputs suggest that road information is being extracted at coarse scale and passed to the high-resolution network. 3 DISCUSSION We have performed "proof-of-concept" simulations on a realistic search problem of two of the key concepts of our Hybrid Pyramid/Neural Network (HPNN) approach. We have shown that: 1. The HPNN can learn to utilize explicitly-provided context data to improve object detection; 2. the networks can learn to use context without being explicitly taught to do so. This second point implies that the person training the system does not need to know of the existence of the context objects. The networks at the different scales are trained in sequence. starting at low resolution. A benefit of this coarse-to-fine training is a simple form of exemplar selection. since we can choose to train a network only in the regions of 988 Clay D. Spence. John C. Pearson. Jim Bergen interest as detected by the already-trained networks at lower resolution. We could also train all of the networks simultaneously, so that the lower-resolution networks learn to extract the most helpful information from the low-resolution image. The resulting system would probably perform better, but it would also be more expensive to train. This approach should work quite well for many automatic target recognition problems, since the targets are frequently quite small. For more extended objects like clusters of buildings, our method is an efficient way of examining the objects on several length scales, but the detection at the finest scale is based on a small part of the potential object's appearance. Extended images of real objects typically have many features at each of several resolutions. We are currently working on techniques for discovering several characteristic features at each resolution by training several networks at each resolution, and integrating their responses to compute an overall probability of detection. Acknowledgments We would like to thank Peter Burt, P. Anandan, and Paul Sajda for many helpful discussions. This was work was funded by the National Information Display Laboratory and under ARPA contract No. NOOOI4-93-C-0202. References D.H. Ballard and L.E. Wixson (1993) Object recognition using steerable filters at multiple scales, Proceedings of the IEEE Workshop on Qualitative Vision, New York, NY. D. Burr and Y. Miyata (1993) Hierarchical recurrent networks for learning musical structure, Proceedings of the IEEE Conference on Neural Networks for Signal Processing Ill, C. Kamm, G. Kuhn, B. Yoon, S.Y. Kung, and R. Chellappa, eds., Piscataway, NJ, pp. 216-225. P. J. Burt and E. H. Adelson. (1983) The Laplacian pyramid as a compact image code. IEEE Transactions, Vol. COM-31:4, April, pp. 532-540. P.J. Burt (1988a) Smart sensing with a pyramid vision machine, Proceedings of the IEEE Vol. 76, pp. 1006-1015. PJ. Burt (1988b) Attention mechanisms for vision in a dynamic world, Proceedings of the 9th International Conference on Pattern Recognition, pp. 977-987. PJ. Burt, J.R. Bergen, R. Kolczynski, R. Hingorani, W.A. Lee, A. Leung, J. Lubin, and H. Shvayster (1989) Object tracking with a moving camera, Proceedings of the IEEE Workshop on Motion, Irvine. W.T. Freeman and E.H. Adelson (1991) The design and use of steerable filters, IEEE Trans. PAMI, 12:9, pp. 891-906. S.H. Lane, J.C. Pearson, and R. Sverdlove (1992) Neural networks for classifying image textures, Proceedings of the Government Applications of Neural Networks Conference, Dayton, Ohio. M.e. Mozer (1994) Neural network music composition by prediction: Exploring the benefits of psychoacoustic constraints and multiscale processing, to appear in Connection Science. PART IX APPLICATIONS 

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On the Computational Utility of Consciousness Donald W. Mathis and Michael C. Mozer mathis@cs.colorado.edu, mozer@cs.colorado.edu Department of Computer Science and Institute of Cognitive Science University of Colorado, Boulder Boulder, CO 80309-0430 Abstract We propose a computational framework for understanding and modeling human consciousness. This framework integrates many existing theoretical perspectives, yet is sufficiently concrete to allow simulation experiments. We do not attempt to explain qualia (subjective experience), but instead ask what differences exist within the cognitive information processing system when a person is conscious of mentally-represented information versus when that information is unconscious. The central idea we explore is that the contents of consciousness correspond to temporally persistent states in a network of computational modules. Three simulations are described illustrating that the behavior of persistent states in the models corresponds roughly to the behavior of conscious states people experience when performing similar tasks. Our simulations show that periodic settling to persistent (i.e., conscious) states improves performance by cleaning up inaccuracies and noise, forcing decisions, and helping keep the system on track toward a solution. 1 INTRODUCTION We propose a computational framework for understanding and modeling consciousness. Though our ultimate goal is to explain psychological and brain imaging data with our theory, and to make testable predictions, here we simply present the framework in the context of previous experimental and theoretical work, and argue that 12 Donald Mathis, Michael C. Mozer it is sensible from a computational perspective. We do not attempt to explain qualia-subjective experience and feelings. It is not clear that qualia are amenable to scientific investigation. Rather, our aim is to understand the mechanisms underlying awareness, and their role in cognition. We address three key questions: ? What are the preconditions for a mental representation to reach consciousness? ? What are the computational consequences of a representation reaching consciousness? Does a conscious state affect processing differently than an unconscious state? ? What is the computational utility of consciousness? That is, what is the computational role of the mechanism(s) underlying consciousness? 2 THEORETICAL FRAMEWORK Modular Cognitive Architecture. We propose that the human cognitive architecture consists of a set of functionally specialized computational modules (e.g., Fodor, 1983). We imagine the modules to be organized at a somewhat coarse level and to implement processes such as visual object recognition, visual word-form recognition, auditory word and sound recognition, computation of spatial relationships, activation of semantic representations of words, sentences, and visual objects, construction of motor plans, etc. Cognitive behaviors require the coordination of many modules. For example, functional brain imaging studies indicate that there are several brain areas used for different subtasks during cognitive tasks such as word recognition (Posner & Carr, 1992). Modules Have Mapping And Cleanup Processes. We propose that modules perform an associative memory function in their domain, and operate via a two-stage process: a fast, essentially feedforward input-output mapping 1 followed by a slower relaxation search (Figure 1). The computational justification for this two stage process is as follows. We assume that, in general, the output space of a module can represent a large number of states relative to the number of states that are meaningful or well formed-i.e., states that are interpretable by other modules or (for output modules) that correspond to sensible motor primitives. If we know which representations are well-formed, we can tolerate an inaccurate feedforward mapping, and "clean up" noise in the output by constraining it to be one of the well-formed states. This is the purpose of the relaxation step: to clean up the output of the feedforward step, resulting in a well-formed state. The cleanup process knows nothing about which output state is the best response to the input; it acts solely to enforce well-formedness. Similar architectures have been used recently to model various neuropsychological data (Hinton & Shallice, 1991; Mozer & Behrmann, 1990; Plaut & Shallice, 1993). The empirical motivation for identifying consciousness with the results of relaxation search comes from studies indicating that the contents of consciousness tend to be coherent, or well-formed (e.g., Baars, 1988; Crick, 1994; Damasio, 1989). Persistent States Enter Consciousness. In our model, module outputs enter consciousness if they persist for a sufficiently long time. What counts as long enough lWe do not propose that this process is feedforward at the neural level. Rather, we mean that any iterative refinement of the output over time is unimportant and irrelevant. On the Computational Utility of Consciousness 13 relaxation search feedforward mapping .00 ?? Figure 1: Modules consist of two components. is not yet determined, but in order to model specific psychological data, we will be required to make this issue precise. At that time a specific commitment will need to be made, and this commitment must be maintained when modeling further data. An important property of our model is that there is no hierarchy of modules with respect to awareness, in contrast to several existing theories that propose that access to some particular module (or neural processing area) is required for consciousness (e.g., Baars, 1988). Rather, information in any module reaches awareness simply by persisting long enough. The persistence hypothesis is consistent with the theoretical perspectives of Smolensky (1988), Rumelhart et al (1986), Damasio (1989), Crick and Koch (1990), and others. 2.1 WHEN ARE MENTAL STATES CONSCIOUS? In our framework, the output of any module will enter consciousness if it persists in time. The persistence of an output state of a module is assured if: (1) it is a point attractor of the relaxation search (i.e., a well-formed state), and (2) the inputs to the module are relatively constant , i.e., they continue to be mapped into the same attractor basin. While our framework appears to make strong claims about the necessary and sufficient conditions for consciousness, without an exact specification of the modules forming the cognitive architecture, it is lacking as a rigorous, testable theory. A complete theory will require not only a specification of the modules, but will also have to avoid arbitrariness in claiming that certain cognitive operations or brain regions are modules while other are not. Ultimately, one must identify the neurophysiological and neuroanatomical properties of the brain that determine the module boundaries (see Crick, 1994, for a promising step in this regard). 3 COMPUTATIONAL UTILITY OF CONSCIOUSNESS For the moment, suppose that our framework provides a sensible account of awareness phenomena (demonstrating this is the goal of ongoing work.) If one accepts this, and hence the notion that a cleanup process and the resulting persistent states are required for awareness, questions about the role of cleanup in the model become quite interesting because they are equivalent to questions about the role of the mechanism underlying awareness in cognition. One question one might ask is whether there is computational utility to achieving conscious states. That is, does a system that achieves persistent states perform better than a system that does not? 14 Donald Mathis, Michael C. Mozer Does a system that encourages settling to well-formed states perform better than a system that does not? We now show that the answer to this question is yes. 3.1 ADDITION SIMULATION To examine the utility of cleanup, we trained a module to perform a simple multistep cognitive task: adding a pair of two-digit numbers in three steps. 2 We tested the system with and without cleanup and compared the generalization performance. The network architecture (Figure 2) consists of a single module. The inputs consist of the problem statement and the current partial solution-state. The output is an updated solution-state. The module's output feeds back into its input. The problem statement is represented by four pools of units, one for each digit of each operand, where each pool uses a local encoding of digits. Partial solution states are represented by five pools, one for each of the three result digits and one for each of the two carry digits. .......... ( ~ ~ jU ~ if) Projection (copy) 1result 11 Iresult 31 Iresult21 tI leany21 + I Input-ot,rtput mapping leany (copy) I hidden units "- .,/ ~[~E::~:::][:iIl[iI] Iresult111resutt 211result311eany 111eany 21 " Figure 2: Network architecture for the addition task Each addition problem was decomposed into three steps (Figure 3), each describing a transformation from one partial solution state to the next, and the mapping net was trained perform each transformation individually. ? ? 48 + 62 ??? step 1 --.. ? 1 48 + 62 ??O step 2 --.. + 1 1 48 62 ?10 step 3 --.. + 1 1 48 62 11 0 Figure 3: The sequence of steps in an example addition problem Step 1 Given the problem statement, activate the rightmost result digit and rightmost carry digit (comprising the first partial solution). 2 Of course, we don't believe that there is a brain module dedicated to addition problems. This choice was made because addition is an intuitive example of a multistep task. On the Computational Utility of Consciousness 15 Step 2 Given the first partial solution, activate the next result and carry digits (second partial solution). Step 3 Given the second partial solution, activate the leftmost result digit (final solution). The set of well-formed states in this domain consists of all possible combinations of digits and "don't knows" among the pools ("don't knows" are denoted by question marks in Figure 3) . Local representations of digits are used within each pool, and "don't knows" are represented by the state in which no unit is active. Thus, the set of well-formed states are those in which either one or no units are active in each pool. To make these states attractors of the cleanup net, the connections were hand-wired such that each pool was a winner-take-all pool with an additional attractor at the zero state. To run the net, a problem statement pattern is clamped on the input units, and the net is allowed to update for 200 iterations. Unit activities were updated using an incremental rule approximating continuous dynamics: ai(t) = TI(L: Wijaj(t - 1)) + (1 - T)ai(t - 1) j where ai(t) is the activity of unit i at time t, T is a time constant in the interval [0,1]' and 10 is the usual sigmoid squashing function. Figure 4 shows the average generalization performance of networks run with and without cleanup, as a function of training set size. Note that, in principle, it is not necessary for the system to have a cleanup process to learn the training set perfectly, or to generalize perfectly. Thus, it is not simply the case that no solutions exist without cleanup. The generalization results were that for any size training set, percent correct on the generalization set is always better with cleanup than without. This indicates that although the mapping network often generalizes incorrectly, the output pattern often falls within the correct attractor basin. This is especially beneficial in multistep tasks because cleanup can correct the inaccuracies introduced by the mapping network, preventing the system from gradually diverging from the desired trajectory. - = Projection No projection training set size (% of all problems) Figure 4: Cleanup improves generalization performance. Figure 5 shows an example run of a trained network. There is one curve for each of the five result and carry pools, showing the degree of "activity" of the ultimate target pattern, t, for that pool as a function of time. Activity is defined to be e-lit-aIl2 where a is the current activity pattern and t is the target. The network /6 Donald Mathis, Michael C. Mozer solves the problem by passing though the correct sequence of intermediate states, each of which are temporarily persistent. This resembles the sequence of conscious states a person might experience while performing this task; each step of the problem is performed by an unconscious process, and the results of each of step appear in conscious awareness. 1.0r----=====~-----_""::--_~""-:-"""""'----------, Activation of target pattern in each pool .8 .6 : .' :'" .' ,:' result digit I, - - carry digit I " " result digit 2, ----- carrydigit2 ...":' ,: ,'. .......... result digit 3 - - - - - - - - - - ---".:!/ ~~~~IO~~~~2~O~--~3~O---~4~O---~50~--~ TIme Figure 5: Network solving the addition task in three steps 3.2 CHOICE POINT SIMULATION In many ordinary situations, people are required to make decisions, e.g., drive straight through an intersection or turn left, order macaroni or a sandwich for lunch. At these choice points, any of the alternative actions are reasonable a priori. Contextual information determines which action is correct, e.g., whether you are trying to drive to work or to the supermarket. Conscious decision making often occurs at these choice points, except when the task is overlearned (Mandler, 1975). We modeled a simple form of a choice point situation. We trained a module to output sequences of states, e.g., ABCD or EFGH, where states were represented by unique activity patterns over a set of units. If the sequences shared no elements, then presenting the first element of any sequence would be sufficient to regenerate the sequence. But when sequences overlap, choice points are created. For example, with the sequences ABCD and AEFG, state A can be followed by either B or E. We show that cleanup allows the module to make a decision and complete one of the two sequences. Figure 6 shows the operation of the module with and without cleanup following presentation of an A after training on the sequence pair ABCD and AEFG. There is one curve for each state, showing the activation of that state (defined as before), as a function of time. When the network is run with cleanup, although both states Band E are initially partially activated, the cleanup process maps this ill-formed state to state B, and the network then correctly completes the sequence ABCD. Without cleanup, the initial activation of states Band E causes a blending of the two sequences ABCD and AEFG and the state degenerates. 3 Although the arithmetic and choice point tasks seem simple in part because we predefined the set of well-formed states. However, because the architecture segre3In this simulation, we are not modeling the role of context in helping to select one sequence or another; we are simply assuming that either sequence is valid in the current context. The nature of the model does not change when we consider context. Assuming that the domain is not highly overlearned, the context will not strongly evoke one alternative action or the other in the feedforward mapping, leading to partial activation of multiple states, and the cleanup process will be needed to force a decision. On the Computational Utility of Consciollsness Activity of state 0 .? 17 stateB __ state C - - - -state D ........ state E o. Figure 6: Decision-point task with and without cleanup gates knowledge of well-formedness from knowledge of how to solve problems in the domain, well-formedness could be learned simultaneously with, or prior to learning the task. One could imagine training the cleanup network to autoassociate states it observes in the domain before or during training using an unsupervised or self-supervised procedure. 4 COMPUTATIONAL CONSEQUENCES OF PERSISTENT STATES In a network of modules, a persistent well-formed state in one module exerts larger influences on the state of other modules than do transient or ill-formed states. As a result the dynamics of the system tends to be dominated by well-formed persistent states. We show this in a final simulation. The network consisted of two modules, A and B, connected in a simple feedforward cascade. Each module's cleanup net was trained to have ten attractors, locally represented in a winner-take-all pool of units. The mapping network of module B was trained to map the attractors of module A to attractors in B in a one-to-one fashion. Thus, state a1 in module A is mapped to {31 in module A, a2 to {32, etc. Module B was initialized to a well-formed state {31, and the output state of module A was varied, creating three conditions. In the persistent well-formed condition, module A was clamped in the well-formed state a2 for 50 time steps. In the transient well-formed condition, module A was clamped in state a2 for only 30 time steps. And in the ill-formed condition, module A was clamped in an ill-formed state in which two states, a2 and a3, were both partially active. Figure 7 shows the subsequent activation of state {32 in module B as a function of time. Module B undergoes a transition from state {31 to state {32 only in the persistent well-formed condition. This indicates that the conjunction of well-formedness and persistence is required to effect a transition from one state to another. 5 CONCLUSIONS Our computational framework and simulation results suggest the following answers to our three key questions: 18 Donald Mathis, Michael C. Mozer well-formed, persistent state <X2 in module A Activity of state ~2 in module B well-formed, transient state <X2 in module A 0' ill-formed, persistent state in module A Figure 7: Well-formeclness and persistence are both required for attractor transitions. ? In order to reach consciousness, the output of a module must be both persistent and semantically well-formed, and must not initiate an overlearned process. ? The computational consequences of conscious (persistent) representations include exerting larger influences on the cognitive system, resulting in increased ability to drive response processes such as verbal report. ? The computational utility of consciousness in our model lies in the ability of cleanup to "focus" cognition, by keeping the system close to states which are semantically meaningful. Because the system has learned to process such states, performance is improved. References Baars, B. J. (1988) A Cognitive Theory of Con8ciou&ne8ll, Cambridge University Press. Crick, F. (1994) The astonishing hypothesis: The scientific search for the soul. Scribner. Crick, F., & Koch, C. (1990) Towards a neurobiological theory of consciousness. Sem. Neuro., 2: 263-275 Damasio, A. (1989) The brain binds entities and events by multiregional activation from convergence zones. Neural Computation, 1, 123-132 Fodor, J. A. (1983) The modularity of mind: An e8llay on faculty p8ychology. Cambridge, MA: MIT Press. Hinton, G. E., & Shallice, T. (1991) Lesioning an attractor network: Investigations of acquired dyslexia., P8ych. Rev., 98: 74-95 Mandler, G. (1975) Consciousness: Respectable, useful and probably necessary. In Information Processing and Cognition, The Loyola Symposium, R. Solso (Ed.). Erlbaum. Mozer, M. C., & Behrmann, M. (1990). On the interaction of spatial attention and lexical knowledge: A connectionist account of neglect dyslexia. Cognitive Neuro8Cience, 2, 96-123. Plaut, D. C., & Shallice, T. (1993) Perseverative and semantic influences on visual object naming errors in optic aphasia: A connectionist account. J. Cog. Neuro.,5(1): 89-117 Posner, M. 1., & Carr, T. (1992) Lexical access and the brain: Anatomical constraints on cognitive models of word recognition. American Journal of P8ychology, 105(1): 1-26 Rumelhart, D. E., Smolensky, P., McClelland, J. L., & Hinton, G. E. (1986) Schemata and sequential thought in PDP models. In J. L. McClelland & D. E. Rumelhart (Eds.), Parallel Di8tributed Proceuing, Vol. 2. Cambridge, MA: MIT Press. Smolensky, P. (1988) On the proper treatment of connectionism. Brain Behav. Sci., 11: 1-74 

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A Convolutional Neural Network Hand Tracker Steven J. Nowlan Synaptics, Inc. 2698 Orchard Parkway San Jose, CA 95134 nowlan@synaptics .com John C. Platt Synaptics, Inc. 2698 Orchard Parkway San Jose, CA 95134 platt@synaptics.com Abstract We describe a system that can track a hand in a sequence of video frames and recognize hand gestures in a user-independent manner. The system locates the hand in each video frame and determines if the hand is open or closed. The tracking system is able to track the hand to within ?10 pixels of its correct location in 99.7% of the frames from a test set containing video sequences from 18 different individuals captured in 18 different room environments. The gesture recognition network correctly determines if the hand being tracked is open or closed in 99.1 % of the frames in this test set . The system has been designed to operate in real time with existing hardware. 1 Introduction We describe an image processing system that uses convolutional neural networks to locate the position of a (moving) hand in a video frame, and to track the position of this hand across a sequence of video frames. In addition , for each frame, the system determines if the hand is currently open or closed. The input to the system is a sequence of black and white, 320 by 240 pixel digitized video frames. We designed the system to operate in a user-independent manner, using video frames from indoor scenes with natural clutter and variable lighting conditions. For ease of hardware implementation, we have restricted the system to use only convolutional networks and simple image filtering operations, such as smoothing and frame differencing. 902 Steven J. Nowlan, John C. PIau Figure 1: Average over all examples of each of the 10 classes of handwritten digits, after first aligning all of the examples in each class before averaging. Our motivation for investigating the hand tracking problem was to explore the limits of recognition capability for convolutional networks. The structure of convolutional networks makes them naturally good at dealing with translation invariance, and with coarse representations at the upper layers, they are also capable of dealing with some degree of size variation. Convolutional networks Qave been successfully applied to machine print OCR (Platt et aI, 1992), machine print address block location (Wolf and Platt, 1994), and hand printed OCR (Le Cun et aI, 1990; Martin and Rashid, 1992). In each of these problems, convolutional networks perform very well on simultaneously segmenting and recognizing two-dimensional objects. In these problems, segmentation is often the most difficult step, and once accomplished the classification is simplified. This can be illustrated by examining the average of all of the examples for each class after alignment and scaling. For the case of hand-printed OCR (see Fig. 1), we can see that the average of all of the examples is quite representative of each class, suggesting that the classes are quite compact, once the issue of translation invariance has been dealt with. This compactness makes nearest neighbor and non-linear template matching classifiers reasonable candidates for good performance. If you perform the same trick of aligning and averaging the open and closed hands from our training database of video sequences, you will see a quite different result (Fig. 2). The extreme variability in hand orientations in both the open and closed cases means that the class averages, even after alignment, are only weakly characteristic of the classes of open and closed hands. This lack of clean structure in the class average images suggested that hand tracking is a challenging recognition problem. This paper examines whether convolutional networks are extendable to hand tracking, and hence possibly to other problems where classification remains difficult even after segmentation and alignment. 2 System Architecture The overall architecture of the system is shown in Fig. 3. There are separate hand tracking and gesture recognition subsystems. For the hand tracking subsystem, each video frame is first sub-sampled and then the previous video frame (stored) is subtracted from the current video frame to produce a difference frame. These difference frames provide a crude velocity signal to the system, since the largest signals in the difference frames tend to occur near objects that are moving (Fig. 5). Independent predictions of hand locations are made by separate convolutional networks, which look at either the intensity frame or the difference frame. A voting scheme then combines the predictions from the intensity and difference networks along with predictions based on the hand trajectory computed from 3 previous frames. A Convolutional Neural Network Hand Tracker 903 Figure 2: Average over all examples of open and closed hands from the database of training video sequences, after first aligning all of the examples in each class before averaging. Previous Subsampled Video Frame Current Video Frame Difference Hanel Locator Network Difference Frame Intensity 1-'----"'" Hand Locator Position Hypotheses Network Position in Previous Frames Voting Procedure Hand PosHlan ~ Frame Cropping ,.. Open/Close Network , Is Hand Open? Figure 3: Architecture of object recognition system for hand tracking and openversus-closed hand identification. 

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Efficient Methods for Dealing with Missing Data in Supervised Learning Volker '!'resp? Siemens AG Central Research Otto-Hahn-Ring 6 81730 Miinchen Germany Ralph Neuneier Siemens AG Central Research Otto-Hahn-Ring 6 81730 Miinchen Germany Subutai Ahmad Interval Research Corporation 1801-C Page Mill R<;l. Palo Alto, CA 94304 Abstract We present efficient algorithms for dealing with the problem of missing inputs (incomplete feature vectors) during training and recall. Our approach is based on the approximation of the input data distribution using Parzen windows. For recall, we obtain closed form solutions for arbitrary feedforward networks. For training, we show how the backpropagation step for an incomplete pattern can be approximated by a weighted averaged backpropagation step. The complexity of the solutions for training and recall is independent of the number of missing features. We verify our theoretical results using one classification and one regression problem. 1 Introduction The problem of missing data (incomplete feature vectors) is of great practical and theoretical interest. In many applications it is important to know how to react if the available information is incomplete, if sensors fail or if sources of information become A.t the time of the research for this paper, a visiting researcher at the Center for Biological and Computational Learning, MIT. E-mail: Volker.Tresp@zfe.siemens.de 690 VoLker Tresp. RaLph Neuneier. Subutai Ahmad unavailable. As an example, when a sensor fails in a production process, it might not be necessary to stop everything if sufficient information is implicitly contained in the remaining sensor data. Furthermore, in economic forecasting, one might want to continue to use a predictor even when an input variable becomes meaningless (for example, due to political changes in a country). As we have elaborated in earlier papers, heuristics such as the substitution of the mean for an unknown feature can lead to solutions that are far from optimal (Ahmad and Tresp, 1993, Tresp , Ahmad, and Neuneier, 1994). Biological systems must deal continuously with the problem of unknown uncertain features and they are certainly extremely good at it. From a biological point of view it is therefore interesting which solutions to this problem can be derived from theory and if these solutions are in any way related to the way that biology deals with this problem (compare Brunelli and Poggio, 1991). Finally, having efficient methods for dealing with missing features allows a novel pruning strategy: if the quality of the prediction is not affected if an input is pruned, we can remove it and use our solutions for prediction with missing inputs or retrain the model without that input (Tresp, Hollatz and Ahmad, 1995). In Ahmad and Tresp (1993) and in Tresp, Ahmad and Neuneier (1994) equations for training and recall were derived using a probabilistic setting (compare also Buntine and Weigend, 1991, Ghahramani and Jordan, 1994). For general feedforward neural networks the solution was in the form of an integral which has to be approximated using numerical integration techniques. The computational complexity of these solutions grows exponentially with the number of missing features. In these two publications, we could only obtain efficient algorithms for networks of normalized Gaussian basis functions. It is of great practical interest to find efficient ways of dealing with missing inputs for general feedforward neural networks which are more commonly used in applications. In this paper we describe an efficient approximation for the problem of missing information that is applicable to a large class of learning algorithms, including feedforward networks. The main results are Equation 2 (recall) and Equation 3 (training). One major advantage of the proposed solution is that the complexity does not increase with an increasing number of missing inputs. The solutions can easily be generalized to the problem of uncertain (noisy) inputs. 2 2.1 Missing Information During Recall Theory We assume that a neural network N N(x) has been trained to predict E(ylx), the expectation of y E !R given x E ~. During recall we would like to know the (XC, XU) where XC network's prediction based on an incomplete input vector x denotes the known inputs and XU the unknown inputs. The optimal prediction given the known features can be written as (Ahmad and Tresp, 1993) = Efficient Methods for Dealing with Missing Data in Supervised Learning 691 u ~=X x;~,a) Yi o Xl: . ? 00 : o 000 o o X c Figure 1: The circles indicate 10 Gaussians approximating the input density distribution. XC = Xl indicates the known input, X2 = XU is unknown. Similarly, for a network trained to estimate class probabilities, N Ni(X) ~ P(classilx), simply substitute P(classdx C) for E(ylx C ) and N Ni(X C , XU) for N N (XC, XU) in the last equation. The integrals in the last equations Can be problematic. In the worst case they have to be approximated numerically (Tresp, Ahmad and Neuneier, 1994) which is costly, since the computation is exponential in the number of missing inputs. For networks of normalized Gaussians, there exist closed form solutions to the integrals (Ahmad and Tresp, 1993). The following section shows how to efficiently approximate the integral for a large class of algorithms. 2.2 An Efficient Approximation Parzen windows are commonly used to approximate densities. Given N training data {(xk, yk)lk = 1, ... , N}, we can approximate 1 N P(x) ~ N L:G(x;xk,O') (1) k=l where k 11k 2 G( X; X ,0') = (211'0'2)D /2 exp( - 20'211x - X II ) is a multidimensional properly normalized Gaussian centered at data xk with variance 0'2. It has been shown (Duda and Hart (1973)) that Parzen windows approximate densities for N - 00 arbitrarily well, if 0' is appropriately scaled. 692 Volker Tresp, Ralph Neuneier, Subutai Ahmad Using Parzen windows we may write where we have used the fact that and where G( xc; xc,le, u) is a Gaussian projected onto the known input dimensions (by simply leaving out the unknown dimensions in the exponent and in the normalization, see Ahmad and Tresp, 1993). xc,le are the components of the training data corresponding to the known input (compare Figure 1). Now, if we assume that the network prediction is approximately constant over the "width" of the Gaussians, u, we can approximate J NN(xC, x") G(XC,x";xle,u) dx" ~ NN(xC,x",Ie) G(XC;xc,le,u) where N N(x C, x",Ie) is the network prediction which we obtain if we substituted the corresponding components of the training data for the unknown inputs. With this approximation, (2) Interestingly, we have obtained a network of normalized Gaussians which are centered at the known components of the data points. The" output weights" N N(xC, x",Ie) consist of the neural network predictions where for the unknown input the corresponding components of the training data points have been substituted. Note, that we have obtained an approximation which has the same structure as the solution for normalized Gaussian basis functions (Ahmad and Tresp, 1994). In many applications it might be easy to select a reasonable value for u using prior knowledge but there are also two simple ways to obtain a good estimate for u using leave-one-out methods. The first method consists of removing the k - th pattern from the training data and calculating P(x le ) ~ N:l L:f:l,l# G(x le ; xl, u). Then select the u for which the log likelihood L:1e log P(x le ) is maximum. The second method consists of treating an input of the k - th training pattern as missing and then testing how well our algorithm (Equation 2) can predict the target. Select the u which gives the best performance. In this way it would even be possible to select input-dimension-specific widths Ui leading to "elliptical", axis-parallel Gaussians (Ahmad and Tresp, 1993). Efficient Methods for Dealing with Missing Data in Supervised Learning 693 Note that the complexity of the solution is independent of the number of missing inputs! In contrast, the complexity of the solution for feedforward networks suggested in Tresp, Ahmad and Neuneier (1994) grows exponentially with the number of missing inputs. Although similar in character to the solution for normalized RBFs, here we have no restrictions on the network architecture which allows us to choose the network most appropriate for the application. If the amount of training data is large, one can use the following approximations: ? Select only the K nearest data points. The distance is determined based on the known inputs. K can probably be reasonably small ? 10). In the extreme case, K = 1 and we obtain a nearest-neighbor solution. Efficient tree-based algorithms exist for computing the K-nearest neighbors. ? Use Gaussian mixtures instead of Parzen windows to estimate the input data distribution . Use the centers and variances of the components in Equation 2. ? Use a clustering algorithm and use the cluster centers instead of the data points in Equation 2. Note that the solution which substitutes the components of the training data closest to the input seems biologically plausible. 2.3 Experimental Results We tested our algorithm using the same data as in Ahmad and Tresp, 1993. The task was to recognize a hand gesture based on its 2D projection. As input, the classifier is given the 2D polar coordinates of the five finger tip positions relative to the 2D center of mass of the hand (the input space is therefore 10-D). A multi-layer perceptron was trained on 4368 examples (624 poses for each gesture) and tested on a similar independent test set. The inputs were normalized to a variance of one and u was set to 0.1. (For a complete description of the task see (Ahmad and Tresp, 1993).) As in (Ahmad & Tresp, 1993) we defined a correct classification as one in which the correct class was either classified as the most probable or the second most probable. Figure 2 shows experimental results. On the horizontal axis, the number of randomly chosen missing inputs is shown. The continuous line shows the performance using Equation 2 where we used only the 10 nearest neighbors in the approximation. Even with 5 missing inputs we obtain a score of over 90 % which is slightly better than the solution we obtained in Ahmad and Tresp (1993) for normalized RBFs. We expect our new solution to perform very well in general since we can always choose the best network for prediction and are not restricted in the architecture. As a benchmark we also included the case where the mean of the missing input was substituted. With 5 missing inputs, the performance is less than 60 %. 694 Volker Tresp. Ralph Neuneier. Subutai Ahmad 3D - Hanel Geetu", R8CC91ftIOn : 10 inpuIa ----.---- 0.9 '. 0.8 '. !II. !.7 .... ?.. . " . 0.8 " . .?... ". " . .?.. ...... 0.5 2 3 4 5 8 runber 01 rristing Inputs ...., 7 Figure 2: Experimental results using a generalization data set. The continuous line indicates the performance using our proposed method. The dotted lines indicate the performance if the mean of the missing input variable is substituted. As a comparison, we included the results obtained in Ahmad and Tresp (1993) using the closed-form solution for RBF -networks (dashed). 3 Training (Backpropagation) For a complete pattern (x.l:, yk), the weight update of a backpropagation step for weight Wj is k A ? (k _ NN.w ( x k))8NN U.w 8 w (x ) . J (X Y Wj Using the approximation of Equation 1, we obtain for an incomplete data point (compare Tresp, Ahmad and Neuneier, 1994) (3) Here, IE compl indicates the sum over complete patterns in the training set, and (111 is the standard deviation of the output noise. Note that the gradient is a network of normalized Gaussian basis functions where the "output-weight" is now Efficient Methods for Dealing with Missing Data in Supervised Learning 695 The derivation of the last equation can be found in the Appendix. Figure 3 shows experimental results. Boston housing data: 13 inputs 0.15r------,r-------,r------,--~--__r--__._--"""T""--...., 0.14 0.13 sub6titute mean 0.12 1 B0.11 28 CQI11)Iete pattems ..... --------------------~~-------------- 0.1 0.09 0.08 O?061'----2'------J3'------J4'-----L5--~6-----'-7----'-8----'9 number of missing inputs Figure 3: In the experiment, we used the Boston housing data set, which consists of 506 samples. The task is to predict the housing price from 13 variables which were thought to influence the housing price in a neighborhood. The network (multilayer perceptron) was trained with 28 complete patterns plus an additional 225 incomplete samples. The horizontal axis indicates how many inputs were missing in these 225 samples. The vertical axis shows the generalization performance. The continuous line indicates the performance of our approach and the dash-dotted line indicates the performance, if the mean is substituted for a missing variable. The dashed line indicates the performance of a network only trained with the 28 complete patterns. 4 Conclusions We have obtained efficient and robust solutions for the problem of recall and training with missing data. Experimental results verified our method. All of our results can easily be generalized to the case of noisy inputs. Acknowledgement Valuable discussions with Hans-Georg Zimmermann, Tomaso Poggio, Michael Jordan and Zoubin Ghahramani are greatfully acknowledged. The first author would like to thank the Center for Biological and Computational Learning (MIT) for providing and excellent research environment during the summer of 1994. 696 5 Volker Tresp, Ralph Neuneier, Subutai Ahmad Appendix Assuming the standard signal-plus-Gaussian-noise model we obtain for a complete sample P(xk,ykl{wi}) =G(yk;NNw(xk),(1y) P(x k) where {Wi} is the set of weights in the network. For an incomplete sample Using the same approximation as in Section 2.2, p(xc,k, ykl{ W;}) ~ L G(yk; N Nw(xc,k, xu,I), (1y) G(xc,k; xc, I, (1) l?compl where I sums over all complete samples. As before, we substitute for the missing components the ones from the complete training data. The log-likelihood C (a function of the network weights {Wi}) can be calculated as (x k can be either complete or incomplete) C Lf=llog p(xk, yk I{ Wi}). The maximum likelihood solution consists of finding weights {Wi} which maximize the log-likelihood. Using the approximation of Equation 1, we obtain for an incomplete sample as gradient Equation 3 (compare Tresp, Ahmad and Neuneier, 1994). = References Ahmad, S. and Tresp, V. (1993). Some Solutions to the Missing Feature Problem in Vision. In S. J. Hanson, J. D. Cowan and C. L. Giles, (Eds.), Advances in Neural Information Processing Systems 5, San Mateo, CA: Morgan Kaufmann . Brunelli, R. and Poggio, T. (1991). HyperBF Networks for Real Object Recognition. IJCAL Buntine, W. L. and Weigend, A. S. (1991). Bayesian Back-Propagation. Complex systems, Vol. 5, pp. 605-643. Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis. Wiley and Sons, New York. John Ghahramani, Z. and Jordan, M. 1. (1994) . Supervised Learning from Incomplete Data via an EM approach. In: Cowan, J. D., Tesauro, G., and Alspector, J., eds., Advances in Neural Information Processing Systems 6, San Mateo, CA, Morgan Kaufman. Tresp, V., Ahmad, S. and Neuneier, R. (1994). Training Neural Networks with Deficient Data. In: Cowan, J. D., Tesauro, G., and Alspector, J., eds., Advances in Neural Information Processing Systems 6, San Mateo, CA, Morgan Kaufman. Tresp, V., Hollatz, J. and Ahmad, S. (1995). Representing Probabilistic Rules with Networks of Gaussian Basis Functions. Accepted for publication in Machine Learning. 

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Grammar Learning by a Self-Organizing Network Michiro Negishi Dept. of Cognitive and Neural Systems, Boston University 111 Cummington Street Boston, MA 02215 email: negishi@cns.bu.edu Abstract This paper presents the design and simulation results of a selforganizing neural network which induces a grammar from example sentences. Input sentences are generated from a simple phrase structure grammar including number agreement, verb transitivity, and recursive noun phrase construction rules. The network induces a grammar explicitly in the form of symbol categorization rules and phrase structure rules. 1 Purpose and related works The purpose of this research is to show that a self-organizing network with a certain structure can acquire syntactic knowledge from only positive (i.e. grammatical) data, without requiring any initial knowledge or external teachers that correct errors. There has been research on supervised neural network models of language acquisition tasks [Elman, 1991, Miikkulainen and Dyer, 1988, John and McClelland, 1988]. Unlike these supervised models, the current model self-organizes word and phrasal categories and phrase construction rules through mere exposure to input sentences, without any artificially defined task goals. There also have been self-organizing models of language acquisition tasks [Ritter and Kohonen, 1990, Scholtes, 1991]. Compared to these models, the current model acquires phrase structure rules in more explicit forms, and it learns wider and more structured contexts, as will be explained below. 2 Network Structure and Algorithm The design of the current network is motivated by the observation that humans have the ability to handle a frequently occurring sequence of symbols (chunk) as an unit of information [Grossberg, 1978, Mannes, 1993]. The network consists of two parts: classification networks and production networks (Figure 1). The classification networks categorize words and phrases, and the production networks 28 Michiro Negishi evaluate how it is likely for a pair of categories to form a phrase. A pair of combined categories is given its own symbol, and fed back to the classifiers. After weights are formed, the network parses a sentence as follows. Input words are incrementally added to the neural sequence memory called the Gradient Field [Grossberg, 1978] (GF hereafter). The top (i.e. most recent) two symbols and the lookahead token are classified by three classification networks. Here a symbol is either a word or a phrase, and the lookahead token is the word which will be read in next. Then the lookahead token and the top symbol in the GF are sent to the right production network, and the top and the second ones are sent to the left production network. If the latter pair is judged to be more likely to form a phrase, the symbol pair reduces to a phrase, and the phrase is fed back to the GF after removing the top two symbols. Otherwise, the lookahead token is added to the sequence memory, causing a shift in the sequence memory. If the input sentence is grammatical, the repetition of this process reduces the whole sentence to a single "5" (sentence) symbol. The sequence of shifts and reductions (annoted with the resultant symbols) amounts to a parse of the sentence. During learning, the operations stated above are carried out as weights are gradually formed. In classification networks, the weights record a distribution pattern with respect to each symbol. That is, the weights record the co-occurrence of up to three adjacent symbols in the corpus. An symbol is classified in terms of this distribution in the classification networks. The production networks keep track of the categories of adjacent symbols. If the occurrence of one category reliably predicts the next or the previous one, the pair of categories forms a phrase, and is given the status of an symbol which is treated just like a word in the sentence. Because the symbols include phrases, the learned context is wider and more structured than the mere bigram, as well as the contexts utilized in [Ritter and Kohonen, 1990, Scholtes, 1991]. 3 Simulation 3.1 The Simulation Task The grammar used to generate input sentences (Table 3) is identical to that used in [Elman,1991], except that it does not include optionally transitive verbs and proper nouns. Lengths of the input sentences are limited to 16 words. To determine the completion of learning, after accepting 200 consecutive sentences with learning, learning is suppressed and other 200 sentences are processed to see if all are accepted. In addition, the network was tested for 44 ungrammatical sentences to see that they are correctly rejected. Ungrammatical sentences are derived by hand from randomly generated grammatical sentences. Parameters used in the simulation are : number of symbol nodes = 30 (words) + 250 (phrases), number of category nodes = ISO, f. = 10- 9, 'Y = 0.25, p = 0.65, 0'1 = 0.00005, /31 = 0.005, f32 = 0.2, 0'3 = 0.0001, /33 = 0.001, and T = 4.0. Grammar Learning by a Self-Organizing NeMork 29 3.2 Acquired Syntax Rules Learning was completed after learning 19800 grammatical sentences. Tables 1 and 2 show the acquired syntax rules extracted from the connection weights. Note that category names such as Ns, VPp, are not given a priori, but assigned by the author for the exposition. Only rules that eventually may reach the "S"(sentence) node are shown. There were a small number of uninterpretable rules, which are marked I/?". These rules might disturb normal parsing for some sentences, but they were not activated while testing for 200 sentences after learning. 3.3 Discussion Recursive noun phrase structures should be learned by finding equivalences of distribution between noun phrases and nouns. However, nouns and noun phrases have the same contextual features only when they are in certain contexts. An examination of the acquired grammar reveals that the network finds equivalence of features not of ''N'' and ''N RC" (where RC is a relative clause) but of "N V" and ''N RC V" (when ''N RC" is subjective), or "V N" and /IV NRC" (when ''N RC" is objective). As an example, let us examine the parsing of the sentence [19912] below. The rule used to reduce FEEDS CATS WHO UVE (''V NRC") is PO, which is classified as category C4, which includes P121 (''V N") where V are the singular forms of transitive verbs, and also includes the ''V'' where V are singular forms of intransitive verbs. Thus, GIRL WHO FEEDS CATS WHO UVE is reduced to GIRL WHO "VPsingle". ***[19912}***********??**********??*******??************ ?? *******.**** +---141--- + I +---88------+ I I +---206------+ I I I +----0----+ I I I I +-219-+ I I +-41-+ I +-36-+ I BOYS CHASE GIRL WHO FEEDS CATS WHO LIVE ?Accepted? Top symbol was 77 4 Conclusion and Future Direction In this paper, a self-organizing neural network model of grammar learning was presented. A basic principle of the network is that all words and phrases are categorized by the contexts in which they appear, and that familiar sequence of categories are chunked. As it stands, the scope of the grammar used in the simulation is extremely limited. Also, considering the poverty of the actual learning environment, the learning of syntax should also be guided by the cognitive competence to comprehend the utterance situations and conversational contexts. However, being a self-organizing network, the current model offers a plausible model of natural language acquisition through mere exposures to only grammatical sentences, not requiring any external teacher or an explicit goal. 30 Miclziro Negislzi Table 1. Acquired categorization rules S := C4 :a C13 C16 C18 C20 C26 .:= ..- := C29 .- C30 := C32 .- C29 r NPs VPs ., I C30r??,1 enr~vpp?, LIVES I ALKS I POrvrsNpRC?' I P74 rvrsNs RC?, I P121 rvrsNs?' I P157 rvrsNpo, GIRL I DOG I CAT I BOY CHASE I FEED WHO CHASES I FEEDS BOYS I CATS I DOGS I GIRLS P93 r Ns RC VPs?' I P138 rNs VPs?' P2 vrp NP\.iPp ., I P94 r vrp N ., I P137 r??, WALK I LIVE I PI r~NpRC?' I P61 pNpo, I P88rvrpNs RC?, I P122 r vrp Ns?' r =rR?, = rvrs?' =rNp?' P41 rNsR?, .- P36 rNpR?, C69 .- C74 .- C77 .- C1l9 C122 C139 := .. rVps?' = /"Ns?, = rvrp?' .- C52 C56 C58 = rNPsvps?, wnere =r? 0, RCs RCp NPp NPs := ..- = = = = P28 r Ns vrs ., I P34 r Np vrVi I P68 r Ns RC so, I P147 rNpRCvrp?, P206 rNs R VPs?' P238 rNsRNvr?, P219 rNpRVpp I P249 rNpRNvr?, P141 rNpVPp?' I P217 rNpRCVpp?, Pl48 P243 PIO rvrs NPs VPs?' I P32 r vrs NPp VPp ? , 0' = rNvr?, rNsRCs?' = = rNpRCp?' = rNPpVPp?, =/"vrsNvr?, = rNsR vrsNvr?, = /" VPs' VPp's RVPs I RNvr RVPp I RNvr Np I NpRCp Ns I NsRCs r = rvpp?' Table 2. Acquired production rules PO P1 P2 PI0 P28 P32 P34 P36 P41 P61 P68 P74 P88 P93 P94 P121 P122 P137 P138 P141 P147 P148 P157 P206 P217 P219 P238 P243 P249 :=C20 rvrs?' :=C16 rvrp?' := C16 rvrp?, := C20 rvrs?' :=C13 rNs?' :=C20 rvrs?' :=C26 rNp?' := C26 ,. Np ., :=C13 rNs?' := C16 rvrp?' :=C69 rNsRCs?, :=C20 rvrs?' :=C16 rVTp?' := C69 rNsRCs?, := C16 rvrp?' := C20 r vrs ? , :=C16 rVTp?' := C122 r NsR vrsNVT?, :=C13 rNs?' := C26 rNp?' :=C74 rNp RCs?, := C20 rvrs :=C20 rvrs?' := C52 r Ns R ., := C74 rNp RCs?, := C56 r Np R ., :=C52rNsR?, := C52 r Ns R :=C56 rNpR?, 0' 0' 0' C74 rNpRCp C74rNpRCp?' en /"NPpVpp?' C29 rNPs VPs?' C20 rvrso, C77/"NPpVPp?' C16/"vrp?' C18 /"R?I C18 /"R?, C26 /"Np?1 C20 /"vrs?' C69 rNs RCs?, C69 r Ns RCs ., C4 r VPs?' C58/"Nvr?, C13/"Ns?, C13 rNs?' C32 /"VPp C4rVps?' C32 /"Vpp?, C16 rvrp?' C58/"Nvr?1 C26/"Np?' C4 rVPs?1 C32r Vpp?' C32r Vpp?' C58 rNvrol C119/"vrsNvr?, C58/"Nvr?, 0' = r vrs Np RCp ., = rvrpNpRCp?' = rvrpNPp VPp?' .. rvrsNPs VPs?' = rNsvrs?' = /"vrsNPpVPp?' = /"Npvrp?' =rNpR?, =rNsR?, =/"vrpNp?' = /"NsRCs vrs?' = vrs Ns RCs ., = rvrpNsRCs?' =r Ns RCs VPs ., = rvrpNvr?, = /"vrs Ns?1 = rvrpNs?, = /"??I = rNsvps, =/"NpVPp?' =rNp RCs vrp?' = rvrsNvr?, = rvrsNp?, = /"NsRVPs?' =r Np RCs VPp ., = /"NpRVPp?' =/"NsRNvr?, = /" (Ns R vrs N) vr ?1 = /"Np RNvr?, r ?, Grammar Learning by a Self-Organizing Network 3/ Acknowledgements The author wishes to thank Prof. Dan Bullock, Prof. Cathy Harris, Prof. Mike Cohen, and Chris Myers of Boston University for valuable discussions. This work was supported in part by the Air Force Office of Scientific Research (AFOSR F49620-92-J-0225). References [Elman, 1991] Elman, J. (1991). Distributed representations, simple recurrent networks, and grammatical structure. Machine Learning, 7. [Grossberg, 1978] Grossberg, S. (1978). A theory of human memory: Selforganization and performance of sensory-motor codes, maps, and plans. Progress in Theoretical Biology,S. (John and McClelland, 1988] John, M. F. S. and McClelland, J. L. (1988). Applying contextual constraints in sentence comprehension. In Touretzky, D. 5., Hinton, G. E., and Sejnowsky, T. J., editors, Proceedings of the Second Connectionist Models Summer School 1988, Los Altos, CA. Morgan Kaufmann Publisher, Inc. [Mannes, 1993] Mannes, C. (1993). Self-organizing grammar induction using a neural network model. In Mitra, J., Cabestany, J., and Prieto, A., editors, New Trends in Neural Computation: Lecture Notes in Computer Science 686. Springer Verlag, New York. [Miikkulainen and Dyer, 1988] Miikkulainen, R. and Dyer, M. G. (1988). Encoding input/output representations in connectionist cognitive systems. In Touretzky, D. D., Hinton, G. E., and Senowsky, T. J., editors, Proceedings of the Second Connectionist Models Summer School 1988, Los Altos, CA. Morgan Kauffman Publisher, Inc. [Ritter and Kohonen, 1990] Ritter, H. and Kohonen, T. (1990). Learning seman totopic maps from context. Proceedings of. ITCNN 90, Washington D.C., 1. [Scholtes, 1991] Scholtes, J. C. (1991). Unsupervised context learning in natural language processing. Proceedings of I[CNN Seattle 1991. Appendix A. Activation and learning equations A.l Classification Network Activities eGradient Field (1 ) where t is a discrete time, i is the symbol id. and Ii(t) is an input symbol. eInput Layer X1Ai(t) = O(2(XOi(t)-O(XOi(t)))), X1Bi(t) = O(XOi(t)), X1ci(t) = Ii(t+1) Where the suffix A ,B, and C the most recent, the next to most recent, and the lookahead symbols, respectively'. Weights in networks A, B, and C are identical. O(x) = 1 if x > 1.-2- M 1. 0 otherwIse J 32 Michiro Negishi Here M is the maximum number of symbols on the gradient field . ? Feature Layer X2;i =L Xl"j Wl"ji, X2;{1 = I(Xl;d(a+ 2: X2;j)), j X2"i = X2;{ /(a+ 2: X2;f) j j I(x) = 2/(1 + exp( -Tx)) - 1 where s is a suffix which is either A, B, or C and T is the steepness of the sigmoid function and a is a small positive constant. Table 4 shows the meaning of above suffix i . ? Category Layer X3pi I ={ 0 if i = min{jl2:k8X2"kW2"kj > p}, or if ? = min{jl2:k" X2"k W2"kj > p} & unreli =ja:r: {unrefJ} otherwise (2) Where p is the least match score required and ure Ii is an unreferenced count. A.2 Classification Learning .Feature Weights LlWl"ij = -alWl"ij +,81Xl i (X2"j - Wl"ij) where al is the forgetting rate, and ,81 is the learning rate . ? Categorization Weights { Ll W2"ij = /32X3"i(X2"i - W2"ij) if the node is selected by the first line of (2) W2"ij = X2"i if the node is selected by the second line of (2) where /32 is the learning rate. A.3 Production Network Activities .Mutual predictiveness = = X3Ai W3ij, X3B j W4 ji , = X3cjW4 ji , = X3Bi W3 ij, = X4ij XSji = X7ij X8ji The phrase identification number for a category pair (i, j) is given algOrithmically in the current version by a cash function cash(i, j). (i) Case in which 'Y 2:ij X6ij ~ 2:ij X9ij X10 j XOi(t ={ 1 0 if i = cash(I, J) Reduce where X6IJ =i]a:r: (X6ij ) otherwise + 1) = 0.5 * pop(pOp(XOi(t))) + XlO, pop(x) = 2(x - 8(x)) Case in which 'Y Eij X6ij < 2:ij X9ij Shift The next input symbol is added on the gradient field, as was expressed in (1). (ii) Grammar Learning by a Self-Organizing Network 33 A.4 Production Learning where X3 Ai and X3 Bj are nodes that receive the next to the most recent symbol i and the most recent symbol j, respectively. Shift I Reduce Controller Production Networks (predlctlveness evaluators) Classification ~ ~ Networks ... ~ (~ 00 eedback hrases ) 0 171o~_70v'ovO~I--+----.J Neural Sequence Me ory Lookahead Token Input words Figure 1. Block diagram of the network NP VP RC N N I NRC V[NP] whoNPV I whoVP boy I girl I cat I dog I boys I girls I cats 1 dogs V _ chase I feed I work I live I chases I feeds I works I lives urn agreement - Agreements between N and V within clause - Agreements between head Nand subordinate V (where a ro riate) er arguments - chase, feed -> require a direct object - walk, live -> preclude a direct object (Observed also for head/verb relations in relative clauses) Table 3. Grammar for generated sentences Table 4. Subfields in a feature layer 34 Michiro Negishi Category (The MOst RlI08nt) Category (The "Next to Most Recent) To the Production r---~------------~~------------------------------"'Ne~k X 3B Category (Lookahead) o C Xo (X,o> '----A--""I.."'::. Gradient Field Nonterm inal s (Reduction Results) From the Production Network Lookahead Terminals Terminals(lnputs) ... ABC ... - - - - -.... Classification Path c===J terminals - - -.. ~ Copy 'or Learning E:::;:;:@:?:~:@ nonterminals category _ prllYious next next to next Figure 2. Classification Network x '0 Nonterminal X 3C Xa V Xg + X7 The ext to Most Recent Category The Most Recent Category Figure 3. Production Network Lookahead Category 

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Recurrent Networks: Second Order Properties and Pruning Morten With Pedersen and Lars Kai Hansen CONNECT, Electronics Institute Technical University of Denmark B349 DK-2800 Lyngby, DENMARK emails:with.lkhansen@ei.dtu.dk Abstract Second order properties of cost functions for recurrent networks are investigated. We analyze a layered fully recurrent architecture, the virtue of this architecture is that it features the conventional feedforward architecture as a special case. A detailed description of recursive computation of the full Hessian of the network cost function is provided. We discuss the possibility of invoking simplifying approximations of the Hessian and show how weight decays iron the cost function and thereby greatly assist training . We present tentative pruning results, using Hassibi et al.'s Optimal Brain Surgeon, demonstrating that recurrent networks can construct an efficient internal memory. 1 LEARNING IN RECURRENT NETWORKS Time series processing is an important application area for neural networks and numerous architectures have been suggested, see e.g. (Weigend and Gershenfeld, 94). The most general structure is a fully recurrent network and it may be adapted using Real Time Recurrent Learning (RTRL) suggested by (Williams and Zipser, 89). By invoking a recurrent network, the length of the network memory can be adapted to the given time series, while it is fixed for the conventional lag-space net (Weigend et al., 90). In forecasting, however, feedforward architectures remain the most popular structures; only few applications are reported based on the Williams&Zipser approach. The main difficulties experienced using RTRL are slow convergence and 674 Morten With Pedersen, Lars Kai Hansen lack of generalization. Analogous problems in feedforward nets are solved using second order methods for training and pruning (LeCun et al., 90; Hassibi et al., 92; Svarer et al., 93). Also, regularization by weight decay significantly improves training and generalization. In this work we initiate the investigation of second order properties for RTRL; a detailed calculation scheme for the cost function Hessian is presented, the importance of weight decay is demonstrated, and preliminary pruning results using Hassibi et al.'s Optimal Brain Surgeon (OBS) are presented. We find that the recurrent network discards the available lag space and constructs its own efficient internal memory. 1.1 REAL TIME RECURRENT LEARNING The fully connected feedback nets studied by Williams&Zipser operate like a state machine, computing the outputs from the internal units according to a state vector z(t) containing previous external inputs and internal unit outputs. Let x(t) denote a vector containing the external inputs to the net at time t, and let y(t) denote a vector containing the outputs of the units in the net. We now arrange the indices on x and y so that the elements of z(t) can be defined as , k EI , kEU where I denotes the set of indices for which Zk is an input, and U denotes the set of indices for which Zk is the output of a unit in the net. Thresholds are implemented using an input permanently clamped to unity. The k'th unit in the net is now updated according to where Wkj denotes the weight to unit k from input/unit j and function of the k'th unit. "'0 is the activation When used for time series prediction, the input vector (excluding threshold) is usually defined as x(t) = [x(t), . .. , x(t - L + 1)] where L denotes the dimension of the lag space. One of the units in the net is designated to be the output unit Yo, and its activating function 10 is often chosen to be linear in order to allow for arbitrary dynamical range. The prediction of x(t + 1) is x(t + 1) = lo[so(t?). Also, if the first prediction is at t = 1, the first example is presented at t = 0 and we 'set y(O) = O. We analyse here a modification of the standard Williams&Zipser construction that is appropriate for forecasting purposes. The studied architecture is layered. Firstly, we remove the external inputs from the linear output unit in order to prevent the network from getting trapped in a linear mode. The output then reads x(t + 1) =Yo(t + 1) = L WojYj(t) + Wthres,o (1) jeU = = Since y(O) 0 we obtain a first prediction yielding x(l) Wthres,o which is likely to be a poor prediction, and thereby introducing a significant error that is fed back into the network and used in future predictions. Secondly, when pruning Recurrent Networks: Second Order Properties and Pruning 675 a fully recurrent feedback net we would like the net to be able to reduce to a simple two-layer feedforward net if necessary. Note that this is not possible with the conventional Williams&Zipser update rule, since it doesn't include a layered feedforward net as a special case. In a layered feedforward net the output unit is disconnected from the external inputs; in this case, cf. (1) we see that x(t + 1) is based on the internal 'hidden' unit outputs Yk(t) which are calculated on the basis of z(t - 1) and thereby x(t -1). Hence, besides the startup problems, we also get a two-step ahead predictor using the standard architecture. In order to avoid the problems with the conventional Williams&Zipser update scheme we use a layered updating scheme inspired by traditional feedforward nets, in which we distinguish between hidden layer units and the output unit. At time t, the hidden units work from the input vector zh(t) , , k EI kE U , k=O where I denotes the input indices, U denotes the hidden layer units and 0 the output unit. Further, we use superscripts hand 0 to distinguish between hidden unit and output units. The activation of the hidden units is calculated according to y~(t) = fr[s~(t)] = fr [ L Wki zJ (t)] , kEU (2) ie1uUuO The hidden unit outputs are forwarded to the output unit, which then sees the input vector zkCt) , kEU OCt) _ { y~(t) Zk - yO(t-1) k=O and is updated according to (3) The cost function is defined as C = E + wTRw. R is a regularization matrix, w is the concatenated set of parameters, and the sum of squared errors is 1 E T = 2 L[e(t)F , e(t) = x(t) - yO(t), (4) t=l where T is the size of the training set series. RTRL is based on gradient descent in the cost function, here we investigate accelerated training using Newton methods. For that we need to compute first and second derivatives of the cost function. The essential difficulty is to determine derivatives of the sum of squared errors: aE = _ {-.. e(t) ayO(t) aw?'3? L...J aw'3.. t=l (5) 676 Morten With Pedersen, Lars Kai Hansen The derivative of the output unit is computed as 8yO(t) -8Wij 8r[sO(t)] 8s0(t) ._8so(t) 8Wij (6) where 8s0(t) _ 1: . O(t) "" . 8yjl(t) -8-- - UO,Zj + L- WOJI 8 Wij j/EU Wij + woo 8 yO(t - 1) 8 Wij (7) where 6j k is the Kronecker delta. This expression contains the derivative of the hidden units (8) where (9) ... ... 132 132 <> <> ~!:'::.3-.a~.2:;:-5---:'.a.'::-2---:.a7..15'--.a-:':.1'--.a7..0:;:-5-~-::0~.05~-::-0.1;---::'!0.1?5 WEIGHT VAlUE ~.3 .a.25 .a.2 .a.15 .a.1 .a.OS WEIGHT VALUE 0.05 0.1 0.15 Figure 1: Cost function dependence of a weight connecting two hidden units for the sunspot benchmark series. Left panel: Cost function with small weight decay, the (local) optimum chosen is marked by an asterix. Right panel: The same slice through the cost function but here retrained with higher weight decay. The complexity of the training problem for the recurrent net using RTRL is demonstrated in figure 1. The important role of weight decay (we have used a simple weight decay R = at) in controlling the complexity of the cost function is evident in the right panel of figure 1. The example studied is the sunspot benchmark problem (see e.g. (Weigend et al., 90) for a definition). First, we trained a network with the small weight decay and recorded the left panel result. Secondly, the network was retrained with increased weight decay and the particular weight connecting two hidden units was varied to produce the right panel result. In both cases all other weights remained fixed at their optimal values for the given weight decay. In addition to the complexity visible in these one-parameter slices of the cost function, the cost function is highly anisotropic in weight space and consequently the network Hessian is ill-conditioned. Hence, gradient descent is hampered by slow con vergen ce. Recurrent Networks; Second Order Properties and Pruning 2 677 SECOND ORDER PROPERTIES OF THE COST FUNCTION To improve training by use of Newton methods and for use in OBS-pruning we compute the second derivative of the error functional: t 8 2E = _ [e(t) 8 2yO(t) _ 8 yO(t) . 8 yO (t)] 8Wij8wpq t=l 8Wij8wpq 8Wij 8wpq (10) The second derivative of the output is 2 8wij 8wpq 8 yO(t) _ --....:.....:.- 8 2 r[sO(t)] 8s0(t) 8s0(t) 8r[sO(t)] 8 2 SO(t) .- - .- - + . ----:...;....;..8so(t)2 8Wij 8wpq 8so(t) 8Wij8wpq (11) with 8 2so(t) _, 8zJ(t) ~ 8 2yj,(t) Ooi+ ~ Woj' 8 Wij 8 Wpq OWpq j'EU 8wij 8wpq + woo 8 2yO(t - 1) 8wij 8wpq 8z~(t) + Dop 8Wij (12) This expression contains the second derivative of the hidden unit outputs 8 2yi(t) _ 8 2 fr[si(t)] . 8si(t) . 8si(t) + 8fr[si(t)]. 02si(t) OWijOWpq - osi(t)2 OWij OWpq osi(t) OWijOWpq (13) with 02si(t) _ ozj(t) ~ 02yj,(t - 1) 02 yO(t - 1) oz~(t) (14) - Dki -0-- + L..J Wkj + Wko + Dkp 0 8WijOWpq Wpq j'EU OWijOWpq OWijOWpq Wij I Recursion in the five index quantity (14) imposes a significant computational burden; in fact the first term of the Hessian in (10), involving the second derivative, is often neglected for computational convenience (LeCun et al., 90). Here we start by analyzing the significance of this term during training. We train a layered architecture to predict the sunspot benchmark problem. In figure 2 the ratio between the largest eigenvalue of the second derivative term in (10) and the largest eigenvalue of the full Hessian is shown. The ratio is presented for two different magnitudes of weight decay. In line with our observations above the second order properties of the "ironed" cost function are manageable, and we can simplify the Hessian calculation by neglecting the second derivative term in (10), i.e., apply the Gauss-Newton approximation. 3 PRUNING BY THE OPTIMAL BRAIN SURGEON Pruning of recurrent networks has been pursued by (Giles and Omlin, 94) using a heuristic pruning technique, and significant improvement in generalization for a sequence recognition problem was demonstrated. Two pruning schemes are based on systematic estimation of weight saliency: the Optimal Brain Damage (OBD) scheme of (LeCun et al., 90) and OBS by (Hassibi et al., 93). OBD is based on the diagonal approximation of the Hessian and is very robust for forecasting (Svarer et al., 93). If an estimate of the full Hessian is available OBS can be used 678 Morten With Pedersen, Lars Kai Hansen 10' ..:.::.: .... :::::;:;:::... :::: . . '" 10...'----!-10--f::20--:30~----! ..:---~50--~---:::70:----=1O .. ITERATION. 1040''-----:'10--f::20--:30~----!..:---~50,----.. ~---:70~---:!IO ITERATION. Figure 2: Ratio between the largest magnitude eigenvalue of the second derivative term of the Hessian (c.f. equation (10)) and the largest magnitude .eigenvalue of the complete Hessian as they appeared during ten training sessions. The connected circles represent the average ratio. Left panel: Training with small weight decay. Right panel: Training with a high weight decay. for estimation of saliencies incorporating linear retraining. In (Hansen and With Pedersen, 94) OBS was generalized to incorporate weight decays; we use these modifications in our experiments. Note that OBS in its standard form only allows for one weight to be eliminated at a time. The result of a pruning session is a nested family of networks. In order to select the optimal network within the family it was suggested in (Svarer et al., 93.) to use the estimated test error. In particular we use Akaike's Final Prediction Error (Akaike, 69) to estimate the network test error Etest = ?(T + N)/(T - N? . 2E/T 1, and N is the number of parameters in the network. In figure 3 we show the results of such a pruning session on the sunspot data starting from a (4-4-1) network architecture. The recurrent network was trained using a damped Gauss-Newton scheme. Note that the training error increases as weights are eliminated, while the test error and the estimated test error both pass through shallow minima showing that generalization is slightly improved by pruning. In fact, by retraining the optimal architecture with reduced weight decay both training and test errors are decreased in line with the observations in (Svarer et al., 93). It is interesting to observe that the network, though starting with access to a lag-space of four delay units, has lost three of the delayed inputs; hence, rely solely on its internal memory, as seen in the right panel of. figure 3. To further illustrate the memory properties of the optimal network, we show in figure 4 the network response to a unit impulse. It is interesting that the response of the network extends for approximately 12 time steps corresponding to the "period" of the sunspot series. lThe use of Akaike's estimate is not well justified for a feedback net, test error estimates for feedback models is a topic of current research. Recurrent Networks: Second Order Properties and Pruning 679 OUTPUT 0.25 . ~0.'5 w 0.1 ,' : .. "': - .- - ... .~:.. __ .... ~ ..; .... ~ ....... ._~_ . __ ~ ...... ~~ .-: :7 0.05 ~~-='0--~'5~~~~~~~~~--~$~-~~~"~~ X(I-I) NUMSER OF PARAMETERS X(I-2) X(I-3) X(I-4) Figure 3: Left panel: OBS pruning of a (4-4-1) recurrent network trained on sunspot benchmark. Development of training error, test error, and Akaike estimated test error (FPE). Right panel: Architecture of the FPE-optimal network. Note that the network discards the available lag space and solely predicts from internal memory. 0.8 0.1 ! .. 0.5 0.4 ., 0.8 .-.. , .. 0.3 w ~ ... 0.2 II! 0.' .0.' .0.20 I . \ .,: V"'0 15 TIME ~ , \ : orf" t .0.8 ..\./ , .0.1 ... : " I .0.4 I I .' :\ I ,, ,, ,-- '0 TIME '5 ~ Figure 4: Left panel: Output of the pruned network after a unit impulse input at t = O. The internal memory is about 12 time units long which is, in fact, roughly the period of the sunspot series. Right panel: Activity of the four hidden units in the pruned network after a unit impUlse at time t = O. 4 CONCLUSION A layered recurrent architecture, which has a feedforward net as a special case, has been investigated. A scheme for recursive estimation of the Hessian of the fully recurrent neural net is devised . It's been shown that weight decay plays a decisive role when adapting recurrent networks. Further, it is shown that the' second order information may be used to train and prune a recurrent network and in this process the network may discard the available lag space. The network builds an efficient 680 Morten With Pedersen, Lars Kai Hansen internal memory extending beyond the lag space that was originally available. Acknowledgments We thank J an Larsen, Sara Solla, and Claus Svarer for useful discuss~ons, and Lee Giles for providing us with a preprint of (Giles and amlin, 94). We thank the anonymous reviewers for valuable comments on the manuscript. This research is supported by the Danish Natural Science and Technical Research Councils through the Computational Neural Network Center (CONNECT). References H. Akaike: Fitting Autoregressive M ode/s for Prediction. Ann. Inst. Stat. Mat. 21, 243-247, (1969). Y. Le Cun, J.S. Denker, and S.A. Solla: Optimal Brain Damage. In Advances in Neural Information Processing Systems 2, (Ed. D.S. Touretzsky) Morgan Kaufmann, 598-605, (1990). C.L. Giles and C.W. amlin: Pruning of Recurrent Neural Networks for Improved Generalization Performance. IEEE Transactions on Neural Networks, to appear. Preprint NEC Research Institute (1994). L.K. Hansen and M. With Pedersen: Controlled Growth of Cascade Correlation Nets, International Conference on Artificial Neural Networks ICANN'94 Sorrento. (Eds. M. Marinaro and P.G. Morasso) Springer, 797-801, (1994). B. Hassibi, D. G. Stork, and G. J. Wolff, Optimal Brain Surgeon and General Network Pruning, in Proceedings of the 1993 IEEE International Conference on Neural Networks, San Francisco (Eds. E.H. Ruspini et al. ) IEEE, 293-299 (1993). C. Svarer, L.K. Hansen, and J. Larsen: On Design and Evaluation of .Tapped Delay Line Networks, In Proceedings ofthe 1993 IEEE International Conference on Neural Networks, San Francisco, (Eds . E.H. Ruspini et al. ) 46-51, (1993). A.S . Weigend, B.A. Huberman, and D.E. Rumelhart: Predicting the future: A Connectionist Approach. Int. J. of Neural Systems 3, 193-209 (1990). A.S. Weigend and N.A. Gershenfeld, Eds.: Times Series Prediction: Forecasting the Future and Understanding the Past. Redwood City, CA: Addison-Wesley (1994). R.J. Williams and D. Zipser: A Learning Algorithm for Continually Running Fully Recurrent Neural Networks, Neural Computation 1, 270-280, (1989). 

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Comparing the prediction accuracy of artificial neural networks and other statistical models for breast cancer survival Harry B. Burke Department of Medicine New York Medical College Valhalla, NY 10595 David B. Rosen Department of Medicine New York Medical College Valhalla, NY 10595 Philip H. Goodman Department of Medicine University of Nevada School of Medicine Reno, Nevada 89520 Abstract The TNM staging system has been used since the early 1960's to predict breast cancer patient outcome. In an attempt to increase prognostic accuracy, many putative prognostic factors have been identified. Because the TNM stage model can not accommodate these new factors, the proliferation of factors in breast cancer has lead to clinical confusion. What is required is a new computerized prognostic system that can test putative prognostic factors and integrate the predictive factors with the TNM variables in order to increase prognostic accuracy. Using the area under the curve of the receiver operating characteristic, we compare the accuracy of the following predictive models in terms of five year breast cancer-specific survival: pTNM staging system, principal component analysis, classification and regression trees, logistic regression, cascade correlation neural network, conjugate gradient descent neural, probabilistic neural network, and backpropagation neural network. Several statistical models are significantly more ac- 1064 Harry B. Burke, David B. Rosen, Philip H. Goodman curate than the TNM staging system. Logistic regression and the backpropagation neural network are the most accurate prediction models for predicting five year breast cancer-specific survival 1 INTRODUCTION For over thirty years measuring cancer outcome has been based on the TNM staging system (tumor size, number of lymph nodes with metastatic disease, and distant metastases) (Beahr et. al., 1992). There are several problems with this model (Burke and Henson , 1993). First, it is not very accurate, for breast cancer it is 44% accurate. Second its accuracy can not be improved because predictive variables can not be added to the model. Third, it does not apply to all cancers. In this paper we compare computerized prediction models to determine if they can improve prognostic accuracy. Artificial neural networks (ANN) are a class of nonlinear regression and discrimination models. ANNs are being used in many areas of medicine, with several hundred articles published in the last year. Representative areas of research include anesthesiology (Westenskow et. al., 1992) , radiology (Tourassi et. al. , 1992) , cardiology (Leong and Jabri, 1982) , psychiatry (Palombo, 1992), and neurology (Gabor and Seyal, 1992). ANNs are being used in cancer research including image processing (Goldberg et. al., 1992) , analysis of laboratory data for breast cancer diagnosis (0 Leary et. al., 1992), and the discovery of chemotherapeutic agents (Weinstein et . al., 1992). It should be pointed out that the analyses in this paper rely upon previously collected prognostic factors. These factors were selected for collection because they were significant in a generalized linear model such as the linear or logistic models . There is no predictive model that can improve upon linear or logistic prediction models when the predictor variables meet the assumptions of these models and there are no interactions. Therefore he objective of this paper is not to outperform linear or logistic models on these data. Rather, our objective is to show that, with variables selected by generalized linear models, artificial neural networks can perform as well as the best traditional models . There is no a priori reason to believe that future prognostic factors will be binary or linear, and that there will not be complex interactions between prognostic factors. A further objective of this paper is to demonstrate that artificial neural networks are likely to outperform the conventional models when there are unanticipated nonmonotonic factors or complex interactions. 2 2.1 METHODS DATA The Patient Care Evaluation (PCE) data set is collected by the Commission on Cancer of the American College of Surgeons (ACS). The ACS, in October 1992, requested cancer information from hospital tumor registries in the United States. The ACS asked for the first 25 cases of breast cancer seen at that institution in 1983, and it asked for follow up information on each of these 25 patients through the date of the request. These are only cases of first breast cancer. Follow-up information included known deaths. The PCE data set contains, at best, eight year follow-up. Prediction Accuracy of Models for Breast Cancer Survival 1065 We chose to use a five year survival end-point. This analysis is for death due to breast cancer, not all cause mortality. For this analysis cases with missing data, and cases censored before five years, are not included so that the prediction models can be compared without putting any prediction model at a disadvantage. We randomly divided the data set into training, hold-out, and testing subsets of 3,100, 2,069, and 3,102 cases, respectively. 2.2 MODELS The TMN stage model used in this analysis is the pathologic model (pTNM) based on the 1992 American Joint Committee on Cancer's Manual for the Staging of Cancer (Beahr et. al., 1992). The pathologic model relies upon pathologically determined tumor size and lymph nodes, this contrasts with clinical staging which relies upon the clinical examination to provide tumor size and lymph node information. To determine the overall accuracy of the TNM stage model we compared the model's prediction for each patient, where the individual patient's prediction is the fraction of all the patients in that stage who survive, to each patient's true outcome. Principal components analysis, is a data reduction technique based on the linear combinations of predictor variables that minimizes the variance across patients (Jollie, 1982). The logistic regression analysis is performed in a stepwise manner, without interaction terms, using the statistical language S-PLUS (S-PLUS, 1992), with the continuous variable age modeled with a restricted cubic spline to avoid assuming linearity (Harrell et. al., 1988). Two types of Classification and Regression Tree (CART) (Breiman et. al., 1984) analyses are performed using S-PLUS. The first was a 9-node pruned tree (with 10-fold cross validation on the deviance), and the second was a shrunk tree with 13.7 effective nodes. The multilayer perceptron neural network training in this paper is based on the maximum likelihood function unless otherwise stated, and backpropagation refers to gradient descent. Two neural networks that are not multilayer perceptrons are tested. They are the Fuzzy ARTMAP neural network (Carpenter et. al., 1991) and the probabilistic neural network (Specht, 1990). 2.3 ACCURACY The measure of comparative accuracy is the area under the curve of the receiver operating characteristic (Az) . Generally, the Az is a nonparametric measure of discrimination. Square error summarizes how close each patient's predicted value is to its true outcome. The Az measures the relative goodness of the set of predictions as a whole by comparing the predicted probability of each patient with that of all other patients. The computational approach to the Az that employs the trapezoidal approximation to the area under the receiver operating characteristic curve for binary outcomes was first reported by Bamber (Bamber, 1975), and later in the medical literature by Hanley (Hanley and McNeil, 1982). This was extended by Harrell (Harrell et. al., 1988) to continuous outcomes. Harry B. Burke, David B. Rosen, Philip H. Goodman 1066 Table 1: PCE 1983 Breast Cancer Data: 5 Year Survival Prediction, 54 Variables. PREDICTION MODEL pTNM Stages Principal Components Analysis CART, pruned CART, shrunk Stepwise Logistic regression Fuzzy ARTMAP ANN Cascade correlation ANN Conjugate gradient descent ANN Probabilistic ANN Backpropagation ANN ACCURACY? SPECIFICATIONS .720 .714 .753 .762 .776 .738 .761 .774 .777 .784 O,I,I1A,I1B,IIIA,I1IB,IV one scaling iteration 9 nodes 13.7 nodes with cubic splines 54-F2a, 128-1 54-21-1 54-30-1 bandwidth = 16s 54-5-1 * The area under the curve of the receiver operating characteristic. 3 RESULTS All results are based on the independent variable sample not used for training (i.e., the testing data set), and all analyses employ the same testing data set. Using the PCE breast cancer data set, we can assess the accuracy of several prediction models using the most powerful of the predictor variables available in the data set (See Table 1). Principal components analysis is not expected to be a very accurate model; with one scaling iteration, its accuracy is .714. Two types of classification and regression trees (CART), pruned and shrunk, demonstrate accuracies of .753 and .762, respectively. Logistic regression with cubic splines for age has an accuracy of .776. In addition to the backpropagation neural network and the probabilistic neural network, three types of neural networks are tested. Fuzzy ARTMAP's accuracy is the poorest at .738. It was too computationally intensive to be a practical model. Cascade-correlation and conjugate gradient descent have the potential to do as well as backpropagation. The PNN accuracy is .777. The PNN has many interesting features, but it also has several drawbacks including its storage requirements. The backpropagation neural network's accuracy is .784.4. 4 DISCUSSION For predicting five year breast cancer-specific survival, several computerized prediction models are more accurate than the TNM stage system, and artificial neural networks are as good as the best traditional statistical models. References Bamber D (1975). The area above the ordinal dominance graph and the area below the receiver operating characteristic. J Math Psych 12:387-415. Beahrs OH, Henson DE, Hutter RVP, Kennedy BJ (1992). Manual for staging of Prediction Accuracy of Models for Breast Cancer Survival 1067 cancer, 4th ed. Philadelphia: JB Lippincott. Burke HB, Henson DE (1993) . Criteria for prognostic factors and for an enhanced prognostic system. Cancer 72:3131-5. Breiman L, Friedman JH, Olshen RA (1984). Classification and Regression Trees. Pacific Grove, CA: Wadsworth and Brooks/Cole. Carpenter GA, Grossberg S, Rosen DB (1991). Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system. Neural Networks 4:759-77l. Gabor AJ, M. Seyal M (1992) . Automated interictal EEG spike detection using artificial neural networks. Electroencephalogr Clin Neurophysiology 83 :271-80. Goldberg V, Manduca A, Ewert DL (1992). Improvement in specificity of ultrasonography for diagnosis of breast tumors by means of artificial intelligence. Med Phys 19:1275-8l. Hanley J A, McNeil BJ (1982). The meaning of the use of the area under the receiver operating characteristic (ROC) curve. Radiology 143:29-36. Harrell FE, Lee KL, Pollock BG (1988). Regression models in clinical studies: determining relationships between predictors and response. J Natl Cancer Instit 80:1198-1202. Jollife IT (1986). Principal Component Analysis. New York: Springer-Verlag, 1986. Leong PH, J abri MA (1982). MATIC - an intracardiac tachycardia classification system. PACE 15:1317-31,1982. O'Leary TJ, Mikel UV, Becker RL (1992). Computer-assisted image interpretation: use of a neural network to differentiate tubular carcinoma from sclerosing adenosis. Modern Pathol 5:402-5. Palombo SR (1992). Connectivity and condensation in dreaming. JAm Psychoanal Assoc 40:1139-59. S-PLUS (1991), v 3.0. Seattle, WA; Statistical Sciences, Inc. Specht DF (1990). Probabilistic neural networks. Neural Networks 3:109-18. Tourassi GD, Floyd CE, Sostman HD, Coleman RE (1993). Acute pulmonary embolism: artificial neural network approach for diagnosis. Radiology 189:555-58. Weinstein IN, Kohn KW, Grever MR et. al. (1992) Neural computing in cancer drug development: predicting mechanism of action. Science 258:447-51. Westenskow DR, Orr JA, Simon FH (1992) . Intelligent alarms reduce anesthesiologist's response time to critical faults. Anesthesiology 77:1074-9, 1992. 

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Convergence Properties of the K-Means Algorithms Leon Bottou Neuristique, 28 rue des Petites Ecuries, 75010 Paris, France leonCneuristique.fr Yoshua Bengio" Dept. LR.O. Universite de Montreal Montreal, Qc H3C-3J7, Canada bengioyCiro.umontreal.ca Abstract This paper studies the convergence properties of the well known K-Means clustering algorithm. The K-Means algorithm can be described either as a gradient descent algorithm or by slightly extending the mathematics of the EM algorithm to this hard threshold case. We show that the K-Means algorithm actually minimizes the quantization error using the very fast Newton algorithm. 1 INTRODUCTION K-Means is a popular clustering algorithm used in many applications, including the initialization of more computationally expensive algorithms (Gaussian mixtures, Radial Basis Functions, Learning Vector Quantization and some Hidden Markov Models). The practice of this initialization procedure often gives the frustrating feeling that K-Means performs most of the task in a small fraction of the overall time. This motivated us to better understand this convergence speed. A second reason lies in the traditional debate between hard threshold (e .g. KMeans, Viterbi Training) and soft threshold (e.g. Gaussian Mixtures, Baum Welch) algorithms (Nowlan, 1991) . Soft threshold algorithms are often preferred because they have an elegant probabilistic framework and a general optimization algorithm named EM (expectation-maximization) (Dempster, Laird and Rubin, 1977). In the case of a gaussian mixture, the EM algorithm has recently been shown to approximate the Newton optimization algorithm (Xu and Jordan, 1994). We prove in this "also, AT&T Bell Labs, Holmdel, NJ 07733 586 Leon Bottou, Yoshua Bengio paper that the corresponding hard threshold algorithm) K-Means) minimizes the quantization error using exactly the Newton algorithm. In the next section) we derive the K-Means algorithm as a gradient descent procedure. Section 3 extends the mathematics of the EM algorithm to the case of K-Means . This second derivation of K-Means provides us with proper values for the learning rates. In section 4 we show that this choice of learning rates optimally rescales the parameter space using Newton)s method. Finally) in section 5 we present and discuss a few experimental results comparing various versions of the K-Means algorithm. The 5 clustering algorithms presented here were chosen for a good coverage of the algorithms related to K-Means) but this paper does not have the ambition of presenting a literature survey on the subject. 2 K-MEANS AS A GRADIENT DESCENT Given a set of P examples (Xi) the K-Means algorithm computes k prototypes = (Wk) which minimize the quantization error, i.e.) the average distance between each pattern and the closest prototype: W E (W ) = def ""' L....J L ( Xi) W ) . t = L....J -1 ml.n( Xi . 2 t def ""' - Wk) 2 (1) Writing Si (w) for the subscript of the closest prototype to example Xi, we have (2) 2.1 GRADIENT DESCENT ALGORITHM We can then derive a gradient descent algorithm for the quantization error: ~W = - ( t 8~;:). This leads to the following batch update equation (updating prototypes after presenting all the examples) : (3) We can also derive a corresponding online algorithm which updates the prototypes after the presentation of each pattern Xi: A OL(Xi) w) _ u.W - -(t OW ) l.e.) if k = Si(W) otherwise. (4) The proper value of the learning rate (t remain to be specified in both batch and online algorithms. Convergence proofs for both algorithms (Bottou) 1991) exist for decreasing values of the learning rates satisfying the conditions E (t = 00 and E (; < 00. Following (Kohonen) 1989), we could choose (t = (o/t . We prove however in this paper that there exist a much better choice of learning rates. Convergence Properties of the K-Means Algorithms 3 3.1 587 K-MEANS AS AN EM STYLE ALGORITHM EM STYLE ALGORITHM The following derivation of K-Means is similar to the derivation of (MacQueen, 1967). We insist however on the identity between this derivation and the mathematics of EM (Liporace, 1976) (Dempster, Laird and Rubin, 1977). Although [{-Means does not fit in a probabilistic framework, this similarity holds for a very deep reason: The semi-ring of probabilies (!R+, +, x) and the idempotent semi-ring of hard-threshold scores (!R, Min, +) share the most significant algebraic properties (Bacceli, Cohen and Olsder, 1992). This assertion completely describes the similarities and the potential differences between soft-threshold and hard-threshold algorithms. A complete discussion however stands outside the scope of this paper. The principle of EM is to introduce additional "hidden" variables whose knowledge would make the optimization problem easier. Since these hidden variables are unknown, we maximize an auxiliary function which averages over the values of the hidden variables given the values of the parameters at the previous iteration. In our case, the hidden variables are .the assignments Si(W) of the patterns to the prototypes. Instead of considering the expected value over the distribution on these hidden variables, we just consider the values of the hidden variables that minimize the cost, given the previous values of the parameters: '"' 1 ( Q( WI ,W ) def = L...J '2 Xi - WI &.(w) )2 i The next step consists then in finding a new set of prototypes Wi which mllllmizes Q( Wi, w) where w is the previous set of prototypes. We can analytically compute the explicit solution of this minimization problem. Solving the equation 8Q(w' , W)/8W'k = 0 yields: 1 = 111 '"' (5) L...J Xi k i:k=&i(w) where Nk is the number of examples assigned to prototype Wk . The algorithm consists in repeatedly replacing w by Wi using update equation (6) until convergence. Since si(w' ) is by definition the best assignment of patterns Xi to the prototypes w~, we have the following inequality: wkI E(w' ) - Q(w' , w) = ~ L(Xi - Wi &.(w,))2 - (Xi - Wi Si(W))2 ::; 0 i Using this result, the identity E(w) = Q(w, w) and the definition of Wi, we can derive the following inequality: E(w' ) - E(w) E(w' ) - Q(w' , w) + Q(w ' , w) - Q(w, w) < Q(w' , w) - Q(w, w) ::; 0 Each iteration of the algorithm thus decreases the otherwise positive quantization error E (equation 1) until the error reaches a fixed point where condition W*' = w* is verified (unicity of the minimum of Q(., w*)). Since the assignment functions Si (w) are discrete, there is an open neighborhood of w* on which the assignments are constant. According to their definition, functions E(.) and Q(., w*) are equal on this neighborhood. Being the minimum of function Q(.,w*), the fixed point w* of this algorithm is also a local minimum of the quantization error E. 0 588 3.2 Leon Bottou, Yoshua Bengio BATCH K-MEANS The above algorithm (5) can be rewritten in a form similar to that of the batch gradient descent algorithm (3). = if k S(Xi' w) otherwise. (6) This algorithm is thus equivalent to a batch gradient descent with a specific, prototype dependent, learning rate k ? J 3.3 ONLINE K-MEANS The online version of our EM style update equation (5) is based on the computation of the mean flt of the examples Xl, ... ,Xt with the following recursive formula: flt+l = t~l (t flt + Xt+l) = flt + t~l (Xt+l - fld Let us introduce new variables nk which count the number of examples so far assigned to prototype Wk. We can then rewrite (5) as an online update applied after the presentation of each pattern Xi: D..nk D..wk { { n\ = 1 if k S(Xi' w) 0 otherwise. (Xi - Wk) 0 if k = S(Xi'W) otherwise. (7) This algorithm is equivalent to an online gradient descent (4) with a specific, prototype dependent, learning rate nlk. Unlike in the batch case, the pattern assignments S(Xi , w) are thus changing after each pattern presentation. Before applying this algorithm, we must of course set nk to zero and Wk to some initial value. Various methods have been proposed including initializing Wk with the first k patterns. 3.4 CONVERGENCE General convergence proofs for the batch and online gradient descent (Bottou, 1991; Driancourt, 1994) directly apply for all four algorithms. Although the derivatives are undefined on a few points, these theorems prove that the algorithms almost surely converge to a local minimum because the local variations of the loss function are conveniently bounded (semi-differentiability). Unlike previous results, the above convergence proofs allow for non-linearity, non-differentiability (on a few points) (Bottou, 1991), and replacing learning rates by a positive definite matrix (Driancourt, 1994). 4 K-MEANS AS A NEWTON OPTIMIZATION We prove in this section that Batch K-Means (6) applies the Newton algorithm. 4.1 THE HESSIAN OF K-MEANS Let us compute the Hessian H of the K-Means cost function (2). This matrix contains the second derivatives of the cost E(w) with respect to each pair of parameters. Since E(w) is a sum of terms L(Xi'W), we can decompose H as the sum 589 Convergence Properties of the K-Means Algorithms of matrices Hi for each term of the cost function: L(Xi'W) = ~in~(xi - Wk)2. Furthermore, the Hi can be decomposed in blocks corresponding to each pair of prototypes. Since L( Xi, w) depends only on the closest prototype to pattern Xi, all these blocks are zero except block (Si (w), Si (w)) which is the identity matrix. Summing the partial Hessian matrices Hi thus gives a diagonal matrix whose diagonal elements are the counts of examples Nk assigned to each prototype. H=(T o We can thus write the Newton update of the parameters as follows: ~w = _H- 1 oE(w) ow which can be exactly rewritten as the batch EM style algorithm (6) presented earlier: ~w k 4.2 =" { 0~k (Xi - Wk) L..J i = if k ~(Xi' w) otherwlse. (8) CONVERGENCE SPEED When optimizing a quadratic function, Newton's algorithm requires only one step . In the case of a non quadratic function, Newton's algorithm is superlinear if we can bound the variations of the second derivatives. Standard theorems that bound this variation using the third derivative are not useful for K-Means because the gradient of the cost function is discontinuous. We could notice that the variations of the second derivatives are however nicely bounded and derive similar proofs for K-Means. For the sake of brevity however, we are just giving here an intuitive argument: we can make the cost function indefinitely differentiable by rounding up the angles around the non differentiable points. We can even restrict this cost function change to an arbitrary small region of the space. The iterations of K-Means will avoid this region with a probability arbitrarily close to 1. In practice, we obtain thus a superlinear convergence. Batch K-Means thus searches for the optimal prototypes at Newton speed. Once it comes close enough to the optimal prototypes (i .e. the pattern assignment is optimal and the cost function becomes quadratic), K-Means jumps to the optimum and terminates. Online K-Means benefits of these optimal learning rates because they remove the usual conditioning problems of the optimization. However , the stochastic noise induced by the online procedure limits the final convergence of the algorithm. Final convergence speed is thus essentially determined by the schedule of the learning rates. Online K-Means also benefits from the redundancies of the training set. It converges significantly faster than batch K-Means during the first training epochs (Darken Leon Bottou, Yoshua Bengio 590 IaI BII Co.t Co.t 2500 2'00 2300 2200 2100 2000 1900 1800 1700 1600 1500 1600 1300 1200 1100 1000 900 800 700 600 500 '00 300 200 100 1 2 3 , 5 6 7 8 9 10 11 12 13 16 15 16 17 18 19 20 Figure 1: E t - Eoo versus t. black circles: online K-Means; black squares: batch K-Means; empty circles: online gradient; empty squares: batch gradient; no mark: EM +Gaussian mixture and Moody, 1991). After going through the first few patterns (depending of the amount of redundancy), online K-Means indeed improves the prototypes as much as a complete batch K-Means epoch. Other researchers have compared batch and online algorithms for neural networks, with similar conclusions (Bengio, 1991). 5 EXPERIMENTS Experiments have been carried out with Fisher's iris data set, which is composed of 150 points in a four dimensional space representing physical measurements on various species of iris flowers. Codebooks of six prototypes have been computed using both batch and online K-Means with the proper learning rates (6) and (7). These results are compared with those obtained using both gradient descent algorithms (3) and (4) using learning rate ft = 0.03/t that we have found optimal. Results are also compared with likelihood maximization with the EM algorithm, applied to a mixture of six Gaussians, with fixed and uniform mixture weights, and fixed unit variance. Inputs were scaled down empirically so that the average cluster variance was around unity. Thus only the cluster positions were learned, as for the K-Means algorithms. Each run of an algorithm consists in (a) selecting a random initial set of prototypes, (b) running the algorithm during 20 epochs and recording the error measure E t after each epoch, (c) running the batch K-Means algorithm! during 40 more epochs in order to locate the local minimum Eoo corresponding to the current initialization of the algorithm. For the four K-Means algorithms, E t is the quantization error (equation 1). For the Gaussian mixture trained with EM, the cost E t is the negative 1 except for the case of the mixture of Gaussians, in which the EM algorithm was applied Convergence Properties of the K-Means Algorithms 1 2 3 ? 5 6 7 8 9 10 11 12 13 U 591 15 16 17 18 19 Figure 2: E~-;~~~oo versus t . black circles: online K-Means; black squares: batch K-Means ; empty circles: online gradient; empty squares: batch gradient ; no mark: EM+Gaussian mixture logarithm of the likelihood of the data given the model. Twenty trials were run for each algorithm. Using more than twenty runs did not improve the standard deviation of the averaged measures because various initializations lead to very different local minima. The value Eoo of the quantization error on the local minima ranges between 3300 and 5800. This variability is caused by the different initializations and not by the different algorithms. The average values of Eoo for each algorithm indeed fall in a very small range (4050 to 4080). Figure 1 shows the average value of the residual error E t - Eoo during the first 20 epochs. Online K-Means (black circles) outperforms all other algorithms during the first five epochs and stabilizes on a level related to the stochastic noise of the online procedure. Batch K-Means (black squares) initially converges more slowly but outperforms all other methods after 5 epochs. All 20 runs converged before the 15th epoch. Both gradients algorithms display poor convergence because they do not benefit of the Newton effect. Again, the online version (white circles) starts faster then the batch version (white square) but is outperformed in the long run. The negative logarithm of the Gaussian mixture is shown on the curve with no point marks , and the scale is displayed on the right of Figure 1. Figure 2 show the final convergence properties of all five algorithms. The evolutions of the ratio (Et+l - Eoo)/(Et - Eoo) characterize the relative improvement of the residual error after each iteration. All algorithms exhibit the same behavior after a few epochs except batch K-Means (black squares). The fast convergence ofthis ratio to zero demonstrates the final convergence of batch K-Means. The EM algorithm displays a better behavior than all the other algorithms except batch K-Means. Clearly, however , its relative improvement ratio doesn't display the fast convergence behavior of batch K-Means. Uon Bottou, Yoshua Bengio 592 The online K-Means curve crosses the batch K-Means curve during the second epoch, suggesting that it is better to run the online algorithm (7) during one epoch and then switch to the batch algorithm (6). 6 CONCLUSION We have shown with theoretical arguments and simple experiments that a well implemented K-Means algorithm minimizes the quantization error using Newton's algorithm. The EM style derivation of K-Means shows that the mathematics of EM are valid well outside the framework of probabilistic models. Moreover the provable convergence properties of the hard threshold K-Means algorithm are superior to those of the EM algorithm for an equivalent soft threshold mixture of Gaussians. Extending these results to other hard threshold algorithms (e .g. Viterbi Training) is an interesting open question. References Bacceli, F., Cohen, G., and Olsder, G. J. (1992). Synchronization and Linearity. Wiley. Bengio, Y. (1991). Artificial Neural Networks and their Application to Sequence Recognition. PhD thesis, McGill University, (Computer Science), Montreal, Qc., Canada. Bottou, 1. (1991). Une approche theorique de l'apprentissage connexioniste; applications a la reconnaissance de la parole. PhD thesis, Universite de Paris XI. Darken, C. and Moody, J. (1991). Note on learning rate schedules for stochastic optimization. In Lippman, R. P., Moody, R., and Touretzky, D. S., editors, Advances in Neural Information Processing Systems 3, pages 832-838, Denver, CO. Morgan Kaufmann, Palo Alto. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum-likelihood from incomplete data via the EM algorithm. Journal of Royal Statistical Society B, 39:1-38. Driancourt, X. (1994). Optimisation par descente de gradient stochastique ... , PhD thesis, Universite de Paris XI, 91405 Orsay cedex, France. Kohonen, T. (1989). Self-Organization and Associative Memory. Springer-Verlag, Berlin, 3 edition. Liporace, L. A . (1976). PTAH on continuous multivariate functions of Markov chains. Technical Report 80193, Institute for Defense Analysis, Communication Research Department. MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the Fifth Berkeley Symposium on Mathematics, Statistics and Probability, Vol. 1, pages 281-296. Nowlan, S. J. (1991). Soft Competitive Adaptation: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. CMU-CS-91-126, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA. Xu, L. and Jordan, M. (1994). Theoretical and experimental studies of convergence properties of the em algorithm for unsupervised learning based on finite mixtures. Presented at the Neural Networks for Computing Conference. 

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695 ANALOG IMPLEMENTATION OF SHUNTING NEURAL NETWORKS Bahram Nabet, Robert B. Darling, and Robert B. Pinter Department of Electrical Engineering, FT-lO University of Washington Seattle, WA 98195 ABSTRACT An extremely compact, all analog and fully parallel implementation of a class of shunting recurrent neural networks that is applicable to a wide variety of FET-based integration technologies is proposed. While the contrast enhancement, data compression, and adaptation to mean input intensity capabilities of the network are well suited for processing of sensory information or feature extraction for a content addressable memory (CAM) system, the network also admits a global Liapunov function and can thus achieve stable CAM storage itself. In addition the model can readily function as a front-end processor to an analog adaptive resonance circuit. INTRODUCTION Shunting neural networks are networks in which multiplicative, or shunting, terms of the form Xi Lj f;(Xj) or Xi Lj Ij appear in the short term memory equations, where Xi is activity of a cell or a cell population or an iso-potential portion of a cell and Ii are external inputs arriving at each site. The first case shows recurrent activity, while the second case is non-recurrent or feed forward. The polarity of these terms signify excitatory or inhibitory interactions. Shunting network equations can be derived from various sources such as the passive membrane equation with synaptic interaction (Grossberg 1973, Pinter 1983), models of dendritic interaction (RaIl 1977), or experiments on motoneurons (Ellias and Grossberg 1975). While the exact mechanisms of synaptic interactions are not known in every individual case, neurobiological evidence of shunting interactions appear in several 696 Nabet, Darling and Pinter areas such as sensory systems, cerebellum, neocortex, and hippocampus (Grossberg 1973, Pinter 1987). In addition to neurobiology, these networks have been used to successfully explain data from disciplines ranging from population biology (Lotka 1956) to psychophysics and behavioral psychology (Grossberg 1983). Shunting nets have important advantages over additive models which lack the extra nonlinearity introduced by the multiplicative terms. For example, the total activity of the network, shown by Li Xi, approaches a constant even as the input strength grows without bound. This normalization in addition to being computationally desirable has interesting ramifications in visual psychophysics (Grossberg 1983). Introduction of multiplicative terms also provides a negative feedback loop which automatically controls the gain of each cell, contributes to the stability of the network, and allows for large dynamic range of the input to be processed by the network. The automatic gain control property in conjunction with properly chosen nonlinearities in the feedback loop makes the network sensitive to small input values by suppressing noise while not saturating at high input values (Grossberg 1973). Finally, shunting nets have been shown to account for short term adaptation to input properties, such as adaptation level tuning and the shift of sensitivity with background strength (Grossberg 1983), dependence of visual size preference and latency of response on contrast and mean luminance, and dependence of temporal and spatial frequency tuning on contrast and mean luminance (Pinter 1985). IMPLEMENTATION The advantages, generality, and applicability of shunting nets as cited in the previous section make their implementation very desirable, but digital implementation of these networks is very inefficient due to the need for analog to digital conversion, multiplication and addition instructions, and implementation of iterative algorithms. A linear feedback class of these networks (Xi Lj !; (Xj) Xi Li J{ijXj), however, can be implemented very efficiently with simple, completely parallel and all analog circuits. = FRAMEWORK Figure 1 shows the design framework for analog implementation of a class of shunting nets. In this design addition (subtraction) is achieved, via Kirchoff's current law by placing transistors in upper (lower) rails, and through the choice of depletion or enhancement mode devices. Multiplicative, or shunting, interconnections are done by one transistor per interconnect, using a field-effect transistor (FET) in the voltage-variable conductance region. Temporal properties are characterized by cell membrane capacitance C, which can be removed, or in effect replaced by the parasitic device capacitances, if higher speed is desired. A buffer stage is necessary for correct polarity of interconnections and the large fan-out associated with high connectivity of neural networks. Analog Implementation of Shunting Neural Networks Vdd I., + x., c -t" '.J . x? J Vss Figure 1. Design framework for implementation of one cell in a shunting network. Voltage output of other cells is connected to the gate of transistors Qi,i' Such a circuit is capable of implementing the general network equation: (1) Excitatory and inhibitory input current sources can also be shunted, with extra circuitry, to implement non-recurrent shunting networks. NMOS, CMOS and GALLIUM ARSENIDE Since the basic cell of Fig. 1 is very similar to a standard logic gate inverter, but with the transistors sized by gate width-to-Iength ratio to operate in the nonsaturated current region, this design is applicable to a variety of FET technologies including NMOS, CMOS, and gallium arsenide (GaAs). A circuit made of all depletion-mode devices such as GaAs MESFET buffered FET logic, can implement all the terms of Eq. (1) except shunting excitatory terms and requires a level shifter in the buffer stage. A design with all enhancement mode devices such as silicon NMOS can do the same but without a level shifter. With the addition of p-channel devices, e.g. Si CMOS, all polarities and all terms of Eq. (1) can be realized. As mentioned previously a buffer stage is necessary for correct polarity of interconnections and fan out/fan in capacity. Figure 2 shows a GaAs MESFET implementation with only depletion mode devices which employs a level shifter as the buffer stage. 697 698 Nabet, Darling and Pinter VDD-------------r------~--------------~-- INPUTS: EXTERNAL OR FROM PREVJOUS LAYER EXCITATORY CONNECTIONS INHIBITORY CONNECTIONS GN~-L--~------~------~-- TUNABLE SELF-RELAXATION CONNECTION OUTPUT TO NEXT LAYER VSS--~--....L....? LEVEL SHIFT AND BUFFER STAGE Figure 2. Gallium arsenide MESFET implementation with level shifter and depletion mode devices. Lower rail transistors produce shunting off-surround terms. Upper transistors can produce additive excitatory connections. SPECIFIC IMPLEMENTATION The simplest shunting network that can be implemented by the general framework of Fig.1 is Fig. 2 with only inhibitory connections (lower rail transistors). This circuit implements the network model dX? d/ = Ii - a,X, + Xi(J(iXi) - " J(ijXj) Xi(L....J j#i (2), The simplicity of the implementation is notable; a linear array with nearest neighbor interconnects consists of only 5 transistors, 1-3 diodes, and if required 1 capacitor per cell. A discrete element version of this implementation has been constructed and shows good agreement with expected properties. Steady state output is proportional to the square root of a uniform input thereby compressing the input data and showing adaptation to mean input intensity (figure 3). The network exhibits contrast enhancement of spatial edges which increases with higher mean input strength (figure 4). A point source input elicits an on-center off-surround response, similar to the difference-of-Gaussians receptive field of many excitable cells. This 'receptive field' becomes more pronounced as the input intensity increases, showing the dependence of spatial frequency tuning on mean input level (figure 5). The temporal response of the network is also input dependent since the time constant of the exponential Analog Implementation of Shunting Neural Networks decay of the impulse response decreases with input intensity. Finally, the dependence of the above properties on mean input strength can be tuned by varying the conductance of the central FET. 700.0 --- 600.0 500.0 ? > E 11.1 II 400.0 !:i ~ 5 300.0 ~ o 200.0 100.0 0.1 0.3 0.5 0 .7 0.9 1. 1 1.3 1.5 1.7 1.11 2.1 INPUT CURRENT rnA Figure 3. Response of network to uniform input. Output is proportional to the square root of the input. DEPENDENCE OF ENHANCEMENT ON MEAN INPUT 1.5 1.4 ~ :J 0 ... 1.3 a 1.2 i 1.1 N II: 0 z 5 ~ 0 a 1.0 0 .11 f ...::l 0.8 IL ~ 0 .7 0.11 2 5 3 7 CELL NUMBER Figure 4. Response of network to spatial edge patterns with the same contrast but increasing mean input level. 699 700 Nabet, Darling and Pinter Imox ~ 2 .0J rnA, llmox - J75.74 mil 1.0 0 .11 ~ g :> 0.11 0 Q ~ 0 .7 !5 II. I iii 0.11 II: 0.5 i 0 Z 0.4 O.J 2 J 4 II 5 7 cEll NUMBER INPUT -x- OUTPUT Figure 5. Response of network to a point source input. Inset shows the receptive field of fly's lamina monopolar cells (LMC of Lucilia sericata). Horizontal axis of inset in visual angle, vertical a.xis relative voltage units of hyperpolarization. Inset from Pinter et al. (in preparation) CONTENT ADDRESSABILITY AND RELATION TO ART Using a theorem by Cohen and Grossberg (1983), it can be shown that the network equa.tion (2) a.dmits the global Liapunov function n V -- - "'(l?ln(xi) ~ 1 >. - a'x 1 1' + K'x~) 1 1 n +.!2 '" ;=1 ~ K??x IJ J'XL .. , (3) j,k=l = where>. is a constant, under the constraints Kij Kji and Xi > O. This shows that in response to an arbitrary input the network always approaches an equilibrium point. The equilibria represent stored patterns and this is Content Addressable Memory (CAM) property. In addition, Eq. (2) is a special case of the feature representation field of an analog adaptive resonance theory ART-2 circuit, (Carpenter and Grossberg 1987), and hence this design can operate as a module in a learning multilayer ART architecture. Analog Implementation of Shunting Neural Networks FUTURE PLANS Due to the very small number of circuit components required to construct a cell, this implementation is quite adaptable to very high integration densities. A solid state implementation of the circuit of figure (2) on a gallium arsenide substrate, chosen for its superiority for opto-electronics applications, is in progress. The chip includes monolithically fabricated photosensors for processing of visual information. All of the basic components of the circuit have been fabricated and tested. With standard 2 micron GaAs BFL design rules, a chip could contain over 1000 cells per cm 2 , assuming an average of 20 inputs per cell. CONCLUSIONS ? ? ? ? ? ? The present work has the following distinguishing features: Implements a mathematically well described and stable model. Proposes a framework for implementation of shunting nets which are biologically feasible, explain variety of psychophysical and psychological data and have many desirable computational properties. Has self-sufficient computational capabilities; especially suited for processing of sensory information in general and visual information in particular (N abet and Darling 1988). Produces a 'good representation' of the input data which is also compatible with the self-organizing multilayer neural network architecture ART-2. Is suitable for implementation in variety of technologies. Is parallel, analog, and has very little overhead circuitry. .. -. 701 702 Nabet, Darling and Pinter REFERENCES Carpenter, G.A. and Grossberg, S. (1987) "ART 2: self organization of stable category recognition codes for analog input patterns,". Applied Optics 26, pp. 49194930. Cohen,M.A. and Grossberg, S. (1983) "Absolute stability of global pattern formation and parallel memory storage by competitive neural networks" , IEEE Transactions on Systems Man and Cybernetics SMC-13, pp. 815-826. Ellias, S.A. and Grossberg, S. (1975) "Pattern formation, contrast control, and oscillations in the short term memory of shunting on-center off-surround networks" Biological Cybernetics, 20, pp. 69-98. Grossberg, S. (1973), "Contour enhancement, Short term memory and constancies in reverberating neural networks," Studies in Applied Mathematics, 52, pp. 217257. Grossberg, S. (1983), "The quantized geometry of visual space: the coherent computation of depth, form, and lightness." The behavioral and brain sciences, 6, pp. 625-692. Lotka, A.J. (1956). Elements of mathematical biology. New York: Dover. Nabet, B. and Darling, R.B. (1988). "Implementation of optical sensory neural networks with simple discrete and monolithic circuits," (Abstract) Neural Networks, Vol.l, Suppl. 1, 1988, pp. 396. Pinter, R.B., (1983). "The electrophysiological bases for linear and nonlinear product term lateral inhibition and the consequences for wide-field textured stimuli" J. Theor. Bioi. 105 pp. 233-243. Pinter, R.B. (1985) " Adaptation of spatial modulation transfer functions via nonlinear lateral inhibition" Bioi. Cybernetics 51, pp. 285-291. Pinter, R.B. (1987) "Visual system neural networks: Feedback and feedforward lateral inhibition" Systems and Control Encyclopedia (ed.M.G. Singh) Oxford: Pergamon Press. pp. 5060-5065. Pinter, R.B., Osorio, D., and Srinivasan, M.V., (in preperation) "Shift of edge preference to scototaxis depends on mean luminance and is predicted by a matched filter hypothesis in fly lamina cells" RaIl, W. (1977). "Core conductor theory and cable properties of neurons" in Handbook of Physiology: The Nervous System vol. I, part I, Ed. E.R. Kandel pp. 39-97. Bethesda, MD: American Physiological Society. 

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Comparing the prediction accuracy of artificial neural networks and other statistical models for breast cancer survival Harry B. Burke Department of Medicine New York Medical College Valhalla, NY 10595 David B. Rosen Department of Medicine New York Medical College Valhalla, NY 10595 Philip H. Goodman Department of Medicine University of Nevada School of Medicine Reno, Nevada 89520 Abstract The TNM staging system has been used since the early 1960's to predict breast cancer patient outcome. In an attempt to increase prognostic accuracy, many putative prognostic factors have been identified. Because the TNM stage model can not accommodate these new factors, the proliferation of factors in breast cancer has lead to clinical confusion. What is required is a new computerized prognostic system that can test putative prognostic factors and integrate the predictive factors with the TNM variables in order to increase prognostic accuracy. Using the area under the curve of the receiver operating characteristic, we compare the accuracy of the following predictive models in terms of five year breast cancer-specific survival: pTNM staging system, principal component analysis, classification and regression trees, logistic regression, cascade correlation neural network, conjugate gradient descent neural, probabilistic neural network, and backpropagation neural network. Several statistical models are significantly more ac- 1064 Harry B. Burke, David B. Rosen, Philip H. Goodman curate than the TNM staging system. Logistic regression and the backpropagation neural network are the most accurate prediction models for predicting five year breast cancer-specific survival 1 INTRODUCTION For over thirty years measuring cancer outcome has been based on the TNM staging system (tumor size, number of lymph nodes with metastatic disease, and distant metastases) (Beahr et. al., 1992). There are several problems with this model (Burke and Henson , 1993). First, it is not very accurate, for breast cancer it is 44% accurate. Second its accuracy can not be improved because predictive variables can not be added to the model. Third, it does not apply to all cancers. In this paper we compare computerized prediction models to determine if they can improve prognostic accuracy. Artificial neural networks (ANN) are a class of nonlinear regression and discrimination models. ANNs are being used in many areas of medicine, with several hundred articles published in the last year. Representative areas of research include anesthesiology (Westenskow et. al., 1992) , radiology (Tourassi et. al. , 1992) , cardiology (Leong and Jabri, 1982) , psychiatry (Palombo, 1992), and neurology (Gabor and Seyal, 1992). ANNs are being used in cancer research including image processing (Goldberg et. al., 1992) , analysis of laboratory data for breast cancer diagnosis (0 Leary et. al., 1992), and the discovery of chemotherapeutic agents (Weinstein et . al., 1992). It should be pointed out that the analyses in this paper rely upon previously collected prognostic factors. These factors were selected for collection because they were significant in a generalized linear model such as the linear or logistic models . There is no predictive model that can improve upon linear or logistic prediction models when the predictor variables meet the assumptions of these models and there are no interactions. Therefore he objective of this paper is not to outperform linear or logistic models on these data. Rather, our objective is to show that, with variables selected by generalized linear models, artificial neural networks can perform as well as the best traditional models . There is no a priori reason to believe that future prognostic factors will be binary or linear, and that there will not be complex interactions between prognostic factors. A further objective of this paper is to demonstrate that artificial neural networks are likely to outperform the conventional models when there are unanticipated nonmonotonic factors or complex interactions. 2 2.1 METHODS DATA The Patient Care Evaluation (PCE) data set is collected by the Commission on Cancer of the American College of Surgeons (ACS). The ACS, in October 1992, requested cancer information from hospital tumor registries in the United States. The ACS asked for the first 25 cases of breast cancer seen at that institution in 1983, and it asked for follow up information on each of these 25 patients through the date of the request. These are only cases of first breast cancer. Follow-up information included known deaths. The PCE data set contains, at best, eight year follow-up. Prediction Accuracy of Models for Breast Cancer Survival 1065 We chose to use a five year survival end-point. This analysis is for death due to breast cancer, not all cause mortality. For this analysis cases with missing data, and cases censored before five years, are not included so that the prediction models can be compared without putting any prediction model at a disadvantage. We randomly divided the data set into training, hold-out, and testing subsets of 3,100, 2,069, and 3,102 cases, respectively. 2.2 MODELS The TMN stage model used in this analysis is the pathologic model (pTNM) based on the 1992 American Joint Committee on Cancer's Manual for the Staging of Cancer (Beahr et. al., 1992). The pathologic model relies upon pathologically determined tumor size and lymph nodes, this contrasts with clinical staging which relies upon the clinical examination to provide tumor size and lymph node information. To determine the overall accuracy of the TNM stage model we compared the model's prediction for each patient, where the individual patient's prediction is the fraction of all the patients in that stage who survive, to each patient's true outcome. Principal components analysis, is a data reduction technique based on the linear combinations of predictor variables that minimizes the variance across patients (Jollie, 1982). The logistic regression analysis is performed in a stepwise manner, without interaction terms, using the statistical language S-PLUS (S-PLUS, 1992), with the continuous variable age modeled with a restricted cubic spline to avoid assuming linearity (Harrell et. al., 1988). Two types of Classification and Regression Tree (CART) (Breiman et. al., 1984) analyses are performed using S-PLUS. The first was a 9-node pruned tree (with 10-fold cross validation on the deviance), and the second was a shrunk tree with 13.7 effective nodes. The multilayer perceptron neural network training in this paper is based on the maximum likelihood function unless otherwise stated, and backpropagation refers to gradient descent. Two neural networks that are not multilayer perceptrons are tested. They are the Fuzzy ARTMAP neural network (Carpenter et. al., 1991) and the probabilistic neural network (Specht, 1990). 2.3 ACCURACY The measure of comparative accuracy is the area under the curve of the receiver operating characteristic (Az) . Generally, the Az is a nonparametric measure of discrimination. Square error summarizes how close each patient's predicted value is to its true outcome. The Az measures the relative goodness of the set of predictions as a whole by comparing the predicted probability of each patient with that of all other patients. The computational approach to the Az that employs the trapezoidal approximation to the area under the receiver operating characteristic curve for binary outcomes was first reported by Bamber (Bamber, 1975), and later in the medical literature by Hanley (Hanley and McNeil, 1982). This was extended by Harrell (Harrell et. al., 1988) to continuous outcomes. Harry B. Burke, David B. Rosen, Philip H. Goodman 1066 Table 1: PCE 1983 Breast Cancer Data: 5 Year Survival Prediction, 54 Variables. PREDICTION MODEL pTNM Stages Principal Components Analysis CART, pruned CART, shrunk Stepwise Logistic regression Fuzzy ARTMAP ANN Cascade correlation ANN Conjugate gradient descent ANN Probabilistic ANN Backpropagation ANN ACCURACY? SPECIFICATIONS .720 .714 .753 .762 .776 .738 .761 .774 .777 .784 O,I,I1A,I1B,IIIA,I1IB,IV one scaling iteration 9 nodes 13.7 nodes with cubic splines 54-F2a, 128-1 54-21-1 54-30-1 bandwidth = 16s 54-5-1 * The area under the curve of the receiver operating characteristic. 3 RESULTS All results are based on the independent variable sample not used for training (i.e., the testing data set), and all analyses employ the same testing data set. Using the PCE breast cancer data set, we can assess the accuracy of several prediction models using the most powerful of the predictor variables available in the data set (See Table 1). Principal components analysis is not expected to be a very accurate model; with one scaling iteration, its accuracy is .714. Two types of classification and regression trees (CART), pruned and shrunk, demonstrate accuracies of .753 and .762, respectively. Logistic regression with cubic splines for age has an accuracy of .776. In addition to the backpropagation neural network and the probabilistic neural network, three types of neural networks are tested. Fuzzy ARTMAP's accuracy is the poorest at .738. It was too computationally intensive to be a practical model. Cascade-correlation and conjugate gradient descent have the potential to do as well as backpropagation. The PNN accuracy is .777. The PNN has many interesting features, but it also has several drawbacks including its storage requirements. The backpropagation neural network's accuracy is .784.4. 4 DISCUSSION For predicting five year breast cancer-specific survival, several computerized prediction models are more accurate than the TNM stage system, and artificial neural networks are as good as the best traditional statistical models. References Bamber D (1975). The area above the ordinal dominance graph and the area below the receiver operating characteristic. J Math Psych 12:387-415. Beahrs OH, Henson DE, Hutter RVP, Kennedy BJ (1992). Manual for staging of Prediction Accuracy of Models for Breast Cancer Survival 1067 cancer, 4th ed. Philadelphia: JB Lippincott. Burke HB, Henson DE (1993) . Criteria for prognostic factors and for an enhanced prognostic system. Cancer 72:3131-5. Breiman L, Friedman JH, Olshen RA (1984). Classification and Regression Trees. Pacific Grove, CA: Wadsworth and Brooks/Cole. Carpenter GA, Grossberg S, Rosen DB (1991). Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system. Neural Networks 4:759-77l. Gabor AJ, M. Seyal M (1992) . Automated interictal EEG spike detection using artificial neural networks. Electroencephalogr Clin Neurophysiology 83 :271-80. Goldberg V, Manduca A, Ewert DL (1992). Improvement in specificity of ultrasonography for diagnosis of breast tumors by means of artificial intelligence. Med Phys 19:1275-8l. Hanley J A, McNeil BJ (1982). The meaning of the use of the area under the receiver operating characteristic (ROC) curve. Radiology 143:29-36. Harrell FE, Lee KL, Pollock BG (1988). Regression models in clinical studies: determining relationships between predictors and response. J Natl Cancer Instit 80:1198-1202. Jollife IT (1986). Principal Component Analysis. New York: Springer-Verlag, 1986. Leong PH, J abri MA (1982). MATIC - an intracardiac tachycardia classification system. PACE 15:1317-31,1982. O'Leary TJ, Mikel UV, Becker RL (1992). Computer-assisted image interpretation: use of a neural network to differentiate tubular carcinoma from sclerosing adenosis. Modern Pathol 5:402-5. Palombo SR (1992). Connectivity and condensation in dreaming. JAm Psychoanal Assoc 40:1139-59. S-PLUS (1991), v 3.0. Seattle, WA; Statistical Sciences, Inc. Specht DF (1990). Probabilistic neural networks. Neural Networks 3:109-18. Tourassi GD, Floyd CE, Sostman HD, Coleman RE (1993). Acute pulmonary embolism: artificial neural network approach for diagnosis. Radiology 189:555-58. Weinstein IN, Kohn KW, Grever MR et. al. (1992) Neural computing in cancer drug development: predicting mechanism of action. Science 258:447-51. Westenskow DR, Orr JA, Simon FH (1992) . Intelligent alarms reduce anesthesiologist's response time to critical faults. Anesthesiology 77:1074-9, 1992. Learning with Product Units Laurens R. Leerink Australian Gilt Securities LTD 37-49 Pitt Street NSW 2000, Australia laurens@sedal.su.oz.au Bill G. Horne NEC Research Institute 4 Independence Way Princeton, NJ 08540, USA horne@research.nj.nec.com C. Lee Giles NEC Research Institute 4 Independence Way Princeton, NJ 08540, USA giles@research.nj.nec.com Marwan A. Jabri Department of Electrical Engineering The University of Sydney NSW 2006, Australia marwan@sedal.su.oz.au Abstract Product units provide a method of automatically learning the higher-order input combinations required for efficient learning in neural networks. However, we show that problems are encountered when using backpropagation to train networks containing these units. This paper examines these problems, and proposes some atypical heuristics to improve learning. Using these heuristics a constructive method is introduced which solves well-researched problems with significantly less neurons than previously reported. Secondly, product units are implemented as candidate units in the Cascade Correlation (Fahlman & Lebiere, 1990) system. This resulted in smaller networks which trained faster than when using sigmoidal or Gaussian units. 1 Introduction It is well-known that supplementing the inputs to a neural network with higher-order combinations ofthe inputs both increases the capacity of the network (Cover, 1965) and the the ability to learn geometrically invariant properties (Giles & Maxwell, 538 Laurens Leerink, C. Lee Giles, Bill G. Home, Marwan A. Jabri 1987). However, there is a combinatorial explosion of higher order terms as the number of inputs to the network increases. Yet in order to implement a certain logical function, in most cases only a few of these higher order terms are required (Redding et al., 1993). The product units (PUs) introduced by (Durbin & Rumelhart, 1989) attempt to make use of this fact. These networks have the advantage that, given an appropriate training algorithm, the units can automatically learn the higher order terms that are required to implement a specific logical function. In these networks the hidden layer units compute the weighted product ofthe inputs, that is N N II X~i instead of i=l 2:XiWi (1) i=l as in standard networks. An additional advantage of PUs is the increased information capacity of these units compared to standard summation networks. It is approximately 3N (Durbin & Rumelhart, 1989), compared to 2N for a single threshold logic function (Cover, 1965), where N is the number of inputs to the unit. The larger capacity means that the same functions can be implemented by networks containing less units. This is important for certain applications such as speech recognition where the data bandwidth is high or if realtime implementations are desired. When PUs are used to process Boolean inputs, best performance is obtained (Durbin & Rumelhart, 1989) by using inputs of {+1, -I}. If the imaginary component is ignored, with these inputs, the activation function is equivalent to a cosine summation function with {-1,+1} inputs mapped {I,D} (Durbin & Rumelhart, 1989). In the remainder of this paper the terms product unit (PU) and cos{ine) unit will be used interchangeably as all the problems examined have Boolean inputs. 2 Learning with Product Units As the basic mechanism of a PU is multiplicative instead of additive, one would expect that standard neural network training methods and procedures cannot be directly applied when training these networks. This is indeed the case. If a neural network simulation environment is available the basic functionality of a PU can be obtained by simply adding the cos function cos( 1(" * input) to the existing list of transfer functions. This assumes that Boolean mappings are being implemented and the appropriate {-1,+1} - {I,D} mapping has been performed on the input vectors. However, if we then attempt to train a network on on the parity-6 problem shown in (Durbin & Rumelhart, 1989), it is found that the standard backpropagat ion (BP) algorithm simply does not work. We have found two main reasons for this. The first is weight initialization. A typical first step in the backpropagation procedure is to initialize all weights to small random values. The main reason for this is to use the dynamic range of the sigmoid function and it's derivative. However, the dynamic range of a PU is unlimited. Initializing the weights to small random 539 Learning with Product Units values results in an input to the unit where the derivative is small. So apart from choosing small weights centered around ml" with n = ?1, ?2, ... this is the worst possible choice. In our simulations weights were initialized randomly in the range [-2,2]. In fact, learning seems insensitive to the size of the weights, as long as they are large enough. The second problem is local minima. Previous reports have mentioned this problem, (Lapedes & Farber, 1987) commented that "using sin's often leads to numerical problems, and nonglobal minima, whereas sigmoids seemed to avoid such problems". This comment summarizes our experience of training with PUs. For small problems (less than 3 inputs) backpropagation provides satisfactory training. However, when the number of inputs are increased beyond this number, even with th: .weight initialization in the correct range, training usually ends up in a local mInIma. 3 Training Algorithms With these aspects in mind, the following training algorithms were evaluated: online and batch versions of Backpropagation (BP), Simulated Annealing (SA), a Random Search Algorithm (RSA) and combinations of these algorithms. BP was used as a benchmark and for use in combination with the other algorithms. The Delta-Bar-Delta learning rate adaptation rule (Jacobs, 1988) was used along with the batch version of BP to accelerate convergence, with the parameters were set to 0 0.35, K, 0.05 and ? 0.90. RSA is a global search method (i.e. the whole weight space is explored during training). Weights are randomly chosen from a predefined distribution, and replaced if this results in an error decrease. SA (Kirkpatrick et aI., 1983) is a standard optimization method. The operation of SA is similar to RSA, with the difference that with a decreasing probability solutions are accepted which increase the training error. The combination of algorithms were chosen (BP & SA, BP & RSA) to combine the benefits of global and local search. Used in this manner, BP is used to find the local minima. If the training error at the minima is sufficiently low, training is terminated. Otherwise, the global method initializes the weights to another position in weight space from which local training can continue. = = = The BP-RSA combination requires further explanation. Several BP-(R)SA combinations were evaluated, but best performance was obtained using a fixed number of iterations of BP (in this case 120) along with one initial iteration of RSA. In this manner BP is used to move to the local minima, and if the training error is still above the desired level the RSA algorithm generates a new set of random weights from which BP can start again. The algorithms were evaluated on two problems, the parity problem and learning all logical functions of 2 and 3 inputs. The infamous parity problem is (for the product unit at least) an appropriate task. As illustrated by (Durbin & Rumelhart, 1989), this problem can be solved by one product unit. The question is whether the training algorithms can find a solution. The target values are {-1, +1}, and the output is taken to be correct if it has the correct sign. The simulation results are shown in Table 1. It should be noted that one epoch of both SA and RSA involves 540 Laurens Leerink, C. Lee Giles, Bill G. Home, Marwan A. Jahri relaxing the network across the training set for every weight, so in computational terms their nepoeh values should be multiplied by a factor of (N + 1). Parity N 6 8 10 Online BP Batch BP neonv nepoeh n eonv 10 8 6 30.4 101.3 203.3 7 2 0 SA RSA nepoeh neonv nepoeh neon v nepoeh 34 700 10 10 10 12.6 52.8 99.9 10 10 10 15.2 45.4 74.1 - Table 1: The parity N problem: The table shows neon v the number of runs out of 10 that have converged and nepoeh' the average number of training epochs required when training converged. For the parity problem it is clear that local learning alone does not provide good convergence. For this problem, global search algorithms have the following advantages: (1) The search space is bounded (all weights are restricted to [-2, +2]) (2) The dimension of search space is low (maximum of 11 weights for the problems examined). (3) The fraction of the weight space which satisfies the parity problem relative to the total bounded weight space is high. In a second set of simulations, one product unit was trained to calculate all 2(2 N ) logical functions of the N input variables. Unfortunately, this is only practical for N E {2,3} . For N = 2 there are only 16 functions, and a product unit has no problem learning all these functions rapidly with all four training algorithms. In comparison a single summation unit can learn 14 (not the XOR & XNOR functions). For N =3, a product unit is able to implement 208 of the 256 functions, while a single summation unit could only implement 104. The simulation results are displayed in Table 2. Online BP Batch BP BP-RSA Table 2: Learning all logical functions of 3 inputs: The rows display nlogie , the average number of logical functions implemented by a product unit and nepoeh, the number of epochs required for convergence. Ten simulations were performed for each of the 256 logical functions , each for a maximum of 1,000 iterations. 4 Constructive Learning with Product Units Selecting the optimal network architecture for a specific application is a nontrivial and time-consuming task, and several algorithms have been proposed to automate this process. These include pruning methods and growing algorithms. In this section a simple method is proposed for adding PUs to the hidden layer of a three layer network. The output layer contains a single sigmoidal unit . Several constructive algorithms proceed by freezing a subset of the weights and limiting training to the newly added units. As mentioned earlier, for PUs a global 541 Learning with Product Units 300 Tiling AI orithm Upstart AI orithm 81M using Pr Units ~ I-t--< >S-t 250 i!: 0 ~c 200 .!; <II C e::l 150 CI) c '15 ~ D E ::l Z 100 50 200 400 600 800 Number of patterns (2"N) 1000 1200 Figure 1: The number of units required for learning the random mapping problems by the 'Tiling', 'Upstart' and SIM algorithms. search is required to solve the local-minima problems. Freezing a subset of the weights restricts the new solution to an affine subset of the existing weight space, often resulting in non-minimal networks (Ash, 1989). For this reason a simple incremental method (SIM) was implemented which retains the global search for all weights during the whole training process. The method used in our simulations is as follows: ? Train a network using the BP-RSA combination on a network with a specified minimum number of hidden PUs. ? If there is no convergence within a specified number of epochs, add a PU to the network. Reinitialize weights and continue training with the BP-RSA combination . ? Repeat process until a solution is found or the network has grown a predetermined maximum size. The method of (Ash , 1989) was also evaluated, where neurons with small weights were added to a network according to certain criteria. The SIM performed better, possibly because of the global search performed by the RSA step. The 'Upstart' (Frean, 1990) and 'Tiling' (Mezard & Nadal, 1989) constructive algorithms were chosen as benchmarks. A constructive PU network was trained on two problems described in these papers, namely the parity problem and the random mapping problem. In (Frean, 1990) it was reported that the Upstart 542 Laurens Leerink, C. Lee Giles, Bill G. Home, Marwan A. Jabri algorithm required N units for all parity N problems, and 1,000 training epochs were sufficient for all values of N except N = 10, which required 10,000. As seen earlier, one PU is able to perform any parity function, and SIM required an an average of 74.1 iterations for N = 6,8,10. The random mapping problem is defined by assigning each of the 2N patterns its target { -1, +I} with 50% probability. This is a difficult problem, due to the absence of correlations and structure in the input. As in (Frean, 1990; Mezard & Nadal, 1989) the average of 25 runs were performed, each on a different training set. The number of units required by SIM is plotted in Figure 1. The values for the Tiling and Upstart algorithms are approximate and were obtained through inspection from a similar graph in (Frean, 1990). 5 U sing Cosine Candidate Units in Cascade Correlation Initially we wanted to compare the performance of SIM with the well-known 'cascade-correlation' (CC) algorithm of (Fahlman & Lebiere, 1990). However, the network architectures differ and a direct comparison between the number of units in the respective architectures does not reflect the efficiency of the algorithms. Instead, it was decided to integrate PUs into the CC system as candidate units. For these simulations a public domain version of CC was used (White, 1993) which supports four different candidate types; the asymmetric sigmoid, symmetric sigmoid, variable sigmoid and gaussian units. Facilities exist for either constructing homogeneous networks by selecting one unit type, or training with a pool of different units allowing the construction of hybrid networks. It was thus relatively simple to add PU candidate units to the system. Table 3 displays the results when CC was trained on the random logic problem using three types of homogeneous candidate units. N 7 8 9 10 CC Sigmoid CC Gauss CC PU nunih nepOCh6 nunih nepoch, nunih n epoch6 6.6 12.1 20.5 32.9 924.5 1630.9 2738.3 4410.9 6.7 11.5 18.4 30.2 642.6 1128.2 1831.1 2967.6 5.7 9.9 16.4 26.6 493.8 833.8 1481.8 2590.8 Table 3: Learning random logic functions of N inputs: The table shows nunih' the average number of units required and nepoch6' the average number of training epochs required for convergence of CC using sigmoidal, Gaussian and PU candidate units . Figures are based on 25 simulations. In a separate experiment the performance of hybrid networks were re-evaluated on the same random logic problem. To enable a fair competition between candidate units of different types, the simulations were run with 40 candidate units, 8 of each type. The simulations were evaluated on 25 trails for each of the random mapping problems (7,8,9 and 10 inputs, a total of 1920 input vectors). In total 1460 hidden units were allocated, and in all cases PU candidate units were chosen above units of the 4 other types during the competitive stage. During this comparison all Learning with Product Units 543 parameters were set to default values, i.e. the weights of the PU candidate units were random numbers initialized in the range of [-1, +1]. As discussed earlier, this puts the PUs at a slight disadvantage as their optimum range is [-2, +2]. 6 Discussion The BP-RSA combination is in effect equivalent to the 'local optimization with random restarts' process discussed by (Karmarkar & Karp, 1982), where the local optimization is this case is performed by the BP algorithm. They reported that for certain problems where the error surface was 'exceedingly mountainous', multiple random-start local optimization outperformed more sophisticated methods. We hypothesize that adding PUs to a network makes the error surface sufficiently mountainous so that a global search is required. As expected, the higher separating capacity of the PU enables the construction of networks with less neurons than those produced by the Tiling and Upstart algorithms. The fact that SIM works this well is mainly a result of the error surface; the surface is so irregular that even training a network of fixed architecture is best done by reinitializing the weights if convergence does not occur within certain bounds. This again is in accordance with the results of (Karmarkar & Karp, 1982) discussed above . When used in CC we hypothesize that there are three main reasons for the choice of PUs above any of the other types during the competitive learning phase. Firstly, the higher capacity (in a information capacity sense) of the PUs allows a better correlation with the error signal. Secondly, having N competing candidate units is equivalent to selecting the best of N random restarts, and performs the required global search. Thirdly, although the error surface of networks with PUs contains more local minima than when using standard transfer functions, the surface is locally smooth. This allows effective use of higher-order error derivatives, resulting in fast convergence by the quickprop algorithm. In (Dawson & Schopflocher, 1992) it was shown that networks with Gaussian units train faster and require less units than networks with standard sigmoidal units. This is supported by our results shown in Table 3. However, for the problem examined, PUs outperform Gaussian units by approximately the same margin as Gaussian units outperform sigmoidal units. It should also be noted that these problems where not chosen for their suitability for PUs. In fact, if the problems are symmetric/regular the difference in performance is expected to increase. 7 Conclusion Of the learning algorithms examined BP provides the fastest training, but is prone to nonglobal minima. On the other hand, global search methods are impractical for larger networks. For the problems examined, a combination of local and global search methods were found to perform best. Given a network containing PUs, there are some atypical heuristics that can be used: (a) correct weight initialization (b) reinitialization of the weights if convergence is not rapidly reached. In addition, the representational power of PUs have enabled us to solve standard problems 544 lAurens Leerink, C. Lee Giles, Bill G. Home, Marwan A. labri using significantly smaller networks than previously reported, using a very simple constructive method. When implemented in the CC architecture, for the problems examined PUs resulted in smaller networks which trained faster than other units. When included in a pool of competing candidate units, simulations showed that in all cases PU candidate units were preferred over candidate units of the other four types. References Ash, T. (1989) . Dynamic node creation in backpropagation networks. Connection Science, 1 (4), 365-375. Cover, T. (1965) . Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition . IEEE Transactions on Electronic Computers, 14, 326-334. Dawson, M. & Schopflocher, D. (1992) . Modifying the generalized delta rule to train networks of nonmonotonic processors for pattern classification. Connection Science, 4, 19-31. Durbin, R. & Rumelhart, D. (1989). Product units: A computationally powerful and biologically plausible extension to backpropagation networks. Neural Computation, 1, 133- 142. Fahlman, S. & Lebiere, C . (1990) . The cascade-correlation learning architecture. In Touretzky, D. (Ed.), Advances in Neural Information Processing Systems, volume 2, (pp. 524-532)., San Mateo. (Denver 1989), Morgan Kaufmann. Frean, M. (1990). The upstart algorithm: A method for constructing and training feedforward neural networks. Neural Computation, 2, 198-209. Giles, C. & Maxwell, T . (1987). Learning, invariance, and generalization in highorder neural networks. Applied Optics, 26(23),4972-4978 . Jacobs, R. (1988). Increased rates of convergence through learning rate adaptation . Neural Networks, 1, 295-307. Karmarkar , N. & Karp, R . (1982). The differencing method of set partitioning. Technical Report UCB/CSD 82/113, Computer Science Division, University of California, Berkeley, California. Kirkpatrick, S., Jr., C. G ., , & Vecchi, M. (1983) . Optimization by simulated annealing. Science, 220. Reprinted in (?) . Lapedes, A. & Farber, R. (1987) . Nonlinear signal processing using neural networks: Prediction and system modelling. Technical Report LA-UR-87-2662, Los Alamos National Laboratory, Los Alamos, NM. Mezard, M. & Nadal, J.-P. (1989). Learning in feedforward layered networks: The tiling algorithm. Journal of Physics A, 22, 2191-2204. Redding, N., Kowalczyk, A., & Downs, T . (1993). A constructive higher order network algorithm that is polynomial-time. Neural Networks, 6,997. White, M. (1993). A public domain C implement ion of the Cascade Correlation algorithm. Department of Computer Science, Carnegie Mellon University, Pittsburgh , PA. 

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Stochastic Dynamics of Three-State Neural Networks Toru Ohira Sony Computer Science Laboratory 3-14-13 Higashi-gotanda, Tokyo 141, Japan ohira@csl.sony.co.jp Jack D. Cowan Depts. of Mathematics and Neurology University of Chicago Chicago, IL 60637 cowan@synapse.uchicago.edu Abstract We present here an analysis of the stochastic neurodynamics of a neural network composed of three-state neurons described by a master equation. An outer-product representation of the master equation is employed. In this representation, an extension of the analysis from two to three-state neurons is easily performed. We apply this formalism with approximation schemes to a simple three-state network and compare the results with Monte Carlo simulations. 1 INTRODUCTION Studies of single neurons or networks under the influence of noise have been a continuing item in neural network modelling. In particular, the analogy with spin systems at finite temprature has produced many important results on networks of two-state neurons. However, studies of networks of three-state neurons have been rather limited (Meunier, Hansel and Verga, 1989). A master equation was introduced by Cowan (1991) to study stochastic neural networks. The equation uses the formalism of "second quantization" for classical many-body systems (Doi, 1976a; Grassberger and Scheunert, 1980), and was used to study networks of of two-state neurons (Ohira and Cowan, 1993, 1994). In this paper, we reformulate the master equation using an outer-product representation of operators and extend our previous analysis to networks of three-state neurons. A hierarchy of moment equations for such networks is derived and approximation schemes are used to obtain equa- 272 Toru Ohira, Jack D. Cowan Figure 1: Transition rates for a three-state neuron. tions for the macroscopic activities of model networks. We compare the behavior of the solutions of these equations with Monte Carlo simulations. 2 THE BASIC NEURAL MODEL We first introduce the network described by the master equation. In this network (Cowan, 1991), neurons at each site, say the ith site, are assumed to cycle through three states: "quiescent", "activated" and "refractory", labelled 'qi', 'aj', and 'ri' respectively. We consider four transitions: q --t a, l' --t a, a --t 1', and " --t q. Two of these, q --t a and r --t a, are functions of the neural input current. We assume these are smoothly increasing functions of the input current and denoted them by fh, and (}2. The other two transition rates, a --t r, and r --t q, are defined as constants a and /3. The resulting stochastic transition scheme is shown in Figure 1. We assume that these transition rates depend only on the current state of the network and not on past states, and that all neural state transitions are asynchronous. This Markovian assumption is essential to the master equation description of this model. We represent the state of each neuron by three-dimensional basis vectors using the Dirac notation lai >, Iri > and Iqi >. They correspond, in more standard vector notation, to: Iq; >= (D, la;>= 0), I h >= ( I ~) o ?. (1) We define the inner product of these states as < adai >=< qdqj >=< 1'jlri >= 1, (2) < qjlaj >=< ajlqi >=< rdai >=< ad"i >=< 1'dqi >=< Qil " j >= O. (3) Let the states (or configurations) of a network be represented by product space of each neuron in the network. {In>}, the direct (4) Let p[n, t] be the probability of finding the network in a particular state n at time t. We introduce the "neural state vector" for N neurons in a network as I~(t) >= L p[n, t]l n >, {fl} (5) 273 Stochastic Dynamics of Three-State Neural Networks where the sum is taken over all possible network states. With these definitions, we can write the master equation for a network with the transition rates shown in Figure 1, using the outer-product representations of operators (Sakurai, 1985). For example: lai >< q;j = ( 0o 0 1) 0 0 0 0 0 (6) . ? The master equation then takes the form of an evolution equation: , 8 - 8t 1<I>(t) >= LI<I>(t) > with the network "Liouvillian" L given by: (7) N N 1 N L = a I)lai}(ad -Iri}(a;j) + L)h}(1'il-l a i}(,'d)e2(= wijlaj}(ajl) . . n . 1=1 .=1 )=1 L +,6L:~1 (Iri}(ril-Iqi}(r;!) + L:~l (lqi}(qil-lai}(qiI)OlUi L:7=1 wijlaj}(ajl).(8) where n is an average number of connections to each neuron, and Wij is the "weight" from the jth to the ith neuron. Thus the weights are normalized with respect to the average number n of connections per neuron. The master equation given here is the same as the one introduced by Cowan using Gell-Mann matrices (Cowan, 1991). However, we note that with the outer-product representation, we can extend the description from two to three-state neurons simply by including one more basis vector. In analogy with the analysis of two-state neurons, we introduce the state vector: N (a r cJI = IT (qi(q;! + ri(rd + ai(ail). (9) i=1 where the product is taken as a direct product, and ai, ri, and qi are parameters. We also introduce the point moments ?ai(t)>>, ?qi(t)>>, and ?ri(t)>> as the probability that the ith neuron is active, quiescent, and refract.ory respectively, at time t. Similarly, we can define the multiple moment, for example, ?aiqp'k .. , (t)>> as the probability that the ith neuron is active, the jth neuron is quiescent, the kth neuron is refractory and so on at time t. Then, it can be shown that they are given by: = (a = r = q= 1lsi}(sd @ ISj)(Sjl @ ISk)(Skl ... 1<I>(t)}, ?SiSjSk ... (t)>> S (10) = a, " ,q For example, ?riqjak(t)>> = (a = r = q = 1I"i}(ril @ Iqj}(qjl @ lak)(akl<I>(t)} (11) We note the following relations, ?ai(t)>> + ?qi(t)>> + ?ri(t)>> =1 (12) and ?aHt)>> = ?ai(t)>>, ?r;(t)>> = ?"i(t)>>, ?q;(t)>> = ?qi(t)>>, (13) Toru Ohira, Jack D. Cowan 274 3 THE HIERARCHY OF MOMENT EQUATIONS We can now obtain an equation of motion for the moments. As is typical in the case of many-body problems, we obtain an analogue of the BBGKY hierarchy of equations (Doi, 1976b). This can be done by using the definition of moments, the master equation, and the a-r-q state vector. We show the hierarchy up to the second order: a --at ?airj? = -a( ?aiaj? - ?airj?) + (3?ai 1'j? 1 N + ?air j(}2(=n~ '"' Wikak)>> k=l -?rirj(}2(~'E~=1 Wikak)>> - ?qjrj(}lUf'E:=l Wikak)>> (19) We note that since (20) ?ai? + ?ri? + ?qi? = 1, one of the parameters can be eliminated. We also note that the equations are coupled into higher orders in this hierarchy. This leads to a need for approximation schemes which can terminate the hierarchy at an appropriate order. In the following, we introduce first and the second moment level approximation schemes. For simplicity, we consider the special case in which (}l and (}2 are linear and equal. With the above simplication the first moment (mean field) approximation leads to: Stochastic Dynamics of Three-State Neural Networks -:t ?ai? = a?ai? - o 275 Wi( ?ri? + ~qi?) (22) j3_ - Ot ?ri? = -a?ai? + ?ri? + Wi?ri?, o - Ot ?qi? _ = - j3?ri? + Wi?qi?, (23) (24) where 1 N WI = iiL wlk?ak?, (25) k=l We also obtain the second moment approximation as: o - Ot ?ai? = a?ai? - 1 N ff~ Wij( ?qiaj? + ?riaj?), (26) )=1 o 1 N - Ot ?ri? = -a?ai? + j3?ri? + ff~ Wij?riaj?, (27) )=1 o - Ot ?qi? = -j3?ri? 1 N + ff~ Wij?qiaj?, (28) )=1 -:t o + ?qiaj?) -Wji( ?airj? + ?aiqj?), ?aiaj? = 2a?aiaj? - Wij( ?riaj? - Ot ?airj? = -a( ?aiaj? - (29) _ ?airj?) + j3?air j? + Wji?airj? -Wij(<<7'i7'j? + ?qi7'j?), (31) where (32) We note that the first moment dynamics obtained via the first approximation differs from that obtained from the second moment approximation, In the next section, we briefly examine this difference by comparing these approximations with Monte Carlo simulations. 276 4 Toru Ohira, Jack D. Cowan COMPARISON WITH SIMULATIONS In this section, we compare first and second moment approximations with Monte Carlo simulation of a one dimensional ring of three-state neurons. This was studied in a previous publication (Ohira and Cowan, 1993) for two-state neurons. As shown there, each three-state neuron in the ring interacts with its two neighbors. More precisely, the Liouville operator is N N L = a 2:)lai}(ail-/ri}(a;!) + f3 L(/ri}{ril-/qi}(ril) i=1 i=1 1 N +2W2 L(/ri}(ri/-l a i}(r;I)(/ai+l}(ai+d + /ai-l}{ai-d) i=1 1 N +2W1 L(/qi}{q;/ - /ai}(qil)(/ai+l}{ai+d + /ai-J}(ai-d) i=1 (33) We now define the dynamical variables of interest as follows: Xa 1 1 1 1 =N L i=1 ?aj?, Xr =N L ?ri?, Xq =N ;=1 N 1]aa = N L ?ai a i+l?, 1]rr ;=1 1 L ?qi?, (34) ;=1 N = N L ?ri r i+l?, 1]ar i=1 1 N =NL ?ai r i+l?. (35) i=1 Then, for this network, the first moment approximation is given by 8 - 8tXa 8 - at Xr Xq - ax - W2XaXr - WIXqXa, = -ax - f3Xr = 1 - Xa - Xr. + W2XqXa, (36) The second moment approximation is given by 8 - at Xa = 8 - 8t Xr = {) + 1) - - 8t1]aa - 2a1]aa - W21]ar(Xa 8 - 8t1]ar = -a('1aa -1]ar) - f31]ar Wl(Xa + 1)(Xa -1]ar -1]aa), 1 + 2 W2 '1ar(Xa + 1), 1 +21]rrXa + WIXa(Xr -1]rr -1]ar)' (37) 277 Stochastic Dynamics of Three-State Neural Networks Monte Carlo simulations of a ring of 10000 neurons were performed and compared with the first and second moment approximation predictions. We fixed the following parameters: (38) a = 1.0, f3 = 0.2, Wl = 0.01? WO, W2 = 0.6? Wo x",. ?.... - .................. __ .............. - ..... .. x. I .' " --------_ ....... -- .. _--x, ?.? x" .? X.'" (A) - -- ..... -- .... -_ .. ------_. ., ... .. .. ........... _----_ ........ _---- (8) Figure 2: Comparison of Monte Carlo simulations (dots) with the first moment (dashed line) and the second moment (solid line) approximations for the three state case with the fraction of total active and refractory state variables Xa (A) and Xr (B). Each graph is labeled by the values of wo/a. We varied Wo and sampled the numerical dynamics of these parameters. Some comparisons are shown in Figure 2 for the time dependence of the total number of active and refractory state variables. We dearly see the improvement of the second over the first moment level approximation. More simulations with different parameter ranges remain to be explored. 5 CONCLUSION We have introduced here a neural network master equation using the outer-product representation. In this representation, the extension from two to three-state neurons is transparent. We have taken advantage of this natural extension to analyse three-state networks. Even though the calculations involved are more intricate, we 278 Torn Ohira, Jack D. Cowan have obtained results indicating that the second moment level approximation is significantly more accurate than the first moment level approximation. We also note that as in the two-state case, the first moment level approximation produces more activation than the simulation. FUrther analytical and theoretical investigations are needed to fully uncover the dynamics of three-state networks described by the master equation introduced above. Acknowledgements This work was supported in part by the Robert R. McCOlmick fellowship at the University of Chicago, and in part by grant No. N0014-89-J-I099 from the US Department of the Navy, Office of Naval Research. References Cowan JD (1991) Stochastic neurodynamics in Advances in Neural Information Processing Systems (D. S. Touretzky, R. P. Lippman, J. E. Moody, ed.), vol. 3, Morgan Kaufmann Publishers, San Mateo Doi M (1976a) Second quantization representation for classical many-particle system. J. Phys. A: Math Gen. 9:1465-1477. Doi M (1976b) Stochastic theory of diffusion-controlled reactions. J. Phys. A: Math. Gen. 9:1479. Grassberger P, Scheunert M (1980) Fock-space methods for identical classical objects. Fortschritte der Physik 28:547 Meunier C, Hansel D, Verga A (1989) Information processing in three-state neural networks. J. Stat. Phys. 55:859 Ohira T, Cowan JD (1993) Master-equation approach to stochastic neurodynamics. Phys. Rev. E 48:2259 Ohira T, Cowan JD (1994) Feynman Diagrams for Stochastic Neurodynamics. In Proceedings of Fifth Australian Conference of Neural Networks, pp218-221 Sakurai JJ (1985) Modern Quantum Mechanics. Benjamin/Cummings, Menlo Park 

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Grouping Components of ? Three-Dimensional Moving Objects In Area MST of Visual Cortex Richard S. Zemel Carnegie Mellon University Department of Psychology Pittsburgh, PA 15213 zemel?lcmu. edu Terrence J. Sejnowski CNL, The Salk Institute P.O. Box 85800 San Diego, CA 92186-5800 terry?lsalk.edu Abstract Many cells in the dorsal part of the medial superior temporal (MST) area of visual cortex respond selectively to spiral flow patterns-specific combinations of expansion/ contraction and rotation motions. Previous investigators have suggested that these cells may represent self-motion. Spiral patterns can also be generated by the relative motion of the observer and a particular object. An MST cell may then account for some portion of the complex flow field, and the set of active cells could encode the entire flow; in this manner, MST effectively segments moving objects. Such a grouping operation is essential in interpreting scenes containing several independent moving objects and observer motion. We describe a model based on the hypothesis that the selective tuning of MST cells reflects the grouping of object components undergoing coherent motion. Inputs to the model were generated from sequences of ray-traced images that simulated realistic motion situations, combining observer motion, eye movements, and independent object motion. The input representation was modeled after response properties of neurons in area MT, which provides the primary input to area MST. After applying an unsupervised learning algorithm, the units became tuned to patterns signaling coherent motion. The results match many of the known properties of MST cells and are consistent with recent studies indicating that these cells process 3-D object motion information. 166 1 Richard S. Zemel, Terrence J. Sejnowski INTRODUCTION A number of studies have described neurons in the dorsal part of the medial superior temporal (MSTd) monkey cortex that respond best to large expanding/contracting, rotating, or shifting patterns (Tanaka et al., 1986; Duffy & Wurtz, 1991a). Recently Graziano et al. (1994) found that MSTd cell responses correspond to a point in a multidimensional space of spiral motions, where the dimensions are these motion types. Combinations of these motions are generated as an animal moves through its environment, which suggests that area MSTd could playa role in optical flow analysis. When an observer moves through a static environment, a singularity in the flow field known as the focus of expansion may be used to determine the direction of heading (Gibson, 1950; Warren & Hannon, 1988). Previous computational models of MSTd (Lappe & Rauschecker, 1993; Perrone & Stone, 1994) have shown how navigational information related to heading may be encoded by these cells. These investigators propose that each MSTd cell represents a potential heading direction and responds to the aspects of the flow that are consistent with that direction. In natural environments, however, MSTd cells are often faced with complex flow patterns produced by the combination of observer motion with other independentlymoving objects. These complex flow fields are not a single spiral pattern, but instead are composed of multiple spiral patterns. This observation that spiral flows are local subpatterns in flow fields suggests that an MSTd cell represents a particular regular subpattern which corresponds to the aspects of the flow field arising from a single cause-the relative motion of the observer and some object or surface in the scene. Adopting this view implies a new goal for MST: the set of MST responses accounts for the flow field based on the ensemble of motion causes. An MST cell that responds to a local subpattern accounts for a portion of the flow field, specifically the portion that arises from a single motion cause. In this manner, MST can be considered to be segmenting motion signals. As in earlier models, the MSTd cell responds to the aspects of flow consistent with its motion hypothesis, but here a cell's motion hypothesis is not a heading direction but instead represents the 3-D motion of a scene element relative to the observer. This encoding may be useful not only in robustly estimating heading detection, but may also facilitate several other tasks thought to occur further down the motion processing stream, such as localizing objects and parsing scenes containing multiple moving objects. In this paper we describe a computational model based on the hypothesis that an MST cell signals those aspects of the flow that arise from a common underlying cause. We demonstrate how such a model can develop response properties from the statistics of natural flow images, such as could be extracted from MT signals, and show that this model is able to capture several known properties of information processing in MST. 2 THE MODEL The input to the system is a movie containing some combination of observer motion, eye movements, and a few objects undergoing independent 3-D motion. An optical Grouping Components of 3-D Moving Objects in Area MST of Visual Cortex 167 flow algorithm is then applied to yield local motion estimates; this flow field is the input to the network, which consists of three layers. The first layer is designed after monkey area MT. The connectivity between this layer and the second layer is based on MST receptive field properties, and the second layer has the same connectivity pattern to the output layer. The weights on all connections are determined by a training algorithm which attempts to force the network to recreate the input pattern on the output units. We discuss the inputs, the network, and the training algorithm in more detail below. 2.1 STIMULI The flow field input to the network is produced from a movie. The various movies are intended to simulate natural motion situations. Sample situations include one where all motion is due to the observer's movement, and the gaze is in the motion direction. Another situation that produces a qualitatively different flow field is when the gaze is not in the motion direction. A third situation includes independent motion of some of the objects in the environment. Each movie is a sequence of images that simulates one of these situations. The images are created using a flexible ray-tracing program, which allows the simulation of many different objects, backgrounds, observer/camera motions, and lighting effects. We currently employ a database of 6 objects (a block of swiss cheese, a snail, a panther, a fish, a ball, and a teapot) and three different backgrounds. A movie is generated by randomly selecting one to four of the objects, and a background. To simulate one of the motion situations, a random selection of motion parameters follows: a). The observer's motion along (x, z) describes her walking; b). The eyes can rotate in (x, y), simulating the tracking of an object during motion; c). Each object can undergo independent 3-D motion. A sequence of 15 images is produced by randomly selecting 3-D initial positions and then updating the pose of the camera and each object in the image based on these motion parameters. Figure 1 shows 3 images selected from a movie generated in this manner. We apply a standard optical flow technique to extract a single flow field from each synthetic image sequence. Nagel's (1987) flow algorithm is a gradient-based scheme which performs spatiotemporal smoothing on the set of images and then uses a multi-resolution second-order derivative technique, in combination with an oriented smoothness relaxation scheme, to produce the flow field. 2.2 MODEL INPUT AND ARCHITECTURE The network input is a population encoding of these optical flow vectors at each location in a 21x31 array by small sets of neurons that share the same receptive field position but are tuned to different directions of motion. The values for each input unit is computed by projecting the local flow vector onto the cell's preferred direction. We are currently using 4 inputs per location, with evenly spaced preferred directions and a tuning half-width of 45?. This popUlation encoding in the input layer is intended to model the response of cells in area MT to a motion sequence. The receptive field (RF) of each model MT unit is determined by the degree of spatial smoothing and subsampling in the flow 

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Visual Speech Recognition with Stochastic Networks Javier R. Movellan Department of Cognitive Science University of California San Diego La Jolla, Ca 92093-0515 Abstract This paper presents ongoing work on a speaker independent visual speech recognition system. The work presented here builds on previous research efforts in this area and explores the potential use of simple hidden Markov models for limited vocabulary, speaker independent visual speech recognition. The task at hand is recognition of the first four English digits, a task with possible applications in car-phone dialing. The images were modeled as mixtures of independent Gaussian distributions, and the temporal dependencies were captured with standard left-to-right hidden Markov models. The results indicate that simple hidden Markov models may be used to successfully recognize relatively unprocessed image sequences. The system achieved performance levels equivalent to untrained humans when asked to recognize the fIrst four English digits. 1 INTRODUCTION Visual articulation is an important source of information in face to face speech perception. Laboratory studies have shown that visual information allows subjects to tolerate an extra 4-dB of noise in the acoustic signal. This is particularly important considering that each decibel of signal to noise ratio translates into a 10-15% error reduction in the intelligibility of entire sentences (McCleod and SummerfIeld, 1990). Lip reading alone provides a basis for understanding for a large majority of the hearing impaired and when supplemented by acoustic or electrical signals it allows fluent understanding of speech in highly trained subjects. However visual information plays more than a simple compensatory role in speech perception. From early on humans are predisposed to integrate acoustic and visual information. Sensitivity to correspondences in auditory and visual information for speech events has been shown in 4 month old infants (Spelke, 1976; Kuhl & Meltzoff, 1982). By 6 years of age, humans consistently use audio visual contingencies to understand speech (Massaro, 1987). By adulthood, visual articulation automatically modulates perception of the acoustic signal. Under laboratory conditions it is possible to create powerful illusions in which subjects mistakenly hear sounds which are biased by visual articulations. Subjects in these experiments are typically unaware cf 852 Javier Movellan the discrepancy between the visual and auditory tracks and their experience is that of a unified auditory percept (McGurk & McDonnald, 1976). Recent years have seen a revival of interest in audiovisual speech perception both in psychology and in the pattern recognition literature. There have been isolated efforts to build synthetic models of visual and audio-visual speech recognition (Petahan, 1985; Nishida, 1986; Yuhas, Goldstein, Sejnowski & Jenkins, 1988; Bregler, Manke, Hild & Waibel, 1993; Wolff, Prassad, Stork, & Hennecke, 1994). The main goal of these efforts has been to explore different architectures and visual processing techniques and to illustrate the potential use of visual information to improve the robustness of current speech recognition systems. Cognitive psychologists have also developed high level models of audio-visual speech perception that describe regularities in the way humans integrate visual and acoustic information (Massaro, 1987). In general these studies support the idea that human responses to visual and acoustic stimuli are conditional independent. This regularity has been used in some synthetic systems to simplify the task of integrating visual and acoustic signals (Wolff, Prassad, Stork, & Hennecke, 1994). Overall, multimodal speech perception is still an emerging field in which a lot cf exploration needs to be done. The work presented here builds on the previous research efforts in this area and explores the potential use of simple hidden Markov models 1ir limited vocabulary, speaker independent visual speech recognition. The task at hand is recognition of the first four English digits, a task with possible applications in car-phone dialing. 2 TRAINING SAMPLE The training sample consisted of 96 digitized movies of 12 undergraduate students (9 males,3 females) from the Cognitive Science Department at UCSD. Video capturing was performed in a windowless room at the Center for Research in Language at UCSD. Subjects were asked to talk into a video camera and to say the first four digits in English twice. Subjects could monitor the digitized images in a small display conveniently located in front of them. They were asked to position themselves so that that their lips be roughly centered in the feed-back display. Gray scale video images were digitized at 30 fps, 100x75 pixels, 8 bits per pixel. The video tracks were hand segmented by selecting a few relevant frames before and after the beginning and end of activity in the acoustic track. Statistics of the entire training sample are shown in table 1. Table 1: Frame number statistics. Digit "One" "Two" "Three" "Four" Average 8.9 9.6 9.7 10.6 S.D. 2.1 2.1 2.3 2.2 3 IMAGE PREPROCESSING There are two different approaches to visual preprocessing in the visual speech recognition literature (Bregler, Manke, Hild & Waibel, 1993). The first approach, represented by the work of Wolff and colleagues (Wolff, Prassad, Stork, & Hennecke, 1994) favors sophisticated image preprocessing techniques to extract a limited set of hand-crafted features (e.g., height and width of the lips). The advantage of this approach is that it Visual Speech Recognition with Stochastic Networks 853 drastically reduces the number of input dimensions. This translates into lower variability of the signal, potentially improved generalization, and large savings in computing time. The disadvantage is that vital information may be lost when compressing the image into a limited set of hand-crafted features. Variability is reduced at the possible expense cf bias. Moreover, tests have shown that subtle holistic features such as the wrinkling and protrusion of the lips may play an important role in human lip-reading (Montgomery & Jackson, 1983). The second approach to visual preprocessing emphasizes preserving the original images as much as possible and letting the recognition engine discover the relevant features in the images. In most cases, images are low-pass filtered and dimension-reduced by using principal component analysis. The results in this papers indicate that good results can be obtained even without the use of principal components. In this investigation image preprocessing consisted of the following phases: 1. Symmetry enforcement: At each time frame the raw images were symmetrized by averaging pixel by pixel the left and right side of each image, using the vertical midline as the axis of symmetry. For convenience from now on we will refer to the raw images as "rho-images" and the symmetrized images as "sigma-images." The potential benefits cf sigma-images are robustness, and compression, since the number of relevant pixels is reduced by half. 2. Temporal differentiaion: At each time frame we calculated the pixel by pixel differences between present sigma-images and immediately past sigma-images. For convenience we refer to the resulting images as "delta-images." One of the potential advantages of deltaimages in the visual domain is their robustness to changes in illumination and the fuct that they emphasize the dynamic aspects of the visual track. 3. Low pass filtering and subsampling: The sigma and delta images were compressed and subsampled using 20x 15 equidistant Gaussian filters. Different values of the standard deviation of the Gaussian filters were tested. 4. Logistic thresholding and scaling: The sigma and delta images were independently thresholded by feeding the output of the Gaussian filters through a according to the following equation Y = 256 j(K 7r r;; -v3a (x - J.L)) where f is the logistic function, and J.L, a, are respectively the average and standard deviation of the gray level distribution of entire image sequences. The constant K controls the sharpness of the logistic function. Assuming an approximately Gaussian distribution of gray levels when K=1 the thresholding function approximates histogram equalization, a standard technique in visual processing. Three different K values were tried: 0.3, 0.6 and 1.2. 5. Composites of the relevant portions of the blurred sigma and delta images were fed to the recognition network. The number of pixels of each processed image was 300 (150 from the blurred sigma images and 150 from the blurred delta images). Figure 1 shows the effect of the different preprocessing stages. 854 Javier Movellan Figure 1: Image Preprocessing. 1) Rho-Image. 2) Sigma-Image. 3) Delta-Image. 4) Filtered and Sharpened Composite. 4 RECOGNITION NETWORK We used the standard approach in limited vocabulary systems: a bank of hidden Markov models, one per word category, independently trained on the corresponding word categories. The images were modeled as mixtures of continuous probability distributions in pixel space. We tried mixtures of Gaussians and mixtures of Cauchy distributions. The mixtures of Cauchy distributions were very stable numerically but they did perform very poorly when compared to the Gaussian mixtures. We believe the reason for their poor performance is the tendency of Cauchy-based maximum-likelihood estimates to focus on individual exemplars. Gaussian-based estimates are much more prone to blend exemplars that belong to the same cluster. The initial state probabilities, transition probabilities, mixture coefficients, mixture centroids and variance parameters were trained using the E-M algorithm. We initially encountered severe numerical underflow problems when using the E-M algorithm with Gaussian mixtures. These instabilities were due to the fact that the probability densities of images rapidly went to zero due to the large dimensionality of the images. Trimming the outputs of the Gaussian and using very small Gaussian gains did not work well. We solved the numerical problems in the following way: 1) Constraining all the variance parameters for all the states and mixtures to be equal. This allowed pulling out a constant in the likelihood-function of the mixtures, avoiding most numerical problems. 2) Initializing the mixture centroids using linear segmentation followed by the K-means clustering algorithm. For example, ifthere were 4 visual frames and 2 states, the first 2 frames were assigned to state 1 and the last 2 frames to state 2. Kmeans was then used independently on each of the states and their assigned frames. This is a standard initialization method in the acoustic domain (Rabiner & Bing-Hwang, 1993). Since K-means can be trapped in local minima, the algorithm was repeated 20 times with different starting point and the best solution was fed as the starting point for the E-M algorithm. 5 RESULTS The main purpose of this study was to fmd simple image preprocessing techniques that would work well with hidden Markov models. We tested a wide variety of architectures and preprocessing parameters. In all cases the results were evaluated in terms cf generalization to new speakers. Since the training sample is small, generalization performance was estimated using the jackknife procedure. Models were trained with 11 Visual Speech Recognition with Stochastic Networks 855 subjects, leaving one subject out for generalization testing. The entire procedure was repeated 12 times, each time leaving a different subject out for testing. Results are thus based on 96 generalization trials (4 digits x 12 subjects x 2 observations per subject). In all cases we tested several preprocessing techniques using 20 different architectures with different number of states (1,3,5,7,9) and mixtures per state (1,3,5,7). To compare the effect of each processing technique we used the average generalization performance of the best 4 architectures out of the 20 architectures tested. 85 65 45 25 +---L._ _ Rho Sigma Delta+Sigma Figure 2: Average performance with the rho, sigma, and delta images. Figure 2 shows the effects of symmetry enforcement and temporal differentiation. Symmetry enforcement had the benefit of reducing the input dimensionality by half and, as the figure show it did not hinder recognition performance. Using delta images had a very positive effect on recognition performance, as the figure shows. Figure 3 shows the effect of varying the thresholding constant and the standard deviation of the Gaussian filters. Best performance was obtained with blurring windows about 4 pixel wide and with thresholding just about histogram equalization. 84 83 82 81 ---'-K=i.2 ---K=O.6 K=O.3 80 3 4 5 Stamdard Deviation of Gaussian Filter. Figure 3: Effect of blurring and sharpening. 856 Javier Movellan Table 2 shows the effects of variations in the number of states (S) and Gaussian mixtures (G) per state. The number within each cell is the percentage of simulations for which a particular combination of states and mixtures performed best out of the 20 architectures tested. Table 2: Effect of varying the the number of states (S) and Gaussian mixtures (G). SI S3 S5 S7 S9 Gl 0.00% 0.00% 3.12% 6.25% 6.25% G3 0.00% 21.87% 9.37% 12.5% 0.00% G5 0.00% 12.5% 15.62% 0.00% 3.12% G7 0.00% 6.25% 0.00% 3.12% 0.00% Best overall performance was obtained with about 3 states and 3 mixtures per state. Peak performance was also obtained with a 3-state, 3-mixture per state network, with a generalization rate of 89.58% correct. To compare these results with human performance, 9 subjects were tested on the same sample. Six subjects were normal hearing adults who were not trained in lip-reading. Three were hearing impaired with profound hearing loss and had received training in lip reading at 2 to 8 years of age. The mean correct response for normal subjects was 89.93 % correct, just about the same rate as the best artificial network. The hearing impaired had an average performance of 95.49% correct, significantly better than our network. Table 3: Confusion matrix of the best artificial system. "One" "Two" "Three" "Four" 1 100.00% 4.17% 12.5% 8.33% 2 0.00% 87.50% 0.00% 4.17% 3 0.00% 4.17% 83.33% 0.00% 4 0.00% 4.17% 4.17% 87.50% Table 4: Average human confusion matrix. "One" "Two" "Three" "Four" 1 89.36% 1.39% 9.25% 4.17% 2 0.46% 98.61% 3.24% 0.46% 3 8.33% 0.00% 85.64% 1.85% 4 1.85% 0.00% 1.87% 93.52% Tables 3 and 4 show the confusion matrices for the best network and the average confusion matrix with all 9 subjects combined. The correlation between these two matrices was 0.99. This means that 98% of the variance in human confusions can be accounted for by the artificial model. This suggests that the representational space learned by the artificial system may be a reasonable model of the representational space used by humans. Figure 5 shows the representations learned by a network with 6 states and 1 mixture per state. Each column is a different digit, starting with "one." Each row Visual Speech Recognition with Stochastic Networks 857 is a different temporal state. The two pictures within each cell are sigma and delta image centroids. As the figure shows, the identity of individual exemplars is lost but the underlying dynamics of the digits are preserved. The digits can be easily recognized when played as a movie. Figure 4: Dynamic representations learned by a simple network. 6 CONCLUSIONS This paper shows that simple stochastic networks, like hidden Markov models, can be successfully applied for visual speech recognition using relatively unprocessed images. The performance level obtained with these networks roughly matches untrained human performance. Moreover, the representational space learned by these networks may be a reasonable model of the representations used by humans. More research should be done to better understand how humans integrate visual and acoustic information in speech perception and to develop practical models for robust audio-visual speech recognition. References Bregler c., Manke S., Hild H. & Waibel A. (1993) Bimodal Sensor Integration on the Example of "Speech-Reading". Proc ICNN-93, 11,667-677. Kuhl P. & Meltzoff A (1982) The Bimodal Development of Speech in Infancy. Science, 218, 1138-1141. MacLeod A. & Summerfield Q. (1990) A Procedure for Measuring Auditory and Audiovisual Speech-Reception Measuring Thresholds for Sentences in Noise: Rationale, Evaluation and Recommendations for Use. British Journal of Audiology. 24,29-43. Massaro D. (1987) Speech Perception by Ear and Eye. In Dodd B. & Campbell R. (ed.) Hearing by Eye: The Psychology of Lip-Reading. London, LEA, 53-83. Massaro D., Cohen M & Getsi (1993) Long-Term Training, Transfer and Retention in Learning to Lip-read. Perception and Psychophysics, 53,549-562. 858 Javier Movellan McGurk H. & MacDonald J. (1976) Hearing Lips and Seeing Voices. Nature, 264, 126130. Montgomery A. & Jackson P. (1983) Physical Characteristics of the Lips Underlying Vowel Lipreading Performance. Journal of the Acoustical Society of America, 73,21342144. Nishida S. (1986) Speech Recognition Enhancement by Lip Information Proceedings of ACMlCHI86, 198-204. Petajan E. (1985) Automatic Lip Reading to Enhance Speech Recognition. IEEE CVPR 85, 40-47. Rabiner L., Bing-Hwang J. (1993) Fundamentals of Speech Recognition. New Jersey, Prentice Hall. Spelke E. (1976) 533-560. Infant's Intermodal Perception of Events. Cognitive Psychology, 8, Yuhas B., Goldstein T., Sejnowski T., Jenkins R. (1988) Neural Network Models cf Sensory Integration for Improved Vowel Recognition. Proceedings IEEE 78, 16551668. Wolff G., Prassad L., Stork D., Hennecke M. (1994) Lipreading by Neural Networks: Visual Preprocessing, Learning and Sensory Integration. In J. Cowan, G. Tesauro, J. Alspector (ed.), Advances in Neural Information Processing Systems 6, 1027-1035. San Mateo, CA: Morgan Kaufinann. 

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Single Transistor Learning Synapses Paul Hasler, Chris Diorio, Bradley A. Minch, Carver Mead California Institute of Technology Pasadena, CA 91125 (818) 395 - 2812 paul@hobiecat.pcmp.caltech.edu Abstract We describe single-transistor silicon synapses that compute, learn, and provide non-volatile memory retention. The single transistor synapses simultaneously perform long term weight storage, compute the product of the input and the weight value, and update the weight value according to a Hebbian or a backpropagation learning rule. Memory is accomplished via charge storage on polysilicon floating gates, providing long-term retention without refresh. The synapses efficiently use the physics of silicon to perform weight updates; the weight value is increased using tunneling and the weight value decreases using hot electron injection. The small size and low power operation of single transistor synapses allows the development of dense synaptic arrays. We describe the design, fabrication, characterization, and modeling of an array of single transistor synapses. When the steady state source current is used as the representation of the weight value, both the incrementing and decrementing functions are proportional to a power of the source current. The synaptic array was fabricated in the standard 21'm double - poly, analog process available from MOSIS. 1 INTRODUCTION The past few years have produced a number of efforts to design VLSI chips which "learn from experience." The first step toward this goal is developing a silicon analog for a synapse. We have successfully developed such a synapse using only Paul Hasler, Chris Diorio, Bradley A. Minch, Carver Mead 818 Gate Aoating Gate ! Drain Source ...I._High ,oltage --~~--'-~------r-~--~~--~ Figure 1: Cross section of the single transistor synapse. Our single transistor synapse uses a separate tunneling voltage terminal The pbase implant results in a larger threshold voltage, which results in all the electrons reaching the top of the Si02 barrier to be swept into the floating gate. a single transistor. A synapse has two functional requirements. First, it must compute the product of the input multiplied by the strength (the weight) of the synapse. Second, the synapse must compute the weight update rule. For a Hebbian synapse, the change in the weight is the time average of the product of the input and output activity. In many supervised algorithms like backpropagation, this weight change is the time average of the product of the input and some fed back error signal. Both of these computations are similar in function. We have developed single transistor synapses which simultaneously perform long term weight storage, compute the product of the input and the weight value, and update the weight value according to a Hebbian or a backpropagation learning rule. The combination of functions has not previously been achieved with floating gate synapses. There are five requirements for a learning synapse. First, the weight should be stored permanently in the absence of learning. Second, the synapse must compute as an output the product of the input signal with the synaptic weight. Third, each synapse should require minimal area, resulting in the maximum array size for a given area. Fourth, each synapse should operate with low power dissipation so that the synaptic array is not power constrained. And finally, the array should be capable of implementing either Hebbian or Backpropagation learning rule for modifying the weight on the floating gate. We have designed, fabricated, characterized, and modeled an array of single transistor synapses which satisfy these five criteria. We believe this is the first instance of a single transistor learning synapse fabricated in a standard process. 2 OVERVIEW Figure 1 shows the cross section for the single transistor synapse. Since the floating gate is surrounded by Si0 2 , an excellent insulator, charge leakage is negligible resulting in nearly permanent storage of the weight value. An advantage of using floating gate devices for learning rules is the timescales required to add and remove charge from the floating gate are well matched to the learning rates of visual and auditory signals. In addition, these learning rates can be electronically controlled. Typical resolution of charge on the floating gate after ten years is four bits (Holler 89). The FETs are in a moderately doped (1 x 10 17 em- 3 ) substrate, to achieve a Single Transistor Learning Synapses Vg1 t 819 Vg2 Vtun1 Vd1 (1,1) t (1,2) ---+----~--------~----~----VS1~ ---+-..-------------+_e_--------- Vtun2 _ Vd2 (2,1) (2,2) ---+--~~---1---~----VS2~ Figure 2: Circuit diagram of the single - transistor synapse array. Each transistor has a floating gate capacitively coupled to an input column line. A tunneling connection (arrow) allows weight increase. Weight decreased is achieved by hot electron injection in the transistor itself. Each synapse is capable of simultaneous feedforward computations and weight updates. A 2 x 2 section of the array allows us to characterize how modifying a single floating gate (such as synapse (1,1)) effects the neighboring floating gate values. The synapse currents are a measure of the synaptic weights, and are summed along each row by the source (Vs) or drain (Vd) lines into some soma circuit. high threshold voltage. The moderately doped substrate is formed in the 2pm MOSIS process by the pbase implant. npn transistor. The implant has the additional benefit of increasing the efficiency of the hot electron injection process by increasing the electric field in the channel. Each synapse has an additional tunneling junction for modifying the charge on the floating gate. The tunneling junction is formed with high quality gate oxide separating a well region from the floating gate. Each synapse in our synaptic array is a single transistor with its weight stored as a charge on a floating silicon gate. Figure 2 shows the circuit diagram of a 2 x 2 array of synapses. The column 'gate' inputs (Vg) are connected to second level polysilicon which capacitively couples to the floating gate. The inputs are shared along a column. The source (Vs), drain (lid), and tunneling (Viun) terminals are shared along a row. These terminals are involved with computing the output current and feeding back 'error' signal voltages. Many other synapses use floating gates to store the weight value, as in (Holler 89), but none of the earlier approaches update the charge on the floating gate during the multiplication of the input and floating gate value. In these previous approaches one must drive the floating gate over large a voltage range to tunnel electrons onto the floating gate. Synaptic computation must stop for this type of weight update. The synapse computes as an output current a product of weight and input signal, 820 Paul Hasler. Chris Diorio. Bradley A. Minch. Carver Mead 10.7 Tunneling Opel1llions synapse (1.2) ... . . . '. Iojec:tion Opel1lliolll 10" synapse (2,1), synapse (2.2) 10" ". "" synapse (1.1) ... ..- 15 S j 10. 10 10.11 10- 12 row 2 background alln:nl row I IJeckarouod alrlent 10-13 "-_--'-_ _-'-_ _-'--_ _'"--_----'_ _--'-_ _...J o SO I 00 I SO 200 2S0 300 3SO Step iteration Number Figure 3: Output currents from a 2 x 2 section of the synapse array, showing 180 injection operations followed by 160 tunneling operations. For the injection operations, the drain (V dl) is pulsed from 2.0 V upto 3.3 V for 0.5s with Vg1 at 8V and Vg2 at OV. For the tunneling operations, the tunneling line (Vtunl) is pulsed from 20 V up to 33.5 V with Vg2 at OV and "91 at 8V. Because our measurements from the 2 x 2 section come from a larger array, we also display the 'background' current from all other synapses on the row. This background current is several orders of magnitude smaller than the selected synapse current, and therefore negligible. and can simultaneously increment or decrement the weight as a function of its input and error voltages. The particular learning algorithm depends on the circuitry at the boundaries of the array; in particular the circuitry connected to each of the source, drain, and tunneling lines in a row. With charge Qlg on the floating gate and Vs equal to 0 the subthreshold source current is described by (1) k;. where Qo is a device dependent parameter, and UT is the thermal voltage The coupling coefficient, 6, of the gate input to the transistor surface potential is typically less than 0.1. From ( 1) We can consider the weight as a current I, defined by ISllnapse = ( ~~) Ioe---qo- e T e 6gVp T = Ie 6gVp T (2) where Vgo is the input voltage bias, and ~ Vg is Vg - "90. The synaptic current is thus the product of the weight, I, and a weak exponential function of the input voltage. The single transistor learning synapses use a combination of electron tunneling and hot electron injection to adapt the charge on the floating gate, and thereby the 821 Single Transistor Learning Synapses weight of the synapse. Hot electron injection adds electrons to the floating gate, thereby decreasing the weight. Injection occurs for large drain voltages; therefore the floating gate charge can be reduced during normal feedforward operation by raising the drain voltage. Electron tunneling removes electrons from the floating gate, thereby increasing the weight. The tunneling line controls the tunneling current; thus the floating gate charge can be increased during normal feedforward operation by raising the tunneling line voltage. The tunneling rate is modulated by both the input voltage and the charge on the floating gate. Figure 3 shows an example the nature of the weight update process. The source current is used as a measure of the synapse weight. The experiment starts with all four synapses set to the same weight current. Then, synapse (1,1) is injected for 180 cycles to preferentially decrease its weight. Finally, synapse (1,1) is tunneled for 160 cycles to preferentially increase its weight. This experiment shows that a synapse can be incremented by applying a high voltage on tunneling terminals and a low voltage on the input, and can be decremented by applying a high voltage on drain terminals and a high voltage on the input. In the next two sections, we consider the nature of these update functions. In section three we examine the dependence of hot electron injection on the source current of our synapses. In section four we examine the dependence of electron tunneling on the source current of our synapses. 3 Hot Electron Injection Hot electron injection gives us a method to add electrons to the floating gate. The underlying physics of the injection process is to give some electrons enough energy and direction in the channel to drain depletion region to surmount the Si0 2 energy barrier. A device must satisfy two requirements to inject an electron on a floating gate. First, we need a region where the potential drops more than 3.1 volts in a distance of less than 0.2pm to allow electrons to gain enough energy to surmount the oxide barrier. Second, we need a field in the oxide in the proper direction to collect electrons after they cross the barrier. The moderate substrate doping level allows us to easily achieve both effects in subthreshold operation. First, the higher substrate doping results in a much higher threshold voltage (6.1 V), which guarantees that the field in the oxide at the drain edge of the channel will be in the proper direction for collecting electrons over the useful range of drain voltages. Second, the higher substrate doping results in higher electric fields which yield higher injection efficiencies. The higher injection efficiencies allow the device to have a wide range of drain voltages substantially below the threshold voltage. Figure 4 shows measured data on the change in source current during injection vs. source current for several values of drain voltage. Because the source current, I, is related to the floating gate charge, Q,g as shown in ( 1) and the charge on the floating gate is related to the tunneling or injection current (I, g) by dQ,g _ I (3) dt - Ig an approximate model for the change of the weight current value is I (4) -dl = -1 dt Qo '9 Paul Hasler, Chris Diorio, Bradley A. Minch, Carver Mead 822 ! 10" i 10.10 ~ 10'" synapse (1,1) i!, J. .S 3.3 10.12 g synapse (1,2) 10'" 3.6 Sour.-e Curren! (Weiaht Value) Figure 4: Source Current Decrement during injection vs. Source Current for several values of drain voltage. The injection operation decreases the synaptic weight. V 92 was held at OV, and V 91 was at 8V during the 0.5s injecting pulses. The change in source current is approximately proportional to the source current to the f3 power, where of f3 is between 1.7 and 1.85 for the range of drain voltages shown. The change in source current in synapse (1,2) is much less than the corresponding change in synapse (1,1) and is nearly independent of drain voltage. The effect of this injection on synapses (2,1) and (2,2) is negligible. The injection current can be approximated over the range of drain voltages shown in Fig. 4 by (Hasler 95) (5) where Vd-c is the voltage from the drain to the drain edge of the channel, Vd is the drain voltage, /0 is a slowly varying function defined in (Hasler 95), and Vini is in the range of 60m V to 100mV. A is device dependent parameter, Since hot electron injection adds electrons to the floating gate, the current into the floating gate (If 9) is negative, which results in IfJ ~ -dI = -A-e .nJ (6) dt Qo The model agrees well with the data in Fig. 4, with f3 in the range of 1. 7 - 1.9. Injection is very selective along a row with a selectivity coefficient between 102 and 107 depending upon drain voltage and weight. The injection operations resulted in negligible changes in source current for synapses (2,1) and (2,2). 4 ELECTRON TUNNELING Electron tunneling gives us a method for removing electrons from the floating gate. Tunneling arises from the fact that an electron wavefunction has finite extent. For a Single Transistor Learning Synapses i 10" I 'Ii i! 10.9 ~ VlUn I a )10. 823 =36.0 ?' 3S.0 ..-- 10 34.0 3S.S 34.S .5 110. 11 33.S 33.0 Source Cumnt (Weig/ll Value) Figure 5: Synapse (1,1) source current increment vs. Source Current for several values of tunneling voltage. The tunneling operation increases the synaptic weight. V 91 was held at OV and V 92 was 8V while the tunneling line was pulsed for 0.5s from 20V to the voltage shown. The change in source current is approximately proportional to the a power of the svurce current where a is between 0.7 and 0.9 for the range of tunneling voltages shown. The effect of this tunneling procedure on synapse (2,1) and (2,2) are negligible. The selectivity ratio of synapses on the same row is typically between 3-7 for our devices. thin enough barrier, this extent is sufficient for an electron to penetrate the barrier. An electric field across the oxide will result in a thinner barrier to the electrons on the floating gate. For a high enough electric field, the electrons can tunnel through the oxide. When traveling through the oxide, some electrons get trapped in the oxide, which changes the barrier profile. To reduce this trapping effect we tunnel through high quality gate oxide, which has far less trapping than interpoly oxide. Both injection and tunneling have very stable and repeatable characteristics. When tunneling at a fixed oxide voltage, the tunneling current decreases only 50 percent after lOnG of charge has passed through the oxide. This quantity of charge is orders of magnitude more than we would expect a synapse to experience over a lifetime of operation. Figure 5 shows measured data on the change in source current during tunneling as a function of source current for several values of tunneling voltage. The functional form of tunneling current is of the form (Lenzlinger 69) (7) where Yo, f Olun are model parameters which roughly correspond with theory. Tunneling removes electrons from the floating gate; therefore the floating gate current is positive. By expanding Vfg for fixed Viun as VfgO + ~ Vfg and inserting ( 1), the 824 Paul Hasler, Chris Diorio, Bradley A. Minch, Carver Mead II Parameter I Typical Values II Parameter I Typical Values II {3 Q' 1.7 - 1.9 0.7 - 0.9 .2 pC 6 0.02 - 0.1 Qo 8.6 x 10 ?~u A 78mV Vinj Table 1: Typical measured values of the parameters in the modeling of the single transistor synapse array. resulting current change is dI I dt = Q:n e- Viu Vg v/ gO Iso ( I Iso )Q (8) where IsO is the bias current corresponding to VfgO. The model qualitatively agrees with the data in Fig. 5, with Q' in the range of 0.7- 0.9. The tunneling selectivity between synapses on different rows is very good, but tunneling selectivity along along a row is poor. We typically measure tunneling selectivity ratios along a row between 3 - 7 for our devices. 5 Model of the Array of Single Transistor Synapses Finally, we present an approximate model of our array of these single transistor synapses. The learning increment of the synapse at position (i,i) can be modeled as 61l.V ISllnapse'j = Iije ~ == Iso WijXj dW. . 1 _ Vq _ ;d; _ (9) =..:.:...!.L =.:.?t.II..A.e VI"n v/ gO W~x~ 1 _....d.... e inj WP.xl? 1 dt Qo I) ) Qo I) ) for the synapse at position (i,i), where Wi,j can be considered the weight value, and x j are the effective inputs network. Typical values for the parameters in ( 9 ) are given in Table 1. Acknowledgments The work was supported by the office of Naval Research, the Advanced Research Projects Agency, and the Beckman Foundation. References P. Hasler, C. Diorio, B. Minch, and C. Mead (1995) "An Analytic model of Hot Electron Injection from Boltzman Transport", Tech. Report 123456 M. Holler, S. Tam, H. Castro, and R. Benson (1989), "An electrically trainable artificial neural network with 10240 'floating gate' synapses", International Joint Conference on Neural Networks, Wasllington, D.C., June 1989, pp. 11-191 - 11-196. M. Lenzlinger and E. H. Snow (1969), "Fowler-Nordheim tunneling into thermally grown Si0 2 ," J. Appl. Phys., vol. 40, pp. 278-283, 1969. PARTVll SPEECH AND SIGNAL PROCESSING 

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Bias, Variance and the Combination of Least Squares Estimators Ronny Meir Faculty of Electrical Engineering Technion, Haifa 32000 Israel rmeirGee.technion.ac.il Abstract We consider the effect of combining several least squares estimators on the expected performance of a regression problem. Computing the exact bias and variance curves as a function of the sample size we are able to quantitatively compare the effect of the combination on the bias and variance separately, and thus on the expected error which is the sum of the two. Our exact calculations, demonstrate that the combination of estimators is particularly useful in the case where the data set is small and noisy and the function to be learned is unrealizable. For large data sets the single estimator produces superior results. Finally, we show that by splitting the data set into several independent parts and training each estimator on a different subset, the performance can in some cases be significantly improved. Key words: Bias, Variance, Least Squares, Combination. 1 INTRODUCTION Many of the problems related to supervised learning can be boiled down to the question of balancing bias and variance. While reducing bias can usually be accomplished quite easily by simply increasing the complexity of the class of models studied, this usually comes at the expense of increasing the variance in such a way that the overall expected error (which is the sum of the two) is often increased. 296 Ronny Meir Thus, many efforts have been devoted to the issue of decreasing variance, while attempting to keep the concomitant increase in bias as small as possible. One of the methods which has become popular recently in the neural network community is variance reduction by combining estimators, although the idea has been around in the statistics and econometrics literature at least since the late sixties (see Granger 1989 for a review). Nevertheless, it seems that not much analytic work has been devoted to a detailed study of the effect of noise and an effectively finite sample size on the bias/variance balance. It is the explicit goal of this paper to study in detail a simple problem of linear regression, where the full bias/variance curve can be computed exactly for any effectively finite sample size and noise level. We believe that this simple and exactly solvable model can afford us insight into more complex non-linear problems, which are at the heart of much of the recent work in neural networks. A further aspect of our work is related to the question of the partitioning of the data set between the various estimators. Thus, while most studies assume the each estimator is trained on the complete data set, it is possible to envisage a situation where the data set is broken up into several subsets, using each subset of data to form a different estimator. While such a scheme seems wasteful from the bias point of view, we will see that in fact it produces superior foreca..,ts in some situations. This, perhaps suprising, result is due to a large decrease in variance resulting from the independence of the estimators, in the case where the data subsets are independent. 2 ON THE COMBINATION OF ESTIMATORS The basic objective of regression is the following: given a finite training set, D, composed of n input/output pairs, D = {(XIJ' YIJ)}~=l' drawn according to an unkown distribution P(x, y), find a function (,estimator'), f(x; D), which 'best' approximates y. Using the popular mean-squared error criterion and taking expectations with respect to the data distribution one finds the well-known separation of the error into a bias and variance terms respectively (Geman et al. 1992) + ED (f(x; D) - EDf(x; D)]2 . (1) We consider a data source of the form y = g(x) + 7], where the 'target' function g(x) ?(x) = (EDf(x; D) - E[ylx])2 is an unknown (and potentially non-linear) function and 7] is a Gaussian random variable with zero mean and variance (72. Clearly then E[Ylx] = g(x). In the usual scenario for parameter estimation one uses the complete data set, D, to form an estimator f(x; D). In this paper we consider the case where the data set D is broken up into I< subsets (not necessarily disjoint), such that D = 1 DC k) , and a separate estimator is found for each subset. The full estimator is then given by the linear combination (Granger 1989) nf= K f(x; D) = L bk/k(x; D(k) (2) k=l The optimal values of the parameters bk can be easily obtained if the data distribution, P(x, y), is known, by simply minimizing the mean-squared error (Granger Bias. Variance and the Combination of Least Squares Estimators 297 1989). In the more typical case where this distribution is unkown, one may resort to other schemes such as least-squares fitting for the parameter vector b = {b l , ... , bK}. The bias and variance of the combined estimator can be simply expressed in this case, and are given by B(x; g) = (t k=l hh:(x) - g(x)) 2 ; V(x; g) = L bkh ' {lkfk'(X) - fk(X)fk'(X)} k,k' (3) where the overbars denote an average with respect to the data. It is immediately apparent that the variance term is composed of two contributions. The first term, corresponding to k = k', simply computes a weighted average of the single estimator variances, while the second term measures the average covariance between the different estimators. While the first term in the variance can be seen to decay as 1/ J{ in the case where all the weights h are of the same order of magnitude, the second term is finite unless the covariances between estimators are very small. It would thus seem beneficial to attempt to make the estimators as weakly correlated as possible in order to decrease the variance. Observe that in the extreme case where the data sets are independent of each other, the second term in the variance vanishes identically. Note that the bias term depends only on single estimator properties and can thus be computed from the theory of the single estimator. As mentioned above, however , the second term in the variance expression explicitly depends on correlations between the different estimators, and thus requires the computation of quantities beyond those of single estimators. 3 THE SINGLE LINEAR ESTIMATOR Before considering the case of a combination of estimators, we first review the case of a single linear estimator, given by f(x; D) = wT . x, where w is estimated from the data set D. Following Bos et al. (1993) we further assume that the data arises through an equation of the form y = g(x) + 1] with 9 = g(w~ . x). Looking back at equations (3) it is clear that the bias and variance are explicit functions of x and the weight vector woo In order to remove the explicit dependence we compute in what follows expectations with respect to the probability distribution of x and Wo, denoted respectively by E p ['] and E o[']' Thus, we define the averaged bias and variance by B = EoEp[B(x; wo)] and V = EoEp[V(x; wo)] and the expected error is then ? = B + V. In this work we consider least-squares estimation which corresponds to minimizing the empirical error, ?emp(w, D) = IIXw - Y112, where X is the n x d data matrix, Y is the n x 1 output vector and w is a d x 1 weight vector. The components of the 'target' vector Yare given by YP = g(w~ . xl-') + 1]p where 1Jp are i.i.d normal random variables with zero mean and variance (J'2 . Note that while we take the estimator itself to be linear we allow the target function g(.) to be non-linear. This is meant to model the common situation where the model we are trying to fit is inadequate, since the correct model (even it exists) is usually unkown. Thus, the least squares estimator is given by wE argminw?emp(w,D). Since in this case the error-function is quadratic it possesses either a unique global minimum 298 Ronny Meir or a degenerate manifold of minima, in the case where the Hessian matrix, XT X, is singular. The solution to the least squares problem is well known (see for example Scharf 1991), and will be briefly summarized. When the number of examples, n, is smaller than the input dimension, d, the problem is underdetermined and there are many solutions with zero empirical error. The solutions can be written out explicitly in the form (n < d), (4) where V is an arbitrary d-dimensional vector. It should be noted that any vector w satisfying this equation (and thus any least-squares estimator) becomes singular as n approaches d from below, since the matrix X XT becomes singular. The minimal norm solution, often referred to as the Moore-Penrose solution, is given in this case by the choice V O. It is common ill the literature to neglect the study of the underdetermined regime since the solution is not unique in this case. We however will pay specific attention to this case, corresponding to the often prevalent situation where the amount of data is small, attempting to show that the combination of estimators approach can significantly improve the quality of predictors in this regime. Moreover, many important inverse problems in signal processing fall into this category (Scharf 1991). = In the overdetermined case, n > d (assuming the matrix X to be of full rank), a zero error solution is possible only if the function g( .) is linear and there is no noise, namely El1]2] = O. In any other case, the problem becomes unrealizable and the minimum error is non-zero. In any event, in this regime the unique solution minimizing the empirical error is given by W = (XT X)-l XTy (n > d). (5) It is eay to see that this estimator is unbiased for linear g( .). In order to compute the bias and variance for this model we use Eqs. (3) with K 1 and bTl: 1. In order to actually compute the expectations with respect to x and the weight vector Wo we assume in what follows that the random vector x is distributed according to a multi-dimensional normal distributions of zero mean and covariance matrix (lid)!. The vector Wo is similarly distributed with unit covariance matrix. The reason for the particular scaling chosen for the covariance matrices will become clear below. In the remainder of the paper we will be concerned with exact calculations in the so called thermodynamic limit: n, d -+ 00 and a = nl d finite. This limit is particularly useful in that the central limit theorem allows one to make precise statements about the behavior of the system, for an effectively finite sample size, a. We note in passing that in the thermodynamic limit, d -+ 00, we have Ei -+ 1 with probability 1 and similarly for (lid) Ei W5i' Using these simple distributions we can, after some algerbra, directly compute the bias and variance. Denoting R = Eo[w T . wo], r = Eollwll 2 , Q = Eollwl1 2 , one can show that the bias and variance are given by = = x; B = 7' - 2ugR + g2 v = Q -r. (6) = In the above equations we have used g2 = f Dug 2(u) and ug f Du ug(u) where the Gaussian measure Du is defined by Du = (e-u~/2 1...j2-i)du. We note in passing Bias, Variance and the Combination of Least Squares Estimators 299 that the same result is obtained for any i.i.d variables, Xi, with zerO mean and variance lid. We thus note that a complete calculation of the expected bias and variance requires the explicit computation of the variables R, rand Q defined above. In principle, with the explicit expressions (4) and (5) at hand one may proceed to compute all the quantities relevant to the evaluation of the bias and variance. Unfortunately, it turns out that a direct computation of r, Rand Q using these expressions is a rather difficult task in the theory of random matrices, keeping in mind the potential non-linearity of the function g{.). A way to solve the problem can be undertaken via a slightly indirect route, using tools from statistical physics. The variables Rand Q above have been recently computed by Bos et al. (1993) using replicas and by Opper and Kinzel (1994) by a direct calculation. The variable r can be computed along similar lines resulting in the following expressions for the bias and variance (given for brevity for the Moore-Penrose solution): 0'< 1 : v= B _0'_ [g2 1-0' + u2 _ a> 1 : 0'(2 _ a )ttg2] (7) We see from this solution that for Q > 1 the bias is constant, while the variance is monotonically decreasing with the sample size Q. For Q < 1, there are of course multiple solutions corresponding to different normalizations Q. It is easy to see, however, that the Moore-Penrose solution, gives rise to the smallest variance of all least-squares estimators (the bias is unaffected by the normalization of the solution) . The expected (or generalization) error is given simply by ? = B + V, and is thus smallest for the Moore-Penrose solution. Note that this latter result is independent of whether the function g(.) is linear or not. We note in passing that in the simple case where the target function g(-) is linear and the data is noise-free (u 2 = 0) one obtains the particularly simple result ? = 1 - a for a < 1 and ? = 0 above Q = 1. Note however that in any other case the expected error is a non-linear function of the normalized sample size a. 4 COMBINING LINEAR ESTIMATORS Having summarized the situation in the case of a single estimator, we proceed to the case of K linear estimators. In this case we assume that the complete data set, D, is broken up into K subsets D(k) of size nk = Qkd each. In particular we consider two extreme situations: (i) The data sets are independent of each other , namely D(k) n D(k') = 0 for all k, and (ii) D(k) = D for all k. We refer to the first situation as the non-overlapping case, while to the second case where all estimators are trained on the same data set as the fully-overlappin~ case. Denoting by w(k) the k'th least-squares estimator, based on training set D( ), we define the following quantities: R(k) = Eo [w(k)T . w o] , r(k,k') = Eo [w(k)T . w(k')] , p(k,k') = Eo [w(k)T . w(k')] . (8) Making use of Eqs. (3) and the probability distribution of Wo one then straighfor- 300 Ronny Meir wardly finds for the mixture estimator L L b~ blebk,r(k,Ie') - 2ug 1e,Ie' L bkR(k) + g2, Ie [p<k,le) - r(k,Ie)] Ie + L: bleb le , [p(k,k') _ r(k,k')] . (9) k#Ie' Now, the computation of the parameters p(le,k) and R(k) is identical to the case of a single estimator and can be directly read from the results of Bos et al. (1993) with the single modification of rescaling the sample size a to ak. The only interaction between estimators arises from the terms p(Ie,k') and r(k ,k') which couple two different estimators. Note however that r(k,k') is independent of the degree of overlap between the data sets D(Ie) since each estimator is averaged independently. In fact , a straighforward, if lengthy, calculation leads to the conclusion that in general r(Ie,Ie') = R(Ie) R(1e'). For the case where the data sets are independent of each other, namely D(k) n D(k') = 0 it is obvious that p(k ,k') r(k,k'). One then finds p(Ie,Ie') = r(Ie,Ie') = R(Ie) R(k') for independent data sets. In the case where the data sets are identical, D(k) = D for all k, the variable p(k,k') has been computed by BOs et al. (1993). At this point we proceed by discussing individually the two data generation scenarios described in section 1. = 4.1 NON?OVERLAPPING DATA SETS As shown above, since the data sets are independent of each other one has in this case p(le,k') = r(k,k'), implying that the second term in the expression for the variance in Eq. (9) vanishes. Specializing now to the case where the sizes of the data sets are equal, namelyak nk/d a/I{ for all k, we expect the variables R(k) and O(k) to be independent of the specific index k. In the particular case of a uniform mixture (b k = 1/1< for all k) we obtain = = (10) where the subscript 1< indicates that the values of the corresponding variables must be taken with respect to aK = a/1<. The specific values of the parameters RK and OK can be found directly from the solution of Bos et al. (1993) by replacing a by a/I{ and dividing the variance term by a further ]{ factor. Since the bias is a non-increasing function a it is always increased (or at best unchanged) by combining estimators from non-overlapping data subsets. It can also be seen from the above equation that as 1< increases the variance can be made arbitrarily small, albeit at the expense of increasing the bias. It can thus be expected that for any sample size a there is an optimal number of estimators minimizing the expected error, ? = B + V. In fact, for small a one may expand the equations for the bias and variance obtaining the expected error from which it is easy to see that the optimal value of 1< is given by 1<* = (g2 + (72)/ug 2. As could be expected we see that the optimal value of I{ scales with the noise level (72, since the effect of combining multiple independent estimators is to reduce the effective noise variance by a factor of I{. As the sample size a increases and outweighs the effect of the noise, we find that the bias increase due to the data partitioning dominates the Bias, Variance and the Combination of Least Squares Estimators 301 decrease in variance and the combined estimator performs more and more poorly relative to the single estimator. Finally, for a > K we find the choice 1< = 1 always yields a lower expected error. Thus, while we have shown that for small sample sizes the effect of splitting the data set into independent subsets is helpful, this is no longer the case if the sample size is sufficiently large, in which case a single estimator based on the complete data set is superior. For a -+ 00, however , one finds that all uniform mixtures converge (to leading order) at the same rate, namely ?K(a) : : : : ?00 + (?00 + (!2)ja , where ?00 = g2 - ug 2. For finite values of cr, however, the value of 1< has a strong effect on the quality of the stimator. 4.2 FULLY OVERLAPPING DATA SETS We focus now on the case where all estimators are formed from the same data set D, namely DUe) = D for all k . Since there is a unique solution to the least-squares estimation problem for a > 1, all least-squares estimators coincide in this regime . Thus, we focus here on the case a < 1, where multiple least-squares estimators coexist. We further assume that only mixtures of estimators of the same norm Q are allowed. We obtain for the uniform mixture Clearly the expression for the bias in this case is identical to that obtained for the single estimator, since all estimators are based On the same data set and the bias term depends only on single estimator properties. The variance term , however , is modified due to the correlation between the estimators expressed through the variable p(k,k'). Since the variance for the case of a single estimator is Q - R2 and since q $ Q it is clear in this case that the variance is reduced while the bias remains unchanged. Thus we conclude that the mixture of estimators in this case indeed produces superior performance to that of the single estimator. However, it can be seen that in the case of the Moore-Penrose solution , corresponding to choosing the smallest possible norm Q, the expected error is minimal. We thus conclude that for a < 1 the Moore-Penrose pseudo-inverse solution yields the lowest expected error , and this cannot be improved on by combining least-squares estimators obtained from the full data set D . Recall that we have shown in the previous section that (for small and noisy data sets) combining estimators formed using non-overlapping data subsets produced results superior to those of any single estimator trained on the complete data set. An interesting conclusion of these results is that splitting the data set into nonoverlapping subsets is a better strategy than training each estimator with the full data. As mentioned previously, the basic reason for this is the independence of the estimators formed in this fashion, which helps to reduce the variance term more drastically than in the case where the estimators are dependent (having been exposed to overlapping data sets). 5 CONCLUSIONS In this paper we have studied the effect of combining different estimators on the performance of linear regression. In particular we have focused on the case of linear 302 Ronny Meir least-squares estimation, computing exactly the full bias and variance curves for the case where the input dimension is very large (the so called thermodynamic limit). While we have focused specifically on the case of linear estimators, it should not be hard to extend these results to simple non-linear functions ofthe form f(w T ?x) (see section 2). The case of a combination of more complex estimators (such as multilayered neural networks) is much more demanding, as even the case of a single such network is rather difficult to analyzes. Several positive conclusions we can draw from our study are the following. First, the general claim that combining experts is always helpful is clearly fallacious. While we have shown that combining estimators is beneficial in some cases (such as small noisy data sets), this is not the case in general. Second, we have shown that in some situations (specifically unrealizable rules and small sample size) it is advantageous to split the data into several non-overlapping subsets. It turns out that in this case the decrease in variance resulting from the independence of the different estimators, is larger than the concomitant increase in bias. It would be interesting to try to generalize our results to the case where the data is split in a more efficient manner. Third, our results agree with the general notion that when attempting to learn an unrealizable function (whether due to noise or to a mismatch with the target function) the best option is to learn with errors. Ultimately one would like to have a general theory for combining empirical estimators. Our work has shown that the effect of noise and finite sample size is expected to produce non-trivial effects which are impossible to observe when considering only the asymptotic limit. Acknowledgements The author thanks Manfred Opper for a very helpful conversation and the Ollendorff center of the Electrical Engineering department at the Technion for financial support. References S. Bos., W. Kinzel and M. Opper 1993, The generalization ability of perceptrons with continuous outputs, Phys. Rev. A 47:1384-1391. S. Geman, E. Bienenstock and R. Dorsat 1992, Neural networks and the bias/variance dilemma, Neural Computation 4:1-58. C.W.J. Granger 1989, Combining forecasts - twenty years later, J. of Forecast. 8:167-173. M. Opper and W. Kinzel 1994, Statistical mechanics of generalization, in Physics of Neural networks, van Hemmen, J.S., E. Domany and K. Schulten eds., SpringerVerlag, Berlin. L.1. Scharf, 1991 Statistical Signal Processing: Detection, Estimation and Time Series Analysis, Addison-Wesley, Massachusetts. 

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A Real Time Clustering CMOS Neural Engine T. Serrano-Gotarredona, B. Linares-Barranco, and J. L. Huertas Dept. of Analog Design, National Microelectronics Center (CNM), Ed. CICA, Av. Reina Mercedes sIn, 41012 Sevilla, SPAIN. Phone: (34)-5-4239923, Fax: (34)-5-4624506, E-mail: bernabe@cnm.us.es Abstract We describe an analog VLSI implementation of the ARTI algorithm (Carpenter, 1987). A prototype chip has been fabricated in a standard low cost 1.5~m double-metal single-poly CMOS process. It has a die area of lcm2 and is mounted in a 12O-pins PGA package. The chip realizes a modified version of the original ARTI architecture. Such modification has been shown to preserve all computational properties of the original algorithm (Serrano, 1994a), while being more appropriate for VLSI realizations. The chip implements an ARTI network with 100 F1 nodes and 18 F2 nodes. It can therefore cluster 100 binary pixels input patterns into up to 18 different categories. Modular expansibility of the system is possible by assembling an NxM array of chips without any extra interfacing circuitry, resulting in an F1 layer with l00xN nodes, and an F2 layer with 18xM nodes. Pattern classification is performed in less than 1.8~s, which means an equivalent computing power of 2.2x109 connections and connection-updates per second. Although internally the chip is analog in nature, it interfaces to the outside world through digital signals, thus having a true asynchrounous digital behavior. Experimental chip test results are available, which have been obtained through test equipments for digital chips. 1 INTRODUCTION The original ARTI algorithm (Carpenter, 1987) proposed in 1987 is a massively parallel architecture for a self-organizing neural binary-pattern recognition machine. In response to arbitrary orderings of arbitrarily many and complex binary input patterns, ARTI is 756 T. Serrano-Gotarredona, B. Linares-Barranco, 1. L. Huertas Initialize weights: zij =1 N L Z/i-LB L Zij+LM Tj = LA i Read input pattern: I = (/1' 12, ... I N ) N =1 i =1 Winner-Take-All: = = Y, 1 if T, maxj {Tj y . = 0 if j:t;l } J '-- - - - - - - - - - - - - - - - - - - l z,1 new = In z,1 old Fig.!: Modified Fast Learning or Type-3 ART! implementation algorithm capable of learning, in an unsupervised way, stable recognition codes. The ARTl architecture is described by a set of Short Term Memory (STM) and another set of Long Term Memory (LTM) time domain nonlinear differential equations. It is valid to assume that the STM equations settle much faster (instantaneously) than the LTM equations, so that the STM differential equations can be substituted by nonlinear algebraic equations that describe the steady-state of the STM differential equations. Furthermore, in the fast-learning mode (Carpenter, 1987), the LTM differential equations are as well substituted by their corresponding steady-state nonlinear algebraic equations. This way, the ARTI architecture can be behaviorally modelled by the sequential application of nonlinear algebraic equations. Three different levels of ARTI implementations (both in software and in hardware) can therefore be distinguished: Type-I: Full Model Implementation: both STM and LTM time-domain differential equations are realized. This implementation is the most expensive (both in software and in hardware), and requires a large amount of computational power. Type-2: STM steady-state Implementation: only the LTM time-domain differential equations are implemented. The STM behavior is governed by nonlinear algebraic equations. This implementation requires less resources than the previous one. However, a proper sequencing of STM events has to be introduced artificially, which is architecturally implicit in the Type-I implementation. Type-3: Fast Learning Implementation: STM and LTM is implemented with algebraic equations. This implementation is computationally the less expensive one. In this case an artificial sequencing of STM and LTM events has to be done. The implementation presented in this paper realizes a modified version of the original ARTI Type-3 algorithm, more suitable for VLSI implementations. Such modified ARTI system has been shown to preserve all computational properties of the original ARTI architecture (Serrano, 1994a). The flow diagram that describes the modified ARTI architecture is shown in Fig. I. Note that there is only one binary-valued weight template (Zij)' instead ofthe two weight templates (one binary-valued and the other real-valued) of the original ARTl. For a more detailed discussion of the modified ARTI algorithm refer to (Serrano, 1994a, 1994b). In the next Section we will provide an analog current-mode based circuit that implements in hardware the flow diagram of Fig. 1. Note that, although internally this circuit is analog in nature, from its input and output signals point of view it is a true asynchronous digital 757 A Real Time Clustering CMOS Neural Engine circuit, easy to interface with any conventional digital machine. Finally, in Section 3 we will provide experimental results measured from the chip using a digital data acquisition test equipment. 2 CIRCUIT DESCRIPTION The ART! chip reported in this paper has an F J layer with 100 neurons and an F2 layer with 18 neurons. This means that it can handle binary input patterns of 100 pixels each, and cluster them into up to 18 different categories; according to a digitally adjustable vigilance parameter p. The circuit architecture of the chip is shown in Fig. 2(a). It consists of an array of 18x100 synapses, a lx100 array of "vigilance synapses", a unity gain 18-outputs current mirror, an adjustable gain 18-outputs current mirror (with p=O.O, 0.1, ... 0.9)1, 18 current-comparator-controlled switches and an 18-input-currents Winner-Take-All (WTA) (Serrano, 1994b). The inputs to the circuit are the 100 binary digital input voltages Ii ' and the outputs of the circuit are the 18 digital output voltages Yj . External control signals allow to change parameters p, LA' L B , and L M . Also, extra circuitry has been added for reading the internal weights Zij while the system is learning. Each row of synapses generates two currents, 100 100 Tj = LA LZ/i-LBLZij+LM i= 1 i= 1 100 (1) Vj = LA LZ/i i =1 while the row of the "vigilance synapses" generates the current 100 Vp = LALli (2) i= 1 Each of the current comparators compares the current V . versus pVp ' and allows current T. to reach the WTA only if pV ~ V . . This way dompetition and vigilance occur srmultaneously and in parallel, speeding tip significantly the search process. Fig. 2(b) shows the content of a synapse in the 18x 100 array. It consists of three current sources with switches, two digital AND gates and a flip-flop. Each synapse receives two input voltages Ii and y . , and two global control voltages <1>/ (to enable/disable learning) and reset (to initializ6 all weights zj" to '1'). Each synapse generates two currents LAlizij-LBzij and LAliz ij ,which will be1summed up for all the synapses in the same row to generate the currents T. and V.. If learning is enabled (<I> / = 1) the value of Zj" will change to Iiz i . if y . = f . The ,lvigilance synapses" consist each of a current-sou~ce of value L A witI~ a s~itch controlled by the input voltage Ii' The current comparators are those proposed in (Dominguez-Castro, 1992), the WTA used is reported in (Lazzaro, 1989), and the digitally adjustable current mirror is based on (Loh, 1989), while its continuous gain fine tuning mechanism has been taken from (Adams, 1991). 1. An additional pin ofthe chip can fine-tune p between 0.9 and 1.0. T. Serrano-Gotarredona, B. Linares-Barranco, J. L. Huertas 758 -------....... -------..... --- 1 1 ,, ,, :l,t.ti~j 11 12 . . . 1\00 (a) (b) Fig. 2: (a) System Diagram of Current-Mode ART! Chipt (b) Circuit Diagram of Synapse Fig. 3: Tree based current-mirror scheme for matched current sources The circuit has been designed in such a way that the WTA operates with a precision around 1.5% (--6 bits). This means that all LA and L8 current sources have to match within an error of less than that. From a circuit implementation point of view this is not easy to achieve, since there are 5500 current sources spread over a die area of lcm2 . Typical mismatch between reasonable size MOS transistors inside such an area extension can be expected to be above 10% (pelgrom, 1989). To overcome this problem we implemented a tree-based current mirror scheme as is shown in Fig. 3. Starting from a unique current reference, and using high-precision lO(or less)-outputs current mirrors (each yielding a precision around 0.2%), only up to four cascades are needed. This way, the current mismatch attained at the synapse current sources was around 1% for currents between LA/8 = 5J.LA and LA/8 = 10~A. This is shown in Fig. 4, where the measured dc output current-error (in %) versus input current of the tree based configuration for 18 of the 3600 LA synapse sources is depicted. A Real Time Clustering CMOS Neural Engine Fig. 4: Measured current mirror cascade missmatch (1 %/div) for LA for currents below 3 759 10~A EXPERIMENTAL RESULTS Fig. 5 shows a microphotograph of a prototype chip fabricated in a standard digital double-metal, single-poly 1.5~m low cost CMOS process. The chip die area is Icm2 , and it is mounted in a 120-pins PGA package. Fig. 6 shows a typical training sequence accomplished by the chip and obtained experimentally using a test equipment for digital chips. The only task performed by the test equipment was to provide the input data patterns I (first column in Fig. 6), detect which of the output nodes became 'I' (pattern with a vertical bar to its right), and extract the learned weights. Each IOxlO square in Fig. 6 represents either a lOa-pixels input vector I, or one row of lOa-pixels synaptic weights z . == ezi" zp ... zIOO?) . Each row of squares in Fig. 6 represents the input pattern (first square)land the 18 1vectors Zj after learning has been performed for this input pattern. The sequence shown in Fig. 6 has been obtained for p = 0.7, LA = JO/lA, LB = 9.5/lA , and LM = 950 /lA. Only two iterations of input patterns presentations were necessary, in this case, for the system to learn and self-organize in response to these 18 input patterns. The last row in Fig. 6 shows the final learned templates. Fig. 7 shows final learned templates for different values of p. The numbers below each square indicate the input patterns that have been clustered into each Zj category. Delay time measurements have been performed for the feedforward action of the chip (establishment of currents T., V. , and Vp ' and their competitions until the WTA settles), and for the updating of weights: The feedforward delay is pattern and bias currents (LA' L B , LM ) dependent, but has been measured to be always below 1.6/ls. The learning time is constant and is around 180ns. Therefore, throughput time is less than 1.8/ls. A digital neuroprocessor able to perform a connections/s, b connection-updates/s, and with a dedicated WTA section with a c seconds delay, must satisfy T. Serrano-Gotarredona. B. Linares-Barranco. J. L Huertas 760 Fig. 5: Microphotograph of ARTl chip 3700 + 100 + c = 1.8J..lS a b (3) to meet the performance of our ~rototype chip. If a = band c = lOOns, the equivalent speed would be a = b = 2.2 x 10 connections and connection-updates per second. 4 CONCLUSIONS A high speed analog current-mode categorizer chip has been built using a standard low cost digital CMOS process. The high performance of the chip is achieved thanks to a simplification of the original ARTl algorithm. The simplifications introduced are such that all the original computational properties are preserved. Experimental chip test results are provided. A Real Time Clustering CMOS Neural Engine Z, Z2 Z3 Z4 Zs z6 Z7 Zg Z9 Z10 Zll Z12 Z13 Z14 ZlS Z16 Z17 Zig 1? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? 13. ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????? U??????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? ?????????????????? 2? 3. 4 ? 5 ? 6 ? 7 ? 8 ? 9? 10 ? 11 ? 12 ? 14 ? 15 ? 16 ? 17 ? 18 ? 1 ? 2 ? 3 ? 4 ? 5 ? 6 ? 7 ? 8 ? 9_ 10 ? 11 ? 12 ? 13 ? 14 II 15 ? 16 ? 17 ? 18 ? Fig. 6: Test sequence obtained experimentally for p=O.7, LA=lO/-lA, LB=9.5/-lA, and LM=950/-lA 761 T. Serrano-Gotarredona, B. Linares-Barranco, J. L. Huertas 762 Fig. 7: Categorization of the input patterns for LA=3.2~A, LB=3.0~, LM=400~A, and different values of p References W. J. Adams and J. Ramimez-Angulo. (1991) "Extended Transconductance AdjustmentlLinearisation Technique," Electronics Letters, vol. 27, No. 10, pp. 842-844, May 1991. G. A. Carpenter and S. Grossberg. (1987) "A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine," Computer VIsion, Graphics, and Image Processing, vol. 37, pp. 54-115, 1987. R. Dominguez-castro, A. Rodrfguez-Vazquez, F. Medeiro, and 1. L. Huertas. (1992) "High Resolution CMOS Current Comparators," Proc. of the 1992 European Solid-State Circuits Conference (ESSCIRC'92), pp. 242-245, 1992. J. Lazzaro, R. Ryckebusch, M. A. Mahowald, and C. Mead. (1989) "Winner-Take-All Networks of O(n) Complexity," in Advances in Neural Information Processing Systems, vol. 1, D. S. Touretzky (Ed.), Los Altos, CA: Morgan Kaufmann, 1989, pp. 703-711. K. Loh, D. L. Hiser, W. J. Adams, and R. L. Geiger. (1989) "A Robust Digitally Programmable and ReconfigurabJe Monolithic Filter Structure," Proc. of the 1989 Int. Symp. on Circuits and Systems (ISCAS'89), Portland, Oregon, vol. 1, pp. 110-113, 1989. M. J. Pelgrom, A. C. J. Duinmaijer, and A. P. G. Welbers. (1989) "Matching Properties of MOS Transistors," IEEE Journal of Solid-State Circuits, vol. 24, No.5, pp. 1433-1440, October 1989. T. Serrano-Gotarredona and B. Linares-Barranco. (1994a) "A Modified ARTl Algorithm more suitable for VLSI Implementations," submitted for publication (journal paper). T. Serrano-Gotarredona and B. Linares-Barranco. (1994b) "A Real-Time Clustering Microchip Neural Engine," submitted for publication (journal paper). 

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Learning direction in global motion: two classes of psychophysically-motivated models V. Sundareswaran Lucia M. Vaina* Intelligent Systems Laboratory, College of Engineering, Boston University 44 Cummington Street, Boston, MA 02215 Abstract Perceptual learning is defined as fast improvement in performance and retention of the learned ability over a period of time. In a set of psychophysical experiments we demonstrated that perceptual learning occurs for the discrimination of direction in stochastic motion stimuli. Here we model this learning using two approaches: a clustering model that learns to accommodate the motion noise, and an averaging model that learns to ignore the noise. Simulations of the models show performance similar to the psychophysical results. 1 Introduction Global motion perception is critical to many visual tasks: to perceive self-motion, to identify objects in motion, to determine the structure of the environment, and to make judgements for safe navigation. In the presence of noise, as in random dot kinematograms, efficient extraction of global motion involves considerable spatial integration. Newsome and Colleagues (1989) showed that neurons in the macaque middle temporal area (MT) are motion direction-selective, and perform global integration of motion in their large receptive fields. Psychophysical studies in humans have characterized the limits of spatial and temporal integration in motion (Watamaniuk et. aI, 1984) and the nature of the underlying motion computations (Vaina et. al 1990). ?Please address all correspondence to Lucia Vaina 918 V. Sundareswaran, Lucia M. Vaina Since the psychophysical and neural substrate of global motion are fairly well understood, we were interested to see whether the perception of direction in such global motion stimuli can improve with practice. Studies specifically addressing this question for other early perceptual tasks have shown that improvements of performance obtained in the first experimental session are preserved in a subsequent session and retained over weeks. This is considered as perceptual learning (Gibson, 1953). Psychophysical studies of perceptual learning show that the beneficial effects of practice are lost if some stimulus parameters are changed significantly, such as orientation, spatial frequency or location in the visual field. Based on the time scale necessary for the improvement to occur, two major learning paradigms have been used in perceptual learning : slow, progressive learning (several thousand trials are required to reach stable performance) and fast !earning (improvement occurs and stabilizes in the first 100-200 trials). The idea of fast learning and the nature of its limits is attractive from a computational point of view because it encourages the exploration of practice-dependent plasticity found in the adult early visual system (Fregnac et. al., 1988, Gilbert and Wiesel 1992). A recent line of research in biologically motivated learning models originated by Poggio (Poggio, 1990) takes perceptual learning as evidence that"the brain may be able to synthesize-possibly in the cortex-appropriate task-specific modules that receive input from retinotopic cells and learn to solve the task after a short training phase in which they are exposed to examples of the task". Poggio and colleagues (1992) have illustrated this approach in learning vernier hyperacuity. Here, we adopted this general framework to study learning of direction in global motion. In contrast to Poggio et. aI's supervised learning paradigm, we used unsupervised learning both in the psychophysical experiments and modeling of learning. We designed a set of psychophysical tasks to study whether fast learning occurs in discrimination of opposite directions of global motion and to explore the limits of this learning. To model the learning, we studied two models that differ in the way they deal with noise. 2 Psychophysics Ball and Sekuler (1982, 1987) showed that discriminability of the direction of motion of two random dot patterns improved with training. In this learning paradigm, more than 2000 trials are required for reaching a stable performance. Such a "slow" learning time scale has been reported for the learning of other perceptual tasks, such as vernier acuity (McKee and Westheimer 1978, Fahle 1994), stereoacuity (Fendick and Westheimer, 1983; Ramachandran and Braddick 1973) and discrimination of line orientation (Vogels and Orban 1985). In contrast to this "slow learning," Fiorentini and Berardi (1981) showed that for learning the discrimination of complex gratings with two harmonics of different spatial phase, 100-200 trials suffice. Similarly Poggio et. al. (1992) show that a small number of trials suffice for significantly improving performance on a vernier hyperacuity task. Both studies discussed the specificity of learning to the stimulus attributes. In our study, we used a two-alternative, forced-choice psychophysical procedure to measure the subject's ability to discriminate between two opposite directions of motion in dynamic random dot patterns in which 25% of the dots provide a correlated Learning Direction in Global Motion 919 Figure 1: (a) Stimulus: a fraction of the dots (filled circles) move coherently in the signal direction; the rest (open circles) move randomly, (b) Fast improvement is observed on the first day of testing, and retained over a period of time. Each block consisted of 40 trials; data points averaged over 4 subjects, and errorbars show standard error. motion signal spatially dispersed in a masking motion noise due to random motion of the reminder 75% of dots (Fig. 1(a)). Each trial lasted 90 msec during which two frames were presented (with inter-frame interval equal to zero). A session consisted of 4-6 blocks of 40 trials each. Feedback was not provided during the experimental sessions. Observers were required to maintain fixation on a fixation mark placed at 2? from the imaginary circumference of the stimulus. To investigate the effects of practice and their retention, the discrimination of leftward vs rightward direction of motion in the display was tested first on each experimental session for 3 consecutive days and repeated 10 days later. The results are presented in Fig. 1(b). For most observers, a fast and dramatic improvement was seen in the first day. Fig. 1(b) shows that learning was maintained in subsequent days, and even ten days later without any training in between. Examination of the individual observers' data revealed that improvement of performance occurred only if they started above chance level. This suggests that this fast learning might imply the improvement of an existing representation of the stimulus. In additional experiments, we did not find transfer to another direction of motion (up/down), indicating that the learning is selective to specific characteristics of the stimulus. Details of experiments testing the limits of the learning appear in Vaina et. al (1995). 3 Modeling We propose two paradigms to model the learning found in psychophysics. Both use directionally-tuned units with properties similar to those of neurons found in MT (Maunsell and Van Essen, 1983). Schematic MT neurons used in our modeling integrate global information by summing over localized responses: Xt ~ -(~(oi-od) = ~e 2<1" ? (1) i=l where (Jh is the standard deviation of the tuning, and information from n local responses is taken into account. The responses of a collection of such units (each 920 V. Sundareswaran, Lucia M. Vaina (a) (b) Figure 2: Architectures of the models: (a) Learning to Accommodate: Cluster Gaussians !I and h operate on input vector X to decide global motion direction, (b) Learning to Ignore: Global motion direction is computed as a weighted combination of units' preferred directions. tuned to a different direction) form the input vector to the models. 3.1 Learning to accommodate This model is based on grouping similar data. In this paradigm, after learning, the network has an estimate of the contribution of noise, and takes it into account when performing the task of discriminating between two different directions of motion; so, we say that the model "learns to accommodate" the noise. Fig 2(a) contains a schematic of this model. The representation vector consists of responses Xl.X2, ..? , Xn of the directionally-tuned units .. Clustering is done in the space of the representation vectors. The model is a combination of HyperBF-like functions and clustering (Poggio et. aI, 1992, Moody and Darken 1989). We use gaussians with mean at the cluster centers, a.nd "move" the cluster centers by a learning algorithm. A cluster gaussian computes a gaussian function of the representation vector for the current stimulus from the current center of the corresponding cluster. We say "current" because the center is moved as the learning proceeds. At any given point, a center is at the current best estimate of the center of the corresponding data cluster. More precisely, the learning rule to modify the jth coordinate of the center is given by: c(t-+:l) = c(t) . + '11 * (x~t) _ c(t) . ) (2) W,) W,) '/ ) W,) , where w is the index of "winning" cluster and Cw,j is the jth coordinate of the center for the wth cluster. This moves the center towards the new data vector X that has been judged to belong to the wth cluster. The parameter TJ controls the learning rate. 921 Learning Direction in Global Motion Learning to ignore 3.2 The learning involved in this approach is Hebbian, and is termed "learning to ignore," because in this weighted averaging scheme, as learning occurs, the weights for the noise response are progressively reduced to zero, leaving only the contribution from the signal. In other words, the network learns to ignore the noise (Vaina et. al 1995). The model's output is a global motion direction. If the weights associated with the responses Xl, X2,? . ? Xn are WI, W2,'" W n , the global motion direction is calculated as II Uo = tan -1 (L:'"w wiXisinOi ) WiXiCOSOj ' where ti is the tuned direction (angle) of the ith unit. A schematic of the model is shown in Fig. 2(b). The global direction is judged to be rightward if +Ot > 80 > -Ot. We have examined two different learning rules: exposure-based learning, and selfsupervised learning. Exposure-based learning: The weight corresponding to a unit is incremented by an amount proportional to the current weight. Only units whose response values are above a certain threshold are allowed to increase their weights. This learning rule favors units that fire consistently: (3) where rt is a threshold, and 1] is a small fraction that controls the learning rate. Self-supervised learning: The weight corresponding to a unit is increased by an amount proportional to the product of the current weight and a decreasing function (exponential) of the angular difference between the calculated global motion direction and the direction of tuning of the unit: Wi ~ Wi + 1]Wi e (- (O; -OO)2/ 2u e) , if Xi> rt . In this case, the model uses its own estimate of the global direction as an internal feedback to determine the learning. An approach similar to this model for learning vernier hyperacuity was proposed by Weiss et. al (1993). 4 Experiments For the simulations, the input (motion) vectors were represented by their angles relative to the positive horizontal axis (Le. , the magnitude is ignored). The responses of the directionally-tuned units can be directly computed from the angles (see Eqn. 1; alternatively, cosine tuning functions were used, and similar results were obtained) . For each trial, the coherent motion direction was randomly decided. In the exper, where nd is the number of unit preferred directions; the iments, O'h was set to ~ nd directions Ot are chosen by uniformly dividing nd in to 211'. In all the experiments, eight preferred directions were used; each trial contained 40 random dots moving with a correlation of25% (the same correlation as in the tests with human subjects); 922 V. Sundareswaran, Lucia M. Vaina 1.00 0.9 I u 0.90 0.8 0.80 - G - Espoeure 8 'f 0.70 0.7 0.60 0.6 ~ Do 14 Block No --.-- Self?npervised O.S 0 s 10 1S 20 Block No Figure 3: Results from typical simulation runs. On the left is the curve for the learning to accommodate model (1} = 0.005; cluster (j = 0.5) and on the right are the curves for the learning to ignore model. each block consisted of 50 trials. The performance is measured by fraction correct, corresponding to the fraction of the inputs that were correctly classified. For both models, we did simulations to justify the architecture of the models by disabling learning, and studying the performance of the models for increasing correlation. The performance was qUalitatively similar to that described in perceptual studies of these stimuli in humans and monkeys. Learning curves from simulations of both models are shown in Fig. 3. These are results from averaging over the performance in ten simulation runs. As can be seen, the models learn to do the discrimination. While both models successfully learned to do the direction discrimination, there are quantitative differences in the learning curves. The learning to accommodate paradigm improved very rapidly (somewhat faster than the human subjects). The exposure-based learning rule for the learning to ignore paradigm learned at a rate comparable with the human observers. However, the final performance in this case was not very stable, and oscillated (not shown), consistent with the observations of Weiss et. al (1993) in learning vernier hyperacuity. The self-supervised rule, on the other hand, reached a stable level of performance, but the learning was slower; the reason is that this learning rule exhibits instability if a high value of learning rate (1}) is used. If either model trained on inputs containing horizontal correlated motion was later presented with inputs containing vertical correlated motion, the performance dropped to pre-training levels. This non-transfer is consistent with psychophysical results mentioned in Section 2. 5 Discussion In this study, we focused on learning of global motion direction, and not on motion perception. Motion perception is well-understood both physiologically and psy- Learning Direction in Global Motion 923 chophysically. In a series of studies of neuronal correlates of the perceptual decision of direction of motion in stochastic motion signals like the stimuli used here, together with studies of performance on these stimuli on monkeys with MT lesions, Newsome and his collaborators (for a review, Newsome et. aI, 1989) provide strong support for the hypothesis that perceptual judgements of motion direction are by and large based on the directional signals carried by MT neurons. Salzman and Newsome (1994) reported that in trained monkeys, the perception of motion in stochastic noise is more likely mediated by a winner-take-all mechanism than by a weighted averaging mechanism. Interestingly, our clustering model, after learning, is behaviorally similar to the winner-take-all mechanism: the dominating component of the representation results in the assignment to the closest cluster. Our psychophysical studies clearly demonstrate that perceptual learning of global motion direction occurs. While significant progress has been made to understand the mechanisms and underlying circuitry of learning (Zohary et. al 1994) as yet there is no satisfactory biologically explanation of how and where this learning may occur (Zohary and Newsome 1994). Our focus in this paper was on computational models of learning direction in global motion. For this, we proposed two approaches, which differ in the way they deal with noise: one learns to accommodates noise, and the other learns to ignore it. We do not advocate that one or the other of the models we proposed here provides the biologically correct choice for this task. However, together with the psychophysics described here these models suggest new experiments which we are now exploring both psychophysically and computationally. We hope that by closely connecting models and psychophysics while keeping in mind the aim of neuronal compatibility, we will make progress in understanding how the cortex learns so fast to discriminate direction of motion in extremely noisy situations. Acknowledgments This research was conducted at the Intelligent Systems Laboratory of Boston University, College of Engineering. LMV and VS were supported in part by grants from the Office of Naval Research (# N00014-93-1-0381) and the National Institute of Health (EY R01- 07861) to LMV. VS was in part supported by the Boston University College of Engineering Dean's postdoctoral fellowship. We gratefully acknowledge additional financial support for VS from the Dean's special research fund. References [1] K. Ball and R. Sekuler. Direction-specific improvement in motion discrimination. Vision Research, 27(6):953-965, 1987. [2] M. Fahle. Human pattern recognition: parallel processing and perceptual learning. Perception, 23:411-427, 1994. [3] M. Fendick and G. Westheimer. Effects of practice and the separation of test targets on foveal and perifoveal hyperacuity. Vision Research, 23: 145-150, 1983. 924 V. Sundareswaran, Lucia M. Vaina [4] A. Fiorentini and N. Berardi. Perceptual learning specific for orientation and spatial frequency. Nature, 287(5777):43-44, September 1980. [5] Y. Fregnac, D. Shulz, S. Thorpe, and E. Bienstock. A cellular analogue of visual cortical plasticity. Nature, 333:367-370, 1988. [6] E. J. Gibson. Improvements in perceptual judgement as a function of controlled practice of training. Psychology bulletin, 50:402-431, 1953. [7] C. D. Gilbert and T. N. Wiesel. Receptive field dynamics in adult primary visual cortex. Nature, 356:150-152, 1992. [8] A. Karni and D. Sagi. Where practice makes perfect in texture discrimination: Evidence for primary visual cortex plasticity. Proc. Natl. Acad. Sci. USA, 88:4966- 4970, June 1991. [9] J. H. R. Maunsell and D. C. Van Essen. Functional properties of neurons in the middle temporal visual area of the macaque monkey i: selectivity for stimulus direction, speed, and orientation. J. Neurophysiology, 49:1127-1147, 1983. [10] S. P. McKee and G. Westheimer. Improvement in venier acuity with practice. Perception ?3 Psychophysics, 24:258-262, 1978. [11] W. T. Newsome, K. H. Britten, and J. A. Movshon. Neuronal correlates of a perceptual decision. Nature, 341 :52-54, 1989. [12] T. Poggio. A theory of how the brain might work. In Cold Spring Harbor Symposia on Quantitative Biology, pages 899-910. Cold Spring Harbor Laboratory Press, 1990. [13] T. Poggio, M. Fahle, and S. Edelman. Fast perceptual learning in visual hyperacuity. Science, 256:1018-1021, 1992. [14] T. Poggio and F. Girosi. A theory of networks for approximation and learning. AI Memo 1140, M.LT, July 1989. [15] V. S. Ramachandran and O. Braddick. Orientation-specific learning in stereopsis. Perception, 2:371-376, 1973. [16] C. D. Salzman and W. T. Newsome. Neural mechanisms for forming a perceptual decision. Science, 264:231-237, 1994. [17] L. M. Vaina, V. Sundareswaran, and J. Harris. Computational learning and natural learning. Cognitive Brain Research, 1995. (in press). [18] L. M. Vaina, N. M. Grzywacz, and M. LeMay. Structure from motion with impaired local-speed and global motion-field computations. Neuml Computation, 2:420-435, 1990. [19] R. Vogels and G . A. Orban. The effect of practice on the oblique effect in line orientation judgements. Vision Research, 25:1679-1687, 1985. [20] S. Watamaniuk, R. Sekuler, and D. Williams. Direction perception in complex dynamic displays: the integration of direction information. Vision Research, 24:55- 62, 1984. [21] Y. Weiss, S. Edelman, and M. Fahle. Models of perceptual learning in vernier hyperacuity. Neural Computation, 5:695-718, 1993. [22] E. Zohary, S. Celebrini, K. H. Britten, and W. T . Newsome. Neuronal plasticity that underlies improvement in perceptual performance. Science, 263:1289-1292, March 1994. [23] E. Zohary and W. T. Newsome. Perceptual learning in a direction discrimination task is not based upon enhanced neuronal sensitivity in the sts. Investigative Ophthalmology and Visual Science supplement, page 1663, 1994. 

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Correlation and Interpolation Networks for Real-time Expression Analysis/Synthesis. Trevor Darrell, Irfan Essa, Alex Pentland Perceptual Computing Group MIT Media Lab Abstract We describe a framework for real-time tracking of facial expressions that uses neurally-inspired correlation and interpolation methods. A distributed view-based representation is used to characterize facial state, and is computed using a replicated correlation network. The ensemble response of the set of view correlation scores is input to a network based interpolation method, which maps perceptual state to motor control states for a simulated 3-D face model. Activation levels of the motor state correspond to muscle activations in an anatomically derived model. By integrating fast and robust 2-D processing with 3-D models, we obtain a system that is able to quickly track and interpret complex facial motions in real-time. 1 INTRODUCTION An important task for natural and artificial vision systems is the analysis and interpretation of faces. To be useful in interactive systems and in other settings where the information conveyed is of a time critical nature, analysis of facial expressions must occur quickly, or be oflittle value. However, many of the traditional computer vision methods for estimating and modeling facial state have proved difficult to perform fast enough for interactive settings. We have therefore investigated neurally inspired mechanisms for the analysis of facial expressions. We use neurally plausible distributed pattern recognition mechanisms to make fast and robust assessments of facial state, and multi-dimensional interpolation networks to connect these measurements to a facial model. There are many potential applications of a system for facial expression analysis. Person- 910 Trevor Darrell. Irfan Essa. Alex Pentland alized interfaces which sense a users emotional state, ultra-low bitrate video conferencing which sends only facial muscle activations, as well as the enhanced recognition systems mentioned above. We have focused on a application in computer graphics which stresses both the analysis and synthesis components of our system: interactive facial animation. In the next sections we develop a computational framework for neurally plausible expression analysis, and the connection to a physically-based face model using a radial basis function method. Finally we will show the results of these methods applied to the interactive animation task, in which an computer graphics model of a face is rendered in real time, and matches the state of the users face as sensed through a conventional video camera. 2 EXPRESSION MODELINGffRACKING The modeling and tracking of expressions and faces has been a topic of increasing interest recently. In the neural network field, several successful models of character expression modeling have been developed by Poggio and colleagues. These models apply multidimensional interpolation techniques, using the radial basis function method, to the task of interpolating 2D images of different facial expression. Librande [4] and Poggio and Brunelli [9] applied the Radial Basis Function (RBF) method to facial expression modeling, using a line drawing representation of cartoon faces. In this model a small set of canonical expressions is defined, and intermediate expressions constructed via the interpolation technique. The representation used is a generic "feature vector", which in the case of cartoon faces consists of the contour endpoints. Recently, Beymer et al. [1] extended this approach to use real images, relying on optical flow and image warping techniques to solve the correspondence and prediction problems, respectively. RBF-based techniques have the advantage of allowing for the efficient and fast computation of intermediate states in a representation. Since the representation is simple and the interpolation computation straight-forward, real-time implementations are practical on conventional systems. These methods interpolate between a set of 2D views, so the need for an explicit 3-D representation is sidestepped. For many applications, this is not a problem, and may even be desirable since it allows the extrapolation to "impossible" figures or expressions, which may be of creative value. However, for realistic rendering and recognition tasks, the use of a 3-D model may be desirable since it can detect such impossible states. In the field of computer graphics, much work has been done on on the 3-D modeling of faces and facial expression. These models focus on the geometric and physical qualities of facial structure. Platt and B adler [7], Pieper [6], Waters [11] and others have developed models of facial structure, skin dynamics, and muscle connections, respectively, based on available anatomical data. These models provide strong constraints for the tracking of feature locations on a face. Williams et. al. [12] developed a method in which explicit feature marks are tracked on a 3-D face by use of two cameras. Terzopoulos and Waters [10] developed a similar method to track linear facial features, estimate corresponding parameters of a three dimensional wireframe face model, and reproduce facial expression. A significant limitation of these systems is that successful tracking requires facial markings. Essa and Pentland [3] applied optical flow methods (see also Mase [5]) for the passive tracking of facial motion, and integrated the flow measurement method into a dynamic system model. Their method allowed for completely passive estimation of facial expressions, using all the constraints provided by a full 3-D model of facial expression. Both the view based method of Beymer et. al. and the 3-D model of Essa and Pentland rely Correlation and Interpolation Networks for Real-Time Expression Analysis/Synthesis ( 911 (b) Figure 1: (a) Frame of video being processed to extract view model. Outlined rectangle indicates area of image used for model. (b) View models found via clustering method on training sequence consisting of neutral, smile, and surprise expressions. on estimates of optic flow, which are difficult to compute reliably, especially in real-time. Our approach here is to combine interpolated view-based measurements with physically based models, to take advantage of the fast interpolation capability of the RBF and the powerful constraints imposed by physically based models. We construct a framework in which perceptual states are estimated from real video sequences and are interpolated to control the motor control states of a physically based face model. 3 VIEW-BASED FACE PERCEPTION To make reliable real-time measurements of a complex dynamic object, we use a distributed representation corresponding to distinct views of that object. Previously, we demonstrated the use of this type of representation for the tracking and recognition of hand gestures [2]. Like faces, hands are complex objects with both non-rigid and rigid dynamics. Direct use of a 3-D model for recognition has proved difficult for such objects, so we developed a view-based method for representation. Here we apply this technique to the problem of facial representation, but extend the scheme to connect to a 3-D model for high-level modeling and generation/animation. With this, we gain the representational power and constraints implied by the 3-D model as a high-level representation; however the 3-D model is only indirectly involved in the perception stage, so we can still have the same speed and reliability afforded by the view-based representation. In our method each view characterizes a particular aspect or pose of the object being represented. The view is stored iconically, that is, it is a literal image or template (but with some point-wise statistics) of the appearance of the object in that aspect or pose. A match criteria is defined between views and input images; usually a normalized correlation function is used, but other criteria are possible. An input image is represented by the ensemble of match scores from that image to the stored views. To achieve invariance across a range of transformations, for example translation, rotation and/or scale, units which compute the match score for each view are replicated at different values of each transformation. I The unit which has maximal response across all values of the transformation is selected, and the ensemble response of the view units which share the 1 In a computer implementation this exhaustive sampling may be impractical due to the number of units needed, in which case this stage may be approximated by methods which are hybrid sampling/search methods. 912 Trevor Darrell, Irfan Essa, Alex Pentland same transformation values as the selected unit is stored as the representation for the input image. We set the perceptual state X to be a vector containing this ensemble response. If the object to be represented is fully known a priori, then methods to generate views can be constructed by analysis of the aspect graph if the object is polyhedral, or in general by rendering images of the object at evenly spaced rotations. However, in practice good 3-D models that are useful for describing image intensity values are rare2 , so we look to data-driven methods of acquiring object views. As described in [2] a simple clustering algorithm can find a set of views that "span" a training sequence of images, in the sense that for each image in the sequence at least one view is within some threshold similarity to that image. The algorithm is as follows. Let V be the current set of views for an object (initially one view is specified manually). For each frame I of a training sequence, if at least one v E V has a match value M (v, 1) that is greater than a threshold (J, then no action is performed and the next frame is processed. If no view is close, then I is used to construct a new view which is added to the view set. A view v' is created using a window of I centered at the location in the previous image where the closest view was located. (All views usually share the same window size, determined by the initial view.) The view set is then augmented to include the new view: V = V u v'. This algorithm will find a set of views which well-characterizes an object across the range of poses or expressions contained in the training sequence. For example, in the domain of hand gestures, inputing a training sequence consisting of a waving hand will yield views which contain images of the hand at several different rotations. In the domain of faces, when input a training sequence consisting of a user performing 3 different expressions, neutral, smile, and surprise, this algorithm (with normalized correlation and (J = 0.7) found three views corresponding to these expressions to represent the face, as shown in Figure l(b). These 3 views serve as a good representation for the face of this user as long as his expression is similar to one in the training set. The major advantage of this type of distributed view-based representation lies in the reduction of the dimensionality of the processing that needs to occur for recognition, tracking, or control tasks. In the gesture recognition domain, this dimensionality reduction allowed for conventional recognition strategies to be applied successfully and in real-time, on examples where it would have been infeasible to evaluate the recognition criteria on the full signal. In the domain explored in this paper it makes the interpolation problem of much lower order: rather than interpolate from thousands of input dimensions as would be required when the input is the image domain, the view domain for expression modeling tasks typically has on the order of a dozen dimensions. 4 3-D MODELINGIMOTOR CONTROL To model the structure of the face and the dynamics of expression performance, we use the physically based model of Essa et. al. This model captures how expressions are generated by muscle actuations and the resulting skin and tissue deformations. The model is capable of controlled nonrigid deformations of various facial regions, in a fashion similar to how humans generate facial expressions by muscle actuations attached to facial tissue. Finite Element methods are used to model the dynamics of the system. 2 As opposed to modeling forces and shape deformations, for which 3-D models are useful and indeed are used in the method presented here. Correlation and Interpolation Networks for Real-Time Expression Analysis/Synthesis 9 J3 (a) 900. 1 8 0<5'> core (b) 700 l\I1odel ParalTleters . o0.2ActuatIons (e) -0.2 (d) Figure 2: (a) Face images used as input, (b) normalized correlation scores X(t) for each view model, (c) resulting muscle control parameters Y(t), (d) rendered images of facial model corresponding to muscle parameters. 914 Trevor Darrell, Irfan Essa, Alex Pentland This model is based on the mesh developed by Platt and Badler [7], extended into a topologically invariant physics-based model through the addition of a dynamic skin and muscle model [6, 11]. These methods give the facial model an anatomically-based facial structure by modeling facial tissue/skin, and muscle actuators, with a geometric model to describe force-based deformations and control parameters. The muscle model provides us with a set of control knobs to drive the facial state, defined to be a vector Y. These serve to define the motor state of the animated face. Our task now is to connect the perceptual states of the observed face to these motor states. 5 CONNECTING PERCEPTION WITH ACTION We need to establish a mapping from the perceptual view scores to the appropriate muscle activations on the 3-D face model. To do this, we use multidimensional interpolation strategies implemented in network form. Interpolation requires a set of control points or exemplars from which to derive the desired mapping. Example pairs of real faces and model faces for different expressions are presented to the interpolation method during a training phase. This can be done in one of two ways, with either a user-driven or model-driven paradigm. In the model-driven case the muscle states are set to generate a particular expression by an animator/programmer and then the user is asked to make the equivalent expression. The resulting perceptual (view-model) scores are then recorded and paired with the muscle activation levels. In the user-driven case, the user makes an expression of hislher own choosing, and the optic flow method of Essa et. al. is used to derive the corresponding muscle activation levels. The modeldriven paradigm is simpler and faster, but the user-driven paradigm yields more detailed and authentic facial expressions. We use the Radial Basis Function (RBF) method presented in [8], and define the interpolated motor controls to be a weighted sum of radial functions centered at each example: n Y = I:ci9(X - Xi) (1) i=1 where Y are the muscle states, X are the observed view-model scores, Xi are the example scores, 9 is an RBF (and in our case was simply a linear ramp 9(?) = II?II), and the weights Ci are computed from the example motor values Yi using the pseudo-inverse method [8]. 6 INTERACTIVE ANIMATION SYSTEM The correlation network, RBF interpolator, and facial model described above have been combined into a single system for interactive animation. The entire system can be updated at over 5 Hz, using a dedicated single board accelerator to compute the correlation network, and an SGI workstation to render the facial mesh. ? Here we present two examples of the processing performed by the system, using different strategies for coupling perceptual and motor state. Figure 2 illustrates one example of real-time facial expression tracking using this system, using a full-coupling paradigm. Across the top, labeled (a), are five frames of a video sequence of a user making a smile expression. This was one of the expressions used in the training sequence for the view models shown in Figure 1(b), so they were applicable to be Correlation and Interpolation Networks for Real-Time Expression Analysis/Synthesis (a) 915 (b) Figure 3: (a) Processing of video frame with independent view model regions for eyes, eye-brows, and mouth region. (b) Overview shot of full system. User is on left, vision system and camera is on right, and animated face is in the center of the scene. The animated face matches the state of the users face in real-time, including eye-blinks (as is the case in this shot.) used here. Figure 2(b) shows the correlation scores computed for each of the 3 view models for each frame of the sequence. This constituted the perceptual state representation, X(t). In this example the full face is coupled with the full suite of motor control parameters. An RBF interpolator was trained using perceptual/motor state pairs for three example full-face expressions (neutral, smile, surprise); the resulting (interpolated) motor control values, yet), for the entire sequence are shown in Figure 2(c). Finally, the rendered facial mesh for five frames of these motor control values is shown in Figure 2(d). When there are only a few canonical expressions that need be tracked/matched, this full-face template approach is robust and simple. However if the user wishes to exercise independent control of the various regions of the face, then the full coupling paradigm will be overly restrictive. For example, if the user trains two expressions, eyes closed and eyes open, and then runs the system and attempts to blink only one eye, the rendered face will be unable to match it. (In fact closing one eye leads to the rendered face half-closing both eyes.) A solution to this is to decouple the regions of the face which are independent geometrically (and to some degree, in terms of muscle effect.) Under this paradigm, separate correlation networks are computed for each facial regions, and multiple RBF interpolations are performed for each system. Each interpolator drives a distinct subset of the motor state vector. Figure 3(a) shows the regions used for decoupled local templates. In these examples independent regions were used for each eye, eyebrow, and the mouth region. Finally, figure 3 (b) shows a picture of the set-up of the system as it is being run in an interactive setting. The animated face mimics the facial state of the user, matching in real time the position of the eyes, eyelids, eyebrows and mouth of the user. In the example shown in this picture, the users eyes are closed, so the animated face's eyes are similarly closed. Realistic performance of animated facial expressions and gestures are are possible 916 Trevor Darrell, Irfan Essa, Alex Pentland through this method, since the timing and levels of the muscle activations react immediately to changes in the users face. 7 CONCLUSION We have explored the use of correlation networks and Radial Basis Function techniques for the tracking of real faces in video sequences. A distributed view-based representation is computed using a network of replicated normalized correlation units, and offers a fast and robust assesment of perceptual state. 3-D constraints on facial shape are achieved through the use of a an anatomically derived facial model, whose muscle activations are controled via interolated perceptual states using the RBF method. With this framework we have been able to acheive the fast and robust analysis and synthesis of facial expressions. A modeled face mimics the expression of a user in real-time, using only a conventional video camera sensor and no special marking on the face of the user. This system has promise as a new approach in the interactive animation, video tele-conferencing, and personalized interface domains. References [1] D. Beymer, A. Shashua, and T. Poggio, Example Based Image Analysis and Synthesis, MIT AI Lab TR-1431, 1993. [2] T. Darrell and A. Pentland. Classification of Hand Gestures using a View-Based Distributed Representation In NIPS-6, 1993. [3] I. Essa and A. Pentland. A vision system for observing and extracting facial action parameters. In Proc. IEEE Conf. Computer Vision and Pattern Recognition, 1994. [4] S. Librande. Example-based Character Drawing. S.M. Thesis, Media Arts and SciencelMedia Lab, MIT. 1992 [5] K. Mase. Recognition of facial expressions for optical flow. IEICE Transactions, Special Issue on Computer Vision and its Applications, E 74( 10), 1991. [6] S. Pieper, J. Rosen, and D. Zeltzer. Interactive graphics for plastic surgery: A task level analysis and implementation. Proc. Siggraph-92, pages 127-134, 1992. [7] S. M. Platt and N. I. Badler. Animating facial expression. ACM SIGGRAPH Conference Proceedings, 15(3):245-252,1981. [8] T. Poggio and F. Girosi. A theory of networks for approximation and learning. MIT AI Lab TR-1140, 1989. [9] T. Poggio and R. Brunelli, A Novel Approach to Graphics, MIT AI Lab TR- 1354. 1992. [10] D. Terzopoulus and K. Waters. Analysis and synthesis offacial image sequences using physical and anatomical models. IEEE Trans. PAMI, 15(6):569-579, June 1993. [11] K. Waters and D. Terzopoulos. Modeling and animating faces using scanned data. The Journal of Visualization and Computer Animation, 2: 123-128, 1991. [12] L. Williams. Performance-driven facial animation. ACM SIGGRAPH Conference Proceedings, 24(4):235-242, 1990. 

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