Often the parameters used in these networks needto be learned from examples. Unfortunately, estimating the parameters via exact probabilistic calculations (i.e, the EMalgorithm) is intractable even for networks with fairly small numbers of hidden units. We propose to avoid the infeasibility of the E step by bounding likelihoods instead of computing them exactly. Weintroduce extended and complementary representations for these networks and show that the estimation of the network parameters can be made fast (reduced to quadratic optimization) by performing the estimation in either of the alternative domains. The complementary networks can be used for continuous density estimation as well. 1 Introduction The appeal of probabilistic networks for knowledge representation, inference, and learning (Pearl, 1988) derives both from the sound Bayesian framework and from the explicit representation of dependencies among the network variables which allows readyincorporation of prior information into the design of the network.

Due to the simplicity and efficiency of its parameter estimation algorithm, the hidden Markov model (HMM) has emerged as one of the basic statistical tools for modeling discrete time series, finding widespread application in the areas of speech recognition (Rabinerand Juang, 1986) and computational molecular biology (Baldi et al., 1994). An HMM is essentially a mixture model, encoding information about the history of a time series in the value of a single multinomial variable (the hidden state). This multinomial assumption allows an efficient parameter estimation algorithm tobe derived (the Baum-Welch algorithm). However, it also severely limits the representational capacity of HMMs.

Tel-Aviv University Tel-Aviv, ISRAEL Abstract The Fourier transform of boolean functions has come to play an important role in proving many important learnability results. We aim to demonstrate that the Fourier transform techniques are also a useful and practical algorithm in addition to being a powerful theoretical tool. We describe the more prominent changes we have introduced to the algorithm, ones that were crucial and without which the performance of the algorithm would severely deteriorate. Oneof the benefits we present is the confidence level for each prediction which measures the likelihood the prediction is correct. 1 INTRODUCTION It has been used mainly to demonstrate the learnability of various classes of functions with respect to the uniform distribution. The work of [5] developed a very powerful algorithmic procedure: given a function and a threshold parameter it finds in polynomial time all the Fourier coefficients ofthe function larger than the threshold.

The center-surround opponent receptive field(CSRF) mechanism represents one such example. Here, analogous CSRFs are shown to be formed in an artificial neural network which learns to localize contours (edges) of the luminance difference. Furthermore, when the input pattern is corrupted by a background noise, the CSRFs of the hidden units becomes shallower andbroader with decrease of the signal-to-noise ratio (SNR). The same kind of SNR-dependent plasticity is present in the CSRF of real visual neurons; in bipolar cells of the carp retina as is shown here experimentally, as well as in large monopolar cells of the fly compound eye as was described by others. Also, analogous SNRdependent plasticityis shown to be present in the biphasic flash responses (BPFR) of these artificial and biological visual systems. Thus, the spatial (CSRF) and temporal (BPFR) filtering properties withwhich a wide variety of creatures see the world appear to be optimized for detectability of changes in space and time. 1 INTRODUCTION A number of learning algorithms have been developed to make synthetic neural machines be trainable to function in certain optimal ways. If the brain and nervous systems that we see in nature are best answers of the evolutionary process, then one might be able to find some common'softwares' in real and artificial neural systems. This possibility is examined in this paper, with respect to a basic visual 160 S.YASUI, T. FURUKAWA, M. YAMADA, T. SAITO

A statistical theory for overtraining is proposed. The analysis treats realizable stochastic neural networks, trained with Kullback Leibler loss in the asymptotic case. It is shown that the asymptotic gain in the generalization error is small if we perform early stopping, evenif we have access to the optimal stopping time. Considering cross-validation stopping we answer the question: In what ratio the examples should be divided into training and testing sets in order toobtain the optimum performance. In the non-asymptotic region cross-validated early stopping always decreases the generalization error.Our large scale simulations done on a CM5 are in nice agreement with our analytical findings.

By applying a likelihood splitting criteria to each expert in the HME we "grow" the tree adaptively during training. Secondly,by considering only the most probable path through the tree we may "prune" branches away, either temporarily, or permanently ifthey become redundant. We demonstrate results for the growing and path pruning algorithms which show significant speed ups and more efficient use of parameters over the standard fixed structure in discriminating between two interlocking spirals and classifying 8-bit parity patterns. INTRODUCTION The HME (Jordan & Jacobs 1994) is a tree structured network whose terminal nodes are simple function approximators in the case of regression or classifiers in the case of classification. The outputs of the terminal nodes or experts are recursively combined upwards towards the root node, to form the overall output of the network, by "gates" which are situated at the non-terminal nodes.

This paper discusses how a robot can learn goal-directed navigation tasksusing local sensory inputs. The emphasis is that such learning tasks could be formulated as an embedding problem of dynamical systems: desired trajectories in a task space should be embedded into an adequate sensory-based internal state space so that an unique mapping from the internal state space to the motor command could be established. The paper shows that a recurrent neural network suffices in self-organizing such an adequate internal state space from the temporal sensory input.

Completely parallel object recognition is NPcomplete. Achieving a recognizer with feasible complexity requires a compromise between paralleland sequential processing where a system selectively focuses on parts of a given image, one after another. Successive fixations are generated to sample the image and these samples are processed and abstracted to generate a temporal context in which results are integrated over time. A computational model based on a partially recurrent feedforward network is proposed and made credible bytesting on the real-world problem of recognition of handwritten digitswith encouraging results.

A new online learning algorithm which minimizes a statistical dependency amongoutputs is derived for blind separation of mixed signals. The dependency is measured by the average mutual information (MI)of the outputs. The source signals and the mixing matrix are unknown except for the number of the sources. The Gram-Charlier expansion instead of the Edgeworth expansion is used in evaluating the MI. The natural gradient approach is used to minimize the MI. A novel activation function is proposed for the online learning algorithm which has an equivariant property and is easily implemented on a neural network like model. The validity of the new learning algorithm are verified by computer simulations.

Search for the largest (or the smallest) among a number of input data, Le., the winner-take-all (WTA) action, is an essential part of intelligent data processing such as data retrieval in associative memories [3], vector quantization circuits [4], Kohonen's self-organizing maps [5] etc. In addition to the maximum or minimum search, data sorting also plays an essential role in a number of signal processing such as median filtering in image processing, evolutionary algorithms in optimizing problems [6] and so forth.