In this paper we examine a perceptron learning task. The task is realizable since it is provided by another perceptron with identical architecture. Both perceptrons have nonlinear sigmoid output functions. The gain of the output function determines the level of nonlinearity of the learning task. It is observed that a high level of nonlinearity leads to overfitting. We give an explanation for this rather surprising observation and develop a method to avoid the overfitting. This method has two possible interpretations, one is learning with noise, the other cross-validated early stopping.

We derive a smoothing regularizer for recurrent network models by requiring robustness in prediction performance to perturbations of the training data. The regularizer can be viewed as a generalization of the first order Tikhonov stabilizer to dynamic models. The closed-form expression of the regularizer covers both time-lagged and simultaneous recurrent nets, with feedforward nets and onelayer linear nets as special cases. We have successfully tested this regularizer in a number of case studies and found that it performs better than standard quadratic weight decay. 1 Introd uction One technique for preventing a neural network from overfitting noisy data is to add a regularizer to the error function being minimized. Regularizers typically smooth the fit to noisy data. Well-established techniques include ridge regression, see (Hoerl & Kennard 1970), and more generally spline smoothing functions or Tikhonov stabilizers that penalize the mth-order squared derivatives of the function being fit, as in (Tikhonov & Arsenin 1977), (Eubank 1988), (Hastie & Tibshirani 1990) and (Wahba 1990). Thes(-ilethods have recently been extended to networks of radial basis functions (Girosi, Jones & Poggio 1995), and several heuristic approaches have been developed for sigmoidal neural networks, for example, quadratic weight decay (Plaut, Nowlan & Hinton 1986), weight elimination (Scalettar & Zee 1988),(Chauvin 1990),(Weigend, Rumelhart & Huberman 1990) and soft weight sharing (Nowlan & Hinton 1992).

Linear threshold elements are the basic building blocks of artificial neural networks. A linear threshold element computes a function that is a sign of a weighted sum of the input variables. The weights are arbitrary integers; actually, they can be very big integers-exponential in the number of the input variables. However, in practice, it is difficult to implement big weights. In the present literature a distinction is made between the two extreme cases: linear threshold functions with polynomial-size weights as opposed to those with exponential-size weights.

A neural network model of 3-D lightness perception is presented which builds upon the FACADE Theory Boundary Contour System/Feature Contour System of Grossberg and colleagues. Early ratio encoding by retinal ganglion neurons as well as psychophysical results on constancy across different backgrounds (background constancy) are used to provide functional constraints to the theory and suggest a contrast negation hypothesis which states that ratio measures between coplanar regions are given more weight in the determination of lightness of the respective regions.

The wake-sleep algorithm (Hinton, Dayan, Frey and Neal 1995) is a relatively efficient method of fitting a multilayer stochastic generative model to high-dimensional data. In addition to the top-down connections in the generative model, it makes use of bottom-up connections for approximating the probability distribution over the hidden units given the data, and it trains these bottom-up connections using a simple delta rule. We use a variety of synthetic and real data sets to compare the performance of the wake-sleep algorithm with Monte Carlo and mean field methods for fitting the same generative model and also compare it with other models that are less powerful but easier to fit. 1 INTRODUCTION Neural networks are often used as bottom-up recognition devices that transform input vectors into representations of those vectors in one or more hidden layers. But multilayer networks of stochastic neurons can also be used as top-down generative models that produce patterns with complicated correlational structure in the bottom visible layer. In this paper we consider generative models composed of layers of stochastic binary logistic units. Given a generative model parameterized by top-down weights, there is an obvious way to perform unsupervised learning. The generative weights are adjusted to maximize the probability that the visible vectors generated by the model would match the observed data.

A Bayesian-Kullback learning scheme, called Ying-Yang Machine, is proposed based on the two complement but equivalent Bayesian representations for joint density and their Kullback divergence. Not only the scheme unifies existing major supervised and unsupervised learnings, including the classical maximum likelihood or least square learning, the maximum information preservation, the EM & em algorithm and information geometry, the recent popular Helmholtz machine, as well as other learning methods with new variants and new results; but also the scheme provides a number of new learning models. 1 INTRODUCTION Many different learning models have been developed in the literature. We may come to an age of searching a unified scheme for them. With a unified scheme, we may understand deeply the existing models and their relationships, which may cause cross-fertilization on them to obtain new results and variants; We may also be guided to develop new learning models, after we get better understanding on which cases we have already studied or missed, which deserve to be further explored. Recently, a Baysian-Kullback scheme, called the YING-YANG Machine, has been proposed as such an effort(Xu, 1995a). It bases on the Kullback divergence and two complement but equivalent Baysian representations for the joint distribution of the input space and the representation space, instead of merely using Kullback divergence for matching un-structuralized joint densities in information geometry type learnings (Amari, 1995a&b; Byrne, 1992; Csiszar, 1975).

The lateral connections learn the correlations of activity between units on the map. The resulting excitatory connections focus the activity into local patches and the inhibitory connections decorrelate redundant activity on the map. The map thus forms internal representations that are easy to recognize with e.g. a perceptron network. The recognition rate on a subset of NIST database 3 is 4.0% higher with LISSOM than with a regular Self-Organizing Map (SOM) as the front end, and 15.8% higher than recognition of raw input bitmaps directly. These results form a promising starting point for building pattern recognition systems with a LISSOM map as a front end. 1 Introduction Handwritten digit recognition has become one of the touchstone problems in neural networks recently.

In this paper, we introduce REMAP, an approach for the training and estimation of posterior probabilities using a recursive algorithm that is reminiscent of the EMbased Forward-Backward (Liporace 1982) algorithm for the estimation of sequence likelihoods. Although very general, the method is developed in the context of a statistical model for transition-based speech recognition using Artificial Neural Networks (ANN) to generate probabilities for Hidden Markov Models (HMMs). In the new approach, we use local conditional posterior probabilities of transitions to estimate global posterior probabilities of word sequences. Although we still use ANNs to estimate posterior probabilities, the network is trained with targets that are themselves estimates of local posterior probabilities. An initial experimental result shows a significant decrease in error-rate in comparison to a baseline system. 1 INTRODUCTION The ultimate goal in speech recognition is to determine the sequence of words that has been uttered.

This paper presents a self-organizing neural network that learns to detect, represent, and predict the visibility and invisibility relationships that arise during occlusion events, after a period of exposure to motion sequences containing occlusion and disocclusion events. The network develops two parallel opponent channels or "chains" of lateral excitatory connections for every resolvable motion trajectory. One channel, the "On" chain or "visible" chain, is activated when a moving stimulus is visible. The other channel, the "Off" chain or "invisible" chain, carries a persistent, amodal representation that predicts the motion of a formerly visible stimulus that becomes invisible due to occlusion. The learning rule uses disinhibition from the On chain to trigger learning in the Off chain.

A new learning algorithm is developed for the design of statistical classifiers minimizing the rate of misclassification. The method, which is based on ideas from information theory and analogies to statistical physics, assigns data to classes in probability. The distributions are chosen to minimize the expected classification error while simultaneously enforcing the classifier's structure and a level of "randomness" measured by Shannon's entropy. Achievement of the classifier structure is quantified by an associated cost. The constrained optimization problem is equivalent to the minimization of a Helmholtz free energy, and the resulting optimization method is a basic extension of the deterministic annealing algorithm that explicitly enforces structural constraints on assignments while reducing the entropy and expected cost with temperature. In the limit of low temperature, the error rate is minimized directly and a hard classifier with the requisite structure is obtained. This learning algorithm can be used to design a variety of classifier structures. The approach is compared with standard methods for radial basis function design and is demonstrated to substantially outperform other design methods on several benchmark examples, while often retaining design complexity comparable to, or only moderately greater than that of strict descent-based methods.