For a detailed rationale the reader is referred to the work of Rissanen (1984) or Wallace and Freeman (1987) and the references therein. Note that the Minimum Description Length (MDL) technique (as Rissanen's approach has become known) is implicitly related to Maximum A Posteriori (MAP) Bayesian estimation techniques if cast in the appropriate framework.

Learning an input-output mapping from a set of examples can be regarded as synthesizing an approximation of a multidimensional function.

M2/n' for sufficiently large values of M. Here, Poo(error) denotes the probability of error in the infinite sample limit, and is at most twice the error of a Bayes classifier. Although the value of the coefficient a depends upon the underlying probability distributions, the exponent of M is largely distribution free. We thus obtain a concise relation between a classifier's ability to generalize from a finite reference sample and the dimensionality of the feature space, as well as an analytic validation of Bellman's well known "curse of dimensionality." 1 INTRODUCTION One of the primary tasks assigned to neural networks is pattern classification. Common applications include recognition problems dealing with speech, handwritten characters, DNA sequences, military targets, and (in this conference) sexual identity. Two fundamental concepts associated with pattern classification are generalization (how well does a classifier respond to input data it has never encountered before?) and scalability (how are a classifier's processing and training requirements affected by increasing the number of features that describe the input patterns?).

Padhraic Smyth Communications Systems Research Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109 Abstract Given some training data how should we choose a particular network classifier froma family of networks of different complexities? In this paper we discuss how the application of stochastic complexity theory to classifier design problems can provide some insights into this problem. In particular we introduce the notion of admissible models whereby the complexity of models under consideration is affected by (among other factors) the class entropy, the amount of training data, and our prior belief. In particular we discuss the implications of these results with respect to neural architectures anddemonstrate the approach on real data from a medical diagnosis task. 1 Introduction and Motivation In this paper we examine in a general sense the application of Minimum Description Length (MDL) techniques to the problem of selecting a good classifier from a large set of candidate models or hypotheses. Pattern recognition algorithms differ from more conventional statistical modeling techniques in the sense that they typically choose from a very large number of candidate models to describe the available data.

Santosh S. Venkatesh Electrical Engineering University of Pennsylvania Philadelphia, PA 19104 If patterns are drawn from an n-dimensional feature space according to a probability distribution that obeys a weak smoothness criterion, we show that the probability that a random input pattern is misclassified by a nearest-neighbor classifier using M random reference patterns asymptotically satisfies a PM(error) "" Poo(error) M2/n' for sufficiently large values of M. Here, Poo(error) denotes the probability of error in the infinite sample limit, and is at most twice the error of a Bayes classifier. Although the value of the coefficient a depends upon the underlying probability distributions, the exponent of M is largely distribution free.We thus obtain a concise relation between a classifier's ability to generalize from a finite reference sample and the dimensionality of the feature space, as well as an analytic validation of Bellman's well known "curse of dimensionality." 1 INTRODUCTION One of the primary tasks assigned to neural networks is pattern classification.

This paper introduces a generalization of the layered neural network that can implement a time-varying nonlinear mapping between its observable input and output. The variation of the network's mapping is due to an additional, hidden control input, while the network parameters remain unchanged. We proposed an algorithm for finding the network parameters and the hidden control sequence from a training set of examples of observable input and output. This algorithm implements an approximate maximum likelihood estimation of parameters of an equivalent statistical model, when only the dominant control sequence is taken into account. The conceptual difference between the proposed model and the HMM is that in the HMM approach, the observable data in each of the states is modeled as though it was produced by a memoryless source, and a parametric description of this source is obtained during training, while in the proposed model the observations in each state are produced by a nonlinear dynamical system driven by noise, and both the parametric form of the dynamics and the noise are estimated. The perfonnance of the model was illustrated for the tasks of nonlinear time-varying system modeling and continuously spoken digit recognition. The reported results show the potential of this model for providing high performance speech recognition capability. Acknowledgment Specialthanks are due to N. Merhav for numerous comments and helpful discussions.

Learning an input-output mapping from a set of examples can be regarded as synthesizing an approximation of a multidimensional function.

For a detailed rationale the reader is referred to the work of Rissanen (1984) or Wallace and Freeman (1987) and the references therein. Note that the Minimum Description Length (MDL) technique (as Rissanen's approach has become known) is implicitly related to Maximum A Posteriori (MAP) Bayesian estimation techniques if cast in the appropriate framework.

In this paper, we prove that the vectors in the LVQ learning algorithm converge. We do this by showing that the learning algorithm performs stochastic approximation. Convergence is then obtained by identifying the appropriate conditions on the learning rate and on the underlying statistics of the classification problem. We also present a modification to the learning algorithm which we argue results in convergence of the LVQ error to the Bayesian optimal error as the appropriate parameters become large.

This paper introduces a generalization of the layered neural network that can implement a time-varying nonlinear mapping between its observable input and output. The variation of the network's mapping is due to an additional, hidden control input, while the network parameters remain unchanged. We proposed an algorithm for finding the network parameters and the hidden control sequence from a training set of examples of observable input and output. This algorithm implements an approximate maximum likelihood estimation of parameters of an equivalent statistical model, when only the dominant control sequence is taken into account. The conceptual difference between the proposed model and the HMM is that in the HMM approach, the observable data in each of the states is modeled as though it was produced by a memoryless source, and a parametric description of this source is obtained during training, while in the proposed model the observations in each state are produced by a nonlinear dynamical system driven by noise, and both the parametric form of the dynamics and the noise are estimated. The perfonnance of the model was illustrated for the tasks of nonlinear time-varying system modeling and continuously spoken digit recognition. The reported results show the potential of this model for providing high performance speech recognition capability. Acknowledgment Special thanks are due to N. Merhav for numerous comments and helpful discussions.