The self-organization of recurrent feature-discovery networks is studied from the perspective of dynamical systems. Bifurcation theory reveals parameter regimes in which multiple equilibria or limit cycles coexist with the equilibrium at which the networks perform principal component analysis.

"vector quantization", and "unsupervised learning" are all words which descn'be the same process: assigning a few exemplars to represent a large set of samples. Perfonning that process is the subject of a substantial body of literature. In this paper, we are concerned with the comparison of various clustering techniques to a particular, practical application: color clustering. The color clustering problem is as follows: an image is recorded in full color -- that is, three components, RED, GREEN, and BLUE, each of which has been measured to 8 bits of precision. Thus, each pixel is a 24 bit quantity. We must find a representation in which 2563 possible colors are represented by only 8 bits per pixel. That is, for a problem with 256000 variables (512 x 512) variables, assign each variable to one of only 256 classes. The color clustering problem is currently of major economic interest since millions of display systems are sold each year which can only store 8 bits per pixel, but on which users would like to be able to display "true" color (or at least as near true color as possible). In this study, we have approached the problem using the standard techniques from the literature (including k-means -- ISODATA clustering[1,3,61, LBG[4]), competitive learning (referred to as CL herein) [2], and Kohonen feature maps [5,7,9].

Seven different neural network and conventional pattern classifiers were compared using artificial and speech recognition tasks. High order polynomial GMDH classifiers typically provided intermediate error rates and often required long training times and large amounts of memory. In addition, the decision regions formed did not generalize well to regions of the input space with little training data. Radial basis function classifiers generalized well in high dimensional spaces, and provided low error rates with training times that were much less than those of back-propagation classifiers (Lee and Lippmann, 1989). Gaussian mixture classifiers provided good performance when the numbers and types of mixtures were selected carefully to model class densities well. Linear tree classifiers were the most computationally ef- 976 Ng and Lippmann ficient but performed poorly with high dimensionality inputs and when the number of training patterns was small. KD-tree classifiers reduced classification time by a factor of four over conventional KNN classifiers for low 2-input dimension problems. They provided little or no reduction in classification time for high 22-input dimension problems. Improved condensed KNN classifiers reduced memory requirements over conventional KNN classifiers by a factor of two to fifteen for all problems, without increasing the error rate significantly.

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?).

We introduce a framework for training architectures composed of several modules. This framework, which uses a statistical formulation of learning systems, provides a unique formalism for describing many classical connectionist algorithms as well as complex systems where several algorithms interact. It allows to design hybrid systems which combine the advantages of connectionist algorithms as well as other learning algorithms.

Sherif M. Botros Christopher G. Atkeson Brain and Cognitive Sciences Department and the Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA 02139 Abstract We examine the ability of radial basis functions (RBFs) to generalize. We compare the performance of several types of RBFs. We use the inverse dynamics of an idealized two-joint arm as a test case. We find that without a proper choice of a norm for the inputs, RBFs have poor generalization properties. A simple global scaling of the input variables greatly improves performance.

This paper is a summary of SPRINT project aims and results. The project focus on the use of neuro-computing techniques to tackle various problems that remain unsolved in speech recognition. First results concern the use of feedforward nets for phonetic units classification, isolated word recognition, and speaker adaptation.

We develop a sequential adaptation algorithm for radial basis function (RBF) neural networks of Gaussian nodes, based on the method of successive F-Projections.

We present and compare learning rate schedules for stochastic gradient descent, a general algorithm which includes LMS, online backpropagation and k-means clustering as special cases. We introduce "search-thenconverge" type schedules which outperform the classical constant and "running average" (1ft) schedules both in speed of convergence and quality of solution.

Bell-shaped response curves are commonly found in biological neurons whenever a natural metric exist on the corresponding relevant stimulus variable (orientation, position in space, frequency, time delay,...). As a result, they are often used in neural models in different context ranging from resolution enhancement and interpolation to learning (see, for instance, Baldi et al. (1988), Moody et al. (1989) *and Division of Biology, California Institute of Technology. The complete title of this paper should read: "Computing with arrays of bell-shaped and sigmoid functions.