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.

We are concerned in this paper with the application of multiple models, specifically the Hierarchical Mixtures of Experts, to time series prediction, specifically the problem of predicting acoustic vectors for use in speech coding. There have been a number of applications of multiple models in time series prediction. A classic example is the Threshold Autoregressive model (TAR) which was used by Tong & 836 S. R. Waterhouse, A. J. Robinson Lim (1980) to predict sunspot activity. More recently, Lewis, Kay and Stevens (in Weigend & Gershenfeld (1994)) describe the use of Multivariate and Regression Splines (MARS) to the prediction of future values of currency exchange rates. Finally, in speech prediction, Cuperman & Gersho (1985) describe the Switched Inter-frame Vector Prediction (SIVP) method which switches between separate linear predictors trained on different statistical classes of speech.

We propose an alternative model for mixtures of experts which uses a different parametric form for the gating network. The modified model is trained by the EM algorithm. In comparison with earlier models-trained by either EM or gradient ascent-there is no need to select a learning stepsize. We report simulation experiments which show that the new architecture yields faster convergence. We also apply the new model to two problem domains: piecewise nonlinear function approximation and the combination of multiple previously trained classifiers. 1 INTRODUCTION For the mixtures of experts architecture (Jacobs, Jordan, Nowlan & Hinton, 1991), the EM algorithm decouples the learning process in a manner that fits well with the modular structure and yields a considerably improved rate of convergence (Jordan & Jacobs, 1994).

The problem of interpolating between specified images in an image sequence is a simple, but important task in model-based vision. We describe an approach based on the abstract task of "manifold learning" and present results on both synthetic and real image sequences. This problem arose in the development of a combined lipreading and speech recognition system.

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

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.

We present a graph-based method for rapid, accurate search through prototypes for transformation-invariant pattern classification. Our method has in theory the same recognition accuracy as other recent methods based on ''tangent distance" [Simard et al., 1994], since it uses the same categorization rule. Nevertheless ours is significantly faster during classification because far fewer tangent distances need be computed. Crucial to the success of our system are 1) a novel graph architecture in which transformation constraints and geometric relationships among prototypes are encoded during learning, and 2) an improved graph search criterion, used during classification. These architectural insights are applicable to a wide range of problem domains. Here we demonstrate that on a handwriting recognition task, a basic implementation of our system requires less than half the computation of the Euclidean sorting method. 1 INTRODUCTION In recent years, the crucial issue of incorporating invariances into networks for pattern recognition has received increased attention, most especially due to the work of 666 Alessandro Sperduti, David G. Stork

The parietal cortex is thought to represent the egocentric positions of objects in particular coordinate systems. We propose an alternative approach to spatial perception of objects in the parietal cortex from the perspective of sensorimotor transformations. The responses of single parietal neurons can be modeled as a gaussian function of retinal position multiplied by a sigmoid function of eye position, which form a set of basis functions. We show here how these basis functions can be used to generate receptive fields in either retinotopic or head-centered coordinates by simple linear transformations. This raises the possibility that the parietal cortex does not attempt to compute the positions of objects in a particular frame of reference but instead computes a general purpose representation of the retinal location and eye position from which any transformation can be synthesized by direct projection. This representation predicts that hemineglect, a neurological syndrome produced by parietal lesions, should not be confined to egocentric coordinates, but should be observed in multiple frames of reference in single patients, a prediction supported by several experiments.

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 EMbased 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.

We present a new method for obtaining local error bars for nonlinear regression, i.e., estimates of the confidence in predicted values that depend on the input. We approach this problem by applying a maximumlikelihood framework to an assumed distribution of errors. We demonstrate our method first on computer-generated data with locally varying, normally distributed target noise. We then apply it to laser data from the Santa Fe Time Series Competition where the underlying system noise is known quantization error and the error bars give local estimates of model misspecification. In both cases, the method also provides a weightedregression effect that improves generalization performance.