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The Whitney Reduction Network: A Method for Computing Autoassociative Graphs

Department of Mathematics, University of Manchester Institute of Science and Technology, Manchester M60 1QD, U.K.

M. J. Kirby

Department of Mathematics, Colorado State University, Fort Collins, CO 80523, U.S.A.

This article introduces a new architecture and associated algorithms ideal for implementing the dimensionality reduction of an m-dimensional manifold initially residing in an n-dimensional Euclidean space where n>>m. Motivated by Whitney's embedding theorem, the network is capable of training the identity mapping employing the idea of the graph of a function. In theory, a reduction to a dimension d that retains the differential structure of the original data may be achieved for some d2m+1. To implement this network, we propose the idea of a good-projection, which enhances the generalization capabilities of the network, and an adaptive secant basis algorithm to achieve it. The effect of noise on this procedure is also considered. The approach is illustrated with several examples.