The Whitney Reduction Network: A Method for Computing Autoassociative Graphs

Abstract

This article introduces a new architecture and associated algorithms ideal for implementing the dimensionality reduction of an m-dimensionalmanifold 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 d ≤ 2m + 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.