A New Approach to Dimensionality Reduction: Theory and Algorithms

Abstract

This paper applies Whitney's embedding theorem to the data reduction problem and introduces a new approach motivated in part by the (constructive) proof of the theorem. The notion of a good projection is introduced which involves picking projections of the high-dimensional system that are optimized such that they are easy to invert. The basic theory of the approach is outlined and algorithms for finding the projections are presented and applied to several test cases. A method for constructing the inverse projection is detailed and its properties, including a new measure of complexity, are discussed. Finally, well-known methods of data reduction are compared with our approach within the context of Whitney's theorem.