A new approach to dimensionality reduction: theory and algorithms

This paper applies Whitney's embedding theorem to the data reduction proble m and introduces a new approach motivated in part by the (constructive) pro of of the theorem. The notion of a good projection is introduced which invo lves picking projections of the high-dimensional system that are optimized such that they are easy to invert. The basic theory of the approach is outl ined and algorithms for finding the projections are presented and applied t o several test cases. A method for constructing the inverse projection is d etailed and its properties, including a new measure of complexity, are disc ussed. Finally, well-known methods of data reduction are compared with our approach within the context of Whitney's theorem.