TESTING THE MANIFOLD HYPOTHESIS

Abstract

We are increasingly confronted with very high dimensional data from speech,images, genomes, and other sources. A collection of methodologies for analyzing high dimensional data based on the hypothesis that data tend to lie near a low dimensional manifold is now called “manifold learning” (see Figure 1). We refer to the underlying hypothesis as the “manifold hypothesis.” Manifold learning, in particular, fitting low dimensional nonlinear manifolds to sampled data points in high dimensional spaces, has been an area of intense activity over the past two decades. These problems have been viewed as optimization problems generalizing the projection theorem in Hilbert space. We refer the interested reader to a limited set of papers associated with this field; see [3, 8, 9, 11, 16, 20, 26, 31, 32, 41, 43, 47,50,52,55] and the references therein. Section 2 contains a brief review of manifoldlearning