Random Projections of Signal Manifolds

Abstract

Random projections have recently found a surprising niche in signal processing. The key revelation is that the relevant structure in a signal can be preserved when that signal is projected onto a small number of random basis functions. Recent work has exploited this fact under the rubric of Compressed Sensing (CS): signals that are sparse in some basis can be recovered from small numbers of random linear projections. In many cases, however, we may have a more specific low-dimensional model for signals in which the signal class forms a nonlinear manifold in R^N. This paper provides preliminary theoretical and experimental evidence that manifold-based signal structure can be preserved using small numbers of random projections. The key theoretical motivation comes from Whitneyâ s Embedding Theorem, which states that a K-dimensional manifold can be embedded in R^{2K+1}. We examine the potential applications of this fact. In particular, we consider the task of recovering a manifold-modeled signal from a small number of random projections. Thanks to our (more specific) model, the ability to recover the signal can be far superior to existing techniques in CS.