Stable grassmann manifold embedding via Gaussian random matrices

Abstract

Compressive Sensing (CS) provides a new perspective for dimensionnality reduction without compromising performance. The theoretical foundation for most of existing studies of CS is a stable embedding (i.e., a distance-preserving property) of certain low-dimensional signal models such as sparse signals or signals in a union of linear subspaces. However, few existing literatures clearly discussed the embedding effect of points on the Grassmann manifold in under-sampled linear measurement systems. In this paper, we explore the stable embedding property of multi-dimensional signals based on Grassmann manifold, which is a topological space with each point being a linear subspace of ℝN (or ℂN), via the Gaussian random matrices. It should be noted that the stability mentioned here is about the volume-preserving instead of distance-preserving, because volume is the key characteristic for linear subspace spanned by multiple vectors. The theorem of the volume-preserving stable embedding property is proposed, and sketched proofs as well as discussions about our theorem is also given.