Stable Embedding of Grassmann Manifold via Gaussian Random Matrices

Abstract

Compressive sensing (CS) provides a new perspective for data reduction without compromising performance when the signal of interest is sparse or has intrinsically low-dimensional structure. The theoretical foundation for most of the existing studies on CS is based on the stable embedding (i.e., a distance-preserving property) of vectors that are sparse or in a union of subspaces via random measurement matrices. To the best of our knowledge, few existing literatures of CS have clearly discussed the stable embedding of linear subspaces via compressive measurement systems. In this paper, we explore a volume-based stable embedding of multidimensional signals based on Grassmann manifold, via Gaussian random measurement matrices. The Grassmann manifold is a topological space, in which each point is a linear vector subspace, and is widely regarded as an ideal model for multidimensional signals generated from linear subspaces. In this paper, we formulate the linear subspace spanned by multidimensional signal vectors as points on the Grassmann manifold, and use the volume and the product of sines of principal angles (also known as the product of principal sines) as the generalized norm and distance measure for the space of Grassmann manifold. We prove a volume-preserving embedding property for points on the Grassmann manifold via Gaussian random measurement matrices, i.e., the volumes of all parallelotopes from a finite set in Grassmann manifold are preserved upon compression. This volume-preserving embedding property is a multidimensional generalization of the conventional stable embedding properties, which only concern the approximate preservation of lengths of vectors in certain unions of subspaces. In addition, we use the volume-preserving embedding property to explore the stable embedding effect on a generalized distance measure of Grassmann manifold induced from volume. It is proved that the generalized distance measure, i.e., the product of principal sines between different points on the Grassmann manifold, is well preserved in the compressed domain via Gaussian random measurement matrices. Numerical simulations are also provided for validation.