Stable Embedding of Grassmann Manifold via Gaussian Random matrices

TL;DR

This paper demonstrates that Gaussian random matrices can stably embed points on the Grassmann manifold by preserving volumes and distances, enabling reliable compression of multi-dimensional subspace signals.

Contribution

It introduces a volume-preserving embedding property for Grassmann manifold points via Gaussian matrices, extending stable embedding concepts to volume and principal angles.

Findings

Volumes of parallelotopes are preserved under Gaussian embedding.

Generalized distances based on principal sines are maintained after compression.

Numerical simulations validate theoretical volume and distance preservation.

Abstract

In this paper, we explore a volume-based stable embedding of multi-dimensional 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 multi-dimensional signals. In this paper, we formulate the linear subspace spanned by multi-dimensional 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 multi-dimensional generalization of the conventional stable embedding properties, which only concern the approximate preservation of lengths of vectors in certain unions of subspaces. Additionally, 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.