Restricted Isometry Property of Subspace Projection Matrix Under Random Compression

TL;DR

This paper investigates the restricted isometry property (RIP) of subspace projection matrices under random orthonormal compression, establishing conditions for stable embedding with a specific measurement complexity.

Contribution

It provides the first analysis of RIP for structured subspace projection matrices under random compression, revealing measurement bounds for stable embedding.

Findings

RIP of subspace projection matrices is guaranteed with O(s(N-s) log N) measurements.

Subspace projection matrices form an s(N-s)-dimensional manifold in matrix space.

Stable embedding of this manifold under random orthonormal compression is achieved with high probability.

Abstract

Structures play a significant role in the field of signal processing. As a representative of structural data, low rank matrix along with its restricted isometry property (RIP) has been an important research topic in compressive signal processing. Subspace projection matrix is a kind of low rank matrix with additional structure, which allows for further reduction of its intrinsic dimension. This leaves room for improving its own RIP, which could work as the foundation of compressed subspace projection matrix recovery. In this work, we study the RIP of subspace projection matrix under random orthonormal compression. Considering the fact that subspace projection matrices of $s$ dimensional subspaces in $\mathbb{R}^N$ form an $s(N-s)$ dimensional submanifold in $\mathbb{R}^{N\times N}$, our main concern is transformed to the stable embedding of such submanifold into $\mathbb{R}^{N\times N}$. The result is that by $O(s(N-s)\log N)$ number of random measurements the RIP of subspace projection matrix is guaranteed.