Joint Sparse Recovery Using Signal Space Matching Pursuit

TL;DR

This paper introduces SSMP, a novel joint sparse recovery algorithm that sequentially identifies support, with proven guarantees for exact reconstruction under RIP conditions, and demonstrates superior performance over existing methods.

Contribution

The paper proposes the SSMP algorithm for joint sparse recovery, providing new theoretical guarantees and showing improved performance compared to conventional methods.

Findings

SSMP guarantees exact recovery under specific RIP conditions.

SSMP outperforms conventional algorithms in noiseless and noisy scenarios.

Performance improves as the rank r increases, with less restrictive RIP requirements.

Abstract

In this paper, we put forth a new joint sparse recovery algorithm called signal space matching pursuit (SSMP). The key idea of the proposed SSMP algorithm is to sequentially investigate the support of jointly sparse vectors to minimize the subspace distance to the residual space. Our performance guarantee analysis indicates that SSMP accurately reconstructs any row $K$-sparse matrix of rank $r$ in the full row rank scenario if the sampling matrix $\mathbf{A}$ satisfies $\text{krank}(\mathbf{A}) \ge K+1$, which meets the fundamental minimum requirement on $\mathbf{A}$ to ensure exact recovery. We also show that SSMP guarantees exact reconstruction in at most $K-r+\lceil \frac{r}{L} \rceil$ iterations, provided that $\mathbf{A}$ satisfies the restricted isometry property (RIP) of order $L(K-r)+r+1$ with $$\delta_{L(K-r)+r+1} < \max \left \{ \frac{\sqrt{r}}{\sqrt{K+\frac{r}{4}}+\sqrt{\frac{r}{4}}}, \frac{\sqrt{L}}{\sqrt{K}+1.15 \sqrt{L}} \right \},$$ where $L$ is the number of indices chosen in each iteration. This implies that the requirement on the RIP constant becomes less restrictive when $r$ increases. Such behavior seems to be natural but has not been reported for most of conventional methods. We further show that if $r=1$, then by running more than $K$ iterations, the performance guarantee of SSMP can be improved to $\delta_{\lfloor 7.8K \rfloor} \le 0.155$. In addition, we show that under a suitable RIP condition, the reconstruction error of SSMP is upper bounded by a constant multiple of the noise power, which demonstrates the stability of SSMP under measurement noise. Finally, from extensive numerical experiments, we show that SSMP outperforms conventional joint sparse recovery algorithms both in noiseless and noisy scenarios.