Subspace Methods for Joint Sparse Recovery

TL;DR

This paper introduces the SA-MUSIC algorithm, enhancing joint sparse recovery in compressed sensing, especially under challenging conditions like rank-defect or high correlation, with proven performance guarantees.

Contribution

The paper develops the SA-MUSIC algorithm and subspace-based greedy methods, providing the first non-asymptotic analysis of signal subspace estimation in this context.

Findings

SA-MUSIC improves support recovery under rank-defect conditions.

The algorithms have proven performance guarantees based on restricted isometry property.

Non-asymptotic perturbation analysis of signal subspace estimation is provided.

Abstract

We propose robust and efficient algorithms for the joint sparse recovery problem in compressed sensing, which simultaneously recover the supports of jointly sparse signals from their multiple measurement vectors obtained through a common sensing matrix. In a favorable situation, the unknown matrix, which consists of the jointly sparse signals, has linearly independent nonzero rows. In this case, the MUSIC (MUltiple SIgnal Classification) algorithm, originally proposed by Schmidt for the direction of arrival problem in sensor array processing and later proposed and analyzed for joint sparse recovery by Feng and Bresler, provides a guarantee with the minimum number of measurements. We focus instead on the unfavorable but practically significant case of rank-defect or ill-conditioning. This situation arises with limited number of measurement vectors, or with highly correlated signal components. In this case MUSIC fails, and in practice none of the existing methods can consistently approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC), which improves on MUSIC so that the support is reliably recovered under such unfavorable conditions. Combined with subspace-based greedy algorithms also proposed and analyzed in this paper, SA-MUSIC provides a computationally efficient algorithm with a performance guarantee. The performance guarantees are given in terms of a version of restricted isometry property. In particular, we also present a non-asymptotic perturbation analysis of the signal subspace estimation that has been missing in the previous study of MUSIC.