Joint-sparse recovery from multiple measurements

TL;DR

This paper studies the recovery of jointly sparse matrices from multiple measurements, comparing sum-of-norm minimization and the ReMBo algorithm, revealing their limitations and how additional measurements affect recovery performance.

Contribution

It provides a theoretical comparison of recovery algorithms for joint sparsity, highlighting differences from single-measurement approaches and analyzing the impact of additional measurements.

Findings

Sum-of-norm minimization matches the uniform recovery rate of sequential SMV.

ReMBo algorithm's performance improves with more measurements.

More measurements than nonzero rows do not enhance theoretical recovery rate.

Abstract

The joint-sparse recovery problem aims to recover, from sets of compressed measurements, unknown sparse matrices with nonzero entries restricted to a subset of rows. This is an extension of the single-measurement-vector (SMV) problem widely studied in compressed sensing. We analyze the recovery properties for two types of recovery algorithms. First, we show that recovery using sum-of-norm minimization cannot exceed the uniform recovery rate of sequential SMV using $\ell_1$ minimization, and that there are problems that can be solved with one approach but not with the other. Second, we analyze the performance of the ReMBo algorithm [M. Mishali and Y. Eldar, IEEE Trans. Sig. Proc., 56 (2008)] in combination with $\ell_1$ minimization, and show how recovery improves as more measurements are taken. From this analysis it follows that having more measurements than number of nonzero rows does not improve the potential theoretical recovery rate.