Joint Block-Sparse Recovery Using Simultaneous BOMP/BOLS

TL;DR

This paper introduces new greedy algorithms for joint recovery of high-dimensional block-sparse signals, providing theoretical guarantees and analysis based on mutual incoherence property, enhancing understanding of block-sparse recovery in compressed sensing.

Contribution

It proposes two novel simultaneous block orthogonal least squares algorithms and analyzes their performance, extending theoretical bounds for reliable recovery under MIP conditions.

Findings

Algorithms successfully recover block-sparse signals under MIP conditions

Theoretical bounds for data volume needed for reliable recovery are derived

Results are applicable to both block and non-block greedy algorithms

Abstract

We consider the greedy algorithms for the joint recovery of high-dimensional sparse signals based on the block multiple measurement vector (BMMV) model in compressed sensing (CS). To this end, we first put forth two versions of simultaneous block orthogonal least squares (S-BOLS) as the baseline for the OLS framework. Their cornerstone is to sequentially check and select the support block to minimize the residual power. Then, parallel performance analysis for the existing simultaneous block orthogonal matching pursuit (S-BOMP) and the two proposed S-BOLS algorithms is developed. It indicates that under the conditions based on the mutual incoherence property (MIP) and the decaying magnitude structure of the nonzero blocks of the signal, the algorithms select all the significant blocks before possibly choosing incorrect ones. In addition, we further consider the problem of sufficient data volume for reliable recovery, and provide its MIP-based bounds in closed-form. These results together highlight the key role of the block characteristic in addressing the weak-sparse issue, i.e., the scenario where the overall sparsity is too large. The derived theoretical results are also universally valid for conventional block-greedy algorithms and non-block algorithms by setting the number of measurement vectors and the block length to 1, respectively.