Compressed Sensing of Block-Sparse Signals: Uncertainty Relations and Efficient Recovery

TL;DR

This paper develops uncertainty relations and efficient algorithms for recovering block-sparse signals in compressed sensing, demonstrating improved reconstruction by leveraging block structure over traditional sparsity assumptions.

Contribution

It introduces a block-coherence measure, derives an uncertainty relation, and proves that block-OMP and mixed $\ ext{l}_2/\text{l}_1$ optimization can reliably recover block-sparse signals under certain conditions.

Findings

Block-OMP recovers block $k$-sparse signals in at most $k$ steps with low block-coherence.

Mixed $\ell_2/\ell_1$ optimization guarantees successful recovery under the same coherence conditions.

Explicit use of block-sparsity improves reconstruction performance over conventional sparsity methods.

Abstract

We consider compressed sensing of block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block $k$-sparse signals in no more than $k$ steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed $\ell_2/\ell_1$-optimization approach. This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.