Soft Recovery Through $\ell_{1,2}$ Minimization with Applications in Recovery of Simultaneously Sparse and Low-Rank Matrice

TL;DR

This paper introduces a new analysis method for $ ext{l}_{1,2}$-norm minimization in compressed sensing, providing approximate recovery guarantees for column-sparse matrices and proposing algorithms for simultaneously sparse and low-rank matrix recovery.

Contribution

It offers the first soft recovery guarantees for $ ext{l}_{1,2}$-norm minimization and suggests heuristic algorithms for recovering matrices that are both sparse and low-rank.

Findings

Established approximate recovery conditions for column-sparse matrices.

Proposed heuristic algorithms leveraging $ ext{l}_{1,2}$-minimization.

Introduced a novel analysis approach in compressed sensing.

Abstract

This article provides a new type of analysis of a compressed-sensing based technique for recovering column-sparse matrices, namely minimization of the $\ell_{1,2}$-norm. Rather than providing conditions on the measurement matrix which guarantees the solution of the program to be exactly equal to the ground truth signal (which already has been thoroughly investigated), it presents a condition which guarantees that the solution is approximately equal to the ground truth. Soft recovery statements of this kind are to the best knowledge of the author a novelty in Compressed Sensing. Apart from the theoretical analysis, we present two heuristic proposes how this property of the $\ell_{1,2}$-program can be utilized to design algorithms for recovery of matrices which are sparse and have low rank at the same time.