Recoverability of Group Sparse Signals from Corrupted Measurements via Robust Group Lasso

TL;DR

This paper introduces a robust group lasso method to accurately recover group sparse signals and sparse errors from corrupted measurements, with theoretical guarantees for exact recovery under broad conditions.

Contribution

The paper proposes a novel robust group lasso model with proven high-probability exact recovery guarantees for a wide class of sensing matrices.

Findings

Exact recovery of signals and errors with high probability.

The method handles general sensing matrices.

Theoretical guarantees extend to broad measurement models.

Abstract

This paper considers the problem of recovering a group sparse signal matrix $\mathbf{Y} = [\mathbf{y}_1, \cdots, \mathbf{y}_L]$ from sparsely corrupted measurements $\mathbf{M} = [\mathbf{A}_{(1)}\mathbf{y}_{1}, \cdots, \mathbf{A}_{(L)}\mathbf{y}_{L}] + \mathbf{S}$, where $\mathbf{A}_{(i)}$'s are known sensing matrices and $\mathbf{S}$ is an unknown sparse error matrix. A robust group lasso (RGL) model is proposed to recover $\mathbf{Y}$ and $\mathbf{S}$ through simultaneously minimizing the $\ell_{2,1}$-norm of $\mathbf{Y}$ and the $\ell_1$-norm of $\mathbf{S}$ under the measurement constraints. We prove that $\mathbf{Y}$ and $\mathbf{S}$ can be exactly recovered from the RGL model with a high probability for a very general class of $\mathbf{A}_{(i)}$'s.