On the optimality of a L1/L1 solver for sparse signal recovery from sparsely corrupted compressive measurements

TL;DR

This paper proves the instance optimality of an / solver for recovering sparse signals from sparsely corrupted compressive measurements, combining existing results to establish theoretical guarantees.

Contribution

It demonstrates the / solver's optimality in sparse signal recovery under sparsely corrupted measurements, extending prior theoretical results.

Findings

Proves / solver's instance optimality.

Establishes theoretical guarantees for sparse signal recovery.

Combines known results to support the optimality claim.

Abstract

This short note proves the $\ell_2-\ell_1$ instance optimality of a $\ell_1/\ell_1$ solver, i.e a variant of \emph{basis pursuit denoising} with a $\ell_1$ fidelity constraint, when applied to the estimation of sparse (or compressible) signals observed by sparsely corrupted compressive measurements. The approach simply combines two known results due to Y. Plan, R. Vershynin and E. Cand\`es.