Coherence-based Partial Exact Recovery Condition for OMP/OLS

TL;DR

This paper derives a new coherence-based condition for the exact support recovery of sparse signals using OMP and OLS, improving upon previous bounds by considering partial support knowledge.

Contribution

It introduces a novel partial recovery condition that relaxes existing coherence bounds for OMP and OLS in noiseless sparse recovery.

Findings

New sufficient and necessary coherence condition derived

Condition accounts for partial support knowledge during iterations

Improves existing bounds for exact support recovery

Abstract

We address the exact recovery of the support of a k-sparse vector with Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS) in a noiseless setting. We consider the scenario where OMP/OLS have selected good atoms during the first l iterations (l<k) and derive a new sufficient and worst-case necessary condition for their success in k steps. Our result is based on the coherence \mu of the dictionary and relaxes Tropp's well-known condition \mu<1/(2k-1) to the case where OMP/OLS have a partial knowledge of the support.