Support Recovery with Orthogonal Matching Pursuit in the Presence of Noise: A New Analysis

TL;DR

This paper analyzes the orthogonal matching pursuit algorithm for support recovery in noisy compressed sensing, establishing conditions under which exact or approximate support recovery is possible depending on the SNR and sparsity level.

Contribution

It provides necessary and sufficient SNR conditions for exact support recovery with OMP and demonstrates the possibility of approximate recovery under certain noise conditions.

Findings

Exact support recovery requires SNR to scale linearly with sparsity level K.

Exact recovery is impossible when SNR is constant and independent of K.

Approximate support recovery with small error fraction is achievable under certain noise conditions.

Abstract

Support recovery of sparse signals from compressed linear measurements is a fundamental problem in compressed sensing (CS). In this paper, we study the orthogonal matching pursuit (OMP) algorithm for the recovery of support under noise. We consider two signal-to-noise ratio (SNR) settings: i) the SNR depends on the sparsity level $K$ of input signals, and ii) the SNR is an absolute constant independent of $K$. For the first setting, we establish necessary and sufficient conditions for the exact support recovery with OMP, expressed as lower bounds on the SNR. Our results indicate that in order to ensure the exact support recovery of all $K$-sparse signals with the OMP algorithm, the SNR must at least scale linearly with the sparsity level $K$. In the second setting, since the necessary condition on the SNR is not fulfilled, the exact support recovery with OMP is impossible. However, our analysis shows that recovery with an arbitrarily small but constant fraction of errors is possible with the OMP algorithm. This result may be useful for some practical applications where obtaining some large fraction of support positions is adequate.