An Improved RIP-Based Performance Guarantee for Sparse Signal Recovery via Orthogonal Matching Pursuit

TL;DR

This paper improves the theoretical performance guarantee for orthogonal matching pursuit in sparse signal recovery by relaxing the restricted isometry constant bounds, thus narrowing the gap towards the conjectured optimal condition.

Contribution

It introduces a new relaxed bound on the restricted isometry constant for OMP, based on approximate orthogonality, enhancing recovery guarantees in compressive sensing.

Findings

Relaxed the RIP constant bound for perfect recovery in OMP

Derived less restrictive conditions for noisy signal recovery

Extended the approach to interference cancellation and subspace pursuit

Abstract

A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations is that the restricted isometry constant of the sensing matrix satisfies delta_K+1<1/(sqrt(delta_K+1)+1). By exploiting an approximate orthogonality condition characterized via the achievable angles between two orthogonal sparse vectors upon compression, this paper shows that the upper bound on delta can be further relaxed to delta_K+1<(sqrt(1+4*delta_K+1)-1)/(2K).This result thus narrows the gap between the so far best known bound and the ultimate performance guarantee delta_K+1<1/(sqrt(delta_K+1)) that is conjectured by Dai and Milenkovic in 2009. The proposed approximate orthogonality condition is also exploited to derive less restricted sufficient conditions for signal reconstruction in several compressive sensing problems, including signal recovery via OMP in a noisy environment, compressive domain interference cancellation, and support identification via the subspace pursuit algorithm.