Improved RIP Analysis of Orthogonal Matching Pursuit

TL;DR

This paper improves theoretical guarantees for Orthogonal Matching Pursuit (OMP) in compressive sensing, extends error bounds to noisy and non-sparse signals, and empirically shows OMP's superior average-case performance.

Contribution

It provides an improved RIP-based performance guarantee for OMP, extends analysis to noisy and non-sparse signals, and demonstrates empirical advantages over other algorithms.

Findings

OMP outperforms other algorithms in average-case scenarios.

Theoretical RIP bounds for OMP are asymptotically improved.

Empirical results challenge worst-case RIP analysis expectations.

Abstract

Orthogonal Matching Pursuit (OMP) has long been considered a powerful heuristic for attacking compressive sensing problems; however, its theoretical development is, unfortunately, somewhat lacking. This paper presents an improved Restricted Isometry Property (RIP) based performance guarantee for T-sparse signal reconstruction that asymptotically approaches the conjectured lower bound given in Davenport et al. We also further extend the state-of-the-art by deriving reconstruction error bounds for the case of general non-sparse signals subjected to measurement noise. We then generalize our results to the case of K-fold Orthogonal Matching Pursuit (KOMP). We finish by presenting an empirical analysis suggesting that OMP and KOMP outperform other compressive sensing algorithms in average case scenarios. This turns out to be quite surprising since RIP analysis (i.e. worst case scenario) suggests that these matching pursuits should perform roughly T^0.5 times worse than convex optimization, CoSAMP, and Iterative Thresholding.