Analysis of Orthogonal Matching Pursuit using the Restricted Isometry Property

TL;DR

This paper shows that the restricted isometry property (RIP) with specific bounds guarantees exact recovery of sparse signals using Orthogonal Matching Pursuit (OMP), providing a simple and intuitive analysis.

Contribution

The paper introduces a straightforward RIP-based analysis for OMP, establishing new bounds on the isometry constant for exact sparse signal recovery.

Findings

RIP of order K+1 with δ < 1/(3√K) guarantees exact recovery of K-sparse signals.

Relaxed RIP bounds apply to highly compressible signals.

Brief analysis of Regularized OMP (ROMP) included.

Abstract

Orthogonal Matching Pursuit (OMP) is the canonical greedy algorithm for sparse approximation. In this paper we demonstrate that the restricted isometry property (RIP) can be used for a very straightforward analysis of OMP. Our main conclusion is that the RIP of order $K+1$ (with isometry constant $\delta < \frac{1}{3\sqrt{K}}$) is sufficient for OMP to exactly recover any $K$-sparse signal. Our analysis relies on simple and intuitive observations about OMP and matrices which satisfy the RIP. For restricted classes of $K$-sparse signals (those that are highly compressible), a relaxed bound on the isometry constant is also established. A deeper understanding of OMP may benefit the analysis of greedy algorithms in general. To demonstrate this, we also briefly revisit the analysis of the Regularized OMP (ROMP) algorithm.