Sparse Recovery with Orthogonal Matching Pursuit under RIP

TL;DR

This paper provides a new RIP-based analysis for orthogonal matching pursuit (OMP), showing it can recover sparse signals efficiently with fewer measurements than previous mutual incoherence-based bounds.

Contribution

It introduces an RIP-based condition under which OMP guarantees sparse signal recovery, improving measurement bounds compared to earlier mutual incoherence-based results.

Findings

OMP can recover sparse signals under RIP at sparsity level O(𝑘̄).

Recovery requires only O(𝑘̄ log d) measurements, fewer than previous bounds.

The analysis broadens understanding of OMP's effectiveness in compressed sensing.

Abstract

This paper presents a new analysis for the orthogonal matching pursuit (OMP) algorithm. It is shown that if the restricted isometry property (RIP) is satisfied at sparsity level $O(\bar{k})$, then OMP can recover a $\bar{k}$-sparse signal in 2-norm. For compressed sensing applications, this result implies that in order to uniformly recover a $\bar{k}$-sparse signal in $\Real^d$, only $O(\bar{k} \ln d)$ random projections are needed. This analysis improves earlier results on OMP that depend on stronger conditions such as mutual incoherence that can only be satisfied with $\Omega(\bar{k}^2 \ln d)$ random projections.