A Sharp Condition for Exact Support Recovery of Sparse Signals With Orthogonal Matching Pursuit

TL;DR

This paper establishes a sharp condition on the restricted isometry constant under which orthogonal matching pursuit (OMP) can exactly recover sparse signals, improving understanding of its performance limits in noisy settings.

Contribution

The paper provides a precise RIP-based condition for exact support recovery with OMP, which is shown to be optimal and weaker than previous constraints.

Findings

OMP recovers support exactly under $ ext{RIC} < 1/\sqrt{K+1}$

The condition is sharp and cannot be improved further

The minimum magnitude constraint on nonzero elements is weaker than prior results

Abstract

Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied in the literature. In this paper, we show that for any $K$-sparse signal $\x$, if the sensing matrix $\A$ satisfies the restricted isometry property (RIP) of order $K + 1$ with restricted isometry constant (RIC) $\delta_{K+1} < 1/\sqrt {K+1}$, then under some constraint on the minimum magnitude of the nonzero elements of $\x$, the OMP algorithm exactly recovers the support of $\x$ from the measurements $\y=\A\x+\v$ in $K$ iterations, where $\v$ is the noise vector. This condition is sharp in terms of $\delta_{K+1}$ since for any given positive integer $K\geq 2$ and any $1/\sqrt{K+1}\leq t<1$, there always exist a $K$-sparse $\x$ and a matrix $\A$ satisfying $\delta_{K+1}=t$ for which OMP may fail to recover the signal $\x$ in $K$ iterations. Moreover, the constraint on the minimum magnitude of the nonzero elements of $\x$ is weaker than existing results.