A Sharp Condition for Exact Support Recovery of with Orthogonal Matching Pursuit

Abstract

Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied. In this paper, we show that for any $K$-sparse signal $\x$, if a sensing matrix $\A$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\delta_{K+1} < 1/\sqrt {K+1}$, then under some constraints on the minimum magnitude of nonzero elements of $\x$, OMP exactly recovers the support of $\x$ from its measurements $\y=\A\x+\v$ in $K$ iterations, where $\v$ is a noise vector that is $\ell_2$ or $\ell_{\infty}$ bounded. This sufficient condition is sharp in terms of $\delta_{K+1}$ since for any given positive integer $K$ and any $1/\sqrt{K+1}\leq \delta<1$, there always exists a matrix $\A$ satisfying the RIP with $\delta_{K+1}=\delta$ for which OMP fails to recover a $K$-sparse signal $\x$ in $K$ iterations. Also, our constraints on the minimum magnitude of nonzero elements of $\x$ are weaker than existing ones. Moreover, we propose worst-case necessary conditions for the exact support recovery of $\x$, characterized by the minimum magnitude of the nonzero elements of $\x$.