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

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.