An Improved RIP-Based Performance Guarantee for Sparse Signal Recovery via Orthogonal Matching Pursuit

Abstract

A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations (when there is no noise) is that the restricted isometry constant (RIC) of the sensing matrix satisfies delta(K+1) < (1/root K + 1). In the noisy case, this RIC upper bound along with a requirement on the minimal signal entry magnitude is known to guarantee exact support identification. In this paper, we show that, in the presence of noise, a relaxed RIC upper bound delta(K+1) < (root 4K + 1 - 1/2K) together with a relaxed requirement on the minimal signal entry magnitude suffices to achieve perfect support identification using OMP. In the noiseless case, our result asserts that such a relaxed RIC upper bound can ensure exact support recovery in K iterations: this narrows the gap between the so far best known bound delta(K+1) < (1/root K + 1) and the ultimate performance guarantee delta(K+1) = (1/root K). Our approach relies on a newly established near orthogonality condition, characterized via the achievable angles between two orthogonal sparse vectors upon compression, and, thus, better exploits the knowledge about the geometry of the compressed space. The proposed near orthogonality condition can be also exploited to derive less restricted sufficient conditions for signal reconstruction in two other compressive sensing problems, namely, compressive domain interference cancellation and support identification via the subspace pursuit algorithm.