Some New Results About Sufficient Conditions for Exact Support Recovery of Sparse Signals via Orthogonal Matching Pursuit

Abstract

Support recovery of sparse signals via orthogonal matching pursuit (OMP) has been extensively studied in recent years. In this paper, by exploiting the knowledge about orthogonal projection matrix and Schur complement, we study the sufficient conditions for exact support recovery of sparse signals with OMP in the framework of restricted isometry property (RIP). In the noisy case, we prove that under some constraints on the minimum magnitude of the nonzero elements of the K-sparse signal, OMP can exactly recover the support of the signal if the restricted isometry constant ${\delta _{K + 1}}$ satisfies ${\delta _{K + 1}} . Our constraints on the minimum magnitude of nonzero elements of the signal are weaker than existing ones. In the noiseless case, although it has been proved that ${\delta _{K + 1}} is a sharp condition for exactly recovering any K-sparse signal with OMP, our result shows that under some constraints on the signal, OMP can also exactly recover the signal if ${\delta _{K + 1}}$ satisfies ${\delta _{K + 1}} .