Perturbation Analysis of Orthogonal Matching Pursuit

Abstract

Orthogonal Matching Pursuit (OMP) is a canonical greedy pursuit algorithm for sparse approximation. Previous studies of OMP have mainly considered the exact recovery of a sparse signal $\bm x$ through $\bm \Phi$ and $\bm y=\bm \Phi \bm x$, where $\bm \Phi$ is a matrix with more columns than rows. In this paper, based on Restricted Isometry Property (RIP), the performance of OMP is analyzed under general perturbations, which means both $\bm y$ and $\bm \Phi$ are perturbed. Though exact recovery of an almost sparse signal $\bm x$ is no longer feasible, the main contribution reveals that the exact recovery of the locations of $k$ largest magnitude entries of $\bm x$ can be guaranteed under reasonable conditions. The error between $\bm x$ and solution of OMP is also estimated. It is also demonstrated that the sufficient condition is rather tight by constructing an example. When $\bm x$ is strong-decaying, it is proved that the sufficient conditions can be relaxed, and the locations can even be recovered in the order of the entries' magnitude.