Perturbation Analysis of Orthogonal Matching Pursuit

TL;DR

This paper analyzes the performance of Orthogonal Matching Pursuit (OMP) under general perturbations of data and measurement matrix, showing conditions for recovering the support of the largest entries of a sparse signal.

Contribution

It extends OMP analysis to perturbed settings using RIP, providing guarantees for support recovery and error bounds under realistic conditions.

Findings

Support of largest entries can be recovered under perturbations.

Error bounds for the OMP solution are established.

Conditions are shown to be tight through examples.

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.