Signal Recovery From Incomplete and Inaccurate Measurements Via Regularized Orthogonal Matching Pursuit

We demonstrate a simple greedy algorithm that can reliably recover a vector v ¿ ¿^d from incomplete and inaccurate measurements x = ¿v + e. Here, ¿ is a N x d measurement matrix with N<e is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to provide the benefits of the two major approaches to sparse recovery. It combines the speed and ease of implementation of the greedy methods with the strong guarantees of the convex programming methods. For any measurement matrix ¿ that satisfies a quantitative restricted isometry principle, ROMP recovers a signal v with O(n) nonzeros from its inaccurate measurements x in at most n iterations, where each iteration amounts to solving a least squares problem. The noise level of the recovery is proportional to ¿{logn} ||e||₂. In particular, if the error term e vanishes the reconstruction is exact.

I. Introduction

The recent massive work in the area of compressed sensing, surveyed in [4], rigorously demonstrated that one can algorithmically recover sparse (and, more generally, compressible) signals from incomplete observations. The simplest model is a -dimensional signal with a small number of nonzeros

$$v\in{\BBR}^d,\quad \vert{\rm supp}(v)\vert\leq n \ll d.$$ Such signals are called -sparse. We collect nonadaptive linear measurements of , given as , where is some by measurement matrix. The sparse recovery problem is to then efficiently recover the signal from its measurements .