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

We demonstrate a simple greedy algorithm that can reliably recover a vector upsilon is an element of R-d from incomplete and inaccurate measurements x =Phi upsilon + e. Here, Phi is a N x d measurement matrix with N << d and 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 Phi that satisfies a quantitative restricted isometry principle, ROMP recovers a signal upsilon 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 root logn parallel to e parallel to(2). In particular, if the error term e vanishes the reconstruction is exact. This stability result extends naturally to the very accurate recovery of approximately sparse signals.