Signal recovery from incomplete and inaccurate measurements via ROMP

We demonstrate a simple greedy algorithm that can reliably recover a vector v ∈ R from incomplete and inaccurate measurements x = Φv + e. HereΦ is a N × d measurement matrix with N ≪ d, ande is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to close the gap between 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 Φ that satisfies a Uniform Uncertainty Principle, ROMP recovers a signal v with O(n) nonzeros from its inaccurate measurements x in at mostn iterations, where each iteration amounts to solving a Least Squares Problem. The noise level of the recovery is proportional to √ log n‖e‖2. In particular, if the error terme vanishes the reconstruction is exact. This stability result extends naturally to the very accurat e recovery of approximately sparse signals.