Signal Recovery from Incomplete and Inaccurate Measurements via Regularized Orthogonal Matching Pursuit

TL;DR

This paper introduces ROMP, a fast greedy algorithm that reliably recovers sparse signals from incomplete, noisy measurements, bridging the gap between greedy and convex optimization methods with strong theoretical guarantees.

Contribution

The paper presents ROMP, a simple greedy algorithm with provable recovery guarantees for sparse signals from incomplete, inaccurate measurements, combining efficiency with robustness.

Findings

ROMP recovers signals with O(n) nonzeros in at most n iterations.

Recovery noise level is proportional to measurement error, up to a log factor.

Exact recovery when measurement error vanishes.

Abstract

We demonstrate a simple greedy algorithm that can reliably recover a d-dimensional vector v from incomplete and inaccurate measurements x. Here our measurement matrix is an N by d matrix with N much smaller than d. 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 that satisfies a Uniform Uncertainty Principle, ROMP recovers a signal 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 the norm of the error, up to a log factor. In particular, if the error vanishes the reconstruction is exact. This stability result extends naturally to the very accurate recovery of approximately sparse signals.