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.
