Online Recovery Guarantees and Analytical Results for OMP

Orthogonal Matching Pursuit (OMP) is a simple, yet empirically competitive algorithm for sparse recovery. Recent developments have shown that OMP guarantees exact recovery of K-sparse signals with K or more than K iterations if the observation matrix satisfies the restricted isometry property (RIP) with some conditions. We develop RIP-based online guarantees for recovery of a K-sparse signal with more than K OMP iterations. Though these guarantees cannot be generalized to all sparse signals a priori, we show that they can still hold online when the state-of-the-art K-step recovery guarantees fail. In addition, we present bounds on the number of correct and false indices in the support estimate for the derived condition to be less restrictive than the K-step guarantees. Under these bounds, this condition guarantees exact recovery of a K-sparse signal within 3K/2 iterations, which is much less than the number of steps required for the state-of-the-art exact recovery guarantees with more than K steps. Moreover, we present phase transitions of OMP in comparison to basis pursuit and subspace pursuit, which are obtained after extensive recovery simulations involving different sparse signal types. Finally, we empirically analyse the number of false indices in the support estimate, which indicates that these do not violate the developed upper bound in practice.