Coherence-based Partial Exact Recovery Condition for OMP/OLS

We address the exact recovery of the support of a k-sparse vector with Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS) in a noiseless setting. We consider the scenario where OMP/OLS have selected good atoms during the first l iterations (l < k) and derive a new sufficient and worst-case necessary condition for their success in k steps. Our result is based on the coherence μ of the dictionary and relaxes Tropp’s well-known condition μ < 1/(2k − 1) to the case where OMP/OLS have a partial knowledge of the support.