On the exponential convergence of Matching Pursuits in quasi-incoherent dictionaries

The purpose of this paper is to extend results by Villemoes and Temlyakov about exponential convergence of Matching Pursuit with some structured dictionaries for ``simple'' functions in finite or infinite dimension. Our results are based on an extension of Tropp's results about Orthogonal Matching Pursuit in finite dimension, with the observation that it does not only work for OMP but also for MP. Our main contribution is a detailed analysis of the approximation and stability properties of MP with quasi-incoherent dictionaries, and a bound on the number of steps sufficient to reach an error no larger than a penalization factor times the best $m$-term approximation error.