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

Rémi Gribonval 1 Pierre Vandergheynst 2
1 METISS - Speech and sound data modeling and processing
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
2 LTS2 - EPFL
Abstract : The purpose of this correspondence is to extend results by Villemoes and Temlyakov about exponential convergence of Matching Pursuit (MP) with some structured dictionaries for "simple" functions in finite or infinite dimension. The results are based on an extension of Tropp's results about Orthogonal Matching Pursuit (OMP) in finite dimension, with the observation that it does not only work for OMP but also for MP. The 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.
Keywords : OMP approximation error exponential convergence finite dimension greedy algorithm nonlinear approximation orthogonal matching pursuit quasiincoherent dictionary sparse representation stability property approximation theory greedy algorithms image processing iterative methods numerical stability sparse matrices time-frequency analysis
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Rémi Gribonval, Pierre Vandergheynst. On the exponential convergence of Matching Pursuits in quasi-incoherent dictionaries. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2006, 52 (1), pp.255--261. ⟨10.1109/TIT.2005.860474⟩. ⟨inria-00544945⟩

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