Joint k-step analysis of Orthogonal Matching Pursuit and Orthogonal Least Squares

Charles Soussen 1 Rémi Gribonval 2, 3 Jérôme Idier 4 Cédric Herzet 5
1 CRAN - Centre de Recherche en Automatique de Nancy
2 METISS - Speech and sound data modeling and processing
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
3 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
4 IRCCyN - Institut de Recherche en Communications et en Cybernétique de Nantes
5 FLUMINANCE - Fluid Flow Analysis, Description and Control from Image Sequences
CEMAGREF, Inria Rennes – Bretagne Atlantique
Abstract : Tropp's analysis of Orthogonal Matching Pursuit (OMP) using the Exact Recovery Condition (ERC) is extended to a first exact recovery analysis of Orthogonal Least Squares (OLS). We show that when the ERC is met, OLS is guaranteed to exactly recover the unknown support in at most k iterations where k denotes the support cardinality. Moreover, we provide a closer look at the analysis of both OMP and OLS when the ERC is not fulfilled. The existence of dictionaries for which some subsets are never recovered by OMP is proved. This phenomenon also appears with basis pursuit where support recovery depends on the sign patterns, but it does not occur for OLS. Finally, numerical experiments show that none of the considered algorithms is uniformly better than the other but for correlated dictionaries, guaranteed exact recovery may be obtained after fewer iterations for OLS than for OMP.
Keywords : Optimized Orthogonal Matching Pursuit (OOMP) Exact recovery condition (ERC) forward selection Orthogonal Matching Pursuit (OMP) Orthogonal Least Squares (OLS) Order Recursive Matching Pursuit (ORMP) Optimized Orthogonal Matching Pursuit (OOMP).
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Article dans une revue
IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2013, 59 (5), pp.3158-3174. 〈10.1109/TIT.2013.2238606〉
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Charles Soussen, Rémi Gribonval, Jérôme Idier, Cédric Herzet. Joint k-step analysis of Orthogonal Matching Pursuit and Orthogonal Least Squares. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2013, 59 (5), pp.3158-3174. 〈10.1109/TIT.2013.2238606〉. 〈hal-00637003v2〉

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