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Constructive Approximation

, Volume 45, Issue 1, pp 113–127 | Cite as

Orthogonal Matching Pursuit Under the Restricted Isometry Property

  • Albert CohenEmail author
  • Wolfgang Dahmen
  • Ronald DeVore
Article

Abstract

This paper is concerned with the performance of orthogonal matching pursuit (OMP) algorithms applied to a dictionary \(\mathcal{D}\) in a Hilbert space \(\mathcal{H}\). Given an element \(f\in \mathcal{H}\), OMP generates a sequence of approximations \(f_n\), \(n=1,2,\ldots \), each of which is a linear combination of n dictionary elements chosen by a greedy criterion. It is studied whether the approximations \(f_n\) are in some sense comparable to best n-term approximation from the dictionary. One important result related to this question is a theorem of Zhang (IEEE Trans Inf Theory 57(9):6215–6221, 2011) in the context of sparse recovery of finite dimensional signals. This theorem shows that OMP exactly recovers n-sparse signals with at most An iterations, provided the dictionary \(\mathcal{D}\) satisfies a restricted isometry property (RIP) of order An for some constant A, and that the procedure is also stable in \(\ell ^2\) under measurement noise. The main contribution of the present paper is to give a structurally simpler proof of Zhang’s theorem, formulated in the general context of n-term approximation from a dictionary in arbitrary Hilbert spaces \(\mathcal{H}\). Namely, it is shown that OMP generates near best n-term approximations under a similar RIP condition.

Keywords

Orthogonal matching pursuit Best n-term approximation Instance optimality Restricted isometry property 

Mathematics Subject Classification

94A12 94A15 68P30 41A46 15A52 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParisFrance
  2. 2.Institut für Geometrie und Praktische MathematikRWTH AachenAachenGermany
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

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