Orthogonal Matching Pursuit under the Restricted Isometry Property *

Albert Cohen (1) , Wolfgang Dahmen (2) , Ronald Devore (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 RWTH - Rheinisch-Westfälische Technische Hochschule Aachen University
3 Texas A&M University [College Station]

This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary D in a Hilbert space H. Given an element f ∈ H, OMP generates a sequence of approximations f n , n = 1, 2,. . ., 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 [14] 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 D satisfies a Restricted Isometry Property (RIP) of order An for some constant A, and that the procedure is also stable in 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 H. Namely, it is shown that OMP generates near best n-term approximations under a similar RIP condition.

Mots clés: 41A46; 15A52 Key Words: Orthogonal matching pursuit; AMS Subject Classification: 94A12; 94A15; 68P30; best n-term approximation; instance optimality; restricted isometry property

Domaines: Analyse numérique [math.NA]

Albert Cohen, Wolfgang Dahmen, Ronald Devore. Orthogonal Matching Pursuit under the Restricted Isometry Property *. Constructive Approximation, 2017, 45, pp.113-127. ⟨10.1007/s00365-016-9338-2⟩. ⟨hal-01239767⟩