In this paper we investigate the efficiency of the Orthogonal Matching Pursuit algorithm (OMP) for random dictionaries. We concentrate on dictionaries satisfying the Restricted Isometry Property. We also introduce a stronger Homogenous Restricted Isometry Property which we show is satisfied with overwhelming probability for random dictionaries used in compressed sensing. For these dictionaries we obtain upper estimates for the error of approximation by OMP in terms of the error of the best n-term approximation (Lebesgue-type inequalities). We also present and discuss some open problems about OMP. This is a development of recent results obtained by D.L. Donoho, M. Elad and V.N. Temlyakov.

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Abstract. Let BV = BV(R d ) be the space of functions of bounded variation on R d with d ≥ 2. Let ψ λ , λ ∈ ∆, be a wavelet system of compactly supported functions normalized in BV, i.e.,is the set of N indicies λ ∈ ∆ for which |c λ (f )| are largest (with ties handled in an arbitrary way), thenwith C a constant independent of f . This answers in the affirmative a conjecture of Meyer (2001).