Error Estimates for Orthogonal Matching Pursuit and Random Dictionaries

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.