On performance of greedy algorithms

In this paper we show that for dictionaries with small coherence in a Hilbert space the Orthogonal Greedy Algorithm (OGA) performs almost as well as the best m−term approximation for all signals with sparsity almost as high as the best theoretically possible threshold s = 1 2 (M−1 + 1) by proving a Lebesgue-type inequality for arbitrary signals. On the other hand, we present an example of a dictionary with coherence M and an s−sparse signal for which OGA fails to pick up any atoms from the support, thus showing that the above threshold is sharp. Also, by proving a Lebesgue-type inequality for Pure Greedy Algorithm (PGA), we show that PGA matches the rate of convergence of the best m−term approximation, even beyond the saturation limit of m− 1 2 .