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This article presents new results on using a greedy ale gorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasie incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function Is Introduced to quantify the level of Incoherence. This analysis unifies all the recent results OMP and extends them to OMR Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP Is an approximation algorithm for the sparse problem over a quasi-Incoherent dictionary That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms. [ABSTRACT FROM AUTHOR]

This note provides a condition under which ℓ1 minimization (also known as basis pursuit) can recover short linear combinations of complex vectors chosen from fixed, over complete collection. This condition has already been established in the real setting by Fuchs, who used convex analysis. The proof given here is more direct. [ABSTRACT FROM AUTHOR]