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Greed is Good: Algorithmic Results for Sparse Approximation (2004)
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@MISC{Tropp04greedis,
author = {Joel A. Tropp},
title = {Greed is Good: Algorithmic Results for Sparse Approximation},
year = {2004}
}
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Abstract
This article presents new results on using a greedy algorithm, 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 quasi-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 on BP and extends them to OMP. 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.
Keyphrases
sparse approximation algorithmic result quasi-incoherent dictionary sufficient condition sparse problem greedy algorithm optimal approximation basis pursuit minimal error cumulative coherence function sparse approximation problem approximation algorithm sparse approximant new result optimal representation nonsparse signal wide class orthogonal matching pursuit redundant dictionary recent result small factor incoherent dictionary input signal natural generalization sparse signal
