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Greed is good: algorithmic results for sparse approximation

Tropp, Joel A. (2004) Greed is good: algorithmic results for sparse approximation. IEEE Transactions on Information Theory, 50 (10). pp. 2231-2242. ISSN 0018-9448. http://resolver.caltech.edu/CaltechAUTHORS:TROieeetit04a

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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.


Item Type:Article
Additional Information:© Copyright 2004 IEEE. Reprinted with permission. Manuscript received March 21, 2003; revised June 6, 2004. [Posted online: 2004-09-27] This work was supported by a National Science Foundation Graduate Fellowship. Communicated by G. Battail, Associate Editor At Large. This paper would never have been possible without the encouragement and patience of Anna Gilbert, Martin Strauss, and Muthu Muthukrishnan.
Subject Keywords:Algorithms, approximation methods, basis pursuit (BP), iterative methods, linear programming, orthogonal matching pursuit (OMP)
Record Number:CaltechAUTHORS:TROieeetit04a
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:TROieeetit04a
Alternative URL:http://dx.doi.org/10.1109/TIT.2004.834793
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9035
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:22 Oct 2007
Last Modified:26 Dec 2012 09:44

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