Sparse Approximation and Recovery by Greedy Algorithms

We study sparse approximation by greedy algorithms. Our contribution is twofold. First, we prove exact recovery with high probability of random K-sparse signals within ΓK(1+ε)l iterations of the orthogonal matching pursuit (OMP). This result shows that in a probabilistic sense, the OMP is almost opt...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 60; no. 7; pp. 3989 - 4000
Main Authors: Livshitz, Eugene D., Temlyakov, Vladimir N.
Format: Journal Article
Language:English
Published: New York IEEE 01.07.2014
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Approximation
Approximation algorithms
Banach space
Banach spaces
Chebyshev approximation
Dictionaries
Greedy algorithms
Hilbert space
Inequalities
Matching
Matching pursuit algorithms
Mathematical analysis
Optimization
Probability distribution
Recovery
Greedy algorithms
lebesgue-type inequality
sparse approximation
probability
orthogonal matching pursuit
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:We study sparse approximation by greedy algorithms. Our contribution is twofold. First, we prove exact recovery with high probability of random K-sparse signals within ΓK(1+ε)l iterations of the orthogonal matching pursuit (OMP). This result shows that in a probabilistic sense, the OMP is almost optimal for exact recovery. Second, we prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm, a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. However, even in the case of a Hilbert space, our results add some new elements to known results on the Lebesgue-type inequalities for the restricted isometry property dictionaries. Our technique is a development of the recent technique created by Zhang.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2014.2320932