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Stable recovery of sparse overcomplete representations in the presence of noise (2006)
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| Venue: | IEEE TRANS. INFORM. THEORY |
| Citations: | 460 - 22 self |
BibTeX
@ARTICLE{Donoho06stablerecovery,
author = {David L. Donoho and Michael Elad and Vladimir N. Temlyakov},
title = {Stable recovery of sparse overcomplete representations in the presence of noise},
journal = {IEEE TRANS. INFORM. THEORY},
year = {2006},
volume = {52},
number = {1},
pages = {6--18}
}
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Abstract
Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimal-sparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
Keyphrases
stable recovery sparse overcomplete representation sparse representation sparse approximation overcomplete system computational effort noise level constant multiple overcomplete representation ideal underlying signal signal processing theory optimal-sparsity method noisy version ideal noiseless signal noisy data ideal noiseless sparse signal favorable structure sufficient sparsity approximation algorithm noisy data differs matching pursuit algorithm sparse decomposition similar stability
