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   -> Volume 13, Issue 1


Preprint: "Beyond sparsity: recovering structured representations with applications to sparse underdetermined ICA", by R. Gribonval and M. Nielsen
 
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Remi Gribonval (remi.gribonval AT irisa.fr)
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PostPosted: Tue Jan 25, 2005 5:25 pm    
Subject: Preprint: "Beyond sparsity: recovering structured representations with applications to sparse underdetermined ICA", by R. Gribonval and M. Nielsen
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The following preprint is now available :

"Beyond sparsity : recovering structured representations by l 1 minimization and greedy algorithms. -- Application to the analysis of sparse underdetermined ICA --"
at
http://www.irisa.fr/bibli/publi/pi/2005/1684/1684.html

AUTHORS:
Rémi Gribonval, IRISA, Rennes, France
and
Morten Nielsen, University of Aalborg, Denmark


ABSTRACT:
In a series of recent results, several authors have shown that both l 1 (Basis Pursuit) and greedy algorithms (Matching Pursuit) can successfully recover a sparse representation of a signal provided that it is sparse enough, that is to say if its support (which indicates where are located the nonzero coefficients) is of sufficiently small size. In this paper we define more general identifiable structures that support signals that can be recovered exactly by l 1 minimization and greedy algorithms. In addition, we obtain that if the output of an arbitrary decomposition algorithm is supported on an identifiable structure, then one can be sure that the representation is optimal within the class of signals supported by the structure. As an application of the theoretical results, we give a detailed study of a family of multichannel dictionaries with a special structure (corresponding to the representation problem X = ASFT) often used in, e.g., under-determined source separation problems or in multichannel signal processing.

KEYWORDS:
Sparse representation, overcomplete dictionary, matching pursuit, basis pursuit, linear programming, greedy algorithm, independent component analysis, multichannel signal processing, identifiability, inverse problem
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