A necessary and sufficient condition for exact recovery by l1 minimization.

Abstract : The minimum $\ell_1$-norm solution to an underdetermined system of linear equations $y = A x$, is often, remarkably, also the sparsest solution to that system. In this paper, we provide a \textit{necessary} and \textit{sufficient} condition for $x$ to be identifiable for a large set of matrices $A$; that is to be the unique sparsest solution to the $\ell_1$-norm minimization problem. Furthermore, we prove that this sparsest solution is stable under a reasonable perturbation of the observations $y$. We also propose an efficient semi-greedy algorithm to check our condition for any vector $x$. We present numerical experiments showing that our condition is able to predict almost perfectly all identifiable solutions $x$, whereas other previously proposed criteria are too pessimistic and fail to identify properly some identifiable vectors $x$. Beside the theoretical proof, this provides empirical evidence to support the sharpness of our condition.
Type de document :
Pré-publication, Document de travail
2011
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https://hal.archives-ouvertes.fr/hal-00164738
Contributeur : Charles Dossal <>
Soumis le : vendredi 25 novembre 2011 - 09:51:14
Dernière modification le : mardi 22 mars 2016 - 01:24:27
Document(s) archivé(s) le : vendredi 16 novembre 2012 - 12:00:41

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  • HAL Id : hal-00164738, version 2

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Charles Dossal. A necessary and sufficient condition for exact recovery by l1 minimization.. 2011. <hal-00164738v2>

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