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Pré-Publication, Document De Travail Année : 2011

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

Charles H Dossal
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Résumé

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.
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Dates et versions

hal-00164738 , version 1 (23-07-2007)
hal-00164738 , version 2 (25-11-2011)

Identifiants

  • HAL Id : hal-00164738 , version 2

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Charles H Dossal. A necessary and sufficient condition for exact recovery by l1 minimization.. 2011. ⟨hal-00164738v2⟩
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