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

Charles H Dossal (1)

1 LaBAG - Laboratoire Bordelais d'Analyse et Géométrie

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.

Keywords: identifiable vectors; Sparse representations; underdetermined linear systems; $\ell_1$-minimization; identifiable vectors.

Domains: Optimization and Control [math.OC]

Charles H Dossal. A necessary and sufficient condition for exact recovery by l1 minimization.. 2007. ⟨hal-00164738v1⟩