On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\ell_1$ Minimization

Anatoli B. Juditsky (1) , Arkadii S. Nemirovski (2)

1 SAM - Statistique Apprentissage Machine
2 ISyE - School of Industrial and Systems Engineering [Georgia Tech]

We propose novel necessary and sufficient conditions for a sensing matrix to be ``$s$-good'' -- to allow for exact $\ell_1$-recovery of sparse signals with $s$ nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect $\ell_1$-recovery (nonzero measurement noise, nearly $s$-sparse signal, near-optimal solution of the optimization problem yielding the $\ell_1$-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse $\ell_1$-recovery and to efficiently computable upper bounds on those $s$ for which a given sensing matrix is $s$-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties.

Mots clés: sparse recovery; compressed sensing; $\ell_1$-recovery

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Optimisation et contrôle [math.OC]

Anatoli B. Juditsky, Arkadii S. Nemirovski. On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\ell_1$ Minimization. Mathematical Programming, 2011, 127 (1), pp.57-88. ⟨10.1007/s10107-010-0417-z⟩. ⟨hal-00321775v2⟩