A Rice method proof of the Null-Space Property over the Grassmannian

Jean-Marc Azaïs (1) , Yohann de Castro (2) , Stéphane Mourareau (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. Although this property is the keystone of $\ell_{1}$-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP.

Mots clés: Rice Method; Null-Space Property; Random processes theory; $\ell_{1}$-minimization; High-dimensional statistics

Domaines: Théorie [stat.TH]; Probabilités [math.PR]; Statistiques [math.ST]

Jean-Marc Azaïs, Yohann de Castro, Stéphane Mourareau. A Rice method proof of the Null-Space Property over the Grassmannian. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2017. ⟨hal-00995902v3⟩