A Rice method proof of the Null-Space Property over the Grassmannian - Archive ouverte HAL Access content directly
Journal Articles Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Year : 2017

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

Jean-Marc Azaïs
Yohann de Castro
• Function : Author
Stéphane Mourareau
• Function : Author

#### Abstract

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.

### Dates and versions

hal-00995902 , version 1 (25-05-2014)
hal-00995902 , version 2 (16-06-2014)
hal-00995902 , version 3 (08-10-2015)

### Identifiers

• HAL Id : hal-00995902 , version 3
• ARXIV :

### Cite

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⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

414 View