A Rice method proof of the Null-Space Property over the Grassmannian - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2014

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

Résumé

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 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 L1-minimization techniques, it is an open problem to construct deterministic matrices satisfying NSP. One can only prove that NSP holds, with an overwhelming probability, when the matrix is chosen at random, in general with a rotation invariant distribution. To the best of our knowledge, the only direct proof of NSP for rotation invariant distribution, initiated by Donoho and Tanner, is based on integral convex geometry theory. In this article, we follow a different path and we provide a short and self-contained proof using random processes theory and the Rice method. Moreover, our analysis gives new explicit non-asymptotic bounds for NSP with respect to rotation invariant distributions.
Fichier principal
Vignette du fichier
Direct_NSP_v1.pdf (298.59 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

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

Identifiants

Citer

Jean-Marc Azaïs, Yohann de Castro, Mourareau Stéphane. A Rice method proof of the Null-Space Property over the Grassmannian. 2014. ⟨hal-00995902v1⟩
462 Consultations
363 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More