Compressed Sensing of Approximately-Sparse Signals: Phase Transitions and Optimal Reconstruction

Abstract : Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only "approximately sparse", i.e. even though the signal contains only a small fraction of relevant (large) components the other components are not strictly equal to zero, but are only close to zero. In this paper we model the approximately sparse signal with a Gaussian distribution of small components, and we study its compressed sensing with dense random matrices. We use replica calculations to determine the mean-squared error of the Bayes-optimal reconstruction for such signals, as a function of the variance of the small components, the density of large components and the measurement rate. We then use the G-AMP algorithm and we quantify the region of parameters for which this algorithm achieves optimality (for large systems). Finally, we show that in the region where the GAMP for the homogeneous measurement matrices is not optimal, a special "seeding" design of a spatially-coupled measurement matrix allows to restore optimality.
Type de document :
Communication dans un congrès
50th annual Allerton conference on communication, control, and computing, Oct 2012, United States. pp.800-807, 2013
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-00873240
Contributeur : Claudine Le Vaou <>
Soumis le : mardi 15 octobre 2013 - 13:17:10
Dernière modification le : jeudi 24 janvier 2019 - 01:13:54

Lien texte intégral

Identifiants

  • HAL Id : hal-00873240, version 1
  • ARXIV : 1207.2079

Collections

Citation

Jean Barbier, Florent Krzakala, Marc Mézard, Lenka Zdeborová. Compressed Sensing of Approximately-Sparse Signals: Phase Transitions and Optimal Reconstruction. 50th annual Allerton conference on communication, control, and computing, Oct 2012, United States. pp.800-807, 2013. 〈hal-00873240〉

Partager

Métriques

Consultations de la notice

461