Sparse Recovery from Extreme Eigenvalues Deviation Inequalities
Abstract
This article provides a new toolbox to derive sparse recovery guarantees from small deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and HighDimensional Statistics as these constants finely assess how a linear operator is conditioned on the set of sparse vectors and hence how it performs in SRSR. While it is an open problem to construct deterministic matrices with apposite RICs, one can prove that such matrices exist using random matrices models. In this paper, we show upper bounds on RICs for Gaussian and Rademacher matrices using stateoftheart small deviation estimates on their extreme eigenvalues. This allows us to derive a lower bound on the probability of getting SRSR. One benefit of this paper is a direct and explicit derivation of upper bounds on RICs and lower bounds on SRSR from small deviations on the extreme eigenvalues given by Random Matrix theory.
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 DOI:
 10.48550/arXiv.1604.01171
 arXiv:
 arXiv:1604.01171
 Bibcode:
 2016arXiv160401171D
 Keywords:

 Mathematics  Statistics Theory;
 Computer Science  Information Theory;
 Mathematics  Probability;
 Statistics  Machine Learning
 EPrint:
 33 pages, 1 figure, final version