Sparse Recovery from Extreme Eigenvalues Deviation Inequalities

TL;DR

This paper introduces a new method to derive sparse recovery guarantees using small deviations in extreme eigenvalues from Random Matrix Theory, focusing on Restricted Isometry Constants for Gaussian and Rademacher matrices.

Contribution

It provides explicit bounds on RICs and SRSR probabilities based on small deviation estimates of extreme eigenvalues, linking Random Matrix Theory with compressed sensing guarantees.

Findings

Upper bounds on RICs for Gaussian and Rademacher matrices.

Lower bounds on the probability of successful sparse recovery.

Explicit derivation connecting eigenvalue deviations to recovery guarantees.

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 High-Dimensional 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 state-of-the-art 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.