Sparse recovery guarantees from extreme eigenvalues small deviations

Sandrine Dallaporta (1) , Yohann de Castro (2)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

This article provides a new toolbox to derive sparse recovery guarantees - that is referred to as “stable and robust sparse regression” (SRSR) - 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.

Mots clés: Restricted Isometry Property; Gaussian matrices; Rademacher matrices; Small Deviations; Sparse regression

Domaines: Statistiques [math.ST]; Théorie de l'information et codage [math.IT]; Probabilités [math.PR]; Machine Learning [stat.ML]

Sandrine Dallaporta, Yohann de Castro. Sparse recovery guarantees from extreme eigenvalues small deviations. 2017. ⟨hal-01309439v3⟩