Restricted Isometry Constants for Gaussian and Rademacher matrices

Restricted Isometry Constants (RICs) are a pivotal notion in Compressed Sensing as these constants finely assess how a linear operator is conditioned on the set of sparse vectors and hence how it performs in stable and robust sparse regression (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 of the benefits of this approach is to introduce a simple tool from Random Matrix Theory to derive upper bounds on RICs and phase transition on SRSR from small deviations on the extreme eigenvalues.