On sparse recovery using finite Gaussian matrices: Rip-based analysis

Abstract

We provide a probabilistic framework for the analysis of the restricted isometry constants (RICs) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distribution of the extreme eigenvalues of Wishart matrices, or on its approximation based on the gamma distribution. In particular, we derive tight lower bounds on the cumulative distribution functions (CDFs) of the RICs. The presented framework provides the tightest lower bound on the maximum sparsity order, based on sufficient recovery conditions on the RICs, which allows signal reconstruction with a given target probability via different recovery algorithms.