On RIC bounds of Compressed Sensing Matrices for Approximating Sparse Solutions

This paper follows the recent discussion on the sparse solution recovery with quasi-norms `q, q ∈ (0, 1) when the sensing matrix possesses a Restricted Isometry Constant δ2k (RIC). Our key tool is an improvement on a version of “the converse of a generalized Cauchy-Schwarz inequality” extended to the setting of quasi-norm. We show that, if δ2k ≤ 1/2, any minimizer of the lq minimization, at least for those q ∈ (0, 0.9181], is the sparse solution of the corresponding underdetermined linear system. Moreover, if δ2k ≤ 0.4931, the sparse solution can be recovered by any lq, q ∈ (0, 1) minimization. The values 0.9181 and 0.4931 improves those reported previously in the literature.