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This paper follows the recent discussion on the sparse solution recovery withquasi-norms $\ell_q,~q\in(0,1)$ when the sensing matrix possesses a RestrictedIsometry Constant $\delta_{2k}$ (RIC). Our key tool is an improvement on aversion of "the converse of a generalized Cauchy-Schwarz inequality" extendedto the setting of quasi-norm. We show that, if $\delta_{2k}\le 1/2$, anyminimizer of the $l_q$ minimization, at least for those $q\in(0,0.9181]$, isthe sparse solution of the corresponding underdetermined linear system.Moreover, if $\delta_{2k}\le0.4931$, the sparse solution can be recovered byany $l_q, q\in(0,1)$ minimization. The values $0.9181$ and $0.4931$ improvesthose reported previously in the literature.

On RIC bounds of Compressed Sensing Matrices for Approximating Sparse Solutions Using $\ell_q$ Quasi Norms