On $q$-ratio CMSV for sparse recovery

Abstract: Sparse recovery aims to reconstruct an unknown spare or approximately sparse signal from significantly few noisy incoherent linear measurements. As a kind of computable incoherence measure of the measurement matrix, $q$-ratio constrained minimal singular values (CMSV) was proposed in Zhou and Yu \cite{zhou2018sparse} to derive the performance bounds for sparse recovery. In this paper, we study the geometrical property of the $q$-ratio CMSV, based on which we establish new sufficient conditions for signal recovery involving both sparsity defect and measurement error. The $\ell_1$-truncated set $q$-width of the measurement matrix is developed as the geometrical characterization of $q$-ratio CMSV. In addition, we show that the $q$-ratio CMSVs of a class of structured random matrices are bounded away from zero with high probability as long as the number of measurements is large enough, therefore satisfy those established sufficient conditions. Overall, our results generalize the results in Zhang and Cheng \cite{zc} from $q=2$ to any $q\in(1,\infty]$ and complement the arguments of $q$-ratio CMSV from a geometrical view.