  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.
