Sparse recovery of weighted nonconvex $\boldsymbol{\ell_q}$-minimization

Abstract

This paper focuses on weighted $\ell_q$-minimization based on infinite-dimensional compressed sensing with weights$\boldsymbol{w}=\{\omega_1,~\omega_2,\ldots,~\omega_N\}^{\rm~T}\in~\mathbb{R}^N$ and $\omega_i~\geq~1$.We first introduce a kind of weighted robust null space propertyand show it is weaker than the weighted restricted isometry property.Furthermore, we propose and elaborate the instance optimality and the quotient property.Also, we show that Gaussian random matrices satisfy the weighted quotient property with high probability.Finally, with the combination of the above mentioned properties,we give some important approximation characterizations of the solutions to weighted $\ell_q$-minimization.Particularly, for arbitrary measurement error, we obtain the robustness estimate of the weighted $\ell_q$-minimization decoderwithout requiring a priori knowledge of noise level, which provides apractical advantage when the estimates of measurement noise levels are difficult to be obtained or absent.The results will be of significant benefit to further theoretical analysis of infinite-dimensional compressed sensing.