Robust recovery of a kind of weighted l1-minimization without noise level

Abstract

This paper studies a kind of weighted [Formula: see text]-minimization that is motivated by function interpolation. Combining with the weighted robust null space property, we first propose a new sufficient condition for robust recovery via the weighted [Formula: see text]-minimization when the measurements are corrupted by arbitrary noise without requiring the proper estimation of noise level. Second, we investigate the instance optimality of the weighted [Formula: see text]-minimization decoder according to the weighted quotient property and the weighted restricted isometry property (RIP). In addition, to give a better error estimation in a general problem to recover noisy compressible signals, we improve the [Formula: see text]-RIP constant [Formula: see text] from [Formula: see text] to [Formula: see text] by using the weighted sparsity and Stechkin-type estimate. Our results show that the weighted [Formula: see text]-minimization remains not only stable but also robust to reconstruct signals with noisy observations.