New Conditions on Stable Recovery of Weighted Sparse Signals via Weighted $$l_1$$ l 1 Minimization

Abstract

A problem of recovering weighted sparse signals via weighted $$l_1$$ minimization has recently drawn considerable attention with application to function interpolation. The weighted robust null space property (NSP) of order s and the weighted restricted isometry property (RIP) with the weighted 3s-RIP constant $$\delta _{\mathbf {w},3s}$$ have been proposed and proved to be sufficient conditions for guaranteeing stable recovery of weighted s-sparse signals. In this paper, we propose two new sufficient conditions, i.e., the weighted $$l_q$$ -robust NSP of order s and the weighted RIP with $$\delta _{\mathbf {w},2s}$$ . Different from the existing results, the weighted $$l_q$$ -robust NSP of order s is more general and weaker than the weighted robust NSP of order s, and the weighted RIP is characterized by $$\delta _{\mathbf {w},2s}$$ instead of $$\delta _{\mathbf {w},3s}$$ . Accordingly, the reconstruction error estimations based on the newly proposed recovery conditions are also derived, respectively. Moreover, we demonstrate that the weighted RIP with small $$\delta _{\mathbf {w},2s}$$ implies the weighted $$l_1$$ -robust NSP of order s.