New Conditions on Stable Recovery of Weighted Sparse Signals via Weighted l1 Minimization

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 3 s -RIP constant δ w , 3 s 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 δ w , 2 s . 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 δ w , 2 s instead of δ w , 3 s . 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 δ w , 2 s implies the weighted l 1 -robust NSP of order s .