A null-space-based weightedl1minimization approach to compressed sensing

Abstract

It has become an established fact that the constrained $\ell_1$ minimization is capable of recovering the sparse solution from a small number of linear observations and the reweighted version can significantly improve its numerical performance. The recoverability is closely related to the Restricted Isometry Constant (RIC) {of order $s$ ($s$ is an integer), often denoted as $\delta_{s}$. A class of sufficient conditions for successful $k$-sparse signal recovery often take the form $\delta_{tk} 1$, such a bound is often called RIC bound of high order.} There exist a number of such bounds of RICs, high order or not. For example, a high order bound {is recently given by Cai and Zhang \cite{CZ14}:} $\delta_{tk} < \sqrt{(t-1)/t}$, and this bound is known sharp for $t \ge 4/3$. In this paper, we propose a new weighted $\ell_1$ minimization which only requires the following RIC bound that is more relaxed (i.e., bigger) than the above mentioned bound: \[\delta_{tk} 1$ and $0< \omega \le 1$ is determined {by two optimizations of a similar type}over the null space of the linear observation operator. {In tackling the combinatorial nature of the two optimization problems,we develop a reweighted $\ell_1$ minimization that yields a sequence of approximate solutions,which enjoy strong convergence properties. Moreover, the numerical performance of the proposed method}is very satisfactory when compared to some of the state of-the-art methods incompressed sensing.