A null-space-based weighted l1 minimization approach to compressed sensing

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 $\delta _{tk} < c$ for some $t \ge 1$ and $c$ being a constant. When $t >1$, 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 & Zhang (2014, IEEE Trans. Inform. Theory, 60, 122–132): $\delta _{tk} < \sqrt {(t-1)/t}$, 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: $$\delta_{tk} < \sqrt{\frac{t-1}{t -(1-\omega^2)}},$$where $t >1$ 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 with some of the state-of-the-art methods in compressed sensing.