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

Abstract

It has become an established fact that the constrained ℓ 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 δ_s. A class of sufficient conditions for successful k-sparse signal recovery often take the form δ_tk < c, for some t ≥ 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 and Zhang [CZ14]: δ_tk < √  (t-1)/t, and this bound is known sharp for t ≥ 4/3. In this paper, we propose a new weighted ℓ₁ minimization which only requires the following RIC bound that is more relaxed (i.e., bigger) than the above mentioned bound:δ_tk < √  { t-1}  /  { t -(1-ω²)} where t > 1 and 0 < ω  ≤  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 ℓ₁ 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.

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Identifiers

ORCID for Shenglong Zhou: orcid.org/0000-0003-2843-1614

ORCID for Hou-Duo Qi: orcid.org/0000-0003-3481-4814

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