Exact recovery of sparse multiple measurement vectors by l 2 , p $l_{2,p}$ -minimization

Abstract

Abstract The joint sparse recovery problem is a generalization of the single measurement vector problem widely studied in compressed sensing. It aims to recover a set of jointly sparse vectors, i.e., those that have nonzero entries concentrated at a common location. Meanwhile l p $l_{p}$ -minimization subject to matrixes is widely used in a large number of algorithms designed for this problem, i.e., l 2 , p $l_{2,p}$ -minimization min X ∈ R n × r ∥ X ∥ 2 , p s.t. A X = B . $$\begin{aligned} \min_{X \in\mathbb {R}^{n\times r}} \Vert X \Vert _{2,p}\quad \text{s.t. }AX=B. \end{aligned}$$ Therefore the main contribution in this paper is two theoretical results about this technique. The first one is proving that in every multiple system of linear equations there exists a constant p ∗ $p^{\ast}$ such that the original unique sparse solution also can be recovered from a minimization in l p $l_{p}$ quasi-norm subject to matrixes whenever 0 < p < p ∗ $0< p<p^{\ast}$ . The other one is showing an analytic expression of such p ∗ $p^{\ast}$ . Finally, we display the results of one example to confirm the validity of our conclusions, and we use some numerical experiments to show that we increase the efficiency of these algorithms designed for l 2 , p $l_{2,p}$ -minimization by using our results.

Keywords

• sparse recovery
• multiple measurement vectors
• joint sparse recovery
• l 2 , p $l_{2,p}$ -minimization