Performance Analysis of Joint-Sparse Recovery from Multiple Measurement Vectors via Convex Optimization: Which Prior Information is Better?

Abstract

In sparse signal recovery of compressive sensing, the phase transition determines the edge, which separates successful recovery and failed recovery. The phase transition can be seen as an indicator and an intuitive way to judge, which recovery performance is better. Traditionally, the multiple measurement vectors (MMVs) problem is usually solved via $\ell _{2,1}$ -norm minimization, which is our first investigation via conic geometry in this paper. Then, we are interested in the same problem but with two common constraints (or prior information): prior information relevant to the ground truth and the inherent low rank within the original signal. To figure out which constraint is most helpful, the MMVs problems are solved via $\ell _{2,1}$ - $\ell _{2,1}$ minimization and $\ell _{2,1}$ -low rank minimization, respectively. By theoretically presenting the necessary and sufficient condition of successful recovery from MMVs, we can have a precise prediction of phase transition to judge, which constraint or prior information is better. All our findings are verified via simulations and show that, under certain conditions, $\ell _{2,1}$ - $\ell _{2,1}$ minimization outperforms $\ell _{2,1}$ -low rank minimization. Surprisingly, $\ell _{2,1}$ -low rank minimization performs even worse than $\ell _{2,1}$ -norm minimization. To the best of our knowledge, we are the first to study the MMVs problem under different prior information in the context of compressive sensing.