Improving M-SBL for Joint Sparse Recovery using a Subspace Penalty

TL;DR

This paper enhances the M-SBL algorithm for joint sparse recovery in MMV problems by integrating a subspace-based rank proxy, leading to improved accuracy and convergence guarantees, validated through extensive simulations.

Contribution

It introduces a novel modification to M-SBL using a Schatten-$p$ quasi-norm rank proxy, linking it to subspace algorithms and proving convergence to the true solution.

Findings

Proposed algorithm outperforms existing methods in various scenarios.

Global minimizer approaches the true solution as p approaches 0.

Convergence to a local minimizer is guaranteed with an alternating minimization.

Abstract

The multiple measurement vector problem (MMV) is a generalization of the compressed sensing problem that addresses the recovery of a set of jointly sparse signal vectors. One of the important contributions of this paper is to reveal that the seemingly least related state-of-art MMV joint sparse recovery algorithms - M-SBL (multiple sparse Bayesian learning) and subspace-based hybrid greedy algorithms - have a very important link. More specifically, we show that replacing the $\log\det(\cdot)$ term in M-SBL by a rank proxy that exploits the spark reduction property discovered in subspace-based joint sparse recovery algorithms, provides significant improvements. In particular, if we use the Schatten-$p$ quasi-norm as the corresponding rank proxy, the global minimiser of the proposed algorithm becomes identical to the true solution as $p \rightarrow 0$. Furthermore, under the same regularity conditions, we show that the convergence to a local minimiser is guaranteed using an alternating minimization algorithm that has closed form expressions for each of the minimization steps, which are convex. Numerical simulations under a variety of scenarios in terms of SNR, and condition number of the signal amplitude matrix demonstrate that the proposed algorithm consistently outperforms M-SBL and other state-of-the art algorithms.