Belief propagation for joint sparse recovery

TL;DR

This paper introduces belief propagation algorithms for joint sparse recovery in compressed sensing, providing theoretical analysis and conditions for exact recovery of signals sharing a common sparse support.

Contribution

It formulates the joint sparse recovery problem within an information theoretic framework and develops belief propagation and approximate message passing algorithms for it.

Findings

Derived belief propagation algorithms for joint sparse recovery

Provided a density evolution-based condition for exact recovery

Extended single signal CS results to multiple signals sharing support

Abstract

Compressed sensing (CS) demonstrates that sparse signals can be recovered from underdetermined linear measurements. We focus on the joint sparse recovery problem where multiple signals share the same common sparse support sets, and they are measured through the same sensing matrix. Leveraging a recent information theoretic characterization of single signal CS, we formulate the optimal minimum mean square error (MMSE) estimation problem, and derive a belief propagation algorithm, its relaxed version, for the joint sparse recovery problem and an approximate message passing algorithm. In addition, using density evolution, we provide a sufficient condition for exact recovery.