A brief introduction to distributed compressed sensing and its typical reconstruction limits

Abstract

A brief introduction to compressed sensing and a discussion of its reconstruction limit analysis are presented. In addition, we briefly introduce the replica method, a technique in statistical mechanics, to show its effectiveness for typical performance analysis on signal processing problems. In this presentation, we especially focus on a signal model called joint sparse model 2 (JSM-2) or the multiple measurement vector problem, in which all sparse signals share their support. The signal model JSM-2 is important for dealing with practical processing problems. We investigate the typical performance of l2,1-norm regularized minimization and the Bayesian optimal reconstruction schemes in terms of mean square error for noiseless and noisy measurement JSM- 2 problems. Employing the replica method, we show that these schemes, which exploit the knowledge that the signal support is shared, can recover the signals more precisely as the number of channels increases. The main focus of our presentation is theoretical analysis; however, we will introduce applications of the signal model JSM-2 for acoustic problems. Most of our presentation is based on papers J. Stat. Mech. (2015) P05029 and J. Stat. Mech. (2016) 063304.