a CS signal estimation framework that often achieves the MMSE asymptotically. A phase transition for BP is identified. This phase transition, verified by numerical results, separates the regions where BP achieves the MMSE and where it is suboptimal. Numerical results also illustrate that more signal vectors in the jointly sparse signal ensemble lead to a better phase transition.
author:
- 'Junan Zhu, , Dror Baron, , and Florent Krzakala [^1] [^2] [^3]'
title: |
    Performance Limits\
    for Noisy Multi-Measurement Vector Problems
---

[*Keywords*]{}:Approximate message passing, multi-measurement vector problem, replica analysis.

Introduction
============

Compressed sensing (CS) [@CandesRUP; @DonohoCS; @BaraniukCS2007] demonstrates that sparse signals can be estimated from under-determined linear measurements. Owing to the potential for radically reduced measurement rates, CS has become an active research area within signal processing. CS has many application areas including magnetic resonance imaging [@JuYeKi07; @JuSuNaKiYe09], communication [@Cotter2002scemp], and remote sensing [@Ma2009deblur].

Distributed CS (DCS) [@HN05; @DuarteWakinBaronSarvothamBaraniuk2013] is based on the premise that joint sparsity within signal ensembles enables a further reduction in the number of measurements. Motivated by sensor networks [@Pottie2000], preliminary work in DCS [@Duarte2006IPSN; @HN05; @DuarteWakinBaronSarvothamBaraniuk2013] showed that the number of measurements required per sensor must account for the minimum features unique to that 