Performance Analysis of Approximate Message Passing for Distributed Compressed Sensing

TL;DR

This paper analyzes the performance and complexity reduction of Bayesian approximate message passing (BAMP) in distributed compressed sensing, focusing on signal correlation, noise, and the impact of joint sparsity.

Contribution

It introduces a decorrelation transform to simplify BAMP, proves its invariance properties, and uses replica analysis to evaluate performance with correlated signals.

Findings

BAMP and SE are equivariant under the decorrelation transform.

The residual noise covariance remains diagonal for Bernoulli-Gauss prior.

Performance depends on signal correlation and number of jointly sparse signals.

Abstract

Bayesian approximate message passing (BAMP) is an efficient method in compressed sensing that is nearly optimal in the minimum mean squared error (MMSE) sense. Bayesian approximate message passing (BAMP) performs joint recovery of multiple vectors with identical support and accounts for correlations in the signal of interest and in the noise. In this paper, we show how to reduce the complexity of vector BAMP via a simple joint decorrelation diagonalization) transform of the signal and noise vectors, which also facilitates the subsequent performance analysis. We prove that BAMP and the corresponding state evolution (SE) are equivariant with respect to the joint decorrelation transform and preserve diagonality of the residual noise covariance for the Bernoulli-Gauss (BG) prior. We use these results to analyze the dynamics and the mean squared error (MSE) performance of BAMP via the replica method, and thereby understand the impact of signal correlation and number of jointly sparse signals.