Performance Analysis of Approximate Message Passing for Distributed Compressed Sensing

Bayesian approximate message passing (BAMP) is an efficient method in compressed sensing that is nearly optimal in the minimum mean squared error (MMSE) sense. Multiple measurement vector (MMV)-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 the corresponding state evolution is equivariant with respect to the joint decorrelation transform and preserves diagonality of the residual noise covariance for the Bernoulli–Gauss 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. Finally, we evaluate an application of MMV-BAMP for single-pixel imaging with correlated color channels and thereby explore the performance gain of joint recovery compared to conventional BAMP reconstruction as well as group lasso.