  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.
