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.
In @cite_5 , the replica method (a statistical physics tool for large disordered systems) is used to calculate the mmse of the cs measurement (note that @cite_5 refers to mmv and dcs as MMV-2 and MMV-1, respectively). The replica trick non-rigorously simplifies the high-dimensional integral for the mmse of the Bayesian estimator of the cs channel, thereby leading to the free energy as a function of the mse . The local maxima in the free energy function correspond to stable fixed points of bp and bamp and thus predict the expected mse of bamp . The replica analysis in @cite_5 is performed for the bg signal prior with uncorrelated isotropic unitary signal and uncorrelated isotropic Gaussian noise distribution, i.e., with a single noise parameter.
@cite_5
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We present a theory, based on statistical mechanics, to evaluate analytically the performance of uncoded, fully synchronous, randomly spread code-division multiple-access (CDMA) multiuser detectors with additive white Gaussian noise (AWGN) channel, under perfect power control, and in the large-system limit. Application of the replica method, a tool developed in the literature of statistical mechanics, allows us to derive analytical expressions for the bit-error rate, as well as the multiuser efficiency, of the individually optimum (IO) and jointly optimum (JO) multiuser detectors over the whole range of noise levels. The information-theoretic capacity of the randomly spread CDMA channel and the performance of decorrelating and linear minimum mean-square error (MMSE) detectors are also derived in the same replica formulation, thereby demonstrating validity of the statistical-mechanical approach.