Exploiting General Multi-Dimensional Priors in Compressed-Sensing Reconstruction

Message passing based algorithms have been shown to perform well in terms of minimum mean-squared error for high-dimensional signals composed of independent and identically distributed one-dimensional and sparse components. These conditions limit the applicability and performance of these algorithms since dependencies among components are not used during recovery. A detailed derivation is given that, as a novelty, extends the known derivation of the conventional Bayesian Approximate Message Passing scheme (BAMP) to general multi-dimensional priors. The proposed algorithms significantly reduce the number of samples required for reconstruction compared to methods which do not exploit dependencies. Applications include multiple-measurement vector (MMV) problems, group sparsity as well as symbol recovery in MIMO systems and reconstruction in the case of general, non-sparse dependencies between components.