Exploiting structured sparsity in Bayesian experimental design

In this paper, we merge Bayesian experimental design with turbo approximate message passing (AMP) algorithms for the purpose of recovering structured-sparse signals using a multi-step adaptive compressive-measurement procedure. First, we show that, when the signal posterior is Gaussian, a waterfilling approach can be used to adapt the measurement matrix in a way that expected information gain is maximized. Next, we propose four methods of approximating AMP´s non-Gaussian marginal posteriors by a Gaussian joint posterior. One of these methods requires only point estimates of the signal, and leads to a novel kernel adaptation scheme that works even with non-Bayesian signal recovery algorithms like LASSO. Finally, we demonstrate (empirically) that our adaptive turbo AMP yields estimation performance very close to the support-oracle bound.