Computationally efficient sparse Bayesian learning via generalized approximate message passing

The sparse Bayesian learning (also referred to as Bayesian compressed sensing) algorithm is a popular approach for sparse signal recovery, and has demonstrated superior performance in several experiments. Nevertheless, the sparse Bayesian learning algorithm has a computational complexity that grows rapidly with the dimension of the signal, which hinders its application to many practical problems even with moderately large data sets. To address this issue, in this paper, we propose a computationally efficient sparse Bayesian learning method by integrating the generalized approximate message passing (GAMP) technique. Specifically, the algorithm is developed within an expectation-maximization (EM) framework, using the GAMP to efficiently compute an approximation of the posterior distribution of hidden variables. The hyperparameters associated with the hierarchical Gaussian prior are learned by iteratively maximizing the Q-function which is calculated based on the posterior approximation obtained from the GAMP. Numerical results are provided to illustrate the computational efficiency and the effectiveness of the proposed algorithm.