  The sparse Beyesian learning (also referred to as Bayesian compressed
sensing) algorithm is one of the most popular approaches for sparse signal
recovery, and has demonstrated superior performance in a series of experiments.
Nevertheless, the sparse Bayesian learning algorithm has computational
complexity that grows exponentially 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 via the generalized approximate
message passing (GAMP) technique. Specifically, the algorithm is developed
within an expectation-maximization (EM) framework, using 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 efficacy and the effectiveness of the proposed
algorithm.
