We derive low complexity versions of a wide range of algorithms for sparse Bayesian learning (SBL) in underdetermined linear systems. The proposed algorithms are obtained by applying the generalized mean field (GMF) inference framework to a generic SBL probabilistic model. In the GMF framework, we constrain the auxiliary function approximating the posterior probability density function of the unknown variables to factorize over disjoint groups of contiguous entries in the sparse vector - the size of these groups dictates the degree of complexity reduction. The original high-complexity algorithms correspond to the particular case when all the entries of the sparse vector are assigned to one single group. Numerical investigations are conducted for both a generic compressive sensing application and for channel estimation in an orthogonal frequency-division multiplexing receiver. They show that, by choosing small group sizes, the resulting algorithms perform nearly as well as their original counterparts but with much less computational complexity.
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