  We present a fast update rule for variational block-sparse Bayesian learning
(SBL) methods. Based on a variational Bayesian approximation, we show that
iterative updates of probability density functions (PDFs) of the prior
precisions and weights can be expressed as a nonlinear first-order recurrence
from one estimate of the parameters of the proxy PDFs to the next. In
particular, for commonly used prior PDFs such as Jeffrey's prior, the
recurrence relation turns out to be a strictly increasing rational function.
This property is the basis for two important analytical results. First, the
determination of fixed points by solving for the roots of a polynomial. Second,
the determination of the limit of the prior precision after an infinite
sequence of update steps. These results are combined into a simplified
single-step check for convergence/divergence of each prior precision.
Consequently, our proposed criterion significantly reduces the computational
complexity of the variational block-SBL algorithm, leading to a remarkable two
orders of magnitude improvement in convergence speed shown by simulations.
Moreover, the criterion provides valuable insights into the sparsity of the
estimators obtained by different prior choices.
