  This paper introduces a new sparse spatio-temporal structured Gaussian
process regression framework for online and offline Bayesian inference. This is
the first framework that gives a time-evolving representation of the
interdependencies between the components of the sparse signal of interest. A
hierarchical Gaussian process describes such structure and the
interdependencies are represented via the covariance matrices of the prior
distributions. The inference is based on the expectation propagation method and
the theoretical derivation of the posterior distribution is provided in the
paper. The inference framework is thoroughly evaluated over synthetic, real
video and electroencephalography (EEG) data where the spatio-temporal evolving
patterns need to be reconstructed with high accuracy. It is shown that it
achieves 15% improvement of the F-measure compared with the alternating
direction method of multipliers, spatio-temporal sparse Bayesian learning
method and one-level Gaussian process model. Additionally, the required memory
for the proposed algorithm is less than in the one-level Gaussian process
model. This structured sparse regression framework is of broad applicability to
source localisation and object detection problems with sparse signals.
