  The so-called block-term decomposition (BTD) tensor model, especially in its
rank-$(L_r,L_r,1)$ version, has been recently receiving increasing attention
due to its enhanced ability of representing systems and signals that are
composed of \emph{blocks} of rank higher than one, a scenario encountered in
numerous and diverse applications. Its uniqueness and approximation have thus
been thoroughly studied. Nevertheless, the challenging problem of estimating
the BTD model structure, namely the number of block terms and their individual
ranks, has only recently started to attract significant attention. In this
work, a Bayesian approach is taken to addressing the problem of
rank-$(L_r,L_r,1)$ BTD model selection and computation, based on the idea of
imposing column sparsity \emph{jointly} on the factors and in a
\emph{hierarchical} manner and estimating the ranks as the numbers of factor
columns of non-negligible energy. Using variational inference in the proposed
probabilistic model results in an iterative algorithm that comprises
closed-form updates. Its Bayesian nature completely avoids the ubiquitous in
regularization-based methods task of hyper-parameter tuning. Simulation results
with synthetic data are reported, which demonstrate the effectiveness of the
proposed scheme in terms of both rank estimation and model fitting.
