  The so-called block-term decomposition (BTD) tensor model 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 paper, a novel method of BTD model selection and computation
is proposed, 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 magnitude. Following a block
successive upper bound minimization (BSUM) approach for the proposed
optimization problem is shown to result in an alternating hierarchical
iteratively reweighted least squares (HIRLS) algorithm, which is fast
converging and enjoys high computational efficiency, as it relies in its
iterations on small-sized sub-problems with closed-form solutions. Simulation
results for both synthetic examples and a hyper-spectral image de-noising
application are reported, which demonstrate the superiority of the proposed
scheme over the state-of-the-art in terms of success rate in rank estimation as
well as computation time and rate of convergence.
