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 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 sparsityjointly on the factors and in a 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  .
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 blocks of rank higher than one, a scenario encountered in numerous and diverse applications. Uniqueness conditions and fitting methods have thus been thoroughly studied. Nevertheless, the challenging problem of estimating the BTD model structure, namely the number of block terms , R, and their individual ranks, L_r, has only recently started to attract significant attention , mainly through regularization-based approaches which entail the need to tune the regularization parameter(s) . In this work, we build on ideas of sparse Bayesian learning (SBL) and put forward a fully automated Bayesian approach. Through a suitably crafted multi-level hierarchical probabilistic model, which gives rise to heavy-tailed prior distributions for the BTD factors, structured sparsity isjointly imposed. Ranks are then estimated from the numbers of blocks (R) and columns (L_r) of non-negligible energy. Approximate posterior inference is implemented, within the variational inference framework. The resulting iterative algorithm completely avoids hyperparameter tuning, which is a significant defect of regularization-based methods . Alternative probabilistic models are also explored and the connections with their regularization-based counterparts are brought to light with the aid of the associated maximum a-posteriori (MAP) estimators. We report simulation results with both synthetic and real-word data , which demonstrate the merits of the proposed method in terms of both rank estimation and model fitting as compared to state-of-the-art relevant methods .
R
Its uniqueness and approximation
Uniqueness conditions and fitting methods
A
, R,
A
L_r,
A
, mainly through regularization-based approaches which entail the need to tune the regularization parameter(s)
D
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
D
jointly
R
on the factors and in a
we build on ideas of sparse Bayesian learning (SBL) and put forward a fully automated Bayesian approach. Through a suitably crafted multi-level
R
manner and estimating the ranks as the numbers of factor columns
probabilistic model, which gives rise to heavy-tailed prior distributions for the BTD factors, structured sparsity is
A
jointly
A
imposed. Ranks are then estimated from the numbers
A
blocks (R) and columns (L_r) of
R
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
Approximate posterior inference is implemented, within the variational inference framework. The resulting iterative algorithm completely avoids hyperparameter tuning, which is a significant defect of
R
task of hyper-parameter tuning. Simulation results with synthetic data are reported
. Alternative probabilistic models are also explored and the connections with their regularization-based counterparts are brought to light with the aid of the associated maximum a-posteriori (MAP) estimators. We report simulation results with both synthetic and real-word data
R
effectiveness
merits
R
scheme
method
A
as compared to state-of-the-art relevant methods