A Bayesian Approach to Block-Term Tensor Decomposition Model Selection and Computation

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.