Block-Term Tensor Decomposition: Model Selection and Computation

The so-called block-term decomposition (BTD) ten- sor model 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. Never- theless, 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 jointly on the factors and in a 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 itera- tively 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.