Rank-Revealing Block-Term Decomposition For Tensor Completion

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. In this paper, BTD is employed for the completion of a tensor from its partially observed entries. A novel method is proposed, which is based on the idea of imposing column sparsity jointly on the BTD factors and in a hierarchical manner. This way the number of block terms and their ranks can also be estimated, as the numbers of factor columns of non-negligible magnitude. Following a block successive upper bound minimization (BSUM) approach with appropriate choice of the surrogate majorizing functions 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 with both synthetic and real data are reported, which demonstrate the effectiveness of the proposed scheme.