Online Rank-Revealing Block-Term Tensor Decomposition

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{block} components of rank higher than one, a scenario encountered in numerous and diverse applications. Its uniqueness and approximation have thus been thoroughly studied. The challenging problem of estimating the BTD model structure, namely the number of block terms (rank) and their individual (block) ranks, is of crucial importance in practice and has only recently started to attract significant attention. In data-streaming scenarios and/or big data applications, where the tensor dimension in one of its modes grows in time or can only be processed incrementally, it is essential to be able to perform model selection and computation in a recursive (incremental/online) manner. To date there is only one such work in the literature concerning the (general rank-$(L,M,N)$) BTD model, which proposes an incremental method, however with the BTD rank and block ranks assumed to be a-priori known and time invariant. In this preprint, a novel approach to rank-$(L_r,L_r,1)$ BTD model selection and tracking is proposed, based on the idea of imposing column sparsity jointly on the factors and estimating the ranks as the numbers of factor columns of nonnegligible magnitude. An online method of the alternating iteratively reweighted least squares (IRLS) type is developed and shown to be computationally efficient and fast converging, also allowing the model ranks to change in time. Its time and memory efficiency are evaluated and favorably compared with those of the batch approach. Simulation results are reported that demonstrate the effectiveness of the proposed scheme in both selecting and tracking the correct BTD model.