Online rank-revealing block-term tensor decomposition

Abstract

The so-called block-term decomposition (BTD) tensor model, especially in its rank-(Lr,Lr,1) version, has been recently receiving increasing attention due to its enhanced ability to represent systems and signals that are composed of 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 size grows in time or the processing can only be done incrementally, it is essential to be able to perform model selection and computation in a recursive (online/incremental) manner. In this paper, a novel approach to rank-(Lr,Lr,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 non-negligible magnitude. In this vein, and using a new rank-revealing batch BTD approximation algorithm as the starting point, an online method of the alternating reweighted least squares (RLS) 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 and a recently reported online BTD scheme that is based on a-priori knowledge of the ranks. Simulation results with both synthetic and real (video) data are reported, that demonstrate the effectiveness of the proposed scheme in both selecting and tracking the correct BTD model.