Probabilistic Block Term Decomposition for the Modelling of Higher-Order Arrays
This work provides a probabilistic method for tensor factorization, which is incremental as it extends Bayesian inference to BTD for improved robustness in multi-linear data analysis.
The authors tackled the problem of modeling higher-order arrays by proposing a probabilistic Block-Term Decomposition (pBTD) using variational Bayesian inference with von-Mises Fisher distributions for orthogonality, and demonstrated its effectiveness on synthetic and real datasets for robust pattern inference and model order quantification.
Tensors are ubiquitous in science and engineering and tensor factorization approaches have become important tools for the characterization of higher order structure. Factorizations includes the outer-product rank Canonical Polyadic Decomposition (CPD) as well as the multi-linear rank Tucker decomposition in which the Block-Term Decomposition (BTD) is a structured intermediate interpolating between these two representations. Whereas CPD, Tucker, and BTD have traditionally relied on maximum-likelihood estimation, Bayesian inference has been use to form probabilistic CPD and Tucker. We propose, an efficient variational Bayesian probabilistic BTD, which uses the von-Mises Fisher matrix distribution to impose orthogonality in the multi-linear Tucker parts forming the BTD. On synthetic and two real datasets, we highlight the Bayesian inference procedure and demonstrate using the proposed pBTD on noisy data and for model order quantification. We find that the probabilistic BTD can quantify suitable multi-linear structures providing a means for robust inference of patterns in multi-linear data.