Probabilistic Block Term Decomposition for the Modelling of Higher-Order Arrays

TL;DR

This paper introduces a variational Bayesian approach for Block-Term Decomposition (BTD) of higher-order tensors, enabling robust modeling and inference of multi-linear structures in noisy data.

Contribution

It proposes an efficient probabilistic BTD method using Bayesian inference and the von-Mises Fisher distribution to impose orthogonality, advancing tensor factorization techniques.

Findings

Effective in noisy data scenarios

Capable of model order quantification

Provides robust inference of multi-linear patterns

Abstract

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.