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.

翻译：张量在科学和工程领域广泛存在，张量分解方法已成为表征高阶结构的重要工具。分解方法包括外积秩的典型多路分解（CPD）以及多线性秩的Tucker分解，其中块项分解（BTD）是一种在这两种表示之间进行插值的结构化中间形式。尽管传统上CPD、Tucker和BTD依赖于极大似然估计，但贝叶斯推断已被用于构建概率型CPD和Tucker。我们提出一种高效的变分贝叶斯概率型BTD，该方法利用von-Mises Fisher矩阵分布在构成BTD的多线性Tucker部分施加正交性。在合成数据集和两个真实数据集上，我们展示了贝叶斯推断过程，并论证了所提出的pBTD在含噪数据中的表现及其用于模型阶数量化的能力。研究发现，概率型BTD能够量化合适的的多线性结构，为多线性数据中模式的稳健推断提供了有效手段。