# Block-Term Tensor Decomposition Model Selection and Computation: The Bayesian Way

arXiv: https://arxiv.org/abs/2101.02931 · HTML: https://mlsanity.com/paper/2101.02931/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 37 references that are in the map.
- First posted on arXiv 8 Jan 2021.

## Builds on (most cited first, 7 of 7)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Bayesian Tensorized Neural Networks with Automatic Rank Selection](https://mlsanity.com/paper/1905.10478/) (2019)
- [Bayesian Robust Tensor Factorization for Incomplete Multiway Data](https://mlsanity.com/paper/1410.2386/) (2014)
- [Simultaneously sparse and low-rank abundance matrix estimation for hyperspectral image unmixing](https://mlsanity.com/paper/1504.01515/) (2015)
- [Bayesian Sparse Tucker Models for Dimension Reduction and Tensor Completion](https://mlsanity.com/paper/1505.02343/) (2015)

## Built on by (most cited first, 4 of 4)

- [Probabilistic Shape Completion by Estimating Canonical Factors with Hierarchical VAE](https://mlsanity.com/paper/2212.03370/) (2022)
- [StyleTRF: Stylizing Tensorial Radiance Fields](https://mlsanity.com/paper/2212.09330/) (2022)
- [Probabilistic Block Term Decomposition for the Modelling of Higher-Order Arrays](https://mlsanity.com/paper/2310.02694/) (2023)
- [Robust Principal Component Completion](https://mlsanity.com/paper/2603.25132/) (2026)

## Abstract (arXiv)

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 of representing systems and signals that are composed of blocks of rank higher than one, a scenario encountered in numerous and diverse applications. Uniqueness conditions and fitting methods have thus been thoroughly studied. Nevertheless, the challenging problem of estimating the BTD model structure, namely the number of block terms, R, and their individual ranks, Lr, has only recently started to attract significant attention, mainly through regularization-based approaches which entail the need to tune the regularization parameter(s). In this work, we build on ideas of sparse Bayesian learning (SBL) and put forward a fully automated Bayesian approach. Through a suitably crafted multi-level hierarchical probabilistic model, which gives rise to heavy-tailed prior distributions for the BTD factors, structured sparsity is jointly imposed. Ranks are then estimated from the numbers of blocks (R) and columns (Lr) of non-negligible energy. Approximate posterior inference is implemented, within the variational inference framework. The resulting iterative algorithm completely avoids hyperparameter tuning, which is a significant defect of regularization-based methods. Alternative probabilistic models are also explored and the connections with their regularization-based counterparts are brought to light with the aid of the associated maximum a-posteriori (MAP) estimators. We report simulation results with both synthetic and real-word data, which demonstrate the merits of the proposed method in terms of both rank estimation and model fitting as compared to state-of-the-art relevant methods.