# Interpretable Bayesian Tensor Network Kernel Machines with Automatic Rank and Feature Selection

arXiv: https://arxiv.org/abs/2507.11136 · HTML: https://mlsanity.com/paper/2507.11136/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 49 references that are in the map.
- First posted on arXiv 15 Jul 2025.

## Builds on (most cited first, 18 of 18)

- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Multilinear tensor regression for longitudinal relational data](https://mlsanity.com/paper/1412.0048/) (2014)
- [Scalable Gaussian Processes with Billions of Inducing Inputs via Tensor Train Decomposition](https://mlsanity.com/paper/1710.07324/) (2017)
- [Bayesian Robust Tensor Factorization for Incomplete Multiway Data](https://mlsanity.com/paper/1410.2386/) (2014)
- [Bayesian Tensor Regression](https://mlsanity.com/paper/1509.06490/) (2015)
- [Gaussian Quadrature for Kernel Features](https://mlsanity.com/paper/1709.02605/) (2017)
- [Large-Scale Learning with Fourier Features and Tensor Decompositions](https://mlsanity.com/paper/2109.01545/) (2021)
- [Tensor Network alternating linear scheme for MIMO Volterra system identification](https://mlsanity.com/paper/1607.00127/) (2016)
- [Tensor Regression Meets Gaussian Processes](https://mlsanity.com/paper/1710.11345/) (2017)
- [Bayesian Sparse Tucker Models for Dimension Reduction and Tensor Completion](https://mlsanity.com/paper/1505.02343/) (2015)
- [Bayesian Conditional Tensor Factorizations for High-Dimensional Classification](https://mlsanity.com/paper/1301.4950/) (2013)
- [Quantized Fourier and Polynomial Features for more Expressive Tensor Network Models](https://mlsanity.com/paper/2309.05436/) (2023)
- [Equivariant and scale-free Tucker decomposition models](https://mlsanity.com/paper/1312.6397/) (2013)
- [Bayesian multi-tensor factorization](https://mlsanity.com/paper/1412.4679/) (2014)

## Built on by (most cited first, 3 of 3)

- [A Fully Probabilistic Tensor Network for Regularized Volterra System Identification](https://mlsanity.com/paper/2511.20457/) (2025)
- [(MPO)2: Multivariate Polynomial Optimization based on Matrix Product Operators](https://mlsanity.com/paper/2607.15916/) (2026)
- [Tensor Network Kernel Machines: A JAX Framework for Machine Learning and Nonlinear System Identification](https://mlsanity.com/paper/2608.07043/) (2026)

## Abstract (arXiv)

Tensor Network (TN) Kernel Machines speed up model learning by representing parameters as low-rank TNs, reducing computation and memory use. However, most TN-based Kernel methods are deterministic and ignore parameter uncertainty. Further, they require manual tuning of model complexity hyperparameters like tensor rank and feature dimensions, often through trial-and-error or computationally costly methods like cross-validation. We propose Bayesian Tensor Network Kernel Machines, a fully probabilistic framework that uses sparsity-inducing hierarchical priors on TN factors to automatically infer model complexity. This enables automatic inference of tensor rank and feature dimensions, while also identifying the most relevant features for prediction, thereby enhancing model interpretability. All the model parameters and hyperparameters are treated as latent variables with corresponding priors. Given the Bayesian approach and latent variable dependencies, we apply a mean-field variational inference to approximate their posteriors. We show that applying a mean-field approximation to TN factors yields a Bayesian ALS algorithm with the same computational complexity as its deterministic counterpart, enabling uncertainty quantification at no extra computational cost. Experiments on synthetic and real-world datasets demonstrate the superior performance of our model in prediction accuracy, uncertainty quantification, interpretability, and scalability.