# Laplace Approximation for Bayesian Tensor Network Kernel Machines

arXiv: https://arxiv.org/abs/2604.26673 · HTML: https://mlsanity.com/paper/2604.26673/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 51 references that are in the map.
- First posted on arXiv 29 Apr 2026.

## Builds on (most cited first, 26 of 26)

- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks](https://mlsanity.com/paper/2002.10118/) (2020)
- [Practical Gauss-Newton Optimisation for Deep Learning](https://mlsanity.com/paper/1706.03662/) (2017)
- [Approximate Inference Turns Deep Networks into Gaussian Processes](https://mlsanity.com/paper/1906.01930/) (2019)
- [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)
- [Do Bayesian Neural Networks Need To Be Fully Stochastic?](https://mlsanity.com/paper/2211.06291/) (2022)
- [Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives](https://mlsanity.com/paper/1708.09165/) (2017)
- [Adapting the Linearised Laplace Model Evidence for Modern Deep Learning](https://mlsanity.com/paper/2206.08900/) (2022)
- [Dropout Inference in Bayesian Neural Networks with Alpha-divergences](https://mlsanity.com/paper/1703.02914/) (2017)
- [Generalized Variational Inference in Function Spaces: Gaussian Measures meet Bayesian Deep Learning](https://mlsanity.com/paper/2205.06342/) (2022)
- [Uncertainty Quantification Metrics for Deep Regression](https://mlsanity.com/paper/2405.04278/) (2024)
- [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)
- [FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning](https://mlsanity.com/paper/2407.13711/) (2024)
- [Quantized Fourier and Polynomial Features for more Expressive Tensor Network Models](https://mlsanity.com/paper/2309.05436/) (2023)
- [The Bayesian Method of Tensor Networks](https://mlsanity.com/paper/2101.00245/) (2021)
- [Bayes without Underfitting: Fully Correlated Deep Learning Posteriors via Alternating Projections](https://mlsanity.com/paper/2410.16901/) (2024)
- [Tensor Network Based Feature Learning Model](https://mlsanity.com/paper/2512.02547/) (2025)
- [Projecting basis functions with tensor networks for Gaussian process regression](https://mlsanity.com/paper/2310.20630/) (2023)
- [Density Regression and Uncertainty Quantification with Bayesian Deep Noise Neural Networks](https://mlsanity.com/paper/2206.05643/) (2022)

## Built on by (most cited first, 1 of 1)

- [Tensor Network Kernel Machines: A JAX Framework for Machine Learning and Nonlinear System Identification](https://mlsanity.com/paper/2608.07043/) (2026)

## Abstract (arXiv)

Uncertainty estimation is essential for robust decision-making in the presence of ambiguous or out-of-distribution inputs. Gaussian Processes (GPs) are classical kernel-based models that offer principled uncertainty quantification and perform well on small- to medium-scale datasets. Alternatively, formulating the weight space learning problem under tensor network assumptions yields scalable tensor network kernel machines. However, these assumptions break Gaussianity, complicating standard probabilistic inference. This raises a fundamental question: how can tensor network kernel machines provide principled uncertainty estimates? We propose a novel Bayesian Tensor Network Kernel Machine (LA-TNKM) that employs a (linearized) Laplace approximation for Bayesian inference. A comprehensive set of numerical experiments shows that the proposed method consistently matches or surpasses Gaussian Processes and Bayesian Neural Networks (BNNs) across diverse UCI regression benchmarks, highlighting both its effectiveness and practical relevance.