Laplace Approximation for Bayesian Tensor Network Kernel Machines

2026-04-29

AI Summary

This work addresses the key challenge of providing principled uncertainty estimates for tensor network kernel machines while maintaining scalability. It introduces, for the first time, the Laplace approximation into this framework, effectively overcoming Bayesian inference obstacles posed by its non-Gaussian structure through a linearization strategy. This enables efficient and theoretically grounded uncertainty quantification. By seamlessly integrating tensor networks, kernel methods, and Bayesian inference, the proposed approach achieves performance on multiple UCI regression benchmarks that matches or even surpasses that of Gaussian processes and Bayesian neural networks, all while preserving both computational scalability and theoretical rigor.

Abstract

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.

Problem

• Uncertainty Estimation
• Tensor Network Kernel Machines
• Gaussianity
• Bayesian Inference
• Laplace Approximation

Innovation

• Laplace Approximation
• Bayesian Tensor Network
• Kernel Machines
• Uncertainty Quantification
• Tensor Network