# Projecting basis functions with tensor networks for Gaussian process regression

arXiv: https://arxiv.org/abs/2310.20630 · HTML: https://mlsanity.com/paper/2310.20630/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 20 references that are in the map.
- First posted on arXiv 31 Oct 2023.

## Builds on (most cited first, 8 of 8)

- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Manifold Gaussian Processes for Regression](https://mlsanity.com/paper/1402.5876/) (2014)
- [Scalable Gaussian Processes with Billions of Inducing Inputs via Tensor Train Decomposition](https://mlsanity.com/paper/1710.07324/) (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)
- [Alternating linear scheme in a Bayesian framework for low-rank tensor approximation](https://mlsanity.com/paper/2012.11228/) (2020)

## Built on by (most cited first, 2 of 2)

- [Laplace Approximation For Tensor Train Kernel Machines In System Identification](https://mlsanity.com/paper/2512.02532/) (2025)
- [Laplace Approximation for Bayesian Tensor Network Kernel Machines](https://mlsanity.com/paper/2604.26673/) (2026)

## Abstract (arXiv)

This paper presents a method for approximate Gaussian process (GP) regression with tensor networks (TNs). A parametric approximation of a GP uses a linear combination of basis functions, where the accuracy of the approximation depends on the total number of basis functions M. We develop an approach that allows us to use an exponential amount of basis functions without the corresponding exponential computational complexity. The key idea to enable this is using low-rank TNs. We first find a suitable low-dimensional subspace from the data, described by a low-rank TN. In this low-dimensional subspace, we then infer the weights of our model by solving a Bayesian inference problem. Finally, we project the resulting weights back to the original space to make GP predictions. The benefit of our approach comes from the projection to a smaller subspace: It modifies the shape of the basis functions in a way that it sees fit based on the given data, and it allows for efficient computations in the smaller subspace. In an experiment with an 18-dimensional benchmark data set, we show the applicability of our method to an inverse dynamics problem.