# Fast Gaussian process inference by exact Mat'ern kernel decomposition

arXiv: https://arxiv.org/abs/2508.01864 · HTML: https://mlsanity.com/paper/2508.01864/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 61 references that are in the map.
- First posted on arXiv 3 Aug 2025.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [GPflow: A Gaussian process library using TensorFlow](https://mlsanity.com/paper/1610.08733/) (2016)
- [Exact Gaussian Processes on a Million Data Points](https://mlsanity.com/paper/1903.08114/) (2019)
- [Kernel Operations on the GPU, with Autodiff, without Memory Overflows](https://mlsanity.com/paper/2004.11127/) (2020)
- [Kernel methods through the roof: handling billions of points efficiently](https://mlsanity.com/paper/2006.10350/) (2020)
- [Distributed Gaussian Processes](https://mlsanity.com/paper/1502.02843/) (2015)
- [FALKON: An Optimal Large Scale Kernel Method](https://mlsanity.com/paper/1705.10958/) (2017)
- [A survey on high-dimensional Gaussian process modeling with application to Bayesian optimization](https://mlsanity.com/paper/2111.05040/) (2021)
- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Local Gaussian process approximation for large computer experiments](https://mlsanity.com/paper/1303.0383/) (2013)
- [A Case Study Competition Among Methods for Analyzing Large Spatial Data](https://mlsanity.com/paper/1710.05013/) (2017)
- [Product Kernel Interpolation for Scalable Gaussian Processes](https://mlsanity.com/paper/1802.08903/) (2018)
- [Gaussian Process Boosting](https://mlsanity.com/paper/2004.02653/) (2020)
- [Preconditioning for Scalable Gaussian Process Hyperparameter Optimization](https://mlsanity.com/paper/2107.00243/) (2021)
- [Conditioning Sparse Variational Gaussian Processes for Online Decision-making](https://mlsanity.com/paper/2110.15172/) (2021)
- [When are Iterative Gaussian Processes Reliably Accurate?](https://mlsanity.com/paper/2112.15246/) (2021)
- [Gaussian Process Learning via Fisher Scoring of Vecchia's Approximation](https://mlsanity.com/paper/1905.08374/) (2019)
- [Exact Gaussian Processes for Massive Datasets via Non-Stationary Sparsity-Discovering Kernels](https://mlsanity.com/paper/2205.09070/) (2022)
- [The Fast Kernel Transform](https://mlsanity.com/paper/2106.04487/) (2021)
- [Understanding and Comparing Scalable Gaussian Process Regression for Big Data](https://mlsanity.com/paper/1811.01159/) (2018)
- [Fast multivariate empirical cumulative distribution function with connection to kernel density estimation](https://mlsanity.com/paper/2005.03246/) (2020)
- [A spectral mixture representation of isotropic kernels with application to random Fourier features](https://mlsanity.com/paper/2411.02770/) (2024)
- [Iterative Methods for Full-Scale Gaussian Process Approximations for Large Spatial Data](https://mlsanity.com/paper/2405.14492/) (2024)
- [A Comparison of Spatial Predictors when Datasets Could be Very Large](https://mlsanity.com/paper/1410.7748/) (2014)
- [Which Parameterization of the Mat'ern Covariance Function?](https://mlsanity.com/paper/2309.12000/) (2023)
- [Large-scale Environmental Data Science with ExaGeoStatR](https://mlsanity.com/paper/1908.06936/) (2019)
- [Giga-scale Kernel Matrix Vector Multiplication on GPU](https://mlsanity.com/paper/2202.01085/) (2022)

## Abstract (arXiv)

To speed up Gaussian process inference, a number of fast kernel matrix-vector multiplication (MVM) approximation algorithms have been proposed over the years. In this paper, we establish an exact fast kernel MVM algorithm based on exact kernel decomposition into weighted empirical cumulative distribution functions, compatible with a class of kernels which includes multivariate Mat'ern kernels with half-integer smoothness parameter. This algorithm uses a divide-and-conquer approach, during which sorting outputs are stored in a data structure. We also propose a new algorithm to take into account some linear fixed effects predictor function. Our numerical experiments confirm that our algorithm is very effective for low-dimensional Gaussian process inference problems with hundreds of thousands of data points. An implementation of our algorithm is available at https://gitlab.com/warin/fastgaussiankernelregression.git.