# Linear cost and exponentially convergent approximation of Gaussian Mat'ern processes on intervals

arXiv: https://arxiv.org/abs/2410.13000 · HTML: https://mlsanity.com/paper/2410.13000/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 60 references that are in the map.
- First posted on arXiv 16 Oct 2024.

## Builds on (most cited first, 10 of 10)

- [Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets](https://mlsanity.com/paper/1406.7343/) (2014)
- [Local Gaussian process approximation for large computer experiments](https://mlsanity.com/paper/1303.0383/) (2013)
- [Sparse Cholesky factorization by Kullback-Leibler minimization](https://mlsanity.com/paper/2004.14455/) (2020)
- [High-Dimensional Gaussian Process Inference with Derivatives](https://mlsanity.com/paper/2102.07542/) (2021)
- [Scaling Gaussian Processes with Derivative Information Using Variational Inference](https://mlsanity.com/paper/2107.04061/) (2021)
- [The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running](https://mlsanity.com/paper/2111.01084/) (2021)
- [Kernel Packet: An Exact and Scalable Algorithm for Gaussian Process Regression with Mat'ern Correlations](https://mlsanity.com/paper/2203.03116/) (2022)
- [Smoothing and mean-covariance estimation of functional data with a Bayesian hierarchical model](https://mlsanity.com/paper/1402.5723/) (2014)
- [Covariance-based rational approximations of fractional SPDEs for computationally efficient Bayesian inference](https://mlsanity.com/paper/2209.04670/) (2022)
- [Statistical inference for Gaussian Whittle-Mat'ern fields on metric graphs](https://mlsanity.com/paper/2304.10372/) (2023)

## Abstract (arXiv)

The computational cost for inference and prediction of statistical models based on Gaussian processes with Mat'ern covariance functions scales cubicly with the number of observations, limiting their applicability to large data sets. The cost can be reduced in certain special cases, but there are currently no generally applicable exact methods with linear cost. Several approximate methods have been introduced to reduce the cost, but most of these lack theoretical guarantees for the accuracy. We consider Gaussian processes on bounded intervals with Mat'ern covariance functions and for the first time develop a generally applicable method with linear cost and with a covariance error that decreases exponentially fast in the order m of the proposed approximation. The method is based on an optimal rational approximation of the spectral density and results in an approximation that can be represented as a sum of m independent Gaussian Markov processes, which facilitates easy usage in general software for statistical inference, enabling its efficient implementation in general statistical inference software packages. Besides the theoretical justifications, we demonstrate the accuracy empirically through carefully designed simulation studies which show that the method outperforms all state-of-the-art alternatives in terms of accuracy for a fixed computational cost in statistical tasks such as Gaussian process regression.