# Sample Path Regularity of Gaussian Processes from the Covariance Kernel

arXiv: https://arxiv.org/abs/2312.14886 · HTML: https://mlsanity.com/paper/2312.14886/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 31 references that are in the map.
- First posted on arXiv 22 Dec 2023.

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

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [The free energy principle made simpler but not too simple](https://mlsanity.com/paper/2201.06387/) (2022)
- [Probabilistic ODE Solvers with Runge-Kutta Means](https://mlsanity.com/paper/1406.2582/) (2014)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Collective behavior from surprise minimization](https://mlsanity.com/paper/2307.14804/) (2023)
- [Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach](https://mlsanity.com/paper/2310.19270/) (2023)

## Built on by (most cited first, 14 of 14)

- [TabICL: A Tabular Foundation Model for In-Context Learning on Large Data](https://mlsanity.com/paper/2502.05564/) (2025)
- [TabICLv2: A better, faster, scalable, and open tabular foundation model](https://mlsanity.com/paper/2602.11139/) (2026)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning](https://mlsanity.com/paper/2407.13711/) (2024)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Finite-Sample-Based Reachability for Safe Control with Gaussian Process Dynamics](https://mlsanity.com/paper/2505.07594/) (2025)
- [Local Linearity: the Key for No-regret Reinforcement Learning in Continuous MDPs](https://mlsanity.com/paper/2410.24071/) (2024)
- [Distributionally Robust Active Learning for Gaussian Process Regression](https://mlsanity.com/paper/2502.16870/) (2025)
- [Scaling up Probabilistic PDE Simulators with Structured Volumetric Information](https://mlsanity.com/paper/2406.05020/) (2024)
- [A Provable Approach for End-to-End Safe Reinforcement Learning](https://mlsanity.com/paper/2505.21852/) (2025)
- [Differentially Private Hyperparameter Tuning using Local Bayesian Optimization](https://mlsanity.com/paper/2502.06044/) (2025)
- [Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators](https://mlsanity.com/paper/2608.06155/) (2026)
- [The BdryMat'ern GP: Reliable incorporation of boundary information on irregular domains for Gaussian process modeling](https://mlsanity.com/paper/2507.09178/) (2025)
- [Optimizing the Unknown: Black Box Bayesian Optimization with Energy-Based Model and Reinforcement Learning](https://mlsanity.com/paper/2510.19530/) (2025)

## Abstract (arXiv)

Gaussian processes (GPs) are the most common formalism for defining probability distributions over spaces of functions. While applications of GPs are myriad, a comprehensive understanding of GP sample paths, i.e. the function spaces over which they define a probability measure, is lacking. In practice, GPs are not constructed through a probability measure, but instead through a mean function and a covariance kernel. In this paper we provide necessary and sufficient conditions on the covariance kernel for the sample paths of the corresponding GP to attain a given regularity. We focus primarily on H"older regularity as it grants particularly straightforward conditions, which simplify further in the cases of stationary and isotropic GPs. We then demonstrate that our results allow for novel and unusually tight characterisations of the sample path regularities of the GPs commonly used in machine learning applications, such as the Mat'ern GPs.