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Sample Path Regularity of Gaussian Processes from the Covariance Kernel

Nathaël Da Costa, Marvin Pförtner, Lancelot Da Costa, Philipp Hennig

TL;DR

This work provides a comprehensive, kernel-centric characterization of Gaussian process sample path regularity, proving necessary and sufficient conditions for Hölder (and Sobolev) smoothness of GP samples directly from the covariance kernel. It unifies stationary, isotropic, and general kernels, deriving sharp results for widely used families such as Matérn and Wendland, and shows how kernel algebra (sums, products, tensor products, and coordinate transforms) preserves regularity. The results support tight prior design for probabilistic numerical methods and PDE solvers, where sample-path regularity must be controlled without overconstraining uncertainty. Overall, the paper offers practical criteria and versatile tools for predicting and manipulating GP sample-path regularity across diverse kernel constructions and manifolds.

Abstract

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 Hölder 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 Matérn GPs.

Sample Path Regularity of Gaussian Processes from the Covariance Kernel

TL;DR

This work provides a comprehensive, kernel-centric characterization of Gaussian process sample path regularity, proving necessary and sufficient conditions for Hölder (and Sobolev) smoothness of GP samples directly from the covariance kernel. It unifies stationary, isotropic, and general kernels, deriving sharp results for widely used families such as Matérn and Wendland, and shows how kernel algebra (sums, products, tensor products, and coordinate transforms) preserves regularity. The results support tight prior design for probabilistic numerical methods and PDE solvers, where sample-path regularity must be controlled without overconstraining uncertainty. Overall, the paper offers practical criteria and versatile tools for predicting and manipulating GP sample-path regularity across diverse kernel constructions and manifolds.

Abstract

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 Hölder 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 Matérn GPs.
Paper Structure (25 sections, 17 theorems, 48 equations, 4 figures)

This paper contains 25 sections, 17 theorems, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Summary of Contributions
  3. Related Work
  4. Preliminaries
  5. Setup
  6. Hölder Regularity
  7. Wiener Kernel
  8. Matérn Kernels
  9. Wendland Kernels
  10. Smooth Kernels
  11. Feature Kernels
  12. Kernel Algebra
  13. Conic Combinations
  14. Products
  15. Tensor Products
  16. ...and 10 more sections

Key Result

Corollary 1.1

Let $k \colon \mathbb R^d \times \mathbb R^d \to \mathbb R$ be a symmetric positive definite kernel. If either then the sample paths of $f \sim \mathcal{GP}(0, k)$ are, up to modification, $n$ times continuously differentiable.

Figures (4)

  • Figure 1: Left: surface plot of a kernel defined as the tensor product of two Wendland kernels (with $d=1$, $n=0$ on the $x_1$-axis and $d=1$, $n=1$ on the $x_2$-axis). Right: surface plot of a sample path from the corresponding centered GP. The sample path has different regularity along each axis (non-differentiable along the $x_1$-axis; once differentiable along the $x_2$-axis), reflecting the rotational asymmetry in the regularity of the kernel.
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (41)

  • Corollary 1.1
  • Corollary 1.2
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1: Local Hölder and almost-Hölder spaces
  • Remark 3.1
  • Theorem 3.1: Sample path Hölder regularity
  • Remark 3.2
  • Remark 3.3
  • ...and 31 more