Small Sample Spaces for Gaussian Processes

TL;DR

This paper introduces a method to identify small, functionally relevant sets containing Gaussian process samples using scaled RKHSs and basis expansions, extending understanding of sample support in function spaces.

Contribution

It proposes a novel approach to define the sample support set for Gaussian processes using scaled RKHSs and basis coefficient bounds, generalizing previous nuclear dominance conditions.

Findings

Sample support set characterized by basis coefficient bounds

Supports functions expandable in RKHS basis with bounded coefficients

Method extends understanding of Gaussian process sample spaces

Abstract

It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process $X$ is controlled by a certain nuclear dominance condition. However, it is less clear how to identify a "small" set of functions (not necessarily a vector space) that contains the samples. This article presents a general approach for identifying such sets. We use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to define the sample support set as the largest set which is contained in every element of full measure under the law of $X$ in the $\sigma$-algebra induced by the collection of scaled RKHS. This potentially non-measurable set is then shown to consist of those functions that can be expanded in terms of an orthonormal basis of the RKHS of the covariance kernel of $X$ and have their squared basis coefficients bounded away from zero and infinity, a result suggested by the Karhunen-Lo\`{e}ve theorem.