Small Sample Spaces for Gaussian Processes
This work addresses a theoretical gap in Gaussian process analysis, providing a foundational framework for understanding sample spaces, but it is incremental as it builds on existing RKHS and Karhunen-Loève theory.
The paper tackles the problem of identifying a small set of functions that contains the samples of a Gaussian process, beyond traditional RKHS membership, by introducing scaled RKHSs and defining a sample support set based on orthonormal basis expansions with bounded coefficients.
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 $σ$-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-Loève theorem.