# Small Sample Spaces for Gaussian Processes

arXiv: https://arxiv.org/abs/2103.03169 · HTML: https://mlsanity.com/paper/2103.03169/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 45 references that are in the map.
- First posted on arXiv 4 Mar 2021.

## Builds on (most cited first, 4 of 4)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Bayesian Learning of Kernel Embeddings](https://mlsanity.com/paper/1603.02160/) (2016)
- [Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions](https://mlsanity.com/paper/2001.10965/) (2020)

## Built on by (most cited first, 5 of 5)

- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Orthonormal Expansions for Translation-Invariant Kernels](https://mlsanity.com/paper/2206.08648/) (2022)
- [Asymptotic Bounds for Smoothness Parameter Estimates in Gaussian Process Interpolation](https://mlsanity.com/paper/2203.05400/) (2022)
- [Dynamic Pricing and Demand Learning on a Large Network of Products: A PAC-Bayesian Approach](https://mlsanity.com/paper/2111.00790/) (2021)

## Abstract (arXiv)

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-Lo`eve theorem.