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Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences

Year: 2025

Type: preprint

Abstract: This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. We study connections and equivalences for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. The monograph serves as a basis to bridge many other methods based on Gaussian processes and reproducing kernels, which are developed in parallel by the two research communities.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2506.17366

Authors: Kanagawa, Motonobu, Hennig, Philipp, Sejdinovic, Dino, Sriperumbudur, Bharath K.

Institutions: EURECOM, Max Planck Institute for Intelligent Systems, Max Planck Institute for Biology, Australian Institute of Business, Adelaide University, Pennsylvania State University

Language: English

Topic: Gaussian Processes and Bayesian Inference

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Alan Turing Institute, Deutsche Forschungsgemeinschaft, Leibniz-Gemeinschaft

Awards: G1309768829, G1562512020, CAREER: Statistical Learning, Inference and Approximation with Reproducing Kernels, G3855435662, G6671297155, Reproducing Kernel Hilbert Space Embedding of Measures: Theory and Applications to Statistical Learning, EXC 2064: Maschinelles Lernen: Neue Perspektiven für die Wissenschaft

OpenAlex ID: W6910747409

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