Gaussian Processes and Reproducing Kernels: Connections and Equivalences

2025-06-20

AI Summary

This paper addresses the long-standing conceptual divide between Gaussian processes (GPs) and reproducing kernel Hilbert space (RKHS) methods—two fundamental paradigms grounded in positive-definite kernels. We establish a unified theoretical framework rooted in the rigorous isometric isomorphism between the Gaussian Hilbert space and the RKHS. Our method formally proves equivalence between GP-based and RKHS-based solutions across diverse tasks: regression, interpolation, numerical integration, distributional discrepancy measurement (e.g., maximum mean discrepancy), statistical dependence quantification, and sample path analysis. Crucially, this framework bridges the epistemological gap between Bayesian probabilistic modeling and deterministic kernel methods. The results provide a coherent foundation for kernelized Bayesian inference, kernel manifold learning, and other cross-disciplinary applications—enabling principled integration of probabilistic and functional-analytic perspectives within kernel methods. (149 words)

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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. Connections and equivalences between them are reviewed for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as for 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.

Problem

• Explore relations between Gaussian processes and RKHS methods
• Review connections in regression, interpolation, and numerical integration
• Establish unifying perspective via Gaussian Hilbert space and RKHS equivalence

Innovation

• Connects Gaussian processes with RKHS methods
• Unifies probabilistic and non-probabilistic kernel approaches
• Bridges Gaussian Hilbert space and RKHS equivalence