Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences

TL;DR

This review explores the deep connections and differences between Gaussian processes and kernel methods, aiming to unify their theoretical foundations and facilitate cross-community understanding in machine learning.

Contribution

It provides a comprehensive comparison of Gaussian processes and kernel methods, highlighting their equivalences, differences, and potential for knowledge transfer.

Findings

Kernel ridge regression estimator equals Gaussian process posterior mean.

Close relationships between the two frameworks are systematically reviewed.

Discussion of philosophical and theoretical distinctions enhances understanding.

Abstract

This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and frequentist kernel methods based on reproducing kernel Hilbert spaces on the other. It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression. However, they have been studied and developed almost independently by two essentially separate communities, and this makes it difficult to seamlessly transfer results between them. Our aim is to overcome this potential difficulty. To this end, we review several old and new results and concepts from either side, and juxtapose algorithmic quantities from each framework to highlight close similarities. We also provide discussions on subtle philosophical and theoretical differences between the two approaches.