Reproducing kernel Hilbert spaces in the mean field limit

TL;DR

This paper investigates the behavior of kernel methods in high-dimensional settings with many variables, establishing a rigorous mean field limit and analyzing the resulting reproducing kernel Hilbert space.

Contribution

It provides a rigorous mathematical framework for understanding the mean field limit of kernels in high-dimensional data, extending kernel theory to interacting particle systems.

Findings

Established the mean field limit of kernels

Analyzed the structure of the limiting RKHS

Presented examples of kernels with a rigorous mean field limit

Abstract

Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so called reproducing kernel Hilbert spaces. Motivated by recent developments of learning approaches in the context of interacting particle systems, we investigate kernel methods acting on data with many measurement variables. We show the rigorous mean field limit of kernels and provide a detailed analysis of the limiting reproducing kernel Hilbert space. Furthermore, several examples of kernels, that allow a rigorous mean field limit, are presented.