Reproducing kernel Hilbert spaces in the mean field limit

Kinetic and Related Models, vol. 16, pp. 850–870

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.

Authors 5

1. Christian Fiedler corresponding Aachen Institute for Data Science in Mechanical Engineering

   RWTH Aachen University

   Affiliation as printed

   Institute for Data Science in Mechanical Engineering , RWTH Aachen University , Germany

   Institute for Data Science in Mechanical Engineering, RWTH Aachen University, Germany

2. Michaël Herty Aachen Institute of Geometry and Practical Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute of Geometry and Practical Mathematics , RWTH Aachen University , Germany

   Institute of Geometry and Practical Mathematics, RWTH Aachen University, Germany

3. Michael Rom Aachen Institute of Geometry and Practical Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute of Geometry and Practical Mathematics , RWTH Aachen University , Germany

   Institute of Geometry and Practical Mathematics, RWTH Aachen University, Germany

4. Chiara Segala Aachen Institute of Geometry and Practical Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute of Geometry and Practical Mathematics , RWTH Aachen University , Germany

   Institute of Geometry and Practical Mathematics, RWTH Aachen University, Germany

5. Sebastian Trimpe Aachen Institute for Data Science in Mechanical Engineering

   RWTH Aachen University

   Affiliation as printed

   Institute for Data Science in Mechanical Engineering , RWTH Aachen University , Germany

   Institute for Data Science in Mechanical Engineering, RWTH Aachen University, Germany