# Reproducing kernel Hilbert spaces in the mean field limit

arXiv: https://arxiv.org/abs/2302.14446 · HTML: https://mlsanity.com/paper/2302.14446/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 58 references that are in the map.
- First posted on arXiv 28 Feb 2023.

## Builds on (most cited first, 17 of 17)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Learning interaction kernels in heterogeneous systems of agents from multiple trajectories](https://mlsanity.com/paper/1910.04832/) (2019)
- [Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories](https://mlsanity.com/paper/2007.15174/) (2020)
- [A consensus-based global optimization method for high dimensional machine learning problems](https://mlsanity.com/paper/1909.09249/) (2019)
- [A consensus-based model for global optimization and its mean-field limit](https://mlsanity.com/paper/1604.05648/) (2016)
- [Kernels over Sets of Finite Sets using RKHS Embeddings, with Application to Bayesian (Combinatorial) Optimization](https://mlsanity.com/paper/1910.04086/) (2019)
- [Consensus-Based Optimization on the Sphere: Convergence to Global Minimizers and Machine Learning](https://mlsanity.com/paper/2001.11988/) (2020)
- [Continuous Limits for Constrained Ensemble Kalman Filter](https://mlsanity.com/paper/1912.08406/) (2019)
- [Performance bounds for the mean-field limit of constrained dynamics](https://mlsanity.com/paper/1511.08364/) (2015)
- [Interacting particles with L'evy strategies: limits of transport equations for swarm robotic systems](https://mlsanity.com/paper/1807.10124/) (2018)
- [From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem](https://mlsanity.com/paper/1404.5485/) (2014)
- [Mean-Field and Kinetic Descriptions of Neural Differential Equations](https://mlsanity.com/paper/2001.04294/) (2020)
- [Binary interaction methods for high dimensional global optimization and machine learning](https://mlsanity.com/paper/2105.02695/) (2021)

## Built on by (most cited first, 3 of 3)

- [On kernel-based statistical learning in the mean field limit](https://mlsanity.com/paper/2310.18074/) (2023)
- [Use of operator defect identities in multi-channel signal plus residual-analysis via iterated products and telescoping energy-residuals: Applications to kernels in machine learning](https://mlsanity.com/paper/2601.18080/) (2026)
- [Data-driven approximation of transfer operators for mean-field stochastic differential equations](https://mlsanity.com/paper/2509.09891/) (2025)

## Abstract (arXiv)

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.