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.

翻译：核方法凭借其完备的理论基础和高效算法，已成为最流行且成功的机器学习技术之一。从数学角度看，这些方法建立在核函数及其生成函数空间（即再生核希尔伯特空间）的概念之上。受相互作用粒子系统领域近期学习方法的启发，我们研究了作用于多测量变量数据的核方法。本文严格证明了核函数的平均场极限，并对极限再生核希尔伯特空间进行了详细分析。此外，还给出了若干可严格实现平均场极限的核函数实例。