In many applications of machine learning, a large number of variables are
considered. Motivated by machine learning of interacting particle systems, we
consider the situation when the number of input variables goes to infinity.
First, we continue the recent investigation of the mean field limit of kernels
and their reproducing kernel Hilbert spaces, completing the existing theory.
Next, we provide results relevant for approximation with such kernels in the
mean field limit, including a representer theorem. Finally, we use these
kernels in the context of statistical learning in the mean field limit,
focusing on Support Vector Machines. In particular, we show mean field
convergence of empirical and infinite-sample solutions as well as the
convergence of the corresponding risks. On the one hand, our results establish
rigorous mean field limits in the context of kernel methods, providing new
theoretical tools and insights for large-scale problems. On the other hand, our
setting corresponds to a new form of limit of learning problems, which seems to
have not been investigated yet in the statistical learning theory literature.

使用机器学习中的大规模问题为背景，本论文针对输入变量个数趋向无穷大的情况展开研究。首先，我们完善了核和其再生核希尔伯特空间均值场极限的现有理论。接下来，我们提供了与这些核在均值场极限下逼近相关的结果，包括一个再现者定理。最后，我们将这些核应用于均值场极限下的统计学习，重点是支持向量机。特别地，我们展示了经验和无穷样本解的均值场收敛，以及相应风险的收敛。一方面，我们的结果在核方法的背景下建立了严谨的均值场极限，为大规模问题提供了新的理论工具和洞见。另一方面，我们提出了一种尚未在统计学习理论文献中研究的学习问题形式。

关于均场极限中的基于核的统计学习

On kernel-based statistical learning in the mean field limit

In this paper we consider the problems of supervised classification and
regression in the case where attributes and labels are functions: a data is
represented by a set of functions, and the label is also a function. We focus
on the use of reproducing kernel Hilbert space theory to learn from such
functional data. Basic concepts and properties of kernel-based learning are
extended to include the estimation of function-valued functions. In this
setting, the representer theorem is restated, a set of rigorously defined
infinite-dimensional operator-valued kernels that can be valuably applied when
the data are functions is described, and a learning algorithm for nonlinear
functional data analysis is introduced. The methodology is illustrated through
speech and audio signal processing experiments.

本文介绍了如何在函数数据上使用再生核希尔伯特空间理论进行有监督学习和回归，扩展了基于核的学习的概念和性质，包括估计函数值函数的算法，阐述了一套严格定义的无限维算子值核，以及非线性函数数据分析的学习算法，并通过语音和音频信号处理实验进行了说明。