This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of probability metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimensional analysis and stands in contrast to classical metrics ({\it e.g.}, the Wasserstein metric). The proposed metrics fall into the category of integral probability metrics, for which we specify criteria of test function spaces to guarantee the property of being free of CoD. Examples of the selected test function spaces include the reproducing kernel Hilbert spaces, Barron space, and flow-induced function spaces. Three applications of the proposed metrics are presented: 1. The convergence of empirical measure in the case of random variables; 2. The convergence of $n$-particle system to the solution to McKean-Vlasov stochastic differential equation; 3. The construction of an $\varepsilon$-Nash equilibrium for a homogeneous $n$-player game by its mean-field limit. As a byproduct, we prove that, given a distribution close to the target distribution measured by our metric and a certain representation of the target distribution, we can generate a distribution close to the target one in terms of the Wasserstein metric and relative entropy. Overall, we show that the proposed class of metrics is a powerful tool to analyze the convergence of empirical measures in high dimensions without CoD.

翻译：本文关注高维空间中经验测度的收敛问题。我们提出了一类新的概率度量，并证明在此类度量下，收敛性不受维数灾难（CoD）的影响。这一特性对于高维分析至关重要，且与经典度量（如Wasserstein度量）形成鲜明对比。所提出的度量属于积分概率度量范畴，我们为该类度量指定了测试函数空间的选择准则，以确保其具备无维数灾难的性质。选定的测试函数空间示例包括再生核希尔伯特空间、Barron空间以及流诱导函数空间。本文展示了所提度量的三个应用：1. 随机变量情形下经验测度的收敛；2. n粒子系统向McKean-Vlasov随机微分方程解的收敛；3. 通过平均场极限构造同质n人博弈的ε-纳什均衡。作为副产品，我们证明：给定一个用我们的度量测度与目标分布接近的分布，以及目标分布的某种表示，我们可以生成一个在Wasserstein度量和相对熵意义上与目标分布接近的分布。总体而言，我们表明所提出的度量类别是分析高维空间中经验测度收敛性且不受维数灾难影响的强大工具。