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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 iscritical for high-dimensional analysis and stands in contrast to classicalmetrics ({\it e.g.}, the Wasserstein metric). The proposed metrics fall intothe category of integral probability metrics, for which we specify criteria oftest function spaces to guarantee the property of being free of CoD. Examplesof the selected test function spaces include the reproducing kernel Hilbertspaces, Barron space, and flow-induced function spaces. Three applications ofthe proposed metrics are presented: 1. The convergence of empirical measure inthe case of random variables; 2. The convergence of $n$-particle system to thesolution to McKean-Vlasov stochastic differential equation; 3. The constructionof an $\varepsilon$-Nash equilibrium for a homogeneous $n$-player game by itsmean-field limit. As a byproduct, we prove that, given a distribution close tothe target distribution measured by our metric and a certain representation ofthe target distribution, we can generate a distribution close to the target onein terms of the Wasserstein metric and relative entropy. Overall, we show thatthe proposed class of metrics is a powerful tool to analyze the convergence ofempirical measures in high dimensions without CoD.

A Class of Dimension-free Metrics for the Convergence of Empirical Measures