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

Understand

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.