A Class of Dimension-free Metrics for the Convergence of Empirical Measures. (arXiv:2104.12036v3 [math.PR] UPDATED)

This paper concerns the convergence of empirical measures in high dimensions.
We propose a new class of 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 distance). The proposed metrics originate
from the maximum mean discrepancy, which we generalize by proposing specific
criteria for selecting test function spaces to guarantee the property of …

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