Hierarchical Integral Probability Metrics: A distance on random probability measures with low sample complexity
arxiv(2024)
摘要
Random probabilities are a key component to many nonparametric methods in
Statistics and Machine Learning. To quantify comparisons between different laws
of random probabilities several works are starting to use the elegant
Wasserstein over Wasserstein distance. In this paper we prove that the
infinite-dimensionality of the space of probabilities drastically deteriorates
its sample complexity, which is slower than any polynomial rate in the sample
size. We thus propose a new distance that preserves many desirable properties
of the former while achieving a parametric rate of convergence. In particular,
our distance 1) metrizes weak convergence; 2) can be estimated numerically
through samples with low complexity; 3) can be bounded analytically from above
and below. The main ingredient are integral probability metrics, which lead to
the name hierarchical IPM.
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