分层积分概率度量：一种具有低样本复杂度的随机概率测度距离
Hierarchical Integral Probability Metrics: A distance on random probability measures with low sample complexity
AI总结：
本文提出分层积分概率度量（Hierarchical IPM），作为随机概率测度间的距离，克服了Wasserstein-over-Wasserstein距离在无限维空间中样本复杂度极慢的问题，实现参数收敛速率，并保留度量弱收敛、可低复杂度估计及解析界等性质。
摘要：
随机概率是统计学和机器学习中许多非参数方法的关键组成部分。为了量化不同随机概率定律之间的比较，一些工作开始使用优雅的Wasserstein-over-Wasserstein距离。在本文中，我们证明了概率空间的无限维性会急剧恶化其样本复杂度，其收敛速度比样本量的任何多项式速率都慢。我们提出了一种新距离，它保留了前者的许多理想性质，同时实现了参数收敛速率。特别是，我们的距离1）度量弱收敛；2）可以通过低复杂度的样本进行数值估计；3）可以从上下方解析地界定。主要成分是积分概率度量，这导致了分层IPM这一名称。
英文摘要：
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 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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