Estimation of smooth densities in Wasserstein distance
Estimation of smooth densities in Wasserstein distance
复制标题
Wasserstein 距离中的平滑密度估计
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Quentin Berthet
中科院分区：
文献类型：
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作者：
J. Weed;Quentin Berthet
Wasserstein距离是$mathbb{R}^d$上支持的概率分布的一组度量，在统计和机器学习中应用。通常，这样的距离被用于变分问题的背景下，其中统计学家采用在独立样本的基础上构建的代理来代替未知的测量。这就提出了一个基本问题，即如何在Wasserstein距离中近似测量。虽然已知包括i.i.d.样本是速率最优的一般措施，没有改进的结果是已知的措施拥有光滑的密度。我们证明了第一个极小极大率估计一般Wasserstein距离的光滑密度，从而显示如何灾难的维数可以减轻足够经常的措施。我们还展示了如何构建离散支持的措施，适合计算的目的，享受提高利率。我们的方法是基于新的Wasserstein距离和合适的Besov规范，这可能是独立的利益之间的界限。
The Wasserstein distances are a set of metrics on probability distributions supported on $mathbb{R}^d$ with applications throughout statistics and machine learning. Often, such distances are used in the context of variational problems, in which the statistician employs in place of an unknown measure a proxy constructed on the basis of independent samples. This raises the basic question of how well measures can be approximated in Wasserstein distance. While it is known that an empirical measure comprising i.i.d. samples is rate-optimal for general measures, no improved results were known for measures possessing smooth densities. We prove the first minimax rates for estimation of smooth densities for general Wasserstein distances, thereby showing how the curse of dimensionality can be alleviated for sufficiently regular measures. We also show how to construct discretely supported measures, suitable for computational purposes, which enjoy improved rates. Our approach is based on novel bounds between the Wasserstein distances and suitable Besov norms, which may be of independent interest.