Minimax estimation of smooth densities in Wasserstein distance
Minimax estimation of smooth densities in Wasserstein distance
复制标题
Wasserstein 距离中平滑密度的极小极大估计
作者：
Jonathan Niles;Quentin Berthet
我们研究非参数密度估计问题，其中误差是在Wasserstein距离中测量的，Wasserstein距离是统计和机器学习许多领域中流行的概率分布度量。我们给出了这个问题的第一个最小最大最优速率一般Wasserstein距离，并表明，与经典的非参数密度估计，这些速率取决于是否有界的密度问题。出于变分问题涉及Wasserstein距离，我们还展示了如何构建离散支持的措施，适合计算的目的，实现极大极小率。我们的主要技术工具是一个不等式，给出了一个几乎紧的双重表征的Wasserstein距离的Besov规范。
We study nonparametric density estimation problems where error is measured in the Wasserstein distance, a metric on probability distributions popular in many areas of statistics and machine learning. We give the first minimax-optimal rates for this problem for general Wasserstein distances, and show that, unlike classical nonparametric density estimation, these rates depend on whether the densities in question are bounded below. Motivated by variational problems involving the Wasserstein distance, we also show how to construct discretely supported measures, suitable for computational purposes, which achieve the minimax rates. Our main technical tool is an inequality giving a nearly tight dual characterization of the Wasserstein distances in terms of Besov norms.
DOI：
10.1093/imaiai/iaz006
发表时间：
2018-06
期刊：
Information and Inference: A Journal of the IMA
影响因子：
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作者：
P. Rigollet;J. Weed
通讯作者：
P. Rigollet;J. Weed