论文标题
瓦斯汀空间中重中心的定量稳定性
Quantitative Stability of Barycenters in the Wasserstein Space
论文作者
论文摘要
Wasserstein Barycenters以几何有意义的方式定义了概率度量的平均值。它们的使用越来越流行，例如图像，几何或语言处理。但是，在这些领域中，人们通常无法全面访问感兴趣的概率度量，并且从业者可能必须处理统计或计算近似值。在本文中，我们量化了这种近似值对相应的barycenters的影响。我们表明，在相对温和的假设下，Wasserstein Barycenters依赖于边际上的h {Ö}连续的方式。我们的证明取决于最近估计的估计，这些估计值量化了双二次最佳运输问题的强凸度，以及一个新的结果，该结果允许在A（不一定是平滑）最佳传输图下控制推杆操作的连续性模量。
Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability measures of interest are often not accessible in their entirety and the practitioner may have to deal with statistical or computational approximations instead. In this article, we quantify the effect of such approximations on the corresponding barycenters. We show that Wasserstein barycenters depend in a H{ö}lder-continuous way on their marginals under relatively mild assumptions. Our proof relies on recent estimates that quantify the strong convexity of the dual quadratic optimal transport problem and a new result that allows to control the modulus of continuity of the push-forward operation under a (not necessarily smooth) optimal transport map.
