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{\"o}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.

翻译：Wasserstein重心以几何意义定义概率测度的平均，在图像处理、几何或语言处理等应用领域中日益流行。然而，在这些领域中，所关注的概率测度往往无法完整获取，研究者可能需要处理统计或计算近似值。本文量化了此类近似对相应重心的影响，证明在相对温和的假设下，Wasserstein重心以Hölder连续方式依赖于其边际分布。我们的证明基于两项关键结果：一是对偶二次最优传输问题强凸性的近期估计，二是控制（非光滑）最优传输映射下前推算子模连续性的新结论。