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重心以几何上有意义的方式定义了概率测度的平均值。其在图像处理、几何处理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，实践者可能需要处理统计或计算近似。本文量化了此类近似对相应重心的影响。我们证明，在相对温和的假设下，Wasserstein重心以其边际分布的Hölder连续方式依赖。该证明依赖于近期对偶二次最优传输问题强凸性的定量估计，以及一个控制（非必然光滑的）最优传输映射下前推操作连续模的新结果。