标题： 最优传输映射的定量稳定性及2-Wasserstein空间的线性化 显示英文标题

标题： Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space

摘要： 这项工作研究了将概率测度集显式嵌入到希尔伯特空间中的方法，该嵌入是通过参考概率密度的最优传输映射定义的。 这种嵌入在一定程度上线性化了2-瓦瑟斯坦空间，并使得可以直接在测度数据上使用通用的监督和非监督学习算法。 我们的主要结果是，当参考密度在凸集中均匀分布时，该嵌入是（双）霍尔德连续的，并且可以等价地表述为最优传输映射的维度无关的霍尔德稳定性结果。 显示英文摘要

摘要： This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)H\"older continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent H\"older-stability results for optimal transport maps.