论文标题
估计最佳运输计划的图形和统一的一致性
Graphical and uniform consistency of estimated optimal transport plans
论文作者
论文摘要
提供了一般理论，可提供最大循环单调映射的收敛性，该映射包含欧几里得空间上可能随机概率测量对序列的耦合度量的支持。该理论基于对这种映射的识别，该映射是欧几里得空间的笛卡尔产物的封闭子集和随机集理论的杠杆作用。在适当的秋季空间中，弱收敛与最大周期性单调性一起自动产生相关映射的局部均匀收敛性。将这些映射视为在平方欧几里德距离之间的概率措施之间的最佳运输计划，因为成本函数将基于最佳运输的多元等级和分位数的概念产生一致性结果，尤其是经验性的中心外向分布和分位数。
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based on the identification of such a mapping with a closed subset of a Cartesian product of Euclidean spaces and leveraging tools from random set theory. Weak convergence in the appropriate Fell space together with the maximal cyclical monotonicity then automatically yields local uniform convergence of the associated mappings. Viewing such mappings as optimal transport plans between probability measures with respect to the squared Euclidean distance as cost function yields consistency results for notions of multivariate ranks and quantiles based on optimal transport, notably the empirical center-outward distribution and quantile functions.
