This paper is concerned with learning principal variations of random probability measures on $\mathbb{R}^m$ under the Wasserstein geometry. We introduce a new dynamical formulation to interpret the log-PCA, a linearized principal geodesic analysis, as a variational approach. Our differentiable version, termed as the Wasserstein Tangential PCA (WT-PCA), captures the local principal modes of geodesic variations of a (weighted) probability measure on the Wasserstein space via its covariance operator at barycenter. Based on the dynamical perspective and leveraging parallel transport structure of the optimal transport problems, we derive a general statistical convergence rate of the empirical WT-PCA when estimated from data in terms of the 2-Wasserstein distance between the population and empirical barycenter reference measures.

翻译：本文研究在Wasserstein几何下随机概率测度主变异的学习问题。我们引入了一种新的动力学公式来诠释对数主成分分析（log-PCA）——一种线性化主测地线分析——将其视为变分方法。所提出的可微分版本称为Wasserstein切向主成分分析（WT-PCA），该版本通过重心处的协方差算子捕获Wasserstein空间中（加权）概率测度的局部主测地线变异模式。基于动力学视角并利用最优传输问题的平行传输结构，我们推导出从数据估计得到的经验WT-PCA在总体分布与经验分布的重心参考测度之间的2-Wasserstein距离上的通用统计收敛速率。