提出新方法学习概率分布的主变化模式，可解释为变分问题。
Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence
- 基于流形动力学视角，将对数PCA重构为变分方法。
- 在2-Wasserstein距离下，给出经验估计的统计收敛速率。
- 适合研究概率分布主成分分析的学者使用。
本文研究在Wasserstein几何下，对定义于ℝᵐ上的随机概率测度进行主成分分析的问题。我们提出一种新的动态表述，将对数PCA（即线性化的主测地线分析）解释为变分方法。所提出的可微版本称为Wasserstein切空间PCA（WT-PCA），通过在质心处的概率测度协方差算子，捕捉加权概率测度在Wasserstein空间中的局部主测地线变化模式。基于动态视角并利用最优传输问题中的平行移动结构，我们推导出当从数据估计时，经验WT-PCA在2-Wasserstein距离下相对于总体与经验质心测度之间的收敛速率。
原文摘要 · Abstract (English)
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.
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