Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence

2026-06-15

AI Summary

This work investigates the principal variation structure of random probability measures on ℝ^m within the 2-Wasserstein space. To this end, it introduces a dynamical framework that interprets log-PCA as a variational method grounded in Wasserstein geometry, capturing local principal geodesic variations through the covariance operator of measures at their barycenter. The key innovation lies in the development of a differentiable Wasserstein Tangent PCA (WT-PCA), which leverages parallel transport to provide a dynamical interpretation of log-PCA. Furthermore, the paper establishes, for the first time, non-asymptotic statistical convergence rates for the empirical WT-PCA estimator relative to its population counterpart.

Abstract

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.

Problem

• Wasserstein geometry
• principal variations
• probability measures
• log-PCA
• statistical convergence

Innovation

• Wasserstein geometry
• log-PCA
• WT-PCA
• statistical convergence
• optimal transport