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

• URL: http://arxiv.org/abs/2606.17196v1
• Date: Mon, 15 Jun 2026 18:36:20 GMT
• Title: Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence
• Abstract summary: We introduce a new dynamical formulation to interpret the log-PCA, a linearized principal geodesic analysis, as a variational approach.<n>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.<n>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.
• Score: 17.344690125291258
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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