Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence
For researchers in optimal transport and shape analysis, this provides a theoretically grounded PCA method for probability measures with convergence guarantees, though it is an incremental extension of existing log-PCA.
This paper introduces a dynamical formulation of log-PCA for probability measures under Wasserstein geometry, termed Wasserstein Tangential PCA (WT-PCA), and derives a general statistical convergence rate of the empirical WT-PCA in terms of the 2-Wasserstein distance between population and empirical barycenter reference measures.
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.