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

A new statistical framework for analyzing random probability distributions under the Wasserstein geometry has been posted to arXiv on 15 Jun 2026. The work introduces Wasserstein Tangential PCA, a differentiable reformulation of log-PCA, and provides the first general convergence rate for the empirical method [1]. The paper interprets the existing log-PCA algorithm—a linearized principal geodesic analysis for probability data—from a dynamical and variational perspective, connecting it to a spectral method without imposing a tangential constraint [3]. When applied to a finite set of input probability distributions, the resulting Wasserstein Tangential PCA (WT-PCA) is expressed as an unconstrained minimization of the Wasserstein covariance operator and is practically equivalent to the log-PCA algorithm [3]. The formulation captures local principal modes of geodesic variation around a barycenter reference measure via its covariance operator [1]. Standard principal component analysis identifies directions of maximum variance in Euclidean data by constructing an orthonormal basis of uncorrelated components [6]. Extending such dimensionality reduction to the space of probability measures requires a geometry that respects the structure of distributions. The Wasserstein geometry provides that structure, and the new work leverages first-order Otto calculus together with second-order parallel transport from optimal transport theory to analyze statistical behavior [3]. The authors derive a general convergence rate for empirical WT-PCA that depends on the 2-Wasserstein distance between the population barycenter and the empirical barycenter estimated from data [1]. For the specific case of n independent Gaussian inputs with the Bures-Wasserstein barycenter as the reference measure, the eigenvalues and eigenfunctions of the aligned Wasserstein covariance operators converge at a rate of O_p(n^{-1/4}) [4]. The proof uses parallel transport to map estimated and true eigenfunctions into a common space, making them directly comparable [4]. While diffusion models—which learn to reverse a noising process and are widely used in image and video generation—operate on a different generative principle, they share a conceptual link in modeling distributions through learned transformations [7]. The WT-PCA framework instead focuses on identifying principal geodesic modes of variation within a population of probability measures, offering a complementary tool for distributional data analysis [1].