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

arXiv: https://arxiv.org/abs/2606.17196 · HTML: https://mlsanity.com/paper/2606.17196/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 33 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Statistical optimal transport](https://mlsanity.com/paper/2407.18163/) (2024)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric](https://mlsanity.com/paper/1506.07944/) (2015)
- [On the Wasserstein Geodesic Principal Component Analysis of probability measures](https://mlsanity.com/paper/2506.04480/) (2025)
- [Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization](https://mlsanity.com/paper/2501.14635/) (2025)
- [Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis](https://mlsanity.com/paper/2505.13660/) (2025)
- [Statistical Inference for Bures-Wasserstein Flows](https://mlsanity.com/paper/2310.13764/) (2023)
- [A Unified Approach for Computing Wasserstein Barycenters of Discrete and Continuous Measures](https://mlsanity.com/paper/2605.11270/) (2026)

## Abstract (arXiv)

This paper is concerned with learning principal variations of random probability measures on ℝm 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.