# On the Wasserstein Geodesic Principal Component Analysis of probability measures

arXiv: https://arxiv.org/abs/2506.04480 · HTML: https://mlsanity.com/paper/2506.04480/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 48 references that are in the map.
- First posted on arXiv 4 Jun 2025.

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

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric](https://mlsanity.com/paper/1506.07944/) (2015)
- [Inexact Riemannian Gradient Descent Method for Nonconvex Optimization](https://mlsanity.com/paper/2409.11181/) (2024)
- [Improved SQP and SLSQP Algorithms for Feasible Path-based Process Optimisation](https://mlsanity.com/paper/2402.10396/) (2024)

## Built on by (most cited first, 3 of 3)

- [Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing](https://mlsanity.com/paper/2510.04579/) (2025)
- [PCA of probability measures: Sparse and Dense sampling regimes](https://mlsanity.com/paper/2602.02190/) (2026)
- [Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence](https://mlsanity.com/paper/2606.17196/) (2026)

## Abstract (arXiv)

This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. We first address the case of a collection of Gaussian distributions, and show how to lift the computations in the space of invertible linear maps. For the more general setting of absolutely continuous probability measures, we leverage a novel approach to parameterizing geodesics in Wasserstein space with neural networks. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.