# Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric

arXiv: https://arxiv.org/abs/1506.07944 · HTML: https://mlsanity.com/paper/1506.07944/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jun 2015.

## Built on by (most cited first, 15 of 15)

- [Curriculum Reinforcement Learning using Optimal Transport via Gradual Domain Adaptation](https://mlsanity.com/paper/2210.10195/) (2022)
- [Wasserstein K-means for clustering probability distributions](https://mlsanity.com/paper/2209.06975/) (2022)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023)
- [Learning to Generate Wasserstein Barycenters](https://mlsanity.com/paper/2102.12178/) (2021)
- [Robust Sampling for Active Statistical Inference](https://mlsanity.com/paper/2511.08991/) (2025)
- [Group-blind optimal transport to group parity and its constrained variants](https://mlsanity.com/paper/2310.11407/) (2023)
- [Geometric Sparse Coding in Wasserstein Space](https://mlsanity.com/paper/2210.12135/) (2022)
- [On the Wasserstein Geodesic Principal Component Analysis of probability measures](https://mlsanity.com/paper/2506.04480/) (2025)
- [Manifold learning in metric spaces](https://mlsanity.com/paper/2503.16187/) (2025)
- [Scalable Learning from Probability Measures with Mean Measure Quantization](https://mlsanity.com/paper/2502.04907/) (2025)
- [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)

Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family. We propose to study this problem under the optimal transport (Wasserstein) geometry, using curves that are restricted to be geodesic segments under that metric. We show that concepts that play a key role in Euclidean PCA, such as data centering or orthogonality of principal directions, find a natural equivalent in the optimal transport geometry, using Wasserstein means and differential geometry. The implementation of these ideas is, however, computationally challenging. To achieve scalable algorithms that can handle thousands of measures, we propose to use a relaxed definition for geodesics and regularized optimal transport distances. The interest of our approach is demonstrated on images seen either as shapes or color histograms.