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arXiv · 2506.04480

On the Wasserstein Geodesic Principal Component Analysis of probability measures

Nina Vesseron·Elsa Cazelles·Alice Le Brigant·Thierry Klein
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Abstract

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.

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Nina Vesseron, Elsa Cazelles, Alice Le Brigant, Thierry Klein. 2025-06-04. On the Wasserstein Geodesic Principal Component Analysis of probability measures. https://arxiv.org/abs/2506.04480

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