The GdT OT-PDE-ML is a research seminar organized by the ParMA team. It brings together researchers to discuss recent theoretical and numerical developments at the crossroads of Optimal Transport theory, Partial Differential Equations and Machine Learning.

Abstract: In this talk we discuss how to perform Principal Component Analysis (PCA) of a dataset whose elements are probability distributions, compared using the Wasserstein distance. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. To achieve this, we leverage the Riemannian interpretation of the Wasserstein metric. We first address the case of a collection of centered Gaussian distributions, and show how to lift the computations in the space of invertible linear maps using Bures-Wasserstein geometry. For the more general setting of absolutely continuous probability measures, we use Otto-Wasserstein geometry and neural networks to parameterize geodesics in Wasserstein space. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.