Communication Dans Un Congrès Année : 2026

Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing

Résumé

The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several sources of data can be conveniently modeled as probability distributions, it is natural to study this function in the Wasserstein space, which carries a rich formal Riemannian structure induced by Optimal Transport metrics. In this work, we investigate the existence and computation of Busemann functions in Wasserstein space, which admits geodesic rays. We establish closed-form expressions in two important cases: one-dimensional distributions and Gaussian measures. These results enable explicit projection schemes for probability distributions on $\mathbb{R}$, which in turn allow us to define novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets. We demonstrate the efficiency of those original schemes on synthetic datasets as well as transfer learning problems.

Dates et versions

hal-05307431 , version 1 (10-10-2025)

Identifiants

Citer

Clément Bonet, Elsa Cazelles, Lucas Drumetz, Nicolas Courty. Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing. In the 29th International Conference on Artificial Intelligence and Statistics (AISTATS), May 2026, Tangier, Morocco. ⟨10.48550/arXiv.2510.04579⟩. ⟨hal-05307431⟩
242 Consultations
0 Téléchargements

Altmetric

Partager

  • More