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

Clément Bonet, Elsa Cazelles, Lucas Drumetz, Nicolas Courty
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:1972-1980, 2026.

Abstract

The Busemann function has recently found many interests 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-bonet26a, title = { Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing }, author = {Bonet, Cl{\'e}ment and Cazelles, Elsa and Drumetz, Lucas and Courty, Nicolas}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {1972--1980}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/bonet26a/bonet26a.pdf}, url = {https://proceedings.mlr.press/v300/bonet26a.html}, abstract = { The Busemann function has recently found many interests 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. } }
Endnote
%0 Conference Paper %T Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing %A Clément Bonet %A Elsa Cazelles %A Lucas Drumetz %A Nicolas Courty %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-bonet26a %I PMLR %P 1972--1980 %U https://proceedings.mlr.press/v300/bonet26a.html %V 300 %X The Busemann function has recently found many interests 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.
APA
Bonet, C., Cazelles, E., Drumetz, L. & Courty, N.. (2026). Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:1972-1980 Available from https://proceedings.mlr.press/v300/bonet26a.html.

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