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

2025-10-06

AI Summary

This work addresses the existence and explicit construction of Busemann functions on the Wasserstein space. Focusing on two fundamental cases—univariate distributions and Gaussian measures—we derive, for the first time, closed-form expressions for their Busemann functions. Leveraging the Riemannian geometry induced by optimal transport, we explicitly compute Busemann projections of probability measures along geodesic rays. Based on this, we propose the Busemann-sliced Wasserstein distance (BSWD), which enriches geometric modeling in distribution spaces. Our approach integrates Wasserstein geometry, geodesic analysis, and Busemann theory. We validate its effectiveness on Gaussian mixture models and labeled datasets, and demonstrate—on synthetic data and transfer learning tasks—that BSWD outperforms the conventional sliced Wasserstein distance in computational efficiency, interpretability, and discriminative power.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Existence and computation of Busemann functions in Wasserstein space
• Establish closed-form expressions for one-dimensional and Gaussian distributions
• Develop novel Sliced-Wasserstein distances for Gaussian mixtures and datasets

Innovation

• Establishes closed-form Busemann functions for distributions
• Enables explicit projection schemes for probability distributions
• Defines novel Sliced-Wasserstein distances for Gaussian mixtures