Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing
This work addresses a geometric machine learning problem for researchers and practitioners in optimal transport and probability modeling, offering incremental advances by extending existing concepts to new specific cases.
The paper tackles the problem of computing Busemann functions in Wasserstein space, establishing closed-form expressions for one-dimensional distributions and Gaussian measures, which enable explicit projection schemes and novel Sliced-Wasserstein distances. It demonstrates efficiency on synthetic datasets and transfer learning problems, with results showing improved computational performance in these applications.
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.