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

Clément Bonet, Elsa Cazelles, Lucas Drumetz, Nicolas Courty

TL;DR

This work studies Busemann functions in the Wasserstein space $(\mathcal{P}_2(\mathbb{R}^d), W_2)$, addressing existence of geodesic rays and practical computation of Busemann functions. It derives closed-form Busemann expressions in 1D and Gaussian (Bures–Wasserstein) settings, enabling explicit projection schemes and the construction of new Sliced-Wasserstein distances for labeled data and Gaussian mixtures. The proposed SWBG and SWB1DG distances provide efficient, scalable surrogates for costly OTDD, and the authors demonstrate dataset-flow capabilities via Wasserstein-over-WoW gradient flows in rings and transfer-learning tasks. The results show strong correlation with OT-based distances and practical gains in transfer learning, highlighting the methodological and computational advantages of Busemann-based slicing in the Wasserstein space.

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.

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

TL;DR

This work studies Busemann functions in the Wasserstein space , addressing existence of geodesic rays and practical computation of Busemann functions. It derives closed-form Busemann expressions in 1D and Gaussian (Bures–Wasserstein) settings, enabling explicit projection schemes and the construction of new Sliced-Wasserstein distances for labeled data and Gaussian mixtures. The proposed SWBG and SWB1DG distances provide efficient, scalable surrogates for costly OTDD, and the authors demonstrate dataset-flow capabilities via Wasserstein-over-WoW gradient flows in rings and transfer-learning tasks. The results show strong correlation with OT-based distances and practical gains in transfer learning, highlighting the methodological and computational advantages of Busemann-based slicing in the Wasserstein space.

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 , 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.

Paper Structure

This paper contains 76 sections, 18 theorems, 123 equations, 13 figures, 4 tables.

Table of Contents

  1. INTRODUCTION
  2. WASSERSTEIN SPACE
  3. Wasserstein Distance
  4. Riemannian Structure.
  5. Riemannian Structure.
  6. Geodesic Rays on $(\mathcal{P}_2(\mathbb{R}^d),\mathrm{W}_2)$
  7. BUSEMANN FUNCTION
  8. Background on the Busemann Function
  9. Busemann on the Wasserstein Space
  10. Closed-forms of the Busemann Function
  11. SLICING DATASETS
  12. Sliced-Wasserstein Distance
  13. Comparing Datasets
  14. Slicing Datasets with Busemann
  15. Gaussian Approximation.
  16. ...and 61 more sections

Key Result

Proposition 1

Let $\mu_0\in\mathcal{P}_{2,\mathrm{ac}}(\mathbb{R}^d),\ \mu_1\in\mathcal{P}_2(\mathbb{R}^d)$, and $\mathrm{T}$ the Monge map between $\mu_0$ and $\mu_1$. The curve $t\mapsto \mu_t = ((1-t)\mathrm{Id} + t \mathrm{T})_\#\mu_0$ is a geodesic ray if and only if $\mathrm{T}$ is the gradient of a 1-conve

Figures (13)

  • Figure 1: Spearman ($\rho_S)$ and Pearson ($\rho_P$) correlation between $\mathrm{SOTDD}$, $\mathrm{SWB1DG}$, $\mathrm{SWBG}$ and OTDD between subdatasets of CIFAR10.
  • Figure 2: Evolution of the WoW gradient flow of SWBG and SOTDD with the three rings dataset as target.
  • Figure 3: Convergence of the flow towards the 3 rings, averaged over 100 random batches of the target.
  • Figure 4: Projections of centered 1D Gaussian $\mathcal{N}(0,\sigma_i^2)$ (blue points) on the geodesic $\mathcal{N}(0,\sigma_t)$ starting at $\sigma_0=1$ and passing through $\sigma_1=\frac{1}{2}$ (Left) and $\sigma_1=\frac{3}{2}$ (Right).
  • Figure 5: Projections of 1D Gaussian $\mathcal{N}(m_i,\sigma_i^2)$ (blue points) on the geodesic $\mathcal{N}(m_t,\sigma_t)$ starting at $\sigma_0=1$, $m_0=0$ and passing through $\sigma_1=\frac{1}{2}$ (Left) and $\sigma_1=\frac{3}{2}$ (Right) with $m_1=-\sqrt{1-(\sigma_1-\sigma_0)^2}$.
  • ...and 8 more figures

Theorems & Definitions (37)

  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Proposition 3
  • Corollary 4
  • Corollary 5
  • Proposition 4
  • Corollary 6
  • ...and 27 more