Hyperbolic Optimal Transport

TL;DR

This work extends optimal transport (OT) to hyperbolic space by formulating a semi-discrete OT problem on $\mathbb{H}^m$ with cost $c(x,y)=\ln \cosh d_M(x,y)$ and establishing a geometric variational principle that connects OT on hyperbolic manifolds to the Minkowski problem via hyperbolic Legendre duality. It develops the theory of $\Gamma$-convex bodies, Fuchsian convexity, Gauss curvature measures, and a hyperbolic power diagram framework that yields a finite-dimensional optimization problem whose solution determines the hyperbolic OT map as a radial projection of a $\Gamma$-convex polyhedron. The authors prove existence and concavity properties of the Kantorovich functional in this setting and provide explicit Newton-based algorithms leveraging hyperbolic power diagrams and area-preserving parametrizations for genus $g>1$ surfaces. Experiments on synthetic data and multi-genus surfaces demonstrate fast convergence (exponential in some cases) and comparable scalability to Euclidean OT, enabling OT computations on complex hyperbolic domains without requiring GPUs. This advances OT methodology for hierarchical and networked data representations that naturally live in hyperbolic geometry and facilitates maps on multi-genus surfaces via area-preserving hyperbolic parametrizations.

Abstract

The optimal transport (OT) problem aims to find the most efficient mapping between two probability distributions under a given cost function, and has diverse applications in many fields such as machine learning, computer vision and computer graphics. However, existing methods for computing optimal transport maps are primarily developed for Euclidean spaces and the sphere. In this paper, we explore the problem of computing the optimal transport map in hyperbolic space, which naturally arises in contexts involving hierarchical data, networks, and multi-genus Riemann surfaces. We propose a novel and efficient algorithm for computing the optimal transport map in hyperbolic space using a geometric variational technique by extending methods for Euclidean and spherical geometry to the hyperbolic setting. We also perform experiments on synthetic data and multi-genus surface models to validate the efficacy of the proposed method.

Figures (21)

• Figure 1: Hyperboloid and Future cone
• Figure 2: Relating the Hyperboloid model and Poincaré disk
• Figure 3: $\Gamma$-convex polyhedron
• Figure 4: Hyperbolic Legendre Dual
• Figure 5: Convex Hull $\mathcal{C}$ and upper envelope $\mathcal{P}$

Theorems & Definitions (53)

• Definition 2.1
• Proposition 2.1: Proposition 1.11 santambrogio2015optimal
• Theorem 2.1: Theorem 1.39 santambrogio2015optimal
• Definition 4.1
• Proposition 4.1: fillastre2013fuchsian bertrand2014prescription
• Proposition 4.2: Proposition 2.15 bertrand2014prescription
• Definition 4.2
• Proposition 4.3: Lemma 2.19 bertrand2014prescription
• Proposition 4.4: Lemma 2.21 bertrand2014prescription
• Definition 4.3