Hyperbolic Optimal Transport

TL;DR

This paper introduces a novel algorithm for computing optimal transport maps in hyperbolic space, addressing the lack of existing methods beyond Euclidean and spherical geometries, with applications in hierarchical data and complex networks.

Contribution

The paper extends optimal transport computation to hyperbolic geometry using a geometric variational approach, filling a gap in existing methods for non-Euclidean spaces.

Findings

Efficient algorithm for hyperbolic optimal transport developed

Validated on synthetic and multi-genus surface data

Demonstrates applicability to hierarchical and network data

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.