将最优传输算法拓展到双曲空间，更好处理层次化数据
Hyperbolic Optimal Transport
- 用几何变分法将欧氏与球面方法推广至双曲空间
- 在合成数据和多亏格曲面模型上验证了算法有效性
- 适合处理树状结构、网络等具有层次特性的数据
最优传输（OT）问题旨在给定代价函数下找到两个概率分布间的最高效映射，在机器学习、计算机视觉和计算机图形学等领域有广泛应用。然而，现有最优传输计算方法主要针对欧几里得空间和球面。本文探索在双曲空间中计算最优传输映射的问题，该空间天然适用于层次化数据、网络及多亏格黎曼曲面等场景。我们提出一种新颖且高效的算法，通过将欧氏与球面几何方法拓展至双曲设定，采用几何变分技术实现。我们在合成数据和多亏格曲面模型上进行了实验，验证了所提方法的有效性。
原文摘要 · Abstract (English)
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.
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