While many Machine Learning methods were developed or transposed on Riemannian manifolds to tackle data with known non Euclidean geometry, Optimal Transport (OT) methods on such spaces have not received much attention. The main OT tool on these spaces is the Wasserstein distance which suffers from a heavy computational burden. On Euclidean spaces, a popular alternative is the Sliced-Wasserstein distance, which leverages a closed-form solution of the Wasserstein distance in one dimension, but which is not readily available on manifolds. In this work, we derive general constructions of Sliced-Wasserstein distances on Cartan-Hadamard manifolds, Riemannian manifolds with non-positive curvature, which include among others Hyperbolic spaces or the space of Symmetric Positive Definite matrices. Then, we propose different applications. Additionally, we derive non-parametric schemes to minimize these new distances by approximating their Wasserstein gradient flows.

翻译：尽管许多机器学习方法已在黎曼流形上开发或移植以处理具有已知非欧几何结构的数据，但此类空间上的最优传输方法尚未受到广泛关注。这些空间上的主要最优传输工具是Wasserstein距离，但其计算负担沉重。在欧几里得空间中，一种流行的替代方案是切片Wasserstein距离，它利用了一维Wasserstein距离的闭式解，但在流形上无法直接应用。本研究在Cartan-Hadamard流形（具有非正曲率的黎曼流形，包括双曲空间或对称正定矩阵空间等）上推导了切片Wasserstein距离的通用构造方法。进而提出多种应用方案，并推导出通过近似其Wasserstein梯度流来最小化这些新距离的非参数化框架。