Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds

TL;DR

This paper introduces Sliced-Wasserstein distances on Cartan-Hadamard manifolds, enabling efficient optimal transport computations on non-Euclidean spaces like hyperbolic spaces and SPD matrices, with applications and gradient flow schemes.

Contribution

It develops the first general construction of Sliced-Wasserstein distances on Cartan-Hadamard manifolds, extending OT tools to non-positive curvature spaces.

Findings

Derived explicit formulas for Sliced-Wasserstein distances on Cartan-Hadamard manifolds.

Proposed non-parametric schemes for minimizing these distances via Wasserstein gradient flows.

Demonstrated applications in machine learning tasks involving non-Euclidean data.

Abstract

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.