Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds
This work addresses the problem of efficient optimal transport on non-Euclidean geometries for researchers in machine learning and data analysis, representing a novel extension rather than an incremental improvement.
The paper tackles the computational burden of Wasserstein distances on Riemannian manifolds by deriving Sliced-Wasserstein distances on Cartan-Hadamard manifolds, such as Hyperbolic spaces, and proposes applications and gradient flow minimization schemes.
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.