Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds
Clément Bonet Lucas Drumetz Nicolas Courty
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.