Sliced Wasserstein on Manifolds: Spherical and Hyperbolical cases

Optimal transport has received a lot of attention into the machine learning and computational geometry communities recently. Many variants of the associated Wasserstein distance have been introduced to reduce its original computational burden. In particular the Sliced-Wasserstein distance (SW), which leverages one-dimensional projections for which a closed-form solution of the Wasserstein distance is available, has received a lot of interest. Yet, it is restricted to data living in Euclidean spaces, while the Wasserstein distance has been studied and used recently on manifolds. In this talk I will discuss novel methodologies to transpose SW to the Riemannian manifold case. By appropriately choosing a proper Radon transform, we show how fast and differentiable algorithms can be designed in two cases: Spherical and Hyperbolic manifolds. After discussing some of the theoretical properties of those novel discrepancies, I will showcase applications in machine learning problems, where data naturally live on those spaces.