Appears in collection : 2022 - T3 - WS3 - Measure-theoretic Approaches and Optimal Transportation in Statistics

Many variants of the 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.