Spherical Sliced-Wasserstein
This work addresses the need for computationally efficient Wasserstein distances on spherical data, which is incremental as it adapts an existing method to a new geometric setting.
The paper tackles the problem of extending the Sliced-Wasserstein distance to manifolds, specifically the sphere, by defining a novel spherical Sliced-Wasserstein discrepancy, enabling efficient computation and demonstrating its utility in machine learning applications like sampling, density estimation, and hyperspherical auto-encoders.
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. We focus more specifically on the sphere, for which we define a novel SW discrepancy, which we call spherical Sliced-Wasserstein, making a first step towards defining SW discrepancies on manifolds. Our construction is notably based on closed-form solutions of the Wasserstein distance on the circle, together with a new spherical Radon transform. Along with efficient algorithms and the corresponding implementations, we illustrate its properties in several machine learning use cases where spherical representations of data are at stake: sampling on the sphere, density estimation on real earth data or hyperspherical auto-encoders.