Spherical Sliced-Wasserstein
- URL: http://arxiv.org/abs/2206.08780v1
- Date: Fri, 17 Jun 2022 13:48:50 GMT
- Title: Spherical Sliced-Wasserstein
- Authors: Cl\'ement Bonet, Paul Berg, Nicolas Courty, Fran\c{c}ois Septier,
Lucas Drumetz, Minh-Tan Pham
- Abstract summary: Sliced-Wasserstein distance (SW) is restricted to data living in Euclidean spaces.
We focus more specifically on the sphere, for which we define a novel SW discrepancy, which we call spherical Sliced-Wasserstein.
Our construction is notably based on closed-form solutions of the Wasserstein distance on the circle, together with a new spherical Radon transform.
- Score: 14.98994743486746
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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:
density estimation on the sphere, variational inference or hyperspherical
auto-encoders.
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