Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections

Published: 18 Jun 2023, Last Modified: 26 Jun 2023TAGML2023 PosterEveryoneRevisions
Keywords: Optimal Transport, Sliced-Wasserstein, Hyperbolic spaces, Lorentz model, Poincaré ball
TL;DR: We compare different sliced discrepancies between distributions supported on hyperbolic spaces.
Abstract: Hyperbolic space embeddings have been shown beneficial for many learning tasks where data have an underlying hierarchical structure. Consequently, many machine learning tools were extended to such spaces, but only few discrepancies to compare probability distributions defined over those spaces exist. Among the possible candidates, optimal transport distances are well defined on such Riemannian manifolds and enjoy strong theoretical properties, but suffer from high computational cost. On Euclidean spaces, sliced-Wasserstein distances, which leverage a closed-form solution of the Wasserstein distance in one dimension, are more computationally efficient, but are not readily available on hyperbolic spaces. In this work, we propose to derive novel hyperbolic sliced-Wasserstein discrepancies. These constructions use projections on the underlying geodesics either along horospheres or geodesics. We study and compare them on different tasks where hyperbolic representations are relevant, such as sampling or image classification.
Supplementary Materials: zip
Type Of Submission: Proceedings Track (8 pages)
Submission Number: 19