Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections

TL;DR

This paper introduces hyperbolic sliced-Wasserstein distances using geodesic and horospherical projections, enabling efficient comparison of probability distributions in hyperbolic spaces with applications in sampling and image classification.

Contribution

It proposes novel hyperbolic sliced-Wasserstein discrepancies based on projections along geodesics and horospheres, extending efficient Wasserstein methods to hyperbolic spaces.

Findings

Effective in tasks like sampling and image classification

Comparable or improved performance over existing methods

Provides computationally efficient hyperbolic distribution comparison

Abstract

It has been shown beneficial for many types of data which present an underlying hierarchical structure to be embedded in hyperbolic spaces. Consequently, many tools of machine learning 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 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.