  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.
