  Tools from optimal transport (OT) theory have recently been used to define a
notion of quantile function for directional data. In practice, regularization
is mandatory for applications that require out-of-sample estimates. To this
end, we introduce a regularized estimator built from entropic optimal
transport, by extending the definition of the entropic map to the spherical
setting. We propose a stochastic algorithm to directly solve a continuous OT
problem between the uniform distribution and a target distribution, by
expanding Kantorovich potentials in the basis of spherical harmonics. In
addition, we define the directional Monge-Kantorovich depth, a companion
concept for OT-based quantiles. We show that it benefits from desirable
properties related to Liu-Zuo-Serfling axioms for the statistical analysis of
directional data. Building on our regularized estimators, we illustrate the
benefits of our methodology for data analysis.
