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.

翻译：最优传输理论中的工具最近被用于定义方向性数据的分位数函数。在实际应用中，对于需要样本外估计的场景，正则化是必不可少的。为此，我们通过将熵最优传输映射的定义扩展至球面情形，构建了一种基于熵最优传输的正则化估计量。我们提出了一种随机算法，通过将Kantorovich势在球谐函数基中展开，直接求解均匀分布与目标分布之间的连续最优传输问题。此外，我们定义了方向性Monge-Kantorovich深度，作为基于最优传输的分位数的一个伴随概念。我们证明该深度具备Liu-Zuo-Serfling公理体系所期望的优良性质，适用于方向性数据的统计分析。基于所提出的正则化估计量，我们通过实例展示了该方法在数据分析中的优势。