球面数据的正则化Monge-Kantorovich量估计
Regularized estimation of Monge-Kantorovich quantiles for spherical data
AI总结：
本文提出了一种基于熵最优传输的正则化方法，用于球面数据的Monge-Kantorovich量估计，并引入了方向Monge-Kantorovich深度以提升方向数据的统计分析能力。
摘要：
最优传输（OT）理论的工具最近被用于定义方向数据的量函数概念。在实践中，对于需要离样估计的应用，正则化是必不可少的。为此，我们引入了一个基于熵最优传输的正则化估计器，通过将熵映射的定义扩展到球面设置来构建。我们提出了一种随机算法，通过将Kantorovich势展开为球面谐波基来直接求解均匀分布与目标分布之间的连续OT问题。此外，我们定义了方向Monge-Kantorovich深度，这是基于OT的量的互补概念。我们展示了它具有与Liu-Zuo-Serfling公理相关的有利性质，这些公理与方向数据的统计分析有关。基于我们的正则化估计器，我们展示了我们的方法在数据分析中的优势。
英文摘要：
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.
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