Regularized estimation of Monge-Kantorovich quantiles for spherical data

TL;DR

This paper introduces a regularized optimal transport-based method for estimating quantiles of spherical data, enabling out-of-sample predictions and defining a new depth measure with favorable statistical properties.

Contribution

We extend entropic optimal transport to spherical data, propose a stochastic algorithm using spherical harmonics, and define a new directional Monge-Kantorovich depth.

Findings

The method effectively estimates spherical data quantiles.

The proposed depth satisfies key statistical axioms.

Applications demonstrate improved data analysis capabilities.

Abstract

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.