# Regularized estimation of Monge-Kantorovich quantiles for spherical data

arXiv: https://arxiv.org/abs/2407.02085 · HTML: https://mlsanity.com/paper/2407.02085/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2024.

## Built on by (most cited first, 1 of 1)

- [ANGLE: Angular Neural Generative Learning via Engression](https://mlsanity.com/paper/2607.12833/) (2026)

## Abstract (arXiv)

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.