Regularized estimation of Monge-Kantorovich quantiles for spherical data
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| Format: | Preprint |
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2024
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| author | Bercu, Bernard Bigot, Jérémie Thurin, Gauthier |
| author_facet | Bercu, Bernard Bigot, Jérémie Thurin, Gauthier |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_02085 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularized estimation of Monge-Kantorovich quantiles for spherical data Bercu, Bernard Bigot, Jérémie Thurin, Gauthier Methodology Computation 62H11, 62G30, 62L20 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. |
| title | Regularized estimation of Monge-Kantorovich quantiles for spherical data |
| topic | Methodology Computation 62H11, 62G30, 62L20 |
| url | https://arxiv.org/abs/2407.02085 |