Regularized estimation of Monge-Kantorovich quantiles for spherical data

• Institut de Mathématiques de Bordeaux et CNRS [bernard.bercu@math.u-bordeaux.fr]

• Institut de Mathématiques de Bordeaux et CNRS [jeremie.bigot@math.u-bordeaux.fr]

• Institut de Mathématiques de Bordeaux et CNRS [gauthier-louis.thurin@math.u-bordeaux.fr]

Keywords: Statistical depth – Spherical data – Optimal transport

Submission for the invited session “Recent advances in statistical depth”.

1 Spherical quantiles

In various situations, data naturally correspond to directions that are modeled as observations belonging to the circle or the unit $d$-sphere ${𝕊}^{d-1}$ for $d\ge 2$. In the present work, we focus on the concepts of quantiles and depth for spherical data, for which a recent definition leverages ideas from measure transportation [Hallin et al., 2022]. The building block is the optimal transport problem between a target distribution and the uniform probability distribution on the sphere. This spherical extension follows the Euclidean definitions from Chernozhukov et al. [2017], which have given rise to a fast-growing field [Hallin, 2023].

Depending on the context, different estimators show different benefits. In the directional setting of Hallin et al. [2022], it is advocated to solve an optimal matching between two discrete distributions, ensuring finite-sample distribution-freeness of MK ranks, crucial for statistical testing. However, regularization is mandatory for applications that require out-of-sample estimates, $i.e.$ when one is willing that the estimators interpolate between observations, as for depth-based contours.

To this end, we propose a new algorithm for computing regularized spherical quantiles, and we illustrate the benefits of our estimator for data analysis.

2 Regularized estimation

The estimation of the MK quantile function for a probability distribution $\nu$ requires to solve an OT problem between $\nu$ and a reference measure ${\mu }_{{𝕊}^2}$, the uniform probability on ${𝕊}^2$. In the field of computational optimal transport [Cuturi and Peyré, 2019] it is well-known that the OT problem can be regularized by entropy (EOT) for faster algorithms. Consequently, we target EOT, both for smoothing and computational purposes. For $\epsilon >0$ a regularization parameter, EOT between ${\mu }_{{𝕊}^2}$ and $\nu$ writes as

$$\underset{u\in L^{\mathrm{\infty }}({𝕊}^2)}{\max }{\int }_{{𝕊}^2}u(x)𝑑{\mu }_{{𝕊}^2}(x)+{\int }_{{𝕊}^2}{u}^{c,\epsilon }(y)𝑑\nu (y),$$ (1)

with $u^{c,\epsilon }$ the smooth conjugate of $u$ defined by

$$u^{c,\epsilon }(y)=-\epsilon \log \left({\int }_{{𝕊}^2}\exp \left(\frac{u(x)-c(x,y)}{\epsilon }\right)𝑑{\mu }_{{𝕊}^2}(x)\right).$$ (2)

Our proposal is to parameterize $u$ in (1) by its spherical harmonics, to perform stochastic optimization on spherical harmonic coefficients. Based on the obtained estimator ${\widehat{𝐮}}_{\epsilon ,n}$, we derive a regularized estimator for the MK quantile function ${𝐐}_{\epsilon }$. This yields, even empirically, smooth maps that are not constrained to belong to the set of observed data.

In addition, we define the directional MK depth, a companion concept for MK quantiles, following Euclidean definitions [Chernozhukov et al., 2017]. 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 inference, from descriptive analysis to depth-based classification.