Regularized estimation of Monge-Kantorovich quantiles for spherical data

TL;DR

The work tackles the challenge of defining and estimating quantiles for directional data on ${ m S}^2$ using OT, with a focus on enabling out-of-sample estimates. It introduces a regularized, entropic OT framework on the sphere, parameterizes Kantorovich potentials by spherical harmonics, and implements a stochastic gradient method to solve the continuous OT problem between the uniform sphere measure and a target distribution. A directional MK depth is developed alongside regularized quantile functions, and theoretical/empirical analyses establish smoothness, adaptivity, and depth properties consistent with directional data, while numerical experiments highlight improved stability, interpolation, and out-of-sample applicability. The approach yields smooth, data-adaptive quantile contours, scalable computations via FFT-based spherical harmonic transforms, and practical tools for depth-based inference and classification in directional settings.

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.

Figures (10)

• Figure 1: Mean squared error $\mathcal{R}_n(\widehat{\mathbf Q}_\varepsilon)$ as a function of the regularization parameter $\varepsilon \in [0,0.2]$. The horizontal dashed-line is the value of $\mathcal{R}_n(\widehat{\mathbf Q}_0)$.
• Figure 2: Empirical regularized quantile contours (blue dots), unregularized ones (orange triangles), and ground truth (dashed green line).
• Figure 3: Time for solving regularized and unregularized optimal transport between two data with $n$ instances, as a function of $n$.
• Figure 4: Empirical MK quantile contours on a mixture of vMF distributions
• Figure 5: Regularized quantile contours of levels $\{0.1,0.25,0.5,0.75,0.9 \}$ and associated sign curves, with $\epsilon= 0.1$.

Theorems & Definitions (23)

• Definition 3.1
• Definition 3.2
• Remark 3.1: Regularity
• Definition 3.3: Quantile contours, regions and signs
• Proposition 4.1
• Proposition 4.2
• Remark 4.1
• Definition 4.1
• Proposition 4.3
• Remark 4.2