Regularized estimation of Monge-Kantorovich quantiles for spherical data

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.

💡 Research Summary

This paper addresses the challenge of defining and estimating quantile functions for directional data that lie on the unit sphere, by leveraging the Monge‑Kantorovich (MK) framework from optimal transport (OT) theory and introducing entropic regularization. Classical quantiles rely on a natural ordering of the real line, which is absent on the sphere, making the construction of meaningful quantile maps non‑trivial. Recent work has proposed MK quantiles based on solving a discrete OT problem between an empirical distribution of the data and a uniform grid on the sphere. While this yields a map that pushes forward the uniform measure to the target distribution, it is piecewise constant and cannot provide out‑of‑sample predictions, a serious limitation for applications such as depth‑based classification or volume estimation of quantile regions.

The authors overcome this limitation by extending the entropic optimal transport (EOT) map—originally defined in Euclidean spaces—to the spherical setting. They first formalize the continuous OT problem on the 2‑sphere S² with the quadratic geodesic cost c(x,y)=½ d²(x,y), where d is the geodesic distance. Assuming the reference measure μ is the uniform surface measure (which belongs to the class B₂ of densities bounded away from zero and infinity) and the target ν has a smooth density, they prove existence and uniqueness of a c‑convex potential ψ and its c‑transform ψᶜ. The MK quantile function is then expressed as Q(x)=Expₓ(−∇ψ(x)), while the associated distribution function is F(x)=Expₓ(−∇ψᶜ(x)). This formulation guarantees that Q pushes forward μ to ν and that F is the inverse of Q almost everywhere.

To make the estimation computationally tractable, the paper expands ψ in the orthonormal basis of spherical harmonics Y_{ℓm}. This yields a set of Fourier‑type coefficients \bar ψ_{ℓm} that fully characterize the potential. The entropic regularization introduces a smoothing parameter ε>0, turning the original OT problem into a strictly convex one whose dual variables are precisely these coefficients. By employing a fast spherical Fourier transform (FFT) on a grid of size O(p²) (p being the maximal harmonic degree), each stochastic gradient step can be performed in O(p² log p) time, a dramatic improvement over the O(n³) or O(n²) complexities of classical discrete OT solvers.

The authors propose a stochastic algorithm that processes data points X₁,…,X_n sequentially. At each iteration, a randomly drawn sample x_i is used to compute stochastic estimates of the gradient of the entropic dual objective with respect to the harmonic coefficients. An adaptive optimizer (e.g., Adam) updates the coefficients, and the updated potential ψ yields an updated quantile map Q. Because the algorithm works directly on the continuous problem, it does not require discretizing the uniform reference distribution; instead, the uniform measure is implicitly represented by the spherical harmonic basis. This online nature makes the method suitable for streaming data and for scenarios where the sample size is large.

Beyond quantile estimation, the paper introduces a directional MK depth. Choosing a central point—specifically the Fréchet median θ_M of the target distribution—the authors define nested spherical caps C_τ centered at Q(θ_M) with μ‑probability τ. The depth D(x) is defined as the τ for which x belongs to C_τ. The authors verify that this depth satisfies the Liu‑Zuo‑Serfling axioms (affine invariance, maximality at the center, monotonicity relative to the deepest point, etc.) adapted to the spherical geometry. Consequently, the depth can be used for robust classification, outlier detection, and visualization of directional data.

Extensive experiments are presented. Synthetic data with asymmetric and multimodal patterns on S² demonstrate that the regularized MK quantile regions accurately capture the underlying structure, whereas Mahalanobis‑based or spatial quantiles either assume rotational symmetry or produce overly smooth contours. Real‑world astronomical data (galaxy positions on the celestial sphere) are analyzed to illustrate depth‑based classification; the MK depth achieves higher area‑under‑curve (AUC) scores than competing depths. Computational benchmarks show that with p=64 (i.e., harmonics up to degree 64) the stochastic algorithm runs in roughly 0.12 seconds per iteration, compared to several seconds for a comparable discrete OT solver.

In summary, the paper makes three major contributions: (1) a rigorous extension of entropic OT maps to the sphere, providing smooth, invertible quantile functions; (2) a scalable stochastic algorithm that leverages spherical harmonics and fast FFT for online estimation; and (3) the definition and validation of a new MK depth that respects spherical geometry and statistical depth axioms. The work opens avenues for further research on higher‑dimensional spheres, other Riemannian manifolds, and deep‑learning‑based parameterizations of OT potentials.