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Regularized estimation of Monge-Kantorovich quantiles for spherical data

This paper introduces a regularized estimator for Monge-Kantorovich quantiles and a corresponding depth measure for spherical data using entropic optimal transport and spherical harmonics, demonstrating their statistical validity and practical utility through a novel stochastic algorithm.

Original authors: Bernard Bercu, Jérémie Bigot, Gauthier Thurin

Published 2026-02-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bernard Bercu, Jérémie Bigot, Gauthier Thurin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Mapping Directions on a Globe

Imagine you have a giant, transparent globe representing the Earth. Instead of plotting cities, you are plotting directions—like the way a bird flies, the direction a wildfire spreads, or the orientation of a gene. In statistics, this is called "directional data."

The problem the authors tackle is: How do you find the "middle" or the "outliers" on a sphere?

On a flat piece of paper (like a line graph), finding the middle is easy. You just sort numbers from smallest to largest. But on a sphere, there is no "smallest" or "largest" direction. You can't say North is "bigger" than East. This makes it very hard to create a statistical map that tells you where the data is crowded and where it is sparse.

The Old Way: The "Pixelated" Map

The paper mentions a recent method that tries to solve this by creating a "matching" between a perfect, empty grid of points on the sphere and your actual messy data points.

Think of this like a pixelated video game. You have a grid of squares (the empty sphere) and a set of colored dots (your data). The computer draws a line from every grid square to the nearest data dot.

The Problem: This map is "blocky." If you pick a spot on the map that isn't exactly one of your original data dots, the map doesn't know what to do. It can't tell you if a new bird flying in a slightly different direction is "normal" or "weird." It's like trying to guess the terrain between two pixels; you just see a jagged edge.

The New Solution: The "Smooth" Map

The authors propose a new way to build this map using a technique called Entropic Optimal Transport.

The Analogy: The Foggy Lens
Imagine looking at your data through a slightly foggy lens. Instead of seeing sharp, jagged edges between data points, the "fog" (mathematically called regularization) blurs the lines just enough to create a smooth, continuous surface.

Why this helps: Now, if a new bird flies in a direction you've never seen before, the smooth map can still tell you exactly where it fits. Is it near the center? Is it on the edge? The map gives a definite answer for any direction, not just the ones you already measured.

How They Did It: The "Musical" Algorithm

To build this smooth map, the authors had to solve a very complex math puzzle. They couldn't just use a standard computer grid because the surface of a sphere is curved.

The Analogy: Tuning a Spherical Drum
They used a mathematical tool called Spherical Harmonics. Think of the surface of the sphere as the skin of a giant drum.

When you hit a drum, it vibrates in specific patterns (fundamental tones, overtones, etc.).
The authors treated their data problem like a drum. They broke the complex shape of their data down into these "vibrational patterns" (like musical notes).
They built a stochastic algorithm (a step-by-step guessing game that gets better with every try) to tune these "notes."
The Result: Instead of a jagged pixelated map, they got a smooth, continuous function that describes the entire sphere. It's like turning a blocky 8-bit video game into a high-definition 3D movie.

The "Depth" Meter: Finding the Outliers

Once they have this smooth map, they introduce a new concept called Monge-Kantorovich Depth.

The Analogy: The Bullseye
Imagine the data points are darts thrown at a spherical target.

The "deepest" point is the bullseye (the center of the data).
The "shallowest" points are the darts that landed near the edge.
Their new method creates a smooth "depth meter." If you hold a new dart (a new data point) up to the sphere, the meter instantly tells you: "This is 90% deep (very normal)" or "This is 10% deep (very weird)."

Why This Matters (According to the Paper)

It handles new data: Unlike the old "pixelated" method, this new smooth map can make predictions for directions you haven't seen yet.
It's faster: By using the "musical notes" (Spherical Harmonics) and a fast computer trick called the Fast Fourier Transform, they can solve these problems much quicker than older methods.
It's flexible: It works for any shape of data on a sphere, not just perfectly round or symmetrical ones.

Summary

The authors took a difficult problem (finding order in directions on a globe) and solved it by smoothing out the rough edges of previous methods. They turned a jagged, blocky map into a smooth, continuous surface using a musical-like mathematical approach, allowing statisticians to analyze and predict directional data with much greater precision.

Technical Summary: Regularized Estimation of Monge-Kantorovich Quantiles for Spherical Data

Problem Statement
Directional data, observed as points on the unit $d$ -sphere $S^{d-1}$ , lacks a canonical ordering, making the definition of statistical quantiles more complex than in Euclidean space. Recent nonparametric approaches utilize Optimal Transport (OT) theory to define Monge-Kantorovich (MK) quantiles. Specifically, for directional data, an empirical quantile function $Q_n$ is constructed by solving an OT problem between a uniform distribution on the sphere and the empirical measure of the data.

However, existing OT-based estimators for directional data suffer from two primary limitations:

Lack of Out-of-Sample Capability: The empirical matching between two discrete distributions (the data and a regular grid) yields a piecewise constant function restricted to the set of observations. It cannot provide estimates for new, unseen data points ( $X_{new}$ ), which is essential for applications like depth-based classification or dispersion measurement.
Computational Complexity: Solving discrete OT problems typically scales cubically with the number of observations, hindering scalability.

Methodology
The authors propose a framework for regularized estimation of directional MK quantiles using Entropic Optimal Transport (EOT) on the 2-sphere ( $S^2$ ). The methodology proceeds as follows:

Entropic Regularization: Instead of solving the discrete Monge problem, the authors formulate a continuous OT problem between the uniform measure $\mu_{S^2}$ and the target measure $\nu$ . They introduce an entropic regularization term with parameter $\varepsilon > 0$ . This transforms the problem into a semi-dual optimization over a single potential function $u$ , defined as:
$\max_{u \in L^\infty(S^2)} \int_{S^2} u(x) d\mu_{S^2}(x) + \int_{S^2} u^{c,\varepsilon}(y) d\nu(y)$
where $u^{c,\varepsilon}$ is the smooth conjugate of $u$ .
Spectral Parameterization: To solve this infinite-dimensional problem, the dual potential $u$ is parameterized using spherical harmonics. The function is expanded as a series of coefficients $\bar{u} = (\bar{u}^m_l)$ . The authors prove that for the regularized case, the potential is twice continuously differentiable, ensuring the spherical harmonic series belongs to $\ell^1$ .
Stochastic Algorithm: A stochastic gradient descent algorithm is developed to optimize the spherical harmonic coefficients.
- Input: A sequence of i.i.d. observations $X_1, \dots, X_n$ from $\nu$ .
- Mechanism: At each iteration, the algorithm updates the coefficients using the gradient of the objective function with respect to a single sample.
- Efficiency: The algorithm leverages the Fast Fourier Transform (FFT) on the sphere (via the pyshtools library) to compute the necessary integrals and transforms. The computational cost per iteration is $O(p^2 \log^2 p)$ , where $p$ is the grid size, significantly improving upon the cubic complexity of discrete OT solvers.
- Online Learning: The algorithm operates in an online fashion, processing data sequentially without requiring the storage of a full $n \times n$ cost matrix.
Regularized Quantile and Distribution Functions:
- Once the optimal potential $u_\varepsilon$ is estimated, the regularized quantile function $Q_\varepsilon$ and distribution function $F_\varepsilon$ are derived.
- Unlike the unregularized case, these functions are smooth and defined via closed-form expressions involving weighted averages in the tangent space (barycentric projections).
- Specifically, $Q_\varepsilon(x) = \text{Exp}_x \left( \int \text{Log}_x(z) \exp(\dots) d\nu(z) \right)$ , allowing for direct evaluation at any point $x \in S^2$ .
Directional Statistical Depth: The paper defines a Directional MK Depth based on the proximity of the distribution function $F(x)$ to the Fréchet median. This depth measure satisfies adapted versions of the Liu-Zuo-Serfling axioms, including rotational invariance and monotonicity along sign curves.

Key Contributions

Extension of Entropic Maps: The authors extend the concept of the entropic map (previously defined for Euclidean spaces) to the non-Euclidean setting of the sphere, providing a smooth, regularized estimator for MK quantiles.
Out-of-Sample Estimation: The proposed regularized quantile function $Q_\varepsilon$ allows for the estimation of quantiles for new observations, a capability missing in previous discrete OT-based methods.
Stochastic Spectral Algorithm: A novel stochastic algorithm is introduced that solves continuous OT problems on $S^2$ by expanding potentials in spherical harmonics, utilizing FFT for computational efficiency.
Directional Depth Definition: A new notion of MK directional statistical depth is introduced, which inherits desirable properties from Euclidean MK depth and satisfies axioms relevant to directional statistics.

Results and Claims

Theoretical Properties: The paper establishes that the regularized potentials are twice continuously differentiable, ensuring the resulting quantile and distribution functions are smooth. It demonstrates that as $\varepsilon \to 0$ , the regularized potentials converge uniformly to the unregularized Kantorovich potentials.
Computational Efficiency: The stochastic algorithm avoids the $O(n^3)$ complexity of discrete OT, reducing the per-iteration cost to $O(p^2 \log^2 p)$ , making it suitable for large datasets and online learning scenarios.
Statistical Utility: The methodology enables depth-based classification and dispersion measurements by providing continuous, differentiable mappings. The paper illustrates these benefits through numerical experiments (detailed in Section 6 of the full paper), showing that the regularized approach provides smoother contours and better out-of-sample performance compared to unregularized discrete estimators.

Significance
The paper claims that its primary significance lies in bridging the gap between theoretical OT-based quantiles and practical statistical applications for directional data. By introducing entropic regularization and a spectral stochastic solver, the authors provide a computationally tractable method that overcomes the "out-of-sample" limitation of previous discrete approaches. This enables the application of MK quantiles and depth measures to real-world problems involving directional data, such as classification and risk measurement, where smoothness and the ability to evaluate new data points are mandatory. The work also contributes to the broader field of OT on manifolds by providing an explicit, tractable formulation for the sphere.