Center-Outward Q-Dominance

• Center-outward Q-dominance is an optimal transport-based order relation that generalizes univariate stochastic dominance via unique center-outward quantile functions.
• It leverages cyclically monotone gradients of convex potentials to compare quantile regions, establishing strong conditions for stochastic dominance.
• Applications span multivariate risk measurement, statistical inference, and stochastic multi-objective optimization, providing a coherent alternative to ad-hoc methods.

Center-outward Q-dominance is a canonical, optimal-transport-based order relation for multivariate distributions, serving as a principled multivariate generalization of univariate first-order stochastic dominance via quantile functions. It leverages the uniqueness and cyclical monotonicity properties of the center-outward distribution and quantile functions, as constructed by the gradient of a convex potential pushing a reference spherical-uniform measure on the unit ball to the distribution of interest. This order is central to recent advances in multivariate risk measurement, statistical inference, and stochastic multi-objective optimization, replacing ad-hoc coordinatewise dominance and scalarization approaches with a globally coherent geometric structure (Beirlant et al., 2019, Barrio et al., 2018, Laag et al., 16 Nov 2025).

1. Center-Outward Quantile Functions: Construction and Properties

Let $P$ be a Borel probability measure on $\mathbb{R}^d$ absolutely continuous with respect to Lebesgue measure. The center-outward quantile function $Q^\pm$ is defined as the unique (almost everywhere) gradient-of-convex-function map $\nabla \phi^*$ transporting the reference measure $U_d$ (the spherical-uniform law on the closed Euclidean unit ball $\overline{S}_d$) to $P$: $$Q^\pm : \overline{S}_d \rightarrow \mathbb{R}^d, \quad Q^\pm_\# U_d = P.$$ The center-outward distribution function $F^\pm$ is its inverse (again a.e.): $$F^\pm : \mathbb{R}^d \rightarrow \overline{S}_d, \quad F^\pm_\# P = U_d.$$ Key properties (all established via optimal transport and convex analysis):

• $\mathbb{R}^d$0 and $\mathbb{R}^d$1 are mutual a.e. inverses and homeomorphisms under mild regularity.
• $\mathbb{R}^d$2 for $\mathbb{R}^d$3, with independent uniform radial and directional parts, generalizing the classical probability integral transform.
• $\mathbb{R}^d$4 is equivariant under orthogonal transformations and captures full multivariate information, not just marginal or coordinatewise features (Barrio et al., 2018, Beirlant et al., 2019).

2. Definition of Center-Outward Q-Dominance

Given two absolutely continuous probability laws $\mathbb{R}^d$5 on $\mathbb{R}^d$6 with corresponding center-outward quantile functions $\mathbb{R}^d$7, center-outward Q-dominance (also called quantile dominance) is defined as: $\mathbb{R}^d$8 where $\mathbb{R}^d$9 denotes the Euclidean norm. Equivalently, for each fixed quantile level $Q^\pm$0,

$Q^\pm$1

that is, every center-outward quantile region for $Q^\pm$2 is contained within that for $Q^\pm$3 (Barrio et al., 2018, Beirlant et al., 2019). In univariate settings ($Q^\pm$4), this reduces to the standard quantile function order.

In the sharpest form, center-outward $Q^\pm$5-dominance also imposes a coordinatewise comparison on quantile contours: $Q^\pm$6 for all $Q^\pm$7, where $Q^\pm$8 is the $Q^\pm$9-quantile contour of $\nabla \phi^*$0 (Laag et al., 16 Nov 2025).

3. Key Theoretical Results

A central theorem is that center-outward $\nabla \phi^*$1–dominance for all $\nabla \phi^*$2 implies strong first-order stochastic dominance (FSD) in the sense: $\nabla \phi^*$3 for all componentwise nondecreasing $\nabla \phi^*$4. This is established by coupling both distributions to the unique common reference (the spherical-uniform $\nabla \phi^*$5) and using the monotonicity of quantile maps (Laag et al., 16 Nov 2025, Barrio et al., 2018). Q-dominance is reflexive, transitive, and, in elliptical or symmetric cases, reduces to joint ordering of radial quantiles and scatter structures (Beirlant et al., 2019).

Further properties:

• The volume of quantile regions is ordered: $\nabla \phi^*$6 for all $\nabla \phi^*$7, with equality implying equality of the norms of quantiles.
• Q-dominance implies ordering of maximal-correlation risk measures and convex potentials (Beirlant et al., 2019).
• Stability under mixtures: convex combinations of dominated pairs remain dominated.

4. Empirical Estimation and Smooth Approximations

Empirical center-outward quantile functions are estimated by optimal assignment between data points and a regular grid approximating $\nabla \phi^*$8, seeking the cyclically monotone bijection minimizing squared Euclidean cost. Solutions are piecewise constant but can be smoothed via techniques such as Moreau–Yosida envelopes or log-sum-exp approximations: $\nabla \phi^*$9 with uniform consistency properties for suitable parameter scaling (Beirlant et al., 2019, Barrio et al., 2018).

A test statistic for empirical $U_d$0-dominance is constructed as the minimum over coordinates and grid points of the differences of mapped quantiles; finite-sample error rates can be controlled by explicit sample size thresholds $U_d$1 under bi-Lipschitz conditions (Laag et al., 16 Nov 2025).

Algorithmic Component	Computational Complexity	Purpose
OT assignment	$U_d$2	Estimating $U_d$3
Pairwise comparisons	$U_d$4	Testing $U_d$5-dominance on grid points

No resampling, parameter tuning, or bootstrapping is required beyond grid selection.

5. Applications to Stochastic Multi-objective Optimization

In multi-objective optimization (SMOOP), center-outward Q-dominance provides a sample-computable, non-scalarized criterion for distributional comparison:

• Hyperparameter Tuning: When the expected hypervolume indicator becomes indistinguishable across Pareto sets, Q-dominance ranks methods by comparing empirical quantile maps on their output distributions, revealing dominance missed by mean-value comparisons.
• Evolutionary Algorithms: Integrating Q-dominance into selection (e.g., in NSGA-II) increases convergence speed and effectiveness on noisy benchmarks, compared to mean-based sorting (Laag et al., 16 Nov 2025).

Q-dominance thus acts as a robust, information-preserving proxy for strong stochastic dominance in high-dimensional, noisy settings where conventional metrics fail.

6. Connections, Sufficient Conditions, and Illustrative Examples

For elliptical distributions $U_d$6 and $U_d$7 with the same radial law, Q-dominance holds precisely when the scatter matrices satisfy $U_d$8 (Loewner order) and radial quantile functions are ordered pointwise. For non-elliptical laws, Q-dominance is empirically checked by plug-in approximation on a mesh in $U_d$9 (Beirlant et al., 2019, Barrio et al., 2018).

Typical examples include:

• Uniform distributions on balls of different radii: quantile contours are concentric, and Q-dominance holds trivially.
• Spherical Gaussians: higher-variance distributions are quantile-dominated by lower-variance ones.

7. Extensions, Limitations, and Open Problems

Center-outward Q-dominance is constrained by assumptions of absolute continuity and regularity (e.g., bi-Lipschitz quantile maps). Challenges include computational cost ($\overline{S}_d$0), curse of dimensionality for grid discretization, and the need for scalable approximate assignment (e.g., entropic regularization). Open questions remain about relations to alternative dominance notions, extension to higher-order dominance, and algorithmic acceleration (Laag et al., 16 Nov 2025). Nonetheless, Q-dominance establishes a flexible geometric framework, unifying theory and practice in multivariate stochastic comparison.

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References:

• (Barrio et al., 2018) Center-Outward Distribution Functions, Quantiles, Ranks, and Signs in $\overline{S}_d$1
• (Beirlant et al., 2019) Center-outward quantiles and the measurement of multivariate risk
• (Laag et al., 16 Nov 2025) Center-Outward q-Dominance: A Sample-Computable Proxy for Strong Stochastic Dominance in Multi-Objective Optimisation