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):

• $F^\pm$ and $Q^\pm$ are mutual a.e. inverses and homeomorphisms under mild regularity.
• $F^\pm(X) \sim U_d$ for $X\sim P$, with independent uniform radial and directional parts, generalizing the classical probability integral transform.
• $Q^\pm$ 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 $P, Q$ on $\mathbb{R}^d$ with corresponding center-outward quantile functions $Q^\pm_P, Q^\pm_Q$, center-outward Q-dominance (also called quantile dominance) is defined as: $$P \succeq_Q Q \iff \| Q^\pm_P(u) \| \leq \| Q^\pm_Q(u) \| \quad \forall u \in \overline{S}_d,$$ where $\| \cdot \|$ denotes the Euclidean norm. Equivalently, for each fixed quantile level $\tau \in [0,1)$,

$$R_P(\tau) := \{ Q^\pm_P(u) : \|u\| \leq \tau \} \subset R_Q(\tau) := \{ Q^\pm_Q(u) : \|u\| \leq \tau \},$$

that is, every center-outward quantile region for $P$ is contained within that for $Q$ (Barrio et al., 2018, Beirlant et al., 2019). In univariate settings ($d=1$), this reduces to the standard quantile function order.

In the sharpest form, center-outward $q$-dominance also imposes a coordinatewise comparison on quantile contours: $$P_1 \succeq_q P_2 \iff \forall y \in \mathcal{C}_{P_2}(q): \quad Q^\pm_1(F^\pm_2(y)) \geq y \quad \text{coordinatewise} \tag{1}$$ for all $q \in [0,1]$, where $\mathcal{C}_{P_2}(q)$ is the $q$-quantile contour of $P_2$ (Laag et al., 16 Nov 2025).

3. Key Theoretical Results

A central theorem is that center-outward $q$–dominance for all $q\in[0,1)$ implies strong first-order stochastic dominance (FSD) in the sense: $$\text{If } P_1 \succeq_q P_2 \ \forall q, \quad \text{then } \mathbb{E}[u(X_1)] \geq \mathbb{E}[u(X_2)]$$ for all componentwise nondecreasing $u : \mathbb{R}^d \to \mathbb{R}$. This is established by coupling both distributions to the unique common reference (the spherical-uniform $U_d$) 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: $V_P(\tau)\leq V_Q(\tau)$ for all $\tau$, 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 $U_d$, 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: $$\Psi_{n,\xi}(u) = \frac{1}{\xi}\log \sum_{i=1}^n \exp(\xi \psi_i(u)), \quad \widehat{Q}_{n,\xi}(u) = \nabla \Psi_{n,\xi}(u),$$ with uniform consistency properties for suitable parameter scaling (Beirlant et al., 2019, Barrio et al., 2018).

A test statistic for empirical $q$-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 $n^*(\delta)$ under bi-Lipschitz conditions (Laag et al., 16 Nov 2025).

Algorithmic Component	Computational Complexity	Purpose
OT assignment	$O(n^3)$	Estimating $\widehat{Q}^\pm$
Pairwise comparisons	$O(d n)$	Testing $q$-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 $P = \text{Ell}(\mu_P, \Sigma_P; F_R)$ and $Q = \text{Ell}(\mu_Q, \Sigma_Q; F_S)$ with the same radial law, Q-dominance holds precisely when the scatter matrices satisfy $\Sigma_P \preceq \Sigma_Q$ (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 $\overline{S}_d$ (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 ($O(n^3)$), 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 $\mathbb{R}^d$
• (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