Multivariate Affine Transport Bounds

• Multivariate affine transport bounds are sharp inequalities that quantify constraints on probability measures and functionals under affine and piecewise affine transformations.
• They integrate algebraic, geometric, and probabilistic methods to derive optimal transport, error bounds, and robust statistical procedures.
• Applications span convex geometry, optimal transport, numerical analysis, and nonparametric testing, offering explicit estimates in high-dimensional settings.

Multivariate affine transport bounds are a collection of sharp inequalities, optimality results, and structural characterizations that quantify constraints on probability measures, functionals, and transport maps under affine or piecewise affine transformations—often in high-dimensional settings. These bounds arise in diverse fields including convex geometry, probability theory, optimal transport, functional inequalities, numerical analysis, and domain adaptation. They incorporate the interplay between algebraic (affine/linear transformations), geometric (curvature, convexity, polytope structure), and probabilistic (measure, concentration, robustness) features to produce explicit quantitative estimates or structure theorems.

1. Affine Transport Equations and Generalized Holonomy

Affine transport equations—of the form $\dot{x}^b \nabla_b \xi^a = \alpha \dot{x}^a$—define affine maps between tangent spaces along curves in a manifold, with solutions decomposed into homogeneous (parallel transport) and inhomogeneous (affine displacement) parts. For closed curves, such as geodesic triangles and parallelogramoids, the transport yields a generalized holonomy comprising a linear term and a translation term: $\xi^a = \Lambda^a{}_{a'} \xi^{a'} + \Delta\xi^a,$ where $\Lambda^a{}_b$ involves the Riemann tensor and its derivatives, and $\Delta\xi^a$ is proportional to curvature and higher-order moments of the loop (e.g., area, area$^{3/2}$). This structure quantifies finite-size corrections to standard holonomy and has direct applications in general relativity for momentum transfer and gravitational memory (Vines et al., 2014).

Loop Type	Linear Holonomy $\Lambda^a{}_b$	Inhomogeneous Part $\Delta\xi^a$
Triangle	$$1/2 R^a_{bcd}v^c u^d + 1/6 R^a_{bcd;e}v^c u^d (v^e + u^e)$$	$1/6 R^a_{bcd} (v^b + u^b) v^c u^d$
Parallelogramoid	$R^a_{bcd} \tau^c \chi^d + ...$	$1/2 R^a_{bcd} (\tau^b + \chi^b) \tau^c \chi^d$

2. Affinely-Invariant Extremal Inequalities

Classical isoperimetric and intersection inequalities in convex geometry are generalized to functional settings with affinely-invariant extremal inequalities (Dann et al., 2015). These typically involve averages over Grassmannians or affine Grassmannians and are invariant under volume-preserving affine transformations: $$\int_{G_{n,k}} \left( \int_E f(x) dx \right)^n / \|f|_E\|_\infty^{n-k} d\mu_{n,k}(E) \leq (\omega_k^n/\omega_n^k) \left(\int_{\mathbb{R}^n} f(x) dx\right)^k$$ Such inequalities control the spread/concentration of measures after affine projections and establish optimality (the extremizers are ellipsoids or balls). This framework enables tight control of marginal densities and small-ball probabilities in high dimensions—central for random matrix theory and geometric analysis.

3. Optimal Transport, Marginal and Dependence Uncertainty

Optimal transport duality and bounds are derived for multivariate Fréchet classes under affine constraints and marginal uncertainty (Bartl et al., 2017). With calibration (prescribed mass on subsets $S$) and stochastic dominance relaxations for the marginals, the supremum over joint distributions for an option (box payoff) equals an infimum over dual hedging positions, yielding explicit improved Fréchet–Hoeffding bounds: $$F(x) \leq \min \left\{ \min_i F_i^*(x_i), \min_{s \in S} \left[ \pi_s + \sum_i (F_i^*(x_i) - F_i^*(s_i))^+ \right] \right\}$$ These bounds are pointwise sharp when marginal uncertainty is present, but not necessarily when marginals are exactly known. The explicit solution—using vertical/horizontal strips and boxes—gives geometric insight into the structure of optimal hedging/transport strategies.

4. Error Bounds and Numerical Methods for Affine Transport

Discretized optimal transport problems with affine costs are analyzed using Dirac measure approximations and multilevel active-set strategies (Bartels et al., 2017, Glunt et al., 23 Feb 2024). Key results include:

• Quadratic convergence of optimal values when approximating measures via convex combinations of Dirac masses, with error scaling as $O(h^{1+\alpha})$;
• Linear growth of effective problem size (number of support points) due to reliable active-set selection;
• Explicit error propagation bounds for compositions of piecewise affine approximations (PWA), establishing that composition errors can be bounded as affine functions of the componentwise errors:

$$\varepsilon_{f \circ g} \leq \tau_f + d_f \cdot \varepsilon_g$$

Methods for intelligent breakpoint placement ensure low complexity and tight control for high-dimensional functional approximations.

5. Bounds for Affine Transformations in Wasserstein Space

Sharp lower and upper bounds for Wasserstein distances between measures under affine transformations are constructed using the Bures metric, which quantifies the effect of covariance structure (Hamm et al., 2023): $$W_2(\mu, \nu)^2 \geq |m_\mu - m_\nu|^2 + \operatorname{tr}[\Sigma_\mu + \Sigma_\nu - 2 (\Sigma_\mu \Sigma_\nu)^{1/2}]$$ Explicit formulas for translations, dilations, and rotations give both lower and upper bounds, facilitating applications in manifold learning (Wassmap, MDS, Isomap) and synthetic data generation. These bounds are exact for Gaussian and elliptically contoured measures, and extend to compositions of affine maps.

6. Statistical and Nonparametric Applications

Multivariate distribution-free nonparametric tests generalize Wilcoxon's ranks and signs via optimal transport maps, yielding distribution-free, robust, and efficient procedures for hypothesis testing (Huang et al., 15 Mar 2025, Deb et al., 2021). In high dimensions, OT-based ranks and signed-ranks inherit the desirable properties of classical tests, including high Pitman efficiency (Hodges-Lehmann/Chernoff-Savage phenomena) and universality when integrated with kernel methods (e.g., MMD, energy test). Kernel-based OT tests for symmetry are both universally consistent and distribution-free.

Test Type	Statistic Construction	Efficiency Properties
Rank-Sum (2-sample)	OT-Brenier mapping to reference grid	ARE $\geq 0.864$ (Gaussian)
Signed-Rank (1-sample)	OT-based signs and ranks via affine assignment	Exact dist.-free, robust

7. Robustness and Breakdown Point of Transport-Based Quantiles

Transport-based quantiles (center-outward rank maps via OT) have maximal robustness; the breakdown point of the transport median equals the maximal Tukey depth of the reference measure (usually $1/2$ for half-space symmetric measures) (Avella-Medina et al., 21 Oct 2024). For transport depth contours at order $\tau$, the breakdown point is exactly $\tau$. This rigorous connection between transport-induced quantiles and geometric depth functions ensures that multivariate affine transport bounds inherit optimal robustness under contamination, paralleling the univariate median.

8. Ergodicity Bounds for Stochastic Processes Under Affine Transport

Recent advances show non-asymptotic, explicit upper and lower multivariate affine transport bounds for Wasserstein-$r$ distances in Schur stable autoregressive (AR, ARMA) processes (Barrera et al., 16 Aug 2025). The affine transport bounds quantify thermalization and ergodic convergence rates—mean convergence decays exponentially, variance with double the rate. Coupling techniques produce sharp exponential bounds for highly general multivariate AR processes, with direct implications for parallel sampling and empirical mean estimation.

Concluding Remarks

Multivariate affine transport bounds interconnect optimal transport theory, affine geometry, robust statistics, and high-dimensional probability, producing explicit, sharp, and computationally tractable inequalities and structure theorems for probabilistic, functional, and statistical analyses. Whether in the geometry of holonomy in curved spaces (Vines et al., 2014), bounding marginal densities and small-ball probabilities (Dann et al., 2015), controlling error in numerical methods (Bartels et al., 2017, Glunt et al., 23 Feb 2024), or ensuring robustness and efficiency in nonparametric testing (Huang et al., 15 Mar 2025, Avella-Medina et al., 21 Oct 2024), affine transport bounds empower rigorous understanding of the impact of transformations in multivariate settings and have pervasive applications across mathematical sciences.