AI 中文总结
该研究针对二次最优运输势估计，通过稳定性不等式推导其快收敛速率，在低维时建立中心极限定理与自助法一致性，还证明熵对偶势和的匹配上下界，为相关推断提供理论支撑。
AI 中文摘要
最优运输势是统计学、经济学和机器学习中的基础对象：其梯度生成最优运输映射，而势本身作为依赖位置的对偶价格和敏感性变量发挥作用。我们研究当固定绝对连续参考分布μ被运输至未知分布ν（通过其经验测度获取）时二次最优运输势的估计问题。我们的核心工具是一个稳定性不等式，该不等式通过(∇φ̃)#μ - (∇φ)#μ的弱对偶范数，结合对数型的二阶Wasserstein余项，控制强凸势φ与凸势φ̃之间在模加性常数下的L¹(μ)距离。主经验过程项与Wasserstein余项的分离使得势的收敛速率快于对应运输映射的收敛速率。在光滑性和一致凸性假设下，精确半离散Brenier势在d≤3时以n⁻¹/²速率在L¹(μ)中收敛，在d=4时以n⁻¹/²(log n)^(5/2)速率收敛，在d≥5时以n⁻²/d(log n)^((d+2)/d)速率收敛，且多项式指数是最优的。在维度d≤3时，我们进一步建立非退化函数空间中心极限定理，并证明非参数自助法的一致性。这些结果为势的每一个固定有限的归一化不变加权对比（包括基于参考的风险问题中的区域影子溢价）提供联合根-n推断。最后，我们证明熵对偶势的归一化不变和的阶为εlog(1/ε)的匹配上下界。
英文摘要
Optimal transport potentials are fundamental objects in statistics, economics, and machine learning: their gradients generate optimal transport maps, while the potentials themselves act as location-dependent dual prices and sensitivity variables. We study the estimation of the quadratic optimal transport potential when a fixed absolutely continuous reference distribution $μ$ is transported to an unknown distribution $ν$, accessed to via its empirical measure. Our main ingredient is a stability inequality that controls the $L^1(μ)$ distance, modulo additive constants, between a strongly convex potential $φ$ and a convex potential $\widetildeφ$ by a weak dual norm of $(\nabla\widetildeφ)_\#μ-(\nablaφ)_\#μ,$ together with a second-order Wasserstein remainder of logarithmic type. This separation between the leading empirical-process term and the Wasserstein remainder yields faster convergence for potentials than for the corresponding transport maps. Under smoothness and uniform convexity assumptions, the exact semidiscrete Brenier potential converges in $L^1(μ)$ at rate $n^{-1/2}$ for $d\leq3$, at rate $n^{-1/2}(\log n)^{5/2}$ for $d=4$, and at rate $n^{-2/d}(\log n)^{(d+2)/d}$ for $d\geq5$. The polynomial exponents are sharp. In dimensions $d\leq3$, we further establish a nondegenerate function-space central limit theorem and prove consistency of the nonparametric bootstrap. These results yield joint root-$n$ inference for every fixed finite collection of normalization-invariant weighted contrasts of the potential, including regional shadow premia in reference-based risk problems. Finally, we prove matching upper and lower bounds of order $\varepsilon\log(1/\varepsilon)$ for the normalization-invariant sum of the entropic dual potentials.
