提出改进的最优传输势函数估计方法，实现更优的统计收敛率。
Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter
- 基于归一化核截面的熵界与Birkhoff-Hopf定理结合，获得统一误差分析。
- 在特定几何条件下，收敛率保持n^{-1/2}且对ε依赖为多项式而非指数。
- 适用于离散成本下具有连通紧边图结构或弱残差交互性的模型。
我们研究了在统一损失下，经验Sinkhorn势函数估计在熵正则最优传输中的统计收敛性。由于势函数仅在加法常数意义下唯一，误差采用商上确界范数 $d_ ty([u],[v]) = \inf_{a\in\mathbb{R}}\|u-v-a\|_\infty$ 衡量。对于固定正则化参数 $\varepsilon>0$，建立了非渐近统计速率 $n^{-1/2}$。该结果通过Birkhoff-Hopf收缩定理与归一化核截面的熵界结合实现，但界中常数随 $1/\varepsilon$ 指数增长。为改进此问题，我们识别出几何条件，使经验估计器维持 $n^{-1/2}$ 收敛率的同时，常数仅以 $1/\varepsilon$ 的多项式形式增长。关键要求是总体Sinkhorn映射的多项式残差稳定性估计。我们提供了充分条件，包括多项式收缩性质和局部逆估计。此外，我们引入两类可严格验证的模型类：经可分离中心化后得到的 $\varepsilon$-弱残差交互类，以及固定离散代价下基于连通紧边图的模型类，其多项式速率无需依赖抽象预解算子假设即可保证。最后，我们建立了匹配的极小极大下界，表明在有界交互情形下，$\varepsilon n^{-1/2}$ 的速率无法被统一改进。
原文摘要 · Abstract (English)
We study the empirical Sinkhorn estimator of the entropic optimal transport potentials under the uniform loss. Since the potentials are only unique up to additive constants, we measure the error using the quotient supremum norm, defined as $d_\infty([u],[v]) = \inf_{a\in\mathbb{R}}\|u-v-a\|_\infty$. For a fixed regularization parameter $\varepsilon>0$, we establish a non-asymptotic statistical rate of $n^{-1/2}$. This is achieved by combining the Birkhoff-Hopf contraction theorem with entropy bounds on normalized kernel sections. However, the constant in this bound grows exponentially with $1/ε$. To improve this, we isolate geometric conditions under which the empirical estimator maintains the $n^{-1/2}$ rate but features polynomial dependence on $1/\varepsilon$. The key requirement is a polynomial residual-stability estimate for the population Sinkhorn map. We provide sufficient criteria for this, including a polynomial contraction property and a local inverse estimate. Furthermore, we introduce two rigorously verifiable model classes an $\varepsilon$-weak residual-interaction class obtained after separable centering and another based on connected tight-edge graphs for fixed discrete costs where the polynomial rate is guaranteed without relying on abstract resolvent assumptions. Finally, we establish matching minimax lower bounds demonstrating that the $\varepsilon n^{-1/2}$ rate cannot be uniformly improved in the bounded-interaction regime.
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