发表机构
University of Duisburg-Essen（杜伊斯堡-埃森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对熵最优运输势的经验Sinkhorn估计量，在一致损失下，通过几何条件改进其关于正则化参数的依赖，建立多项式依赖的统计速率并给出匹配极小极大下界。
AI 中文摘要
我们在一致损失下研究熵最优运输势的经验Sinkhorn估计量。由于这些势仅在加性常数下唯一，我们采用商上确界范数衡量误差，定义为$d_\infty([u],[v]) = \inf_{a\in\mathbb{R}}\\\\|u-v-a\\\\|_\infty$。对于固定正则化参数$\varepsilon>0$，我们建立了非渐近统计速率$n^{-1/2}$，这通过结合Birkhoff-Hopf压缩定理与归一化核截面的熵界实现。但该界中的常数随$1/\varepsilon$指数增长，为改进这一点，我们分离出几何条件，在该条件下经验估计量仍保持$n^{-1/2}$速率，且具有$1/\varepsilon$的多项式依赖，关键要求是总体Sinkhorn映射的多项式残差稳定性估计。我们为此提供了充分准则，包括多项式压缩性质和局部逆估计。此外，我们引入两类严格可验证的模型类：经可分离中心化得到的$\varepsilon$-弱残差交互类，以及基于固定离散代价的连通紧边图的模型类，后者无需依赖抽象预解式假设即可保证多项式速率。最后，我们建立了匹配的极小极大下界，表明在有界交互机制下，$\varepsilon n^{-1/2}$速率无法被一致提升。
英文摘要
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.
