发表机构
Carnegie Mellon University（卡内基梅隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究半离散熵最优传输的统计速率，提出参数收敛速率且首项无维度依赖，使重心投影达到 $n^{-1}$ 速率，并证明 Sinkhorn-EM 在混合模型中的 $\sqrt{n}$ 一致性。
AI 中文摘要
我们研究了半离散区间中熵最优传输的统计速率，其中一个测度具有有限支撑，另一个测度为次高斯分布。我们的主要结果确立了经验对偶势函数向其总体对应物收敛的参数速率，且首项不依赖于维度。我们的结果依赖于对半对偶目标函数的定制强凹性分析，并结合针对半离散势函数的专门界。因此，我们获得了由最优耦合导出的下游量的快速速率。主要地，经验重心投影实现了平方误差速率 $n^{-1}$，与完全紧致情形相匹配，并优于完全次高斯设置中已知的不太有利的 $n^{-1/2}$ 速率。总之，这些结果可能表明一种较低复杂度适应现象，即重心投影的统计复杂度由离散测度控制。作为应用，我们分析了 Sinkhorn-EM，一种 EM 型算法，其中 E 步被熵最优传输问题替代。在设定良好且平衡的双分量高斯混合模型中，我们证明了对于任何固定的迭代次数，经验迭代对其总体对应物的 $\sqrt{n}$ 一致性，匹配经典 EM 速率，最多相差 $\sqrt{\log n}$ 因子。模拟支持该理论。
英文摘要
We study statistical rates in entropic optimal transport in the semi-discrete regime where one measure has finite support and the other is subGaussian. Our main result establishes parametric convergence rates for the empirical dual potentials to their population counterparts, with no dimension dependence in the leading term. Our result relies on tailored strong concavity analysis of the semi-dual objective, coupled with specialized bounds for the semi-discrete potentials. As a consequence, we obtain fast rates for downstream quantities derived from the optimal coupling. Chiefly, the empirical barycentric projection achieves a squared-error rate $n^{-1}$, matching the fully compact case and improving over the less favorable $n^{-1/2}$ rate known for fully subGaussian settings. Altogether, these results may indicate a lower complexity adaptation phenomenon whereby the statistical complexity of the barycentric projection is governed by the discrete measure. As an application, we analyze Sinkhorn-EM, an EM-type algorithm in which the E-step is replaced by an entropic optimal transport problem. In a well-specified and balanced two-component Gaussian mixture model, we prove $\sqrt{n}$-consistency of the empirical iterates to their population counterparts for any fixed number of iterations, matching classical EM rates up to a $\sqrt{\log n}$ factor. Simulations support the theory.
