Westudythestatisticalpropertiesoftheentropicoptimal(self) transport problem for smooth probability measures. We provide an accurate description of the limit distribution for entropic (self-)potentials and plans for shrinking regularization parameter, which strongly contrasts prior work where the regularization parameter is held fix. Additionally, we show that a rescaling of the barycentric projection of the empirical entropic optimal self-transport plans converges to the score function, a central object for diffusion models, and characterize the asymptotic fluctuations both pointwise and in L2. Finally, we describe under what conditions the methods used enable to derive (pointwise) limiting distribution results for the empirical entropic optimal transport potentials in the case of two different measures and appropriately chosen shrinking regularization parameter. This endeavour requires better understanding the composition of Sinkhorn operators, a result of independent interest.

翻译：我们研究了光滑概率测度的熵最优（自）输运问题的统计性质。针对逐渐缩小的正则化参数，我们给出了熵（自）势和输运方案极限分布的精确描述，这与先前研究中固定正则化参数的情形形成鲜明对比。此外，我们证明了经验熵最优自输运方案的重心投影经过重新缩放后，会收敛到扩散模型的核心对象——得分函数，并在点态和L2范数下刻画了其渐近波动性。最后，我们阐明了在何种条件下，所采用的方法能够推导出两个不同测度情形下经验熵最优输运势的（点态）极限分布结果，其中正则化参数需适当选择递减速率。这项研究需要更深入地理解Sinkhorn算子的复合性质，该结果本身也具有独立的理论价值。