标题： 熵最优（自）运输问题：减少正则化下的极限分布及其在得分函数估计中的应用 显示英文标题

标题： The entropic optimal (self-)transport problem: Limit distributions for decreasing regularization with application to score function estimation

摘要： 我们研究了平滑概率测度的熵最优（自）运输问题的统计性质。 我们提供了对收缩正则化参数下熵（自）势和计划的极限分布的准确描述，这与之前固定正则化参数的工作形成强烈对比。 此外，我们表明经验熵最优自运输计划的重心投影的缩放收敛到得分函数，这是扩散模型中的核心对象，并在点态和L2中表征了渐近波动。 最后，我们描述了在两种不同测度和适当选择的收缩正则化参数情况下，所使用的方法在经验熵最优运输势上推导（点态）极限分布结果的条件。 这一努力需要更好地理解Sinkhorn算子的组合，这是一个独立感兴趣的结果。 显示英文摘要

摘要： 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.