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

Abstract: 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.