2024/12/16 by Gilles Mordant, Mordant, Gilles · 5 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Applied mathematics #Artificial intelligence #Biology #Computer science #Econometrics #Economic theories and models #FOS: Mathematics #Function (biology) #Limit (mathematics) #Mathematical analysis #Mathematical optimization #Mathematics #Physics #Regularization (linguistics) #Risk and Portfolio Optimization #Statistical physics #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2412.12007
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2024/12/16 · openalex created_date 2024/12/19 · openalex updated_date 2026/08/01
We study the statistical properties of the entropic optimal (self) transport problem for smooth probability measures. We provide an accurate description of the limit distribution for entropic (self-)potentials and plans as the regularization parameter shrinks with the sample size; this regime is largely unexplored in the prior statistical literature, where ε is typically held fixed. 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 a better understanding of the composition of Sinkhorn operators in the small \eps-limit, a result of independent interest.