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

TL;DR

This paper analyzes the statistical behavior of entropic optimal transport as regularization diminishes, revealing limit distributions and connections to score functions in diffusion models.

Contribution

It provides new limit distribution results for entropic (self-)potentials and plans under shrinking regularization, and links barycentric projections to score functions.

Findings

Limit distributions for entropic potentials and plans as regularization shrinks.

Convergence of barycentric projections to score functions in diffusion models.

Conditions for pointwise distribution results for entropic potentials with different measures.

Abstract

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 $\epsilon$ 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 $L^2$. 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.