Central Limit Theorems for Smooth Optimal Transport Maps

TL;DR

This paper establishes a pointwise central limit theorem for estimators of Brenier maps in optimal transport on the flat torus for dimensions three and higher, revealing convergence properties and limitations.

Contribution

It provides the first pointwise CLT for Brenier map estimators and demonstrates non-convergence in high dimensions, advancing understanding of optimal transport estimation.

Findings

Establishes a pointwise CLT for Brenier map estimators in dimension ≥ 3.

Shows non-weak convergence of estimators in high dimensions.

Uses linearization of Monge-Ampère equation to analyze estimator behavior.

Abstract

One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law onto any other given law. A line of recent work has analyzed $L^2$ convergence rates of plugin estimators of Brenier maps, which are defined as the Brenier map between density estimators of the underlying distributions. In this work, we show that such estimators satisfy a pointwise central limit theorem when the underlying laws are supported on the flat torus of dimension $d \geq 3$. We also derive a negative result, showing that these estimators do not converge weakly in $L^2$ when the dimension is sufficiently large. Our proofs hinge upon a quantitative linearization of the Monge-Amp\`ere equation, which may be of independent interest. This result allows us to reduce our problem to that of deriving limit laws for the solution of a uniformly elliptic partial differential equation with a stochastic right-hand side, subject to periodic boundary conditions.