Central Limit Theorems for Smooth Optimal Transport Maps
arxiv(2023)
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.
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