This was part of Applied Optimal Transport

Uniform consistency of estimated optimal transport plans

Johan Segers, UCLouvain
Thursday, May 19, 2022

Abstract: Despite the long history of the theory of optimal transport, its potential for the development of distribution-free hypothesis testing in statistics has only been realized recently. In multivariate statistics, optimal transport plans play the role of the cumulative distribution function and the quantile function in univariate statistics, a prime example being Marc Hallin's center-outward distribution function. Often, mass is transported between a target distribution and a reference distribution, with one direction yielding "ranks" and the inverse one yielding "quantiles". If the target measure needs to be estimated from data and/or the reference distribution is replaced by a (randomized) approximation, the optimal transport plan needs to be estimated too. It is then of interest to show the estimator to be consistent, preferably in a uniform way: this yields consistent estimation of quantiles and helps to prove the consistency of a hypothesis test against alternatives. A general theory is provided delivering uniform consistency for optimal transport plans between estimated probability measures. The data-generating process does not matter; time series dependence is allowed, for instance. Moreover, there are only few assumptions on the true distributions: absolute continuity is already sufficient, but there are no conditions on the form of the support nor any upper or lower bounds on the densities. The theory is based on the fact that an optimal transport plan can be identified with a maximal cyclically monotone subset of a Cartesian product of Euclidean spaces. An estimated transport plan is seen as a random closed set. Weak convergence in the appropriate Fell space together with the maximal cyclical monotonicity then automatically imply a kind of local uniform convergence of the associated (potentially multi-valued) mappings.