A short proof on the rate of convergence of the empirical measure for the Wasserstein distance

We provide a short proof that the Wasserstein distance between the empirical measure of a n-sample and the estimated measure is of order n^−1/d , if the measure has a lower and upper bounded density on the d-dimensional flat torus

Universal consistency of Wasserstein kk-NN classifier

The Wasserstein distance provides a notion of dissimilarities between probability measures, which has recent applications in learning of structured data with varying size such as images and text documents. In this work, we analyze the $k$-nearest neighbor classifier ($k$-NN) under the Wasserstein distance and establish the universal consistency on families of distributions. Using previous known results on the consistency of the $k$-NN classifier on infinite dimensional metric spaces, it suffices to show that the families is a countable union of finite dimensional components. As a result, we are able to prove universal consistency of $k$-NN on spaces of finitely supported measures, the space of finite wavelet series and the spaces of Gaussian measures with commuting covariance matrices.Comment: 12 page

Minimax estimation of smooth optimal transport maps

Brenier's theorem is a cornerstone of optimal transport that guarantees the existence of an optimal transport map $T$ between two probability distributions $P$ and $Q$ over $\mathbb{R}^d$ under certain regularity conditions. The main goal of this work is to establish the minimax estimation rates for such a transport map from data sampled from $P$ and $Q$ under additional smoothness assumptions on $T$. To achieve this goal, we develop an estimator based on the minimization of an empirical version of the semi-dual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semi-dual and a complementary minimax lower bound. Furthermore, we provide numerical experiments on synthetic data supporting our theoretical findings and highlighting the practical benefits of smoothness regularization. These are the first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure

Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space

This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)H\"older continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent H\"older-stability results for optimal transport maps.Comment: 21 page

On data-driven Wasserstein distributionally robust Nash equilibrium problems with heterogeneous uncertainty

We study stochastic Nash equilibrium problems subject to heterogeneous uncertainty on the cost functions of the individual agents. In our setting, we assume no prior knowledge of the underlying probability distributions of the uncertain variables. To account for this lack of knowledge, we consider an ambiguity set around the empirical probability distribution under the Wasserstein metric. We then show that, under mild assumptions, finite-sample guarantees on the probability that any resulting distributionally robust Nash equilibrium is also robust with respect to the true probability distributions with high confidence can be obtained. Furthermore, by recasting the game as a distributionally robust variational inequality, we establish asymptotic convergence of the set of data-driven distributionally robust equilibria to the solution set of the original game. Finally, we recast the distributionally robust Nash game as a finite-dimensional Nash equilibrium problem. We illustrate the proposed distributionally robust reformulation via numerical experiments of stochastic Nash-Cournot games