The Wasserstein distance Wp(µ, ν) is a measure of similarity between two probability distributions µ and ν, that is defined as the cost of some optimal matching (also called an optimal transport map) between µ and ν. Such distances have found numerous applications in machine learning: for example, Wasserstein Generative Adversarial Networks are able to generate realistic fake images by approximating some training sample with respect to the metric W1. From a statistical perspective, the question of the estimation of quantities related to the optimal transport problem in a possibly high dimensional regime is then raised. We will present two methods that are suited to such a setting. First, in the case where the target distribution is supported on a low-dimensional manifold, one can exploit the underlying structure to attain fast rates of convergence. This can be done by building a kernel density estimator on some approximation of the manifold. Second, we will also investigate the case where the target distribution µ is the pushforward of a fixed source measure by some input convex neural network T. In this situation,
we show that one can exploit the structure of the map T to design computable estimators attaining parametric rates of convergence.