  Optimal transport maps define a one-to-one correspondence between probability
distributions, and as such have grown popular for machine learning
applications. However, these maps are generally defined on empirical
observations and cannot be generalized to new samples while preserving
asymptotic properties. We extend a novel method to learn a consistent estimator
of a continuous optimal transport map from two empirical distributions. The
consequences of this work are two-fold: first, it enables to extend the
transport plan to new observations without computing again the discrete optimal
transport map; second, it provides statistical guarantees to machine learning
applications of optimal transport. We illustrate the strength of this approach
by deriving a consistent framework for transport-based counterfactual
explanations in fairness.
