  This paper presents a novel two-step approach for the fundamental problem of
learning an optimal map from one distribution to another. First, we learn an
optimal transport (OT) plan, which can be thought as a one-to-many map between
the two distributions. To that end, we propose a stochastic dual approach of
regularized OT, and show empirically that it scales better than a recent
related approach when the amount of samples is very large. Second, we estimate
a \textit{Monge map} as a deep neural network learned by approximating the
barycentric projection of the previously-obtained OT plan. This
parameterization allows generalization of the mapping outside the support of
the input measure. We prove two theoretical stability results of regularized OT
which show that our estimations converge to the OT plan and Monge map between
the underlying continuous measures. We showcase our proposed approach on two
applications: domain adaptation and generative modeling.
