  Optimal transport (OT)-based methods have a wide range of applications and
have attracted a tremendous amount of attention in recent years. However, most
of the computational approaches of OT do not learn the underlying transport
map. Although some algorithms have been proposed to learn this map, they rely
on kernel-based methods, which makes them prohibitively slow when the number of
samples increases. Here, we propose a way to learn an approximate transport map
and a parametric approximation of the Wasserstein barycenter. We build an
approximated transport mapping by leveraging the closed-form of Gaussian
(Bures-Wasserstein) transport; we compute local transport plans between matched
pairs of the Gaussian components of each density. The learned map generalizes
to out-of-sample examples. We provide experimental results on simulated and
real data, comparing our proposed method with other mapping estimation
algorithms. Preliminary experiments suggest that our proposed method is not
only faster, with a factor 80 overall running time, but it also requires fewer
components than state-of-the-art methods to recover the support of the
barycenter. From a practical standpoint, it is straightforward to implement and
can be used with a conventional machine learning pipeline.
