  An adaptive, adversarial methodology is developed for the optimal transport
problem between two distributions $\mu$ and $\nu$, known only through a finite
set of independent samples $(x_i)_{i=1..N}$ and $(y_j)_{j=1..M}$. The
methodology automatically creates features that adapt to the data, thus
avoiding reliance on a priori knowledge of data distribution. Specifically,
instead of a discrete point-bypoint assignment, the new procedure seeks an
optimal map $T(x)$ defined for all $x$, minimizing the Kullback-Leibler
divergence between $(T(xi))$ and the target $(y_j)$. The relative entropy is
given a sample-based, variational characterization, thereby creating an
adversarial setting: as one player seeks to push forward one distribution to
the other, the second player develops features that focus on those areas where
the two distributions fail to match. The procedure solves local problems
matching consecutive, intermediate distributions between $\mu$ and $\nu$. As a
result, maps of arbitrary complexity can be built by composing the simple maps
used for each local problem. Displaced interpolation is used to guarantee
global from local optimality. The procedure is illustrated through synthetic
examples in one and two dimensions.
