  We provide a framework to approximate the 2-Wasserstein distance and the
optimal transport map, amenable to efficient training as well as statistical
and geometric analysis. With the quadratic cost and considering the Kantorovich
dual form of the optimal transportation problem, the Brenier theorem states
that the optimal potential function is convex and the optimal transport map is
the gradient of the optimal potential function. Using this geometric structure,
we restrict the optimization problem to different parametrized classes of
convex functions and pay special attention to the class of input-convex neural
networks. We analyze the statistical generalization and the discriminative
power of the resulting approximate metric, and we prove a restricted
moment-matching property for the approximate optimal map. Finally, we discuss a
numerical algorithm to solve the restricted optimization problem and provide
numerical experiments to illustrate and compare the proposed approach with the
established regularization-based approaches. We further discuss practical
implications of our proposal in a modular and interpretable design for GANs
which connects the generator training with discriminator computations to allow
for learning an overall composite generator.
