  In this paper, we present a novel and principled approach to learn the
optimal transport between two distributions, from samples. Guided by the
optimal transport theory, we learn the optimal Kantorovich potential which
induces the optimal transport map. This involves learning two convex functions,
by solving a novel minimax optimization. Building upon recent advances in the
field of input convex neural networks, we propose a new framework where the
gradient of one convex function represents the optimal transport mapping.
Numerical experiments confirm that we learn the optimal transport mapping. This
approach ensures that the transport mapping we find is optimal independent of
how we initialize the neural networks. Further, target distributions from a
discontinuous support can be easily captured, as gradient of a convex function
naturally models a {\em discontinuous} transport mapping.
