  Monge map refers to the optimal transport map between two probability
distributions and provides a principled approach to transform one distribution
to another. Neural network based optimal transport map solver has gained great
attention in recent years. Along this line, we present a scalable algorithm for
computing the neural Monge map between two probability distributions. Our
algorithm is based on a weak form of the optimal transport problem, thus it
only requires samples from the marginals instead of their analytic expressions,
and can accommodate optimal transport between two distributions with different
dimensions. Our algorithm is suitable for general cost functions, compared with
other existing methods for estimating Monge maps using samples, which are
usually for quadratic costs. The performance of our algorithms is demonstrated
through a series of experiments with both synthetic and realistic data,
including text-to-image generation and image inpainting tasks.
