Scalable Computation of Monge Maps with General Costs

Abstract

Monge map refers to the optimal transport map between two probability distributions and provides a principled approach to transform one distribution to another. In spite of the rapid developments of the numerical methods for optimal transport problems, computing the Monge maps remains challenging, especially for high dimensional problems. In this paper, we present a scalable algorithm for computing the 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.