A new method for determining Wasserstein 1 optimal transport maps from Kantorovich potentials, with deep learning applications

Abstract: Wasserstein 1 optimal transport maps provide a natural correspondence between points from two probability distributions, $\mu$ and $\nu$, which is useful in many applications. Available algorithms for computing these maps do not appear to scale well to high dimensions. In deep learning applications, efficient algorithms have been developed for approximating solutions of the dual problem, known as Kantorovich potentials, using neural networks (e.g. [Gulrajani et al., 2017]). Importantly, such algorithms work well in high dimensions. In this paper we present an approach towards computing Wasserstein 1 optimal transport maps that relies only on Kantorovich potentials. In general, a Wasserstein 1 optimal transport map is not unique and is not computable from a potential alone. Our main result is to prove that if $\mu$ has a density and $\nu$ is supported on a submanifold of codimension at least 2, an optimal transport map is unique and can be written explicitly in terms of a potential. These assumptions are natural in many image processing contexts and other applications. When the Kantorovich potential is only known approximately, our result motivates an iterative procedure wherein data is moved in optimal directions and with the correct average displacement. Since this provides an approach for transforming one distribution to another, it can be used as a multipurpose algorithm for various transport problems; we demonstrate through several proof of concept experiments that this algorithm successfully performs various imaging tasks, such as denoising, generation, translation and deblurring, which normally require specialized techniques.