Multilevel Optimal Transport: a Fast Approximation of Wasserstein-1 distances

We propose a fast algorithm for the calculation of the Wasserstein-1 distance, which is a particular type of optimal transport distance with homogeneous of degree one transport cost. Our algorithm is built on multilevel primal-dual algorithms. Several numerical examples and a complexity analysis are provided to demonstrate its computational speed. On some commonly used image examples of size 512×512, the proposed algorithm gives solutions within 0.2~ 1.5 seconds on a single CPU, which is much faster than the state-of-the-art algorithms.