Fast Algorithms for Computational Optimal Transport and Wasserstein Barycenter

We provide theoretical complexity analysis for new algorithms to compute the optimal transport (OT) distance between two discrete probability distributions, and demonstrate their favorable practical performance compared to state-of-art primal-dual algorithms. First, we introduce the accelerated primal-dual randomized coordinate descent (APDRCD) algorithm for computing the OT distance. We show that its complexity is (O) over tildeO(n(5/2)/epsilon), where n stands for the number of atoms of these probability measures and epsilon > 0 is the desired accuracy. This complexity bound matches the best known complexities of primal-dual algorithms for the OT problems, including the adaptive primal-dual accelerated gradient descent (APDAGD) and the adaptive primal-dual accelerated mirror descent (APDAMD) algorithms. Then, we demonstrate the improved practical efficiency of the APDRCD algorithm through comparative experimental studies. We also propose a greedy version of APDRCD, which we refer to as accelerated primal-dual greedy coordinate descent (APDGCD), to further enhance practical performance. Finally, we generalize the APDRCD and APDGCD algorithms to distributed algorithms for computing the Wasserstein barycenter for multiple probability distributions.