Multimarginal Optimal Transport by Accelerated Alternating Minimization

We study multimarginal optimal transport (MOT) problems, which include, as a particular case, the Wasserstein barycenter problem. In MOT problems, one has to find an optimal coupling between m probability measures, which amounts to finding a tensor of order m. We propose a method based on accelerated alternating minimization and estimate the complexity to find an approximate solution. We use entropic regularization with a sufficiently small regularization parameter and apply accelerated alternating minimization to the dual problem. A novel primal-dual analysis is used to reconstruct the approximately optimal coupling tensor. Our algorithm exhibits a better computational complexity than the state-of-the-art methods for some regimes of the problem parameters.