  We develop a fast and reliable method for solving large-scale optimal
transport (OT) problems at an unprecedented combination of speed and accuracy.
Built on the celebrated Douglas-Rachford splitting technique, our method
tackles the original OT problem directly instead of solving an approximate
regularized problem, as many state-of-the-art techniques do. This allows us to
provide sparse transport plans and avoid numerical issues of methods that use
entropic regularization. The algorithm has the same cost per iteration as the
popular Sinkhorn method, and each iteration can be executed efficiently, in
parallel. The proposed method enjoys an iteration complexity $O(1/\epsilon)$
compared to the best-known $O(1/\epsilon^2)$ of the Sinkhorn method. In
addition, we establish a linear convergence rate for our formulation of the OT
problem. We detail an efficient GPU implementation of the proposed method that
maintains a primal-dual stopping criterion at no extra cost. Substantial
experiments demonstrate the effectiveness of our method, both in terms of
computation times and robustness.
