On the Complexity of Approximating Multimarginal Optimal Transport.

We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here be-tween m discrete probability distributions supported each on n support points. First, we show that the standard linear programming (LP) representation of the MOT problem is not a minimum-cost flow problem when m >= 3. This negative result implies that some combi-natorial algorithms, e.g., network simplex method, are not suitable for approximating the MOT problem, while the worst-case complexity bound for the deterministic interior-point algorithm remains a quantity of Oe(n3m). We then propose two simple and deterministic algorithms for approximating the MOT problem. The first algorithm, which we refer to as multimarginal Sinkhorn algorithm, is a provably efficient multimarginal generalization of the Sinkhorn algorithm. We show that it achieves a complexity bound of Oe(m3nm epsilon-2) for a tolerance epsilon is an element of (0, 1). This provides a first near-linear time complexity bound guarantee for approximating the MOT problem and matches the best known complexity bound for the Sinkhorn algorithm in the classical OT setting when m = 2. The second algorithm, which we refer to as accelerated multimarginal Sinkhorn algorithm, achieves the acceleration by incorporating an estimate sequence and the complexity bound is Oe(m3nm+1/3 epsilon-4/3). This bound is better than that of the first algorithm in terms of 1/epsilon, and accelerated alternating minimization algorithm (Tupitsa et al., 2020) in terms of n. Finally, we compare our new algorithms with the commercial LP solver GUROBI. Preliminary results on synthetic data and real images demonstrate the effectiveness and efficiency of our algorithms.