# On the Complexity of Approximating Multimarginal Optimal Transport

Tianyi Lin, Nhat Ho, Marco Cuturi, Michael I. Jordan.

Year: 2022, Volume: 23, Issue: 65, Pages: 1−43

#### Abstract

We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here between $m$ 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 $m \geq 3$. This negative result implies that some combinatorial 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 $\tilde{\mathcal{O}}(n^{3m})$. 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 $\tilde{\mathcal{O}}(m^3n^m\varepsilon^{-2})$ for a tolerance $\varepsilon \in (0, 1)$. 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 $\tilde{\mathcal{O}}(m^3n^{m+1/3}\varepsilon^{-4/3})$. This bound is better than that of the first algorithm in terms of $1/\varepsilon$, 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.