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 $\tilde{O}(n^{3m})$. We then propose two simple and \textit{deterministic} algorithms for approximating the MOT problem. The first algorithm, which we refer to as \textit{multimarginal Sinkhorn} algorithm, is a provably efficient multimarginal generalization of the Sinkhorn algorithm. We show that it achieves a complexity bound of $\tilde{O}(m^3n^m\varepsilon^{-2})$ for a tolerance $\varepsilon \in (0, 1)$. This provides a first \textit{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 \textit{accelerated multimarginal Sinkhorn} algorithm, achieves the acceleration by incorporating an estimate sequence and the complexity bound is $\tilde{O}(m^3n^{m+1/3}\varepsilon^{-4/3})$. This bound is better than that of the first algorithm in terms of $1/\varepsilon$, and accelerated alternating minimization algorithm~\citep{Tupitsa-2020-Multimarginal} in terms of $n$. Finally, we compare our new algorithms with the commercial LP solver \textsc{Gurobi}. Preliminary results on synthetic data and real images demonstrate the effectiveness and efficiency of our algorithms.
翻译:我们研究多离子最佳运输(MOT)距离的复杂度, 以及传统最优运输距离的常规化。 首先, 我们发现, 当 $\ geq 3 时, MOT 问题的标准线性编程( LP) 代表不是一个最低成本流问题。 这个负面结果意味着, 一些组合算法, 比如网络简单度方法, 不适合 接近 MOT 问题, 而确定性内部点算法的最坏情况性复杂度仍然是 $\ tille{ (n) 3m} 美元。 我们然后提出两个简单的和 text{ 3q 3 美元 。 第一个算法, 我们称之为 textitle{ compilalginal Sinkhorn} 算算算法, 是一个可以调和的多离子化方法, 用来确定Sinkhorn 算法。 我们通过这个算法, 这个算法可以实现一个复杂的Sncredial=x 美元 。