Numerically solving multi-marginal optimal transport (MMOT) problems is computationally prohibitive, even for moderate-scale instances involving $l\ge4$ marginals with support sizes of $N\ge1000$. The cost in MMOT is represented as a tensor with $N^l$ elements. Even accessing each element once incurs a significant computational burden. In fact, many algorithms require direct computation of tensor-vector products, leading to a computational complexity of $O(N^l)$ or beyond. In this paper, inspired by our previous work [$Comm. \ Math. \ Sci.$, 20 (2022), pp. 2053 - 2057], we observe that the costly tensor-vector products in the Sinkhorn Algorithm can be computed with a recursive process by separating summations and dynamic programming. Based on this idea, we propose a fast tensor-vector product algorithm to solve the MMOT problem with $L^1$ cost, achieving a miraculous reduction in the computational cost of the entropy regularized solution to $O(N)$. Numerical experiment results confirm such high performance of this novel method which can be several orders of magnitude faster than the original Sinkhorn algorithm.

翻译：数值求解多边际最优传输（MMOT）问题在计算上是极其困难的，即使对于涉及$l\ge4$个边际且支撑集大小为$N\ge1000$的中等规模问题也是如此。MMOT中的代价表示为一个具有$N^l$个元素的张量。即使仅访问每个元素一次也会带来巨大的计算负担。事实上，许多算法需要直接计算张量-向量乘积，导致计算复杂度达到$O(N^l)$或更高。在本文中，受我们先前工作[$Comm. \ Math. \ Sci.$, 20 (2022), pp. 2053 - 2057]的启发，我们观察到Sinkhorn算法中计算代价高昂的张量-向量乘积可以通过分离求和与动态规划，利用递归过程进行计算。基于这一思想，我们提出了一种快速张量-向量乘积算法来求解具有$L^1$代价的MMOT问题，将熵正则化解的计算成本奇迹般地降低至$O(N)$。数值实验结果证实了这种新方法的高性能，其计算速度可比原始Sinkhorn算法快数个数量级。