We consider the numerical solution of the discrete multi-marginal optimal transport (MOT) by means of the Sinkhorn algorithm. In general, the Sinkhorn algorithm suffers from the curse of dimensionality with respect to the number of marginals. If the MOT cost function decouples according to a tree or circle, its complexity is linear in the number of marginal measures. In this case, we speed up the convolution with the radial kernel required in the Sinkhorn algorithm by non-uniform fast Fourier methods. Each step of the proposed accelerated Sinkhorn algorithm with a tree-structured cost function has a complexity of $\mathcal O(K N)$ instead of the classical $\mathcal O(K N^2)$ for straightforward matrix-vector operations, where $K$ is the number of marginals and each marginal measure is supported on at most $N$ points. In case of a circle-structured cost function, the complexity improves from $\mathcal O(K N^3)$ to $\mathcal O(K N^2)$. This is confirmed by numerical experiments.

翻译：本文考虑通过Sinkhorn算法数值求解离散多边缘最优传输问题。一般而言，Sinkhorn算法在边缘数量上存在维度灾难。若MOT代价函数按树状或环状结构解耦，其复杂度与边缘测度数量呈线性关系。在此情形下，我们采用非均匀快速傅里叶方法加速Sinkhorn算法中所需的径向核卷积运算。对于树状代价函数，所提出的加速Sinkhorn算法每一步的计算复杂度为$\mathcal O(K N)$，而传统直接矩阵-向量运算的复杂度为$\mathcal O(K N^2)$，其中$K$为边缘数量，且每个边缘测度至多支持$N$个点。对于环状代价函数，复杂度从$\mathcal O(K N^3)$降至$\mathcal O(K N^2)$。数值实验验证了该方法的有效性。