Accelerating the Sinkhorn algorithm for sparse multi-marginal optimal transport by fast Fourier transforms

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.