In this paper, we address the numerical solution to the multimarginal optimal transport (MMOT) with pairwise costs. MMOT, as a natural extension from the classical two-marginal optimal transport, has many important applications including image processing, density functional theory and machine learning, but yet lacks efficient and exact numerical methods. The popular entropy-regularized method may suffer numerical instability and blurring issues. Inspired by the back-and-forth method introduced by Jacobs and L\'{e}ger, we investigate MMOT problems with pairwise costs. First, such problems have a graphical representation and we prove equivalent MMOT problems that have a tree representation. Second, we introduce a noval algorithm to solve MMOT on a rooted tree, by gradient based method on the dual formulation. Last, we obtain accurate solutions which can be used for the regularization-free applications.

翻译：在本文中,我们用双向成本来解决多边最佳运输(MMOT)的数字解决方案。MMOT是传统双边最佳运输的一种自然延伸,它具有许多重要应用,包括图像处理、密度功能理论和机器学习,但缺乏高效和精确的数字方法。流行的环球正规化方法可能受到数字不稳定和模糊问题的影响。在Jacobs和L\'{e}ger采用后方和前方方法的启发下,我们用双向成本来调查MMOT问题。首先,这些问题具有图形代表,我们证明与MMMOT问题相当,具有树面代表。第二,我们采用无价算法,用双向配方的梯度方法在根树上解决MMOT。最后,我们获得了可用于无规范应用的准确解决方案。