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：本文研究成对代价下多边际最优输运(MMOT)的数值求解方法。MMOT作为经典双边际最优输运的自然推广，在图像处理、密度泛函理论和机器学习等领域具有重要应用，但尚缺乏高效精确的数值方法。流行的熵正则化方法可能存在数值不稳定性和模糊问题。受Jacobs和Léger提出的往返法启发，我们研究了成对代价的MMOT问题。首先，这类问题具有图表示形式，我们证明了等价的具有树表示形式的MMOT问题；其次，我们通过基于对偶形式的梯度方法，提出了一种在有根树上求解MMOT的新算法；最后，我们获得了可用于无正则化应用的精确解。