Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured quadratic cost, i.e., a function that can be written as a sum of pairwise cost functions between the nodes of a tree. To address this problem, we develop Tree-based Diffusion Schr\"odinger Bridge (TreeDSB), an extension of the Diffusion Schr\"odinger Bridge (DSB) algorithm. TreeDSB corresponds to a dynamic and continuous state-space counterpart of the multimarginal Sinkhorn algorithm. A notable use case of our methodology is to compute Wasserstein barycenters which can be recast as the solution of a mOT problem on a star-shaped tree. We demonstrate that our methodology can be applied in high-dimensional settings such as image interpolation and Bayesian fusion.

翻译：多边缘最优传输（mOT）作为最优传输（OT）的推广，旨在最小化代价函数关于具有指定边缘分布的积分。本文针对具有树结构二次代价函数（即可表示为树节点间成对代价函数之和）的熵正则化mOT问题展开研究。为此，我们提出了树基扩散薛定谔桥（TreeDSB）算法，这是扩散薛定谔桥（DSB）算法的扩展。TreeDSB对应多边缘Sinkhorn算法的动态连续状态空间版本。该方法的重要应用场景之一是计算Wasserstein重心，该问题可转化为星型树结构上的mOT问题求解。实验证明，该方法可有效应用于高维场景，如图像插值与贝叶斯融合。