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：多边际最优传输（mOT）是经典最优传输（OT）的推广，旨在最小化代价函数相对于具有指定边际分布的测度积分。本文考虑带树结构二次代价的mOT的熵正则化版本，即代价函数可表示为树节点间成对代价函数之和。针对该问题，我们提出基于树的扩散薛定谔桥（TreeDSB），这是扩散薛定谔桥（DSB）算法的扩展。TreeDSB对应于多边际Sinkhorn算法在动态连续状态空间中的推广。该方法的一个典型应用场景是计算Wasserstein重心，该问题可重构为星形树结构上的mOT问题求解。实验表明，本方法可应用于高维场景，如图像插值与贝叶斯融合。