  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.
