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

Tree-Based Diffusion Schr\"odinger Bridge with Applications to Wasserstein Barycenters