Tree-Based Diffusion Schrödinger Bridge with Applications to Wasserstein Barycenters
Main Conference Track
Authors: Maxence Noble, Valentin De Bortoli, Arnaud Doucet, Alain Durmus

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.