  The optimal transport problem has recently developed into a powerful
framework for various applications in estimation and control. Many of the
recent advances in the theory and application of optimal transport are based on
regularizing the problem with an entropy term, which connects it to the
Schr\"odinger bridge problem and thus to stochastic optimal control. Moreover,
the entropy regularization makes the otherwise computationally demanding
optimal transport problem feasible even for large scale settings. This has lead
to an accelerated development of optimal transport based methods in a broad
range of fields. Many of these applications have a underlying graph structure,
for instance information fusion and tracking problems can be described by
trees. In this work we consider multi-marginal optimal transport problems with
a cost function that decouples according to a tree structure. The entropy
regularized multi-marginal optimal transport problem can be viewed as a
generalization of the Schr\"odinger bridge problem with the same
tree-structure, and by utilizing these connections we extend the computational
methods for the classical optimal transport problem in order to solve
structured multi-marginal optimal transport problems in an efficient manner. In
particular, the algorithm requires only matrix-vector multiplications of
relatively small dimensions. We show that the multi-marginal regularization
introduces less diffusion, compared to the commonly used pairwise
regularization, and is therefore more suitable for many applications. Numerical
examples illustrate this, and we finally apply the proposed framework for
tracking of an ensemble of indistinguishable agents.
