Graph-structured multi-marginal optimal transport : Theory, applications, and efficient methods using entropy regularization
Abstract: This thesis deals with a class of multi-marginal optimal transport problems, which we call graph-structured multi-marginal optimal transport. The aim of the thesis is to work towards a unified framework for this class of problems. The included papers explore theoretical, computational, and practical aspects of the novel framework, e.g., connections to related areas, numerical algorithms for solving the problem, and applications in various fields. The first paper treats multi-marginal optimal transport problems with cost functions that decouple according to a tree. We show that the entropy regularized version of this problem can be seen as a generalization of the Schrödinger bridge problem to the same underlying tree. This connection is utilized to derive an efficient method for approximately solving the optimal transport problem. The second paper applies the framework to inverse problems in radar imaging. In this setting only partial information of the marginals in the optimal transport problem are available. We extend the methods from the first paper to handle the case of partial information. Several numerical examples illustrate that the framework can be used to find robust estimates of spatial spectra in information fusion and tracking applications. In the third paper we model network flow problems by utilizing graphstructured multi-marginal optimal transport. In particular, we show that the dynamic minimum-cost multi-commodity network flow problem can be formulated as a multi-marginal optimal transport problem that has an underlying structure, which can be described by a graph that contains cycles. The computational methods from the previous papers are extended to efficiently solve network flow problems. The fourth paper applies the graph-structured multi-marginal optimal transport framework to estimation and control problems for multi-agent systems. Moreover, the framework is utilized to prove a duality result between estimation and control of multi-agent systems. The fifth paper explores connections between graph-structured multimarginal optimal transport and probabilistic graphical models. We show that the entropy regularized optimal transport problem is equivalent to an inference problem for probabilistic graphical models with constraints on some of the marginal distributions. The sixth paper proves complexity results for graph-structured multimarginal optimal transport. Moreover, based on a junction tree factorization of the underlying graph we provide a general algorithm for approximately solving graph-structured multi-marginal optimal transport. Our complexity bounds match the best known results for the optimal transport barycenter problem and the general multi-marginal optimal transport problem.
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