具有树结构费用的多边际最优输运与薛定谔桥问题
Multi-marginal Optimal Transport with a Tree-structured cost and the Schrödinger Bridge Problem
摘要：
最优输运问题最近已发展成为一个强大的框架，各种应用的估计和控制。最优输运理论和应用的许多最新进展都是基于用熵项正则化问题，将其与薛定谔桥问题联系起来，从而与随机最优控制联系起来。此外，熵正则化使得即使对于大规模设置，在其他方面计算要求高的最优输运问题也是可行的。这导致了在广泛领域中基于最优输运的方法的加速发展。这些应用中的许多都具有底层图结构，例如信息融合和跟踪问题可以用树来描述。在这项工作中，我们考虑多边际最优输运问题的费用函数，根据一个树结构。熵正则化的多边际最优输运问题可以看作是薛定谔桥问题的一个推广，它具有相同的树结构，利用这些联系，我们扩展了经典最优输运问题的计算方法，以便有效地求解结构化的多边际最优输运问题.特别是，该算法只需要相对较小维度的矩阵向量乘法。我们表明，多边缘正则化引入较少的扩散，相比常用的成对正则化，因此更适合于许多应用。数值例子说明了这一点，我们最后应用所提出的框架跟踪一个不可区分的代理人的合奏。
英文摘要：
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 Schrödinger 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 Schrödinger 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.
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