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Multimarginal Optimal Transport with a Tree-Structured Cost and the Schrödinger Bridge Problem
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2021-07-06 , DOI: 10.1137/20m1320195 Isabel Haasler , Axel Ringh , Yongxin Chen , Johan Karlsson
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2021-07-06 , DOI: 10.1137/20m1320195 Isabel Haasler , Axel Ringh , Yongxin Chen , Johan Karlsson
SIAM Journal on Control and Optimization, Volume 59, Issue 4, Page 2428-2453, January 2021.
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 led to an accelerated development of optimal transport based methods in a broad range of fields. Many of these applications have an underlying graph structure, for instance, information fusion and tracking problems can be described by trees. In this work we consider multimarginal optimal transport problems with a cost function that decouples according to a tree structure. The entropy regularized multimarginal 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 multimarginal optimal transport problems in an efficient manner. In particular, the algorithm requires only matrix-vector multiplications of relatively small dimensions. We show that the multimarginal 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 the tracking of an ensemble of indistinguishable agents.
中文翻译：
具有树结构成本和薛定谔桥问题的多边际最优传输
SIAM Journal on Control and Optimization，第 59 卷，第 4 期，第 2428-2453 页，2021 年 1 月。
最优传输问题最近已发展成为用于估计和控制中各种应用的强大框架。最优传输理论和应用的许多最新进展都是基于用熵项对问题进行正则化，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，即使对于大规模设置，熵正则化使原本需要计算的最优传输问题变得可行。这导致在广泛的领域加速开发基于最佳传输的方法。许多这些应用程序都有一个底层的图结构，例如，信息融合和跟踪问题可以用树来描述。在这项工作中，我们考虑了具有根据树结构解耦的成本函数的多边际最优传输问题。熵正则化多边际最优输运问题可以看作是树结构相同的薛定谔桥问题的推广，通过利用这些联系，我们扩展了经典最优输运问题的计算方法，以解决结构化多边际最优输运问题以有效的方式。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，与常用的成对正则化相比，多边际正则化引入的扩散更少，因此更适合许多应用。数值例子说明了这一点，
更新日期：2021-07-07
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 led to an accelerated development of optimal transport based methods in a broad range of fields. Many of these applications have an underlying graph structure, for instance, information fusion and tracking problems can be described by trees. In this work we consider multimarginal optimal transport problems with a cost function that decouples according to a tree structure. The entropy regularized multimarginal 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 multimarginal optimal transport problems in an efficient manner. In particular, the algorithm requires only matrix-vector multiplications of relatively small dimensions. We show that the multimarginal 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 the tracking of an ensemble of indistinguishable agents.
中文翻译：
具有树结构成本和薛定谔桥问题的多边际最优传输
SIAM Journal on Control and Optimization，第 59 卷，第 4 期，第 2428-2453 页，2021 年 1 月。
最优传输问题最近已发展成为用于估计和控制中各种应用的强大框架。最优传输理论和应用的许多最新进展都是基于用熵项对问题进行正则化，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，即使对于大规模设置，熵正则化使原本需要计算的最优传输问题变得可行。这导致在广泛的领域加速开发基于最佳传输的方法。许多这些应用程序都有一个底层的图结构，例如，信息融合和跟踪问题可以用树来描述。在这项工作中，我们考虑了具有根据树结构解耦的成本函数的多边际最优传输问题。熵正则化多边际最优输运问题可以看作是树结构相同的薛定谔桥问题的推广，通过利用这些联系，我们扩展了经典最优输运问题的计算方法，以解决结构化多边际最优输运问题以有效的方式。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，与常用的成对正则化相比，多边际正则化引入的扩散更少，因此更适合许多应用。数值例子说明了这一点，
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