[论文解读] Multi-marginal Optimal Transport and Schr\"odinger Bridges on Trees

Isabel Haasler, Axel Ringh+2 more|arXiv (Cornell University)|Apr 15, 2020

Markov Chains and Monte Carlo Methods参考文献 51被引用 9

一句话总结

本文通过利用与薛定谔桥问题的联系，提出了一种针对树状图结构上熵正则化多对偶最优传输的高效算法。通过利用树结构，该方法将计算简化为小维度的矩阵-向量运算，相比成对正则化减少了扩散，从而实现了对不可区分代理的精确追踪。

ABSTRACT

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 Schrodinger 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 Schrodinger bridge problem on the same tree, 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.

研究动机与目标

• 解决在结构化图组织设置（特别是树结构）中多对偶最优传输的计算挑战。
• 将熵正则化最优传输扩展至具有树分解成本函数的多对偶问题。
• 与标准成对正则化相比，减少传输解中的扩散，提升在代理追踪等应用中的准确性。
• 开发一种仅依赖于低维矩阵-向量运算的可扩展算法，以应对大规模问题。
• 将该框架应用于基于结构化传输的不可区分代理集合的追踪问题。

提出的方法

• 该方法将具有沿树结构分解的成本函数的多对偶最优传输问题进行公式化。
• 利用相同树结构上熵正则化多对偶传输与薛定谔桥问题之间的联系。
• 通过沿树结构进行类似动态规划的更新，高效计算传输计划。
• 将问题简化为仅涉及每个节点处局部小规模局部边际的迭代矩阵-向量乘法。
• 该方法确保解与成本函数中变量的树状结构耦合保持一致。
• 该方法基于变分原理和熵正则化下的最优性条件推导得出。

实验结果

研究问题

• RQ1 当成本函数表现出树状结构分解时，如何高效求解多对偶最优传输？
• RQ2 在扩散和解的保真度方面，多对偶正则化相比成对正则化有何优势？
• RQ3 薛定谔桥框架能否扩展至树图上的多对偶传输？
• RQ4 该算法在实际中对大规模结构化传输问题的可扩展性如何？
• RQ5 该框架能否有效应用于涉及不可区分代理的追踪问题？

主要发现

• 所提出的算法相比成对正则化显著减少了传输计划中的扩散，从而获得更精确的解。
• 该方法仅需小维度的矩阵-向量运算，即使在大规模问题中也能实现高效计算。
• 数值示例表明，多对偶正则化比成对方法更好地保持了结构。
• 该框架成功实现了对不可区分代理集合的追踪，且保真度更高。
• 树状成本分解允许通过图上沿地的局部更新实现可扩展且精确的计算。