论文标题
随机控制联络人：理查德·斯霍恩（Richard Sinkhorn）
Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge
论文作者
论文摘要
在1931/32年，施罗丁格（Schroedinger）研究了一种热气体gedankenexperiment，这是经验分布的巨大偏差的实例，以及所谓的最大熵推理方法的早期例子。这个所谓的Schroedinger桥梁问题（SBP）最近被认为是Monge-Kantorovich最佳大众传输（OMT）的正则化，从而有效地计算了后者。具体而言，具有二次成本的OMT可以看作是SBP的零温度极限，这相当于最小化Helmholtz的自由能，而不是概率分布，而概率分布约为具有给定边缘的概率。该问题的特征是由温度参数介导的精致折衷，在最小化内部能量和最大化熵之间。这些概念是现代科学快速扩展领域的核心，该领域涉及所谓的{\ em sndhorn算法}，该算法是法国分析师罗伯特·福特特（Robert Fortet）在1938/40年首次研究的算法的特殊情况，专门针对Schroedinger Bridges。由于终点分布的限制，动态编程不是攻击这些问题的合适工具。取而代之的是，Fortet的迭代算法及其离散的对应物，即sindhorn迭代，可以通过迭代求解所谓的{\ em schroedinger System}来计算。在连续的以及离散的时间和空间设置中，{\ em随机控制}都提供了这些问题的重新制定和动态版本。这些控制问题背后的形式主义引起了人们的注意，因为它们导致了航天器指导，机器人或生物群的控制，感应，主动冷却，网络路由以及计算机和数据科学中的各种新应用。这个多方体和多功能框架（交织的SBP和OMT）为本文所述的该领域的历史和技术概述提供了基材。
In 1931/32, Schroedinger studied a hot gas Gedankenexperiment, an instance of large deviations of the empirical distribution and an early example of the so-called maximum entropy inference method. This so-called Schroedinger bridge problem (SBP) was recently recognized as a regularization of the Monge-Kantorovich Optimal Mass Transport (OMT), leading to effective computation of the latter. Specifically, OMT with quadratic cost may be viewed as a zero-temperature limit of SBP, which amounts to minimization of the Helmholtz's free energy over probability distributions constrained to possess given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called {\em Sinkhorn algorithm} which appears as a special case of an algorithm first studied by the French analyst Robert Fortet in 1938/40 specifically for Schroedinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation by iteratively solving the so-called {\em Schroedinger system}. In both the continuous as well as the discrete-time and space settings, {\em stochastic control} provides a reformulation and dynamic versions of these problems. The formalism behind these control problems have attracted attention as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, network routing as well as in computer and data science. This multifacet and versatile framework, intertwining SBP and OMT, provides the substrate for a historical and technical overview of the field taken up in this paper.
