随机控制联络：理查德·辛霍恩在薛定谔桥上遇到了加斯帕德·蒙吉
Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge
摘要：
在1931年至1932年，薛定谔研究了一个热气体Gedanken实验，这是经验分布大偏差的一个例子，也是所谓的最大熵推断方法的早期例子。这个所谓的Schroedinger桥问题（SBP）最近被认为是Monge-Kantorovich最优质量输运（OMT）的正则化，导致后者的有效计算。具体地说，OMT与二次成本可以被看作是一个零温度限制SBP，这相当于最小化的亥姆霍兹自由能超过概率分布约束，以拥有给定的边缘。这个问题的特点是在最小化内能和最大化熵之间有一个微妙的妥协，由温度参数调节。这些概念是核心的迅速扩大领域的现代科学处理所谓的{ \em Sinkhorn算法}出现作为一个特殊情况下的算法首先研究了法国分析师罗伯特Fortet在1938年至1940年专门为薛定谔桥。由于对端点分布的约束，动态规划不是解决这些问题的合适工具。相反，Fortet的迭代算法及其离散对应物Sinkhorn迭代允许通过迭代求解所谓的{ \em Schroedinger系统}来计算。在连续和离散的时间和空间设置，{ \em随机控制}提供了一个重新制定和动态版本的这些问题。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制导，机器人或生物群控制，传感，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。这个多方面和多功能的框架，交织SBP和OMT，提供了基板的历史和技术概述本文所采取的领域。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
