Optimal transport: from two to many marginals

该建议围绕最优运输，耦合两个概率测度（边际）的变分问题，以最小化指定成本函数的期望值。 这个问题可以追溯到1781年的Gaspard Monge，他感兴趣的是将一堆泥土移动到相同体积的洞中，以最小化泥土移动的平均距离（这里的概率密度分别由堆和洞的高度和深度给出，成本函数是欧几里得距离）。 这个领域自80年代后期以来蓬勃发展;它有许多应用，无论是在数学内部还是数学之外，并涉及分析，偏微分方程，几何和概率。 解的结构现在已经很好地理解了;极小值落在其中一个变量上的图上并且是唯一的条件也是众所周知的。 这些最优映射可以通过某些Monge-Ampere型偏微分方程的解来表征，并且已经发展了一个深正则性（或光滑性）理论。 解的特殊结构在许多应用中起着关键作用。 大多数拟议的研究涉及一种称为多边际最优运输的变体;这是一个以最大效率对齐多个概率分布的问题，同样是相对于给定的成本函数。 对多边缘问题的兴趣相对较新，但在过去几年中呈指数级增长，主要是由于新兴应用的惊人多样性：在密度泛函理论中排列电子以最小化相互作用能，在数据科学中在分布之间插值，在经济学中在多边市场中匹配代理等。虽然已经有了显著的进展，要全面了解解决办法的结构，仍有许多工作要做。 一个重要的一般主题是理解两个边际问题的哪些性质可以延续到多边际环境中。 答案取决于成本函数的方式是微妙的，仍然只有部分理解。 一个二分法已经开始出现之间的好的成本函数，其中的解决方案的行为很像在两个边缘的情况下（解决方案是唯一的，集中在图形上的一个变量）和那些他们表现出更多的异国情调和意想不到的行为（解决方案可能集中在高维集和非唯一）。 分类仍然是粗略的，双方的解决方案必须得到更好的理解。

This proposal revolves around optimal transport, the variational problem of coupling two probability measures (marginals) in order to minimize the expected value of a prescribed cost function. The problem dates back to Gaspard Monge in 1781, who was interested in moving a pile of dirt into a hole of the same volume in order to minimize the average distance that the dirt moves (here the probability densities are given by the height and depth of the pile and hole, respectively, and the cost function is the Euclidean distance). This field has flourished since the late 80's; it has many applications, both within and beyond mathematics, and touches on analysis, partial differential equations, geometry and probability. The structure of solutions is now quite well understood; conditions under which the minimizer lies on a graph over one of the variables, and is unique, are well known. These optimal maps can be characterized by solutions to certain Monge-Ampere type partial differential equations and a deep regularity (or smoothness) theory has been developed. The particular structure of solutions plays a key role in many applications. Much of the proposed research involves a variant known as multi-marginal optimal transport; this is the problem of aligning several probability distributions with maximal efficiency, again relative to a given cost function. Interest in multi-marginal problems is relatively new, but has increased exponentially over the past few years, due largely to a surprisingly diverse collection of emerging applications: aligning electrons to minimize interaction energy in density functional theory, interpolating between distributions in data science, matching agents in multi-sided markets in economics, etc. Though there has been significant progress, much remains to be done in order to have a complete understanding of the structure of solutions. An important general theme is to understand which properties of the two marginal problem carry over to the multi-marginal setting. The answer depends on the cost function in ways which are subtle and still only partially understood. A dichotomy has begun to emerge between nice cost functions, for which the solution behaves much like in the two marginal case (solutions are unique and concentrate on graphs over one of the variables) and those for which they exhibit much more exotic and unexpected behaviour (solutions may concentrate on high dimensional sets and be non-unique). The classification remains crude, and solutions on both sides must be better understood.