Optimal transport: from two to many marginals

这一建议围绕着最优运输，即耦合两个概率度量(边际)以最小化指定成本函数的期望值的变分问题。这个问题可以追溯到1781年的Gaspard Monge，他对将一堆泥土移动到相同体积的洞中以使泥土移动的平均距离最小化感兴趣(这里的概率密度分别由桩和洞的高度和深度给出，成本函数是欧几里德距离)。这一领域自80年代末S以来蓬勃发展；它在数学内外有许多应用，涉及分析、偏微分方程式、几何和概率。解的结构现在已经很清楚了；极小值位于一个变量上的图上并且是唯一的条件是众所周知的。这些最优映射可以用某些Monge-Ampere型偏微分方程解来刻画，并且发展了一个深层正则性(或光滑性)理论。解决方案的特殊结构在许多应用中起着关键作用。 许多拟议的研究涉及一种称为多边际最优运输的变量；这是以最大效率对齐几个概率分布的问题，也是相对于给定的成本函数。人们对多边际问题的兴趣相对较新，但在过去几年里呈指数级增长，这主要是由于新兴应用的惊人多样性：密度泛函理论中的排列电子以最小化相互作用能量，数据科学中的分布之间的内插，经济学中的多边市场中的匹配代理，等等。虽然已经取得了重大进展，但要完全理解解决方案的结构，还有很多工作要做。 一个重要的一般主题是理解两个边缘问题的哪些性质延续到多边缘环境中。答案取决于成本函数，这种方式很微妙，但仍然只有部分人了解。一种两分法已经开始出现，一种是好的成本函数，其解的表现与两个边际情况下的很相似(解是唯一的，集中在其中一个变量的图形上)，另一种是它们表现出更奇异和意想不到的行为(解可能集中在高维集合上，但不是唯一的)。这种分类仍然很粗糙，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.