Optimal transport: from two to many marginals

最优运输：从两个边际到多个边际

• 批准号: RGPIN-2018-04658
• 负责人: Pass, Brendan
• 金额: $ 1.68万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=661663
• 关键词: Optimal; transport; two; many; marginals

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.

该提案围绕最优传输，即耦合两个概率度量（边际）的变分问题，以便最小化规定成本函数的期望值。 这个问题可以追溯到 1781 年的加斯帕德·蒙日 (Gaspard Monge)，他有兴趣将一堆泥土移入相同体积的孔中，以最小化泥土移动的平均距离（这里的概率密度由泥土的高度和深度给出）分别是桩和洞，成本函数是欧几里得距离）。 该领域自20世纪80年代末以来一直蓬勃发展；它在数学内外都有许多应用，涉及分析、偏微分方程、几何和概率。 解决方案的结构现在已经很好理解了；最小化器位于某个变量的图上并且是唯一的条件是众所周知的。 这些最优映射可以通过某些 Monge-Ampere 型偏微分方程的解来表征，并且已经开发了深层正则性（或平滑性）理论。 解决方案的特定结构在许多应用中起着关键作用。***拟议的大部分研究涉及一种称为多边缘最优传输的变体；这是以最大效率对齐多个概率分布的问题，同样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，这主要是由于新兴应用的多样性令人惊讶：排列电子以最小化密度泛函理论中的相互作用能、数据科学中的分布之间的插值、匹配尽管已经取得了重大进展，但为了全面了解解决方案的结构，仍有许多工作要做。 ***一个重要的一般主题是了解两个边际问题的哪些属性可以延续到多边际设置。 答案取决于成本函数，其方式很微妙，而且仍然只有部分被理解。 在良好的成本函数之间已经开始出现一种二分法，对于这种情况，解决方案的行为非常类似于两个边际情况（解决方案是唯一的，并且集中在一个变量的图形上），而对于那些它们表现出更奇特和意想不到的行为的情况（解决方案可能集中在高维集上并且不是唯一的）。 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。