"High-dimensional random phenomena and rare events"

《高维随机现象和罕见事件》

基本信息

批准号：
1713032
负责人：
Kavita Ramanan
金额：
$ 36万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1713032&HistoricalAwards=false
关键词：
dimensional random phenomena rare events

项目摘要

Applications in diverse fields, including data analysis, convex geometry, biology, physics, economics, engineering, operations research, and computer science, give rise to questions about random phenomena in high dimensions. For example, given data that lives in a high-dimensional space, what information can be obtained by studying lower-dimensional projections of the data? Given a large number of interacting agents, who strategically make choices based only on their own state and the distribution of states of the other agents, what do their equilibria (or optimal strategies) look like? This award supports the development of diverse mathematical techniques for the analysis of such questions. The focus will be on characterizing large deviations from typical behavior, which though rare, are often of crucial importance in applications. The investigator will help train new mathematics researchers, mentor early career researchers, and be involved in outreach efforts. The research will address three topics concerning high-dimensional random phenomena. The first involves the study of random projections of probability measures in high-dimensional Euclidean spaces. This is relevant both for the statistical analysis of high-dimensional data, as well as for asymptotic convex geometry and asymptotic functional analysis. The focus is on characterizing the large deviation behavior of such random projections, as the dimension goes to infinity. This is of interest because, unlike fluctuations, the large deviation behavior is non-universal and thus distinguishes between different high-dimensional distributions. The second topic considers properties of high-dimensional Gibbs distributions on a class of hard-core configurations associated with a graph. Whereas classical models assume discrete spins, the focus of this research is on versions with continuous spins, which exhibit quite different behavior and require new techniques for their analysis. The last topic concerns the analysis of fluctuations, large deviations and concentration inequalities for Nash equilibria in both static and dynamic games of mean-field type with a large number of rational agents. This has implications for a broad range of applications and entails the development of new mathematical techniques, including large deviations principles for set-valued random elements, and analysis of solutions to a master equation for mean-field games.

数据分析、凸几何、生物学、物理学、经济学、工程、运筹学和计算机科学等不同领域的应用引发了有关高维随机现象的问题。例如，给定存在于高维空间中的数据，通过研究数据的低维投影可以获得什么信息？给定大量相互作用的智能体，它们仅根据自己的状态和其他智能体的状态分布进行战略选择，那么它们的均衡（或最优策略）是什么样的？该奖项支持开发用于分析此类问题的多种数学技术。重点是描述与典型行为的巨大偏差，虽然这种情况很少见，但在应用中通常至关重要。调查员将帮助培训新的数学研究人员，指导早期职业研究人员，并参与外展工作。该研究将解决有关高维随机现象的三个主题。第一个涉及高维欧几里得空间中概率测度的随机投影的研究。这对于高维数据的统计分析以及渐近凸几何和渐近泛函分析都相关。重点是当维度趋向无穷大时，表征此类随机投影的大偏差行为。这是很有趣的，因为与波动不同，大偏差行为是非普遍的，因此可以区分不同的高维分布。第二个主题考虑与图相关的一类硬核配置的高维吉布斯分布的属性。虽然经典模型假设离散自旋，但本研究的重点是具有连续自旋的版本，它们表现出截然不同的行为，需要新技术进行分析。最后一个主题涉及大量理性主体平均场型静态和动态博弈中纳什均衡的波动、大偏差和集中不等式的分析。这对广泛的应用产生了影响，并需要开发新的数学技术，包括集值随机元素的大偏差原理，以及平均场博弈主方程解的分析。