"High-dimensional random phenomena and rare events"

《高维随机现象和罕见事件》

• 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

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.

在不同领域的应用，包括数据分析，凸几何，生物学，物理学，经济学，工程学，运筹学和计算机科学，引起了关于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，它们只根据自己的状态和其他智能体的状态分布进行策略性选择，它们的均衡（或最优策略）是什么样子的？该奖项支持开发用于分析此类问题的各种数学技术。 重点将是表征与典型行为的大偏差，这虽然罕见，但在应用中往往至关重要。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，并参与推广工作。 这项研究将涉及三个主题的高维随机现象。第一个涉及的随机投影的概率措施在高维欧氏空间的研究。 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是相关的。重点是表征这种随机投影的大偏差行为，因为尺寸趋于无穷大。这很有趣，因为与波动不同，大偏差行为是非普适的，因此可以区分不同的高维分布。第二个主题考虑高维吉布斯分布的一类硬核配置与图的属性。而经典模型假设离散自旋，本研究的重点是与连续自旋的版本，表现出相当不同的行为，并需要新的技术进行分析。最后一个主题涉及的波动分析，大偏差和集中不等式的纳什均衡在静态和动态的平均场类型的游戏与大量的理性代理。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的解的分析。

项目成果

Uniqueness of Gibbs measures for continuous hardcore models

连续硬核模型吉布斯测量的独特性

• DOI: 10.1214/18-aop1298
• 发表时间: 2019
• 期刊: The Annals of Probability
• 作者: Gamarnik, David;Ramanan, Kavita
• 通讯作者: Ramanan, Kavita

A conditional limit theorem for high-dimensional ℓᵖ-spheres

高维α球的条件极限定理

• DOI: 10.1017/jpr.2018.71
• 发表时间: 2018
• 期刊: Journal of Applied Probability
• 影响因子: 1
• 作者: Kim, Steven S.;Ramanan, Kavita
• 通讯作者: Ramanan, Kavita

From the master equation to mean field game limit theory: a central limit theorem

• DOI: 10.1214/19-ejp298
• 发表时间: 2018-04
• 期刊: Electronic Journal of Probability
• 影响因子: 1.4
• 作者: F. Delarue;D. Lacker;K. Ramanan

From the master equation to mean field game limit theory: Large deviations and concentration of measure

• DOI: 10.1214/19-aop1359
• 发表时间: 2018-04
• 期刊: The Annals of Probability
• 作者: F. Delarue;D. Lacker;K. Ramanan

Rare Nash Equilibria and the Price of Anarchy in Large Static Games

• DOI: 10.1287/moor.2018.0929
• 发表时间: 2017-02
• 期刊: Math. Oper. Res.
• 作者: D. Lacker;K. Ramanan