Measure Transportation And Notions Of Dimensionality In High Dimensional Probability

在高维概率中测量传输和维数概念

• 批准号: 2331920
• 负责人: Yair Shenfeld
• 金额: $ 13.78万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2331920&HistoricalAwards=false
• 关键词: Measure; Transportation; Notions; Dimensionality; Dimensional

High dimensional probability is a mathematical theory which aims to explain the behavior of systems with very large number of degrees of freedom. Such systems are of utmost importance in the physical sciences and in a world where social media and communication networks constantly generate huge data sets. This project aims to develop theoretical tools in high dimensional probability via two methods. The first method will exploit the intuition that in many practical situations, although the system contains numerous degrees of freedom, the relevant information is contained in a much smaller set of variables. The project will aim to quantify this intuition by developing the notion of intrinsic dimensionality. The second method aims to represent a given complex system as the transformation of a much simpler system. The project also includes educational efforts, including mentoring undergraduate and graduate students, and the dissemination of the work in professional meetings and to the public. The long-term goal is to improve understanding of high-dimensional probability measures via two approaches. The first approach is to develop an original concept of intrinsic dimensionality in the context of functional inequalities. The notion of dimension plays a crucial role in functional inequalities via the curvature-dimension condition, but it ignores the fact that the measures of interest can live on lower-dimensional spaces. This omission can lead to inefficient functional inequalities; this project hopes to bridge this gap. The second approach is to develop new measure transportation methods based on stochastic processes and probabilistic flows. For example, Lipschitz transport maps provide a powerful way to transfer functional inequalities from simple source measures to complicated target measures. Only a few such Lipschitz transport maps are known to exist, which limits the applicability of the transport method. The project will utilize the tools of stochastic analysis and renormalization group methods to build transport maps with desirable regularity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

高维概率是一种数学理论，旨在解释具有非常大量自由度的系统的行为。在物理科学和社交媒体和通信网络不断生成庞大的数据集的世界中，这种系统至关重要。该项目旨在通过两种方法以高维概率开发理论工具。第一种方法将利用以下直觉，即在许多实际情况下，尽管该系统包含许多自由度，但相关信息包含在一组较小的变量中。该项目的目的是通过发展内在维度的概念来量化这一直觉。第二种方法旨在将给定的复杂系统表示为更简单的系统的转换。该项目还包括教育工作，包括指导本科生和研究生，以及在专业会议和公众中传播这项工作。长期目标是通过两种方法提高对高维概率度量的理解。 第一种方法是在功能不平等的背景下发展一个内在维度的原始概念。维度的概念通过曲率维度条件在功能不平等中起着至关重要的作用，但它忽略了感兴趣的度量可以生活在较低维度的空间上的事实。这种遗漏会导致功能不平等效率低下。这个项目希望弥合这一差距。第二种方法是基于随机过程和概率流开发新的测量运输方法。例如，Lipschitz传输图提供了一种有力的方法，可以将功能不平等从简单的源指标转移到复杂的目标度量。已知只有少数这样的Lipschitz运输图存在，这限制了传输方法的适用性。该项目将利用随机分析和重新规定的方法的工具来建立具有理想规律性的运输图。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准来通过评估来支持的。