Analysis of High-Dimensional Stochastic Systems

高维随机系统分析

基本信息

批准号：
1954351
负责人：
Kavita Ramanan
金额：
$ 30万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954351&HistoricalAwards=false
关键词：
Analysis Dimensional Stochastic Systems

项目摘要

A common theme that arises in many domains of application is that data is high-dimensional and various techniques have to be used to study and analyze such data in a computationally tractable way. The random projection of high-dimensional data is a simple and computationally efficient technique to reduce the dimensionality of a data set by trading a controlled amount of error for faster processing times and smaller model sizes. While several properties of random projections have been studied, the question of what random projections do to outliers in the data, which appear in the tails of the data distribution, is not well understood. This project is to rigorously characterize the tails of random projections of high-dimensional distributions. Understanding such tail behavior will also provide insight into how to distinguish between high-dimensional distributions by looking at their lower-dimensional projections. This has potential applications in a variety of fields including computer science, data analysis, statistics, and convex geometry. Another set of data analysis techniques used for data classification include spectral clustering and correlation clustering. Both these techniques are related to certain operator norms of associated matrices. This project will characterize the asymptotics of operator norms, in the limit of high dimensions, and study potential applications to the stability of numerical methods (for example, matrix condition number estimation) as well as clustering problems. The project has a strong educational component, with provisions for math outreach, research training of graduate students, and development of new courses.The project has two themes. The first theme relates to the study of large deviations or the tail behavior of random projections of high-dimensional measures. These are of interest in high-dimensional statistics and probability, as well as asymptotic convex geometry, where the object of interest is the volume or surface measure of a convex body in high dimensions. While fluctuations of random projections have been well studied, culminating in the celebrated central limit theorem for convex sets, large deviations or tail probabilities of random projections are less well understood. A goal of the project is to establish large deviations principles, both averaged over the direction of projection (the annealed setting) and conditioned on the direction of projection, as well as sharp large deviation estimates, and understand their ramifications for high-dimensional statistics and asymptotic convex geometry. The second theme relates to the study of the asymptotics of operator norms for high-dimensional random matrices, which are relevant in a variety of contexts, including optimization theory, theoretical computer science and functional analysis, with applications to machine learning and data analysis. While the two-to-two norm, which coincides with the singular value, has been well studied, the focus will be to study more general r-to-p norms, where spectral theory can no longer be used and thus will require the development of fundamentally new techniques, involving a combination of tools from algebra, analysis and probability.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.

在许多应用领域中出现的一个共同主题是数据是高维的，并且必须使用各种技术来以计算方式研究和分析此类数据。高维数据的随机投影是一种简单且计算上有效的技术，可以通过交易受控的误差来减少数据集的维度，以换取更快的处理时间和较小的模型尺寸。尽管已经研究了随机预测的几种属性，但对数据分布尾部出现的随机预测对数据中的异常值的问题尚未得到充分了解。该项目将严格地描述高维分布的随机预测的尾巴。了解这种尾巴行为还将提供有关如何通过查看其较低维度预测来区分高维分布的洞察力。这在包括计算机科学，数据分析，统计和凸几何的各个领域具有潜在的应用。用于数据分类的另一组数据分析技术包括光谱群集和相关聚类。这两种技术都与相关矩阵的某些操作员规范有关。该项目将以高维度的限制来表征操作员规范的渐近学，并研究潜在的应用数值方法的稳定性（例如，矩阵条件号估计）以及聚类问题。该项目具有强大的教育组成部分，包括数学外展，研究生的研究培训以及新课程的开发规定。该项目有两个主题。第一个主题涉及研究大偏差或高维措施随机投影的尾巴行为。这些在高维统计和概率以及渐近凸几何形状中引起了人们的关注，其中感兴趣的对象是凸体的体积或表面量度高。尽管对随机预测的波动进行了充分的研究，但最终以凸套件的著名中心限制定理，但对随机投影的较大偏差或尾巴概率的了解程度较低。该项目的一个目的是建立大偏差原则，既平均在投影方向（退火设置），并以投影方向以及急剧的较大偏差估计为条件，又了解其对高维统计的后果和高度统计的后果。第二个主题涉及对高维随机矩阵的操作员规范的渐近学研究，这些矩阵在各种情况下与包括优化理论，理论计算机科学和功能分析有关，以及在机器学习和数据分析中的应用。虽然已经对与单一价值相吻合的两到两个规范进行了充分的研究，但重点将是研究更一般的R到P规范，在这种情况下，光谱理论无法再使用，因此需要开发从根本上进行新技术的开发，涉及代数的工具组合，这些工具来自代数，分析和概率，该奖项反映了NSF的Infort Infort and Internition的构建，并构建了构建的构建，其构建的价值是对eym的构建范围，这些奖项的价值是eym eyt the eyt eym eyr eym eyt yy，影响审查标准。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Large deviation principles for lacunary sums

缺额金额大偏差原则

DOI：
10.1090/tran/8788
发表时间：
2022
期刊：
Transactions of the American Mathematical Society
影响因子：
1.3
作者：
Aistleitner, Christoph;Gantert, Nina;Kabluchko, Zakhar;Prochno, Joscha;Ramanan, Kavita
通讯作者：
Ramanan, Kavita

Marginal dynamics of interacting diffusions on unimodular Galton–Watson trees