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.

许多应用领域出现的一个共同主题是数据是高维的，必须使用各种技术以计算上易于处理的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高效的技术，通过以受控的误差量换取更快的处理时间和更小的模型大小来降低数据集的维数。虽然已经研究了随机投影的几个属性，但随机投影对出现在数据分布尾部的数据中的异常值有何作用的问题尚不清楚。该项目旨在严格表征高维分布随机投影的尾部。了解这种尾部行为还可以深入了解如何通过查看低维投影来区分高维分布。这在计算机科学、数据分析、统计学和凸几何等多个领域都有潜在的应用。用于数据分类的另一组数据分析技术包括谱聚类和相关聚类。这两种技术都与关联矩阵的某些算子范数有关。该项目将在高维限制下表征算子范数的渐近性，并研究数值方法稳定性（例如矩阵条件数估计）以及聚类问题的潜在应用。该项目具有很强的教育成分，包括数学推广、研究生研究培训和新课程开发。该项目有两个主题。第一个主题涉及高维度量随机投影的大偏差或尾部行为的研究。这些对高维统计和概率以及渐近凸几何感兴趣，其中感兴趣的对象是高维凸体的体积或表面测量。虽然随机投影的波动已经得到了很好的研究，最终得出了著名的凸集中心极限定理，但随机投影的大偏差或尾部概率还不太清楚。该项目的目标是建立大偏差原理，在投影方向（退火设置）上进行平均并以投影方向为条件，以及尖锐的大偏差估计，并了解它们对高维统计和预测的影响。渐近凸几何。第二个主题涉及高维随机矩阵算子范数渐近性的研究，这与多种背景相关，包括优化理论、理论计算机科学和泛函分析，以及在机器学习和数据分析中的应用。虽然与奇异值一致的二到二范数已经得到了很好的研究，但重点将是研究更一般的 r 到 p 范数，其中谱理论不能再使用，因此需要开发该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

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：
2022-10
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

单模高尔顿沃森树上相互作用扩散的边际动力学