Analysis of High-Dimensional Stochastic Systems

在许多应用程序领域中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可处理的方式研究和分析这些数据。高维数据的随机投影是一种简单且计算效率高的技术，它通过控制误差量来换取更快的处理时间和更小的模型尺寸，从而降低数据集的维数。虽然人们已经研究了随机预测的几个特性，但随机预测对数据中出现在数据分布尾部的离群值有什么影响的问题还没有得到很好的理解。这个项目是严格表征高维分布的随机投影的尾部。对这种尾部行为的理解也将有助于我们了解如何通过观察低维投影来区分高维分布。这在许多领域都有潜在的应用，包括计算机科学、数据分析、统计学和凸几何。用于数据分类的另一组数据分析技术包括光谱聚类和相关聚类。这两种技术都与相关矩阵的算子范数有关。这个项目将描述算子规范的渐近性，在高维极限下，并研究数值方法稳定性的潜在应用（例如，矩阵条件数估计）以及聚类问题。该项目具有很强的教育成分，包括数学推广、研究生的研究培训和新课程的开发。该项目有两个主题。第一个主题与研究大偏差或高维测量随机投影的尾部行为有关。这些在高维统计和概率以及渐近凸几何中都很有趣，其中感兴趣的对象是高维凸体的体积或表面测量。虽然人们对随机投影的波动进行了很好的研究，最终得出了著名的凸集中心极限定理，但人们对随机投影的大偏差或尾部概率却知之甚少。该项目的目标是建立大偏差原则，既在投影方向上平均（退火设置），也以投影方向为条件，以及尖锐的大偏差估计，并了解它们对高维统计和渐近凸几何的影响。第二个主题涉及高维随机矩阵算子规范的渐近性研究，这与各种上下文相关，包括优化理论、理论计算机科学和泛函分析，以及机器学习和数据分析的应用。虽然与奇异值相一致的二对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-to-p范数，在这些范数中，谱理论不能再使用，因此需要开发基本的新技术，包括代数、分析和概率工具的组合。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.