Modelling Covariance Structure Randomly, with Applications in Bootstrapping, Robust Statistics, and Deep Learning

具有随机协方差结构的高维数据自然而然地出现在许多领域，从遗传学和流行病学到大气和环境科学、医学、社会科学和人工智能。在大数据时代，深入了解这些大数据矩阵是迫切需要的。然而，目前的文献大多集中在总体协方差矩阵是确定性的情况下。这个项目将通过建立具有随机总体协方差结构的数据矩阵的新结果来改变这一点。研究结果将展示随机协方差结构如何有助于增强对复杂和海量数据集的理解。此外，该项目将产生新的和更好的建模和分析高维噪声数据集的工具，可以提供更有意义和可解释的信息。本项目旨在研究一类具有随机总体协方差结构的一般高维矩阵，并解决高维统计和深度学习中的几个挑战。首次对随机矩阵理论和极值理论的交集进行了全面的研究。目标包括：(1)。建立了具有随机协方差结构的大维样本或可分离协方差矩阵的顶本征值和主成分的新的理论和结果；回答了Bootstrapping是否适用于高维推理，以及当标准Bootstrapping过程对海量数据无效时如何修改；(3)为涉及高维椭圆分布数据(包括重尾数据集)的统计推断问题构造新的统计量；对完全连接的两层神经网络的相变提供了新的见解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

High dimensional data with random covariance structure arises naturally in many fields, ranging from genetics and epidemiology to atmospheric and environmental sciences, medical sciences, social sciences and artificial intelligence. A thorough understanding of these large data matrices is in urgent demand in the era of Big Data. However, the current literature has mostly focused on the case when the population covariance matrix is deterministic. This project will change this by establishing novel results for data matrices with random population covariance structure. The research findings will demonstrate how the random covariance structure can help to enhance the understanding of complex and massive data sets. Moreover, this project will result in novel and better modeling and tools for analyzing high dimensional noisy data sets, which can provide more meaningful and interpretable information.This project aims to study a general class of large dimensional matrices with random population covariance structure and address several challenges in high dimensional statistics and deep learning. It is the first time that the intersection between random matrix theory and extreme value theory is studied in full generality. The goals include: (1). Establishing novel theory and results for the top eigenvalues and principal components of large dimensional sample or separable covariance matrix with random covariance structure; (2). Answering the question whether bootstrapping is suitable for high dimensional inference, and how we can modify the standard bootstrapping procedure when it fails for massive data; (3). Constructing new statistics for statistical inference problems involving high dimensional elliptically distributed data in full generality, including heavy tailed data sets; (4). Providing novel insights on the phase transitions of fully connected two-layer neural networks.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.