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在许多领域中自然出现，从遗传学和流行病学到大气和环境科学，医学科学，社会科学和人工智能。在大数据时代，迫切需要对这些大数据矩阵进行透彻的理解。然而，目前的文献大多集中在人口协方差矩阵是确定性的情况下。该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，该项目将为高维噪声数据集的分析提供更好的建模和工具，从而提供更有意义和可解释的信息。该项目旨在研究一类具有随机总体协方差结构的高维矩阵，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与极值理论的交叉进行了全面的研究。 目标包括：（1）. 对具有随机协方差结构的高维样本或可分离协方差矩阵的顶特征值和主成分建立了新的理论和结果;讨论了自举方法是否适用于高维推理，以及当标准的自举方法在海量数据中失效时如何改进;（3）.构造新的统计量用于高维椭圆分布数据的统计推断问题，包括重尾数据集;（4）。为全连接双层神经网络的相变提供新的见解。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。

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.