Random Matrices and Related Problems

近年来，高维数据和大型随机矩阵的统计分析方法受到越来越多的关注。本项目旨在为这些问题开发新的方法，并扩展现有方法的适用性。高维数据处理新方法的研究既包括超高维数据处理工具的研究，也包括对高维固定样本量数据处理方法统计性质的理论研究。在这个项目中，PIS将开发新的统计方法，并将现有方法扩展到许多统计、物理和生物领域的新应用。这些方法的效率将通过模拟和对真实数据集的应用来证明。此外，研究成果还将促进教与学的活动。研究成果将通过会议报告和出版物传播。该项目为高维数据和大型随机矩阵的相关性结构测试开发了新的方法。PIS将调查一系列与高维数据密切相关的主题，包括高维数据的相关性结构测试、高维均值向量和高维协方差矩阵的测试，以及高维数据完全独立性的非参数测试。此外，PI还将研究大型随机矩阵的谱半径，例如样本协方差矩阵的主子式的最大特征值的极限分布、马尔可夫矩阵的最大特征值以及截断酉阵的乘积的最大绝对值的极限分布。该项目的研究将扩大高维统计方法和大型随机矩阵的应用范围。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。

Methods for statistical analysis of high dimensional data and large random matrices have receive increased attention in recent years. This project aims to develop new methods for these problems and extend the applicability of existing methods. The research on new methodology for high dimensional data includes both tools for ultra-high dimensional data, and the theoretical study of statistical properties for methods applicable to data with high dimensionality and fixed sample sizes. In this project, the PIs will develop new statistical methods and extend existing methods with novel applications in many fields of statistics, physics, and biology. The efficiency of these methodologies will be demonstrated via simulations and applications to real data sets. In addition, the research results will promote teaching and learning activities. The research results will be disseminated through conference presentations and publications. This project develops new methodologies for tests on the dependence structure for high dimensional data and large random matrices. The PIs will investigate a series of topics that are closely related to high dimensional data, including tests on dependence structure for high dimensional data, tests on high-dimensional mean vectors and high-dimensional covariance matrices, and nonparametric tests for complete independence for high-dimensional data. In addition, the PIs will study the spectral radii of large random matrices such as the limiting distribution for the maximum eigenvalue from principal minors of sample covariance matrices, the largest eigenvalues of Markov matrices, and the limiting distribution for the largest absolute value for the products of truncated unitary matrices. The research in this project will expand the scope of the application of high-dimensional statistical methods and large random matrices.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.