Random Matrices with Application to Quantum Computing and Econometrics

该项目开发了新的统计和概率方法，并扩展了一些现有的方法在统计和计量经济学中的新应用。这些学科的核心课题是随机矩阵理论。所提出的方法的效率将通过模拟和应用到真实的数据集来证明。该项目的成果将增强方法对高维数据设置的适用性，研究成果将应用于统计学和计量经济学。该项目将促进明尼苏达大学的教学、培训和学习活动。主要教育目标是：（a）在明尼苏达大学培养博士生;（B）促进专家和学生之间的合作。研究结果将通过会议介绍和出版物传播。研究人员计划开发新的方法来研究几种类型的随机矩阵的属性。它们包括Haar正交/酉矩阵、样本相关矩阵、麦克唐纳测度和循环正交系综。然后，研究人员将应用它们来回答统计和计量经济学问题。该项目由以下主要主题组成：（1）研究者计划研究独立范数对Haar不变正交/酉矩阵的逼近。当近似误差由一些已知的距离（包括总变差距离）度量时，解是已知的。 除了在理论研究中的相关性外，它们还应用于数据存储。研究者计划在Wasserstein距离下研究相同的近似问题;（2）研究者计划探索Macdonald测度并证明特征值渐近为高斯自由场。这些特征值的函数收敛到高斯乘性混沌。近年来，在量子力学的许多分支中，如二维Liouville量子引力中Liouville测度的构造、高斯自由场的厚点等都出现了一些活跃的研究。受此理解的启发，研究者将研究Macdonald测度;（3）研究者计划研究依赖数据的样本相关矩阵的最大条目。对于独立数据，同样的问题也是众所周知的。虽然从属情况更适用，但分析它们的技术步骤更密集。研究者提出了解决这一问题的新方法;（4）利用最近建立的相依数据和与极大值渐近独立的新方法，计划研究处理高-该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的学术价值和更广泛的影响审查标准。

This project develops new statistical and probabilistic methods and extends some existing methods with novel applications in statistics and econometrics. The core topic of these subjects is random matrix theory. The efficiency of the proposed methodologies will be demonstrated via simulations and applications to real data sets. The outcomes of the project will enhance the applicability of methods for high-dimensional data settings, and the research results will be applied in statistics and econometrics. The project will promote teaching, training, and learning activities at the University of Minnesota. The main educational goals are (a) to train PhD students at the University of Minnesota; and (b) to promote collaboration among experts and students. The research results will be disseminated through conference presentations and publications. The investigator plans to develop new methodologies to investigate properties of a few types of random matrices. They include Haar orthogonal/unitary matrices, sample correlation matrices, Macdonald measures and circular orthogonal ensembles. The investigator will then apply them to answer statistics and econometrics problems. The project consists of the following main themes: (1) the investigator plans to study the approximation of Haar-invariant orthogonal/unitary matrices by independent normals. The solutions are known when the approximation errors are measured by some well-known distances, including the total variation distance. Besides their relevance in theoretical research, they are also applied to data storage. The investigator plans to study the same approximation question under the Wasserstein distance; (2) the investigator plans to explore the Macdonald measure and show the eigenvalues are asymptotically a Gaussian Free Field. A function of these eigenvalues is shown to converge to a Gaussian multiplicative chaos. Nowadays active investigations such as the construction of the Liouville measure in 2d-Liouville quantum gravity or thick points of the Gaussian Free Field have appeared in many branches. Inspired by this understanding, the investigator will study the Macdonald measure; (3) the investigator plans to study the largest entries of sample correlation matrices for dependent data. The same problem for independent data is well-understood. Although the dependent case is more applicable, the technical steps to analyze them are more intensive. The investigator proposes new methods to solve this question; and (4) by using a new method on asymptotic independence between sums and maxima of dependent data, which was established recently by the investigator and his co-authors, it is planned to study problems dealing with high-dimensional panel data arising in econometrics.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.