Application of Random Matrix Theory to Structured High-dimensional Data

这个应用程序的主要目标是利用频谱分析技术来处理高维推理问题。随机矩阵理论的技术，特别是Stieltjes变换的频谱措施，将被用来加强理解的影响，常用的统计程序在高维设置的观测之间的依赖关系。作为一个关键组成部分，大型随机矩阵之间的行和列的依赖性的频谱特性的调查将进行。此外，新的正则化方案将被开发，调整到数据的特性，包括可能的非平稳性的观察，并利用内在的简约结构的data.The拟议的应用程序是由在广泛的科学领域，如无线通信，光谱，基因组学，环境建模，大气科学，脑成像和计量经济学的问题。该提案的重点是发展理论理解和实用工具，以分析这些学科中出现的复杂和大规模数据。预计该项目的研究成果将使各学科的科学家和从业人员更广泛地获得处理高维数据的现代统计工具和概念。此外，通过该项目开发的工具和想法可能有助于下游技术，这些技术需要复杂的时变信号的复杂实时数据分析技术。

The main goal of this application is to utilize spectral analysis techniques for dealing with high-dimensional inferential problems. Techniques of random matrix theory, especially Stieltjes transforms of spectral measures, will be utilized to enhance understanding the effects of dependencies among observations on commonly used statistical procedures in high-dimensional settings. As a key component, investigations on the spectral characteristics of large random matrices with dependencies among both rows and columns will be carried out. In addition, new regularization schemes will be developed that are tuned to the characteristics of the data, including possible non-stationarity of the observations, and make use of the intrinsic parsimonious structures in the data.The proposed application is motivated by problems in a wide range of scientific fields such as wireless communication, spectrometry, genomics, environmental modeling, atmospheric science, brain imaging and econometrics. The emphasis of this proposal is to develop theoretical understanding and practical tools for analyzing complex and large-scale data arising in these disciplines. The research outputs from this project are expected to give wider access among scientists and practitioners in various disciplines to modern statistical tools and concepts for dealing with high-dimensional data. In addition, the tools and ideas developed through this project are likely to contribute towards downstream technologies that require sophisticated real-time data analysis techniques for complex time-varying signals.