Statistical Methods and Theory in Some High-Dimensional Problems

该研究项目将侧重于开发实用的方法，高效的算法和坚实的理论，用于选择重要特征，估计未知参数和预测高维数据的响应，特别是在特征数量远远大于样本数量的情况下。它将进一步发展最近提出的线性回归特征选择的方法和算法，将它们扩展到更一般的高维统计模型，研究它们在选择和估计中的一致性和最优性。该项目中开发的方法将与许多应用直接相关。该项目将专门研究两个重要领域的应用。第一个是信号处理，包括数据对象的有效采样、表示、传输和恢复。第二个是通信网络，包括检测和估计数据流的数量和变化的重要模式。 随着信息技术的飞速发展以及在现代科学实验中的应用，高维数据是统计研究和实践中的一个热点。具有丰富高维数据的重要领域包括生物信息学、信号处理、神经成像、通信网络等。在许多这样的科学和工程应用中，问题的大小是通过特征的数量来衡量的：生物信息学中的遗传成分，神经成像中的大脑区域或体素，或者互联网中的计算机和路由器。高维数据的一个主要挑战是问题的大小通常比要使用的数据的大小大得多。该项目的动机，并将直接适用于信号处理和监测通信网络。由于涉及高维数据的数学和统计问题的共性，该项目也将直接适用于生物信息学，神经成像和现代信息技术繁荣的许多学科。此外，该项目将产生重大的教育影响。

The research project will focus on developing practical methods, efficient algorithms and solid theory for the selection of important features, estimation of unknown parameters and prediction of responses with high-dimensional data, especially in the case where the number of features is much larger than the number of samples. It will further develop recently proposed methodologies and algorithms for feature selection in linear regression, extend them to more general high-dimensional statistical models, investigate their consistency and optimality properties in selection and estimation. The methodologies developed in the project will be directly relevant to many applications. The project will specifically investigate applications in two important areas. The first one is signal processing, including efficient sampling, representation, transmission and recovery of data objects. The second one is communications networks, including detection and estimation of significant patterns in volume and changes in data streams. High-dimensional data is an area of intense current interest in statistical research and practice due to the rapid development of information technologies and their applications to modern scientific experiments. Important fields with an abundance of high-dimensional data include bioinformatics, signal processing, neural imaging, communications networks and more. In many suchscientific and engineering applications, the size of the problem is measured by the number of features: genetic components in bioinformatics, brain regions or voxels in neural imaging, or computers and routers in theInternet. A main challenge in high-dimensional data is that the size of the problem is often much larger than the size of the data to be used. The project is motivated and will be directly applicable to signal processing and monitoring communications networks. Due to mathematical and statistical commonalities of problems involving high-dimensional data, the project will also be directly applicable to bioinformatics, neural imaging and many more disciplines where modern information technologies prosper. Furthermore, the project will have significant educational impact.