CAREER: Sparse Modeling and Estimation with High-dimensional Data

随着近年来科学技术的进步，高维数据在各个领域变得越来越普遍。这项拟议研究的目标是为与这类数据相关的几种基本统计问题开发方法和理论。其中的核心问题是不同环境下的稀疏性的本质，以及它如何决定我们处理高维数据的能力或无能。研究者研究了一种基于再现核希尔伯特空间的框架，以利用一般预测问题的稀疏性。该框架巩固了鼓励稀疏性的各种流行方法之间的联系，并提供了以统一的方式研究它们的机会，这反过来将促进改进方法和算法的发展。研究者还将考虑协方差矩阵的估计和选择问题。主要研究了大协方差矩阵稀疏性的各种概念的本质和联系，以及它们与高斯图模型的关系。从世界上最强大的望远镜到最精密的原子力显微镜，从繁荣的金融市场到快速发展的万维网，高维海量数据正以惊人的速度产生。即时获取大量有趣和重要的信息带来了前所未有的机遇，但也带来了独特的挑战，对数学家和统计学家来说尤其如此。统计理论的发展，以理解其基本特征的本质，以及解决相关问题的方法，包括本提案中讨论的问题，将推进我们的智力探索和知识，毫无疑问，有益于众多科学和技术领域——基因组学、医学成像、通信网络和金融只是几个众所周知的例子。

With the recent advances in science and technology, high dimensional data are becoming a commonplace in diverse fields. The goal of this proposed research is to develop methods and theory for several basic classes of statistical problems associated with this type of data. Among the central questions are the nature of sparsity in different contexts, and how it determines our ability or inability to deal with high dimensional data. The investigator studies a reproducing kernel Hilbert space based framework to exploit sparsity for general predictive problems. The framework underpins the connections among various popular methods that encourage sparsity, and provides an opportunity to study them in a unified fashion, which in turn will foster the development of improved methods and algorithms. The investigator will also consider the problem of covariance matrix estimation and selection. The research concentrates on understanding the nature of and connection among various notions of sparsity for large covariance matrix, and their relationship with Gaussian graphical models.From the world's most powerful telescopes to the finest atomic force microscopes, from the flourishing financial market to the fast-growing World-Wide Web, high dimensional and massive data are being produced at an astonishing rate. Immediate access to copious amount of interesting and important information presents unprecedented opportunities, but also creates unique challenges, to mathematicians in general and statisticians in particular. Development of statistical theory to understand the nature of their fundamental characteristics, and methodology to address the associated issues, including those discussed in this proposal, will advance our intellectual exploration and knowledge, and undoubtedly benefit a multitude of scientific and technological fields -- genomics, medical imaging, communication networks, and finance are just a few well known examples.