CAREER: Sparse Modeling and Estimation with High-dimensional Data

随着科学技术的进步，高维数据在各个领域中变得越来越普遍。这项研究的目标是为与这类数据相关的几类基本统计问题开发方法和理论。其中的核心问题是稀疏性在不同背景下的性质，以及它如何决定我们处理高维数据的能力或能力。研究人员研究了再生核希尔伯特空间的框架，利用稀疏性一般预测问题。该框架支持鼓励稀疏的各种流行方法之间的联系，并提供了一个以统一的方式研究它们的机会，这反过来又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估计和选择的问题。从世界上最强大的望远镜到最精细的原子力显微镜，从蓬勃发展的金融市场到快速发展的万维网，高维和海量数据正以惊人的速度产生。立即获得大量有趣和重要的信息提供了前所未有的机会，但也创造了独特的挑战，一般数学家，特别是统计学家。发展统计理论以了解其基本特征的性质，以及解决相关问题的方法，包括本提案中讨论的问题，将促进我们的智力探索和知识，无疑有利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.