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.
