Inference in high-dimension: statistics, computation and information theory

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This research proposal consists of two related research thrusts, all centered around the common goal of an integrated treatment of statistical and computational issues. The first research thrust concerns various issues associated with the use of structured regularization methods in high-dimensional inference. Such types of regularization are natural in different settings, including estimation of structured covariance matrices, graphical model selection, and hierarchical data modeling. The researchers propose to provide sharp characterizations of when structured regularization, with its typically higher computational costs, is guaranteed to yield improvements in statistical efficiency, or conversely, when structured regularization might impair statistical efficiency. The second research thrust addresses the role of statistical stability in optimization, and the development of new methodology for choosing path length parameters in iterative algorithms.Statistical inference problems of a high-dimensional nature---meaning where the number of observations n is similar to or even smaller than the number of parameters p---are ubiquitous throughout various areas of science and engineering, among them genomics, neuroscience, remote sensing, natural language processing, data compression, financial time series, and statistical signal processing. As a concrete instances, consider the problem of estimating the structure of a social network consisting of a large number of individuals (p could be 10,000 or larger) based on a relatively small number of snapshots (n could be 100 to 1,000). The overarching theme of the proposal is the development of new methodology and theory for high-dimensional data. Given the ubiquity of such data, such developments have the potential to impact a variety of fields making use of statistical modeling and tools, among them information technology (IT), neuroscience, remote sensing, and data compression. Moreover, the proposal is inter-disciplinary in nature, and so has the potential to strengthen bridges between statisticians and researchers in other departments (e.g., computer science, electrical and civil engineering) also working on IT applications. Via this type of intellectual unification, the proposed research is likely to have broader impact---much beyond any specific technical contributions---in terms of bridging different research communities, and providing broad training to graduate students and postdoctoral researchers.