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

这项研究提案包括两个相关的研究推动力，都围绕着综合处理统计和计算问题的共同目标。第一个研究重点涉及与在高维推理中使用结构化正则化方法相关的各种问题。这种类型的正则化在不同的设置中是自然的，包括结构化协方差矩阵的估计、图形模型选择和分层数据建模。研究人员建议提供结构正则化的明确特征，当结构化正则化具有典型的较高计算成本时，可以保证提高统计效率，或者相反，当结构化正则化可能损害统计效率时。第二个研究重点是统计稳定性在优化中的作用，以及在迭代算法中选择路径长度参数的新方法的开发。高维性质的统计推断问题-意味着观测数量n类似于甚至小于参数数量p-普遍存在于科学和工程的各个领域，其中包括基因组学、神经科学、遥感、自然语言处理、数据压缩、金融时间序列和统计信号处理。作为具体实例，考虑基于相对较少数量的快照(n可以是100到1,000)来估计由大量个人(p可以是10,000或更大)组成的社交网络的结构的问题。该提案的主要主题是为高维数据开发新的方法和理论。鉴于这种数据无处不在，利用统计建模和工具，包括信息技术(IT)、神经科学、遥感和数据压缩，这种发展有可能对各种领域产生影响。此外，该提案具有跨学科性质，因此有可能加强统计学家与其他部门(例如计算机科学、电气和土木工程)也从事信息技术应用工作的研究人员之间的桥梁。通过这种类型的知识统一，拟议的研究可能会产生更广泛的影响-远远超出任何具体的技术贡献-在连接不同研究社区和为研究生和博士后研究人员提供广泛培训方面。

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.