CAREER: Modernizing Classical Nonparametric and Multivariate Theory for Large-scale, High-dimensional Data Analysis

现代数据的维数和复杂性不断增加，促使各领域出现了许多新的数据分析工具，迫切需要进行严格的理论研究，如对不同来源的模型错规范的鲁棒性、分类和预测中的不确定性量化、常规方法在非标准设置下的统计性能保证等。虽然大多数经典理论不能直接适用于复杂数据的方法，部分原因是高度专业化的模型假设和多样化的算法，但这些长期建立的结果中所蕴含的深刻的统计思维仍然可以提供深刻的理论见解。当与随机矩阵理论、矩阵集中和凸几何等现代背景下的前沿成果相结合时，这些经典理论将为从高维回归和分类到网络数据分析和子空间学习的一般问题提供新的原则方法。在拟议的研究中开发的所有方法将作为免费提供的标准R软件包实施，并将具有很高的教学价值，并将用于开发新课程。提出的研究在天文学和医学筛选数据方面具有应用价值。该建议还为遗传学、精神病学、脑科学等应用领域提供了新的推理工具。综合教育活动包括设计有关非参数统计和现代多元分析新视角的课程。本文将进一步将经典的非参数和多元分析理论与四个主要统计研究领域的现代元素结合起来，包括高维回归中的无假设预测带；集值多类分类的广义Neyman-Pearson框架一些贪婪算法在网络社区检测中的统计性能保证及网络模型选择的拟合优度检验并将结构化子空间估计的统一奇异值分解框架表述为一个凸优化问题。这些研究活动将导致现代化的非参数和多元分析课程，以新的理论框架为特色，如计算约束的极大极小分析，额外的主题，如功能数据分析，以及遗传学，脑成像，交通和天文学的前沿例子。

The constantly increasing dimensionality and complexity of modern data has motivated many new data analysis tools in various fields, and urgently call for rigorous theoretical investigation, such as robustness against different sources of model misspecification,uncertainty quantification in classification and prediction, and statistical performance guarantee of conventional methods under non-standard settings. Although most classical theory are not directly applicable to methods developed for complex data, partially due to highly specialized model assumptions and diversified algorithms, the profound statistical thinking carried in these long-established results can still provide deep theoretical insights. When combined with cutting-edge results in modern context such as random matrix theory, matrix concentration, and convex geometry, these classical theory will lead to novel principled methods for a general class of problems ranging from high dimensional regression and classification to network data analysis and subspace learning. All methods developed in the proposed research will be implemented as standard R packages freely available and will have high pedagogical value and will be used to develop new courses. The proposed research has applications in astronomy and medical screening data. The proposal also provides new inference tools for applied areas in genetics, psychiatry, brain sciences. Integrated educational activities include designing courses on new perspectives in nonparametric statistics and modern multivariate analysis.The proposed work will further integrate classical nonparametric and multivariate analysis theory with modern elements in four major areas of statistical research, including assumption-free prediction bands in high dimensional regression; a generalized Neyman-Pearson framework for set-valued multi-class classification; statistical performance guarantee of some greedy algorithms in network community detection as well as goodness-of-fit tests for network model selection; and a unified singular value decomposition framework for structured subspace estimation formulated as a convex optimization problem. These research activities will lead to modernized nonparametric and multivariate analysis courses, featuring new theoretical frameworks such as computationally constrained minimax analysis, additional topics such as functional data analysis, and cutting-edge examples in genetics, brain imaging, traffic, and astronomy.