CAREER: Theory and Methods for Simultaneous Variable Selection and Rank Reduction

科学各个领域的数据爆炸产生了对分析高维多变量数据的方法的迫切需求。该项目通过取得以下重大科学成果，深化和拓宽了现有的稀疏性和低秩数统计理论和方法：(A)通过同时选择变量和投影的创新的可选择降阶方法，在理论上保证了比现有变量选择和降秩率更低的错误率，为高维统计和信息论的新前沿铺平了道路；(B)快速但易于实现的算法，可以处理计算中所有流行的惩罚函数(可能是非凸的)，并保证全局收敛和局部最优性，以确保所提方法在大数据应用中的实用性；(C)能够考虑到多变量响应之间相关性的非高斯模型的一般推广，基于流形优化的通用算法设计；(D)能够识别并适应真实数据中频繁出现的粗大异常值的统一鲁棒性方案，以克服许多传统多变量工具的非稳健性；(E)用于变量选择和/或降阶的通用模型选择方法，并在理论上保证有限样本的最优预测错误率。从高维多变量噪声数据中恢复低维信号的需求渗透到科学和工程的各个领域。因此，这种性质的项目旨在开发用于同时选择变量和降阶的变革性理论和方法，在机器学习、信号处理和生物统计学等广泛的学科和领域中找到了应用。通过交叉融合统计学、数学、工程学和计算机科学的思想，综合研究和教育通过跨学科训练帮助学生发展批判性思维，帮助学生成为终身学习者。调查人员用这个项目中丰富的主题来激发公众和各个年龄段的学生学习和发现的兴趣。教育计划包括课程开发、学生指导、外展和招收代表性不足的学生。

The data explosion in all fields of science creates an urgent need for methodologies for analyzing high dimensional multivariate data. The project deepens and broadens existing sparsity and low rank statistical theories and methods by making the following major scientific achievements: (a) an innovative selectable reduced rank methodology through simultaneous variable selection and projection, with guaranteed lower error rate than existing variable selection and rank reduction rates in theory, which paves the way to new frontiers in high dimensional statistics and information theory; (b) fast but simple-to-implement algorithms that can deal with all popular penalty functions (possibly nonconvex) in computation with guaranteed global convergence and local optimality, to ensure the practicality of the proposed approaches in big data applications; (c) a generic extension to non-Gaussian models capable of taking into account the correlation between multivariate responses, with a universal algorithm design based on manifold optimization; (d) a unified robustification scheme that can both identify and accommodate gross outliers occurring frequently in real data, to overcome the non-robustness of many conventional multivariate tools; (e) general-purpose model selection methods serving variable selection and/or rank reduction and achieving the finite-sample optimal prediction error rate with theoretical guarantee. The need to recover low-dimensional signals from high dimensional multivariate noisy data permeates all fields of science and engineering. Hence a project of this nature, designed to develop transformative theory and methods for simultaneous variable selection and rank reduction, finds applications in a wide range of disciplines and areas such as machine learning, signal processing, and biostatistics, among others. By cross-fertilizing ideas from statistics, mathematics, engineering, and computer science, the integrated research and education help students develop critical thinking through cross-disciplinary training, and assist students in becoming life-long learners. The investigator uses the rich topics in this project to inspire the learning and discovery interest of the public and students of all ages. The educational plan consists of course development, student mentoring, outreach, and recruiting underrepresented students.