CIF: Small: Collaborative Research: Scalable Nonconvex Optimization with Statistical Guarantees for Information Computing in High Dimensions

近年来，随着科学领域的数据爆炸，迫切需要一种方法来分析具有低维结构的高维数据。该研究致力于开发可扩展非凸优化的变革性理论和方法，在噪声环境中具有统计保证，在信号处理，机器学习，运筹学和计算机视觉等广泛的学科中具有应用。研究人员提出了几何学指导的非凸优化和优化辅助统计分析的方法，以研究具有混合类型结构简约性的信号的迭代非凸算法的收敛速度，加速和统计精度。该项目从统计学，运筹学，工程学和计算机科学交叉施肥的想法，并与研究紧密结合的教育。 整合的研究和教育通过跨学科的训练帮助学生发展批判性思维，并帮助学生成为终身学习者。研究人员利用该项目中丰富的主题来激发公众和所有年龄段学生的学习和发现兴趣;此外，外展活动有助于吸引少数民族和女性学生从事科学事业。该研究对一般的优劣化算法进行了精确度指导的算法分析，以实现快速稳定的信号恢复。在大数据应用中，提出并研究了可扩展的随机加速方案。研究人员开发了一种创新的基于优化的统计方法，用于分析高维中的多正则稀疏估计。该项目将这些技术应用于鲁棒主成分估计、分层建模、网络学习等。 该研究为高维信息计算创造了优化与统计的融合，深化和拓展了现有的压缩感知和非凸优化理论和方法。

The data explosion in all fields of science recently creates an urgent need for methodologies for analyzing high dimensional data with low-dimensional structure. The research, devoted to developing transformative theory and methods for scalable nonconvex optimization with statistical guarantees in noisy settings, has applications in a wide range of disciplines such as signal processing, machine learning, operations research, and computer vision. The investigators propose methodologies of statistics-guided nonconvex optimization and optimization-assisted statistical analysis to study convergence rate, acceleration, and statistical accuracy of iterative nonconvex algorithms for signals with mixed types of structural parsimony. The project cross-fertilizes ideas from statistics, operations research, engineering, and computer science and has education tightly coupled with research. The integrated research and education help students develop critical thinking through cross-disciplinary training, and assist students in becoming life-long learners. The investigators use the rich topics in this project to inspire the learning and discovery interest of the public and students of all ages; in addition, the outreach activities help attract minority and female students to careers in science.The research performs statistical-accuracy guided algorithmic analysis of general majorization-minorization algorithms for fast and stable signal recovery. Scalable and randomized acceleration schemes are proposed and studied in big-data applications. The investigators develop an innovative optimization-based statistical methodology for analyzing multi-regularized sparse estimators in high dimensions. The project applies the techniques to robust principle component estimation, hierarchical modeling, network learning, among others. The research creates a fusion of optimization and statistics for information computing in high dimensions, and deepens and broadens existing compressed sensing and nonconvex optimization theories and methods.