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.