Course Description 

This graduate course introduces basic geometric and statistical concepts and principles of low- dimensional models for high-dimensional signal and data analysis, spanning basic theory, efficient algorithms, and diverse applications. We will discuss recovery theory, based on high-dimensional geometry and non-asymptotic statistics, for sparse, low-rank, and low-dimensional models – including compressed sensing theory, matrix completion, robust principal component analysis, and dictionary learning etc. We will introduce principled methods for developing efficient optimization algorithms for recovering low-dimensional structures, with an emphasis on scalable and efficient first-order methods, for solving the associated convex and nonconvex problems. We will illustrate the theory and algorithms with numerous application examples, drawn from computer vision, im- age processing, audio processing, communications, scientific imaging, bioinformatics, information retrieval etc. The course will provide ample mathematical and programming exercises with sup- porting algorithms, codes, and data. A final course project will give students additional hands-on experience with an application area of their choosing. Throughout the course, we will discuss strong conceptual, algorithmic, and theoretical connections between low-dimensional models with other popular data-driven methods such as deep neural networks (DNNs), providing new perspectives to understand deep learning. 

The course includes 3 hours of lectures (by the instructor) and 1 hour discussion session (by a GSI) per week. Homework includes both written exercises and programming exercises. A final course project includes a midterm proposal and final presentation and report.  

Prerequisites 

We require students to have prior knowledge in undergraduate linear algebra, statistics, and probability. Background in signal processing, optimization, machine learning, and computer vision may allow you to appreciate better certain aspects of the course material, but not necessary all at once.