Since the breakthrough from the two papers [Donoho 2006], [Candes- Romberg-Tao 2006], Compressive Sensing (CS) has been becoming a very hot research topic in the fields of statistics, computer science, electrical engineering, applied mathematics, and many others. Its philosophy is the follows: many data that we are interested are indeed sparse, if they are properly represented, therefore, it is possible to measure them “compressively.” So, the issues of compressive sensing are: how to represent them in sparse way, how to measure them efficiently (compressed sensing), and how to recover them from compressed measure- ments? The new emerging area, Data Science, is closely related to this research area, which is so-called “finding a needle in a forest.” Its mathematical theory is still under developping. Its applications include machine learning, medical imaging, computational biology, geophysical data analysis, compressive radar, remote sensing, ..., etc. See the webpage “compressive sens- ing resources” (http://http://dsp.rice.edu/cs) for more informations. One of the purpose of this course is to seek for possible applications in the fields of numerical (stochastic) partial differential equations and inverse problems.
This course will consist of three parts: (1) mathematical foundation of compressive sens- ing, (2) numerical optimization algorithms, (3) applications. For the mathematical foundation, I will select several chapters from the book: Simon Foucart and Holger Rauhut, “A Mathematical Introduction to Compressive Sensing,” Birkhauser, 2013. For the numerical optimization algo- rithms, I will choose Boyd and Vandenberghe’s book, Convex Optimization. For applications, I will organize three workshops: image science + CS, brain imaging + CS, Data Science + CS. In addition, students are required to reported on assigned application articles (mainly partial differential equations and inverse problems) and report at end of this course.
Theory
• An invitation to Compressive Sensing
• Basic Algorithms: Optimization algorithms, Greedy algorithms, Thresholding algorithms • Basic Pursuit, Mutual Incoherence and Restricted Isometry Property
• Basic Probability Theory
• Sparse Recovery with Random Matrices.
Numerical Optimization Algorithms
• Gradient descent method
• Proximal gradient method, Nestrorov acceleration method • Primal-dual methods
• Argumented Lagrangian methods, ADMM