In the era of data deluge, the development of methods for discovering structure in high-dimensional data is becoming increasingly important. This course will cover state-of-the-art methods from algebraic geometry, sparse and low-rank representations, and statistical learning for modeling and clustering high-dimensional data. The first part of the course will cover methods for modeling data with a single low-dimensional subspace, such as PCA, Robust PCA, Kernel PCA, and manifold learning techniques. The second part of the course will cover methods for modeling data with multiple subspaces, such as algebraic, statistical, sparse and low-rank subspace clustering techniques. The third part of the course will cover applications of these methods in image processing, computer vision, and biomedical imaging.

1. Introduction
• Class Overview (Chapter 1)
• Review of Linear Algebra
• Review of Statistics (Appendix A)
• Review of Optimization (Appendix B)
2. Modeling Data with a Single Subspace (Part I)
• Principal Component Analysis (Chapter 2)
• Robust Principal Component Analysis (Chapter 3)
• Nonlinear Principal Component Analysis (Chapter 4)
3. Modeling Data with Multiple Subspaces (Part II)
• Algebraic Subspace Clustering (Chapter 5)
• Statistical Subspace Clustering (Chapter 6)
• Sparse Subspace Clustering (Chapter 7)
4. Applications
• Face Clustering
• Image Representation (Chapter 8)
• Image Segmentation (Chapter 9)
• Motion Segmentation (Chapter 10)

R. Vidal, Y. Ma, and S. Sastry. Generalized Principal Component Analysis. Springer Verlag, 2014. (In preparation)

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Appendices A and B

Appendix C

Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

Project

Homework 7

Slides on Algebraic Subspace Clustering

Slides on Sparse Subspce Clustering

1. Homeworks (40%): There will homeworks every other week (approximately). Homework problems will include both analytical exercises as well as programming assignments in MATLAB.
2. Exams (40%): There will be two in-class exams.
3. Project (20%): There will be a final project to be done in teams of two students. Each team will write a 6-page report and give a 10 minute presentation (including 3 minutes for questions) on the scheduled exam day.

• Late policy:
• Homeworks and projects are due on the specified dates.
• No late homeworks or projects will be accepted.
• Honor policy:

  The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

• Homeworks and exams are strictly individual
• Projects can be done in teams of two students