Topological Data Analysis
Spring' 2022, Tue/Tue: 2:00pm -- 3:20pm
Location: HSS 2154
Topological data analysis: algorithm and applications
Topological methods provide powerful tools for characterizing structures/features behind data and analyzing diverse complex data (images, graphs, point sets, etc). This course introduces basic concepts and topological structures, as well as recent topological tools, algorithms, as well as examples to applications. Some topics include: basics in topology, simplicial complexes to model data, persistent homology, discrete Morse theory, topology inference, the Mapper methodology, hierarchical clustering, and integration of topological methods with machine learning. A tentative list of topics is on the right.
This course is for graduate students and advanced undergraduate students.
Prerequisite: Linear algebra, algorithms (DSC40B or equivalent)
Much of the material is from a up-coming book Computational Topology for Data Analysis co-authored by Dr. T. K. Dey and myself, and you can download an e-version of the book here (for academic use), or on my website.
I will distribute (in Canvas) class notes and slides throughout the course.
Other references that might be useful:
There is one project. The final grade is based on: Project / survey (proposal, project, report, short presentation): 100%.
Project requirement: Each student is required to meet briefly with me around 3-4th week, so that we can discuss to find a good potential topic for your final project. After a topic is chosen, you are required to submit a short proposal about what you intend to work on. Later, after you finish your project, you will submit a final report (4+ pages, single spacing), which should clearly describe the motivation of your project, what you have done, and your findings. A 5-min presentation is also needed. A survey needs to be 10+ pages (single spacing).
Week 2-5: Students meet with me to discuss project topic .
Week 6: Selection of final project topic done. Project proposal due by end of May 8th, 2022 (Sunday) (extended from previous May 6th deadline).
Week 10: Finish up your final project
Thursday June 9th by 11:59pm : Final project report as well as your recorded presentation are due (there is no late submission accepted as I need to assign final grades). Please submit the video to the google drive I shared with you (information will be available in canvas then), and send the report to my email.
Lecture materials: (Lecture notes are distributed in canvas. Note that all animations are lost in pdf slides. )
Topic 0: Introduction to computational topology (slides: pptx)
Topic 1: Basics. (slides: pptx ; Optional reading: Chapter 1 of CTDA textbook.)
Topic 2: Simplicial complex (aka: how do we model space of interest) (slides: pptx; Optional reading: First half of Chapter 2 of CTDA textbook)
Topic 3: Homology (aka: how do we quantify topological information) (slides: pptx; Optional reading: Second half of Chapter 2 of CTDA textbook)
Topic 4: Persistent homology (aka: a modern much more powerful extension of homology)
4-A: Introduction to persistent homology (Optional reading: Chapter 3.1--3.3 of CTDA textbook)
4-B: Persistent homology of PCDs and functions (Optional reading: Chapter 3.4-3.5 of CTDA textbook)
4-C: More on persistent modules: distance and extensions (Optional reading: Chapter 4 of CTDA textbook)
Topic 5: Using persistent homology in practice (Optional reading: Chapter 6 and Chapter 8 of CTDA textbook)
Other course materials: (Optional assignments, potential project topics, announcement etc. )
See Canvas for a list of potential course project topics.
Software / Tools resources