Topological Data Analysis (TDA) is an emerging trend in exploratory data analysis and data mining. It has known a growing interest and some notable successes (such as the identification of a new type of breast cancer, or the classification of NBA players) in the recent years. Indeed, with the explosion in the amount and variety of available data, identifying, extracting and exploiting their underlying structure has become a problem of fundamental importance. Many such data come in the form of point clouds, sitting in potentially high-dimensional spaces, yet concentrated around low-dimensional geometric structures that need to be uncovered. The non-trivial topology of these structures is challenging for classical exploration techniques such as dimensionality reduction. The goal is therefore to develop novel methods that can reliably capture geometric or topological information (connectivity, loops, holes, curvature, etc) from the data without the need for an explicit mapping to lower-dimensional space. The objective of this course is to familiarize the students with these new methods, lying at the interface between pure mathematics, applied mathematics, and computer science.
Before you start the first lab session, you are advised to:
Course grading:
Course introduction (video file) | |||
Session 1: Dimensionality Reduction (video file) | Notes, TeXified notes, slides | TD 1 | Sept. 24 2021 |
Session 2: Clustering (video file) |
Slides intro clustering, Slides mode-seeking Notes mode-seeking, Notes degree-0 persistence ToMATo's webpage |
TD 2 | Oct. 1 2021 |
Session 3: Homology I (video file) | Notes homology, TeXified notes homology, book homology | PC 3-4, solution | Oct. 8 2021 |
Session 4: Homology II (video file) | Oct. 15 2021 | ||
Session 5: Persistence I (video file) |
Notes persistence,
Slides persistence
book persistence 1, book persistence 2 |
TD 5 | Oct. 22 2021 |
Session 6: Persistence II (video file) / Topological Inference (video file) | Notes inference, Slides inference | TD 6 | Nov. 5 2021 |
Session 7: Topological descriptors for geometric data (video file) | Notes descriptors, Slides descriptors, Notes on stability | TD 7 | Nov. 12 2021 |
Session 8: Learning with topological descriptors (video file) | Slides learning | PC 8 | Nov. 19 2021 |
Session 9: Statistics with topological descriptors (video file) | Slides satistics | PC 9 | Nov. 26 2021 |
Final exam: | Text of the exam |
Feel free to come and discuss with us if you are looking for an internship in TDA.