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.
Important: bring pens and paper to take notes during the lectures, and your laptop for the lab sessions.
Before you come to the first lab session, you are advised to:
|Session 1: Dimensionality Reduction||Notes, Short notes, slides||TD 1||Sept. 21 2018|
|Session 2: Clustering||
Slides intro clustering, Slides mode-seeking
Notes mode-seeking, Notes degree-0 persistence
|TD 2||Sept. 28 2018|
|Session 3: Homology I||Notes homology, book homology||PC 3-4, solution||Oct. 5 2018|
|Session 4: Homology II||Oct. 12 2018|
|Session 5: Persistence I||
book persistence 1, book persistence 2
|TD 5 (graded, due Oct. 25)||Oct. 19 2018|
|Session 6: Persistence II / Topological Inference||Notes inference, Slides inference||TD 6||Oct. 26 2018|
|Session 7: Topological descriptors for geometric data||Notes descriptors, Slides descriptors, Notes on stability||TD 7-8||Nov. 9 2018|
|Session 8: Learning with topological descriptors||Slides learning||Nov. 23 2018|
|Session 9: Statistics with topological descriptors||Slides satistics, Slides satistics via optimal transport||PC 10||Nov. 30 2018|
|Final exam: TBA||last year's exam|
Feel free to come and ask us directly if you are looking for an internship in TDA.
The subject has known a steady development throughout the last decade or so. Here is a snapshot of the current TDA community: