# INF556 -- Topological Data Analysis (2023-24)

general introduction - practical aspects - documents - schedule

## General Introduction

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.

### News

• The course will start on September 22nd, at 8:30am, in Amphi Curie.
 Your opinion about the course in general, or any specific aspect of it:

## Practical Aspects

#### Where and when:

• lectures: Fridays 8:30 - 10:30, Amphi Curie
• practical sessions: Fridays 10:45 - 12:45, Amphi Curie

Before you start the first lab session, you are advised to:

• Install a Java compiler like the Java Development Kit, or a C++ compiler like gcc, or a Python interpreter. You can use whatever IDE you prefer, e.g. Eclipse or Code::Blocks.

• Midterm (graded TD) + Final (written exam, 3 hours)

## Documents

Books:
• H. Edelsbrunner, J. Harer. Computational Topology: An Introduction. AMS Press, 2009. A good introduction to applied topology, including TDA. Well-suited for this course. This book is not available at the library, however it was compiled from the following course notes, which you can download instead.
• S. Oudot. Persistence Theory: From Quiver Representations to Data Analysis. AMS Surveys and Monographs, Vol. 209, 2015. A comprehensive treatment of persistence theory, perhaps too advanced for this course, but in principle you should be able to read it by the end of the course! Five printed copies are available at the library, otherwise you can download a pdf version here.
• James R. Munkres. Elements of Algebraic Topology. Perseus, 1984. A general introduction to algebraic topology, which you can consult (one printed copy is available at the library). Especially relevant to the course is its first chapter on homology theory.
• Trevor Hastie, Robert Tibshirani and Jerome Friedman. The Elements of Statistical Learning (2nd edition). Springer-Verlag, 2009. An excellent reference, relevant to the ML aspects of the course. You can download a pdf version of the book here.

## Schedule

 Session 1: Clustering   (video file) Slides intro clustering, Slides mode-seeking Notes mode-seeking, Notes degree-0 persistence ToMATo's webpage TD 1 Sept. 22 2023 Session 2: Homology I   (video file) Notes homology, book homology PC 2-3, solution Sept. 29 2023 Session 3: Homology II   (video file) Oct. 6 2023 Session 4: Persistence I  (video file) Notes persistence, Slides persistence book persistence 1, book persistence 2 TD 4 Oct. 13 2023 Session 5: Persistence II  (video file)  / Topological Inference  (video file) Notes inference, Slides inference TD 5 Oct. 20 2023 Session 6: Topological descriptors for geometric data  (video file) Notes descriptors, Slides descriptors, Notes on stability TD 6 Oct. 27 2023 Session 7: Learning with topological descriptors  (video file) Slides learning PC 7 Nov. 10 2023 Session 8: Statistics with topological descriptors   (video file) Slides satistics PC 8 Nov. 17 2023 Session 9: Reeb graphs and Mapper Notes Reeb and Mapper, Slides Reeb and Mapper TD 9 Nov. 24 2023 Final exam (Dec. 22, 9:00am - 12:00pm): previous exams: 2021 and 2022

Last update: Sept. 21 2023.