Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Introduction to Topological Data Analysis (2024)

Lecturers: Patrick Schnider (OAT Z 13.1)
Assistants: Simon Weber (OAT Z 18), contact assistant
Shengzhe Wang
Lectures: Thu 13-14, CAB G 51,
Fri 12-14, CAB G 61
Exercise: Wed 10-12, CHN F 46
Language: English
Contents: Fundamental concepts, techniques and results in Topological Data Analysis. Preliminary list of topics:
  • Mathematical Background (Topology, Simplicial complexes, Homology)
  • Persistent Homology
  • Complexes on point clouds
  • The TDA pipeline
  • Reeb Graphs
  • Mapper Algorithm
Literature:

The complete lecture notes from last year can be found here: [L] Lecture Notes. These are a collection of the scribe notes from each lecture of 2023. The up-to-date chapters for this year's edition are published here as the course progresses:

We additionally suggest the following supporting textbooks:
Prerequisites: The course assumes knowledge of discrete mathematics, algorithms and data structures and linear algebra, as supplied in the first semesters of Bachelor Studies at ETH.
Special Assignment 1: 21.3. - 11.4. 12:15 Exercise Sheet 1
Special Assignment 2: 2.5. - 17.5. 12:15 Exercise Sheet 2

Absence. If there are compelling reasons why you cannot hand in a special assignment on the due date foreseen, please contact us post-haste and we can look for a solution. Note, however, that we will grant requests of this type only in very exceptional cases (doctor's note, military service, funeral, etc.). Private events (vacation, sports, career fairs, etc.) are never sufficient grounds. As far as special assignments are concerned, you can hand them in to the contact assistant by email at any time before the due date, so there is no need to be present in person. Regarding the final exam, ETH regulations apply.

Exchange students. Exchange students enrolled for TDA in spring might want to return to their home university prior to the date of the final exam. If this applies to you and you nonetheless wish to be evaluated and given credits for TDA, then please take a look at the following website for current regulations.

Schedule

In the table below you can find the lecture dates and the preliminary topics. The exercises and their solutions will be published here.


Date Content Exercises and links Relevant chapters

#1 22.02.2024 Information about the course, Intro to Topological Spaces Slides, Colab [L] 1
#2 23.02.2024 Mathematical Background Exercises 1 (Video for Exercise 6) [L] 2.1, 2.2, 2.3
#3 29.02.2024 Algebra [L] 2.4
#4 01.03.2024 Simplicial Complexes, Intro to Homology Slides, Exercises 2 [L] 3.1, 3.2.1, 3.2.2
#5 07.03.2024 Homology [L] 3.2.1-3.2.3
#6 08.03.2024 Homology Exercises 3 Colab [L] 3.2.4-3.2.6
#7 14.03.2024 Homology of spheres [L] 3.2.7
#8 15.03.2024 Induced Homology, Brouwer fixed-point theorem Exercises 4 [L] 3.2.8, 3.2.9
#9 21.03.2024 Persistent Homology Slides [L] 4.1, 4.2
#10 22.03.2024 Persistent Homology, Algorithms Slides
Exercises 5
[L] 4.3
#11 28.03.2024 Complexes on Point Clouds Slides [L] 5.1, 5.2
#12 11.04.2024 Complexes on Point Clouds (cont'd) Exercises 6 SPA 1 [L] 5.2, 5.3
#13 12.04.2024 Bottleneck distance, Stability [L] 6.1.1
#14 18.04.2024 Wasserstein distance Exercises 7 Colab [L] 6.1.2
#15 19.04.2024 Guest Lecture: Bastian Rieck (Helmholtz München) Slides
#16 25.04.2024 Guest Lecture: Bernadette Stolz (EPFL) Exercises 8
#17 26.04.2024 Interleaving Distance [L] 6.2
#18 02.05.2024 Stability for Cech Complexes Exercises 9 Colab [L] 6.2.3
#19 03.05.2024 Guest Lecture: Marius Huber (UZH)
#20 10.05.2024 Interval Decompositions, Reeb graphs Exercises 10 Slides [L] 6.3, 7.1, 7.3
#21 16.05.2024 Homology of Reeb graphs [L] 7.1.1
#22 17.05.2024 Interleaving Distance for Reeb graphs Exercises 11 Colab SPA 2 [L] 7.2.1
#23 23.05.2024 Multi-scale Mapper [L] 7.4
#24 24.05.2024 Optimal Generators Exercises 12 Slides [L] 8.1, 8.2
#25 30.05.2024
#26 31.05.2024

Procedures, Exercises, Exam

Every week we provide you with exercises. The students are split into small groups, and the members of each group work together. At the end of the session, for each exercise, a student from a group presents their solution to the rest of the students. In addition to the exercise sessions, we encourage you to solve the exercises in written form and to hand in your solutions to the teaching assistant. Your solutions are thoroughly commented, but they do not count towards your final grade. The motivation to work on the exercises stems from your interest in the topic (and possibly also the desire to succeed in the exam).

In addition, you receive two special assignments during the semester. The special assignment is to be solved in written form and you have two weeks of time to return your solutions/reports, typeset in LaTeX. In contrast to the exercises, these special assignments do count towards the final grade: Your two grades will account for 20% of your final grade each. Solving the special assignments in teams is not allowed. Besides one or two exercises, the special assignments may include a small research project, or you are asked to give a short talk about your last small research project. The format of this talk will be determined by the number of students who register for the course.

There is an oral exam of 30 minutes during the examination period. Your final grade consists to 60% of the grade for the exam and to 40% of the grade for the homework assignments.
You are expected to learn proofs discussed in the lecture, should be able to explain their basic ideas and reproduce more details on demand. You should also be able to give a short presentation on any topic treated throughout the course. One of the questions given to you during the exam is to solve one of the exercises posed throughout the semester. Roughly half an hour before the exam you get to know the exercise to be solved and one topic that you will be questioned about in particular, that is, you have 30 minutes preparation time. For this preparation, paper and pens will be provided. You may not use any other material, like books or notes.

For PhD students, the same rules apply for obtaining credit points as for all other participants. Taking the exam and achieving an overall grade of at least 4.0 (computed as a weighted average of grades for homework and the oral final exam as detailed above) qualifies for receiving credits. In order to comply with new regulations recently issued by the department, merely attending the course and/or handing in exercises is no longer sufficient.

Complementary Courses & Semester/Master/Diploma Theses

This course is complemented by a special type of seminar Projects in Topological Data Analysis in the following fall semester. In this seminar, students from four different universities will collaborate on projects in Topological Data Analysis in an international setting. After having completed the course, it is possible to do a semester, master or diploma thesis in the area.