Theory of Machine Learning Group University of Tübingen

Mathematics for Machine Learning (Winter term 2026/27)

Eckart–Young–Mirsky theorem in the original handwriting on an office whiteboard: truncated SVD gives a best rank-at-most-k approximation in both spectral and Frobenius norms. The errors are sigma k+1 and the square root of the sum of squared singular values from k+1 to min(m,n), respectively.

Quick links and news

  • ILIAS page (Registration)
  • On Tuesday, Nov 24, the lecture takes place in lecture hall N10 on Morgenstelle!!! In the "Botanik" building indicated by Number 2 on the following map
  • The tutorial on Nov 24 (8:30-10:00 takes place in room 1B01, in the basement of Hoersaalzentrum Morgenstelle, indicated by number 7 on this map !!!

Lecture notes

Current notes
Linear algebra: pdf (updated 2026-10-29)
Complete notes of 2024
The latex version (converted in hindsight, not yet perfectly proof-read)
Videos
Videos of the course as taught in 2020. Please note that in the meantime, I added quite some material in optimziation and removed some other contents, so the videos are not completely in sync with this year's course.

Assignments

Presence sheets

Tutorials

We have weekly tutorial sessions in small groups of about 30 students, where you can ask questions and interact with other students, and work on assignments.
Current slots are:
Tue 8:30-10:00, lecture hall A2 MvL1 (Rishab-Pradeep)
Tue 14:15-15:45, lecture hall A1 MvL1 (Kanchan)
Wed 10:15-11:45, lecture hall ground floor MvL6 (Baisu)
Thu 8:30-10:00, lecture hall A2 MvL1 (Gabriel)
Thu 14:15-15:45, lecture hall A1 MvL1 (Andrei-Cartalin)
Thu 16:30-18:00, Morgenstelle, GUZ room 3U03 (Tai)
The teaching assistants are:
Eric Guenther (organizes the assignments)
Karolin Frohnapfel (organizes the assignments)
Andrei-Catalin Raduc
Baisu Zhou
Gabriel Scherer
Kanchan Poudel
Rishab-Pradeep Mudliar
Tai Thai

General information

Exams

To be admitted to the final exam, you have to participate in at least 3 of the weekly quizzes that take place in the tutorials. By passing those quizzes you can also earn bonus points for the exam. If you got admitted last year, you can also take part in the exam (but please send us an email when you register to the exam to let us know, so we can check whether you are really elegible). If you got admitted before last year, you need to pass the assignments again this year.

The exam is written. There will be two exams, one at the beginning of the semester break and one at the end. The dates are not known yet, they are determined by a central process. You can choose which exam to take, both will have the same difficulty. But note that there won't be a third exam nor oral exams: if you skip the first exam and fail the second one, you would need to wait for next winter term to take the exam again.
To see how the exam might look like, you can have a look at the following Mock exam (it might not cover all the topics, but you can see the style of questions)

For more details, check out the info sheet (as of oct 13).

Literature:

General: More specific:
  • For linear algebra, I recommend: Sheldon Axler: Linear Algebra Done Right. Third edition, 2015. There are also online videos by the author if you want to get longer explanations than the ones I will provide.
  • Calculus (Integration, Measures, Metric spaces and their topology): Sheldon Axler: Measure, Integration & Real Analysis. 2019
  • Calculus (Differential calculus in R^n): Here I haven't found my one favorite textbook for this course yet (some are too recipe-like, the others a bit too abstract)
    • My current favorite textbook, mathematically rigorous (but not easy to read):
      Terence Tao, Analysis 1 and 2.
    • Books with many figures, but partly informal or recipe-like (might be good as a start if you need to get the intuition before diving deeper):
      Stanley Miklavcic: An Illustrative Guide to Multivariable and Vector Calculus.
      Charles Pugh: Real Mathematical Analysis
    • A classic: Rudin: Principles of Mathematical Analysis.
    • If you are looking for a german book, I like: Walter: Analysis 1 and Analysis 2. The second one covers everything that we have been discussing.
  • Probability theory: Jacod, Protter: Probability essentials. Short and to the point, tries to avoid measure theory whereever possible, yet is rigorous. Good compromise.
  • Statistics:
    • For a very short overview over all the topics we cover: Wasserman: All of statistics, a concise course in statisticial inferece.
    • A bit more details: Casella/Berger, Statistical Inference.
    • Testing, rigorously: Lehmann/Romano: Testing statistical hypotheses.
  • Optimization
    • For convex optimization: Boyd/Vandenberghe: Optimization
    • Non-convex optimization basics, in the new book by Francis Bach, Learning Theory from first principles, pdf
  • For high-dimensional probability and statistics there are several good books, but they go much deeper than our lecture:
    • Wainwritght: High-dimensional statistics
    • Vershynin: High-dimensional probability
    • Bühlmann, van de Geer: Statistics for High-dimensional data (this is from the more traditional statitics point of view)

Lecture

What: Mathematics for Machine Learning, 9 CP
Lecturer: Prof. Ulrike von Luxburg
When and where: Tuesdays 10:15 - 11:45, lecture hall A2, Maria von Lindenstrasse 1; Thusdays 10:15 - 11:45, lecture hall A2, Maria von Lindenstrasse 1; First lecture is on Oct 13.
Language: English.

Background information

This course is intended for the students of the master program in machine learning. Depending on your background, some of the material might be a recap - or not. Contents of the course are Linear algebra, Mulitvariate analysis, Probability Theory, Statistics, Optimization. Note that the course requires a solid basis in mathematics, similar to what students would have attended in our Bachelor program in computer science. The course is not recommended for students without this background.
Info sheet about the setup of the lecture: pdf (as of oct 13)

Start-of-term-FAQ

  • Q: Do I need to register? A: Yes, you need to register for this course in Ilias (link to ILIAS page). Registration will be possible until after the first lecture.
  • Q: I am an exchange student, or a student from a different degree program. Can I take part? A: Yes, if you have enough background knowledge. Q: How do I know? A: if in doubt, please approach me at the end of the first lecture (not by email).
  • Q: Can I also take a 6CP-Version of the course? A: no, sorry.
  • Q: I already took the class last year and got admitted to the exam. Do I need to get the admission again? A: No, not if you got admitted last year, but yes if your admission happened before last year.
  • I have another question ... sure, but please ask me at the end of the first lecture and not by email.