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227-0434-10L 8 Credits DR , MSC D-ITET , D-MATH , D-INFK , D-PHYS

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Mathematics of Information

Lecturers & Examiners: Prof. Dr. Helmut Bölcskei

VVZ CR 4.0

Last Updated: 2026-02-05 15:41:55

Abstract

The class focuses on mathematical aspects of1. Information science: Sampling theorems, frame theory, compressed sensing, sparsity, super-resolution, spectrum-blind sampling, subspace algorithms, dimensionality reduction2. Learning theory: Approximation theory, uniform laws of large numbers, Rademacher complexity, Vapnik-Chervonenkis dimension

Objective

The aim of the class is to familiarize the students with the most commonly used mathematical theories in data science, high-dimensional data analysis, and learning theory. The class consists of the lecture, exercise sessions with homework problems, and of a research project, which can be carried out either individually or in groups. The research project consists of either 1. software development for the solution of a practical signal processing or machine learning problem or 2. the analysis of a research paper or 3. a theoretical research problem of suitable complexity. Students are welcome to propose their own project at the beginning of the semester. The outcomes of all projects have to be presented to the entire class at the end of the semester.

Content

Mathematics of Information 1. Signal representations: Frame theory, wavelets, Gabor expansions, sampling theorems, density theorems 2. Sparsity and compressed sensing: Sparse linear models, uncertainty relations in sparse signal recovery, matching pursuits, super-resolution, spectrum-blind sampling, subspace algorithms (MUSIC, ESPRIT, matrix pencil), estimation in the high-dimensional noisy case, Lasso 3. Dimensionality reduction: Random projections, the Johnson-Lindenstrauss Lemma Mathematics of Learning 4. Approximation theory: Nonlinear approximation theory, fundamental limits on compressibility of signal classes, Kolmogorov-Tikhomirov epsilon-entropy of signal classes, optimal compression of signal classes, recovery from incomplete data, information-based complexity, curse of dimensionality 5. Uniform laws of large numbers: Rademacher complexity, Vapnik-Chervonenkis dimension, classes with polynomial discrimination, blessings of dimensionality

Resources

Lecture Notes

Detailed lecture notes will be provided at the beginning of the semester and as we go along.

Learning Materials (Links)

Main link
Course Website

General Information

Language: English
Levels: DR , MSC
Frequency: Yearly recurring

Examination

Type: session examination
Mode: written 180 minutes
Aids: Lecture and exercise notes allowed. Electronic devices (laptops, calculators, cellphones, etc...) NOT allowed.

The written final exam (180 minutes) will contribute 75% towards the final grade. The remaining 25% will be awarded for a graded student project in the form of a group literature review. A pass grade in this project is a prerequisite for admission to the exam (compulsory continuous performance assessment). Students re-sitting the exam can decide at the beginning of the semester if they want to also repeat the student project (if previously passed).

Course Components

Type	Title	Time & Place	Hours
lecture	Mathematics of Information	Thu 09:15-12:00 (ETZ E 6)	3 h weekly
exercise	Mathematics of Information	Mon 13:15-15:00 (ETZ E 6)	2 h weekly
independent project	Mathematics of Information	No time listed	2 h weekly

Offered In

Computational Science and Engineering Master
- Electives (In the ‘electives’ subcategory, at least two course units must be successfully completed.)
Mathematics Master
- Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
  - Electives: Applied Mathematics and Further Application-Oriented Fields (¬)
    - Selection: Further Realms
Electrical Engineering and Information Technology Master
- Master Studies (Programme Regulations 2018)
  - Communication (The core courses and specialization courses below are a selection for students who wish to specialize in the area of "Communication", see . The individual study plan is subject to the tutor's approval.)
    - Specialization Courses (These specialization courses are particularly recommended for the area of "Communication", but you are free to choose courses from any other field in agreement with your tutor. A minimum of 40 credits must be obtained from specialization courses during the Master's Programme.)
  - Signal Processing and Machine Learning (The core courses and specialization courses below are a selection for students who wish to specialize in the area of "Signal Processing and Machine Learning ", see . The individual study plan is subject to the tutor's approval.)
    - Core Courses (These core courses are particularly recommended for the field of "Signal Processing and Machine Learning". You may choose core courses form other fields in agreement with your tutor. A minimum of 24 credits must be obtained from core courses during the MSc EEIT.)
      - Advanced Core Courses
- Master Studies (Programme Regulations 2008)
  - Major Courses (A total of 42 CP must be achieved form courses during the Master Program. The individual study plan is subject to the tutor's approval.)
    - Communication
      - Recommended Subjects (These courses are recommended, but you are free to choose courses from any other special field. Please consult your tutor.)
    - Signal Processing and Machine Learning
      - Core Subjects
Statistics Master (The following courses belong to the curriculum of the Master's Programme in Statistics. The corresponding credits do not count as external credits even for course units where an enrolment at ETH Zurich is not possible.)
- Specialization Areas and Electives
  - Statistical and Mathematical Courses
Physics Master
- Electives
  - General Electives (Students may choose General Electives from the entire course programme of ETH Zurich - with the following restrictions: courses that belong to the first or second year of a Bachelor curriculum at ETH Zurich as well as courses from GESS "Science in Perspective" are not eligible here. The following courses are explicitly recommended to physics students by their lecturers. (Courses in this list may be assigned to the category "General Electives" directly in myStudies. For the category assignment of other eligible courses keep the choice "no category" and take contact with the Study Administration ( ) after having received the credits.))
Doctoral Dep. of Information Technology and Electrical Engineering (More Information at: )
- Doctoral and Post-Doctoral Courses (A minimum of 12 ECTS credit points must be obtained during doctoral studies. The courses on offer below are but a small selection out of a much larger available number of courses. Please discuss your course selection with your PhD supervisor.)
Data Science Master
- Core Courses
  - Data Analysis
    - Information and Learning
Quantum Engineering Master
- Electives (This is a selection of courses particularly suitable for the MSc QE. In agreement with the tutor, students may choose other courses from the ETH course catalogue.)