# Mathematics of Information

#### Offered in:

- Data Science Master: Information and Learning, Data Analysis
- Computer Science Master: Elective Courses (Wahlfächer)
- CAS in Computer Science: Focus Courses and Electives
- Electrical Engineering and Information Technology Master: Core subjects (Kernfächer), Specialization courses (Vertiefungsfächer), Advanced core courses, Recommended Subjects
- Mathematics Master: Selection: Further Realms (Auswahl: Weitere Gebiete)
- Physics Master: General Electives (Allgemeine Wahlfächer)
- Quantum Engineering Master: Electives (Wahlfächer)
- Computational Science and Engineering Master: Electives (Wahlfächer)
- Statistics Master: Statistical and Mathematical Courses (Statistische und mathematische Fächer)
- Cyber Security Master: Elective Courses (Wahlfächer)
- Doctoral and Post-Doctoral Studies: Department of Information Technology and Electrical Engineering

#### Basic information:

Lecture: | Thursday, 9:15-12:00, in room ML F 36. The first lecture takes place on Thursday, February 22, 2024. |

Exercise session: | Monday, 14:15-16:00, in room ML E 12. The first exercise session takes place on Monday, February 26, 2024. |

Instructor: | Prof. Dr. Helmut Bölcskei |

Teaching assistants: | Clemens Hutter, Konstantin Häberle |

Office hours: | Monday, 16:15-17:15 in ETF E 118. Please contact the TAs if you are planning to attend. |

Lecture notes: | Detailed lecture notes can be found below in the section "Lecture notes and prerequisite material". Exercise problem sets along with detailed solutions will be made available as we go along, see the section "Exercise sessions". The access information for the notes will be sent by email on the first day of the spring semester to all students registered for the class. |

Credits: | 9 ECTS credits |

#### Lecture recordings:

The lectures will not be recorded and no recordings from previous years are available.#### Course structure

The class will be taught in English. There will be a written exam in English of duration 180 minutes.#### News

We will post important announcements, links, and other information here during the semester, so please check back often!

- There will be no exercise session in the first week of the semester. The class will start with the lecture on February 22, 2024 and the first exercise session will be held on February 26, 2024.
- There will be no lecture on April 11, 2024. Please watch this recording instead (same username/password as for the lecture notes).
- The lecture of April 25, 2024 will take place via zoom. Please find the meeting details on this page (same username/password as for the lecture notes).
- The handwritten notes from the zoom lecture of April 25, 2024 can be found here.
- The lecture of May 23, 2024 will take place via zoom. Please find the meeting details on this page (same username/password as for the lecture notes).
- The handwritten notes from the zoom lecture of May 23, 2024 can be found here.

#### Course information

The class focuses on mathematical aspects of information science and learning theory.

- Mathematics of Information:
**Signal representations:**Frame theory, wavelets, Gabor expansions, sampling theorems, density theorems**Sparsity and compressed sensing:**Sparse linear models, uncertainty relations in sparse signal recovery, super-resolution, spectrum-blind sampling, subspace algorithms (ESPRIT), estimation in the high-dimensional noisy case, Lasso**Dimensionality reduction:**Random projections, the Johnson-Lindenstrauss Lemma- Mathematics of Learning:
**Approximation theory:**Linear and nonlinear approximation, best M-term approximation, greedy algorithms, fundamental limits on compressibility of signal classes, Kolmogorov-Tikhomirov epsilon-entropy of signal classes, optimal compression of signal classes**Uniform laws of large numbers:**Rademacher complexity, Vapnik-Chervonenkis dimension, classes with polynomial discrimination

H. Bölcskei and A. Bandeira

#### Prerequisites

This course is aimed at students with a background in linear algebra, probability, and basic functional analysis. In particular, familiarity with Hilbert spaces on the level of the "Hilbert spaces" chapter posted below (excluding the appendices) is expected.

#### Lecture notes and prerequisite material

- Lecture notes
- Prerequisite material

Hilbert spaces - Handouts

Comprehensive summary of linear algebra

Comprehensive summary of functional analysis

Notes on compact sets

#### Homework assignments

There will be exercise session notes posted every Thursday after the lecture. The TAs will inform you by email which of the problems in these notes will be solved in the exercise session the following Monday. The expectation is that you prepare these problems before the exercise session.

Date | Problems | Solutions | Extra Notes |
---|---|---|---|

26.02.2024 | Set 1 | Solutions 1 | Complements on the Fourier Transform |

04.03.2024 | Set 2 | Solutions 2 | |

11.03.2024 | Set 3 | Solutions 3 | Complements on Wavelets |

18.03.2024 | Set 4 | Solutions 4 | |

25.03.2024 | Set 5 | Solutions 5 | |

08.04.2024 | Set 6 | Solutions 6 | |

22.04.2024 | Set 7 | Solutions 7 | |

29.04.2024 | Set 8 | Solutions 8 | |

06.05.2024 | Set 9 | Solutions 9 | Complements on Compactness |

13.05.2024 | Set 10 | Solutions 10 | |

27.05.2024 | Set 11 | Solutions 11 |

#### Previous years' exams and solutions

Year | Exam | Handout | Solutions |
---|---|---|---|

Summer 2018 | Exam 2018 | - | Solutions 2018 |

Summer 2019 | Exam 2019 | Handout 2019 | Solutions 2019 |

Summer 2020 | Exam 2020 | Handout 2020 | Solutions 2020 |

Winter 2020/2021 | Exam 2020/2021 | Handout 2020/2021 | Solutions 2020/2021 |

Summer 2021 | Exam 2021 | Handout 2021 | Solutions 2021 |

Winter 2021/2022 | Exam 2021/2022 | Handout 2021/2022 | Solutions 2021/2022 |

Summer 2022 | Exam 2022 | Handout 2022 | Solutions 2022 |

Summer 2023 | Exam 2023 | Handout 2023 | Solutions 2023 |

#### Material for the exam

All the material that has been covered during the lectures and the exercise sessions will be relevant for the exam.

You are allowed to bring a summary of 10 handwritten or printed A4 pages (or 5 A4 pages on both sides). Electronic devices (laptops, calculators, cellphones, etc...) are not allowed.

#### Recommended reading

If you want to go into more depth or if you need additional background material, please check out these books:

- S. Mallat, "A wavelet tour of signal processing: The sparse way", 3rd ed., Elsevier, 2009
- M. Vetterli and J. Kovačević, "Wavelets and subband coding", Prentice Hall, 1995
- I. Daubechies, "Ten lectures on wavelets", SIAM, 1992
- O. Christensen, "An introduction to frames and Riesz bases", Birkhäuser, 2003
- K. Gröchenig, "Foundations of time-frequency analysis", Springer, 2001
- M. Elad, "Sparse and redundant representations — From theory to applications in signal and image processing", Springer, 2010
- M. Vetterli, J. Kovačević, and V. K. Goyal, "Foundations of signal processing", 3rd ed., Cambridge University Press, 2014
- S. Foucart and H. Rauhut, "A mathematical introduction to compressive sensing", Springer, 2013
- M. J. Wainwright, "High-dimensional statistics: A non-asymptotic viewpoint", Vol. 48, Cambridge University Press, 2019
- R. Vershynin, "High-dimensional probability: An introduction with applications in data science", Vol. 47, Cambridge University Press, 2018
- G. Grimmett and D. Stirzaker, "Probability and random processes", 3rd ed., Oxford University Press, 2001