# 227-0434-10L Mathematics of Information

Semester | Spring Semester 2021 |

Lecturers | H. Bölcskei |

Periodicity | yearly recurring course |

Language of instruction | English |

### Courses

Number | Title | Hours | Lecturers | ||||
---|---|---|---|---|---|---|---|

227-0434-10 V | Mathematics of Information | 3 hrs |
| H. Bölcskei | |||

227-0434-10 U | Mathematics of Information | 2 hrs |
| H. Bölcskei | |||

227-0434-10 A | Mathematics of Information | 2 hrs | H. Bölcskei |

### Catalogue data

Abstract | The class focuses on mathematical aspects of 1. Information science: Sampling theorems, frame theory, compressed sensing, sparsity, super-resolution, spectrum-blind sampling, subspace algorithms, dimensionality reduction 2. Learning theory: Approximation theory, greedy algorithms, 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, super-resolution, spectrum-blind sampling, subspace algorithms (ESPRIT), 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, 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 5. Uniform laws of large numbers: Rademacher complexity, Vapnik-Chervonenkis dimension, classes with polynomial discrimination |

Lecture notes | Detailed lecture notes will be provided at the beginning of the semester. |

Prerequisites / Notice | This course is aimed at students with a background in basic linear algebra, analysis, statistics, and probability. We encourage students who are interested in mathematical data science to take both this course and "401-4944-20L Mathematics of Data Science" by Prof. A. Bandeira. The two courses are designed to be complementary. H. Bölcskei and A. Bandeira |

### Performance assessment

Performance assessment information (valid until the course unit is held again) | |

Performance assessment as a semester course | |

ECTS credits | 8 credits |

Examiners | H. Bölcskei |

Type | session examination |

Language of examination | English |

Repetition | The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit. |

Mode of examination | written 180 minutes |

Additional information on mode of examination | 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). |

Written aids | Lecture and exercise notes allowed. Electronic devices (laptops, calculators, cellphones, etc...) NOT allowed. |

This information can be updated until the beginning of the semester; information on the examination timetable is binding. |

### Learning materials

Main link | Course Website |

Only public learning materials are listed. |

### Groups

No information on groups available. |

### Restrictions

There are no additional restrictions for the registration. |