# 227-0434-10L Mathematics of Information

Semester | Spring Semester 2022 |

Lecturers | H. Bölcskei |

Periodicity | yearly recurring course |

Language of instruction | English |

### Courses

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

227-0434-10 V | Mathematics of Information Hybride Veranstaltung, die Veranstaltung kann in Präsenz oder online verfolgt werden. | 3 hrs |
| H. Bölcskei | |||

227-0434-10 U | Mathematics of Information Hybride Veranstaltung, die Veranstaltung kann in Präsenz oder online verfolgt werden. | 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 and exercise sessions with homework problems. |

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 only offered in the session after the course unit. Repetition only possible after re-enrolling. |

Mode of examination | written 180 minutes |

Written aids | 10 handwritten or printed A4 pages summary (or 5 A4 pages on both sides). Electronic devices (laptops, calculators, cellphones, etc...) are 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. |