227-0434-10L  Mathematics of Information

SemesterSpring Semester 2022
LecturersH. Bölcskei
Periodicityyearly recurring course
Language of instructionEnglish


AbstractThe 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
ObjectiveThe 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.
ContentMathematics 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 notesDetailed lecture notes will be provided at the beginning of the semester.
Prerequisites / NoticeThis 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