227-0434-10L Mathematics of Information
Semester | Frühjahrssemester 2019 |
Dozierende | H. Bölcskei |
Periodizität | jährlich wiederkehrende Veranstaltung |
Lehrsprache | Englisch |
Kurzbeschreibung | The class focuses on fundamental aspects of mathematical information science: Frame theory, sampling theory, sparsity, compressed sensing, uncertainty relations, spectrum-blind sampling, dimensionality reduction and sketching, randomized algorithms for large-scale sparse FFTs, inverse problems, (Kolmogorov) approximation theory, and information theory (lossless and lossy compression). |
Lernziel | After attending this lecture, participating in the exercise sessions, and working on the homework problem sets, students will have acquired a working knowledge of the most commonly used mathematical theories in information science. Students will also have to carry out a research project, either individually or in groups, with presentations at the end of the semester. |
Inhalt | 1. Signal representations: Frames in finite-dimensional spaces, frames in Hilbert spaces, wavelets, Gabor expansions 2. Sampling theorems: The sampling theorem as a frame expansion, irregular sampling, multi-band sampling, density theorems, spectrum-blind sampling 3. Sparsity and compressed sensing: Uncertainty relations in sparse signal recovery, recovery algorithms, Lasso, matching pursuit algorithms, compressed sensing, super-resolution 4. High-dimensional data and dimensionality reduction: Random projections, the Johnson-Lindenstrauss Lemma, sketching 5. Randomized algorithms for large-scale sparse FFTs 6. Approximation theory: Fundamental limits on compressibility of signal classes, Kolmogorov epsilon-entropy of signal classes, optimal encoding and decoding of signal classes 7. Information theory: Entropy, mutual information, lossy compression, rate-distortion theory, lossless compression, arithmetic coding, Lempel-Ziv compression |
Skript | Detailed lecture notes will be provided at the beginning of the semester. |
Voraussetzungen / Besonderes | This course is aimed at students with a background in basic linear algebra, analysis, and probability. We will, however, review required mathematical basics throughout the semester in the exercise sessions. |