Markus Schweighofer - Aktuelle Lehrveranstaltungen

SS 2026: Topics in Mathematics of Data Science
9 ECTS credits, 4V+2Ü
Content. This course provides a mathematically rigorous treatment of high-dimensional phenomena, exploring the concentration of measure, random matrix theory, and convex optimization. We begin with the geometry of high-dimensional space and the "blessing of dimensionality" through concentration inequalities and tail bounds. The curriculum develops a spectral perspective on data, covering the Singular Value Decomposition (SVD) and PCA in the regime of high-dimensional noise, including the Marčenko–Pastur law and BBP phase transitions. We then analyze graph-based learning through spectral clustering, diffusion maps, and the manifold hypothesis, alongside randomized numerical linear algebra and the Johnson–Lindenstrauss lemma. The course concludes with community detection via Semidefinite Programming (SDP) and an advanced study of compressive sensing, establishing the theoretical foundations for exact signal recovery from underdetermined systems.
Target audience. Students of mathematics, mathematical finance, or physics with a firm background in linear algebra and stochastics. The course is primarily designed for students in Master's programs or advanced students in the final year of their Bachelor's programs.
Credit eligibilities.

The lecture can be taken as an elective module (Wahlmodul) for students of mathematics (Bachelor's and Master's programs).
The lecture is admissible in Block 1 (basics) of the Advanced Data and Information Literacy Track (ADILT).
Students of mathematical finance or physics are highly welcome but should verify how this lecture can be credited toward their specific study programs.

Literature. This lecture closely follows the recent textbook draft Topics in Mathematics of Data Science by Afonso Bandeira, Amit Singer, and Thomas Strohmer.
Prerequisites. We will assume that students have a solid knowledge of linear algebra, stochastics, and basic analysis. For students from Konstanz, this ideally means having passed the courses Linear Algebra I and II, Wahrscheinlichkeitstheorie, Statistik, and Analysis I for mathematicians without major difficulty. Alternatively, students should be prepared to work through any unfamiliar parts of the book independently.
Plan of the course. This plan is subject to change but is currently scheduled as follows:

Curses, Blessings, and Surprises in High Dimensions
Lecture 01:	Apr 08,	Geometry of spheres and balls in high dimension
Lecture 02:	Apr 10,	Basic concepts from probability
Lecture 03:	Apr 15,	Concentration of measure, a "blessing of dimensionality"
Singular Value Decomposition and Principal Component Analysis
Lecture 04:	Apr 17,	Singular Value Decomposition
Lecture 05:	Apr 22,	Principal Component Analysis and dimension reduction
Lecture 06:	Apr 24,	PCA in high dimensions and Marčenko–Pastur law
Graphs, Networks, and Clustering
Lecture 07:	Apr 29,	PageRank
Lecture 08:	May 01,	Graph theory
Lecture 09:	May 06,	Clustering
Nonlinear Dimension Reduction and Diffusion Maps
Lecture 10:	May 08,	Diffusion Maps
Lecture 11:	May 13,	Connections between Diffusion Maps and Spectral Clustering
Lecture 12:	May 15,	Semi-supervised learning
Linear Dimension Reduction via Random Projections
Lecture 13:	May 20,	The Johnson–Lindenstrauss Lemma
Lecture 14:	May 22,	Randomized SVD
Community Detection and the Power of Convex Relaxation
Lecture 15:	May 27,	Max-Cut, lifting, and approximation algorithms
Lecture 16:	May 29,	Community Detection
Concentration of Measure and Gaussian Analysis
Lecture 17:	Jun 10,	Gaussian integration by parts and Wick's formula
Lecture 18:	Jun 12,	Gaussian interpolation, Poincaré, and concentration
Lecture 19:	Jun 17,	Gaussian comparison principles
Lecture 20:	Jun 19,	Gordon's Theorem
Matrix Concentration Inequalities
Lecture 21:	Jun 24,	Non-commutative Khintchine inequality
Lecture 22:	Jun 26,	Matrix concentration inequalities
Lecture 23:	Jul 01,	Improvements leveraging intrinsic freeness
Compressive Sensing and Sparsity
Lecture 24:	Jul 03,	Sparse recovery
Lecture 25:	Jul 08,	Null space property and exact recovery
Lecture 26:	Jul 10,	The Restricted Isometry Property
Lecture 27:	Jul 15,	Duality and exact recovery
Lecture 28:	Jul 17,	Compressive sensing: from theory to practice

Assessment Methods. Active and frequent participation in the weekly tutorial sessions ("Übung") is a basic requirement for the successful completion of the module. Exams will be oral and can be scheduled at individually agreed times.
Assignments. Weekly written homework must be submitted electronically. Students are expected to be able to discuss and explain their submitted solutions during the exercise classes.
Weekly Schedule. We will meet twice per week for lectures and once for the tutorial session. Lectures are scheduled for Wednesdays and Fridays from 11:45 to 13:15 in room D406. Please contact me immediately if this schedule presents a conflict. The time slot for the exercise course will be determined during the first session.
Communication. Announcements concerning the course will be sent via ILIAS. I strongly recommend subscribing to the course on ZEUS, which usually results in an automatic registration on ILIAS.