Prof. Dr. Holger Rauhut

Deep learning methods form the basis for modern applications of machine learning and artificial intelligence. Despite many successes and advances in a wide range of areas, the mathematical foundations are still only partially understood, and it is often unclear why the methods work or when they work. In particular, there is often a lack of rigorous mathematical guarantees for the success of learning algorithms. In my research, I aim to advance the mathematical understanding of deep learning.

I focus on several subfields of the mathematics of deep learning:

• Convergence theory and impicit regularization for training algorithms
• Deep learning methods for inverse problems
• Uncertainty quantification for deep learning

Compressive sensing is a subfield of mathematical signal processing. The basic problem is to reconstruct signals using as few measurements as possible. Mathematically, this leads to solving an underdetermined system of equations. To make reconstruction possible, it is used that in practice signals are often compressible, i.e., they can be approximated well by a sparse vector. The theory of compressive sensing provides efficient reconstruction algorithms (e.g., l1 minimization). Interestingly, provably optimal measurement matrices (which describe the linear measurement process) are random matrices. For such matrices, guarantees can be given for the minimum number of measurements required (depending on the signal length and sparsity, i.e., the number of non-vanishing coefficients). Tools from high-dimensional probability theory are used for the mathematical analysis.