Update of the website. Added new information about my professorship at the KU Eichstätt-Ingolstadt.

Update of the website. Added two preprints and updated my biography.

Update of the website. Added two papers. In the first one, Andrei Caragea, Philipp Petersen, and I study the performance of neural networks for high-dimensional classification problems with structured class boundaries. In a nutshell, we show that if these boundaries are locally of Barron-type, then one obtains learning and approximation bounds with rates independent of the underlying dimension.
In the second one, I generalize the classical universal approximation theorem to the setting of complex-valued neural networks. Under very mild continuity assumptions on the activation function, I show for shallow networks that universality holds if and only if the real- or the imaginary part of the activation function is not polyharmonic. For networks with at least two hidden layers, universality holds if and only if the activation function is neither a polynomial, nor holomorphic, nor antiholomorphic.
Added two talks, one of which is available on Youtube.
Continued the endless fight against link rot.

Update of the website and of the CV. Added several new preprints and talks. Updated photograph.

Update of the website and of the CV. Added several new preprints and talks.

First update of the website since I moved from Berlin to Eichstätt. The CV is still out of date, however...

After my talk about the approximation properties of ReLU neural networks at the Research seminar "Mathematics of Computation", I was asked for the slides to my talk. These can be found here.

Many thanks to Tino Ullrich for inviting me to give this talk!

I just added to my list of publications two new preprints that I recently uploaded to the arXiv.

The first one establishes a rather general version of Price's theorem: It gives a simple formula for computing the partial derivatives of the map ρ ↦ 𝔼[g(Xᵨ)], where Xᵨ is a normally distributed random variable with covariance matrix ρ. Price published this result in 1958, but did not precisely state the required assumptions on g. In the paper, I show that one can in fact take every tempered distribution g. This result is used in the paper ℓ¹-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed? written by three of my colleagues.

The second preprint, written jointly with my colleague Philipp Petersen analyzes the approximation power of neural networks that use the ReLU activation function. We analyze how deep and wide such a neural network needs to be, in order to approximate any "piecewise smooth" function. We also show that these bounds are sharp.

I also added several talks that I gave in the last months, including the lecture notes for the lecture series "Sparsity Properties of Frames via Decomposition Spaces" that I gave at the Summer School on Applied Harmonic Analysis in Genoa.

I just added a new preprint to my list of publications that I uploaded to the arXiv in December.

Furthermore, I am very happy to be a speaker at the Summer School on Applied Harmonic Analysis that will take place in Genoa from July 24-28, 2017.

Finally, I want to mention that my toy problem (about whether completeness of spaces can be characterized by the convergence of Neumann series) has been solved (negatively) quite a while ago.

First version of this website was uploaded :)