Photograph by Lutz Voigtlaender

associated to the group of Felix Krahmer

TU Munich

Email: felix@voigtlaender.xyz

CV (last updated on 23 May 2020)

If you trust in yourself. . .and believe in your dreams. . .and follow your star. . . you'll still get beaten by people who spent their time working hard and learning things and weren't so lazy.

—Terry Pratchett, The Wee Free Men

- Neural networks and their properties, in particular their expressiveness and approximation properties
- Information based complexity, in particular related to neural networks
- Sampling
- Solvability Complexity Index
- Approximation theory
- Function spaces and their embeddings
- Multiscale systems (wavelets, curvelets, shearlets and generalizations)
- Banach frames and atomic decompositions
- Coorbit theory
- Time-frequency analysis
- Fourier analysis

I am currently an Emmy Noether independent junior research group leader at the TU Munich, associated to the group of Felix Krahmer, funded through my project Stability and Solvability in Deep Learning.

From June 2020 until May 2021, I was a Senior Scientist in the Department of Mathematics at the University of Vienna, part of the group of Philipp Grohs.

Before that, from February 2018 to May 2020, I worked as an "Akademischer Rat" as part of the Scientific Computing Group at KU Eichstätt, lead by Götz Pfander.

From April 2016 until January 2018, I was a postdoctoral researcher at TU Berlin in the group of Gitta Kutyniok, where I worked on the DEDALE project. As part of that project, and with support by Anne Pein, I developed the DEDALE α-shearlet transform.

As a Ph.D. student, I worked in the group of Hartmut Führ at RWTH Aachen University, where I studied the approximation theoretic properties of different multiscale systems.
With my Ph.D. thesis *'Embedding Theorems for Decomposition Spaces with applications to Wavelet Coorbit Spaces'*, I graduated with distinction in November 2015.
In November 2016, I was awarded the Friedrich-Wilhelm-Award 2016
for my thesis.

I studied mathematics (Bachelor + Master) and computer science (Bachelor) at RWTH Aachen University, Germany, where I graduated with the Master degree with distinction in 2013.

I highly enjoy teaching mathematics and am committed to explaining carefully and presenting the material in an enjoyable way. At RWTH Aachen University, I have been teaching assistant for Analysis I and III, and for Harmonic analysis. Therefore, I am very proud to have received the Teaching award of the student council of mathematics at RWTH Aachen University.

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 :)