16 citations · 69 across the 34 of their papers we have counts for
40 papers
Wavelet orthonormal bases for nonexpansive dilations
Jordy Timo van Velthoven, Felix Voigtlaender
We construct wavelet orthonormal bases for nonexpansive dilations and integer translations that do not admit a wavelet set. In addition, these orthonormal bases do not satisfy the…
Maximally Spread Out Measures and Implications for Phase Transitions in Approximation Theory
Hannes Matt, Erwin Riegler, Felix Voigtlaender
We establish the existence of a "maximally spread out" Borel probability measure on a totally bounded subset $\SC$ of a (…
Linear dependence of time-frequency shifts of a Schwartz function
Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven +1
We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwal…
Smoothness spaces for warped time-frequency representations -- Decomposition spaces and embedding relations
Nicki Holighaus, Felix Voigtlaender
In a recent paper, we have shown that warped time-frequency representations provide a rich framework for the construction and study of smoothness spaces matched to very general pha…
Classification of anisotropic local Hardy spaces and inhomogeneous Triebel-Lizorkin spaces
Jordy Timo van Velthoven, Felix Voigtlaender
This paper provides a characterization of when two expansive matrices yield the same anisotropic local Hardy and inhomogeneous Triebel-Lizorkin spaces. The characterization is in t…
Upper and lower bounds for the Lipschitz constant of random neural networks
Paul Geuchen, Dominik Stöger, Thomas Telaar +1
Empirical studies have widely demonstrated that neural networks are highly sensitive to small, adversarial perturbations of the input. The worst-case robustness against these so-ca…
Optimal approximation using complex-valued neural networks
Paul Geuchen, Felix Voigtlaender
Complex-valued neural networks (CVNNs) have recently shown promising empirical success, for instance for increasing the stability of recurrent neural networks and for improving the…
Classification of anisotropic Triebel-Lizorkin spaces
Sarah Koppensteiner, Jordy Timo van Velthoven, Felix Voigtlaender
This paper provides a classification theorem for expansive matrices generating the same anisotropic homogeneous Triebel-Lizorkin space $\dot{\mat…
sampling numbers for the Fourier-analytic Barron space
Felix Voigtlaender
In this paper, we consider Barron functions of smoothness , which are functions that can be written as \[ f(x) = \int_{\mathbb{R}^d} F(ξ) \, e^{2…
Coorbit theory of warped time-frequency systems in
Nicki Holighaus, Felix Voigtlaender
Warped time-frequency systems have recently been introduced as a class of structured continuous frames for functions on the real line. Herein, we generalize this framework to the s…
Learning ReLU networks to high uniform accuracy is intractable
Julius Berner, Philipp Grohs, Felix Voigtlaender
Statistical learning theory provides bounds on the necessary number of training samples needed to reach a prescribed accuracy in a learning problem formulated over a given target c…
Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II
Sarah Koppensteiner, Jordy Timo van Velthoven, Felix Voigtlaender
Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces for the endpoint case of and…
Coorbit spaces associated to quasi-Banach function spaces and their molecular decomposition
Jordy Timo van Velthoven, Felix Voigtlaender
This paper provides a self-contained exposition of coorbit spaces associated to integrable group representations and quasi-Banach function spaces, and at the same time extends and…
A fractal uncertainty principle for Bergman spaces and analytic wavelets
Luis Daniel Abreu, Zouhair Mouayn, Felix Voigtlaender
Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor sets and by similar results of Knutsen for joint time-frequency representations (i.e., the short-time F…
Optimal learning of high-dimensional classification problems using deep neural networks
Philipp Petersen, Felix Voigtlaender
We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish…
Sobolev-type embeddings for neural network approximation spaces
Philipp Grohs, Felix Voigtlaender
We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in ) by ReLU neural networ…
A note on the invertibility of the Gabor frame operator on certain modulation spaces
Dae Gwan Lee, Friedrich Philipp, Felix Voigtlaender
We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space , the weighted…
Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces
Philipp Grohs, Felix Voigtlaender
We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by…
Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I
Sarah Koppensteiner, Jordy Timo van Velthoven, Felix Voigtlaender
This paper provides maximal function characterizations of anisotropic Triebel-Lizorkin spaces associated to general expansive matrices for the full range of parameters $p \in (0,\i…
The universal approximation theorem for complex-valued neural networks
Felix Voigtlaender
We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a co…
Neural network approximation and estimation of classifiers with classification boundary in a Barron class
Andrei Caragea, Philipp Petersen, Felix Voigtlaender
We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guara…
Phase Transitions in Rate Distortion Theory and Deep Learning
Philipp Grohs, Andreas Klotz, Felix Voigtlaender
Rate distortion theory is concerned with optimally encoding a given signal class using a budget of bits, as . We say that can be compres…
Schur-type Banach modules of integral kernels acting on mixed-norm Lebesgue spaces
Nicki Holighaus, Felix Voigtlaender
Schur's test states that if satisfies and , then the associated integral operator acts boundedly…
On dual molecules and convolution-dominated operators
José Luis Romero, Jordy Timo van Velthoven, Felix Voigtlaender
We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the siz…