131 citations · 174 across the 21 of their papers we have counts for
30 papers
The frame set of the first Hermite function
Markus Faulhuber, Philipp Petersen
We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gröchenig, Rome…
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…
VC dimensions of group convolutional neural networks
Philipp Christian Petersen, Anna Sepliarskaia
We study the generalization capacity of group convolutional neural networks. We identify precise estimates for the VC dimensions of simple sets of group convolutional neural networ…
Limitations of neural network training due to numerical instability of backpropagation
Clemens Karner, Vladimir Kazeev, Philipp Christian Petersen
We study the training of deep neural networks by gradient descent where floating-point arithmetic is used to compute the gradients. In this framework and under realistic assumption…
Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems
Andrés Felipe Lerma Pineda, Philipp Christian Petersen
We study the problem of reconstructing solutions of inverse problems when only noisy measurements are available. We assume that the problem can be modeled with an infinite-dimensio…
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…
Deep Microlocal Reconstruction for Limited-Angle Tomography
Héctor Andrade-Loarca, Gitta Kutyniok, Ozan Öktem +1
We present a deep learning-based algorithm to jointly solve a reconstruction problem and a wavefront set extraction problem in tomographic imaging. The algorithm is based on a rece…
The Modern Mathematics of Deep Learning
Julius Berner, Philipp Grohs, Gitta Kutyniok +1
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of…
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…
Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
Carlo Marcati, Joost A. A. Opschoor, Philipp C. Petersen +1
We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in for weighted analytic function classes in certain polytopal domains , in space dimensio…
Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks
Moritz Geist, Philipp Petersen, Mones Raslan +2
We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysi…
Efficient Approximation of Solutions of Parametric Linear Transport Equations by ReLU DNNs
Fabian Laakmann, Philipp Petersen
We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For…
Approximation in with deep ReLU neural networks
Felix Voigtlaender, Philipp Petersen
We discuss the expressive power of neural networks which use the non-smooth ReLU activation function by analyzing the approximation theoretic properties…
A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
Gitta Kutyniok, Philipp Petersen, Mones Raslan +1
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge…
Error bounds for approximations with deep ReLU neural networks in norms
Ingo Gühring, Gitta Kutyniok, Philipp Petersen
We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU net…
The Oracle of DLphi
Dominik Alfke, Weston Baines, Jan Blechschmidt +24
We present a novel technique based on deep learning and set theory which yields exceptional classification and prediction results. Having access to a sufficiently large amount of l…
Extraction of digital wavefront sets using applied harmonic analysis and deep neural networks
Héctor Andrade-Loarca, Gitta Kutyniok, Ozan Öktem +1
Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences. Of particular importanc…
Gamma-convergence of a shearlet-based Ginzburg--Landau energy
Philipp Christian Petersen, Endre Süli
We introduce two shearlet-based Ginzburg--Landau energies, based on the continuous and the discrete shearlet transform. The energies result from replacing the elastic energy term o…
Equivalence of approximation by convolutional neural networks and fully-connected networks
Philipp Petersen, Felix Voigtlaender
Convolutional neural networks are the most widely used type of neural networks in applications. In mathematical analysis, however, mostly fully-connected networks are studied. In t…
Topological properties of the set of functions generated by neural networks of fixed size
Philipp Petersen, Mones Raslan, Felix Voigtlaender
We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size. Surprisingly, this set has many undesirable properties. It…
Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces
Philipp Petersen, Mones Raslan
In this paper, we study a newly developed shearlet system on bounded domains which yields frames for for some , . We will derive ap…
Optimal approximation of piecewise smooth functions using deep ReLU neural networks
Philipp Petersen, Felix Voigtlaender
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^…
Optimal Approximation with Sparsely Connected Deep Neural Networks
Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok +1
We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes i…
Bendlets: A Second-Order Shearlet Transform with Bent Elements
Christian Lessig, Philipp Petersen, Martin Schäfer
We introduce bendlets, a shearlet-like system that is based on anisotropic scaling, translation, shearing, and bending of a compactly supported generator. With shearing being linea…