Publications (70)
Robust and Fast Training via Per-Sample Clipping
Davide Nobile, Philipp Grohs
We propose a robust gradient estimator based on per-sample gradient clipping and analyze its properties both theoretically and empirically. We show that the resulting method, per-s…
Stable Phase Retrieval in Infinite Dimensions
Rima Alaifari, Ingrid Daubechies, Philipp Grohs +1
The problem of phase retrieval is to determine a signal , with a Hilbert space, from intensity measurements , where $F(Ï):=\langle f , Ï_…
Accurate Ab-initio Neural-network Solutions to Large-Scale Electronic Structure Problems
Michael Scherbela, Nicholas Gao, Philipp Grohs +1
We present finite-range embeddings (FiRE), a novel wave function ansatz for accurate large-scale ab-initio electronic structure calculations. Compared to contemporary neural-networ…
From completeness of discrete translates to phaseless sampling of the short-time Fourier transform
Philipp Grohs, Lukas Liehr, Irina Shafkulovska
We study the uniqueness problem in short-time Fourier transform phase retrieval by exploring a connection to the completeness problem of discrete translates. Specifically, we prove…
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…
Parabolic Molecules
Philipp Grohs, Gitta Kutyniok
Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significantly increased attention due to t…
Is Variational Monte Carlo Robust? Sharp Moment Thresholds and Heavy-tailed Stochastic Optimization
Philipp Grohs, Davide Nobile
Variational Monte Carlo (VMC) is a central algorithm in electronic structure theory and has gained renewed importance through modern neural-network ansätze such as FermiNet. At it…
FakET: Simulating Cryo-Electron Tomograms with Neural Style Transfer
Pavol Harar, Lukas Herrmann, Philipp Grohs +1
In cryo-electron microscopy, accurate particle localization and classification are imperative. Recent deep learning solutions, though successful, require extensive training data se…
The Information-Theoretic Benefit of Shared Representations under Orthogonality Constraints
Thomas Dittrich, Oliver Potocki, Philipp Grohs
Modern deep learning architectures are increasingly multi-task and multi-modal, using a pretrained foundation model combined with task-specific, fine-tuned models. Empirically, exp…
Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
Julius Berner, Philipp Grohs, Arnulf Jentzen
The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolu…
Gold-standard solutions to the Schrödinger equation using deep learning: How much physics do we need?
Leon Gerard, Michael Scherbela, Philipp Marquetand +1
Finding accurate solutions to the Schrödinger equation is the key unsolved challenge of computational chemistry. Given its importance for the development of new chemical compounds…
Discrete Deep Feature Extraction: A Theory and New Architectures
Thomas Wiatowski, Michael Tschannen, Aleksandar StaniÄ +2
First steps towards a mathematical theory of deep convolutional neural networks for feature extraction were made---for the continuous-time case---in Mallat, 2012, and Wiatowski and…
Deep neural network approximations for Monte Carlo algorithms
Philipp Grohs, Arnulf Jentzen, Diyora Salimova
Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are…
Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality
Philipp Grohs, Lukas Liehr
We study the phase reconstruction of signals belonging to complex Gaussian shift-invariant spaces from spectrogram measurements where $\math…
On the Approximation of Functions with Line Singularities by Ridgelets
Axel Obermeier, Philipp Grohs
In [GO15], the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the…
Deep Neural Network Approximation Theory
Dennis Elbrächter, Dmytro Perekrestenko, Philipp Grohs +1
This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount…
Phase Retrieval: Uniqueness and Stability
Philipp Grohs, Sarah Koppensteiner, Martin Rathmair
The problem of phase retrieval, i.e., the problem of recovering a function from the magnitudes of its Fourier transform, naturally arises in various fields of physics, such as astr…
The universal approximation power of finite-width deep ReLU networks
Dmytro Perekrestenko, Philipp Grohs, Dennis Elbrächter +1
We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (Bölcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensio…
Continuous Shearlet Frames and Resolution of the Wavefront Set
Philipp Grohs
In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transform…
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…
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…
Topology Reduction in Deep Convolutional Feature Extraction Networks
Thomas Wiatowski, Philipp Grohs, Helmut Bölcskei
Deep convolutional neural networks (CNNs) used in practice employ potentially hundreds of layers and ,s of nodes. Such network sizes entail significant computational compl…
Variational Monte Carlo on a Budget -- Fine-tuning pre-trained Neural Wavefunctions
Michael Scherbela, Leon Gerard, Philipp Grohs
Obtaining accurate solutions to the Schrödinger equation is the key challenge in computational quantum chemistry. Deep-learning-based Variational Monte Carlo (DL-VMC) has recently…
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…
Transferable Neural Wavefunctions for Solids
Leon Gerard, Michael Scherbela, Halvard Sutterud +2
Deep-Learning-based Variational Monte Carlo (DL-VMC) has recently emerged as a highly accurate approach for finding approximate solutions to the many-electron Schrödinger equation…
Energy Propagation in Deep Convolutional Neural Networks
Thomas Wiatowski, Philipp Grohs, Helmut Bölcskei
Many practical machine learning tasks employ very deep convolutional neural networks. Such large depths pose formidable computational challenges in training and operating the netwo…
Continuous Shearlet Tight Frames
Philipp Grohs
Based on the shearlet transform we present a general construction of continuous tight frames for from any sufficiently smooth function with anisotropic moments.…
Sampling Complexity of Deep Approximation Spaces
Ahmed Abdeljawad, Philipp Grohs
While it is well-known that neural networks enjoy excellent approximation capabilities, it remains a big challenge to compute such approximations from point samples. Based on tools…
Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames
Rima Alaifari, Ingrid Daubechies, Philipp Grohs +1
In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions that co…
Solving the Kolmogorov PDE by means of deep learning
Christian Beck, Sebastian Becker, Philipp Grohs +2
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and…
Gabor phase retrieval is severely ill-posed
Rima Alaifari, Philipp Grohs
The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This…
Optimal Adaptive Ridgelet Schemes for Linear Transport Equations
Philipp Grohs, Axel Obermeier
In this paper we present a novel method for the numerical solution of linear transport equations, which is based on ridgelets. Such equations arise for instance in radiative transf…
Non-uniqueness theory in sampled STFT phase retrieval
Philipp Grohs, Lukas Liehr
The reconstruction of a function from its spectrogram (i.e., the absolute value of its short-time Fourier transform (STFT)) arises as a key problem in several important application…
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…
Space-time error estimates for deep neural network approximations for differential equations
Philipp Grohs, Fabian Hornung, Arnulf Jentzen +1
Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including comp…
DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing
Dennis Elbrächter, Philipp Grohs, Arnulf Jentzen +1
We analyze approximation rates by deep ReLU networks of a class of multi-variate solutions of Kolmogorov equations which arise in option pricing. Key technical devices are deep ReL…
Uniform error estimates for artificial neural network approximations for heat equations
Lukas Gonon, Philipp Grohs, Arnulf Jentzen +2
Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly r…
Definability and stability of multiscale decompositions for manifold-valued data
Philipp Grohs, Johannes Wallner
We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold-analogues of biorthogonal wavelets to possess perfect…
Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality
Philipp Grohs, Shokhrukh Ibragimov, Arnulf Jentzen +1
Artificial neural networks (ANNs) have become a very powerful tool in the approximation of high-dimensional functions. Especially, deep ANNs, consisting of a large number of hidden…
Stable Gabor phase retrieval for multivariate functions
Philipp Grohs, Martin Rathmair
In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Communications on Pure and Applied Mathematics (2018)] the instabilities of the Gabo…
Towards a Foundation Model for Neural Network Wavefunctions
Michael Scherbela, Leon Gerard, Philipp Grohs
Deep neural networks have become a highly accurate and powerful wavefunction ansatz in combination with variational Monte Carlo methods for solving the electronic Schrödinger equa…
How degenerate is the parametrization of neural networks with the ReLU activation function?
Julius Berner, Dennis Elbrächter, Philipp Grohs
Neural network training is usually accomplished by solving a non-convex optimization problem using stochastic gradient descent. Although one optimizes over the networks parameters,…
Projection-Based Finite Elements for Nonlinear Function Spaces
Philipp Grohs, Hanne Hardering, Oliver Sander +1
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in…
Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations
Philipp Grohs, Lukas Herrmann
The approximation of solutions to second order Hamilton--Jacobi--Bellman (HJB) equations by deep neural networks is investigated. It is shown that for HJB equations that arise in t…
Intrinsic Localization of Anisotropic Frames II: -Molecules
Philipp Grohs, Stefano Vigogna
This article is a continuation of the recent paper [Grohs, Intrinsic localization of anisotropic frames, ACHA, 2013], where off-diagonal-decay properties (often referred to as 'loc…
-stability analysis for Gabor phase retrieval
Philipp Grohs, Martin Rathmair
We consider the problem of reconstructing the missing phase information from spectrogram data with $$ \mathcal{G}f(x,y)=\int_\mathbb{R} f(t) e^{-Ï(t-x)^2}e^{-2Ï…
-Molecules
Philipp Grohs, Sandra Keiper, Gitta Kutyniok +1
Within the area of applied harmonic analysis, various multiscale systems such as wavelets, ridgelets, curvelets, and shearlets have been introduced and successfully applied. The ke…
Ruled Laguerre minimal surfaces
Mikhail Skopenkov, Helmut Pottmann, Philipp Grohs
A Laguerre minimal surface is an immersed surface in the Euclidean space being an extremal of the functional \int (H^2/K - 1) dA. In the present paper, we prove that the only ruled…
Phase retrieval in Fock space and perturbation of Liouville sets
Philipp Grohs, Lukas Liehr, Martin Rathmair
We study the determination of functions in Fock space from samples of their absolute value, known as the phase retrieval problem in Fock space. An important finding in this researc…
Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions
Philipp Grohs, Lukas Herrmann
In recent work it has been established that deep neural networks are capable of approximating solutions to a large class of parabolic partial differential equations without incurri…
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…
Integral representations of shallow neural network with Rectified Power Unit activation function
Ahmed Abdeljawad, Philipp Grohs
In this effort, we derive a formula for the integral representation of a shallow neural network with the Rectified Power Unit activation function. Mainly, our first result deals wi…
Towards a regularity theory for ReLU networks -- chain rule and global error estimates
Julius Berner, Dennis Elbrächter, Philipp Grohs +1
Although for neural networks with locally Lipschitz continuous activation functions the classical derivative exists almost everywhere, the standard chain rule is in general not app…
A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
Philipp Grohs, Fabian Hornung, Arnulf Jentzen +1
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recogniti…
The sampling complexity of learning invertible residual neural networks
Yuanyuan Li, Philipp Grohs, Philipp Petersen
In recent work it has been shown that determining a feedforward ReLU neural network to within high uniform accuracy from point samples suffers from the curse of dimensionality in t…
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…
Phase retrieval in the general setting of continuous frames for Banach spaces
Rima Alaifari, Philipp Grohs
We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of…
Approximations with deep neural networks in Sobolev time-space
Ahmed Abdeljawad, Philipp Grohs
Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that c…
Stable Gabor Phase Retrieval and Spectral Clustering
Philipp Grohs, Martin Rathmair
We consider the problem of reconstructing a signal from its spectrogram, i.e., the magnitudes of its Gabor transform $$V_Ïf (x,y):=\int_{\mathbb{R}}f(t)e^{-Ï(t-x)^2…
Limitations of Learning Tanh Neural Networks with Finite Precision
Philipp Grohs, MatÄj Trödler
We investigate limitations of learning neural networks from point evaluations under finite-precision computations and accuracy guarantees, building on Berner, Grohs,…
Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning
Julius Berner, Markus Dablander, Philipp Grohs
We present a deep learning algorithm for the numerical solution of parametric families of high-dimensional linear Kolmogorov partial differential equations (PDEs). Our method is ba…
Phaseless sampling on square-root lattices
Philipp Grohs, Lukas Liehr
Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval…
Theory-to-Practice Gap for Neural Networks and Neural Operators
Philipp Grohs, Samuel Lanthaler, Margaret Trautner
This work studies the sampling complexity of learning with ReLU neural networks and neural operators. For mappings belonging to relevant approximation spaces, we derive upper bound…
Multi-window STFT phase retrieval: lattice uniqueness
Philipp Grohs, Lukas Liehr, Martin Rathmair
Short-time Fourier transform (STFT) phase retrieval refers to the reconstruction of a function from its spectrogram, i.e., the magnitudes of its short-time Fourier transform $V…
On foundational discretization barriers in STFT phase retrieval
Philipp Grohs, Lukas Liehr
We prove that there exists no window function and no lattice such that every is determined up to…
Deep Convolutional Neural Networks on Cartoon Functions
Philipp Grohs, Thomas Wiatowski, Helmut Bölcskei
Wiatowski and Bölcskei, 2015, proved that deformation stability and vertical translation invariance of deep convolutional neural network-based feature extractors are guaranteed by…
Solving the electronic Schrödinger equation for multiple nuclear geometries with weight-sharing deep neural networks
Michael Scherbela, Rafael Reisenhofer, Leon Gerard +2
Accurate numerical solutions for the Schrödinger equation are of utmost importance in quantum chemistry. However, the computational cost of current high-accuracy methods scales po…
Injectivity of Gabor phase retrieval from lattice measurements
Philipp Grohs, Lukas Liehr
We establish novel uniqueness results for the Gabor phase retrieval problem: if denotes the Gabor transform then every $f \in…
Cartoon Approximation with -Curvelets
Philipp Grohs, Sandra Keiper, Gitta Kutyniok +1
It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise -functions, separated by a singularity curve.…
Anisotropic Multiscale Systems on Bounded Domains
Philipp Grohs, Gitta Kutyniok, Jackie Ma +2
We provide a construction of multiscale systems on a bounded domain coined boundary shearlet systems, which satisfy several properties advantageous for app…