arXiv subjects

Philipp Petersen

Publications and source records attributed to Philipp Petersen.

At least 37 records · Page 2Linked to original sources

arXiv2023

The Modern Mathematics of Deep Learning

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 learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.

cs.LG↗

arXiv2022

Neural network approximation and estimation of classifiers with classification boundary in a Barron class

We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guarantees for empirical risk minimization using networks of a suitable size, depending on the number of training samples available. The obtained approximation and estimation rates are independent of the dimension of the input, showing that the curse of dimensionality can be overcome in this setting; in fact, the input dimension only enters in the form of a polynomial factor. Regarding the regularity of the target classification function, we assume the interfaces between the different classes to be locally of Barron-type. We complement our results by studying the relations between various Barron-type spaces that have been proposed in the literature. These spaces differ substantially more from each other than the current literature suggests.

math.FA↗

arXiv2021

Optimal learning of high-dimensional classification problems using deep neural networks

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 universal lower bounds for this estimation problem, for general classes of continuous decision boundaries. For the class of locally Barron-regular decision boundaries, we find that the optimal estimation rates are essentially independent of the underlying dimension and can be realized by empirical risk minimization methods over a suitable class of deep neural networks. These results are based on novel estimates of the $L^1$ and $L^\infty$ entropies of the class of Barron-regular functions.

math.FA↗

arXiv2021

Deep Microlocal Reconstruction for Limited-Angle Tomography

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 recently developed digital wavefront set extractor as well as the well-known microlocal canonical relation for the Radon transform. We use the wavefront set information about x-ray data to improve the reconstruction by requiring that the underlying neural networks simultaneously extract the correct ground truth wavefront set and ground truth image. As a necessary theoretical step, we identify the digital microlocal canonical relations for deep convolutional residual neural networks. We find strong numerical evidence for the effectiveness of this approach.

cs.LG↗

arXiv2021

Equivalence of approximation by convolutional neural networks and fully-connected networks

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 this paper, we establish a connection between both network architectures. Using this connection, we show that all upper and lower bounds concerning approximation rates of {fully-connected} neural networks for functions $f \in \mathcal{C}$ -- for an arbitrary function class $\mathcal{C}$ -- translate to essentially the same bounds concerning approximation rates of convolutional neural networks for functions $f \in {\mathcal{C}^{equi}}$, with the class ${\mathcal{C}^{equi}}$ consisting of all translation equivariant functions whose first coordinate belongs to $\mathcal{C}$. All presented results consider exclusively the case of convolutional neural networks without any pooling operation and with circular convolutions, i.e., not based on zero-padding.

math.FA↗

arXiv2020

A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

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 of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.

math.NA↗

arXiv2020

Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks

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 analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory predicts that the performance of the model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation. We use various methods to establish comparability between test-cases by minimizing the effect of the choice of test-cases on the optimization and sampling aspects of the learning problem. We find strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis.

math.NA↗

arXiv2020

Efficient Approximation of Solutions of Parametric Linear Transport Equations by ReLU DNNs

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 non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension.

math.NA↗

arXiv2020

Topological properties of the set of functions generated by neural networks of fixed size

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 is highly non-convex, except possibly for a few exotic activation functions. Moreover, the set is not closed with respect to $L^p$-norms, $0 < p < \infty$, for all practically-used activation functions, and also not closed with respect to the $L^\infty$-norm for all practically-used activation functions except for the ReLU and the parametric ReLU. Finally, the function that maps a family of weights to the function computed by the associated network is not inverse stable for every practically used activation function. In other words, if $f_1, f_2$ are two functions realized by neural networks and if $f_1, f_2$ are close in the sense that $\|f_1 - f_2\|_{L^\infty} \leq \varepsilon$ for $\varepsilon > 0$, it is, regardless of the size of $\varepsilon$, usually not possible to find weights $w_1, w_2$ close together such that each $f_i$ is realized by a neural network with weights $w_i$. Overall, our findings identify potential causes for issues in the training procedure of deep learning such as no guaranteed convergence, explosion of parameters, and slow convergence.

math.GN↗

arXiv2019

Extraction of digital wavefront sets using applied harmonic analysis and deep neural networks

Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences. Of particular importance is the analysis of wavefront sets and the correct extraction of those. In this paper, we introduce the first algorithmic approach to extract the wavefront set of images, which combines data-based and model-based methods. Based on a celebrated property of the shearlet transform to unravel information on the wavefront set, we extract the wavefront set of an image by first applying a discrete shearlet transform and then feeding local patches of this transform to a deep convolutional neural network trained on labeled data. The resulting algorithm outperforms all competing algorithms in edge-orientation and ramp-orientation detection.

eess.IV↗

arXiv2019

Approximation in $L^p(μ)$ with deep ReLU neural networks

We discuss the expressive power of neural networks which use the non-smooth ReLU activation function $\varrho(x) = \max\{0,x\}$ by analyzing the approximation theoretic properties of such networks. The existing results mainly fall into two categories: approximation using ReLU networks with a fixed depth, or using ReLU networks whose depth increases with the approximation accuracy. After reviewing these findings, we show that the results concerning networks with fixed depth--- which up to now only consider approximation in $L^p(λ)$ for the Lebesgue measure $λ$--- can be generalized to approximation in $L^p(μ)$, for any finite Borel measure $μ$. In particular, the generalized results apply in the usual setting of statistical learning theory, where one is interested in approximation in $L^2(\mathbb{P})$, with the probability measure $\mathbb{P}$ describing the distribution of the data.

math.FA↗

arXiv2019

Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms

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 networks, artificial neural networks with ReLU activation functions that achieve certain approximation rates. Second, we establish lower bounds for the approximation by ReLU neural networks for classes of Sobolev-regular functions. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations.

math.FA↗

arXiv2019

Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces

In this paper, we study a newly developed shearlet system on bounded domains which yields frames for $H^s(Ω)$ for some $s\in \mathbb{N}$, $Ω\subset \mathbb{R}^2$. We will derive approximation rates with respect to $H^s(Ω)$ norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-based algorithm for the solution of an elliptic PDE and analyze its computational complexity and convergence properties.

math.FA↗

arXiv2018

Optimal approximation of piecewise smooth functions using deep ReLU neural networks

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 $L^2$. As a model class, we consider the set $\mathcal{E}^β(\mathbb R^d)$ of possibly discontinuous piecewise $C^β$ functions $f : [-1/2, 1/2]^d \to \mathbb R$, where the different smooth regions of $f$ are separated by $C^β$ hypersurfaces. For dimension $d \geq 2$, regularity $β> 0$, and accuracy $\varepsilon > 0$, we construct artificial neural networks with ReLU activation function that approximate functions from $\mathcal{E}^β(\mathbb R^d)$ up to $L^2$ error of $\varepsilon$. The constructed networks have a fixed number of layers, depending only on $d$ and $β$, and they have $O(\varepsilon^{-2(d-1)/β})$ many nonzero weights, which we prove to be optimal. In addition to the optimality in terms of the number of weights, we show that in order to achieve the optimal approximation rate, one needs ReLU networks of a certain depth. Precisely, for piecewise $C^β(\mathbb R^d)$ functions, this minimal depth is given---up to a multiplicative constant---by $β/d$. Up to a log factor, our constructed networks match this bound. This partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions. Finally, we analyze approximation in high-dimensional spaces where the function $f$ to be approximated can be factorized into a smooth dimension reducing feature map $τ$ and classifier function $g$---defined on a low-dimensional feature space---as $f = g \circ τ$. We show that in this case the approximation rate depends only on the dimension of the feature space and not the input dimension.

math.FA↗

arXiv2018

Optimal Approximation with Sparsely Connected Deep Neural Networks

We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb R^d)$. In other words, we establish a connection between the complexity of a function class and the complexity of deep neural networks approximating functions from this class to within a prescribed accuracy. Additionally, we prove that our lower bounds are achievable for a broad family of function classes. Specifically, all function classes that are optimally approximated by a general class of representation systems---so-called \emph{affine systems}---can be approximated by deep neural networks with minimal connectivity and memory requirements. Affine systems encompass a wealth of representation systems from applied harmonic analysis such as wavelets, ridgelets, curvelets, shearlets, $α$-shearlets, and more generally $α$-molecules. Our central result elucidates a remarkable universality property of neural networks and shows that they achieve the optimum approximation properties of all affine systems combined. As a specific example, we consider the class of $α^{-1}$-cartoon-like functions, which is approximated optimally by $α$-shearlets. We also explain how our results can be extended to the case of functions on low-dimensional immersed manifolds. Finally, we present numerical experiments demonstrating that the standard stochastic gradient descent algorithm generates deep neural networks providing close-to-optimal approximation rates. Moreover, these results indicate that stochastic gradient descent can actually learn approximations that are sparse in the representation systems optimally sparsifying the function class the network is trained on.

cs.LG↗

arXiv2017

Anisotropic Multiscale Systems on Bounded Domains

We provide a construction of multiscale systems on a bounded domain $Ω\subset \mathbb{R}^2$ coined boundary shearlet systems, which satisfy several properties advantageous for applications to imaging science and the numerical analysis of partial differential equations. More precisely, we construct boundary shearlet systems that form frames for the Sobolev spaces $H^s(Ω),s\in \mathbb{N} \cup \{0\},$ with controllable frame bounds and admit optimally sparse approximations for functions, which are smooth apart from a curve-like discontinuity. We show that the constructed systems allow incorporating boundary conditions. Furthermore, for $s \geq 0$ and $f\in H^s(Ω)$ we prove that weighted $\ell^2$ norms of the $L^2-$analysis coefficients of $f$ are equivalent to its $H^s(Ω)$ norm. This yields in particular, that the reweighted systems are frames also for $H^{-s}(Ω)$. Moreover, we demonstrate numerically, that the associated $L^2-$synthesis operator is also stable as a map to $H^s(Ω)$ which, in combination with the previous result, strongly indicates that these systems constitute so-called Gelfand frames for $(H^s(Ω), L^2(Ω), H^{-s}(Ω))$.

math.FA↗

arXiv2017

Bendlets: A Second-Order Shearlet Transform with Bent Elements

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 linear and bending quadratic in spatial coordinates, bendlets provide what we term a second-order shearlet system. As we show in this article, the decay rates of the associated transform enable the precise characterization of location, orientation and curvature of discontinuities in piecewise constant images. These results yield an improvement over existing directional representation systems where curvature only controls the constant of the decay rate of the transform. We also detail the construction of shearlet systems of arbitrary order. A practical implementation of bendlets is provided as an extension of the ShearLab toolbox, which we use to verify our theoretical classification results.

math.FA↗

arXiv2016

Regularization and Numerical Solution of the Inverse Scattering Problem using Shearlet Frames

Regularization techniques for the numerical solution of inverse scattering problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions. Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for regularization in a Tikhonov method. We analyze two approaches, namely solvers for the nonlinear problem and for the linearized problem obtained by the Born approximation technique. As example for the first class we study the acoustic inverse scattering problem, and for the second class, the inverse scattering problem of the Schrödinger equation. In both cases, we derive analytical results for our approaches. Whereas our emphasis for the linearized problem is more on the theoretical side due to the standardness of associated solvers, we provide numerical examples for the nonlinear problem that highlight the effectiveness of our algorithmic approach.

math.FA↗