arXiv subjects

Leon Bungert

Publications and source records attributed to Leon Bungert.

At least 37 records · Page 2Linked to original sources

arXiv2022

The inhomogeneous $p$-Laplacian equation with Neumann boundary conditions in the limit $p\to\infty$

We investigate the limiting behavior of solutions to the inhomogeneous $p$-Laplacian equation $-Δ_p u = μ_p$ subject to Neumann boundary conditions. For right hand sides which are arbitrary signed measures we show that solutions converge to a Kantorovich potential associated with the geodesic Wasserstein-$1$ distance. In the regular case with continuous right hand sides we characterize the limit as viscosity solution to an infinity Laplacian / eikonal type equation.

math.AP↗

arXiv2022

CLIP: Cheap Lipschitz Training of Neural Networks

Despite the large success of deep neural networks (DNN) in recent years, most neural networks still lack mathematical guarantees in terms of stability. For instance, DNNs are vulnerable to small or even imperceptible input perturbations, so called adversarial examples, that can cause false predictions. This instability can have severe consequences in applications which influence the health and safety of humans, e.g., biomedical imaging or autonomous driving. While bounding the Lipschitz constant of a neural network improves stability, most methods rely on restricting the Lipschitz constants of each layer which gives a poor bound for the actual Lipschitz constant. In this paper we investigate a variational regularization method named CLIP for controlling the Lipschitz constant of a neural network, which can easily be integrated into the training procedure. We mathematically analyze the proposed model, in particular discussing the impact of the chosen regularization parameter on the output of the network. Finally, we numerically evaluate our method on both a nonlinear regression problem and the MNIST and Fashion-MNIST classification databases, and compare our results with a weight regularization approach.

cs.LG↗

arXiv2022

Complete Deterministic Dynamics and Spectral Decomposition of the Linear Ensemble Kalman Inversion

The ensemble Kalman inversion (EKI) for the solution of Bayesian inverse problems of type $y = A u +\varepsilon$, with $u$ being an unknown parameter, $y$ a given datum, and $\varepsilon$ measurement noise, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles $\{u^j(t)\}_{j=1}^J$, whose initial empirical measure is sampled from the prior, evolving over an artificial time $t$ towards an approximate solution of the inverse problem, with $t=1$ emulating the posterior, and $t\to\infty$ corresponding to the under-regularized minimum-norm solution of the inverse problem. Using spectral techniques, we provide a complete description of the deterministic dynamics of EKI and its asymptotic behavior in parameter space. In particular, we analyze the dynamics of naive EKI and mean-field EKI with a special focus on their time asymptotic behavior. Furthermore, we show that -- even in the deterministic case -- residuals in parameter space do not decrease monotonously in the Euclidean norm and suggest a problem-adapted norm, where monotonicity can be proved. Finally, we derive a system of ordinary differential equations governing the spectrum and eigenvectors of the covariance matrix. While the analysis is aimed at the EKI, we believe that it can be applied to understand more general particle-based dynamical systems.

math.NA↗

arXiv2022

Eigenvalue Problems in $\mathrm{L}^\infty$: Optimality Conditions, Duality, and Relations with Optimal Transport

In this article we characterize the $\mathrm{L}^\infty$ eigenvalue problem associated to the Rayleigh quotient $\left.{\|\nabla u\|_{\mathrm{L}^\infty}}\middle/{\|u\|_\infty}\right.$ and relate it to a divergence-form PDE, similarly to what is known for $\mathrm{L}^p$ eigenvalue problems and the $p$-Laplacian for $p<\infty$. Contrary to existing methods, which study $\mathrm{L}^\infty$-problems as limits of $\mathrm{L}^p$-problems for $p\to\infty$, we develop a novel framework for analyzing the limiting problem directly using convex analysis and geometric measure theory. For this, we derive a novel fine characterization of the subdifferential of the Lipschitz-constant-functional $u\mapsto\|\nabla u\|_{\mathrm{L}^\infty}$. We show that the eigenvalue problem takes the form $λνu =-\operatorname{div}(τ\nabla_τu)$, where $ν$ and $τ$ are non-negative measures concentrated where $|u|$ respectively $|\nabla u|$ are maximal, and $\nabla_τu$ is the tangential gradient of $u$ with respect to $τ$. Lastly, we investigate a dual Rayleigh quotient whose minimizers solve an optimal transport problem associated to a generalized Kantorovich--Rubinstein norm. Our results apply to all stationary points of the Rayleigh quotient, including infinity ground states, infinity harmonic potentials, distance functions, etc., and generalize known results in the literature.

math.AP↗

arXiv2022

The Geometry of Adversarial Training in Binary Classification

We establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. The resulting regularized risk minimization problems admit exact convex relaxations of the type $L^1+$ (nonlocal) $\operatorname{TV}$, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows us to establish a series of properties of optimal solutions of the original problem, including the existence of minimal and maximal solutions (interpreted in a suitable sense), and the existence of regular solutions (also interpreted in a suitable sense). In addition, we highlight how the connection between adversarial training and perimeter minimization problems provides a novel, directly interpretable, statistical motivation for a family of regularized risk minimization problems involving perimeter/total variation. The majority of our theoretical results are independent of the distance used to define adversarial attacks.

cs.LG↗

arXiv2022

Uniform Convergence Rates for Lipschitz Learning on Graphs

Lipschitz learning is a graph-based semi-supervised learning method where one extends labels from a labeled to an unlabeled data set by solving the infinity Laplace equation on a weighted graph. In this work we prove uniform convergence rates for solutions of the graph infinity Laplace equation as the number of vertices grows to infinity. Their continuum limits are absolutely minimizing Lipschitz extensions with respect to the geodesic metric of the domain where the graph vertices are sampled from. We work under very general assumptions on the graph weights, the set of labeled vertices, and the continuum domain. Our main contribution is that we obtain quantitative convergence rates even for very sparsely connected graphs, as they typically appear in applications like semi-supervised learning. In particular, our framework allows for graph bandwidths down to the connectivity radius. For proving this we first show a quantitative convergence statement for graph distance functions to geodesic distance functions in the continuum. Using the "comparison with distance functions" principle, we can pass these convergence statements to infinity harmonic functions and absolutely minimizing Lipschitz extensions.

math.NA↗

arXiv2022

Improving Robustness against Real-World and Worst-Case Distribution Shifts through Decision Region Quantification

The reliability of neural networks is essential for their use in safety-critical applications. Existing approaches generally aim at improving the robustness of neural networks to either real-world distribution shifts (e.g., common corruptions and perturbations, spatial transformations, and natural adversarial examples) or worst-case distribution shifts (e.g., optimized adversarial examples). In this work, we propose the Decision Region Quantification (DRQ) algorithm to improve the robustness of any differentiable pre-trained model against both real-world and worst-case distribution shifts in the data. DRQ analyzes the robustness of local decision regions in the vicinity of a given data point to make more reliable predictions. We theoretically motivate the DRQ algorithm by showing that it effectively smooths spurious local extrema in the decision surface. Furthermore, we propose an implementation using targeted and untargeted adversarial attacks. An extensive empirical evaluation shows that DRQ increases the robustness of adversarially and non-adversarially trained models against real-world and worst-case distribution shifts on several computer vision benchmark datasets.

cs.LG↗

arXiv2022

The lion in the attic -- A resolution of the Borel--Kolmogorov paradox

The Borel--Kolmogorov paradox of conditioning with respect to events of prior probability zero has fascinated students and researchers since its discovery more than 100 years ago. Classical conditioning is only valid with respect to events of positive probability. If we ignore this constraint and condition on such sets, for example events of type $\{Y=y\}$ for a continuously distributed random variable $Y$, almost any probability measure can be chosen as the conditional measure on such sets. There have been numerous descriptions and explanations of the paradox' appearance in the setting of conditioning on a subset of probability zero. However, most treatments don't supply explicit instructions on how to avoid it. We propose to close this gap by defining a version of conditional measure which utilizes the Hausdorff measure. This makes the choice canonical in the sense that it only depends on the geometry of the space, thus removing any ambiguity. We describe the set of possible measures arising in the context of the Borel--Kolmogorov paradox and classify those coinciding with the canonical measure. The objective of this manuscript is to provide a manual for singular conditional probability: We give an explicit explanation in which settings ambiguity arises (and where not) and how to get rid of this ambiguity once and for all by a canonical choice.

math.PR↗

arXiv2022

A Bregman Learning Framework for Sparse Neural Networks

We propose a learning framework based on stochastic Bregman iterations, also known as mirror descent, to train sparse neural networks with an inverse scale space approach. We derive a baseline algorithm called LinBreg, an accelerated version using momentum, and AdaBreg, which is a Bregmanized generalization of the Adam algorithm. In contrast to established methods for sparse training the proposed family of algorithms constitutes a regrowth strategy for neural networks that is solely optimization-based without additional heuristics. Our Bregman learning framework starts the training with very few initial parameters, successively adding only significant ones to obtain a sparse and expressive network. The proposed approach is extremely easy and efficient, yet supported by the rich mathematical theory of inverse scale space methods. We derive a statistically profound sparse parameter initialization strategy and provide a rigorous stochastic convergence analysis of the loss decay and additional convergence proofs in the convex regime. Using only 3.4% of the parameters of ResNet-18 we achieve 90.2% test accuracy on CIFAR-10, compared to 93.6% using the dense network. Our algorithm also unveils an autoencoder architecture for a denoising task. The proposed framework also has a huge potential for integrating sparse backpropagation and resource-friendly training.

cs.LG↗

arXiv2021

Continuum Limit of Lipschitz Learning on Graphs

Tackling semi-supervised learning problems with graph-based methods has become a trend in recent years since graphs can represent all kinds of data and provide a suitable framework for studying continuum limits, e.g., of differential operators. A popular strategy here is $p$-Laplacian learning, which poses a smoothness condition on the sought inference function on the set of unlabeled data. For $p<\infty$ continuum limits of this approach were studied using tools from $Γ$-convergence. For the case $p=\infty$, which is referred to as Lipschitz learning, continuum limits of the related infinity-Laplacian equation were studied using the concept of viscosity solutions. In this work, we prove continuum limits of Lipschitz learning using $Γ$-convergence. In particular, we define a sequence of functionals which approximate the largest local Lipschitz constant of a graph function and prove $Γ$-convergence in the $L^\infty$-topology to the supremum norm of the gradient as the graph becomes denser. Furthermore, we show compactness of the functionals which implies convergence of minimizers. In our analysis we allow a varying set of labeled data which converges to a general closed set in the Hausdorff distance. We apply our results to nonlinear ground states, i.e., minimizers with constrained $L^p$-norm, and, as a by-product, prove convergence of graph distance functions to geodesic distance functions.

cs.LG↗

arXiv2021

Gradient Flows and Nonlinear Power Methods for the Computation of Nonlinear Eigenfunctions

This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how convergence of (discretized) approximations can be verified. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and prove their convergence to nonlinear eigenfunctions. Finally, we prove that $Γ$-convergence of functionals implies convergence of their ground states, which is important for discrete approximations.

math.NA↗

arXiv2021

Nonlinear Spectral Decompositions by Gradient Flows of One-Homogeneous Functionals

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total variation flow have been proven previously, and by recent results on finite-dimensional polyhedral semi-norms, where gradient flows can yield spectral decompositions into eigenvectors. We provide a geometric characterization of eigenvectors via a dual unit ball and prove them to be subgradients of minimal norm. This establishes the connection to gradient flows, whose time evolution is a decomposition of the initial condition into subgradients of minimal norm. If these are eigenvectors, this implies an interesting orthogonality relation and the equivalence of the gradient flow to a variational regularization method and an inverse scale space flow. Indeed we verify that all scenarios where these equivalences were known before by other arguments - such as one-dimensional total variation, multidimensional generalizations to vector fields, or certain polyhedral semi-norms - yield spectral decompositions, and we provide further examples. We also investigate extinction times and extinction profiles, which we characterize as eigenvectors in a very general setting, generalizing several results from literature.

math.AP↗

arXiv2021

Neural Architecture Search via Bregman Iterations

We propose a novel strategy for Neural Architecture Search (NAS) based on Bregman iterations. Starting from a sparse neural network our gradient-based one-shot algorithm gradually adds relevant parameters in an inverse scale space manner. This allows the network to choose the best architecture in the search space which makes it well-designed for a given task, e.g., by adding neurons or skip connections. We demonstrate that using our approach one can unveil, for instance, residual autoencoders for denoising, deblurring, and classification tasks. Code is available at https://github.com/TimRoith/BregmanLearning.

cs.LG↗

arXiv2021

Nonlinear Power Method for Computing Eigenvectors of Proximal Operators and Neural Networks

Neural networks have revolutionized the field of data science, yielding remarkable solutions in a data-driven manner. For instance, in the field of mathematical imaging, they have surpassed traditional methods based on convex regularization. However, a fundamental theory supporting the practical applications is still in the early stages of development. We take a fresh look at neural networks and examine them via nonlinear eigenvalue analysis. The field of nonlinear spectral theory is still emerging, providing insights about nonlinear operators and systems. In this paper we view a neural network as a complex nonlinear operator and attempt to find its nonlinear eigenvectors. We first discuss the existence of such eigenvectors and analyze the kernel of ReLU networks. Then we study a nonlinear power method for generic nonlinear operators. For proximal operators associated to absolutely one-homogeneous convex regularization functionals, we can prove convergence of the method to an eigenvector of the proximal operator. This motivates us to apply a nonlinear method to networks which are trained to act similarly as a proximal operator. In order to take the non-homogeneity of neural networks into account we define a modified version of the power method. We perform extensive experiments for different proximal operators and on various shallow and deep neural networks designed for image denoising. Proximal eigenvectors will be used for geometric analysis of graphs, as clustering or the computation of distance functions. For simple neural nets, we observe the influence of training data on the eigenvectors. For state-of-the-art denoising networks, we show that eigenvectors can be interpreted as (un)stable modes of the network, when contaminated with noise or other degradations.

math.SP↗

arXiv2021

Identifying Untrustworthy Predictions in Neural Networks by Geometric Gradient Analysis

The susceptibility of deep neural networks to untrustworthy predictions, including out-of-distribution (OOD) data and adversarial examples, still prevent their widespread use in safety-critical applications. Most existing methods either require a re-training of a given model to achieve robust identification of adversarial attacks or are limited to out-of-distribution sample detection only. In this work, we propose a geometric gradient analysis (GGA) to improve the identification of untrustworthy predictions without retraining of a given model. GGA analyzes the geometry of the loss landscape of neural networks based on the saliency maps of their respective input. To motivate the proposed approach, we provide theoretical connections between gradients' geometrical properties and local minima of the loss function. Furthermore, we demonstrate that the proposed method outperforms prior approaches in detecting OOD data and adversarial attacks, including state-of-the-art and adaptive attacks.

cs.LG↗

arXiv2020

Robust Image Reconstruction with Misaligned Structural Information

Multi-modality (or multi-channel) imaging is becoming increasingly important and more widely available, e.g. hyperspectral imaging in remote sensing, spectral CT in material sciences as well as multi-contrast MRI and PET-MR in medicine. Research in the last decades resulted in a plethora of mathematical methods to combine data from several modalities. State-of-the-art methods, often formulated as variational regularization, have shown to significantly improve image reconstruction both quantitatively and qualitatively. Almost all of these models rely on the assumption that the modalities are perfectly registered, which is not the case in most real world applications. We propose a variational framework which jointly performs reconstruction and registration, thereby overcoming this hurdle. Our approach is the first to achieve this for different modalities and outranks established approaches in terms of accuracy of both reconstruction and registration. Numerical results on simulated and real data show the potential of the proposed strategy for various applications in multi-contrast MRI, PET-MR, and hyperspectral imaging: typical misalignments between modalities such as rotations, translations, zooms can be effectively corrected during the reconstruction process. Therefore the proposed framework allows the robust exploitation of shared information across multiple modalities under real conditions.

eess.IV↗

arXiv2020

Variational regularisation for inverse problems with imperfect forward operators and general noise models

We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and convex duality for general data fidelity terms and regularisation functionals. Both for a-priori and a-posteriori parameter choice rules, we obtain convergence rates of the regularized solutions in terms of Bregman distances. Our results apply to fidelity terms such as Wasserstein distances, f-divergences, norms, as well as sums and infimal convolutions of those.

math.NA↗

arXiv2020

Structural analysis of an $L$-infinity variational problem and relations to distance functions

In this work we analyse the functional ${\cal J}(u)=\|\nabla u\|_\infty$ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth $p$-norms. We prove that its ground states coincide with multiples of the distance function to the boundary of the domain. Furthermore, we compute the $L^2$-subdifferential of ${\cal J}$ and characterize the distance function as unique non-negative eigenfunction of the subdifferential operator. We also study properties of general eigenfunctions, in particular their nodal sets. Furthermore, we prove that the distance function can be computed as asymptotic profile of the gradient flow of ${\cal J}$ and construct analytic solutions of fast marching type. In addition, we give a geometric characterization of the extreme points of the unit ball of ${\cal J}$. Finally, we transfer many of these results to a discrete version of the functional defined on a finite weighted graph. Here, we analyze properties of distance functions on graphs and their gradients. The main difference between the continuum and discrete setting is that the distance function is not the unique non-negative eigenfunction on a graph.

math.AP↗