arXiv subjects

Martin Burger

Publications and source records attributed to Martin Burger.

At least 37 records · Page 2Linked to original sources

arXiv2025

Hypergraph $p$-Laplacian regularization on point clouds for data interpolation

As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the $\varepsilon_n$-ball hypergraph and the $k_n$-nearest neighbor hypergraph on a point cloud and study the $p$-Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph $p$-Laplacian regularization and the continuum $p$-Laplacian regularization in a semisupervised setting when the number of points $n$ goes to infinity while the number of labeled points remains fixed. A key improvement compared to the graph case is that the results rely on weaker assumptions on the upper bound of $\varepsilon_n$ and $k_n$. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal-dual hybrid gradient algorithm. Numerical experiments on data interpolation verify that the hypergraph $p$-Laplacian regularization outperforms the graph $p$-Laplacian regularization in preventing the development of spikes at the labeled points.

math.NA↗

arXiv2024

Covariance-modulated optimal transport and gradient flows

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.

math.AP↗

arXiv2024

Analysis of Primal-Dual Langevin Algorithms

We analyze a recently proposed class of algorithms for the problem of sampling from probability distributions $μ^\ast$ in $\mathbb{R}^d$ with a Lebesgue density of the form $μ^\ast(x) \propto \exp(-f(Kx)-g(x))$, where $K$ is a linear operator and $f,g$ convex and non-smooth. The method is a generalization of the primal-dual hybrid gradient optimization algorithm to a sampling scheme. We give the iteration's continuous time limit, a stochastic differential equation in the joint primal-dual variable, and its mean field limit Fokker-Planck equation. Under mild conditions, the scheme converges to a unique stationary state in continuous and discrete time. Contrary to purely primal overdamped Langevin diffusion, the stationary state in continuous time does not have $μ^\ast$ as its primal marginal. Thus, further analysis is carried out to bound the bias induced by the partial dualization, and potentially correct for it in the diffusion. Time discretizations of the diffusion lead to implementable algorithms, but, as is typical in Langevin Monte Carlo methods, introduce further bias. We prove bounds for these discretization errors, which allow to give convergence results relating the produced samples to the target. We demonstrate our findings numerically first on small-scale examples in which we can exactly verify the theoretical results, and subsequently on typical examples of larger scale from Bayesian imaging inverse problems.

math.OC↗

arXiv2024

Lane formation and aggregation spots in a model of ants

We investigate an interacting particle model to simulate a foraging colony of ants, where each ant is represented as an active Brownian particle. The interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of the pheromone field. Unlike conventional models, our study introduces a parameter that enables the reproduction of two distinctive behaviors: the well-known Keller--Segel aggregation into spots and the formation of traveling clusters, without relying on external constraints such as food sources or nests. We consider the associated mean-field limit partial differential equation (PDE) of this system and establish the analytical and numerical foundations for understanding these particle behaviors. Remarkably, the mean-field PDE not only supports aggregation spots and lane formation but also unveils a bistable region where these two behaviors compete. The patterns associated with these phenomena are elucidated by the shape of the growing eigenfunctions derived from linear stability analysis. This study not only contributes to our understanding of complex ant colony dynamics but also introduces a novel parameter-dependent perspective on pattern formation in collective systems.

math.AP↗

arXiv2024

Learned Regularization for Inverse Problems: Insights from a Spectral Model

In this chapter we provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization methods and their convergence in terms of the underlying data distributions, which paves the way for future theoretical studies. Based on a simple spectral learning model previously introduced for supervised learning, we investigate some key properties of different learning paradigms for inverse problems, which can be formulated independently of specific architectures. In particular we investigate the regularization properties, bias, and critical dependence on training data distributions. Moreover, our framework allows to highlight and compare the specific behavior of the different paradigms in the infinite-dimensional limit.

math.NA↗

arXiv2024

Optimal transport on gas networks

This paper models gas networks as metric graphs, with isothermal Euler equations at the edges, Kirchhoff's law at interior vertices and time-(in)dependent boundary conditions at boundary vertices. For this setup, a generalized $p$-Wasserstein metric in a dynamic formulation is introduced and utilized to derive $p$-Wasserstein gradient flows, specifically focusing on the non-standard case $p = 3$.

math.AP↗

arXiv2024

Spectral Total-Variation Processing of Shapes: Theory and Applications

We present an analysis of total-variation (TV) on non-Euclidean parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work explains recent experimental findings in shape spectral TV [Fumero et al., 2020] and adaptive anisotropic spectral TV [Biton and Gilboa, 2022]. A new way to generalize set convexity from the plane to surfaces is derived by characterizing the TV eigenfunctions on surfaces. Relationships between TV, area, eigenvalue, eigenfunctions and their discontinuities are discovered. Further, we expand the shape spectral TV toolkit to include versatile zero-homogeneous flows demonstrated through smoothing and exaggerating filters. Last but not least, we propose the first TV-based method for shape deformation, characterized by deformations along geometrical bottlenecks. We show these bottlenecks to be aligned with eigenfunction discontinuities. This research advances the field of spectral TV on surfaces and its application in 3D graphics, offering new perspectives for shape filtering and deformation.

cs.CG↗

arXiv2023

Convergent Data-driven Regularizations for CT Reconstruction

The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naive) solution does not depend on the measured data continuously, regularization is needed to re-establish a continuous dependence. In this work, we investigate simple, but yet still provably convergent approaches to learning linear regularization methods from data. More specifically, we analyze two approaches: One generic linear regularization that learns how to manipulate the singular values of the linear operator in an extension of our previous work, and one tailored approach in the Fourier domain that is specific to CT-reconstruction. We prove that such approaches become convergent regularization methods as well as the fact that the reconstructions they provide are typically much smoother than the training data they were trained on. Finally, we compare the spectral as well as the Fourier-based approaches for CT-reconstruction numerically, discuss their advantages and disadvantages and investigate the effect of discretization errors at different resolutions.

math.NA↗

arXiv2023

Learning a Sparse Representation of Barron Functions with the Inverse Scale Space Flow

This paper presents a method for finding a sparse representation of Barron functions. Specifically, given an $L^2$ function $f$, the inverse scale space flow is used to find a sparse measure $μ$ minimising the $L^2$ loss between the Barron function associated to the measure $μ$ and the function $f$. The convergence properties of this method are analysed in an ideal setting and in the cases of measurement noise and sampling bias. In an ideal setting the objective decreases strictly monotone in time to a minimizer with $\mathcal{O}(1/t)$, and in the case of measurement noise or sampling bias the optimum is achieved up to a multiplicative or additive constant. This convergence is preserved on discretization of the parameter space, and the minimizers on increasingly fine discretizations converge to the optimum on the full parameter space.

stat.ML↗

arXiv2023

Hypergraph $p$-Laplacians and Scale Spaces

This paper introduces gradient, adjoint, and $p$-Laplacian definitions for oriented hypergraphs as well as differential and averaging operators for unoriented hypergraphs. These definitions are used to define gradient flows in the form of diffusion equations with applications in modelling group dynamics and information flow in social networks as well as performing local and non-local image processing.

cs.SI↗

arXiv2023

Ill-posedness of time-dependent inverse problems in Lebesgue-Bochner spaces

We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.

math.OC↗

arXiv2023

Boltzmann mean-field game model for knowledge growth: limits to learning and general utilities

In this paper we investigate a generalisation of a Boltzmann mean field game (BMFG) for knowledge growth, originally introduced by the economists Lucas and Moll. In BMFG the evolution of the agent density with respect to their knowledge level is described by a Boltzmann equation. Agents increase their knowledge through binary interactions with others; their increase is modulated by the interaction and learning rate: Agents with similar knowledge learn more in encounters, while agents with very different levels benefit less from learning interactions. The optimal fraction of time spent on learning is calculated by a Bellman equation, resulting in a highly nonlinear forward-backward in time PDE system. The structure of solutions to the Boltzmann and Bellman equation depends strongly on the learning rate in the Boltzmann collision kernel as well as the utility function in the Bellman equation. In this paper we investigate the monotonicity behavior of solutions for different learning and utility functions, show existence of solutions and investigate how they impact the existence of so-called balanced growth path solutions, that relate to exponential growth of the overall economy. Furthermore we corroborate and illustrate our analytical results with computational experiments.

math.OC↗

arXiv2023

Well-posedness and stationary states for a crowded active Brownian system with size-exclusion

We prove the existence of solutions to a non-linear, non-local, degenerate equation which was previously derived as the formal hydrodynamic limit of an active Brownian particle system, where the particles are endowed with a position and an orientation. This equation incorporates diffusion in both the spatial and angular coordinates, as well as a non-linear non-local drift term, which depends on the angle-independent density. The spatial diffusion is non-linear degenerate and also comprises diffusion of the angle-independent density, which one may interpret as cross-diffusion with infinitely many species. Our proof relies on interpreting the equation as the perturbation of a gradient flow in a Wasserstein-type space. It generalizes the boundedness-by-entropy method to this setting and makes use of a gain of integrability due to the angular diffusion. For this latter step, we adapt a classical interpolation lemma for function spaces depending on time. We also prove uniqueness in the particular case where the non-local drift term is null, and provide existence and uniqueness results for stationary equilibrium solutions.

math.AP↗

arXiv2023

Resolution-Invariant Image Classification based on Fourier Neural Operators

In this paper we investigate the use of Fourier Neural Operators (FNOs) for image classification in comparison to standard Convolutional Neural Networks (CNNs). Neural operators are a discretization-invariant generalization of neural networks to approximate operators between infinite dimensional function spaces. FNOs - which are neural operators with a specific parametrization - have been applied successfully in the context of parametric PDEs. We derive the FNO architecture as an example for continuous and Fréchet-differentiable neural operators on Lebesgue spaces. We further show how CNNs can be converted into FNOs and vice versa and propose an interpolation-equivariant adaptation of the architecture.

cs.CV↗

arXiv2023

Sharp interface analysis of a diffuse interface model for cell blebbing with linker dynamics

We investigate the convergence of solutions of a recently proposed diffuse interface/phase field model for cell blebbing by means of matched asymptotic expansions. It is a biological phenomenon that increasingly attracts attention by both experimental and theoretical communities. Key to understanding the process of cell blebbing mechanically are proteins that link the cell cortex and the cell membrane. Another important model component is the bending energy of the cell membrane and cell cortex which accounts for differential equations up to sixth order. Both aspects pose interesting mathematical challenges that will be addressed in this work like showing non-singularity formation for the pressure at boundary layers, deriving equations for asymptotic series coefficients of uncommonly high order, and dealing with a highly coupled system of equations.

math.AP↗

arXiv2022

Analysis of Kinetic Models for Label Switching and Stochastic Gradient Descent

In this paper we provide a novel approach to the analysis of kinetic models for label switching, which are used for particle systems that can randomly switch between gradient flows in different energy landscapes. Besides problems in biology and physics, we also demonstrate that stochastic gradient descent, the most popular technique in machine learning, can be understood in this setting, when considering a time-continuous variant. Our analysis is focusing on the case of evolution in a collection of external potentials, for which we provide analytical and numerical results about the evolution as well as the stationary problem.

math.AP↗

arXiv2022

Active Crowds

This chapter focuses on the mathematical modelling of active particles (or agents) in crowded environments. We discuss several microscopic models found in literature and the derivation of the respective macroscopic partial differential equations for the particle density. The macroscopic models share common features, such as cross diffusion or degenerate mobilities. We then take the diversity of macroscopic models to a uniform structure and work out potential similarities and differences. Moreover, we discuss boundary effects and possible applications in life and social sciences. This is complemented by numerical simulations that highlight the effects of different boundary conditions.

cond-mat.stat-mech↗

arXiv2022

Porous medium equation and cross-diffusion systems as limit of nonlocal interaction

This paper studies the derivation of the quadratic porous medium equation and a class of cross-diffusion systems from nonlocal interactions. We prove convergence of solutions of a nonlocal interaction equation, resp. system, to solutions of the quadratic porous medium equation, resp. cross-diffusion system, in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use gradient flow techniques to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme, which can be slightly modified in order to extend our result to PDEs without gradient flow structure. In particular, it does not require convexity of the associated energies. Our analysis allows to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy.

math.AP↗