arXiv subjects

Ronny Bergmann

Publications and source records attributed to Ronny Bergmann.

At least 19 recordsLinked to original sources

arXiv2026

A modified Riemannian Levenberg-Marquardt Algorithm for robust or constraint optimization on manifolds

We extend the Levenberg-Marquardt method on Riemannian manifolds to a robust variant that allows to tackle problems from applications where outliers are to be expected. We formally state the framework for a block-wise variant of the Robust Riemannian Levenberg-Marquardt algorithm and discuss how known convergence results can be applied here as well. We further discuss several alternatives for phrasing the sub problem that has to be solved. Finally we illustrate how the accompanying open source implementation in Manopt$.$jl can be used to efficiently solve problems such as geodesic regression, Procrustes analysis, subspace Procrustes analysis and bundle adjustment robustly and compare the Levenberg-Marquardt solver to other solvers for nonsmooth Riemannian optimization.

math.OC↗

arXiv2026

A Riemannian quasi-Newton algorithm for optimization with Euclidean bounds

We propose a Riemannian limited-memory BFGS method for optimization problems with Euclidean bounds. The method combines a limited-memory quasi-Newton update in the tangent space with a Riemannian adaptation of the generalized Cauchy point strategy from classical L-BFGS-B, enabling efficient handling of Euclidean bounds while exploiting the geometric structure of the optimization domain. This setting is important in several applications, including covariance matrix estimation with bounded variance, neuroimaging, EEG signal classification, and other signal processing or computer-vision tasks that couple manifold variables with constrained Euclidean parameters. We provide a generic algorithmic framework and an implementation of the algorithm in the Manopt.jl library. Numerical experiments on benchmark problems indicate only minor reduction in performance on Euclidean problems compared to the classical L-BFGS-B method, while outperforming interior-point methods. Furthermore, the algorithm was tested on two mixed manifold and bounded Euclidean problems: amplitude-limited blind source separation with Gaussianity penalization and bounded-variance maximum likelihood common principal components analysis. The proposed method outperforms existing methods by several orders of magnitude.

math.OC↗

arXiv2025

Procrustes Problems on Random Matrices

Meaningful comparison between sets of observations often necessitates alignment or registration between them, and the resulting optimization problems range in complexity from those admitting simple closed-form solutions to those requiring advanced and novel techniques. We compare different Procrustes problems in which we align two sets of points after various perturbations by minimizing the norm of the difference between one matrix and an orthogonal transformation of the other. The minimization problem depends significantly on the choice of matrix norm; we highlight recent developments in nonsmooth Riemannian optimization and characterize which choices of norm work best for each perturbation. We show that in several applications, from low-dimensional alignments to hypothesis testing for random networks, when Procrustes alignment with the spectral or robust norm is the appropriate choice, it is often feasible to replace the computationally more expensive spectral and robust minimizers with their closed-form Frobenius-norm counterpart. Our work reinforces the synergy between optimization, geometry, and statistics.

stat.ME↗

arXiv2025

The Intrinsic Riemannian Proximal Gradient Method for Convex Optimization

We consider a class of (possibly strongly) geodesically convex optimization problems on Hadamard manifolds, where the objective function splits into the sum of a smooth and a possibly nonsmooth function. We introduce an intrinsic convex Riemannian proximal gradient (CRPG) method that employs the manifold proximal map for the nonsmooth step, without operating in the embedding or tangent space. A sublinear convergence rate for convex problems and a linear convergence rate for strongly convex problems is established, and we derive fundamental proximal gradient inequalities that generalize the Euclidean case. Our numerical experiments on hyperbolic spaces and manifolds of symmetric positive definite matrices demonstrate substantial computational advantages over existing methods.

math.OC↗

arXiv2025

Newton's method for nonlinear mappings into vector bundles Part II: Application to variational problems

We consider the solution of variational equations on manifolds by Newton's method. These problems can be expressed as root finding problems for mappings from infinite dimensional manifolds into dual vector bundles. We derive the differential geometric tools needed for the realization of Newton's method, equipped with an affine covariant damping strategy. We apply Newton's method to a couple of variational problems and show numerical results.

math.NA↗

arXiv2025

Total Generalized Variation of the Normal Vector Field and Applications to Mesh Denoising

We propose a novel formulation for the second-order total generalized variation (TGV) of the normal vector on an oriented, triangular mesh embedded in $\R^3$. The normal vector is considered as a manifold-valued function, taking values on the unit sphere. Our formulation extends previous discrete TGV models for piecewise constant scalar data that utilize a Raviart-Thomas function space. To extend this formulation to the manifold setting, a tailor-made tangential Raviart-Thomas type finite element space is constructed in this work. The new regularizer is compared to existing methods in mesh denoising experiments.

cs.CV↗

arXiv2025

p-Laplacians for Manifold-valued Hypergraphs

Hypergraphs extend traditional graphs by enabling the representation of N-ary relationships through higher-order edges. Akin to a common approach of deriving graph Laplacians, we define function spaces and corresponding symmetric products on the nodes and edges to derive hypergraph Laplacians. While this has been done before for Euclidean features, this work generalizes previous hypergraph Laplacian approaches to accommodate manifold-valued hypergraphs for many commonly encountered manifolds.

math.DG↗

arXiv2025

The Intrinsic Riemannian Proximal Gradient Method for Nonconvex Optimization

We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses proximal maps defined on the manifold and therefore does not require or work in the embedding. We investigate its convergence properties and illustrate its numerical performance, particularly for nonconvex or nonembedded problems that are hence out of reach for other methods.

math.OC↗

arXiv2025

A geometric perspective of state estimation using Kalman filters

Geometry of the state space is known to play a crucial role in many applications of Kalman filters, especially robotics and motion tracking. The Lie group-centric approach is currently very common, although a Riemannian approach has also been developed. In this work we explore the relationship between these two approaches and develop a novel description of Kalman filters based on affine connections that generalizes both commonly encountered descriptions. We illustrate the results on two test problems involving the special Euclidean group and the tangent bundle of a sphere in which the state is tracked by geometric variants of the extended Kalman filter and the unscented Kalman filter. The examples use a newly developed library GeometricKalman.jl. The new approach provides a greater freedom in selecting the structure of the state space for state estimation and can be easily integrated with standard techniques such as parameter estimation or covariance matrix estimation.

math.OC↗

arXiv2025

On projection mappings and the gradient projection method on hyperbolic space forms

This paper presents several new properties of the intrinsic $\kappa$-projection into $\kappa$-hyperbolically convex sets of $\kappa$-hyperbolic space forms, along with closed-form formulas for the intrinsic $\kappa$-projection into specific $\kappa$-hyperbolically convex sets. It also discusses the relationship between the intrinsic $\kappa$-projection, the Euclidean orthogonal projection, and the Lorentz projection. These properties lay the groundwork for analyzing the gradient projection method and hold importance in their own right. Additionally, new properties of the gradient projection method to solve constrained optimization problems in $\kappa$-hyperbolic space forms are established, considering both constant and backtracking step sizes in the analysis. It is shown that every accumulation point of the sequence generated by the method for both step sizes is a stationary point for the given problem. Additionally, an iteration complexity bound is provided that upper bounds the number of iterations needed to achieve a suitable measure of stationarity for both step sizes. Finally, the properties of the constrained Fermat-Weber problem are explored, demonstrating that the sequence generated by the gradient projection method converges to its unique solution. Numerical experiments on solving the Fermat-Weber problem are presented, illustrating the theoretical findings and demonstrating the effectiveness of the proposed methods.

math.OC↗

arXiv2024

Two Models for Surface Segmentation using the Total Variation of the Normal Vector

We consider the problem of surface segmentation, where the goal is to partition a surface represented by a triangular mesh. The segmentation is based on the similarity of the normal vector field to a given set of label vectors. We propose a variational approach and compare two different regularizers, both based on a total variation measure. The first regularizer penalizes the total variation of the assignment function directly, while the second regularizer penalizes the total variation in the label space. In order to solve the resulting optimization problems, we use variations of the split Bregman (ADMM) iteration adapted to the problem at hand. While computationally more expensive, the second regularizer yields better results in our experiments. In particular it removes noise more reliably in regions of constant curvature. In order to mitigate the computational cost, we present a manifold Newton scheme for the most expensive subproblem, which is related to the Riemannian center of mass on a sphere. This significantly improves the computational cost.

cs.CV↗

arXiv2024

Nonlinear Fenchel Conjugates

The classical concept of Fenchel conjugation is tailored to extended real-valued functions defined on linear spaces. In this paper we generalize this concept to functions defined on arbitrary sets that do not necessarily bear any structure at all. This generalization is obtained by replacing linear test functions by general nonlinear ones. Thus, we refer to it as nonlinear Fenchel conjugation. We investigate elementary properties including the Fenchel-Moreau biconjugation theorem. Whenever the domain exhibits additional structure, the restriction to a suitable subset of test functions allows further results to be derived. For example, on smooth manifolds, the restriction to smooth test functions allows us to state the Fenchel-Young theorem for the viscosity Fr\'echet subdifferential. On Lie groups, the restriction to real-valued group homomorphisms relates nonlinear Fenchel conjugation to infimal convolution and yields a notion of convexity.

math.FA↗

arXiv2024

The Riemannian Convex Bundle Method

We introduce the convex bundle method to solve convex, non-smooth optimization problems on Riemannian manifolds of bounded sectional curvature. Each step of our method is based on a model that involves the convex hull of previously collected subgradients, parallelly transported into the current serious iterate. This approach generalizes the dual form of classical bundle subproblems in Euclidean space. We prove that, under mild conditions, the convex bundle method converges to a minimizer. Several numerical examples implemented using Manopt$.$jl illustrate the performance of the proposed method and compare it to the subgradient method, the cyclic proximal point algorithm, as well as the proximal bundle method.

math.OC↗

arXiv2022

Multivariate Hermite interpolation of manifold-valued data

In this paper, we propose two methods for multivariate Hermite interpolation of manifold-valued functions. On the one hand, we approach the problem via computing suitable weighted Riemannian barycenters. To satisfy the conditions for Hermite interpolation, the sampled derivative information is converted into a condition on the derivatives of the associated weight functions. It turns out that this requires the solution of linear systems of equations, but no vector transport is necessary. This approach treats all given sample data points equally and is intrinsic in the sense that it does not depend on local coordinates or embeddings. As an alternative, we consider Hermite interpolation in a tangent space. This is a straightforward approach, where one designated point, for example one of the sample points or (one of) their center(s) of mass, is chosen to act as the base point at which the tangent space is attached. The remaining sampled locations and sampled derivatives are mapped to said tangent space. This requires a vector transport between different tangent spaces. The actual interpolation is then conducted via classical vector space operations. The interpolant depends on the selected base point. The validity and performance of both approaches is illustrated by means of numerical examples.

math.NA↗

arXiv2022

Total Generalized Variation for Piecewise Constant Functions on Triangular Meshes with Applications in Imaging

We propose a novel discrete concept for the total generalized variation (TGV), which has originally been derived to reduce the staircasing effect in classical total variation (TV) regularization, in image denoising problems. We describe discrete, second-order TGV for piecewise constant functions on triangular meshes, thus allowing the TGV functional to be applied to more general data structures than pixel images, and in particular in the context of finite element discretizations. Particular attention is given to the description of the kernel of the TGV functional, which, in the continuous setting, consists of linear polynomials. We discuss how to take advantage of this kernel structure using piecewise constant functions on triangular meshes. Numerical experiments include denoising and inpainting problems for images defined on non-standard grids, including data from a 3D scanner.

math.NA↗

arXiv2021

The difference of convex algorithm on Hadamard manifolds

In this paper, we propose a Riemannian version of the difference of convex algorithm (DCA) to solve a minimization problem involving the difference of convex (DC) function. We establish the equivalence between the classical and simplified Riemannian versions of the DCA. We also prove that, under mild assumptions, the Riemannian version of the DCA is well-defined, and every cluster point of the sequence generated by the proposed method, if any, is a critical point of the objective DC function. Additionally, we establish some duality relations between the DC problem and its dual. To illustrate the effectiveness of the algorithm, we present some numerical experiments.

math.OC↗

arXiv2021

Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds

We present the Julia package Manifolds$.$jl, providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric positive definite matrices, or one of the models for hyperbolic spaces. We introduce a common interface, available in ManifoldsBase$.$jl, with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds$.$jl using B\'ezier splines, an optimization task on manifolds, and principal component analysis on nonlinear data. In a benchmark, Manifolds$.$jl outperforms all comparable packages for low-dimensional manifolds in speed; over Python and Matlab packages, the improvement is often several orders of magnitude, while over C/C++ packages, the improvement is two-fold. For high-dimensional manifolds, it outperforms all packages except for Tensorflow-Riemopt, which is specifically tailored for high-dimensional manifolds.

cs.MS↗

arXiv2021

Fenchel Duality and a Separation Theorem on Hadamard Manifolds

In this paper, we introduce a definition of Fenchel conjugate and Fenchel biconjugate on Hadamard manifolds based on the tangent bundle. Our definition overcomes the inconvenience that the conjugate depends on the choice of a certain point on the manifold, as previous definitions required. On the other hand, this new definition still possesses properties known to hold in the Euclidean case. It even yields a broader interpretation of the Fenchel conjugate in the Euclidean case itself. Most prominently, our definition of the Fenchel conjugate provides a Fenchel-Moreau Theorem for geodesically convex, proper, lower semicontinuous functions. In addition, this framework allows us to develop a theory of separation of convex sets on Hadamard manifolds, and a strict separation theorem is obtained.

math.OC↗