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Researcher profile

Johannes Hertrich

Johannes Hertrich contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine Learningmath.NANumerical Analysiseess.IVmath.OCmath.STStatistics TheoryComputer Vision

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Published work

7 published item(s)

preprint2026arXiv

A Stability Benchmark of Generative Regularizers for Inverse Problems

Generative (diffusion) priors demonstrate remarkable performance in addressing inverse problems in imaging. Yet, for scientific and medical imaging, it is crucial that reconstruction techniques remain stable and reliable under imperfect settings. Typical definitions of stability encompass the notion of ''convergent regularization'', robustness to out-of-distribution data, and to inaccuracies in the forward operator or noise model. We evaluate these properties numerically. Furthermore, we benchmark generative approaches against modern optimization-based methods inspired by the widely used variational techniques. Our results give insights for which settings and applications generative priors can deliver state-of-the-art reconstructions, and on those in which they fall short or may even be problematic.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Learning Regularization Functionals for Inverse Problems: A Comparative Study

In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Tessellations of Semi-Discrete Flow Matching

We study Flow Matching in a semi-discrete setting where a Gaussian source is transported toward a discrete target supported on finitely many points. This semi-discrete regime is the theoretical setting behind the use of Flow Matching for generative modeling, where the target distribution is represented by a finite dataset. In this semi-discrete regime, the exact Flow Matching velocity field is available in closed form, which makes it possible to analyze the geometry induced by the terminal flow map independently of optimization and approximation effects. We investigate the terminal assignment regions, namely the preimages of the target atoms under the terminal flow. We show that these regions are open, simply connected and, under an additional assumption, homeomorphic to the unit ball. At the same time, a planar four-point example shows that these cells can differ sharply from Laguerre cells arising in semi-discrete optimal transport: they may be non-convex, have curved boundaries, and exhibit different boundedness and adjacency patterns. These results clarify the geometry intrinsically induced by the exact semi-discrete Flow Matching objective before neural approximation enters the picture.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

PCA Reduced Gaussian Mixture Models with Applications in Superresolution

Despite the rapid development of computational hardware, the treatment of large and high dimensional data sets is still a challenging problem. This paper provides a twofold contribution to the topic. First, we propose a Gaussian Mixture Model in conjunction with a reduction of the dimensionality of the data in each component of the model by principal component analysis, called PCA-GMM. To learn the (low dimensional) parameters of the mixture model we propose an EM algorithm whose M-step requires the solution of constrained optimization problems. Fortunately, these constrained problems do not depend on the usually large number of samples and can be solved efficiently by an (inertial) proximal alternating linearized minimization algorithm. Second, we apply our PCA-GMM for the superresolution of 2D and 3D material images based on the approach of Sandeep and Jacob. Numerical results confirm the moderate influence of the dimensionality reduction on the overall superresolution result.

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preprint2021arXiv

Wasserstein Patch Prior for Image Superresolution

In this paper, we introduce a Wasserstein patch prior for superresolution of two- and three-dimensional images. Here, we assume that we have given (additionally to the low resolution observation) a reference image which has a similar patch distribution as the ground truth of the reconstruction. This assumption is e.g. fulfilled when working with texture images or material data. Then, the proposed regularizer penalizes the $W_2$-distance of the patch distribution of the reconstruction to the patch distribution of some reference image at different scales. We demonstrate the performance of the proposed regularizer by two- and three-dimensional numerical examples.

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preprint2020arXiv

Alternatives to the EM Algorithm for ML-Estimation of Location, Scatter Matrix and Degree of Freedom of the Student-$t$ Distribution

In this paper, we consider maximum likelihood estimations of the degree of freedom parameter $ν$, the location parameter $μ$ and the scatter matrix $Σ$ of the multivariate Student-$t$ distribution. In particular, we are interested in estimating the degree of freedom parameter $ν$ that determines the tails of the corresponding probability density function and was rarely considered in detail in the literature so far. We prove that under certain assumptions a minimizer of the negative log-likelihood function exists, where we have to take special care of the case $ν\rightarrow \infty$, for which the Student-$t$ distribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixed $ν$, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix $ν$, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for $ν$. We show how the objective function behaves for the different updates of $ν$ and prove for all three algorithms that it decreases in each iteration step. We compare the algorithms as well as some accelerated versions by numerical simulation and apply one of them for estimating the degree of freedom parameter in images corrupted by Student-$t$ noise.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Inertial Stochastic PALM (iSPALM) and Applications in Machine Learning

Inertial algorithms for minimizing nonsmooth and nonconvex functions as the inertial proximal alternating linearized minimization algorithm (iPALM) have demonstrated their superiority with respect to computation time over their non inertial variants. In many problems in imaging and machine learning, the objective functions have a special form involving huge data which encourage the application of stochastic algorithms. While algorithms based on stochastic gradient descent are still used in the majority of applications, recently also stochastic algorithms for minimizing nonsmooth and nonconvex functions were proposed. In this paper, we derive an inertial variant of a stochastic PALM algorithm with variance-reduced gradient estimator, called iSPALM, and prove linear convergence of the algorithm under certain assumptions. Our inertial approach can be seen as generalization of momentum methods widely used to speed up and stabilize optimization algorithms, in particular in machine learning, to nonsmooth problems. Numerical experiments for learning the weights of a so-called proximal neural network and the parameters of Student-t mixture models show that our new algorithm outperforms both stochastic PALM and its deterministic counterparts.

0 citations0 reviewsNo code linkedOpen access