Martin Holler

Erion Morina, Martin Holler

We show that suitably regular functions can be approximated in the $\mathcal{C}^1$-norm both with rational functions and rational neural networks, including approximation rates with respect to width and depth of the network, and degree of the rational functions. As consequence of our results, we further obtain $\mathcal{C}^1$-approximation results for rational neural networks with the $\text{EQL}^÷$ and ParFam architecture, both of which are important in particular in the context of symbolic regression for physical law learning.

Dominik Narnhofer, Andreas Habring, Martin Holler et al.

In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, also a novel primal-dual Langevin algorithm for sampling from non-smooth distributions is introduced in this work.

Abdullah Abdullah, Martin Holler, Karl Kunisch et al.

Isolating different types of motion in video data is a highly relevant problem in video analysis. Applications can be found, for example, in dynamic medical or biological imaging, where the analysis and further processing of the dynamics of interest is often complicated by additional, unwanted dynamics, such as motion of the measurement subject. In this work, it is empirically shown that a representation of video data via untrained generator networks, together with a specific technique for latent space disentanglement that uses minimal, one-dimensional information on some of the underlying dynamics, allows to efficiently isolate different, highly non-linear motion types. In particular, such a representation allows to freeze any selection of motion types, and to obtain accurate independent representations of other dynamics of interest. Obtaining such a representation does not require any pre-training on a training data set, i.e., all parameters of the generator network are learned directly from a single video.

Wolfgang J. Kern, Simon Orlob, Andreas Bohn et al.

Objective: Exploit accelerometry data for an automatic, reliable, and prompt detection of spontaneous circulation during cardiac arrest, as this is both vital for patient survival and practically challenging. Methods: We developed a machine learning algorithm to automatically predict the circulatory state during cardiopulmonary resuscitation from 4-second-long snippets of accelerometry and electrocardiogram (ECG) data from pauses of chest compressions of real-world defibrillator records. The algorithm was trained based on 422 cases from the German Resuscitation Registry, for which ground truth labels were created by a manual annotation of physicians. It uses a kernelized Support Vector Machine classifier based on 49 features, which partially reflect the correlation between accelerometry and electrocardiogram data. Results: Evaluating 50 different test-training data splits, the proposed algorithm exhibits a balanced accuracy of 81.2%, a sensitivity of 80.6%, and a specificity of 81.8%, whereas using only ECG leads to a balanced accuracy of 76.5%, a sensitivity of 80.2%, and a specificity of 72.8%. Conclusion: The first method employing accelerometry for pulse/no-pulse decision yields a significant increase in performance compared to single ECG-signal usage. Significance: This shows that accelerometry provides relevant information for pulse/no-pulse decisions. In application, such an algorithm may be used to simplify retrospective annotation for quality management and, moreover, to support clinicians to assess circulatory state during cardiac arrest treatment.

Andreas Habring, Martin Holler

The regularity of images generated by convolutional neural networks, such as the U-net, generative networks, or the deep image prior, is analyzed. In a resolution-independent, infinite dimensional setting, it is shown that such images, represented as functions, are always continuous and, in some circumstances, even continuously differentiable, contradicting the widely accepted modeling of sharp edges in images via jump discontinuities. While such statements require an infinite dimensional setting, the connection to (discretized) neural networks used in practice is made by considering the limit as the resolution approaches infinity. As practical consequence, the results of this paper in particular provide analytical evidence that basic L2 regularization of network weights might lead to over-smoothed outputs.

Christian Aarset, Andreas Habring, Martin Holler et al.

In this work, a method for unsupervised energy disaggregation in private households equipped with smart meters is proposed. This method aims to classify power consumption as active or passive, granting the ability to report on the residents' activity and presence without direct interaction. This lays the foundation for applications like non-intrusive health monitoring of private homes. The proposed method is based on minimizing a suitable energy functional, for which the iPALM (inertial proximal alternating linearized minimization) algorithm is employed, demonstrating that various conditions guaranteeing convergence are satisfied. In order to confirm feasibility of the proposed method, experiments on semi-synthetic test data sets and a comparison to existing, supervised methods are provided.

Erion Morina, Martin Holler

This paper addresses the problem of learning reaction-diffusion (RD) systems from data while ensuring physical consistency and well-posedness of the learned models. Building on a regularization-based framework for structured model learning, we focus on learning parameterized reaction terms and investigate how to incorporate key physical properties, such as mass conservation and quasipositivity, directly into the learning process. Our main contributions are twofold: First, we propose techniques to systematically modify a given class of parameterized reaction terms such that the resulting terms inherently satisfy mass conservation and quasipositivity, ensuring that the learned RD systems preserve non-negativity and adhere to physical principles. These modifications also guarantee well-posedness of the resulting PDEs under additional regularity and growth conditions. Second, we extend existing theoretical results on regularization-based model learning to RD systems using these physically consistent reaction terms. Specifically, we prove that solutions to the learning problem converge to a unique, regularization-minimizing solution of a limit system even when conservation laws and quasipositivity are enforced. In addition, we provide approximation results for quasipositive functions, essential for constructing physically consistent parameterizations. These results advance the development of interpretable and reliable data-driven models for RD systems that align with fundamental physical laws.

Erion Morina, Philipp Scholl, Martin Holler

Models based on partial differential equations (PDEs) are powerful for describing a wide range of complex relationships in the natural sciences. Accurately identifying the PDE model, which represents the underlying physical law, is essential for a proper understanding of the problem. This reconstruction typically relies on indirect and noisy measurements of the system's state and, without specifically tailored methods, rarely yields symbolic expressions, thereby hindering interpretability. In this work, we address this issue by considering existing neural network architectures based on rational functions for the symbolic representation of physical laws. These networks leverage the approximation power of rational functions while also benefiting from their flexibility in representing arithmetic operations. Our main contribution is an identifiability result, showing that, in the limit of noiseless, complete measurements, such symbolic networks can uniquely reconstruct the simplest physical law within the PDE model. Specifically, reconstructed laws remain expressible within the symbolic network architecture, with regularization-minimizing parameterizations promoting interpretability and sparsity in case of $L^1$-regularization. In addition, we provide regularity results for symbolic networks. Empirical validation using the ParFam architecture supports these theoretical findings, providing evidence for the practical reconstructibility of physical laws.

Martin Holler, Erion Morina

This paper addresses the problem of uniqueness in learning physical laws for systems of partial differential equations (PDEs). Contrary to most existing approaches, it considers a framework of structured model learning, where existing, approximately correct physical models are augmented with components that are learned from data. The main result of the paper is a uniqueness result that covers a large class of PDEs and a suitable class of neural networks used for approximating the unknown model components. The uniqueness result shows that, in the idealized setting of full, noiseless measurements, a unique identification of the unknown model components is possible as regularization-minimizing solution of the PDE system. Furthermore, the paper provides a convergence result showing that model components learned on the basis of incomplete, noisy measurements approximate the ground truth model component in the limit. These results are possible under specific properties of the approximating neural networks and due to a dedicated choice of regularization. With this, a practical contribution of this analytic paper is to provide a class of model learning frameworks different to standard settings where uniqueness can be expected in the limit of full measurements.

Bruno Viti, Elias Karabelas, Martin Holler

Most machine learning-based image segmentation models produce pixel-wise confidence scores that represent the model's predicted probability for each class label at every pixel. While this information can be particularly valuable in high-stakes domains such as medical imaging, these scores are heuristic in nature and do not constitute rigorous quantitative uncertainty estimates. Conformal prediction (CP) provides a principled framework for transforming heuristic confidence scores into statistically valid uncertainty estimates. However, applying CP directly to image segmentation ignores the spatial correlations between pixels, a fundamental characteristic of image data. This can result in overly conservative and less interpretable uncertainty estimates. To address this, we propose CONSIGN (Conformal Segmentation Informed by Spatial Groupings via Decomposition), a CP-based method that incorporates spatial correlations to improve uncertainty quantification in image segmentation. Our method generates meaningful prediction sets that come with user-specified, high-probability error guarantees. It is compatible with any pre-trained segmentation model capable of generating multiple sample outputs. We evaluate CONSIGN against two CP baselines across three medical imaging datasets and two COCO dataset subsets, using three different pre-trained segmentation models. Results demonstrate that accounting for spatial structure significantly improves performance across multiple metrics and enhances the quality of uncertainty estimates.

Bruno Viti, Franz Thaler, Kathrin Lisa Kapper et al.

Segmentation of cardiac magnetic resonance images (MRI) is crucial for the analysis and assessment of cardiac function, helping to diagnose and treat various cardiovascular diseases. Most recent techniques rely on deep learning and usually require an extensive amount of labeled data. To overcome this problem, few-shot learning has the capability of reducing data dependency on labeled data. In this work, we introduce a new method that merges few-shot learning with a U-Net architecture and Gaussian Process Emulators (GPEs), enhancing data integration from a support set for improved performance. GPEs are trained to learn the relation between the support images and the corresponding masks in latent space, facilitating the segmentation of unseen query images given only a small labeled support set at inference. We test our model with the M&Ms-2 public dataset to assess its ability to segment the heart in cardiac magnetic resonance imaging from different orientations, and compare it with state-of-the-art unsupervised and few-shot methods. Our architecture shows higher DICE coefficients compared to these methods, especially in the more challenging setups where the size of the support set is considerably small.

Erion Morina, Martin Holler

This work focuses on the analysis of fully connected feed forward ReLU neural networks as they approximate a given, smooth function. In contrast to conventionally studied universal approximation properties under increasing architectures, e.g., in terms of width or depth of the networks, we are concerned with the asymptotic growth of the parameters of approximating networks. Such results are of interest, e.g., for error analysis or consistency results for neural network training. The main result of our work is that, for a ReLU architecture with state of the art approximation error, the realizing parameters grow at most polynomially. The obtained rate with respect to a normalized network size is compared to existing results and is shown to be superior in most cases, in particular for high dimensional input.