Commonly used $f$-divergences of measures, e.g., the Kullback-Leibler divergence, are subject to limitations regarding the support of the involved measures. A remedy is regularizing the $f$-divergence by a squared maximum mean discrepancy (MMD) associated with a characteristic kernel $K$. We use the kernel mean embedding to show that this regularization can be rewritten as the Moreau envelope of some function on the associated reproducing kernel Hilbert space. Then, we exploit well-known results on Moreau envelopes in Hilbert spaces to analyze the MMD-regularized $f$-divergences, particularly their gradients. Subsequently, we use our findings to analyze Wasserstein gradient flows of MMD-regularized $f$-divergences. We provide proof-of-the-concept numerical examples for flows starting from empirical measures. Here, we cover $f$-divergences with infinite and finite recession constants. Lastly, we extend our results to the tight variational formulation of $f$-divergences and numerically compare the resulting flows.

State-of-the-art image reconstruction often relies on complex, highly parameterized deep architectures. We propose an alternative: a data-driven reconstruction method inspired by the classic Tikhonov regularization. Our approach iteratively refines intermediate reconstructions by solving a sequence of quadratic problems. These updates have two key components: (i) learned filters to extract salient image features, and (ii) an attention mechanism that locally adjusts the penalty of filter responses. Our method achieves performance on par with leading plug-and-play and learned regularizer approaches while offering interpretability, robustness, and convergent behavior. In effect, we bridge traditional regularization and deep learning with a principled reconstruction approach.

Modelling the temperature of Electric Vehicle (EV) batteries is a fundamental task of EV manufacturing. Extreme temperatures in the battery packs can affect their longevity and power output. Although theoretical models exist for describing heat transfer in battery packs, they are computationally expensive to simulate. Furthermore, it is difficult to acquire data measurements from within the battery cell. In this work, we propose a data-driven surrogate model (LiFe-net) that uses readily accessible driving diagnostics for battery temperature estimation to overcome these limitations. This model incorporates Neural Operators with a traditional numerical integration scheme to estimate the temperature evolution. Moreover, we propose two further variations of the baseline model: LiFe-net trained with a regulariser and LiFe-net trained with time stability loss. We compared these models in terms of generalization error on test data. The results showed that LiFe-net trained with time stability loss outperforms the other two models and can estimate the temperature evolution on unseen data with a relative error of 2.77 % on average.

The classification of pixel spectra of hyperspectral images, i.e. spectral classification, is used in many fields ranging from agricultural, over medical to remote sensing applications and is currently also expanding to areas such as autonomous driving. Even though for full hyperspectral images the best-performing methods exploit spatial-spectral information, performing classification solely on spectral information has its own advantages, e.g. smaller model size and thus less data required for training. Moreover, spectral information is complementary to spatial information and improvements on either part can be used to improve spatial-spectral approaches in the future. Recently, 1D-Justo-LiuNet was proposed as a particularly efficient model with very few parameters, which currently defines the state of the art in spectral classification. However, we show that with limited training data the model performance deteriorates. Therefore, we investigate MiniROCKET and HDC-MiniROCKET for spectral classification to mitigate that problem. The model extracts well-engineered features without trainable parameters in the feature extraction part and is therefore less vulnerable to limited training data. We show that even though MiniROCKET has more parameters it outperforms 1D-Justo-LiuNet in limited data scenarios and is mostly on par with it in the general case

Human Activity Recognition (HAR) has become one of the leading research topics of the last decade. As sensing technologies have matured and their economic costs have declined, a host of novel applications, e.g., in healthcare, industry, sports, and daily life activities have become popular. The design of HAR systems requires different time-consuming processing steps, such as data collection, annotation, and model training and optimization. In particular, data annotation represents the most labor-intensive and cumbersome step in HAR, since it requires extensive and detailed manual work from human annotators. Therefore, different methodologies concerning the automation of the annotation procedure in HAR have been proposed. The annotation problem occurs in different notions and scenarios, which all require individual solutions. In this paper, we provide the first systematic review on data annotation techniques for HAR. By grouping existing approaches into classes and providing a taxonomy, our goal is to support the decision on which techniques can be beneficially used in a given scenario.

We investigate the stationary diffusion equation with a coefficient given by a (transformed) L\'evy random field. L\'evy random fields are constructed by smoothing L\'evy noise fields with kernels from the Mat\'ern class. We show that L\'evy noise naturally extends Gaussian white noise within Minlos' theory of generalized random fields. Results on the distributional path spaces of L\'evy noise are derived as well as the amount of smoothing to ensure such distributions become continuous paths. Given this, we derive results on the pathwise existence and measurability of solutions to the random boundary value problem (BVP). For the solutions of the BVP we prove existence of moments (in the $H^1$-norm) under adequate growth conditions on the L\'evy measure of the noise field. Finally, a kernel expansion of the smoothed L\'evy noise fields is introduced and convergence in $L^n$ ($n\geq 1$) of the solutions associated with the approximate random coefficients is proven with an explicit rate.

The hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 804092 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law--tailed synchronization durations, with $\tau_t \simeq 1.2(1)$, away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: $\tau_t \simeq 1.6(1)$. However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < \tau_t \le 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.

We propose a one-step person detector for topview omnidirectional indoor scenes based on convolutional neural networks (CNNs). While state of the art person detectors reach competitive results on perspective images, missing CNN architectures as well as training data that follows the distortion of omnidirectional images makes current approaches not applicable to our data. The method predicts bounding boxes of multiple persons directly in omnidirectional images without perspective transformation, which reduces overhead of pre- and post-processing and enables real-time performance. The basic idea is to utilize transfer learning to fine-tune CNNs trained on perspective images with data augmentation techniques for detection in omnidirectional images. We fine-tune two variants of Single Shot MultiBox detectors (SSDs). The first one uses Mobilenet v1 FPN as feature extractor (moSSD). The second one uses ResNet50 v1 FPN (resSSD). Both models are pre-trained on Microsoft Common Objects in Context (COCO) dataset. We fine-tune both models on PASCAL VOC07 and VOC12 datasets, specifically on class person. Random 90-degree rotation and random vertical flipping are used for data augmentation in addition to the methods proposed by original SSD. We reach an average precision (AP) of 67.3 % with moSSD and 74.9 % with resSSD onthe evaluation dataset. To enhance the fine-tuning process, we add a subset of HDA Person dataset and a subset of PIROPOdatabase and reduce the number of perspective images to PASCAL VOC07. The AP rises to 83.2 % for moSSD and 86.3 % for resSSD, respectively. The average inference speed is 28 ms per image for moSSD and 38 ms per image for resSSD using Nvidia Quadro P6000. Our method is applicable to other CNN-based object detectors and can potentially generalize for detecting other objects in omnidirectional images.

We analyze the Ensemble and Polynomial Chaos Kalman filters applied to nonlinear stationary Bayesian inverse problems. In a sequential data assimilation setting such stationary problems arise in each step of either filter. We give a new interpretation of the approximations produced by these two popular filters in the Bayesian context and prove that, in the limit of large ensemble or high polynomial degree, both methods yield approximations which converge to a well-defined random variable termed the analysis random variable. We then show that this analysis variable is more closely related to a specific linear Bayes estimator than to the solution of the associated Bayesian inverse problem given by the posterior measure. This suggests limited or at least guarded use of these generalized Kalman filter methods for the purpose of uncertainty quantification.

When propagating uncertainty in the data of differential equations, the probability laws describing the uncertainty are typically themselves subject to uncertainty. We present a sensitivity analysis of uncertainty propagation for differential equations with random inputs to perturbations of the input measures. We focus on the elliptic diffusion equation with random coefficient and source term, for which the probability measure of the solution random field is shown to be Lipschitz-continuous in both total variation and Wasserstein distance. The result generalizes to the solution map of any differential equation with locally H\"older dependence on input parameters. In addition, these results extend to Lipschitz continuous quantities of interest of the solution as well as to coherent risk functionals of these applied to evaluate the impact of their uncertainty. Our analysis is based on the sensitivity of risk functionals and pushforward measures for locally H\"older mappings with respect to the Wasserstein distance of perturbed input distributions. The established results are applied, in particular, to the case of lognormal diffusion and the truncation of series representations of input random fields.

The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to the conjecture. Firstly, we investigate the frequencies of all elements among a union closed family and pose a question generalizing the Union Closed Sets Conjecture. Secondly, we investigate structures equivalent to union closed families and obtain a weakening of the Union Closed Sets Conjecture. We pose some new open questions about union closed families and related structures and hint at some further directions of research regarding the conjecture.

How hard is it to find a local optimum? If we are given a graph and want to find a locally maximal cut--meaning that the number of edges in the cut cannot be improved by moving a single vertex from one side to the other--then just iterating improving steps finds a local maximum since the size of the cut can increase at most $|E|$ times. If, on the other hand, the edges are weighted, this problem becomes hard for the class PLS (Polynomial Local Search)[16]. We are interested in optimization problems with lexicographic costs. For Max-Cut this would mean that the edges $e_1,\dots, e_m$ have costs $c(e_i) = 2^{m-i}$. For such a cost function, it is easy to see that finding a global Max-Cut is easy. In contrast, we show that it is PLS-complete to find an assignment for a 4-CNF formula that is locally maximal (when the clauses have lexicographic weights); and also for a 3-CNF when we relax the notion of local by allowing to switch two variables at a time. We use these results to answer a question in Scheder and Tantow[15], who showed that finding a lexicographic local minimum of a string $s \in \{0,1\}^n$ under the action of a list of given permutations $\pi_1, \dots, \pi_k \in S_{n}$ is PLS-complete. They ask whether the problem stays PLS-complete when the $\pi_1,\dots,\pi_k$ commute, i.e., generate an Abelian subgroup $G$ of $S_n$. In this work, we show that it does, and in fact stays PLS-complete even (1) when every element in $G$ has order two and also (2) when $G$ is cyclic, i.e., all $\pi_1,\dots,\pi_k$ are powers of a single permutations $\pi$.

GNSS localization is an important part of today's autonomous systems, although it suffers from non-Gaussian errors caused by non-line-of-sight effects. Recent methods are able to mitigate these effects by including the corresponding distributions in the sensor fusion algorithm. However, these approaches require prior knowledge about the sensor's distribution, which is often not available. We introduce a novel sensor fusion algorithm based on variational Bayesian inference, that is able to approximate the true distribution with a Gaussian mixture model and to learn its parametrization online. The proposed Incremental Variational Mixture algorithm automatically adapts the number of mixture components to the complexity of the measurement's error distribution. We compare the proposed algorithm against current state-of-the-art approaches using a collection of open access real world datasets and demonstrate its superior localization accuracy.

Cellular micromotors are attractive for locally delivering high concentrations of drug and targeting hard-to-reach disease sites such as cervical cancer and early ovarian cancer lesions by non-invasive means. Spermatozoa are highly efficient micromotors perfectly adapted to traveling up the female reproductive system. Indeed, bovine sperm-based micromotors have recently been reported as a potential candidate for the drug delivery toward gynecological cancers of clinical unmet need. However, due to major differences in the molecular make-up of bovine and human sperm, a key translational bottleneck for bringing this technology closer to the clinic is to transfer this concept to human sperm. Here, we successfully load human sperm with a chemotherapeutic drug and perform treatment of relevant 3D cervical cancer and patient-representative 3D ovarian cancer cell cultures, resulting in strong anti-cancer effects. Additionally, we show the subcellular localization of the chemotherapeutic drug within human sperm heads and assess drug effects on sperm motility and viability over time. Finally, we demonstrate guidance and release of human drug-loaded sperm onto cancer cell cultures by using streamlined microcap designs capable of simultaneously carrying multiple human sperm towards controlled drug dosing by transporting known numbers of sperm loaded with defined amounts of chemotherapeutic drug.

In this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential equation is a drift-diffusion model. Computational methods for quantum graphs require a careful discretization of the differential operator that also incorporates the node conditions, in our case Kirchhoff-Neumann conditions. Traditional numerical schemes are rather mature but have to be tailored manually when the differential equation becomes the constraint in an optimization problem. Recently, physics informed neural networks (PINNs) have emerged as a versatile tool for the solution of partial differential equations from a range of applications. They offer flexibility to solve parameter identification or optimization problems by only slightly changing the problem formulation used for the forward simulation. We compare several PINN approaches for solving the drift-diffusion on the metric graph.

This work deals with uncertainty quantification for a generic input distribution to some resource-intensive simulation, e.g., requiring the solution of a partial differential equation. While efficient numerical methods exist to compute integrals for high-dimensional Gaussian and other separable distributions based on sparse grids (SG), input data arising in practice often does not fall into this class. We therefore employ transport maps to transform complex distributions to multivatiate standard normals. In generative learning, a number of neural network architectures have been introduced that accomplish this task approximately. Examples are affine coupling flows (ACF) and ordinary differential equation-based networks such as conditional flow matching (CFM). To compute the expectation of a quantity of interest, we numerically integrate the composition of the inverse of the learned transport map with the simulation code output. As this map is integrated over a multivariate Gaussian distribution, SG techniques can be applied. Viewing the images of the SG quadrature nodes as learned quadrature nodes for a given complex distribution motivates our title. We demonstrate our method for monomials of total degrees for which the unmapped SG rules are exact. We also apply our approach to the stationary diffusion equation with coefficients modeled by exponentiated L\'evy random fields, using a Karhunen-Lo\`eve-like modal expansions with 9 and 25 modes. In a series of numerical experiments, we investigate errors due to learning accuracy, quadrature, statistical estimation, truncation of the modal series of the input random field, and training data size for three normalizing flows (ACF, conditional Flow Matching and Optimal transport Flow Matching) We discuss the mathematical assumptions on which our approach is based and demonstrate its shortcomings when these are violated.

Koopman-based methods leverage a nonlinear lifting to enable linear regression techniques. Consequently, data generation, learning and prediction is performed through the lens of this lifting, giving rise to a nonlinear manifold that is invariant under the Koopman operator. In data-driven approximation such as Extended Dynamic Mode Decomposition, this invariance is typically lost due to the presence of (finite-data) approximation errors. In this work, we show that reprojections are crucial for reliable predictions. We provide an approach via closest-point projections that ensure consistency with this nonlinear manifold, which is strongly related to a Riemannian metric and maximum likelihood estimates. While these results are already novel for autonomous systems, we present our approach for parametric systems, providing the basis for data-driven bifurcation analysis and control applications.

Medical imaging plays an important role in diagnosis and treatment of multiple diseases. It is a field under continuous development which seeks for improved sensitivity and spatiotemporal resolution to allow the dynamic monitoring of diverse biological processes that occur at the micro- and nanoscale. Emerging technologies for targeted diagnosis and therapy such as nanotherapeutics, micro-implants, catheters and small medical tools also need to be precisely located and monitored while performing their function inside the human body. In this work, we show for the first time the real-time tracking of moving single micro-objects below centimeter thick tissue-mimicking phantoms, using multispectral optoacoustic tomography (MSOT). This technique combines the advantages of ultrasound imaging regarding depth and resolution with the molecular specificity of optical methods, thereby facilitating the discrimination between the spectral signatures of the micro-objects from those of intrinsic tissue molecules. The resulting MSOT signal is further improved in terms of contrast and specificity by coating the micro-objects surface with gold nanorods, possessing a unique absorption spectrum, which will allow their discrimination from surrounding biological tissues when translated to in vivo settings.