Search

In adaptive dynamical networks, the dynamics of the nodes and the edges influence each other. We show that we can treat such systems as a closed feedback loop between edge and node dynamics. Using recent advances on the stability of feedback systems from control theory, we derive local, sufficient conditions for steady states of such systems to be linearly stable. These conditions are local in the sense that they are written entirely in terms of the (linearized) behavior of the edges and nodes. We apply these conditions to the Kuramoto model with inertia written in adaptive form, and the adaptive Kuramoto model. For the former we recover a classic result, for the latter we show that our sufficient conditions match necessary conditions where the latter are available, thus completely settling the question of linear stability in this setting. The method we introduce can be readily applied to a vast class of systems. It enables straightforward evaluation of stability in highly heterogeneous systems.

In this paper, we provide a tutorial overview and an extension of a recently developed framework for data-driven control of unknown nonlinear systems with rigorous closed-loop guarantees. The proposed approach relies on the Koopman operator representation of the nonlinear system, for which a bilinear surrogate model is estimated based on data. In contrast to existing Koopman-based estimation procedures, we state guaranteed bounds on the approximation error using the stability- and certificate-oriented extended dynamic mode decomposition (SafEDMD) framework. The resulting surrogate model and the uncertainty bounds allow us to design controllers via robust control theory and sum-of-squares optimization, guaranteeing desirable properties for the closed-loop system. We present results on stabilization both in discrete and continuous time, and we derive a method for controller design with performance objectives. The benefits of the presented framework over established approaches are demonstrated with a numerical example.

We construct a version of geodesic normal coordinates adapted to a submanifold of a pseudo-Riemannian manifold and show that the Taylor coefficients of the metric in these coordinates can be expressed as universal polynomials in the components of the covariant derivatives of the background curvature tensor and the covariant derivatives of the second fundamental form. We formulate a definition of natural submanifold tensors and show that these are linear combinations of contractions of covariant derivatives of the background curvature tensor and covariant derivatives of the second fundamental form. We also describe how this result gives a similar characterization of natural submanifold differential operators., Comment: 12 pages

Cardio-mechanical models can be used to support clinical decision-making. Unfortunately, the substantial computational effort involved in many cardiac models hinders their application in the clinic, despite the fact that they may provide valuable information. In this work, we present a probabilistic reduced-order modeling (ROM) framework to dramatically reduce the computational effort of such models while providing a credibility interval. In the online stage, a fast-to-evaluate generalized one-fiber model is considered. This generalized one-fiber model incorporates correction factors to emulate patient-specific attributes, such as local geometry variations. In the offline stage, Bayesian inference is used to calibrate these correction factors on training data generated using a full-order isogeometric cardiac model (FOM). A Gaussian process is used in the online stage to predict the correction factors for geometries that are not in the training data. The proposed framework is demonstrated using two examples. The first example considers idealized left-ventricle geometries, for which the behavior of the ROM framework can be studied in detail. In the second example, the ROM framework is applied to scan-based geometries, based on which the application of the ROM framework in the clinical setting is discussed. The results for the two examples convey that the ROM framework can provide accurate online predictions, provided that adequate FOM training data is available. The uncertainty bands provided by the ROM framework give insight into the trustworthiness of its results. Large uncertainty bands can be considered as an indicator for the further population of the training data set.

Deep neural networks have achieved remarkable success in diverse applications, prompting the need for a solid theoretical foundation. Recent research has identified the minimal width $\max\{2,d_x,d_y\}$ required for neural networks with input dimensions $d_x$ and output dimension $d_y$ that use leaky ReLU activations to universally approximate $L^p(\mathbb{R}^{d_x},\mathbb{R}^{d_y})$ on compacta. Here, we present an alternative proof for the minimal width of such neural networks, by directly constructing approximating networks using a coding scheme that leverages the properties of leaky ReLUs and standard $L^p$ results. The obtained construction has a minimal interior dimension of $1$, independent of input and output dimensions, which allows us to show that autoencoders with leaky ReLU activations are universal approximators of $L^p$ functions. Furthermore, we demonstrate that the normalizing flow LU-Net serves as a distributional universal approximator. We broaden our results to show that smooth invertible neural networks can approximate $L^p(\mathbb{R}^{d},\mathbb{R}^{d})$ on compacta when the dimension $d\geq 2$, which provides a constructive proof of a classical theorem of Brenier and Gangbo. In addition, we use a topological argument to establish that for FNNs with monotone Lipschitz continuous activations, $d_x+1$ is a lower bound on the minimal width required for the uniform universal approximation of continuous functions $C^0(\mathbb{R}^{d_x},\mathbb{R}^{d_y})$ on compacta when $d_x\geq d_y$., Comment: 56 pages in the main text, 70 pages with Appendix

We present a variation of the Front Tracking (FT) method for modeling solutions to hyperbolic systems in one space dimension, a Modified FT scheme. Instead of using non-entropic shocks, we approximate simple waves by jumps which exactly match the states, while approximating the wave speed. Our construction makes use of compression curve as well. We work with weak* solutions introduced in \cite{}. Consequently, we are able to analyze residuals rather than errors, and obtain cleaner convergence results. Through this paper, we intend to set up a scheme that is capable of handling strong shocks consistently. We develop this scheme primarily to prove (in a future work) existence of large amplitude solutions of the p-system. Therefore, we treat the p-system more carefully and construct a completely self-sufficient scheme, which has a limit as long as the approximate weak* solutions have uniformly bounded variation., Comment: 43 pages

We combine the power of blind integral field spectroscopy from the Hobby-Eberly Telescope (HET) Dark Energy Experiment (HETDEX) with sources detected by the Low Frequency Array (LOFAR) to construct the HETDEX-LOFAR Spectroscopic Redshift Catalog. Starting from the first data release of the LOFAR Two-metre Sky Survey (LoTSS), including a value-added catalog with photometric redshifts, we extracted 28,705 HETDEX spectra. Using an automatic classifying algorithm, we assigned each object a star, galaxy, or quasar label along with a velocity/redshift, with supplemental classifications coming from the continuum and emission line catalogs of the internal, fourth data release from HETDEX (HDR4). We measured 9,087 new redshifts; in combination with the value-added catalog, our final spectroscopic redshift sample is 9,710 sources. This new catalog contains the highest substantial fraction of LOFAR galaxies with spectroscopic redshift information; it improves archival spectroscopic redshifts, and facilitates research to determine the [O II] emission properties of radio galaxies from $0.0 < z < 0.5$, and the Ly$\alpha$ emission characteristics of both radio galaxies and quasars from $1.9 < z < 3.5$. Additionally, by combining the unique properties of LOFAR and HETDEX, we are able to measure star formation rates (SFR) and stellar masses. Using the Visible Integral-field Replicable Unit Spectrograph (VIRUS), we measure the emission lines of [O III], [Ne III], and [O II] and evaluate line-ratio diagnostics to determine whether the emission from these galaxies is dominated by AGN or star formation and fit a new SFR-L$_{150MHz}$ relationship., Comment: 21 pages, 17 figures, submitted to ApJ

DNSSEC, a DNS security extension, is essential to accurately translating domain names to IP addresses. Digital signatures provide the foundation for this reliable translation, however, the evolution of 'Quantum Computers' has made traditional digital signatures vulnerable. In light of this, NIST has recently selected potential post-quantum digital signatures that can operate on conventional computers and resist attacks made with Quantum Computers. Since these post-quantum digital signatures are still in their early stages of development, replacing pre-quantum digital signature schemes in DNSSEC with post-quantum candidates is risky until the post-quantum candidates have undergone a thorough security analysis. Given this, herein, we investigate the viability of employing 'Double-Signatures' in DNSSEC, combining a post-quantum digital signature and a classic one. The rationale is that double-signatures will offer protection against quantum threats on conventional signature schemes as well as unknown non-quantum attacks on post-quantum signature schemes, hence even if one fails the other provides security guarantees. However, the inclusion of two signatures in the DNSSEC response message doesn't bode well with the maximum allowed size of DNSSEC responses (i.e., 1232B, a limitation enforced by MTU of physical links). To counter this issue, we leverage a way to do application-layer fragmentation of DNSSEC responses with two signatures. We implement our solution on top of OQS-BIND and through experiments show that the addition of two signatures in DNSSEC and application-layer fragmentation of all relevant resource records and their reassembly does not have any substantial impact on the efficiency of the resolution process and thus is suitable for the interim period at least until the quantum computers are fully realized.

In biomechanics and robotics, elasticity plays a crucial role in enhancing locomotion efficiency and stability. Traditional approaches in legged robots often employ series elastic actuators (SEA) with discrete rigid components, which, while effective, add weight and complexity. This paper presents an innovative alternative by integrating continuously compliant structures into the lower legs of a bipedal robot, fundamentally transforming the SEA concept. Our approach replaces traditional rigid segments with lightweight, deformable materials, reducing overall mass and simplifying the actuation design. This novel design introduces unique challenges in modeling, sensing, and control, due to the infinite dimensionality of continuously compliant elements. We address these challenges through effective approximations and control strategies. The paper details the design and modeling of the compliant leg structure, presents low-level force and kinematics controllers, and introduces a high-level posture controller with a gait scheduler. Experimental results demonstrate successful bipedal walking using this new design., Comment: 7 pages, 4 figures, submitted to 2025 IEEE International Conference on Robotics and Automation, 19-23 May, Atlanta, USA

Evidence of excess filamentarity is considered for two spatial point process applications: local minima in whole earth precipitation modelling and locations of cold clumps in the Milky Way. A diagnostic test using the number of aligned triads and tetrads is developed. A Poisson filament process is proposed based on a parent Poisson process with correlated random walk offspring locations. Filaments are initially identified using an arc search method, with ABC for subsequent inference. Simulations indicate good performance. In both applications there is strong evidence of filamentarity. The method successfully identifies two outlying precipitation data sets., Comment: 23 pages, 8 figures

We consider the problem of estimating a high-dimensional covariance matrix from a small number of observations when covariates on pairs of variables are available and the variables can have spatial structure. This is motivated by the problem arising in demography of estimating the covariance matrix of the total fertility rate (TFR) of 195 different countries when only 11 observations are available. We construct an estimator for high-dimensional covariance matrices by exploiting information about pairwise covariates, such as whether pairs of variables belong to the same cluster, or spatial structure of the variables, and interactions between the covariates. We reformulate the problem in terms of a mixed effects model. This requires the estimation of only a small number of parameters, which are easy to interpret and which can be selected using standard procedures. The estimator is consistent under general conditions, and asymptotically normal. It works if the mean and variance structure of the data is already specified or if some of the data are missing. We assess its performance under our model assumptions, as well as under model misspecification, using simulations. We find that it outperforms several popular alternatives. We apply it to the TFR dataset and draw some conclusions., Comment: 23 pages, 5 figures, submitted to the Annals of Applied Statistics

We study quantum algorithms for verifying properties of the output probability distribution of a classical or quantum circuit, given access to the source code that generates the distribution. We consider the basic task of uniformity testing, which is to decide if the output distribution is uniform on $[d]$ or $\epsilon$-far from uniform in total variation distance. More generally, we consider identity testing, which is the task of deciding if the output distribution equals a known hypothesis distribution, or is $\epsilon$-far from it. For both problems, the previous best known upper bound was $O(\min\{d^{1/3}/\epsilon^{2},d^{1/2}/\epsilon\})$. Here we improve the upper bound to $O(\min\{d^{1/3}/\epsilon^{4/3}, d^{1/2}/\epsilon\})$, which we conjecture is optimal., Comment: 21 pages

How many T gates are needed to approximate an arbitrary $n$-qubit quantum state to within error $\varepsilon$? Improving prior work of Low, Kliuchnikov, and Schaeffer, we show that the optimal asymptotic scaling is $\Theta\left(\sqrt{2^n\log(1/\varepsilon)}+\log(1/\varepsilon)\right)$ if we allow ancilla qubits. We also show that this is the optimal T-count for implementing an arbitrary diagonal $n$-qubit unitary to within error $\varepsilon$. We describe applications in which a tensor product of many single-qubit unitaries can be synthesized in parallel for the price of one.

Equivariant neural networks have shown great success in reinforcement learning, improving sample efficiency and generalization when there is symmetry in the task. However, in many problems, only approximate symmetry is present, which makes imposing exact symmetry inappropriate. Recently, approximately equivariant networks have been proposed for supervised classification and modeling physical systems. In this work, we develop approximately equivariant algorithms in reinforcement learning (RL). We define approximately equivariant MDPs and theoretically characterize the effect of approximate equivariance on the optimal Q function. We propose novel RL architectures using relaxed group convolutions and experiment on several continuous control domains and stock trading with real financial data. Our results demonstrate that approximate equivariance matches prior work when exact symmetries are present, and outperforms them when domains exhibit approximate symmetry. As an added byproduct of these techniques, we observe increased robustness to noise at test time., Comment: Preprint

We consider the quantum magic in systems of dense neutrinos undergoing coherent flavor transformations, relevant for supernova and neutron-star binary mergers. Mapping the three-flavor-neutrino system to qutrits, the evolution of quantum magic is explored in the single scattering angle limit for a selection of initial tensor-product pure states for $N_\nu \le 8$ neutrinos. For $|\nu_e\rangle^{\otimes N_\nu}$ initial states, the magic, as measured by the $\alpha=2$ stabilizer Renyi entropy $M_2$, is found to decrease with radial distance from the neutrino sphere, reaching a value that lies below the maximum for tensor-product qutrit states. Further, the asymptotic magic per neutrino, $M_2/N_\nu$, decreases with increasing $N_\nu$. In contrast, the magic evolving from states containing all three flavors reaches values only possible with entanglement, with the asymptotic $M_2/N_\nu$ increasing with $N_\nu$. These results highlight the connection between the complexity in simulating quantum physical systems and the parameters of the Standard Model., Comment: 15 pages, 8 figures

Topology Bench is a comprehensive topology dataset designed to accelerate benchmarking studies in optical networks. The dataset, focusing on core optical networks, comprises publicly accessible and ready-to-use topologies, including (a) 105 georeferenced real-world optical networks and (b) 270,900 validated synthetic topologies. Prior research on real-world core optical networks has been characterised by fragmented open data sources and disparate individual studies. Moreover, previous efforts have notably failed to provide synthetic data at a scale comparable to our present study. Topology Bench addresses this limitation, offering a unified resource and represents a 61.5% increase in spatially-referenced real world optical networks. To benchmark and identify the fundamental nature of optical network topologies through the lens of graph-theoretical analysis, we analyse both real and synthetic networks using structural, spatial and spectral metrics. Our comparative analysis identifies constraints in real optical network diversity and illustrates how synthetic networks can complement and expand the range of topologies available for use. Currently, topologies are selected based on subjective criteria, such as preference, data availability, or perceived suitability, leading to potential biases and limited representativeness. Our framework enhances the generalisability of optical network research by providing a more objective and systematic approach to topology selection. A statistical and correlation analysis reveals the quantitative range of all of these graph metrics and the relationships between them. Finally, we apply unsupervised machine learning to cluster real-world topologies into distinctive groups using nine optimal graph metrics using K-means. We conclude the analysis by providing guidance on how to use such clusters to select a diverse set of topologies for future studies.

Species distribution models (SDMs) aim to predict the distribution of species by relating occurrence data with environmental variables. Recent applications of deep learning to SDMs have enabled new avenues, specifically the inclusion of spatial data (environmental rasters, satellite images) as model predictors, allowing the model to consider the spatial context around each species' observations. However, the appropriate spatial extent of the images is not straightforward to determine and may affect the performance of the model, as scale is recognized as an important factor in SDMs. We develop a modular structure for SDMs that allows us to test the effect of scale in both single- and multi-scale settings. Furthermore, our model enables different scales to be considered for different modalities, using a late fusion approach. Results on the GeoLifeCLEF 2023 benchmark indicate that considering multimodal data and learning multi-scale representations leads to more accurate models., Comment: Published at the CV4Ecology workshop at ECCV 2024 (https://cv4e.netlify.app/papers/06.pdf)

In this paper, we propose a novel controller design approach for unknown nonlinear systems using the Koopman operator. In particular, we use the recently proposed stability- and certificate-oriented extended dynamic mode decomposition (SafEDMD) architecture to generate a data-driven bilinear surrogate model with certified error bounds. Then, by accounting for the obtained error bounds in a controller design based on the bilinear system, one can guarantee closed-loop stability for the true nonlinear system. While existing approaches over-approximate the bilinearity of the surrogate model, thus introducing conservatism and providing only local guarantees, we explicitly account for the bilinearity by using sum-of-squares (SOS) optimization in the controller design. More precisely, we parametrize a rational controller stabilizing the error-affected bilinear surrogate model and, consequently, the underlying nonlinear system. The resulting SOS optimization problem provides explicit data-driven controller design conditions for unknown nonlinear systems based on semidefinite programming. Our approach significantly reduces conservatism by establishing a larger region of attraction and improved data efficiency. The proposed method is evaluated using numerical examples, demonstrating its advantages over existing approaches.

A proliferation of Large Language Models (the GPT series, BLOOM, LLaMA, and more) are driving forward novel development of multipurpose AI for a variety of tasks, particularly natural language processing (NLP) tasks. These models demonstrate strong performance on a range of tasks; however, there has been evidence of brittleness when applied to more niche or narrow domains where hallucinations or fluent but incorrect responses reduce performance. Given the complex nature of scientific domains, it is prudent to investigate the trade-offs of leveraging off-the-shelf versus more targeted foundation models for scientific domains. In this work, we examine the benefits of in-domain pre-training for a given scientific domain, chemistry, and compare these to open-source, off-the-shelf models with zero-shot and few-shot prompting. Our results show that not only do in-domain base models perform reasonably well on in-domain tasks in a zero-shot setting but that further adaptation using instruction fine-tuning yields impressive performance on chemistry-specific tasks such as named entity recognition and molecular formula generation.

Ensuring robust model performance across diverse real-world scenarios requires addressing both transportability across domains with covariate shifts and extrapolation beyond observed data ranges. However, there is no formal procedure for statistically evaluating generalizability in machine learning algorithms, particularly in causal inference. Existing methods often rely on arbitrary metrics like AUC or MSE and focus predominantly on toy datasets, providing limited insights into real-world applicability. To address this gap, we propose a systematic and quantitative framework for evaluating model generalizability under covariate distribution shifts, specifically within causal inference settings. Our approach leverages the frugal parameterization, allowing for flexible simulations from fully and semi-synthetic benchmarks, offering comprehensive evaluations for both mean and distributional regression methods. By basing simulations on real data, our method ensures more realistic evaluations, which is often missing in current work relying on simplified datasets. Furthermore, using simulations and statistical testing, our framework is robust and avoids over-reliance on conventional metrics. Grounded in real-world data, it provides realistic insights into model performance, bridging the gap between synthetic evaluations and practical applications., Comment: 17 pages, 10 figures, Under review at AISTATS 2025

Understanding how information can efficiently spread in distributed systems under noisy communications is a fundamental question in both biological research and artificial system design. When agents are able to control whom they interact with, noise can often be mitigated through redundancy or other coding techniques, but it may have fundamentally different consequences on well-mixed systems. Specifically, Boczkowski et al. (2018) considered the noisy $\mathcal{PULL}(h)$ model, where each message can be viewed as any other message with probability $\delta$. The authors proved that in this model, the basic task of propagating a bit value from a single source to the whole population requires $\Omega(\frac{n\delta}{h(1-\delta|\Sigma|)^2})$ (parallel) rounds. The current work shows that the aforementioned lower bound is almost tight. In particular, when each agent observes all other agents in each round, which relates to scenarios where each agent senses the system's average tendency, information spreading can reliably be achieved in $\mathcal{O}(\log n)$ time, assuming constant noise. We present two simple and highly efficient protocols, thus suggesting their applicability to real-life scenarios. Notably, they also work in the presence of multiple conflicting sources and efficiently converge to their plurality opinion. The first protocol we present uses 1-bit messages but relies on a simultaneous wake-up assumption. By increasing the message size to 2 bits and removing the speedup in the information spreading time that may result from having multiple sources, we also present a simple and highly efficient self-stabilizing protocol that avoids the simultaneous wake-up requirement. Overall, our results demonstrate how, under stochastic communication, increasing the sample size can compensate for the lack of communication structure by linearly accelerating information spreading time., Comment: 41 pages, 1 figure

Fully supervised deep models have shown promising performance for many medical segmentation tasks. Still, the deployment of these tools in clinics is limited by the very timeconsuming collection of manually expert-annotated data. Moreover, most of the state-ofthe-art models have been trained and validated on moderately homogeneous datasets. It is known that deep learning methods are often greatly degraded by domain or label shifts and are yet to be built in such a way as to be robust to unseen data or label distributions. In the clinical setting, this problematic is particularly relevant as the deployment institutions may have different scanners or acquisition protocols than those from which the data has been collected to train the model. In this work, we propose to address these two challenges on the detection of clinically significant prostate cancer (csPCa) from bi-parametric MRI. We evaluate the method proposed by (Kervadec et al., 2018), which introduces a size constaint loss to produce fine semantic cancer lesions segmentations from weak circle scribbles annotations. Performance of the model is based on two public (PI-CAI and Prostate158) and one private databases. First, we show that the model achieves on-par performance with strong fully supervised baseline models, both on in-distribution validation data and unseen test images. Second, we observe a performance decrease for both fully supervised and weakly supervised models when tested on unseen data domains. This confirms the crucial need for efficient domain adaptation methods if deep learning models are aimed to be deployed in a clinical environment. Finally, we show that ensemble predictions from multiple trainings increase generalization performance.

Recent developments in Multiple-Input-Multiple-Output (MIMO) technology include packing a large number of antenna elements in a compact array to access the bandwidth benefits provided by higher mutual coupling (MC). The resulting super-wideband (SW) systems require a circuit-theoretic framework to handle the MC and channel models which span extremely large bands. Hence, in this paper, we make two key contributions. First, we develop a physically-consistent Rician channel model for use with SW systems. Secondly, we express the circuit-theoretic models in terms of a standard MIMO model, so that insights into the effects of antenna layouts, MC, and bandwidth can be made using standard communication theory. For example, we show the bandwidth widening resulting from the new channel model. In addition, we show that MC distorts line-of-sight paths which has beamforming implications. We also highlight the interaction between spatial correlation and MC and show that tight coupling reduces spatial correlations at low frequencies., Comment: This paper has been submitted to the IEEE for possible publication

The eROSITA instrument on board Spectrum-Roentgen-Gamma has completed four scans of the X-ray sky, leading to the detection of almost one million X-ray sources in eRASS1 only, including multiple new X-ray binary candidates. We report on analysis of the X-ray binary 1eRASS J085039.9-421151, using a ~55\,ks long NuSTAR observation, following its detection in each eROSITA scan. Analysis of the eROSITA and NuSTAR X-ray spectra in combination with X-shooter data of the optical counterpart provide evidence of an X-ray binary with a red supergiant (RSG) companion, confirming previous results, although we determine a cooler spectral type of M2-3, owing to the presence of TiO bands in the optical and near infrared spectra. The X-ray spectrum is well-described by an absorbed power law with a high energy cutoff typically applied for accreting high mass X-ray binaries. In addition, we detect a strong fluorescent neutral iron line with an equivalent width of ~700\,eV and an absorption edge, the latter indicating strong absorption by a partial covering component. It is unclear if the partial absorber is ionised. There is no significant evidence of a cyclotron resonant scattering feature. We do not detect any pulsations in the NuSTAR lightcurves, possibly on account of a large spin period that goes undetected due to insufficient statistics at low frequencies or potentially large absorption that causes pulsations to be smeared out. Even so, the low persistent luminosity, the spectral parameters observed (photon index, $\Gamma<1.0$), and the minuscule likelihood of detection of RSG-black hole systems, suggest that the compact object is a neutron star., Comment: 17 pages, 11 figures, accepted by A&A

This survey paper explores the transformative role of Machine Learning (ML) and Artificial Intelligence (AI) in Cardiopulmonary Resuscitation (CPR). It examines the evolution from traditional CPR methods to innovative ML-driven approaches, highlighting the impact of predictive modeling, AI-enhanced devices, and real-time data analysis in improving resuscitation outcomes. The paper provides a comprehensive overview, classification, and critical analysis of current applications, challenges, and future directions in this emerging field.

Investigating the marginal causal effect of an intervention on an outcome from complex data remains challenging due to the inflexibility of employed models and the lack of complexity in causal benchmark datasets, which often fail to reproduce intricate real-world data patterns. In this paper we introduce Frugal Flows, a novel likelihood-based machine learning model that uses normalising flows to flexibly learn the data-generating process, while also directly inferring the marginal causal quantities from observational data. We propose that these models are exceptionally well suited for generating synthetic data to validate causal methods. They can create synthetic datasets that closely resemble the empirical dataset, while automatically and exactly satisfying a user-defined average treatment effect. To our knowledge, Frugal Flows are the first generative model to both learn flexible data representations and also exactly parameterise quantities such as the average treatment effect and the degree of unobserved confounding. We demonstrate the above with experiments on both simulated and real-world datasets., Comment: 23 pages, 10 figures, Accepted as a Poster at NeurIPS 2024

We explore the gravitational-wave emission from head-on collisions of equal-mass solitonic boson-star binaries from simulations spanning a two-dimensional parameter space, consisting of the central scalar-field amplitude of the stars and the solitonic potential parameter. We report the gravitational-wave energies emitted by boson-star binaries which, due to their combination of moderately high compactness with significant deformability, we often find to be louder by up to an order of magnitude than analogous black-hole collisions. The dependence of the radiated energy on the boson-star parameters exhibits striking needle-sharp features and discontinuous jumps to the value emitted by black-hole binaries. We explain these features in terms of the solitonic potential and the stability properties of the respective individual stars., Comment: 17 pages, 10 figures

The discovery and identification of molecules in biological and environmental samples is crucial for advancing biomedical and chemical sciences. Tandem mass spectrometry (MS/MS) is the leading technique for high-throughput elucidation of molecular structures. However, decoding a molecular structure from its mass spectrum is exceptionally challenging, even when performed by human experts. As a result, the vast majority of acquired MS/MS spectra remain uninterpreted, thereby limiting our understanding of the underlying (bio)chemical processes. Despite decades of progress in machine learning applications for predicting molecular structures from MS/MS spectra, the development of new methods is severely hindered by the lack of standard datasets and evaluation protocols. To address this problem, we propose MassSpecGym -- the first comprehensive benchmark for the discovery and identification of molecules from MS/MS data. Our benchmark comprises the largest publicly available collection of high-quality labeled MS/MS spectra and defines three MS/MS annotation challenges: \textit{de novo} molecular structure generation, molecule retrieval, and spectrum simulation. It includes new evaluation metrics and a generalization-demanding data split, therefore standardizing the MS/MS annotation tasks and rendering the problem accessible to the broad machine learning community. MassSpecGym is publicly available at \url{https://github.com/pluskal-lab/MassSpecGym}.

The decarbonisation of the energy system is crucial for achieving climate goals and is inherently linked to the decarbonisation of industry. Despite this, few studies explore the simultaneous impacts of decarbonising both sectors. This paper aims to examine how industrial decarbonisation in Europe affects the energy system and vice versa. To address this, an industry model incorporating key heavy industry sectors across six European countries is combined with an energy system model for electricity and hydrogen covering fifteen European regions, refered to as the EU-15, divided into eleven zones. The study evaluates various policy scenarios under different conditions.The results demonstrate that industrial decarbonisation leads to a significant increase in electricity and hydrogen demand. This additional demand for electricity is largely met through renewable energy sources, while hydrogen supply is predominantly addressed by blue hydrogen production when fossil fuels are authorized and the system lacks renewable energy. This increased demand results in higher prices with considerable regional disparities. Furthermore, the findings reveal that, regardless of the scenario, the electricity mix in the EU-15 remains predominantly renewable, exceeding 85%.A reduction in carbon taxes lowers the prices of electricity and hydrogen, but does not increase consumption, as the lower carbon tax makes the continued use of fossil fuels more attractive to industry. In scenarios that enforce a phase-out of fossil fuels, electricity prices rise, leading to a greater reliance on imports of low-carbon hydrogen and methanol. Results also suggest that domestic hydrogen production benefits from synergies between electrolytic hydrogen and blue hydrogen, helping to maintain competitive prices.

Understanding the physics of the integrable spin-1/2 XXZ chain has witnessed substantial progress, due to the development and application of sophisticated analytical and numerical techniques. In particular, infinite-temperature magnetization transport has turned out to range from ballistic, over superdiffusive, to diffusive behavior in different parameter regimes of the anisotropy. Since integrability is rather the exception than the rule, a crucial question is the change of transport under integrability-breaking perturbations. This question includes the stability of superdiffusion at the isotropic point and the change of diffusion constants in the easy-axis regime. In our work, we study this change of diffusion constants by a variety of methods and cover both, linear response theory in the closed system and the Lindblad equation in the open system, where we throughout focus on periodic boundary conditions. In the closed system, we compare results from the recursion method to calculations for finite systems and find evidence for a continuous change of diffusion constants over the full range of perturbation strengths. In the open system weakly coupled to baths, we find diffusion constants in quantitative agreement with the ones in the closed system in a range of nonweak perturbations, but disagreement in the limit of weak perturbations. Using a simple model in this limit, we point out the possibility of a diverging diffusion constant in such an open system., Comment: 6 pages, 4 figures (+ 5 pages, 7 figures)

We introduce CODES, a benchmark for comprehensive evaluation of surrogate architectures for coupled ODE systems. Besides standard metrics like mean squared error (MSE) and inference time, CODES provides insights into surrogate behaviour across multiple dimensions like interpolation, extrapolation, sparse data, uncertainty quantification and gradient correlation. The benchmark emphasizes usability through features such as integrated parallel training, a web-based configuration generator, and pre-implemented baseline models and datasets. Extensive documentation ensures sustainability and provides the foundation for collaborative improvement. By offering a fair and multi-faceted comparison, CODES helps researchers select the most suitable surrogate for their specific dataset and application while deepening our understanding of surrogate learning behaviour., Comment: 12 pages, 10 figures, accepted for the Machine Learning and the Physical Sciences workshop at NeurIPS 2024, source code available on GitHub at https://github.com/robin-janssen/CODES-Benchmark

As the role of variable renewables in electricity markets expands, new market products help system operators manage imbalances caused by uncertainty and variability. Whereas work in the last decade has focused on constructing demand curves for central procurement of those products, little attention has been paid to designing their settlement scheme and understanding the connections between the economic value of these products, the schedule of variable resources, and the cost of flexibility. In this article, we compare a new product called Flexibility Options, which addresses these gaps, with a traditional reserve product using a case study similar to the 2019 Texas (ERCOT) system. Our findings suggest that both products are equally effective in managing imbalances, but Flexibility Options have superior risk management properties and keep the system operator revenue adequate.

Reconfigurable intelligent surfaces (RISs) offer enhanced control over propagation through phase and amplitude manipulation but face practical challenges like cost and power usage, especially at high frequencies. This is specifically a major problem at high frequencies (Ka- and V-band) where the high cost of semiconductor components (i.e., diodes, varactors, MEMSs) can make RISs prohibitively costly. In recent years, it is shown that liquid crystals (LCs) are low-cost and low-energy alternative which can address the aforementioned challenges but at the cost of lower response time. In LiquiRIS, we enable leveraging LC-based RIS in mobile networks. Specifically, we devise techniques that minimize the beam switching time of LC-based RIS by tapping into the physical properties of LCs and the underlying mathematical principles of beamforming. We achieve this by modeling and optimizing the beamforming vector to account for the rotation characteristics of LC molecules to reduce their transition time from one state to another. In addition to prototyping the proposed system, we show via extensive experimental analysis that LiquiRIS substantially reduces the response time (up to 70.80%) of liquid crystal surface (LCS).

The detection of low abundances of $^{135}$Cs in environmental samples is of significant interest in different fields of environmental sciences, especially in combination with its shorter-lived sister isotope $^{137}$Cs. The method of Ion-Laser InterAction Mass Spectrometry (ILIAMS) for barium separation at the Vienna Environmental Research Accelerator (VERA) was investigated and further improved for low abundance cesium detection. The difluorides BaF$_2^-$ and CsF$_2^-$ differ in their electron detachment energies and make isobar suppression with ILIAMS by more than 7 orders of magnitude possible. By this method, samples with ratios down to the order of $^{135,137}$Cs/$^{133}$Cs $\approx 10^{-11}$ are measurable and the $^{135}$Cs/$^{137}$Cs ratios of first environmental samples were determined by AMS., Comment: We acknowledge funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101008324 (ChETEC-INFRA)

The presence of variable renewable energy resources with uncertain outputs in day-ahead electricity markets results in additional balancing needs in real-time. Addressing those needs cost-effectively and reliably within a competitive market with unbundled products is challenging as both the demand for and the availability of flexibility depends on day-ahead energy schedules. Existing approaches for reserve procurement usually rely either on oversimplified demand curves that do not consider how system conditions that particular day affect the value of flexibility, or on bilateral trading of hedging instruments that are not co-optimized with day-ahead schedules. This article proposes a new product, `Flexibility Options', to address these two limitations. The demand for this product is endogenously determined in the day-ahead market and it is met cost-effectively by considering real-time supply curves for product providers, which are co-optimized with the energy supply. As we illustrate with numerical examples and mathematical analysis, the product addresses the hedging needs of participants with imbalances, provides a less intermittent revenue stream for participants with flexible outputs, promotes convergence between day-ahead and real-time energy prices, and ensures that the system operator is revenue-neutral. This article provides a comprehensive design that can be further tested and applied in large-scale systems.

This article proposes a simple formalism which unifies Hamiltonian simulation techniques from different fields. This formalism leads to a competitive method to construct the Hamiltonian simulation with a comprehensible, simple-to-implement circuit generation technique. It leads to a gate decomposition and a scaling different from the usual strategy based on a Linear Combination of Unitaries (LCU) reformulation of the problem. It can significantly reduce the quantum circuit number of rotational gates, multi-qubit gates, and the circuit depth. This method leads to one exact Hamiltonian simulation for each summed term and Trotter step. Each of these Hamiltonian simulation unitary matrices also allows the construction of the non-exponential terms with a maximum of six unitary matrices to be Block-encoding (BE). The formalism is easy to apply to the widely studied High-order Unconstrained Binary Optimization (HUBO), fermionic transition Hamiltonian, and basic finite difference method instances. For the HUBO, our implementation exponentially reduces the number of gates for high-order cost functions with respect to the HUBO order. The individual electronic transitions are implemented without error for the second-quantization Fermionic Hamiltonian. Finite difference proposed matrix decompositions are straightforward, very versatile, and scale as the state-of-the-art proposals.

Magnetic Resonance Spectroscopic Imaging (MRSI) is a non-invasive imaging technique for studying metabolism and has become a crucial tool for understanding neurological diseases, cancers and diabetes. High spatial resolution MRSI is needed to characterize lesions, but in practice MRSI is acquired at low resolution due to time and sensitivity restrictions caused by the low metabolite concentrations. Therefore, there is an imperative need for a post-processing approach to generate high-resolution MRSI from low-resolution data that can be acquired fast and with high sensitivity. Deep learning-based super-resolution methods provided promising results for improving the spatial resolution of MRSI, but they still have limited capability to generate accurate and high-quality images. Recently, diffusion models have demonstrated superior learning capability than other generative models in various tasks, but sampling from diffusion models requires iterating through a large number of diffusion steps, which is time-consuming. This work introduces a Flow-based Truncated Denoising Diffusion Model (FTDDM) for super-resolution MRSI, which shortens the diffusion process by truncating the diffusion chain, and the truncated steps are estimated using a normalizing flow-based network. The network is conditioned on upscaling factors to enable multi-scale super-resolution. To train and evaluate the deep learning models, we developed a 1H-MRSI dataset acquired from 25 high-grade glioma patients. We demonstrate that FTDDM outperforms existing generative models while speeding up the sampling process by over 9-fold compared to the baseline diffusion model. Neuroradiologists' evaluations confirmed the clinical advantages of our method, which also supports uncertainty estimation and sharpness adjustment, extending its potential clinical applications., Comment: Accepted by Medical Image Analysis (MedIA)

Recent studies of human migration have focused on modern issues of international economics, politics, urbanization, or commuting. Here we make use of very large anonymized genealogies which offer quantitative metrics and models before census data became available. In European and North American data from 1400 to 1950 we find two distinct patterns of lifetime migration. The steady-state pattern shows a universal power-law distribution of migration distance; by its early appearance it cannot be dependent on post-industrial technology. The frontier pattern, in contrast, is not scale-free with its much longer average distances. All migration distances are well fit by a three parameter model; the temporal and geographic patterns of the fitted parameters give new insight to American internal expansion 1620-1950. Frontier migration is also highly directional and asymmetric; gravity models do not apply. The American frontier pattern arose from the colonial-era steady-state within a generation, plateaued for three generations, then returned to a more mobile steady-state, a sequence paralleled by the Steppe migrations that brought the Bronze Age to Neolithic Europe. The transient frontier pattern is enabled by large-scale technological or numeric imbalance and geographic opportunity; when these forces abate, a new steady-state begins., Comment: 23 pages of manuscript, 18 pages of supplementary material, 7 figures, 1 table

Assuming X is a random vector and A a non-invertible matrix, one sometimes need to perform inference while only having access to samples of Y = AX. The corresponding likelihood is typically intractable. One may still be able to perform exact Bayesian inference using a pseudo-marginal sampler, but this requires an unbiased estimator of the intractable likelihood. We propose saddlepoint Monte Carlo, a method for obtaining an unbiased estimate of the density of Y with very low variance, for any model belonging to an exponential family. Our method relies on importance sampling of the characteristic function, with insights brought by the standard saddlepoint approximation scheme with exponential tilting. We show that saddlepoint Monte Carlo makes it possible to perform exact inference on particularly challenging problems and datasets. We focus on the ecological inference problem, where one observes only aggregates at a fine level. We present in particular a study of the carryover of votes between the two rounds of various French elections, using the finest available data (number of votes for each candidate in about 60,000 polling stations over most of the French territory). We show that existing, popular approximate methods for ecological inference can lead to substantial bias, which saddlepoint Monte Carlo is immune from. We also present original results for the 2024 legislative elections on political centre-to-left and left-to-centre conversion rates when the far-right is present in the second round. Finally, we discuss other exciting applications for saddlepoint Monte Carlo, such as dealing with aggregate data in privacy or inverse problems., Comment: 27 pages, 9 figures, 3 tables

In this work, we propose a large-graph limit estimate of the matching coverage for several matching algorithms, on general graphs generated by the configuration model. For a wide class of {\em local} matching algorithms, namely, algorithms that only use information on the immediate neighborhood of the explored nodes, we propose a joint construction of the graph by the configuration model, and of the resulting matching on the latter graph. This leads to a generalization in infinite dimension of the differential equation method of Wormald: We keep track of the matching algorithm over time by a measure-valued CTMC, for which we prove the convergence, to the large-graph limit, to a deterministic hydrodynamic limit, identified as the unique solution of a system of ODE's in the space of integer measures. Then, the asymptotic proportion of nodes covered by the matching appears as a simple function of that solution. We then make this solution explicit for three particular local algorithms: the classical {\sc greedy} algorithm, and then the {\sc uni-min} and {\sc uni-max} algorithms, two variants of the greedy algorithm that select, as neighbor of any explored node, its neighbor having the least (respectively largest) residual degree.

From a monarchy to a democracy, to a dictatorship and back to a democracy -- the German political landscape has been constantly changing ever since the first German national state was formed in 1871. After World War II, the Federal Republic of Germany was formed in 1949. Since then every plenary session of the German Bundestag was logged and even has been digitized over the course of the last few years. We analyze these texts using a time series variant of the topic model LDA to investigate which events had a lasting effect on the political discourse and how the political topics changed over time. This allows us to detect changes in word frequency (and thus key discussion points) in political discourse., Comment: 7 pages, 6 figures

The application of natural language processing on political texts as well as speeches has become increasingly relevant in political sciences due to the ability to analyze large text corpora which cannot be read by a single person. But such text corpora often lack critical meta information, detailing for instance the party, age or constituency of the speaker, that can be used to provide an analysis tailored to more fine-grained research questions. To enable researchers to answer such questions with quantitative approaches such as natural language processing, we provide the SpeakGer data set, consisting of German parliament debates from all 16 federal states of Germany as well as the German Bundestag from 1947-2023, split into a total of 10,806,105 speeches. This data set includes rich meta data in form of information on both reactions from the audience towards the speech as well as information about the speaker's party, their age, their constituency and their party's political alignment, which enables a deeper analysis. We further provide three exploratory analyses, detailing topic shares of different parties throughout time, a descriptive analysis of the development of the age of an average speaker as well as a sentiment analysis of speeches of different parties with regards to the COVID-19 pandemic., Comment: 10 pages, 3 figures

Low-mass emission-line stars belong to various evolutionary stages, from pre-main-sequence young stars to evolved stars. In this work, we present a catalog of late-type (F0 to M9) emission-line stars from the LAMOST Data Release 6. Using the scipy package, we created a Python code that finds the emission peak at H-alpha in all late-type stellar spectra. A dataset of 38,152 late-type emission-line stars was obtained after a rigorous examination of the photometric quality flags and the signal-to-noise ratio of the spectra. Adopting well-known photometric and spectroscopic methods, we classified our sample into 438 infrared excess sources, 4,669 post-main-sequence candidates, 9,718 Fe/Ge/Ke sources, and 23,264 dMe sources. From a cross-match with known databases, we found that 29,222 sources, comprising 65 IR excess sources, 7,899 Fe/Ge/Ke stars, 17,533 dMe stars, and 3,725 PtMS candidates, are new detections. We measured the equivalent width of the major emission lines observed in the spectra of our sample of emission-line stars. Furthermore, the trend observed in the line strengths of major emission lines over the entire late-type spectral range is analyzed. We further classified the sample into 4 groups based on the presence of Hydrogen and Calcium emission lines. This work presents a large dataset of late-type emission-line stars, which can be used to study active phenomena in late-type stars., Comment: Accepted for publication in RAA, 24 pages, 12 figures, 2 tables

High-fidelity mid-circuit measurements, which read out the state of specific qubits in a multiqubit processor without destroying them or disrupting their neighbors, are a critical component for useful quantum computing. They enable fault-tolerant quantum error correction, dynamic circuits, and other paths to solving classically intractable problems. But there are almost no methods to assess their performance comprehensively. We address this gap by introducing the first randomized benchmarking protocol that measures the rate at which mid-circuit measurements induce errors in many-qubit circuits. Using this protocol, we detect and eliminate previously undetected measurement-induced crosstalk in a 20-qubit trapped-ion quantum computer. Then, we use the same protocol to measure the rate of measurement-induced crosstalk error on a 27-qubit IBM Q processor, and quantify how much of that error is eliminated by dynamical decoupling., Comment: 7 pages, 4 figures, plus supplemental notes

The Central Molecular Zone (CMZ) is the largest reservoir of dense molecular gas in the Galaxy and is heavily obscured in the optical and near-IR. We present an overview of the far-IR dust continuum, where the molecular clouds are revealed, provided by Herschel in the inner 40\deg($|l| <$ 20\deg) of the Milky Way with a particular focus on the CMZ. We report a total dense gas ($N$(H$_2$) $> 10^{23}$ cm$^{-2}$) CMZ mass of M=$2\substack{+2 \\ -1} \times 10^7$ M$_{\odot}$ and confirm that there is a highly asymmetric distribution of dense gas, with about 70-75% at positive longitudes. We create and publicly release complete fore/background-subtracted column density and dust temperature maps in the inner 40\deg ($|l| <$ 20\deg) of the Galaxy. We find that the CMZ clearly stands out as a distinct structure, with an average mass per longitude that is at least $3\times$ higher than the rest of the inner Galaxy contiguously from 1.8\deg $> \ell >$ -1.3\deg. This CMZ extent is larger than previously assumed, but is consistent with constraints from velocity information. The inner Galaxy's column density peaks towards the SgrB2 complex with a value of about 2 $\times$ 10$^{24}$ cm$^{-2}$, and typical CMZ molecular clouds are about N(H$_2$)=10$^{23}$ cm$^{-2}$. Typical CMZ dust temperatures range from about $12-35$ K with relatively little variation. We identify a ridge of warm dust in the inner CMZ that potentially traces the base of the northern Galactic outflow seen with MEERKAT., Comment: Submitted to ApJ, 16 pages, project website: https://centralmolecularzone.github.io/3D_CMZ/

The Central Molecular Zone (CMZ) is the way station at the heart of our Milky Way Galaxy, connecting gas flowing in from Galactic scales with the central nucleus. Key open questions remain about its 3-D structure, star formation properties, and role in regulating this gas inflow. In this work, we identify a hierarchy of discrete structures in the CMZ using column density and dust temperature maps from Paper I (Battersby et al., submitted). We calculate the physical ($N$(H$_2$), $T_{\rm{dust}}$, mass, radius) and kinematic (HNCO, HCN, and HC$_3$N moments) properties of each structure as well as their bolometric luminosities and star formation rates (SFRs). We compare these properties with regions in the Milky Way disk and external galaxies. We perform power-law fits to the column density probability distribution functions (N-PDFs) of the inner 100 pc, SgrB2, and the outer 100 pc of the CMZ as well as several individual molecular cloud structures and find generally steeper power-law slopes ($-9<\alpha<-2$) compared with the literature ($-6 < \alpha < -1$). We find that individual CMZ structures require a large external pressure ($P_e$/k$_B$ $> 10^{7-9}$ K cm$^{-3}$) to be considered bound. Despite the fact that the CMZ overall is well below the Gao-Solomon dense gas star-formation relation (and in modest agreement with the Schmidt-Kennicutt relation), individual structures on the scale of molecular clouds generally follow these star-formation relations and agree well with other Milky Way and extragalactic regions., Comment: Submitted to ApJ, 22 pages, project website: https://centralmolecularzone.github.io/3D_CMZ/

The Milky Way's Central Molecular Zone (CMZ) is the largest concentration of dense molecular gas in the Galaxy, the structure of which is shaped by the complex interplay between Galactic-scale dynamics and extreme physical conditions. Understanding the 3-D geometry of this gas is crucial as it determines the locations of star formation and subsequent feedback. We present a catalogue of clouds in the CMZ using Herschel data. Using archival data from the APEX and MOPRA CMZ surveys, we measure averaged kinematic properties of the clouds at 1mm and 3mm. We use archival ATCA data of the H$_{2}$CO (1$_{1,0}$ - 1$_{1,1}$) 4.8 GHz line to search for absorption towards the clouds, and 4.85 GHz GBT C-band data to measure the radio continuum emission. We measure the absorption against the continuum to provide new constraints for the line-of-sight positions of the clouds relative to the Galactic centre, and find a highly asymmetric distribution, with most clouds residing in front of the Galactic centre. The results are compared with different orbital models, and we introduce a revised toy model of a vertically-oscillating closed elliptical orbit. We find that most models describe the PPV structure of the gas reasonably well, but find significant inconsistencies in all cases regarding the near vs. far placement of individual clouds. Our results highlight that the CMZ is likely more complex than can be captured by these simple geometric models, along with the need for new data to provide further constraints on the true 3-D structure of the CMZ., Comment: Submitted to ApJ, 30 pages, project website: https://centralmolecularzone.github.io/3D_CMZ/

We propose a general pipeline to automate the extraction of labels from radiology reports using large language models, which we validate on spinal MRI reports. The efficacy of our labelling method is measured on five distinct conditions: spinal cancer, stenosis, spondylolisthesis, cauda equina compression and herniation. Using open-source models, our method equals or surpasses GPT-4 on a held-out set of reports. Furthermore, we show that the extracted labels can be used to train imaging models to classify the identified conditions in the accompanying MR scans. All classifiers trained using automated labels achieve comparable performance to models trained using scans manually annotated by clinicians. Code can be found at https://github.com/robinyjpark/AutoLabelClassifier., Comment: Accepted to Medical Image Computing and Computer Assisted Intervention (MICCAI 2024, Spotlight). 11 pages plus appendix

We describe observations of low energy excess (LEE) events (background events observed in all light dark matter direct detection calorimeters) and noise in a two-channel silicon athermal phonon detector with 375 meV baseline energy resolution. We measure two distinct LEE populations: ``shared'' multichannel events with a pulse shape consistent with athermal phonon events, and sub-eV events which couple nearly exclusively to a single channel with a significantly faster pulse shape. These ``singles'' are consistent with events occurring within the aluminum athermal phonon collection fins. Similarly, our measured detector noise is higher than the theoretical expectation. Measured noise can be split into an uncorrelated component, consistent with shot noise from small energy depositions within the athermal phonon sensor itself, and a correlated component, consistent with shot noise from energy depositions within the silicon crystal's phonon system., Comment: 6 pages, 5 figures

Restricted Boltzmann machines (RBM) are generative models capable to learn data with a rich underlying structure. We study the teacher-student setting where a student RBM learns structured data generated by a teacher RBM. The amount of structure in the data is controlled by adjusting the number of hidden units of the teacher and the correlations in the rows of the weights, a.k.a. patterns. In the absence of correlations, we validate the conjecture that the performance is independent of the number of teacher patters and hidden units of the student RBMs, and we argue that the teacher-student setting can be used as a toy model for studying the lottery ticket hypothesis. Beyond this regime, we find that the critical amount of data required to learn the teacher patterns decreases with both their number and correlations. In both regimes, we find that, even with an relatively large dataset, it becomes impossible to learn the teacher patterns if the inference temperature used for regularization is kept too low. In our framework, the student can learn teacher patterns one-to-one or many-to-one, generalizing previous findings about the teacher-student setting with two hidden units to any arbitrary finite number of hidden units., Comment: 51 pages, 21 figures