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We consider a boundary value problem (BVP) modelling one-dimensional heat-conduction with radiation, which is derived from the Stefan-Boltzmann law. The problem strongly depends on the parameters, making difficult to estimate the solution. We use an analytical approach to determine upper and lower bounds to the exact solution of the BVP, which allows estimating the latter. Finally, we support our theoretical arguments with numerical data, by implementing them into the MAPLE computer program.

We assume to be given structural equations over discrete variables inducing a directed acyclic graph, namely, a structural causal model, together with data about its internal nodes. The question we want to answer is how we can compute bounds for partially identifiable counterfactual queries from such an input. We start by giving a map from structural casual models to credal networks. This allows us to compute exact counterfactual bounds via algorithms for credal nets on a subclass of structural causal models. Exact computation is going to be inefficient in general given that, as we show, causal inference is NP-hard even on polytrees. We target then approximate bounds via a causal EM scheme. We evaluate their accuracy by providing credible intervals on the quality of the approximation; we show through a synthetic benchmark that the EM scheme delivers accurate results in a fair number of runs. In the course of the discussion, we also point out what seems to be a neglected limitation to the trending idea that counterfactual bounds can be computed without knowledge of the structural equations. We also present a real case study on palliative care to show how our algorithms can readily be used for practical purposes.

The log-conformation formulation, although highly successful, was from the beginning formulated as a partial differential equation that contains an, for PDEs unusual, eigenvalue decomposition of the unknown field. To this day, most numerical implementations have been based on this or a similar eigenvalue decomposition, with Knechtges et al. (2014) being the only notable exception for two-dimensional flows. In this paper, we present an eigenvalue-free algorithm to compute the constitutive equation of the log-conformation formulation that works for two- and three-dimensional flows. Therefore, we first prove that the challenging terms in the constitutive equations are representable as a matrix function of a slightly modified matrix of the log-conformation field. We give a proof of equivalence of this term to the more common log-conformation formulations. Based on this formulation, we develop an eigenvalue-free algorithm to evaluate this matrix function. The resulting full formulation is first discretized using a finite volume method, and then tested on the confined cylinder and sedimenting sphere benchmarks.

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

Student attention is an indispensable input for uncovering their goals, intentions, and interests, which prove to be invaluable for a multitude of research areas, ranging from psychology to interactive systems. However, most existing methods to classify attention fail to model its complex nature. To bridge this gap, we propose AttentioNet, a novel Convolutional Neural Network-based approach that utilizes Electroencephalography (EEG) data to classify attention into five states: Selective, Sustained, Divided, Alternating, and relaxed state. We collected a dataset of 20 subjects through standard neuropsychological tasks to elicit different attentional states. The average across-student accuracy of our proposed model at this configuration is 92.3% (SD=3.04), which is well-suited for end-user applications. Our transfer learning-based approach for personalizing the model to individual subjects effectively addresses the issue of individual variability in EEG signals, resulting in improved performance and adaptability of the model for real-world applications. This represents a significant advancement in the field of EEG-based classification. Experimental results demonstrate that AttentioNet outperforms a popular EEGnet baseline (p-value < 0.05) in both subject-independent and subject-dependent settings, confirming the effectiveness of our proposed approach despite the limitations of our dataset. These results highlight the promising potential of AttentioNet for attention classification using EEG data.

Conditional copulas are useful tools for modeling the dependence between multiple response variables that may vary with a given set of predictor variables. Conditional dependence measures such as conditional Kendall's tau and Spearman's rho that can be expressed as functionals of the conditional copula are often used to evaluate the strength of dependence conditioning on the covariates. In general, semiparametric estimation methods of conditional copulas rely on an assumed parametric copula family where the copula parameter is assumed to be a function of the covariates. The functional relationship can be estimated nonparametrically using different techniques but it is required to choose an appropriate copula model from various candidate families. In this paper, by employing the empirical checkerboard Bernstein copula (ECBC) estimator we propose a fully nonparametric approach for estimating conditional copulas, which doesn't require any selection of parametric copula models. Closed-form estimates of the conditional dependence measures are derived directly from the proposed ECBC-based conditional copula estimator. We provide the large-sample consistency of the proposed estimator as well as the estimates of conditional dependence measures. The finite-sample performance of the proposed estimator and comparison with semiparametric methods are investigated through simulation studies. An application to real case studies is also provided.

Kernel ridge regression, KRR, is a generalization of linear ridge regression that is non-linear in the data, but linear in the parameters. The solution can be obtained either as a closed-form solution, which includes a matrix inversion, or iteratively through gradient descent. Using the iterative approach opens up for changing the kernel during training, something that is investigated in this paper. We theoretically address the effects this has on model complexity and generalization. Based on our findings, we propose an update scheme for the bandwidth of translational-invariant kernels, where we let the bandwidth decrease to zero during training, thus circumventing the need for hyper-parameter selection. We demonstrate on real and synthetic data how decreasing the bandwidth during training outperforms using a constant bandwidth, selected by cross-validation and marginal likelihood maximization. We also show theoretically and empirically that using a decreasing bandwidth, we are able to achieve both zero training error in combination with good generalization, and a double descent behavior, phenomena that do not occur for KRR with constant bandwidth but are known to appear for neural networks.

Machine-learned interatomic potentials (MLIPs) are typically trained on datasets that encompass a restricted subset of possible input structures, which presents a potential challenge for their generalization to a broader range of systems outside the training set. Nevertheless, MLIPs have demonstrated impressive accuracy in predicting forces and energies in simulations involving intricate and complex structures. In this paper we aim to take steps towards rigorously explaining the excellent observed generalisation properties of MLIPs. Specifically, we offer a comprehensive theoretical and numerical investigation of the generalization of MLIPs in the context of dislocation simulations. We quantify precisely how the accuracy of such simulations is directly determined by a few key factors: the size of the training structures, the choice of training observations (e.g., energies, forces, virials), and the level of accuracy achieved in the fitting process. Notably, our study reveals the crucial role of fitting virials in ensuring the consistency of MLIPs for dislocation simulations. Our series of careful numerical experiments encompassing screw, edge, and mixed dislocations, supports existing best practices in the MLIPs literature but also provides new insights into the design of data sets and loss functions.

The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.

While it is nearly effortless for humans to quickly assess the perceptual similarity between two images, the underlying processes are thought to be quite complex. Despite this, the most widely used perceptual metrics today, such as PSNR and SSIM, are simple, shallow functions, and fail to account for many nuances of human perception. Recently, the deep learning community has found that features of the VGG network trained on the ImageNet classification task has been remarkably useful as a training loss for image synthesis. But how perceptual are these so-called "perceptual losses"? What elements are critical for their success? To answer these questions, we introduce a new Full Reference Image Quality Assessment (FR-IQA) dataset of perceptual human judgments, orders of magnitude larger than previous datasets. We systematically evaluate deep features across different architectures and tasks and compare them with classic metrics. We find that deep features outperform all previous metrics by huge margins. More surprisingly, this result is not restricted to ImageNet-trained VGG features, but holds across different deep architectures and levels of supervision (supervised, self-supervised, or even unsupervised). Our results suggest that perceptual similarity is an emergent property shared across deep visual representations.