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The scattering of electromagnetic waves by three--dimensional periodic structures is important for many problems of crucial scientific and engineering interest. Due to the complexity and three-dimensional nature of these waves, the fast, accurate, and reliable numerical simulations of these are indispensable for engineers and scientists alike. For this, High Order Spectral methods are frequently employed and here we describe an algorithm in this class. Our approach is perturbative in nature where we view the deviation of the permittivity from a constant value as the deformation and we pursue regular perturbation theory. This work extends our previous contribution regarding the Helmholtz equation to the full vector Maxwell equations, by providing a rigorous analyticity theory, both in deformation size and spatial variable (provided that the permittivity is, itself, analytic).

Machine learning methods have significantly improved in their predictive capabilities, but at the same time they are becoming more complex and less transparent. As a result, explainers are often relied on to provide interpretability to these black-box prediction models. As crucial diagnostics tools, it is important that these explainers themselves are robust. In this paper we focus on one particular aspect of robustness, namely that an explainer should give similar explanations for similar data inputs. We formalize this notion by introducing and defining explainer astuteness, analogous to astuteness of prediction functions. Our formalism allows us to connect explainer robustness to the predictor's probabilistic Lipschitzness, which captures the probability of local smoothness of a function. We provide lower bound guarantees on the astuteness of a variety of explainers (e.g., SHAP, RISE, CXPlain) given the Lipschitzness of the prediction function. These theoretical results imply that locally smooth prediction functions lend themselves to locally robust explanations. We evaluate these results empirically on simulated as well as real datasets.

A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based representations, the misfit is instead minimized in a small subspace spanned by the first few eigenfunctions of a judicious elliptic operator, which itself depends on the previous iteration. By repeatedly adapting both the dimension and the basis of the search space, regularization is inherently incorporated at each iteration without the need for extra Tikhonov penalization. Convergence is proved under an angle condition, which is included into the resulting \emph{Adaptive Spectral Inversion} (ASI) algorithm. The ASI approach compares favorably to standard grid-based inversion using $L^2$-Tikhonov regularization when applied to an elliptic inverse problem. The improved accuracy resulting from the newly included angle condition is further demonstrated via numerical experiments from time-dependent inverse scattering problems.

Fine-grained classification is a particular case of a classification problem, aiming to classify objects that share the visual appearance and can only be distinguished by subtle differences. Fine-grained classification models are often deployed to determine animal species or individuals in automated animal monitoring systems. Precise visual explanations of the model's decision are crucial to analyze systematic errors. Attention- or gradient-based methods are commonly used to identify regions in the image that contribute the most to the classification decision. These methods deliver either too coarse or too noisy explanations, unsuitable for identifying subtle visual differences reliably. However, perturbation-based methods can precisely identify pixels causally responsible for the classification result. Fill-in of the dropout (FIDO) algorithm is one of those methods. It utilizes the concrete dropout (CD) to sample a set of attribution masks and updates the sampling parameters based on the output of the classification model. A known problem of the algorithm is a high variance in the gradient estimates, which the authors have mitigated until now by mini-batch updates of the sampling parameters. This paper presents a solution to circumvent these computational instabilities by simplifying the CD sampling and reducing reliance on large mini-batch sizes. First, it allows estimating the parameters with smaller mini-batch sizes without losing the quality of the estimates but with a reduced computational effort. Furthermore, our solution produces finer and more coherent attribution masks. Finally, we use the resulting attribution masks to improve the classification performance of a trained model without additional fine-tuning of the model.

We establish a connection between stochastic optimal control and generative models based on stochastic differential equations (SDEs), such as recently developed diffusion probabilistic models. In particular, we derive a Hamilton-Jacobi-Bellman equation that governs the evolution of the log-densities of the underlying SDE marginals. This perspective allows to transfer methods from optimal control theory to generative modeling. First, we show that the evidence lower bound is a direct consequence of the well-known verification theorem from control theory. Further, we can formulate diffusion-based generative modeling as a minimization of the Kullback-Leibler divergence between suitable measures in path space. Finally, we develop a novel diffusion-based method for sampling from unnormalized densities -- a problem frequently occurring in statistics and computational sciences. We demonstrate that our time-reversed diffusion sampler (DIS) can outperform other diffusion-based sampling approaches on multiple numerical examples.

Most applications of Artificial Intelligence (AI) are designed for a confined and specific task. However, there are many scenarios that call for a more general AI, capable of solving a wide array of tasks without being specifically designed for them. The term General-Purpose Artificial Intelligence Systems (GPAIS) has been defined to refer to these AI systems. To date, the possibility of an Artificial General Intelligence, powerful enough to perform any intellectual task as if it were human, or even improve it, has remained an aspiration, fiction, and considered a risk for our society. Whilst we might still be far from achieving that, GPAIS is a reality and sitting at the forefront of AI research. This work discusses existing definitions for GPAIS and proposes a new definition that allows for a gradual differentiation among types of GPAIS according to their properties and limitations. We distinguish between closed-world and open-world GPAIS, characterising their degree of autonomy and ability based on several factors such as adaptation to new tasks, competence in domains not intentionally trained for, ability to learn from few data, or proactive acknowledgment of their own limitations. We then propose a taxonomy of approaches to realise GPAIS, describing research trends such as the use of AI techniques to improve another AI or foundation models. As a prime example, we delve into generative AI, aligning them with the terms and concepts presented in the taxonomy. Through the proposed definition and taxonomy, our aim is to facilitate research collaboration across different areas that are tackling general-purpose tasks, as they share many common aspects. Finally, we discuss the current state of GPAIS, its challenges and prospects, implications for our society, and the need for responsible and trustworthy AI systems and regulation, with the goal of providing a holistic view of GPAIS.

Within the field of Requirements Engineering (RE), the increasing significance of Explainable Artificial Intelligence (XAI) in aligning AI-supported systems with user needs, societal expectations, and regulatory standards has garnered recognition. In general, explainability has emerged as an important non-functional requirement that impacts system quality. However, the supposed trade-off between explainability and performance challenges the presumed positive influence of explainability. If meeting the requirement of explainability entails a reduction in system performance, then careful consideration must be given to which of these quality aspects takes precedence and how to compromise between them. In this paper, we critically examine the alleged trade-off. We argue that it is best approached in a nuanced way that incorporates resource availability, domain characteristics, and considerations of risk. By providing a foundation for future research and best practices, this work aims to advance the field of RE for AI.

Knowledge graph embedding (KGE) is a increasingly popular technique that aims to represent entities and relations of knowledge graphs into low-dimensional semantic spaces for a wide spectrum of applications such as link prediction, knowledge reasoning and knowledge completion. In this paper, we provide a systematic review of existing KGE techniques based on representation spaces. Particularly, we build a fine-grained classification to categorise the models based on three mathematical perspectives of the representation spaces: (1) Algebraic perspective, (2) Geometric perspective, and (3) Analytical perspective. We introduce the rigorous definitions of fundamental mathematical spaces before diving into KGE models and their mathematical properties. We further discuss different KGE methods over the three categories, as well as summarise how spatial advantages work over different embedding needs. By collating the experimental results from downstream tasks, we also explore the advantages of mathematical space in different scenarios and the reasons behind them. We further state some promising research directions from a representation space perspective, with which we hope to inspire researchers to design their KGE models as well as their related applications with more consideration of their mathematical space properties.

In recent years, Graph Neural Networks have reported outstanding performance in tasks like community detection, molecule classification and link prediction. However, the black-box nature of these models prevents their application in domains like health and finance, where understanding the models' decisions is essential. Counterfactual Explanations (CE) provide these understandings through examples. Moreover, the literature on CE is flourishing with novel explanation methods which are tailored to graph learning. In this survey, we analyse the existing Graph Counterfactual Explanation methods, by providing the reader with an organisation of the literature according to a uniform formal notation for definitions, datasets, and metrics, thus, simplifying potential comparisons w.r.t to the method advantages and disadvantages. We discussed seven methods and sixteen synthetic and real datasets providing details on the possible generation strategies. We highlight the most common evaluation strategies and formalise nine of the metrics used in the literature. We first introduce the evaluation framework GRETEL and how it is possible to extend and use it while providing a further dimension of comparison encompassing reproducibility aspects. Finally, we provide a discussion on how counterfactual explanation interplays with privacy and fairness, before delving into open challenges and future works.

This paper aims at revisiting Graph Convolutional Neural Networks by bridging the gap between spectral and spatial design of graph convolutions. We theoretically demonstrate some equivalence of the graph convolution process regardless it is designed in the spatial or the spectral domain. The obtained general framework allows to lead a spectral analysis of the most popular ConvGNNs, explaining their performance and showing their limits. Moreover, the proposed framework is used to design new convolutions in spectral domain with a custom frequency profile while applying them in the spatial domain. We also propose a generalization of the depthwise separable convolution framework for graph convolutional networks, what allows to decrease the total number of trainable parameters by keeping the capacity of the model. To the best of our knowledge, such a framework has never been used in the GNNs literature. Our proposals are evaluated on both transductive and inductive graph learning problems. Obtained results show the relevance of the proposed method and provide one of the first experimental evidence of transferability of spectral filter coefficients from one graph to another. Our source codes are publicly available at: //github.com/balcilar/Spectral-Designed-Graph-Convolutions