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For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that: (i) they are highly sensitive to numerical errors in practice, and; (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles. Here we propose solutions for problems (i) and (ii), backed up by theory and numerical experiments. Problem (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Problem (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the space of valid coefficients vectors. The coefficients vectors can then be selected using either a least squares approach or column subset selection.

The paper is concerned with the mathematical theory and numerical approximation of systems of partial differential equations (pde) of hyperbolic, pseudo-parabolic type. Some mathematical properties of the initial-boundary-value problem (ibvp) with Dirichlet boundary conditions are first studied. They include the weak formulation, well-posedness and existence of traveling wave solutions connecting two states, when the equations are considered as a variant of a conservation law. Then, the numerical approximation consists of a spectral approximation in space based on Legendre polynomials along with a temporal discretization with strong stability preserving (SSP) property. The convergence of the semidiscrete approximation is proved under suitable regularity conditions on the data. The choice of the temporal discretization is justified in order to guarantee the stability of the full discretization when dealing with nonsmooth initial conditions. A computational study explores the performance of the fully discrete scheme with regular and nonregular data.

The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which captures both topological signal and noise. To that effect, Chazel and Divol (2019) proved that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it -- called the persistence density function. We present a class of kernel-based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence surfaces. Interestingly, the persistence density function delivers stronger statistical guarantees.

Discovery of mathematical descriptors of physical phenomena from observational and simulated data, as opposed to from the first principles, is a rapidly evolving research area. Two factors, time-dependence of the inputs and hidden translation invariance, are known to complicate this task. To ameliorate these challenges, we combine Lagrangian dynamic mode decomposition with a locally time-invariant approximation of the Koopman operator. The former component of our method yields the best linear estimator of the system's dynamics, while the latter deals with the system's nonlinearity and non-autonomous behavior. We provide theoretical estimators (bounds) of prediction accuracy and perturbation error to guide the selection of both rank truncation and temporal discretization. We demonstrate the performance of our approach on several non-autonomous problems, including two-dimensional Navier-Stokes equations.

Feature attribution is a fundamental task in both machine learning and data analysis, which involves determining the contribution of individual features or variables to a model's output. This process helps identify the most important features for predicting an outcome. The history of feature attribution methods can be traced back to General Additive Models (GAMs), which extend linear regression models by incorporating non-linear relationships between dependent and independent variables. In recent years, gradient-based methods and surrogate models have been applied to unravel complex Artificial Intelligence (AI) systems, but these methods have limitations. GAMs tend to achieve lower accuracy, gradient-based methods can be difficult to interpret, and surrogate models often suffer from stability and fidelity issues. Furthermore, most existing methods do not consider users' contexts, which can significantly influence their preferences. To address these limitations and advance the current state-of-the-art, we define a novel feature attribution framework called Context-Aware Feature Attribution Through Argumentation (CA-FATA). Our framework harnesses the power of argumentation by treating each feature as an argument that can either support, attack or neutralize a prediction. Additionally, CA-FATA formulates feature attribution as an argumentation procedure, and each computation has explicit semantics, which makes it inherently interpretable. CA-FATA also easily integrates side information, such as users' contexts, resulting in more accurate predictions.

Knowledge graphs contain rich semantic relationships related to items and incorporating such semantic relationships into recommender systems helps to explore the latent connections of items, thus improving the accuracy of prediction and enhancing the explainability of recommendations. However, such explainability is not adapted to users' contexts, which can significantly influence their preferences. In this work, we propose CA-KGCN (Context-Aware Knowledge Graph Convolutional Network), an end-to-end framework that can model users' preferences adapted to their contexts and can incorporate rich semantic relationships in the knowledge graph related to items. This framework captures users' attention to different factors: contexts and features of items. More specifically, the framework can model users' preferences adapted to their contexts and provide explanations adapted to the given context. Experiments on three real-world datasets show the effectiveness of our framework: modeling users' preferences adapted to their contexts and explaining the recommendations generated.

For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local best-approximation error of the finite element function by piecewise polynomial functions of the degree determining the expected approximation order, which need not coincide with the maximal polynomial degree of the element, for example if bubble functions are used. The error estimator is shown to be reliable and locally efficient up to this polynomial best-approximation error and oscillations of the right-hand side.

Quantum computing promises transformational gains for solving some problems, but little to none for others. For anyone hoping to use quantum computers now or in the future, it is important to know which problems will benefit. In this paper, we introduce a framework for answering this question both intuitively and quantitatively. The underlying structure of the framework is a race between quantum and classical computers, where their relative strengths determine when each wins. While classical computers operate faster, quantum computers can sometimes run more efficient algorithms. Whether the speed advantage or the algorithmic advantage dominates determines whether a problem will benefit from quantum computing or not. Our analysis reveals that many problems, particularly those of small to moderate size that can be important for typical businesses, will not benefit from quantum computing. Conversely, larger problems or those with particularly big algorithmic gains will benefit from near-term quantum computing. Since very large algorithmic gains are rare in practice and theorized to be rare even in principle, our analysis suggests that the benefits from quantum computing will flow either to users of these rare cases, or practitioners processing very large data.

Meta-analysis is the aggregation of data from multiple studies to find patterns across a broad range relating to a particular subject. It is becoming increasingly useful to apply meta-analysis to summarize these studies being done across various fields. In meta-analysis, it is common to use the mean and standard deviation from each study to compare for analysis. While many studies reported mean and standard deviation for their summary statistics, some report other values including the minimum, maximum, median, and first and third quantiles. Often, the quantiles and median are reported when the data is skewed and does not follow a normal distribution. In order to correctly summarize the data and draw conclusions from multiple studies, it is necessary to estimate the mean and standard deviation from each study, considering variation and skewness within each study. In past literature, methods have been proposed to estimate the mean and standard deviation, but do not consider negative values. Data that include negative values are common and would increase the accuracy and impact of the me-ta-analysis. We propose a method that implements a generalized Box-Cox transformation to estimate the mean and standard deviation accounting for such negative values while maintaining similar accuracy.

Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.