We present a generalization of Nesterov's accelerated gradient descent algorithm. Our algorithm (AGNES) provably achieves acceleration for smooth convex and strongly convex minimization tasks with noisy gradient estimates if the noise intensity is proportional to the magnitude of the gradient at every point. Nesterov's method converges at an accelerated rate if the constant of proportionality is below 1, while AGNES accommodates any signal-to-noise ratio. The noise model is motivated by applications in overparametrized machine learning. AGNES requires only two parameters in convex and three in strongly convex minimization tasks, improving on existing methods. We further provide clear geometric interpretations and heuristics for the choice of parameters.
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A rapidly developing application of LLMs in XAI is to convert quantitative explanations such as SHAP into user-friendly narratives to explain the decisions made by smaller prediction models. Evaluating the narratives without relying on human preference studies or surveys is becoming increasingly important in this field. In this work we propose a framework and explore several automated metrics to evaluate LLM-generated narratives for explanations of tabular classification tasks. We apply our approach to compare several state-of-the-art LLMs across different datasets and prompt types. As a demonstration of their utility, these metrics allow us to identify new challenges related to LLM hallucinations for XAI narratives.
Finite element discretization of Stokes problems can result in singular, inconsistent saddle point linear algebraic systems. This inconsistency can cause many iterative methods to fail to converge. In this work, we consider the lowest-order weak Galerkin finite element method to discretize Stokes flow problems and study a consistency enforcement by modifying the right-hand side of the resulting linear system. It is shown that the modification of the scheme does not affect the optimal-order convergence of the numerical solution. Moreover, inexact block diagonal and triangular Schur complement preconditioners and the minimal residual method (MINRES) and the generalized minimal residual method (GMRES) are studied for the iterative solution of the modified scheme. Bounds for the eigenvalues and the residual of MINRES/GMRES are established. Those bounds show that the convergence of MINRES and GMRES is independent of the viscosity parameter and mesh size. The convergence of the modified scheme and effectiveness of the preconditioners are verified using numerical examples in two and three dimensions.
Approximating field variables and data vectors from sparse samples is a key challenge in computational science. Widely used methods such as gappy proper orthogonal decomposition and empirical interpolation rely on linear approximation spaces, limiting their effectiveness for data representing transport-dominated and wave-like dynamics. To address this limitation, we introduce quadratic manifold sparse regression, which trains quadratic manifolds with a sparse greedy method and computes approximations on the manifold through novel nonlinear projections of sparse samples. The nonlinear approximations obtained with quadratic manifold sparse regression achieve orders of magnitude higher accuracies than linear methods on data describing transport-dominated dynamics in numerical experiments.
Polynomials known as Multiple Orthogonal Polynomials in a single variable are polynomials that satisfy orthogonality conditions concerning multiple measures and play a significant role in several applications such as Hermite-Pad\'e approximation, random matrix theory or integrable systems. However, this theory has only been studied in the univariate case. We give a generalization of Multiple Orthogonal Polynomials for two variables. Moreover, an extended version of some of the main properties are given. Additionally, some examples are given along the paper.
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a new way of deriving coefficients of implicit Runge-Kutta methods. This approach based on repeated integrals yields both new and well-known Butcher's tableaux. We discuss the properties of newly derived methods and compare them with standard collocation implicit Runge-Kutta methods in a series of numerical experiments, including the Prothero-Robinson problem.
The multilevel Schwarz preconditioner is one of the most popular parallel preconditioners for enhancing convergence and improving parallel efficiency. However, its parallel implementation on arbitrary unstructured triangular/tetrahedral meshes remains challenging. The challenges mainly arise from the inability to ensure that mesh hierarchies are nested, which complicates parallelization efforts. This paper systematically investigates the non-nested unstructured case of parallel multilevel algorithms and develops a highly parallel non-nested multilevel smoothed Schwarz preconditioner. The proposed multilevel preconditioner incorporates two key techniques. The first is a new parallel coarsening algorithm that preserves the geometric features of the computational domain. The second is a corresponding parallel non-nested interpolation method designed for non-nested mesh hierarchies. This new preconditioner is applied to a broad range of linear parametric problems, benefiting from the reusability of the same coarse mesh hierarchy for problems with different parameters. Several numerical experiments validate the outstanding convergence and parallel efficiency of the proposed preconditioner, demonstrating effective scalability up to 1,000 processors.
We give a new construction of binary quantum codes that enables the generation of a CSS-T code from any given CSS code. Using this construction, we prove the existence of asymptotically good binary CSS-T codes, resolving a previously open problem in the literature. Furthermore, we demonstrate that the same result holds for binary quantum low-density parity check CSS-T codes, and establish the existence of asymptotically good CSS codes that support any given $Z$ rotation transversally. Finally, we analyze the structure of the logical operators corresponding to certain non-Clifford gates supported by the quantum codes obtained from our construction.
3D Gaussian Splatting (3DGS) demonstrates unparalleled superior performance in 3D scene reconstruction. However, 3DGS heavily relies on the sharp images. Fulfilling this requirement can be challenging in real-world scenarios especially when the camera moves fast, which severely limits the application of 3DGS. To address these challenges, we proposed Spike Gausian Splatting (SpikeGS), the first framework that integrates the spike streams into 3DGS pipeline to reconstruct 3D scenes via a fast-moving bio-inspired camera. With accumulation rasterization, interval supervision, and a specially designed pipeline, SpikeGS extracts detailed geometry and texture from high temporal resolution but texture lacking spike stream, reconstructs 3D scenes captured in 1 second. Extensive experiments on multiple synthetic and real-world datasets demonstrate the superiority of SpikeGS compared with existing spike-based and deblur 3D scene reconstruction methods. Codes and data will be released soon.
It has been widely observed that language models (LMs) respond in predictable ways to algorithmically generated prompts that are seemingly unintelligible. This is both a sign that we lack a full understanding of how LMs work, and a practical challenge, because opaqueness can be exploited for harmful uses of LMs, such as jailbreaking. We present the first thorough analysis of opaque machine-generated prompts, or autoprompts, pertaining to 3 LMs of different sizes and families. We find that machine-generated prompts are characterized by a last token that is often intelligible and strongly affects the generation. A small but consistent proportion of the previous tokens are fillers that probably appear in the prompt as a by-product of the fact that the optimization process fixes the number of tokens. The remaining tokens tend to have at least a loose semantic relation with the generation, although they do not engage in well-formed syntactic relations with it. We find moreover that some of the ablations we applied to machine-generated prompts can also be applied to natural language sequences, leading to similar behavior, suggesting that autoprompts are a direct consequence of the way in which LMs process linguistic inputs in general.
A directive known as NIS2 was enacted in the European Union (EU) in late 2022. It deals particularly with European critical infrastructures, enlarging their scope substantially from an older directive that only considered the energy and transport sectors as critical. The directive's focus is on cyber security of critical infrastructures, although together with other new EU laws it expands to other security domains as well. Given the importance of the directive and most of all the importance of critical infrastructures, the paper presents a systematic literature review on academic research addressing the NIS2 directive either explicitly or implicitly. According to the review, existing research has often framed and discussed the directive with the EU's other cyber security laws. In addition, existing research has often operated in numerous contextual areas, including industrial control systems, telecommunications, the energy and water sectors, and infrastructures for information sharing and situational awareness. Despite the large scope of existing research, the review reveals noteworthy research gaps and worthwhile topics to examine in further research.
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