analyze quadratic vector equation

This work addresses the lack of theoretical characterization in existing kernel methods for machine learning regarding the residual structure and energy stability of multichannel signals in complex systems. The authors propose an analytical framework grounded in operator defect identities, introducing the novel concept of “telescopic energy residuals.” By integrating iterative products with a λₙ-relaxed Kaczmarz scheme, they establish admissibility conditions for residuals and derive prior energy bounds. For the first time, this framework incorporates operator defect theory into kernel methods and kernel principal component analysis (KPCA), rigorously proving explicit convergence of generalized algorithms, a residual energy decomposition theorem, and stability criteria under noise. The approach significantly extends infinite-dimensional Kaczmarz theory to broader applications in machine learning.

Existing quadratic constraint approaches for characterizing neural network activation functions are overly conservative, limiting the precision of reachability and safety analyses. This work proposes a domain-dependent framework for verifiable quadratic inequalities: it generates candidate constraints via local sampling and employs sum-of-squares (SOS) certificates to ensure global validity, yielding tight and sound quadratic representations for scalar nonlinearities. The method transcends the limitations of conventional sector or slope bounds by incorporating neuron-wise dependencies and local bound refinement—particularly for ReLU networks—to reduce conservatism. It is compatible with convex quadratic programming, semialgebraic set descriptions, and integral quadratic constraint (IQC) techniques. Experiments demonstrate that the framework significantly improves analysis accuracy for smooth activations such as tanh and extends effectively to systems involving saturation-type nonlinearities.

This work proposes a unified framework for solving systems of multivariate polynomials and rectangular multiparameter eigenvalue problems, implemented in the open-source MATLAB toolbox MacaulayLab. The approach leverages numerical linear algebra and Macaulay matrix constructions without relying on any specific polynomial basis or monomial ordering. It is the first method capable of efficiently handling both problem classes within a single framework while accurately characterizing positive-dimensional solution components at infinity. Numerical experiments demonstrate that the proposed method matches or surpasses the performance of established software packages such as PHCpack, PNLA, and MultiParEig. To support reproducible research, the authors provide an extensive suite of test cases alongside the toolbox.

This paper addresses the overly conservative degree bounds in computing polynomial-coefficient linear differential operators—particularly those arising from pessimistic estimates induced by Cramer’s rule. We propose the first unified degree-bound framework, which precisely characterizes the exact degree growth of linear relations among vectors under iterative pseudo-linear mappings, covering fundamental operators including least common left multiples and symmetric products. Methodologically, our approach integrates rational function matrix realization theory, denominator analysis of determinants, differential algebra, and linear systems theory—thereby eliminating reliance on Cramer’s rule. Theoretical error analysis shows a tenfold reduction in bound overestimation compared to state-of-the-art methods. Our framework automatically recovers optimal known degree bounds for multiple classical operators, thereby bridging a longstanding gap between generality and tightness in differential operator degree estimation.

本文通过介绍数值线性代数在偏微分方程、机器学习和数据同化中的应用，展示了如何使用少量核心概念解决大规模稀疏系统问题。

This work addresses the limitations of traditional iterative methods for solving large-scale systems of equations, including low efficiency, poor robustness, and difficulties in parameter selection. To overcome these challenges, it proposes a "hybrid iteration" paradigm that integrates classical numerical algorithms with machine learning. Building upon conventional iterative schemes such as Newton's method, this approach incorporates deep learning-based optimization strategies to enable adaptive hyperparameter tuning, thereby combining the flexibility of data-driven methods with the theoretical reliability and interpretability of traditional algorithms. Furthermore, this project systematically reviews state-of-the-art methodologies in this domain and identifies key open challenges. Ultimately, it establishes a clear research trajectory for developing efficient and robust solvers for scientific computing.

本文介绍了一种通过计算交换算子的联合特征向量来解决多项式根和张量分解问题的方法，并分析了其多重结构。

This work addresses the challenge of characterizing the highly complex loss landscape in large language model (LLM) pretraining, where existing theories struggle to balance analytical tractability with accurate dynamic prediction. By performing Taylor expansions of both the model and loss function at mid-training, the authors construct a local quadratic approximation and combine it with Lanczos quadrature and Hessian spectral estimation. For the first time, they validate this approach on a 150M-parameter LLM trained on 3B tokens, demonstrating predictive accuracy over a training window spanning 10% of total steps. Their analysis reveals that the quadratic model faithfully captures optimization trajectories, that the tail structure of the Hessian spectrum is strongly influenced by batch size, preconditioning, and training stage, and that optimization typically resides in a stochastic edge-of-stability regime dictated by batch size—uncovering a deep connection between local stability and hyperparameter choice.

This work addresses the problem of high-accuracy signal approximation and prediction under non-uniform periodic sampling by proposing a unified framework based on sampling Kantorovich operators. It extends the classical Bernoulli–Strang–Fix condition for the first time to vector-valued generators in the context of non-uniform sampling and rigorously establishes that the proposed operator possesses both exact and asymptotic polynomial reproduction capabilities. Consequently, the method simultaneously achieves accurate function approximation and signal prediction from local average samples. Theoretical analysis, complemented by numerical experiments using Gaussian kernels and B-splines, demonstrates that the proposed approach exhibits superior performance in both approximation accuracy and prediction capability.

Existing anchoring algorithms suffer from a lack of unified perspective due to their reliance on method-specific anchor constructions, hindering systematic understanding and generalization. This work proposes an operator-side Tikhonov regularization framework that unifies diverse anchoring methods by simply incorporating a vanishing regularizer into the base operator and running the original algorithm unchanged. The framework reveals the intrinsic commonality among methods such as Halpern iteration and extrapolated anchored gradient schemes, and naturally yields new variants. Theoretically, it recovers Halpern iteration with an $O(1/k)$ residual convergence rate, establishes a novel $O(1/\sqrt{k})$ guarantee for forward-step methods, and—under unconstrained monotone Lipschitz settings—achieves, for the first time, an $O(1/k)$ convergence rate for both the extragradient (EG) and past extragradient (PEG) methods.