apply m-matrix theory

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.

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 paper addresses the weak theoretical foundations of matrix decomposition in machine learning by systematically constructing a self-consistent, comprehensive, and modern-application-oriented pedagogical framework. Methodologically, it grounds the exposition in numerical linear algebra and matrix analysis, unifying classical decompositions—including LU, QR, SVD, and block triangular factorizations—while integrating numerical stability analysis and Hermitian/Hilbert space theory. Crucially, it bridges traditional numerical analysis with deep learning’s backpropagation setting, emphasizing differentiability and computational robustness of decompositions in algorithm design and model optimization. The primary contribution is a compact, dual-purpose (teaching and research) knowledge system that fills critical gaps in both theoretical coherence and machine-learning relevance present in existing literature, thereby providing rigorous mathematical foundations for high-dimensional data modeling and efficient training.

This work investigates the impact of matrix concatenation operations on singular value spectra, aiming to bridge a theoretical gap in the structural stability of SVDs under concatenation. Addressing the central question—“How are the singular values of a concatenated matrix determined by those of its constituent submatrices?”—we extend Weyl’s inequality to block-wise concatenation for the first time, establishing a quantitative analytical framework grounded in matrix perturbation theory and norm inequalities. We derive computable upper bounds on singular value deviations and rigorously prove that dominant singular values remain stable when the operator norms of the submatrices are comparable. This result provides theoretical guarantees and principled design guidance for low-rank approximation, robust matrix clustering, and compression algorithms.

This work addresses efficiency bottlenecks in solving large-scale linear systems and approximating matrix norms. We propose a multilevel randomized sketching preconditioned iterative method, integrating Nyström low-rank approximation, sparse random sketching, and multilevel preconditioning. It establishes the first multilevel sketched preconditioning framework grounded in the natural average condition number. Theoretical contributions include: (1) optimal complexity $ ilde{O}(n^2 + d_lambda^omega)$ for solving regularized linear systems; (2) accelerated complexity $ ilde{O}(n^{2.065} + k^omega)$ for systems with $k$ outlying singular values; and (3) Schatten-$p$ norm approximation—particularly the nuclear norm—at $ ilde{O}(n^{2.11})$, improving upon the prior best $ ilde{O}(n^{2.18})$. These advances significantly enhance computational efficiency for key subproblems in applications such as Gaussian process regression.

This study addresses the computational expense of traditional methods and the training inefficiency and instability of purely neural approaches for solving high-dimensional operator eigenvalue problems. To overcome these challenges, we propose the Stable Inverse Power Method Neural Network (SIPMNN). This method introduces a novel low-fidelity numerical spectral guidance mechanism, employing coarse-grid finite difference approximations to generate approximate eigenvalues as fixed shifts that guide and constrain the deep neural network training process. Experimental results on ten-dimensional benchmark problems demonstrate that SIPMNN achieves superior solution accuracy compared to purely neural baselines while reducing the required number of iterations by eight- to tenfold. Consequently, this work effectively resolves the bottlenecks associated with high-dimensional eigenvalue search difficulties and training instability.

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。

本文研究了关系数据库中线性代数运算的数值稳定性问题，提出投影向后稳定性和数据库条件数概念以解决传统方法失效的问题。

本文提出了一种从数据中学习控制系统的线性算子的结构化方法，利用(半)群框架和反问题框架分析算法，以获得收敛估计器。

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.