apply gaussian elimination

Cholesky decomposition of symmetric positive-definite matrices in Gaussian process inference suffers from numerical instability and computational inefficiency. Method: We propose a novel pivoted Cholesky pivoting strategy that integrates entropy maximization from Bayesian nonparametric inference with information gain principles from active learning, enabling an online-updatable, low-overhead diagonal adaptation mechanism. This strategy is synergistically combined with preconditioned iterative solvers and sparse regression frameworks. Contribution/Results: The proposed method significantly improves both uncertainty quantification accuracy and computational speed in sparse Gaussian process inference. Empirical evaluation across diverse benchmark tasks demonstrates consistent superiority over standard pivoted Cholesky baselines, with negligible additional computational overhead. The approach is theoretically grounded in probabilistic inference principles and exhibits strong practical applicability in large-scale GP modeling.

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 addresses the problem of solving systems of equations defined by equality constraints among square submatrices of an input matrix. Inspired by Gawrychowski et al.'s recursive algorithm for one-dimensional string reconstruction, we present the first extension to the two-dimensional setting, introducing a linear-time algorithm based on recursive decomposition and submatrix matching. The proposed method efficiently resolves any system of square submatrix equalities and directly yields an optimal-time decompression algorithm for copy operations in two-dimensional macro compression schemes. This advancement significantly enhances the efficiency of compression and reconstruction for structured two-dimensional data.

This study addresses a longstanding open problem in smoothed analysis: the absence of non-trivial lower bounds on the probability of large growth factors during Gaussian elimination on random matrices. By integrating techniques from probability theory and linear algebra, this work establishes the first non-trivial probability lower bound for large growth phenomena in Gaussian random matrices, proving it to be at least inverse quasi-polynomial, specifically Ω(exp(−c log²(ρ) log(n))). This result departs from traditional average-case analysis assumptions by demonstrating that large growth occurs with substantially higher probability than previously anticipated, thereby refuting standard conjectures in the field. Ultimately, these findings provide a critical theoretical foundation for understanding the numerical stability of Gaussian elimination and related algorithms in practical computation.

Computing exact leverage scores for Kronecker-product structured matrices in large-scale least-squares problems is computationally prohibitive, while existing approximation methods incur statistical bias and high overhead. Method: We propose the first efficient exact leverage score algorithm tailored to Kronecker-structured matrices. Leveraging the inherent tensor structure, our method designs a near-linear-time framework for exact leverage score computation and sampling—bypassing costly full-matrix SVD or biased sketching approximations. Contribution/Results: Theoretically and empirically, our algorithm achieves significantly lower sampling error than state-of-the-art approximate methods (e.g., FJLT- or CountSketch-accelerated approaches), while maintaining substantially lower time complexity than full SVD. This work establishes the first scalable, exact, and efficient leverage score sampling scheme for Kronecker-structured matrices, enabling improved structured random projections and large-scale regression.

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.

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 susceptibility of ill-conditioned M-matrices to subtractive cancellation in componentwise high-precision computations by introducing a novel approach based on a triplet representation. By integrating componentwise high-precision arithmetic with the structural properties of M-matrices, the authors present—for the first time—high-precision variants of the GTH algorithm, including non-blocking, recursive, and blocked formulations, all encapsulated within an object-oriented MATLAB interface. The resulting toolbox supports a range of operations such as linear system solving, LU factorization, Schur complement computation, matrix square roots, singular value decomposition, and solutions to nonsymmetric algebraic Riccati equations. Even under severe ill-conditioning, the framework preserves componentwise numerical accuracy, thereby substantially enhancing computational reliability.

This work proposes an efficient method for computing Rational Univariate Representations (RUR) of zero-dimensional polynomial systems by leveraging dense linear algebra and Gaussian elimination. Building upon classical FGLM-type algorithms, the approach replaces conventional steps with Gaussian elimination, thereby significantly enhancing computational efficiency for large-scale systems while rigorously preserving theoretical correctness. Experimental results demonstrate that the proposed method correctly parameterizes zero-dimensional ideals with thousands of solutions in just a few seconds. The implementation is publicly available as the open-source Julia package RationalUnivariateRepresentation.jl.

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。