Article
Mathematics, Applied
Akbar Shirilord, Mehdi Dehghan
Summary: In this study, a new single-step iterative method is proposed for solving complex linear systems. The convergence of the method is analyzed and the optimal parameter is discussed. Numerical examples are provided to demonstrate the efficiency of the method in actual computation.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Information Systems
Hao Xu
Summary: This paper presents an unsupervised manifold learning algorithm on the SPD matrix manifold for data dimensional reduction. By constructing polynomial kernel matrix, weight matrix, and sparsity preserving matrix, and utilizing polynomial mapping with geodesic distance, the proposed approach achieves dimensional reduction of SPD matrix data.
INFORMATION SCIENCES
(2022)
Article
Mathematics, Applied
Xin Xing, Hua Huang, Edmond Chow
Summary: This paper presents a general algorithm for constructing an SPD HSS approximation as a preconditioner for solving linear systems with dense, ill-conditioned, symmetric positive definite (SPD) kernel matrices. The algorithm uses the H-2 representation of the SPD matrix to reduce computational complexity from quadratic to quasilinear, and numerical experiments demonstrate its performance.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Mehdi Dehghan, Akbar Shirilord
Summary: In this paper, we propose two new lopsided methods based on the two-scale-splitting method, LTSCSP1 and LTSCSP2, which can increase the convergence speed of the TSCSP iteration method without adding any additional parameters under certain conditions. The convergence analysis and quasi-optimal parameter determination for the new methods are provided. Additionally, numerical examples are presented to demonstrate the effectiveness of the proposed framework.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics
Wei Chu, Yao Zhao, Hua Yuan
Summary: This paper presents a new approach to parallelize the Bisection iteration algorithm, achieving efficient computation of eigenvalues while reducing the time cost by over 35-70% compared to the traditional Bisection algorithm.
Article
Mathematics, Applied
Jie Xue, Ruifang Liu, Jinlong Shu
Summary: This paper discusses the distance eigenvalues of chain graphs and characterizes all connected graphs with a third largest distance eigenvalue of at most -1 using clique extension. It also proves that a graph is determined by its distance spectrum if its third largest distance eigenvalue is less than -1.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Computer Science, Artificial Intelligence
Sung Woo Park, Junseok Kwon
Summary: This study introduces a novel Riemannian submanifold framework for log-Euclidean metric learning on symmetric positive definite manifolds. The approach surpasses state-of-the-art methods in synthetic, material categorization, and action recognition problems.
EXPERT SYSTEMS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Xiaoxia Wu, Jianguo Qian, Haigen Peng
Summary: In this article, we characterize the simple connected graphs with the second largest eigenvalue less than 1/2 and identify 13 specific classes of graphs. These 13 classes imply that c(2) belongs to [1/2, root 2 + root 5], where c(2) is the minimum real number for which every real number greater than c is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as an open problem.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Mathematics
Xiaomin Duan, Xueting Ji, Huafei Sun, Hao Guo
Summary: This study presents a non-iterative method for calculating the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix. By employing log-Euclidean metrics, the calculations are simplified and accelerated. The method does not require computing the mean matrix and retains the usual Euclidean operations.
Article
Neurosciences
Kisung You, Hae-Jeong Park
Summary: Functional networks are commonly represented using symmetric positive definite matrices on the Riemannian manifold, considering all pairwise interactions. Despite the geometric properties of the SPD manifold, there are limited studies focusing on connectivity analysis.
Article
Mathematics
Fan Yuan, Shengguo Li, Hao Jiang, Hongxia Wang, Cheng Chen, Lei Du, Bo Yang
Summary: A novel algorithm is proposed in this paper to reduce a banded symmetric generalized eigenvalue problem to a banded symmetric standard eigenvalue problem, using the sequentially semiseparable (SSS) matrix techniques. The algorithm requires linear storage cost and offers potential for parallelism.
Article
Mathematics, Applied
Sk Safique Ahmad, Prince Kanhya
Summary: This work focuses on the structured backward error analysis of Hankel and symmetric-Toeplitz matrix pencils, showcasing their additional properties and applications in solving inverse eigenvalue problems. The perturbation analysis of these structured matrix pencils in Frobenius norm helps maintain sparsity while ensuring exact eigenpairs. This framework proves useful in various numerical linear algebra tasks, particularly in solving inverse eigenvalue problems.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Hussam Al Daas, Tyrone Rees, Jennifer Scott
Summary: The study explores the use of randomized methods in constructing high-quality preconditioners for large-scale sparse symmetric positive definite linear systems. A new and efficient approach is proposed, utilizing Nystrom's method for low rank approximations to develop robust algebraic two-level preconditioners. Numerical experiments demonstrate that the inner system can be solved cheaply using block conjugate gradients and that a large convergence tolerance does not negatively impact the quality of the resulting Nystrom-Schur two-level preconditioner.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Nguyen Thanh Son, P-A Absil, Bin Gao, Tatjana Stykel
Summary: This paper addresses the problem of computing the smallest symplectic eigenvalues and corresponding eigenvectors of symmetric positive-definite matrices. It formulates the problem as minimizing a trace cost function over the symplectic Stiefel manifold. The authors propose a numerical procedure for computing the symplectic eigenpairs and discuss the connection of the sought solution with the eigenvalues of a special class of Hamiltonian matrices.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2021)
Article
Computer Science, Artificial Intelligence
Fengzhen Tang, Haifeng Feng, Peter Tino, Bailu Si, Daxiong Ji
Summary: This paper develops a new classification method for manifold-valued data in the framework of probabilistic learning vector quantization. By generalizing the algorithm to symmetric positive definite matrices equipped with Riemannian natural metric, the proposed method demonstrates superior performance on synthetic data, image data, and motor imagery EEG data. Through utilizing Riemannian distance and gradient descent, the probabilistic learning Riemannian space quantization algorithm is derived.
Article
Mathematics, Applied
Muhammad Syifa'ul Mufid, Ebrahim Patel, Sergei Sergeev
Summary: This paper presents an approach to solve maxmin-omega linear systems by performing normalization and generating a principal order matrix. The possible solution indices can be identified using the principal order matrix and the parameter omega, and the fully active solutions can be obtained from these indices. Other solutions can be found by applying a relaxation to the fully active solutions. This approach can be seen as a generalization of solving max-plus or min-plus linear systems. The paper also highlights the unusual feature of maxmin-omega linear systems having a finite number of solutions when the solution is non-unique.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
E. Mainar, J. M. Pena, B. Rubio
Summary: A bidiagonal decomposition of quantum Hilbert matrices is obtained and the total positivity of these matrices is proved. This factorization is used for accurate algebraic computations and the numerical errors caused by imprecise computer arithmetic or perturbed input data are analyzed. Numerical experiments demonstrate the accuracy of the proposed methods.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhong-Zhi Bai
Summary: This study explores the algebraic structures and computational properties of Wasserstein-1 metric matrices. It shows that these matrices can be expressed using the Neumann series of nilpotent matrices and can be accurately and stably computed by solving unit bidiagonal triangular systems of linear equations.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Bogdan Nica
Summary: This study investigates the relationship between the independence number and chromatic number in a graph of non-singular matrices over a finite field, and obtains an upper bound for the former and a lower bound for the latter.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Dijian Wang, Yaoping Hou, Deqiong Li
Summary: In this paper, a Turán-like problem in signed graphs is studied. The properties of signed graphs are proven in the context of the problem.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Tyler Chen, Thomas Trogdon
Summary: This study focuses on the stability of the Lanczos algorithm when applied to problems with eigenvector empirical spectral distribution close to a reference measure characterized by well-behaved orthogonal polynomials. The analysis reveals that the Lanczos algorithm is forward stable on many large random matrix models, even in finite precision arithmetic, which indicates that random matrices differ significantly from general matrices and caution must be exercised when using them to test numerical algorithms.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Constantin Costara
Summary: This passage discusses linear mappings on matrices and the relationship between subsets of the spectrum, providing corresponding characterization conditions.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Amir Hossein Ghodrati, Mohammad Ali Hosseinzadeh
Summary: This paper presents tight upper bounds for all signless Laplacian eigenvalues of a graph with prescribed order and minimum degree, improving upon previously known bounds. Additionally, the relationship between the number of signless Laplacian eigenvalues falling within specific intervals and various graph parameters such as independence, clique, chromatic, edge covering, and matching numbers is explored.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Ya-Lei Jin, Jie Zhang, Xiao-Dong Zhang
Summary: This paper investigates the relationship between the spectral radius of a symmetric matrix and its principal submatrices, and uses these relationships to obtain upper bounds of the spectral radius of graphs.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Davide Bolognini, Paolo Sentinelli
Summary: We introduce immanant varieties associated with simple characters of a finite group and discuss the features of one-dimensional characters and trivial characters.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
A. S. Gordienko
Summary: We introduce the concept of a graded group action on a graded algebra, or equivalently, a group action by graded pseudoautomorphisms. We study the properties of groups of graded pseudoautomorphisms and prove several important theorems and conjectures regarding graded algebras with a group action.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Jiaqi Gu, Shenghao Feng, Yimin Wei
Summary: We propose a tensor product structure compatible with the hypergraph structure and define the algebraic connectivity of the hypergraph in this product, establishing its relationship with vertex connectivity. We introduce connectivity optimization problems into the hypergraph and solve them using algebraic connectivity. Additionally, we apply the Laplacian eigenmap algorithm to the hypergraph under our tensor product.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Samuel Lichtenberg, Abiy Tasissa
Summary: This paper explores a dual basis approach to Classical Multidimensional Scaling (CMDS) and provides explicit formulas for the dual basis vectors. It also characterizes the spectrum of an essential matrix in the dual basis framework. Connections to a related problem in metric nearness are made.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)