Article
Computer Science, Information Systems
Mohammad M. Salut, David Anderson
Summary: This paper proposes a new online robust PCA algorithm that preserves the multi-dimensional structure of data when dealing with higher-order data arrays. It shows good robustness and effectiveness.
Article
Engineering, Electrical & Electronic
Yi Yang, Lixin Han, Yuanzhen Liu, Jun Zhu, Hong Yan
Summary: Inspired by the accuracy and efficiency of the gamma-norm of a matrix, the study generalizes the gamma-norm to tensors and proposes a new tensor completion approach within the tensor singular value decomposition framework. An efficient algorithm, combining the augmented Lagrange multiplier and closed-resolution of a cubic equation, is developed to solve the associated nonconvex tensor multi-rank minimization problem. Experimental results demonstrate that the proposed approach outperforms current state of the art algorithms in recovery accuracy.
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2021)
Article
Mathematics, Applied
Michael Stewart
Summary: This paper discusses the structure and computation of an optimal orthogonal product decomposition of a pair of matrices (A, B) to reveal locally minimal perturbations, and solving the problem through minimization of specific projections.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Multidisciplinary Sciences
Misha E. Kilmer, Lior Horesh, Haim Avron, Elizabeth Newman
Summary: This study demonstrates that compressing high-dimensional datasets as tensors using tensor-SVD is more efficient than representing them using matrices. By proving that a tensor-tensor representation is superior to its matrix counterpart in terms of representation efficiency, it is found that the compressed representation provided by the truncated tensor SVD is related to its two closest tensor-based analogs.
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
(2021)
Article
Computer Science, Artificial Intelligence
Shiqiang Du, Yuqing Shi, Guangrong Shan, Weilan Wang, Yide Ma
Summary: In this paper, a robust tensor low-rank sparse representation (TLRSR) method is proposed for subspace learning on three-dimensional tensors. The method effectively captures the global and local structure of sample data using dual constraints and noise characterization with tensor l(2,1)-norm. The denoised tensor is used as the dictionary for finding low-rank sparse representation, and an iterative update algorithm is proposed for optimization. The method shows good performance in tensor subspace learning, demonstrated through clustering on face images and denoising on real images.
Article
Computer Science, Artificial Intelligence
Miaowen Shi, Fan Zhang, Suwei Wang, Caiming Zhang, Xuemei Li
Summary: This paper proposes a novel image denoising method that reconstructs high and low frequency components separately to retain original image details. Experimental results demonstrate that the method excels in both numerical precision and visual performance.
COMPUTER VISION AND IMAGE UNDERSTANDING
(2021)
Article
Mathematics, Applied
Hongjin He, Chen Ling, Wenhui Xie
Summary: In this paper, a regularized tensor completion model based on tensor-tensor product is proposed, along with an implementable alternating minimization algorithm. The proposed method does not require singular value decomposition (SVD), resulting in shorter computation time and higher accuracy compared to other methods.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Artificial Intelligence
Shuqin Wang, Yongyong Chen, Yigang Cen, Linna Zhang, Hengyou Wang, Viacheslav Voronin
Summary: This paper proposes a nonconvex low-rank and sparse tensor representation method, which can preserve the similarity information of multi-view data from global and local perspectives, and an effective algorithm is proposed to solve the proposed method. Experimental results demonstrate the superiority of the proposed method on multiple datasets.
APPLIED INTELLIGENCE
(2022)
Article
Mathematics, Applied
Ying Wang, Yuning Yang
Summary: This paper considers generalizing the t-SVD of third-order tensors to tensors of arbitrary order N and introduces the Hot-SVD as a tensor-tensor product version of the higher order SVD (HOSVD). The existence of Hot-SVD is proved by introducing the small-t transpose for third-order tensors. Various properties of Hot-SVD and its truncated versions are established, and numerical examples are provided to validate them.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Computer Science, Information Systems
Shicheng Yang, Ying Wen, Lianghua He, Mengchu Zhou, Abdullah Abusorrah
Summary: This work introduces a sparse individual low-rank component-based representation (SILR) method that effectively addresses the impact of undersampled training datasets and same intrasubject variations on classification performance by applying l(2)-norm constraint to intrasubject coefficients.
IEEE INTERNET OF THINGS JOURNAL
(2021)
Article
Automation & Control Systems
Botao Hao, Boxiang Wang, Pengyuan Wang, Jingfei Zhang, Jian Yang, Will Wei Sun
Summary: In this paper, a sparse tensor additive regression (STAR) model is proposed to effectively utilize the sparse and low-rank structures in tensor additive regression, with an efficient parameter estimation algorithm. A non-asymptotic error bound for the estimator obtained from each iteration of the proposed algorithm is established, revealing the interplay between optimization error and statistical rate of convergence, demonstrating the efficacy of STAR in various applications.
JOURNAL OF MACHINE LEARNING RESEARCH
(2021)
Article
Computer Science, Artificial Intelligence
Guoqing Liu, Hongwei Ge, Ting Li, Shuzhi Su, Shuangxi Wang
Summary: In this paper, a multi-view subspace enhanced representation of manifold regularization and low-rank tensor constraint (MSERMLRT) method is proposed to extract manifold information from multi-view data and improve clustering performance. A tensor is utilized to explore correlations between views and reduce redundant information. The model also incorporates manifold information and enforces a sparse constraint to enhance the diagonal block structure of the subspace representation, improving clustering effects. The effectiveness of the MSERMLRT model is demonstrated through experiments on challenging datasets.
INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS
(2023)
Article
Computer Science, Artificial Intelligence
Shiqiang Du, Baokai Liu, Guangrong Shan, Yuqing Shi, Weilan Wang
Summary: In this paper, a novel approach called enhanced tensor LRR (ETLRR) is proposed, which recovers clean tensor data by considering two types of noise and utilizes an iterative update method for optimization. Experiments show that ETLRR performs well in obtaining low-rank tensor subspace structures and recovering tensor data.
KNOWLEDGE-BASED SYSTEMS
(2022)
Article
Computer Science, Software Engineering
Azadeh Montazeri, Mahboubeh Shamsi, Rouhollah Dianat
Summary: This study aims to improve the classification efficiency of advanced methods through a multilayered dictionary learning framework, introducing the concept of multilayered K-singular value decomposition (MLK-SVD) dictionary learning as a multilayer method of classification. By associating class label information with sparse codes during dictionary learning process, it enforces discrimination and utilizes a series of sparse coding and pooling steps to efficiently represent data for classification.
Article
Mathematics, Applied
Hussam Al Daas, Grey Ballard, Peter Benner
Summary: We present efficient and scalable parallel algorithms for performing mathematical operations on low-rank tensors in the TT format. These algorithms include addition, elementwise multiplication, computing norms and inner products, orthonormalization, and rounding. The proposed algorithms demonstrate good scalability and efficiency on various computing systems.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Martin Eigel, Claude Jeffrey Gittelson, Christofh Schwab, Elmar Zander
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2015)
Article
Engineering, Multidisciplinary
Martin Eigel, Claude Jeffrey Gittelson, Christoph Schwab, Elmar Zander
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2014)
Proceedings Paper
Mechanics
Hermann G. Matthies, Elmar Zander, Bojana V. Rosic, Alexander Litvinenko, Oliver Pajonk
COMPUTATIONAL METHODS FOR SOLIDS AND FLUIDS: MULTISCALE ANALYSIS, PROBABILITY ASPECTS AND MODEL REDUCTION
(2016)
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)