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Mathematics
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Spectral Theory of Tensors, Tensor (Rank) Decompositions, and Their Applications
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Spectral Theory of Tensors, Tensor (Rank) Decompositions, and Their Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "A: Algebra and Logic".

Deadline for manuscript submissions: 31 January 2026 | Viewed by 4025

Special Issue Editors

Prof. Dr. Tan Zhang

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Guest Editor
Department of Mathematics & Statistics, Murray State University, Murray, KY 42071-0009, USA
Interests: linear and multilinear algebra; differential geometry; algebraic topology
Prof. Dr. Shenglong Hu

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Guest Editor
Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
Interests: tensor eigenvalue problems; hypergraph theory; numerical linear algebra

Special Issue Information

Dear Colleagues,

The study of higher-order tensors (multi-dimensional arrays) has been an active research area for over a decade. Numerous significant progresses have been made in the front on eigenvalue problems for tensors, tensor rank problems, tensor decompositions problems, and spectral hypergraph theory,  just to name a few, while new discoveries are being made as we speak. Alongside theoretical development, a wide range of applications found their way in Numerical Multilinear Algebra, Image Processing, Statistical Data Analysis, Multi-relational data mining, best rank-one approximation, Convex Optimizations, Higher-order Markov chains, and Quantum entanglement problems, etc.

Prof. Dr. Tan Zhang
Prof. Dr. Shenglong Hu
Guest Editors

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Keywords

  • tensor eigenvalues
  • tensor ranks
  • tensor decompositions
  • spectral hypergraph theory
  • positive semi-definite programming
  • higher-order Markov chains

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Published Papers (4 papers)

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Research

20 pages, 424 KB  
Article
Exploiting Generalized Cyclic Symmetry to Find Fast Rectangular Matrix Multiplication Algorithms Easier
by Charlotte Vermeylen, Nico Vervliet, Lieven De Lathauwer and Marc Van Barel
Mathematics 2025, 13(19), 3064; https://doi.org/10.3390/math13193064 - 23 Sep 2025
Viewed by 645
Abstract
The quest to multiply two large matrices as fast as possible is one that has already intrigued researchers for several decades. However, the ‘optimal’ algorithm for a certain problem size is still not known. The fast matrix multiplication (FMM) problem can be formulated [...] Read more.
The quest to multiply two large matrices as fast as possible is one that has already intrigued researchers for several decades. However, the ‘optimal’ algorithm for a certain problem size is still not known. The fast matrix multiplication (FMM) problem can be formulated as a non-convex optimization problem—more specifically, as a challenging tensor decomposition problem. In this work, we build upon a state-of-the-art augmented Lagrangian algorithm, which formulates the FMM problem as a constrained least squares problem, by incorporating a new, generalized cyclic symmetric (CS) structure in the decomposition. This structure decreases the number of variables, thereby reducing the large search space and the computational cost per iteration. The constraints are used to find practical solutions, i.e., decompositions with simple coefficients, which yield fast algorithms when implemented in hardware. For the FMM problem, usually a very large number of starting points are necessary to converge to a solution. Extensive numerical experiments for different problem sizes demonstrate that including this structure yields more ‘unique’ practical decompositions for a fixed number of starting points. Uniqueness is defined relative to the known scale and trace invariance transformations that hold for all FMM decompositions. Making it easier to find practical decompositions may lead to the discovery of faster FMM algorithms when used in combination with sufficient computational power. Lastly, we show that the CS structure reduces the cost of multiplying a matrix by itself. Full article
(This article belongs to the Special Issue Spectral Theory of Tensors, Tensor (Rank) Decompositions, and Their Applications)
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10 pages, 257 KB  
Article
Quasi-Irreducibility of Nonnegative Biquadratic Tensors
by Liqun Qi, Chunfeng Cui and Yi Xu
Mathematics 2025, 13(13), 2066; https://doi.org/10.3390/math13132066 - 22 Jun 2025
Cited by 1 | Viewed by 382
Abstract
While the adjacency tensor of a bipartite 2-graph is a nonnegative biquadratic tensor, it is inherently reducible. To address this limitation, we introduce the concept of quasi-irreducibility in this paper. The adjacency tensor of a bipartite 2-graph is quasi-irreducible if that bipartite 2-graph [...] Read more.
While the adjacency tensor of a bipartite 2-graph is a nonnegative biquadratic tensor, it is inherently reducible. To address this limitation, we introduce the concept of quasi-irreducibility in this paper. The adjacency tensor of a bipartite 2-graph is quasi-irreducible if that bipartite 2-graph is not bi-separable. This new concept reveals important spectral properties: although all M+-eigenvalues are M++-eigenvalues for irreducible nonnegative biquadratic tensors, the M+-eigenvalues of a quasi-irreducible nonnegative biquadratic tensor can be either M0-eigenvalues or M++-eigenvalues. Furthermore, we establish a max-min theorem for the M-spectral radius of a nonnegative biquadratic tensor. Full article
(This article belongs to the Special Issue Spectral Theory of Tensors, Tensor (Rank) Decompositions, and Their Applications)
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9 pages, 230 KB  
Article
The ασ-Approximation Property and Its Related Operator Ideals
by Ju Myung Kim
Mathematics 2024, 12(13), 2006; https://doi.org/10.3390/math12132006 - 28 Jun 2024
Viewed by 1075
Abstract
In this paper, we study the σ-tensor norm (ασ), the absolutely τ-summing operator and the σ-nuclear operator. We characterize the ασ-approximation property in terms of some density of the space of absolutely τ-summing operators. [...] Read more.
In this paper, we study the σ-tensor norm (ασ), the absolutely τ-summing operator and the σ-nuclear operator. We characterize the ασ-approximation property in terms of some density of the space of absolutely τ-summing operators. When X* or Y*** has the approximation property, we prove that an operator T from X to Y is σ-nuclear if the adjoint of T is σ-nuclear. Full article
(This article belongs to the Special Issue Spectral Theory of Tensors, Tensor (Rank) Decompositions, and Their Applications)
18 pages, 287 KB  
Article
The Nearest Zero Eigenvector of a Weakly Symmetric Tensor from a Given Point
by Kelly Pearson and Tan Zhang
Mathematics 2024, 12(5), 705; https://doi.org/10.3390/math12050705 - 28 Feb 2024
Viewed by 1044
Abstract
We begin with a degree m real homogeneous polynomial in n indeterminants and bound the distance from a given n-dimensional real vector to the real vanishing of the homogeneous polynomial. We then apply these bounds to the real homogeneous polynomial associated with [...] Read more.
We begin with a degree m real homogeneous polynomial in n indeterminants and bound the distance from a given n-dimensional real vector to the real vanishing of the homogeneous polynomial. We then apply these bounds to the real homogeneous polynomial associated with a nonzero m-order n-dimensional weakly symmetric tensor which has zero as an eigenvalue. We provide “nested spheres” conditions to bound the distance from a given n-dimensional real vector to the nearest zero eigenvector. Full article
(This article belongs to the Special Issue Spectral Theory of Tensors, Tensor (Rank) Decompositions, and Their Applications)
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