Singular values and eigenvalues of tensors: a variational approach

We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem

• Publication year

  2005

• Venue

  IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing

• Publication date

  2005-12-13

• Fields of study

  Mathematics, Computer Science

• Identifiers
  DOI 10.1109/CAMAP.2005.1574201arXiv math/0607648
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• L1-norm projection pursuit principal component analysis

  2006cited by this paper
• Transposition invariant principal component analysis in L1 for long tailed data

  2005cited by this paper
• Eigenvalues of a real supersymmetric tensor

  2005influential reference
• From Vectors to Tensors

  2005cited by this paper
• Tensor decomposition and approximation schemes for constraint satisfaction problems

  2005cited by this paper
• On the Best Rank-1 and Rank-(

  2004cited by this paper
• Random sampling and approximation of MAX-CSPs

  2003cited by this paper
• Multivariate Polynomial Minimization and Its Application in Signal Processing

  2003cited by this paper
• On the Best Rank-1 and Rank-(R1 , R2, ... , RN) Approximation of Higher-Order Tensors

  2000cited by this paper
• A Multilinear Singular Value Decomposition

  2000cited by this paper
• Nonnegative Matrices in the Mathematical Sciences

  1979cited by this paper

• t-Hermitian Forms of Arbitrary Degree, Their Spectral Structure, and Positivity

  2026cites this paper
• Two conjectures in spectral hypergraph theory

  2026influential citation
• Several bounds on the spectral radius of uniform hypergraphs

  2026cites this paper
• Randomized Kaczmarz method for solving total least squares solution of multilinear equations

  2026cites this paper
• Multidimensional convolution matrices and perfect colorings of subspace hypergraphs applied for bent functions and related designs

  2026cites this paper
• A two-steps tensor eigenvector centrality for nodes and hyperedges in hypergraphs

  2026cites this paper
• All eigenvalues of the blowup of a graph

  2026cites this paper
• The signless Laplacian spectral Tur\'an problems for hypergraphs

  2026influential citation
• Tensor-based Dinkelbach method for computing generalized tensor eigenvalues and its applications

  2026cites this paper
• A study on the C-eigenvalue of the third-order tensors

  2025cites this paper
• On the second-largest modulus among the eigenvalues of a power hypergraph

  2025cites this paper
• The Nondecreasing Rank

  2025influential citation
• A reduction formula for the Laplacian characteristic polynomial of a uniform hypertree

  2025cites this paper
• The Laplacian spectral moments of uniform hypergraphs

  2025cites this paper
• Orthogonal eigenvectors and singular vectors of tensors

  2025cites this paper
• Riemannian Conjugate Gradient Algorithms for Solving λ$$ \boldsymbol{\lambda} $$ ‐Approximation of Stochastic Tensors and Applications

  2025influential citation
• Unscented and Higher-Order Linear Covariance Fidelity Checks and Measures of Non-Gaussianity

  2025cites this paper
• A note on eigenvalue of tensors and its application

  2025cites this paper
• A novel hesitant fuzzy tensor-based group decision-making approach with application to heterogeneous wireless network evaluation

  2025cites this paper
• Perron-Frobenius theorem for dual tensors and its applications

  2025cites this paper
• The $k$-core of a graph and its high-order spectra

  2025cites this paper
• Evaluating Pancreatic Cancer Treatment Strategies Using a Novel Polytopic Fuzzy Tensor Approach

  2025cites this paper
• Positive Definiteness and Stability of Interval Tensors

  2025cites this paper
• The characteristic polynomial of sunflowers

  2025cites this paper
• Comparison theorems for splittings of different M-tensors

  2025cites this paper
• The characteristic polynomials of $r$-uniform hypercycles with length $l$

  2025cites this paper
• FastPSA: A Fast Version of the Principal Skewness Analysis

  2025cites this paper
• Characterizing upper bounds of Z-spectral radius of uniform hypergraphs

  2025cites this paper
• Beyond independent component analysis: identifiability and algorithms

  2025cites this paper
• An α-triangle eigenvector centrality of graphs

  2025cites this paper
• Renormalization group-like flows in randomly connected tensor networks

  2025cites this paper
• Lexicographical ordering by spectral moments of bicyclic hypergraphs

  2025cites this paper
• High-order spectral characterizations of graphs

  2025cites this paper
• Saito's theorem revisited and application to free pencils of hypersurfaces

  2025cites this paper
• Sign-Enhanced Semidefinite Programming Algorithm and its Application to Independent Component Analysis

  2025cites this paper
• A vector bundle approach to Nash equilibria

  2025cites this paper
• On Third Order Tensors and Their Products

  2025cites this paper
• Developments on the Hoffman program of graphs

  2025cites this paper
• The ordering of hypertrees and unicyclic hypergraphs by the traces of $\mathcal{A}_{\alpha}$-tensor

  2025cites this paper
• $\hat{H}$-eigenvalues of Hermitian tensors and some applications

  2025cites this paper
• Responsible Gaussian Model: Matrix-Based Approximation of Gaussian Mixture Model

  2025cites this paper
• On the geometry of flat minima

  2025cites this paper
• Fuzzy soft tensor based group decision making approach with application to heterogeneous wireless network evaluation

  2025cites this paper
• A Spectral Version of the Theorem of Zykov and Erdős

  2025cites this paper
• Eigenvalues and equivalence classes of third-order symmetric tensors

  2025cites this paper
• A Variational Characterization and A Line Search Newton-Noda Method for the unifying spectral problem of nonnegative tensors

  2025influential citation
• The extremal problems for the spectral radius of hypergraphs without the Berge-graphs

  2025cites this paper
• Qi's problems on classifications of third- and fourth-order symmetric tensors by eigenvalues

  2025cites this paper
• Multi-drawing P\'olya urns via labelled random DAGs

  2025cites this paper
• Dominant H-Eigenvectors of Tensor Kronecker Products Do Not Decouple

  2025cites this paper
• Osculating Geometry and Higher-Order Distance Loci

  2025influential citation
• A general preconditioner for a class of vertical tensor complementarity problems

  2025influential citation
• Fourth Order Random Tensors and Their Applications in Statistics

  2025cites this paper
• Phase transition detection in the community detection for hypergraph network via tensor method

  2025cites this paper
• Minimizing Communication for Parallel Symmetric Tensor Times Same Vector Computation

  2025cites this paper
• On universal eigenvalues and eigenvectors of hypermatrices

  2025cites this paper
• Multi-subspace power method for decomposing all tensors

  2025influential citation
• Pseudospectrum of Kronecker product and stability analysis of tensor equations

  2025cites this paper
• Nonlinear Rayleigh quotient optimization

  2025cites this paper
• The high order spectral radius of graphs without long cycles or paths

  2025cites this paper
• Three Cases of Complex Eigenvalue/Vector Distributions of Symmetric Order-Three Random Tensors

  2024cites this paper
• Applications of Induced Tensor Norms to Guidance Navigation and Control

  2024influential citation
• Eigenpoint collinearities of plane cubics

  2024cites this paper
• Real analytic nonexpansive maps on polyhedral normed spaces

  2024cites this paper
• A power-like method for finding the spectral radius of a weakly irreducible nonnegative symmetric tensor

  2024cites this paper
• Convergence analysis of the largest and smallest H-eigenvalues for a class of tensor sequences

  2024cites this paper
• An Iterative Method for Horizontal Tensor Complementarity Problems

  2024cites this paper
• On the tensor spectral p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}$$\end{document}-norm and its high

  2024cites this paper
• An Efficient GIPM Algorithm for Computing the Smallest V-Singular Value of the Partially Symmetric Tensor

  2024cites this paper
• A new preconditioned Gauss-Seidel method for solving M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$

  2024cites this paper
• An Introduction to Complex Random Tensors

  2024influential citation
• Usefulness of signed eigenvalue/vector distributions of random tensors

  2024cites this paper
• Spectral bipartite Turán problems on linear hypergraphs

  2024cites this paper
• Signed eigenvalue/vector distribution of complex order-three random tensor

  2024cites this paper
• Sketch-based multiplicative updating algorithms for symmetric nonnegative tensor factorizations with applications to face image clustering

  2024influential citation
• Locally Optimal Eigenpairs of Orthogonally Decomposable Tensors: A Generalized Proof

  2024cites this paper
• HOSCF: Efficient decoupling algorithms for finding the best rank-one approximation of higher-order tensors

  2024cites this paper
• Extrapolation methods for multilinear PageRank

  2024cites this paper
• Krylov subspace methods for large multidimensional eigenvalue computation

  2024cites this paper
• Normalized Newton method to solve generalized tensor eigenvalue problems

  2024cites this paper
• A tensor Alternating Anderson-Richardson method for solving multilinear systems with M-tensors

  2024cites this paper
• Tight and Efficient Upper Bound on Spectral Norm of Convolutional Layers

  2024cites this paper
• Spectrum Situation Awareness for Space–Air–Ground Integrated Networks Based on Tensor Computing

  2024cites this paper
• An algorithm for calculating spectral radius of $ s $-index weakly positive tensors

  2024cites this paper
• Tradeoffs of Diagonal Fisher Information Matrix Estimators

  2024cites this paper
• An algorithm for the spectral radius of weakly essentially irreducible nonnegative tensors

  2024cites this paper
• Algebraic and geometric multiplicities for some hypergraphs

  2024cites this paper
• Moment-Sos and Spectral Hierarchies for Polynomial Optimization on the Sphere and Quantum de Finetti Theorems

  2024cites this paper
• Preconditioned inexact fixed point iteration method for solving tensor absolute value equation

  2024influential citation
• Brualdi-type Inequalities on Spectral Radius for the Hadamard Product of Nonnegative Tensors

  2024cites this paper
• A tensor Alternating Anderson-Richardson method for solving multilinear systems with M-tensors

  2024cites this paper
• A Modulus-Based Formulation for the Vertical Tensor Complementarity Problem

  2024cites this paper
• Refined Geršhgorin disks for tensors and GH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G{\mathcal {H}}$$\end{docum

  2024influential citation
• The Algebraic Multiplicity of the Spectral Radius of a Uniform Hypertree

  2024cites this paper
• Perfect colourings of hypergraphs

  2024cites this paper
• A high dimensional Cramer's rule connecting homogeneous multilinear equations to hyperdeterminants

  2024cites this paper
• The uniform supertrees with the extremal spectral radius

  2023cites this paper
• Largest and least H-eigenvalues of symmetric tensors and hypergraphs

  2023cites this paper
• Symmetry & Critical Points for Symmetric Tensor Decomposition Problems

  2023cites this paper
• On a relationship between the characteristic and matching polynomials of a uniform hypertree

  2023cites this paper