Algorithms in invariant theory
Algorithms in invariant theory
On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors
SIAM Journal on Matrix Analysis and Applications
Multivariate Polynomial Minimization and Its Application in Signal Processing
Journal of Global Optimization
D-eigenvalues of diffusion kurtosis tensors
Journal of Computational and Applied Mathematics
On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function
MICCAI '08 Proceedings of the 11th international conference on Medical Image Computing and Computer-Assisted Intervention - Part I
Extreme diffusion values for non-Gaussian diffusions
Optimization Methods & Software - THE JOINT EUROPT-OMS CONFERENCE ON OPTIMIZATION, 4-7 JULY, 2007, PRAGUE, CZECH REPUBLIC, PART I
An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor
Journal of Computational and Applied Mathematics
Finding the Largest Eigenvalue of a Nonnegative Tensor
SIAM Journal on Matrix Analysis and Applications
Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations
SIAM Journal on Optimization
SIAM Journal on Optimization
Higher Order Positive Semidefinite Diffusion Tensor Imaging
SIAM Journal on Imaging Sciences
Computational Optimization and Applications
Further Results for Perron-Frobenius Theorem for Nonnegative Tensors
SIAM Journal on Matrix Analysis and Applications
On the best rank-1 approximation to higher-order symmetric tensors
Mathematical and Computer Modelling: An International Journal
Maximum Block Improvement and Polynomial Optimization
SIAM Journal on Optimization
Further Results for Perron-Frobenius Theorem for Nonnegative Tensors II
SIAM Journal on Matrix Analysis and Applications
Gradient skewness tensors and local illumination detection for images
Journal of Computational and Applied Mathematics
Algebraic connectivity of an even uniform hypergraph
Journal of Combinatorial Optimization
On determinants and eigenvalue theory of tensors
Journal of Symbolic Computation
On solving biquadratic optimization via semidefinite relaxation
Computational Optimization and Applications
Nonnegative Diffusion Orientation Distribution Function
Journal of Mathematical Imaging and Vision
Eigenvectors of tensors and algorithms for Waring decomposition
Journal of Symbolic Computation
Most Tensor Problems Are NP-Hard
Journal of the ACM (JACM)
Criterions for the positive definiteness of real supersymmetric tensors
Journal of Computational and Applied Mathematics
An equi-directional generalization of adaptive cross approximation for higher-order tensors
Applied Numerical Mathematics
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In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An mth-order n-dimensional supersymmetric tensor where m is even has exactly n(m-1)^n^-^1 eigenvalues, and the number of its E-eigenvalues is strictly less than n(m-1)^n^-^1 when m=4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m-1)^n^-^1. The n(m-1)^n^-^1 eigenvalues are distributed in n disks in C. The centers and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations.