Singular values and eigenvalues of tensors: a variational approach
Lek-Heng Lim
- pp 129-132
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TLDR
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 eigen values is proposed.Abstract:
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 theoremread more
Citations
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Tensor Decompositions and Applications
Tamara G. Kolda,Brett W. Bader +1 more
TL;DR: This survey provides an overview of higher-order tensor decompositions, their applications, and available software.
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Most Tensor Problems Are NP-Hard
TL;DR: In this paper, it was shown that determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm, approximating an eigen value, eigenvector, singular vector, or the spectral norm is NP-hard and computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.
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Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging
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Tensor decompositions for learning latent variable models
TL;DR: In this article, the authors consider a wide class of latent variable models, including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation, which exploit a certain tensor structure in their low-order observable moments (typically, of second and third-order).
References
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Book
Nonnegative Matrices in the Mathematical Sciences
TL;DR: 1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of non negative matrices 4. Symmetric nonnegativeMatrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative methods for linear systems 8. Finite Markov Chains
Journal ArticleDOI
A Multilinear Singular Value Decomposition
TL;DR: There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed.
Journal ArticleDOI
On the Best Rank-1 and Rank-( R 1 , R 2 ,. . ., R N ) Approximation of Higher-Order Tensors
TL;DR: A multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher-order tensor, in an optimal least-squares sense, by a tensor that has prespecified column rank value, rowRank value, etc.
Journal ArticleDOI
Eigenvalues of a real supersymmetric tensor
TL;DR: It is shown that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvaluesare roots of another one- dimensional polynomials associated with the symmetric hyperdeterminant.
Book
Tensor Norms and Operator Ideals
Andreas Defant,Klaus Floret +1 more
TL;DR: Tensor Norms and Operator Ideals as mentioned in this paper are a generalization of the algebraic theory of Tensor Products and have been studied extensively in the literature, e.g. in the context of algebraic theories of tensors.