Best rank one approximation of real symmetric tensors can be chosen symmetric
TLDR
In this paper, it was shown that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric if the tensor does not lie on a certain real algebraic variety.Abstract:
We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.read more
Citations
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Journal ArticleDOI
Tensors : A brief introduction
TL;DR: Tensor decompositions are at the core of many blind source separation (BSS) algorithms, either explicitly or implicitly, and plays a central role in the identification of underdetermined mixtures.
Journal ArticleDOI
The Euclidean Distance Degree of an Algebraic Variety
TL;DR: A theory of such nearest point maps of a real algebraic variety with respect to Euclidean distance from the perspective of computational algebraic geometry is developed.
Journal ArticleDOI
Nuclear norm of higher-order tensors
Shmuel Friedland,Lek-Heng Lim +1 more
TL;DR: In this paper, it was shown that computing tensor nuclear norm is NP-hard in several sense, such as determining weak membership in the nuclear norm unit ball of 3-tensors, as well as finding an ε-varepsilon-approximation of nuclear norm for 3-Tensors.
Journal ArticleDOI
The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors
TL;DR: The uniqueness of best approximations for almost all real tensors in the following cases is shown: rank-one approximation; rank- one approximation for partially symmetric tensors (this approximation is also partially asymmetric); rank-(r_1,\ldots ,r_d) approximation for d-mode tensors.
Journal ArticleDOI
Uniqueness of Nonnegative Tensor Approximations
TL;DR: In this article, it was shown that for a nonnegative tensor, a best nonnegative rank-$r$ approximation is almost always unique, its best rank-one approximation may always be chosen to be a best nonsmooth rank-1 approximation, and the set of non-negative tensors with non-unique best-rank-one approximations forms an algebraic hypersurface.
References
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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.
Proceedings ArticleDOI
Singular values and eigenvalues of tensors: a variational approach
TL;DR: 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.
Journal ArticleDOI
Maximum Block Improvement and Polynomial Optimization
TL;DR: It is established that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem; thus the MBI approach can maximize a homogeneity Polynomial function over a Sphere by its Tensor relaxation via the MBO approach.
Journal ArticleDOI
Inverse eigenvalue problems
TL;DR: In this article, the classical inverse additive and multiplicative inverse eigenvalue problems for matrices are studied using general results on the solvability of polynomial systems and it is shown that in the complex case these problems are always solvable by a finite number of solutions.