Best rank one approximation of real symmetric tensors can be chosen symmetric

By Shmuel Friedland. 2011

63 citations recorded by the source corpus.

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.

Subfields: Biomedical Engineering, Computational Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/bcf3b7391ba5dd18da4e0cd9ef70e96b18cc50a5

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In Biomedical Engineering

In Computational Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Tensors : A brief introduction (2014, unknown): “It has been proved recently that the best rank-1 approximation of a symmetric tensor is symmetric [34]; a shorter proof can be found in [35], as well as uniqueness issues.”
Most Tensor Problems Are NP-Hard (2009, result): “First, we state a recent technical result in [Friedland 2011] saying that the best rank1 approximation of a symmetric tensor (by definition, a tensor x ⊗ y ⊗ z) and its best symmetric-rank-1 approximation (by definition, a symmetric tensor x ⊗ x ⊗ x) may be chosen to be the same.”
Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors (2013, background): “Whether it is possible to construct similar examples for (super)symmetric tensors remains an open question; see [22].”