Spectral norm of a symmetric tensor and its computation

Friedland, Shmuel; Wang, Li

doi:10.1090/mcom/3525

Spectral norm of a symmetric tensor and its computation
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by Shmuel Friedland and Li Wang;

Math. Comp. 89 (2020), 2175-2215

DOI: https://doi.org/10.1090/mcom/3525

Published electronically: May 15, 2020

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Abstract:

We show that the spectral norm of a $d$-mode real or complex symmetric tensor in $n$ variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed points is finite, and we give upper and lower bounds for the number of fixed points. For $n=2$ we show that these fixed points are the roots of a corresponding univariate polynomial of degree at most $(d-1)^2+1$, except certain cases, which are completely analyzed. In particular, for $n=2$ the spectral norm of $d$-symmetric tensor is polynomially computable in $d$ with a given relative precision. For a fixed $n>2$ we show that the spectral norm of a $d$-mode symmetric tensor is polynomially computable in $d$ with a given relative precision with respect to the Hilbert-Schmidt norm of the tensor. These results show that the geometric measure of entanglement of $d$-mode symmetric qunits on $\mathbb {C}^n$ are polynomially computable for a fixed $n$.

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