# Spectral norm of a symmetric tensor and its computation

arXiv: https://arxiv.org/abs/1808.03864 · HTML: https://mlsanity.com/paper/1808.03864/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Aug 2018.

## Built on by (most cited first, 3 of 3)

- [Tight and Efficient Upper Bound on Spectral Norm of Convolutional Layers](https://mlsanity.com/paper/2409.11859/) (2024)
- [How to sketch a learning algorithm](https://mlsanity.com/paper/2604.07328/) (2026)
- [Data Understanding Survey: Pursuing Improved Dataset Characterization Via Tensor-based Methods](https://mlsanity.com/paper/2510.14161/) (2025)

## Abstract (arXiv)

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 mathbbCn are polynomially computable for a fixed n.