Upper bounds for the spectral norm of symmetric tensors

TL;DR

This paper introduces two sequences of upper bounds for the spectral norm of symmetric tensors, based on roots of Hilbert-Schmidt norms, and proves their convergence to the minimal upper bound.

Contribution

It presents novel sequences of upper bounds for tensor spectral norms and establishes their convergence properties, extending to iterates of polynomial maps.

Findings

Sequences converge to the minimal upper bound

Bounds are expressed via roots of Hilbert-Schmidt norms

Generalizations to polynomial map iterates are discussed

Abstract

The maximum of the absolute value of a real homogeneous polynomial of degree $d\ge 3$ on the unit sphere corresponds to the spectral norm of the induced real $d$-symmetric tensor $\mathcal{S}$. We give two sequences of upper bounds on the spectral norm of $\mathcal{S}$, which are stated in terms of certain roots of the Hilbert-Schmidt norms of corresponding iterates. We show that these sequences are converging to a limit, which is the minimal value of these upper bounds. Some generalizations to iterates of homogeneous polynomial maps are discussed.