Probabilistic bounds on best rank-one approximation ratio

TL;DR

This paper establishes new bounds on the ratio of spectral to Frobenius norms for symmetric and general tensors, revealing asymptotic behaviors as tensor dimensions and order vary.

Contribution

It provides novel upper and lower bounds on the spectral-Frobenius norm ratio for tensors, improving understanding of tensor approximation limits.

Findings

For general tensors, recovers known upper bounds.

For symmetric tensors, shows the ratio matches the order of the trivial lower bound as dimension grows.

When tensor order increases with fixed dimension, the lower bound outperforms the trivial bound.

Abstract

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric tensors our upper bound unveils that the ratio of norms has the same order of magnitude as the trivial lower bound $1/\sqrt{n^{d-1}}$, when the order of a tensor $d$ is fixed and the dimension of the underlying vector space $n$ tends to infinity. However, when $n$ is fixed and $d$ tends to infinity, our lower bound is better than $1/\sqrt{n^{d-1}}$.