Maximum relative distance between real rank-two and rank-one tensors

TL;DR

This paper establishes tight bounds on how close a real symmetric rank-two tensor can be to a rank-one tensor in Frobenius norm, revealing fundamental limits on tensor approximation quality.

Contribution

It derives the first explicit upper bound on the relative Frobenius distance for rank-two tensors and characterizes the minimal spectral-to-Frobenius norm ratio for symmetric tensors.

Findings

Bound on Frobenius distance: . . .

Minimal spectral-to-Frobenius ratio: . .

Bounds verified for arbitrary rank-two tensors.

Abstract

It is shown that the relative distance in Frobenius norm of a real symmetric order-$d$ tensor of rank two to its best rank-one approximation is upper bounded by $\sqrt{1-(1-1/d)^{d-1}}$. This is achieved by determining the minimal possible ratio between spectral and Frobenius norm for symmetric tensors of border rank two, which equals $\left(1-{1}/{d}\right)^{(d-1)/{2}}$. These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case.