Search

MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint
27/2021

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

Henrik Eisenmann and André Uschmajew

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.

Received:
24.11.2021
Published:
24.11.2021

Related publications

inJournal
2023 Journal Open Access
Henrik Eisenmann and André Uschmajew

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

In: Annali di matematica pura ed applicata, 202 (2023) 2, pp. 993-1009