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

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