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arXiv · 2111.12611

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

Henrik Eisenmann·André Uschmajew
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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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Henrik Eisenmann, André Uschmajew. 2021-11-24. Maximum relative distance between real rank-two and rank-one tensors. https://arxiv.org/abs/2111.12611

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