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Probabilistic bounds on best rank-one approximation ratio

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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 $1/\sqrt{n^{d-1}}$.
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Dates and versions

hal-03517267 , version 1 (07-01-2022)

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Khazhgali Kozhasov, Josué Tonelli-Cueto. Probabilistic bounds on best rank-one approximation ratio. 2022. ⟨hal-03517267⟩
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