Probabilistic bounds on best rank-one approximation ratio

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 \(1/\sqrt{n^{d-1}}\).