Probabilistic bounds on best rank-1 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 $ \def\xmlYUcy{\unicode{x042E}}1/n<^>{\frac {d-1}{2}} $ 1/nd-12, 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/n<^>{\frac {d-1}{2}} $ 1/nd-12.