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 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 (1−1/d)(d−1)/2. These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case.