Best rank-one approximation ratio of higher-order tensors

For a higher-order tensor, its distance to the set of rank-one tensors is an important quantity of interest from several perspectives. Similar as for matrices, the relative distance is determined by the ratio of the spectral and Frobenius norm of the tensor. The minimal possible ratio of these norms hence measures the largest possible relative distance, and is called the best rank-one approximation ratio of the tensor space under consideration. Its precise value is only known in special cases and hard to determine in general. In the talk, we review this problem and present some recent results on the minimal norm ratio.