On the Uniqueness and Perturbation to the Best Rank-One Approximation of a Tensor

The uniqueness of the best rank-one approximation of a tensor under the Frobenius norm is a basic and important studying subject in the tensor theory. By introducing the quasi-singular value of a tensor, we present a new sufficient condition under which the best rank-one approximation of a nonzero tensor $\underline{X}\in\mathbb{R}^{d_{1}\times d_{2}\times d_{3}}$ is unique and obtain that the set consisting of all tensors satisfying the sufficient condition is an open and dense set in $\mathbb{R}^{d_{1}\times d_{2}\times d_{3}}$. In addition, we present a necessary and sufficient condition under which the best rank-one approximation of the sum tensor $\underline{\tilde X}=\underline{X} + {\underline E}$ under some assumption on the tensor $\underline{\tilde X}\in\mathbb{R}^{d_{1}\times d_{2}\times d_{3}}$ is equal to the best rank-one approximation of ${\underline X}$. Meanwhile, a numerical algorithm is proposed for computing the quasi-singular value of a tensor. Finally, several testing examples are pr...