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 X∈ ℝ^d₁×^d₂×^d₃ is unique and obtain that the set consisting of all tensors satisfying the sufficient condition is an open and dense set in ℝ ^d₁ × d₂ × ^d₃. In addition, we present a necessary and sufficient condition under which the best rank-one approximation of the sum tensor X = X+E under some assumption on the tensor X ∈ R ^d₁ × ^d₂×^d₃ is equal to the best rank-one approximation of X. Meanwhile, a numerical algorithm is proposed for computing the quasi-singular value of a tensor. Finally, several testing examples are presented to illustrate the feasibility of the computational process and the correctness of the theoretical results obtained in the paper.