Subtracting a best rank-1 approximation from p × p × 2(p ≥2) tensors

SUMMARY We introduce one special form of the ptimesp × 2 (p≥2) tensors by multilinear orthonormal transformations, and present some interesting properties of the special form. Through discussing on the special form, we provide a solution to one conjecture proposed by Stegeman and Comon in a conference paper (Proceedings of the EUSIPCO 2009 Conference, Glasgow, Scotland, 2009), and reveal an important conclusion about subtracting a best rank-1 approximations from p × p × 2 tensors of the special form. All of these confirm that consecutively subtracting the best rank-1 approximations may not lead to a best low rank approximation of a tensor. Numerical examples show the correctness of our theory. Copyright © 2011 John Wiley & Sons, Ltd.