The properties and the low-rank approximation of a class of unitary decomposable fourth-order conjugate partial symmetric tensors

In this paper, we extend the notion of unitary decomposable tensors to general unitary decomposable tensors, and study the properties and the low-rank approximation of fourth-order general unitary decomposable tensors with conjugate partial symmetric (CPS) property. We show that the CPS rank one approximation problem for fourth-order general unitary decomposable CPS tensors is equivalent to maximizing a complex quadratic form with 1-norm constraint. For two classes of fourth-order general unitary decomposable CPS tensors, we show that the successive rank one approximation algorithm along with the successive rank two approximation algorithm can exactly recover the original tensor. For fourth-order partial symmetric (PS) tensors, we prove that when the rank of the tensor is smaller than its order, then the CP rank and the PS rank coincide. Based on this, we characterize the shape of all rank 2 fourth-order CPS tensors.