CANDECOMP/PARAFAC Decomposition of High-Order Tensors Through Tensor Reshaping

In general, algorithms for order-3 CANDECOMP/PARAFAC (CP), also coined canonical polyadic decomposition (CPD), are easy to implement and can be extended to higher order CPD. Unfortunately, the algorithms become computationally demanding, and they are often not applicable to higher order and relatively large scale tensors. In this paper, by exploiting the uniqueness of CPD and the relation of a tensor in Kruskal form and its unfolded tensor, we propose a fast approach to deal with this problem. Instead of directly factorizing the high order data tensor, the method decomposes an unfolded tensor with lower order, e.g., order-3 tensor. On the basis of the order-3 estimated tensor, a structured Kruskal tensor, of the same dimension as the data tensor, is then generated, and decomposed to find the final solution using fast algorithms for the structured CPD. In addition, strategies to unfold tensors are suggested and practically verified in the paper.