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

TL;DR

This paper introduces a fast method for high-order tensor decomposition by unfolding tensors into lower order, then applying efficient algorithms to the unfolded tensor and reconstructing the original tensor's factors.

Contribution

The paper proposes a novel approach that leverages tensor unfolding and structured CPD to efficiently decompose high-order tensors, reducing computational complexity.

Findings

Effective tensor unfolding strategies verified experimentally

Significant reduction in computational time for high-order tensor decomposition

Accurate factorization achieved comparable to direct methods

Abstract

In general, algorithms for order-3 CANDECOMP/-PARAFAC (CP), also coined canonical polyadic decomposition (CPD), are easily 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 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.