Efficient QR-Based CP Decomposition Acceleration via Dimension Tree and Extrapolation

Abstract: The canonical polyadic (CP) decomposition is one of the most widely used tensor decomposition techniques. The conventional CP decomposition algorithm combines alternating least squares (ALS) with the normal equation. However, the normal equation is susceptible to numerical ill-conditioning, which can adversely affect the decomposition results. To mitigate this issue, ALS combined with QR decomposition has been proposed as a more numerically stable alternative. Although this method enhances stability, its iterative process involves tensor-times-matrix (TTM) operations, which typically result in higher computational costs. To reduce this cost, we propose branch reutilization of dimension tree, which increases the reuse of intermediate tensors and reduces the number of TTM operations. This strategy achieves a $33\%$ reduction in computational complexity for third and fourth order tensors. Additionally, we introduce a specialized extrapolation method in CP-ALS-QR algorithm, leveraging the unique structure of the matrix $\mathbf{Q}_0$ to further enhance convergence. By integrating both techniques, we develop a novel CP decomposition algorithm that significantly improves efficiency. Numerical experiments on five real-world datasets show that our proposed algorithm reduces iteration costs and enhances fitting accuracy compared to the CP-ALS-QR algorithm.