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.

翻译：规范多元（CP）分解是最广泛使用的张量分解技术之一。传统的CP分解算法将交替最小二乘（ALS）与正规方程相结合。然而，正规方程易受数值病态条件影响，这可能对分解结果产生不利影响。为缓解此问题，已有研究提出将ALS与QR分解结合，作为一种数值稳定性更高的替代方案。尽管该方法增强了稳定性，但其迭代过程涉及张量-矩阵乘积（TTM）运算，通常会导致较高的计算成本。为降低此成本，我们提出维度树的分支复用策略，该策略增加了中间张量的重用率并减少了TTM运算次数。此策略对三阶和四阶张量实现了计算复杂度降低$33\%$的效果。此外，我们在CP-ALS-QR算法中引入了一种专门的外推方法，利用矩阵$\mathbf{Q}_0$的独特结构以进一步提升收敛性。通过整合这两种技术，我们开发了一种新型CP分解算法，显著提升了计算效率。在五个真实数据集上的数值实验表明，与CP-ALS-QR算法相比，我们提出的算法降低了迭代成本并提高了拟合精度。