Best rank-one approximation is one of the most fundamental tasks in tensor computation. In order to fully exploit modern multi-core parallel computers, it is necessary to develop decoupling algorithms for computing the best rank-one approximation of higher-order tensors at large scales. In this paper, we first build a bridge between the rank-one approximation of tensors and the eigenvector-dependent nonlinear eigenvalue problem (NEPv), and then develop an efficient decoupling algorithm, namely the higher-order self-consistent field (HOSCF) algorithm, inspired by the famous self-consistent field (SCF) iteration frequently used in computational chemistry. The convergence theory of the HOSCF algorithm and an estimation of the convergence speed are further presented. In addition, we propose an improved HOSCF (iHOSCF) algorithm that incorporates the Rayleigh quotient iteration, which can significantly accelerate the convergence of HOSCF. Numerical experiments show that the proposed algorithms can efficiently converge to the best rank-one approximation of both synthetic and real-world tensors and can scale with high parallel scalability on a modern parallel computer.

翻译：最佳秩一逼近是张量计算中最基础的问题之一。为充分利用现代多核并行计算机的计算能力，需要开发用于大规模计算高阶张量最佳秩一逼近的分解算法。本文首先建立了张量秩一逼近与本征向量依赖的非线性本征值问题（NEPv）之间的联系，进而受计算化学中常用的自洽场（SCF）迭代方法的启发，提出了一种高效的分解算法——高阶自洽场（HOSCF）算法。进一步给出了HOSCF算法的收敛性理论及收敛速度估计。此外，我们提出了一种改进的HOSCF（iHOSCF）算法，该算法融合了Rayleigh商迭代，能显著加速HOSCF的收敛。数值实验表明，所提算法能够高效收敛到合成张量与真实世界张量的最佳秩一逼近，并在现代并行计算机上展现出优异的并行可扩展性。