Modeling complex multiway relationships in large-scale networks is becoming more and more challenging in data science. The multilinear PageRank problem, arising naturally in the study of higher-order Markov chains, is a powerful framework for capturing such interactions, with applications in web ranking, recommendation systems, and social network analysis. It extends the classical Google PageRank model to a tensor-based formulation, leading to a nonlinear system that captures multi-way dependencies between states. Newton-based methods can achieve local quadratic convergence for this problem, but they require solving a large linear system at each iteration, which becomes too costly for large-scale applications. To address this challenge, we present an accelerated Newton-GMRES method that leverages Krylov subspace techniques to approximate the Newton step without explicitly forming the large Jacobian matrix. We further employ vector extrapolation methods, including Minimal Polynomial Extrapolation (MPE), Reduced Rank Extrapolation (RRE), and Anderson Acceleration (AA), to improve the convergence rate and enhance numerical stability. Extensive experiments on synthetic and real-world data demonstrate that the proposed approach significantly outperforms classical Newton-based solvers in terms of efficiency, robustness, and scalability.

翻译：在大规模网络中建模复杂的多向关系正成为数据科学领域日益严峻的挑战。多线性PageRank问题，作为高阶马尔可夫链研究中自然产生的框架，是捕捉此类交互的强大工具，广泛应用于网页排序、推荐系统和社交网络分析。它将经典的谷歌PageRank模型扩展为基于张量的表述，从而导出一个能够捕捉状态间多向依赖关系的非线性系统。基于牛顿的方法对此问题可实现局部二次收敛，但每次迭代都需要求解一个大型线性系统，这在大规模应用中计算成本过高。为应对这一挑战，我们提出了一种加速牛顿-GMRES方法，该方法利用Krylov子空间技术来近似牛顿步，而无需显式构造大型雅可比矩阵。我们进一步采用向量外推法，包括最小多项式外推（MPE）、降秩外推（RRE）和安德森加速（AA），以提高收敛速度并增强数值稳定性。在合成数据和真实数据上进行的大量实验表明，所提出的方法在效率、鲁棒性和可扩展性方面均显著优于经典的基于牛顿的求解器。