The multilinear Pagerank model [Gleich, Lim and Yu, 2015] is a tensor-based generalization of the Pagerank model. Its computation requires solving a system of polynomial equations that contains a parameter $\alpha \in [0,1)$. For $\alpha \approx 1$, this computation remains a challenging problem, especially since the solution may be non-unique. Extrapolation strategies that start from smaller values of $\alpha$ and `follow' the solution by slowly increasing this parameter have been suggested; however, there are known cases where these strategies fail, because a globally continuous solution curve cannot be defined as a function of $\alpha$. In this paper, we improve on this idea, by employing a predictor-corrector continuation algorithm based on a more general representation of the solutions as a curve in $\mathbb{R}^{n+1}$. We prove several global properties of this curve that ensure the good behavior of the algorithm, and we show in our numerical experiments that this method is significantly more reliable than the existing alternatives.
翻译:多线性Pagerank模型[Gleich, Lim和Y, 2015] 是一个基于“Perchank”模型的强效概括。它的计算需要解决一个包含一个参数$\alpha\ in[ 0, 1]$的多元方程式系统。对于$\alpha\ approx 1美元,这一计算仍是一个具有挑战性的问题,特别是因为解决办法可能是非单一的。有人建议了从较小值$\alpha$开始的外推法策略和缓慢增加这一参数的“遵循”解决方案;然而,有些已知的情况是,这些策略失败了,因为无法将全球持续解决方案曲线定义为$\alpha$的函数。在本文中,我们改进了这一想法,采用了一种预测或校正或连续的算法,其基础是更笼统地表示解决办法的曲线值为$\\mathb{R ⁇ n+1}。我们证明了这一曲线的一些全球特性,可以确保算法的良好行为,我们在数字实验中表明,这种方法比现有的替代方法可靠得多。