Sparse power methods for large-scale higher-order PageRank problems

A commonly used technique for the higher-order PageRank problem is the power method that is computationally intractable for large-scale problems. The truncated power method proposed recently provides us with another idea to solve this problem, however, its accuracy and efficiency can be poor in practical computations. In this work, we revisit the higher-order PageRank problem and consider how to solve it efficiently. The contribution of this work is as follows. First, we accelerate the truncated power method for high-order PageRank. In the improved version, it is neither to form and store the vectors arising from the dangling states, nor to store an auxiliary matrix. Second, we propose a truncated power method with partial updating to further release the overhead, in which one only needs to update some important columns of the approximation in each iteration. On the other hand, the truncated power method solves a modified high-order PageRank model for convenience, which is not mathematically equivalent to the original one. Thus, the third contribution of this work is to propose a sparse power method with partial updating for the original higher-order PageRank problem. The convergence of all the proposed methods are discussed. Numerical experiments on large and sparse real-world and synthetic data sets are performed. The numerical results show the superiority of our new algorithms over some state-of-the-art ones for large and sparse higher-order PageRank problems.

翻译：高阶 PageRank 问题通常使用的一种技术。高阶 PageRank 问题通常使用的方法是计算无法解决大规模问题的电源方法。最近提出的短压电力方法为我们提供了另一种解决问题的方法。但是,在实际计算中,其准确性和效率可能较差。在这项工作中,我们重新研究高阶PageRank 问题,并考虑如何有效地解决这个问题。这项工作的贡献如下。首先, 我们加快高阶PageRank 的短压电源方法。在经过改进的版本中,它既不形成和储存来自相交国家的矢量,也不储存辅助矩阵。其次,我们提出一种松动的电源方法,部分更新以进一步释放间接费用。在这项工作中,我们只需更新每层的近似值的一些重要栏目。另一方面, 松动的电法解决了经过修改的高阶PageRank 的方便性高压电源模型,这在数学上不等同于原来的模式。因此, 这项工作的第三项贡献是提出一种稀疏的电源方法, 部分更新了原始的精度的精度方法, 并且正在讨论的是原高阶的数学模拟的数学模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟模拟