Abstract SimRank is popularly used for evaluating similarities between nodes in a graph. Though some recent work has transformed the SimRank into a linear system which can be solved by using conjugate gradient (CG) algorithm. However, the diagonal preconditioner used in the algorithm still left some room for improvement. There is no work showing what should be an optimal preconditioner for the linear system. For such a sake, in this paper, we theoretically study the preconditioner problem and propose a polynomial preconditioner which can improve the computation speed significantly. In our investigation, we also modify the original linear system to adapt to all situations of the graph. We further prove that the condition number of a polynomial preconditioned matrix is bounded in a narrow interval. It implies that the new preconditioner can effectively accelerate the convergence rate of the CG algorithm. The experimental results show the proposed algorithm outperforms the state-of-the-art algorithms for the all-pairs SimRank computation.

中文翻译：

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通过多项式预处理器加速 SimRank 计算

摘要 SimRank 广泛用于评估图中节点之间的相似性。尽管最近的一些工作已将 SimRank 转换为可以使用共轭梯度 (CG) 算法求解的线性系统。然而，算法中使用的对角预处理器仍然有一些改进的空间。没有工作说明什么应该是线性系统的最佳预处理器。为此，在本文中，我们从理论上研究了预处理器问题，并提出了一种可以显着提高计算速度的多项式预处理器。在我们的调查中，我们还修改了原始线性系统以适应图的所有情况。我们进一步证明多项式预处理矩阵的条件数在一个窄区间内是有界的。这意味着新的预处理器可以有效地加快CG算法的收敛速度。实验结果表明，所提出的算法优于所有对 SimRank 计算的最新算法。