
清華大學、交通大學 
統 計 學 研 究 所 
專 題 演 講 
講 題：  An Integrated ShiftInvert Residual Arnoldi Method for Graph Laplacian Matrices  
演講者：  黃韋強博士（國立交通大學應用數學系）  
時 間：  107年04月27日（星期五）10：00  10：50（10:50  11:10 茶會於統計所821室舉行）  
地 點：  綜合三館837室  
摘 要： 
The eigenvalue problem of a graph Laplacian matrix arising from a simple, connected and undirected graph has been given more attention due to its extensive applications in the field of the machine learning. The associated graph Laplacian matrix is symmetric, positive semidefinite, and is usually large and sparse. Computing some smallest positive eigenvalues and corresponding eigenvectors is often of interest for either clustering or dimensionality reduction. However, its singularity makes the classical eigensolvers inefficient since we need to solve related linear systems. Moreover, for largescaled networks from real world, such as social media, transactional databases and sensor systems, there are in general not only local connections. Therefore, it is usually timeconsuming, or even unable, to directly find the matrix factorization for solving involved linear systems exactly. In this talk, we propose an innerouter iterative eigensolver, iSIRA, based on the residual Arnoldi method together with an implicit remedy of the singularity and an effective deflation for convergent eigenvalues. Numerical experiments demonstrate that the integrated eigensolver outperforms the classical methods especially in the case when the matrix factorization is not available. 