標題: iSIRA: Integrated shift-invert residual Arnoldi method for graph Laplacian matrices from big data
作者: Huang, Wei-Qiang
Lin, Wen-Wei
Lu, Henry Horng-Shing
Yau, Shing-Tung
交大名義發表
應用數學系
丘成桐中心
大數據研究中心
National Chiao Tung University
Department of Applied Mathematics
Shing-Tung Yau Center
Big Data Res Ctr
關鍵字: Graph Laplacian matrix;Eigenvalue problem;Trimming;Deflation;Shift-invert residual Arnoldi;Inexact eigensolver
公開日期: 15-Jan-2019
摘要: The eigenvalue problem of a graph Laplacian matrix L arising from a simple, connected and undirected graph has been given more attention due to its extensive applications, such as spectral clustering, community detection, complex network, image processing and so on. The associated matrix L is symmetric, positive semi-definite, and is usually large and sparse. It is often of interest for finding some smallest positive eigenvalues and corresponding eigenvectors. However, the singularity of L makes the classical eigensolvers inefficient since we need to factorize L for the purpose of solving large and sparse linear systems exactly. The next difficulty is that it is usually prohibitive to factorize L generated by real network problems from big data such as social media transactional databases, and sensor systems because there is in general not only local connections. In this paper, we propose a trimming to cure the singularity of L according to its special property: zero row/column sum. This remedy technique leads us to solve a positive definite linear system reduced in one dimension and then recover the result to get a suitable solution of the original system involved in an eigensolver. Besides, we apply a deflating approach to exclude the influence of converged eigenvalues. We show how to apply the idea of trimming to the graph Laplacian eigenvalue problem together with a deflated term and a target shift. Accordingly, based on the inexact residual Arnoldi (Lee, 2007; Lee and Stewart, 2007) method, we propose an integrated eigensolver for this kind of L combining with the implicit remedy of the singularity, an effective deflation for convergent eigenvalues and the shift-invert enhancement. Numerical experiments reveal that the integrated eigensolver outperforms the classical Arnoldi/Lanczos method for computing some smallest positive eigeninformation especially when the LU factorization is not available. (C) 2018 Elsevier B.V. All rights reserved.
URI: http://dx.doi.org/10.1016/j.cam.2018.07.031
http://hdl.handle.net/11536/148408
ISSN: 0377-0427
DOI: 10.1016/j.cam.2018.07.031
期刊: JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
Volume: 346
起始頁: 518
結束頁: 531
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