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Accelerating large partial EVD/SVD calculations by filtered block Davidson methods
Zhou Yunkai1; Wang Zheng1; Zhou Aihui2
AbstractPartial eigenvalue decomposition (PEVD) and partial singular value decomposition (PSVD) of large sparse matrices are of fundamental importance in a wide range of applications, including latent semantic indexing, spectral clustering, and kernel methods for machine learning. The more challenging problems are when a large number of eigenpairs or singular triplets need to be computed. We develop practical and efficient algorithms for these challenging problems. Our algorithms are based on a filter-accelerated block Davidson method. Two types of filters are utilized, one is Chebyshev polynomial filtering, the other is rational-function filtering by solving linear equations. The former utilizes the fastest growth of the Chebyshev polynomial among same degree polynomials; the latter employs the traditional idea of shift-invert, for which we address the important issue of automatic choice of shifts and propose a practical method for solving the shifted linear equations inside the block Davidson method. Our two filters can efficiently generate high-quality basis vectors to augment the projection subspace at each Davidson iteration step, which allows a restart scheme using an active projection subspace of small dimension. This makes our algorithms memory-economical, thus practical for large PEVD/PSVD calculations. We compare our algorithms with representative methods, including ARPACK, PROPACK, the randomized SVD method, and the limited memory SVD method. Extensive numerical tests on representative datasets demonstrate that, in general, our methods have similar or faster convergence speed in terms of CPU time, while requiring much lower memory comparing with other methods. The much lower memory requirement makes our methods more practical for large-scale PEVD/PSVD computations.
Keywordpartial EVD/SVD polynomial filter rational filter kernel graph
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000380212100013
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Document Type期刊论文
Corresponding AuthorZhou Yunkai
Affiliation1.Southern Methodist Univ, Dept Math, Dallas, TX 75275 USA
2.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, LSEC, Beijing 100190, Peoples R China
Recommended Citation
GB/T 7714
Zhou Yunkai,Wang Zheng,Zhou Aihui. Accelerating large partial EVD/SVD calculations by filtered block Davidson methods[J]. SCIENCE CHINA-MATHEMATICS,2016,59(8):1635-1662.
APA Zhou Yunkai,Wang Zheng,&Zhou Aihui.(2016).Accelerating large partial EVD/SVD calculations by filtered block Davidson methods.SCIENCE CHINA-MATHEMATICS,59(8),1635-1662.
MLA Zhou Yunkai,et al."Accelerating large partial EVD/SVD calculations by filtered block Davidson methods".SCIENCE CHINA-MATHEMATICS 59.8(2016):1635-1662.
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