• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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Verlag der Österreichischen Akademie der Wissenschaften
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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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Flip-flop spectrum-revealing QR factorization and its applications to singular value decomposition

    Yuehua Feng, Jianwei Xiao, Ming Gu

ETNA - Electronic Transactions on Numerical Analysis, pp. 469-494, 2019/12/10

doi: 10.1553/etna_vol51s469

doi: 10.1553/etna_vol51s469


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doi:10.1553/etna_vol51s469



doi:10.1553/etna_vol51s469

Abstract

We present the Flip-Flop Spectrum-Revealing QR (Flip-Flop SRQR) factorization, a significantly faster and more reliable variant of the QLP factorization of Stewart for low-rank matrix approximations. Flip-Flop SRQR uses SRQR factorization to initialize a partial column-pivoted QR factorization and then computes a partial LQ factorization. As observed by Stewart in his original QLP work, Flip-Flop SRQR tracks the exact singular values with “considerable fidelity”. We develop singular value lower bounds and residual error upper bounds for the Flip-Flop SRQR factorization. In situations where singular values of the input matrix decay relatively quickly, the low-rank approximation computed by Flip-Flop SRQR is guaranteed to be as accurate as the truncated SVD. We also perform a complexity analysis to show that Flip-Flop SRQR is faster than the randomized subspace iteration for approximating the SVD, the standard method used in the Matlab tensor toolbox. We additionally compare Flip-Flop SRQR with alternatives on two applications, a tensor approximation and a nuclear norm minimization, to demonstrate its efficiency and effectiveness.

Keywords: QR factorization, randomized algorithm, low-rank approximation, approximate SVD, higher-order SVD, nuclear norm minimization

Published Online: 2019/12/10 12:55:18

Object Identifier: 0xc1aa5572 0x003b14b6

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