Deviation maximization for rank-revealing QR factorizations

In this paper, we introduce a new column selection strategy, named here “Deviation Maximization”, and apply it to compute rank-revealing QR factorizations as an alternative to the well-known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented i...

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Published in:	Numerical algorithms Vol. 91; no. 3; pp. 1047 - 1079
Main Authors:	Dessole, Monica, Marcuzzi, Fabio
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.11.2022 Springer Nature B.V
Subjects:	Algebra Algorithms Computer Science Decomposition Deviation Eigenvalues Linear algebra Maximization Numeric Computing Numerical Analysis Optimization Original Paper Theory of Computation Column pivoting Correlation Rank revealing Block algorithm QR factorization
ISSN:	1017-1398, 1572-9265
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Abstract	In this paper, we introduce a new column selection strategy, named here “Deviation Maximization”, and apply it to compute rank-revealing QR factorizations as an alternative to the well-known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented in LAPACK’s xgeqp3 routine. We show that the resulting algorithm, named QRDM, has similar rank-revealing properties of QP3 and better execution times. We present experimental results on a wide data set of numerically singular matrices, which has become a reference in the recent literature.
AbstractList	In this paper, we introduce a new column selection strategy, named here “Deviation Maximization”, and apply it to compute rank-revealing QR factorizations as an alternative to the well-known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented in LAPACK’s xgeqp3 routine. We show that the resulting algorithm, named QRDM, has similar rank-revealing properties of QP3 and better execution times. We present experimental results on a wide data set of numerically singular matrices, which has become a reference in the recent literature. In this paper, we introduce a new column selection strategy, named here “Deviation Maximization”, and apply it to compute rank-revealing QR factorizations as an alternative to the well-known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented in LAPACK’s routine. We show that the resulting algorithm, named QRDM, has similar rank-revealing properties of QP3 and better execution times. We present experimental results on a wide data set of numerically singular matrices, which has become a reference in the recent literature.
Author	Dessole, Monica Marcuzzi, Fabio
Author_xml	– sequence: 1 givenname: Monica orcidid: 0000-0002-2727-9123 surname: Dessole fullname: Dessole, Monica email: monica.dessole.ext@leonardo.com organization: Department of Mathematics Tullio Levi Civita, University of Padova – sequence: 2 givenname: Fabio surname: Marcuzzi fullname: Marcuzzi, Fabio organization: Department of Mathematics Tullio Levi Civita, University of Padova
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Cites_doi	10.1145/290200.287638 10.1137/1.9780898719604 10.14658/PUPJ-DRNA-2020-1-3 10.1016/j.laa.2017.10.014 10.1137/1029112 10.1137/S1064827595296732 10.1007/BF02139475 10.1137/13092157X 10.1007/BF01436084 10.1007/BF01436075 10.1016/0024-3795(75)90112-3 10.1137/0917055 10.1137/0727045 10.1145/290200.287637 10.1016/0024-3795(86)90115-1 10.1145/1377612.1377616 10.1137/0910005 10.3390/math8071122 10.1137/S0895479891223781 10.1145/76263.76290 10.4153/CMB-1966-083-2 10.1137/15M1044680 10.1137/1.9781611971217 10.1016/0024-3795(87)90103-0 10.1016/0024-3795(72)90013-4 10.1109/HiPC.2017.00035
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Copyright_xml	– notice: The Author(s) 2022 – notice: The Author(s) 2022. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
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References_xml	– reference: KahanWNumerical linear algebraCan. Math. Bull.1966975780110.4153/CMB-1966-083-2 – reference: GuMEisenstatSCEfficient algorithms for computing a strong Rank-Revealing QR factorizationSIAM J. Sci. Comput.1996174848869139535110.1137/0917055https://doi.org/10.1137/0917055 – reference: HansenPCRank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion1999USASociety for Industrial and Applied MathematicsISBN 0898714036 – reference: ChandrasekaranSIpsenICFOn Rank-Revealing factorisationsSIAM J. Matrix Anal. Appl.1994152592622126660610.1137/S0895479891223781https://doi.org/10.1137/S0895479891223781 – reference: Bischof, J.R.: A block QR factorization algorithm using restricted pivoting. In: Supercomputing ’89:Proceedings of the 1989 ACM/IEEE Conference on Supercomputing, pp. 248–256. https://doi.org/10.1145/76263.76290 (1989) – reference: BischofCQuintana-OrtíGComputing rank-revealing QR factorizations of dense matricesACM Trans. Math. Softw.199824226253, 06166395310.1145/290200.287637https://doi.org/10.1145/290200.287637 – reference: HongYPPanC-TRank-revealing QR factorizations and the singular value decompositionMath. Comput.19925819721323211069700743.65037ISSN 00255718, 10886842. http://www.jstor.org/stable/2153029 – reference: DuerschJAGuMRandomized QR with column pivotingSIAM J. Sci. Comput.2017394C263C291368137410.1137/15M1044680https://doi.org/10.1137/15M1044680 – reference: DessoleMMarcuzziFVianelloMAccelerating the Lawson-Hanson NNLS solver for large-scale Tchakaloff regression designsDolomites Research Notes on Approximation20201320294115742ISSN 2035-6803. https://doi.org/10.14658/PUPJ-DRNA-2020-1-3. https://drna.padovauniversitypress.it/2020/1/3 – reference: BusingerPGolubGHLinear Least Squares Solutions by Householder TransformationsNumer. Math.19657326927617659010.1007/BF01436084ISSN 0029-599X. https://doi.org/10.1007/BF01436084 – reference: BarlowJDemmelJComputing accurate eigensystems of scaled diagonally dominant matricesSIAM J. Numer. Anal.19902711104126210.1137/0727045https://doi.org/10.1137/0727045 – reference: Quintana-OrtíGSunXBischofCHA BLAS-3 version of the QR factorization with column pivotingSIAM J. Sci. Comput.199819514861494161879210.1137/S1064827595296732https://doi.org/10.1137/S1064827595296732 – reference: BischofCHansenPA block algorithm for computing rank-revealing QR factorizationsNumer. Algo.19922371391,10118482210.1007/BF02139475https://doi.org/10.1007/BF02139475 – reference: GolubGKlemaVStewartGWRank degeneracy and least squares problems. Technical Report STAN-CS-76-5591976StanfordDepartment of Computer Science Stanford University – reference: GolubGVan LoanCMatrix Computations (4th ed.). Johns Hopkins Studies in the Mathematical Sciences2013BaltimoreJohns Hopkins University PressISBN 9781421407944 – reference: LawsonCLHansonRJSolving least squares problems, vol. 151995BangkokSIAM10.1137/1.9781611971217 – reference: DessoleMMarcuzziFVianelloMDCATCH—a numerical package for d-variate near g-optimal Tchakaloff regression via fast NNLSMathematics20208710.3390/math8071122https://doi.org/10.3390/math8071122 – reference: DemmelJGrigoriLGuMXiangHCommunication avoiding rank revealing QR factorization with column pivotingSIAM J. Matrix Anal. 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Snippet	In this paper, we introduce a new column selection strategy, named here “Deviation Maximization”, and apply it to compute rank-revealing QR factorizations as...
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SubjectTerms	Algebra Algorithms Computer Science Decomposition Deviation Eigenvalues Linear algebra Maximization Numeric Computing Numerical Analysis Optimization Original Paper Theory of Computation
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Title	Deviation maximization for rank-revealing QR factorizations
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