Rectangular maximum-volume submatrices and their applications

Handle URI:
http://hdl.handle.net/10754/625919
Title:
Rectangular maximum-volume submatrices and their applications
Authors:
Mikhalev, Aleksandr ( 0000-0002-9274-7237 ) ; Oseledets, I.V.
Abstract:
We introduce a definition of the volume of a general rectangular matrix, which is equivalent to an absolute value of the determinant for square matrices. We generalize results of square maximum-volume submatrices to the rectangular case, show a connection of the rectangular volume with an optimal experimental design and provide estimates for a growth of coefficients and an approximation error in spectral and Chebyshev norms. Three promising applications of such submatrices are presented: recommender systems, finding maximal elements in low-rank matrices and preconditioning of overdetermined linear systems. The code is available online.
KAUST Department:
King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
Citation:
Mikhalev A, Oseledets IV (2017) Rectangular maximum-volume submatrices and their applications. Linear Algebra and its Applications. Available: http://dx.doi.org/10.1016/j.laa.2017.10.014.
Publisher:
Elsevier BV
Journal:
Linear Algebra and its Applications
Issue Date:
18-Oct-2017
DOI:
10.1016/j.laa.2017.10.014
Type:
Article
ISSN:
0024-3795
Sponsors:
Work on the problem setting and numerical examples was supported by Russian Foundation for Basic Research grant 16-31-60095 mol_a_dk. Work on theoretical estimations of approximation error and the practical algorithm was supported by Russian Foundation for Basic Research grant 16-31-00351 mol_a.
Additional Links:
http://www.sciencedirect.com/science/article/pii/S0024379517305931
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorMikhalev, Aleksandren
dc.contributor.authorOseledets, I.V.en
dc.date.accessioned2017-10-22T11:48:14Z-
dc.date.available2017-10-22T11:48:14Z-
dc.date.issued2017-10-18en
dc.identifier.citationMikhalev A, Oseledets IV (2017) Rectangular maximum-volume submatrices and their applications. Linear Algebra and its Applications. Available: http://dx.doi.org/10.1016/j.laa.2017.10.014.en
dc.identifier.issn0024-3795en
dc.identifier.doi10.1016/j.laa.2017.10.014en
dc.identifier.urihttp://hdl.handle.net/10754/625919-
dc.description.abstractWe introduce a definition of the volume of a general rectangular matrix, which is equivalent to an absolute value of the determinant for square matrices. We generalize results of square maximum-volume submatrices to the rectangular case, show a connection of the rectangular volume with an optimal experimental design and provide estimates for a growth of coefficients and an approximation error in spectral and Chebyshev norms. Three promising applications of such submatrices are presented: recommender systems, finding maximal elements in low-rank matrices and preconditioning of overdetermined linear systems. The code is available online.en
dc.description.sponsorshipWork on the problem setting and numerical examples was supported by Russian Foundation for Basic Research grant 16-31-60095 mol_a_dk. Work on theoretical estimations of approximation error and the practical algorithm was supported by Russian Foundation for Basic Research grant 16-31-00351 mol_a.en
dc.publisherElsevier BVen
dc.relation.urlhttp://www.sciencedirect.com/science/article/pii/S0024379517305931en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Linear Algebra and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Linear Algebra and its Applications, [, , (2017-10-18)] DOI: 10.1016/j.laa.2017.10.014 . © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectMaximum volume submatricesen
dc.subjectPseudo-skeleton approximationsen
dc.subjectCGR-approximationsen
dc.subjectRecommender systemsen
dc.subjectPreconditioningen
dc.subjectOptimal experimental designen
dc.titleRectangular maximum-volume submatrices and their applicationsen
dc.typeArticleen
dc.contributor.departmentKing Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabiaen
dc.identifier.journalLinear Algebra and its Applicationsen
dc.eprint.versionPost-printen
dc.contributor.institutionInstitute of Numerical Mathematics, Russian Academy of Sciences, Gubkina St. 8, 119333 Moscow, Russiaen
dc.contributor.institutionSkolkovo Institute of Science and Technology, Novaya St. 100, Skolkovo, Odintsovsky district, 143025, Russiaen
kaust.authorMikhalev, Aleksandren
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