Efficient Low-Rank Approximation of Matrices Based on Randomized Pivoted Decomposition
Given a matrix <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula> with numerical rank <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>, the two-sided orthogonal decomposition (TSOD)...
Saved in:
| Published in: | IEEE transactions on signal processing Vol. 68; pp. 3575 - 3589 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
IEEE
2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Subjects: | |
| ISSN: | 1053-587X, 1941-0476 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | Given a matrix <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula> with numerical rank <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>, the two-sided orthogonal decomposition (TSOD) computes a factorization <inline-formula><tex-math notation="LaTeX">{\bf A} = {\bf UDV}^T</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">{\bf U}</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\bf V}</tex-math></inline-formula> are orthogonal, and <inline-formula><tex-math notation="LaTeX">{\bf D}</tex-math></inline-formula> is (upper/lower) triangular. TSOD is rank-revealing as the middle factor <inline-formula><tex-math notation="LaTeX">{\bf D}</tex-math></inline-formula> reveals the rank of <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula>. The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm. |
|---|---|
| AbstractList | Given a matrix <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula> with numerical rank <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>, the two-sided orthogonal decomposition (TSOD) computes a factorization <inline-formula><tex-math notation="LaTeX">{\bf A} = {\bf UDV}^T</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">{\bf U}</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">{\bf V}</tex-math></inline-formula> are orthogonal, and <inline-formula><tex-math notation="LaTeX">{\bf D}</tex-math></inline-formula> is (upper/lower) triangular. TSOD is rank-revealing as the middle factor <inline-formula><tex-math notation="LaTeX">{\bf D}</tex-math></inline-formula> reveals the rank of <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula>. The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm. Given a matrix [Formula Omitted] with numerical rank [Formula Omitted], the two-sided orthogonal decomposition (TSOD) computes a factorization [Formula Omitted], where [Formula Omitted] and [Formula Omitted] are orthogonal, and [Formula Omitted] is (upper/lower) triangular. TSOD is rank-revealing as the middle factor [Formula Omitted] reveals the rank of [Formula Omitted]. The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the [Formula Omitted] approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm. |
| Author | Chen, Jie Kaloorazi, Maboud F. |
| Author_xml | – sequence: 1 givenname: Maboud F. orcidid: 0000-0003-3385-9244 surname: Kaloorazi fullname: Kaloorazi, Maboud F. email: kaloorazi@nwpu.edu.cn organization: Center of Intelligent Acoustics and Immersive Communications, School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, China – sequence: 2 givenname: Jie orcidid: 0000-0003-2306-8860 surname: Chen fullname: Chen, Jie email: dr.jie.chen@ieee.org organization: Center of Intelligent Acoustics and Immersive Communications, School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, China |
| BookMark | eNp9kMtPAjEQhxuDiYDeTbxs4nlxui-2R0R8JBiJovHWdLvTpAjbtS2-_nqLEA8ePM20-X2dztcjncY0SMgxhQGlwM7mD7NBAgkMUgCaMrZHupRlNIZsWHRCD3ka5-Xw-YD0nFuETJaxokueJkppqbHx0dS8x_eieYlGbWvNh14Jr00TGRXdCm-1RBedC4d1FC5DrjYr_RVOM_1mfKgXKM2qNU5vqEOyr8TS4dGu9snj5WQ-vo6nd1c349E0lgmjPlYSCiiEymuFyGqsRJVQKZmCsqSISa5KUFXNhKxpLmpWSakEZinkKBhimfbJ6fbd8OPXNTrPF2ZtmzCSJxkdZpTmJQsp2KakNc5ZVLy1YT37ySnwjT0e7PGNPb6zF5DiDyK1_xHirdDL_8CTLagR8XcOozTNSki_AXkXgOQ |
| CODEN | ITPRED |
| CitedBy_id | crossref_primary_10_1049_sil2_12035 crossref_primary_10_1016_j_ins_2020_12_066 crossref_primary_10_1016_j_ins_2023_119464 crossref_primary_10_1016_j_dsp_2024_104472 crossref_primary_10_1109_ACCESS_2023_3288889 crossref_primary_10_1109_TVT_2023_3243244 |
| Cites_doi | 10.1137/18M1179432 10.1017/S0962492916000076 10.1137/S0895479896305696 10.1145/1255443.1255449 10.1137/18M1163658 10.1109/TPAMI.2008.79 10.1109/DSW.2018.8439907 10.1561/0400000060 10.1214/11-AOS949 10.1137/1.9780898719697 10.1137/140977898 10.1142/S1793536911000787 10.1137/080731992 10.1137/060673096 10.1137/16M1081270 10.1137/1.9781611970739 10.1109/JSTSP.2018.2869363 10.1016/0024-3795(93)90493-8 10.1109/TSP.2018.2853137 10.1016/S0024-3795(96)00301-1 10.1007/BF02142742 10.1109/ICASSP.2017.7952952 10.1137/S1064827597319519 10.1137/1.9781611971446 10.1137/17M1111590 10.1137/090761793 10.1137/080736417 10.1137/S0036144501387517 10.1137/S0097539704442696 10.1007/s10444-016-9494-8 10.1007/978-3-319-49340-4_23 10.1007/11830924_28 10.1137/120874540 10.1007/978-1-4612-0653-8 10.1006/jcss.2000.1711 10.1090/S0025-5718-1973-0348991-3 10.1016/j.laa.2011.01.017 10.1137/090771806 10.1137/0917055 10.1007/s10208-012-9135-7 10.1137/13092157X 10.1016/0024-3795(87)90103-0 10.1109/JSTSP.2018.2867448 10.1137/080738970 10.1109/TIT.2018.2816685 10.1137/16M1091745 10.1137/110853996 10.1109/TIP.2011.2159730 10.1137/030602678 10.1137/15M1044680 10.1137/130938700 10.1145/3019134 10.1109/FOCS.2006.37 10.1002/nla.404 10.1137/1.9781611971408 10.1145/1039488.1039494 10.1090/pcms/025/04 10.1109/FOCS.2013.21 10.1109/JSTSP.2017.2671789 10.1109/ICASSP.2019.8683284 |
| ContentType | Journal Article |
| Copyright | Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2020 |
| Copyright_xml | – notice: Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2020 |
| DBID | 97E RIA RIE AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D |
| DOI | 10.1109/TSP.2020.3001399 |
| DatabaseName | IEEE All-Society Periodicals Package (ASPP) 2005–Present IEEE All-Society Periodicals Package (ASPP) 1998–Present IEEE Electronic Library (IEL) CrossRef Computer and Information Systems Abstracts Electronics & Communications Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional |
| DatabaseTitle | CrossRef Technology Research Database Computer and Information Systems Abstracts – Academic Electronics & Communications Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional |
| DatabaseTitleList | Technology Research Database |
| Database_xml | – sequence: 1 dbid: RIE name: IEEE Electronic Library (IEL) url: https://ieeexplore.ieee.org/ sourceTypes: Publisher |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering |
| EISSN | 1941-0476 |
| EndPage | 3589 |
| ExternalDocumentID | 10_1109_TSP_2020_3001399 9113480 |
| Genre | orig-research |
| GrantInformation_xml | – fundername: National Natural Science Foundation of China; NSFC grantid: 61671382 funderid: 10.13039/501100001809 – fundername: National Basic Research Program of China (973 Program); National Key Research and Development Program of China grantid: 2018AAA0102200 funderid: 10.13039/501100012166 – fundername: Higher Education Discipline Innovation Project; 111 project grantid: B18041 funderid: 10.13039/501100013314 |
| GroupedDBID | -~X .DC 0R~ 29I 3EH 4.4 53G 5GY 5VS 6IK 85S 97E AAJGR AARMG AASAJ AAWTH ABAZT ABFSI ABQJQ ABVLG ACGFO ACIWK ACKIV ACNCT AENEX AETIX AGQYO AGSQL AHBIQ AI. AIBXA AJQPL AKJIK AKQYR ALLEH ALMA_UNASSIGNED_HOLDINGS ASUFR ATWAV BEFXN BFFAM BGNUA BKEBE BPEOZ CS3 E.L EBS EJD F5P HZ~ H~9 ICLAB IFIPE IFJZH IPLJI JAVBF LAI MS~ O9- OCL P2P RIA RIE RNS TAE TN5 VH1 AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D |
| ID | FETCH-LOGICAL-c291t-fc0606af5dfee9debab21cc9f0881ee25f80fbd9acd15ad9bccfae4305ea9ee83 |
| IEDL.DBID | RIE |
| ISICitedReferencesCount | 7 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000543713500004&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 1053-587X |
| IngestDate | Mon Jun 30 10:22:13 EDT 2025 Tue Nov 18 21:24:09 EST 2025 Sat Nov 29 04:10:51 EST 2025 Wed Aug 27 02:41:20 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Language | English |
| License | https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html https://doi.org/10.15223/policy-029 https://doi.org/10.15223/policy-037 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c291t-fc0606af5dfee9debab21cc9f0881ee25f80fbd9acd15ad9bccfae4305ea9ee83 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0003-3385-9244 0000-0003-2306-8860 |
| PQID | 2417411589 |
| PQPubID | 85478 |
| PageCount | 15 |
| ParticipantIDs | ieee_primary_9113480 crossref_citationtrail_10_1109_TSP_2020_3001399 crossref_primary_10_1109_TSP_2020_3001399 proquest_journals_2417411589 |
| PublicationCentury | 2000 |
| PublicationDate | 20200000 2020-00-00 20200101 |
| PublicationDateYYYYMMDD | 2020-01-01 |
| PublicationDate_xml | – year: 2020 text: 20200000 |
| PublicationDecade | 2020 |
| PublicationPlace | New York |
| PublicationPlace_xml | – name: New York |
| PublicationTitle | IEEE transactions on signal processing |
| PublicationTitleAbbrev | TSP |
| PublicationYear | 2020 |
| Publisher | IEEE The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Publisher_xml | – name: IEEE – name: The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| References | ref57 ref13 ref56 ref12 ref58 ref14 ref53 stewart (ref59) 1990 ref52 ref55 ref11 ref54 ref10 ref17 ref16 ref19 ref18 scott (ref65) 2015 musco (ref41) 0 ref51 larsen (ref67) 1998 ref50 golub (ref1) 1996 ref46 ref45 ref48 ref47 ref42 ref44 ref43 ref49 ref8 ref7 ref9 ref4 ref3 ref6 ref5 ref40 ref35 ref34 ref37 ref36 ref31 ref30 ref33 ref32 ref2 ref39 ref38 ref23 ref26 ref25 ref64 ref20 ref63 ref66 ref22 ref21 ref28 ref27 ref29 wright (ref24) 0 ref60 ref62 ref61 martinsson (ref15) 0 |
| References_xml | – ident: ref48 doi: 10.1137/18M1179432 – ident: ref32 doi: 10.1017/S0962492916000076 – ident: ref37 doi: 10.1137/S0895479896305696 – ident: ref52 doi: 10.1145/1255443.1255449 – ident: ref49 doi: 10.1137/18M1163658 – ident: ref28 doi: 10.1109/TPAMI.2008.79 – ident: ref38 doi: 10.1109/DSW.2018.8439907 – year: 2015 ident: ref65 article-title: An I/O-complexity lower bound for all recursive matrix multiplication algorithms by path-routing – ident: ref56 doi: 10.1561/0400000060 – ident: ref33 doi: 10.1214/11-AOS949 – ident: ref43 doi: 10.1137/1.9780898719697 – ident: ref17 doi: 10.1137/140977898 – ident: ref64 doi: 10.1142/S1793536911000787 – ident: ref19 doi: 10.1137/080731992 – ident: ref55 doi: 10.1137/060673096 – ident: ref8 doi: 10.1137/16M1081270 – ident: ref57 doi: 10.1137/1.9781611970739 – ident: ref39 doi: 10.1109/JSTSP.2018.2869363 – ident: ref62 doi: 10.1016/0024-3795(93)90493-8 – ident: ref18 doi: 10.1109/TSP.2018.2853137 – ident: ref9 doi: 10.1016/S0024-3795(96)00301-1 – ident: ref4 doi: 10.1007/BF02142742 – ident: ref30 doi: 10.1109/ICASSP.2017.7952952 – ident: ref5 doi: 10.1137/S1064827597319519 – year: 1998 ident: ref67 article-title: Efficient algorithms for helioseismic inversion – ident: ref40 doi: 10.1137/1.9781611971446 – ident: ref66 doi: 10.1137/17M1111590 – ident: ref25 doi: 10.1137/090761793 – ident: ref12 doi: 10.1137/080736417 – start-page: 1396 year: 0 ident: ref41 article-title: Randomized block krylov methods for stronger and faster approximate singular value decomposition publication-title: Proc Conf Neural Inf Process Syst – ident: ref27 doi: 10.1137/S0036144501387517 – ident: ref11 doi: 10.1137/S0097539704442696 – ident: ref34 doi: 10.1007/s10444-016-9494-8 – ident: ref20 doi: 10.1007/978-3-319-49340-4_23 – ident: ref51 doi: 10.1007/11830924_28 – year: 1990 ident: ref59 publication-title: Matrix Perturbation Theory – ident: ref14 doi: 10.1137/120874540 – ident: ref58 doi: 10.1007/978-1-4612-0653-8 – ident: ref53 doi: 10.1006/jcss.2000.1711 – ident: ref63 doi: 10.1090/S0025-5718-1973-0348991-3 – ident: ref60 doi: 10.1016/j.laa.2011.01.017 – ident: ref13 doi: 10.1137/090771806 – ident: ref46 doi: 10.1137/0917055 – ident: ref31 doi: 10.1007/s10208-012-9135-7 – ident: ref6 doi: 10.1137/13092157X – ident: ref2 doi: 10.1016/0024-3795(87)90103-0 – ident: ref26 doi: 10.1109/JSTSP.2018.2867448 – ident: ref22 doi: 10.1137/080738970 – ident: ref23 doi: 10.1109/TIT.2018.2816685 – ident: ref42 doi: 10.1137/16M1091745 – ident: ref35 doi: 10.1137/110853996 – ident: ref29 doi: 10.1109/TIP.2011.2159730 – ident: ref44 doi: 10.1137/030602678 – ident: ref7 doi: 10.1137/15M1044680 – ident: ref16 doi: 10.1137/130938700 – ident: ref21 doi: 10.1145/3019134 – ident: ref47 doi: 10.1109/FOCS.2006.37 – ident: ref61 doi: 10.1002/nla.404 – ident: ref3 doi: 10.1137/1.9781611971408 – year: 0 ident: ref15 – year: 1996 ident: ref1 publication-title: Matrix Computations – ident: ref10 doi: 10.1145/1039488.1039494 – ident: ref45 doi: 10.1090/pcms/025/04 – ident: ref54 doi: 10.1109/FOCS.2013.21 – ident: ref36 doi: 10.1109/JSTSP.2017.2671789 – start-page: 2080 year: 0 ident: ref24 article-title: Robust principal component analysis: Exact recovery of corrupted low-rank matrices publication-title: Proc Conf Neural Inf Process Syst – ident: ref50 doi: 10.1109/ICASSP.2019.8683284 |
| SSID | ssj0014496 |
| Score | 2.3755298 |
| Snippet | Given a matrix <inline-formula><tex-math notation="LaTeX">\bf A</tex-math></inline-formula> with numerical rank <inline-formula><tex-math... Given a matrix [Formula Omitted] with numerical rank [Formula Omitted], the two-sided orthogonal decomposition (TSOD) computes a factorization [Formula... |
| SourceID | proquest crossref ieee |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 3575 |
| SubjectTerms | Acoustics Algorithms Approximation Approximation algorithms Decomposition dimension reduction image recovery low-rank approximation Machine learning algorithms Mathematical analysis Matrix decomposition Randomization randomized numerical linear algebra rank-revealing factorization Signal processing algorithms Sparse matrices Subspaces Surges Upper bounds |
| Title | Efficient Low-Rank Approximation of Matrices Based on Randomized Pivoted Decomposition |
| URI | https://ieeexplore.ieee.org/document/9113480 https://www.proquest.com/docview/2417411589 |
| Volume | 68 |
| WOSCitedRecordID | wos000543713500004&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVIEE databaseName: IEEE Electronic Library (IEL) customDbUrl: eissn: 1941-0476 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0014496 issn: 1053-587X databaseCode: RIE dateStart: 19910101 isFulltext: true titleUrlDefault: https://ieeexplore.ieee.org/ providerName: IEEE |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1LT8MwDI5g4gAH3ojBQDlwQaKs7do1OY6XOACaeGm3Kk1sqQJWxMZA_HqctKtAICRubWVXld3En-PkM2N7oRbSRy08ozJKUChgeYLCoocEZSOUSCGmbDaRXF2JwUD2Z9hBfRYGANzmMzi0l66Wbwr9apfK2jQwO5GgBH02SbrlWa26YhBFrhcXwYWOF4tkMC1J-rJ9e9OnRDCk_NQBHvktBLmeKj8mYhddzpb-913LbLFCkbxXun2FzcBwlS184RZcY_enjhyCFPlF8eZdq-ED71kC8fe8PK3IC-SXjqAfRvyIgpnh9JDkTPGUf9BdP58UhEf5Cdh959XmrnV2d3Z6e3zuVU0UPE1mHnuofcpRFMYGAaSBTGVhoLVEml4CgDBG4WNmpNImiJWRmdaowBKBgZIAorPBGsNiCJuMo8hkDDJQXY0RGCmyEOLM7wCqpIsomqw9tWuqK4Zx2-jiMXWZhi9T8kRqPZFWnmiy_VrjuWTX-EN2zVq-lquM3mStqevSaviNUoIlhJSCWMit37W22bx9d7mW0mKN8csr7LA5PRnno5dd92d9Au-ozf4 |
| linkProvider | IEEE |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV3da9swED9KWtj2sHbLSrN2nR76MpgX27ET6bEfCRlLQuiykTcjS3cQtsYjST_oX9-T7ISNlULfbHOHzZ2l-51O-h3ASWykCsnIwOqcExQOWIHksBgQQ9mEFHGIKZtNdEYjOZ2q8RZ83pyFQUS_-Qy_uEtfy7eFuXZLZU0emK1EcoK-nSZJHJantTY1gyTx3bgYMLSCVHam66JkqJqT72NOBWPOUD3kUf8EId9V5b-p2MeX3u7zvmwPXlc4UpyWjn8DWzh_C6_-Yhesw8-up4dgRTEoboNLPf8lTh2F-N2sPK8oChJDT9GPS3HG4cwKfshytria3fPdeHZTMCIVF-h2nlfbu97Bj153ct4PqjYKgWFDrwIyIWcpmlJLiMpirvM4MkYRTzARYpySDCm3Shsbpdqq3BjS6KjAUCtE2dqH2ryY4wEIkrlKUUW6bShBq2QeY5qHLSTdaRPJBjTXds1MxTHuWl38znyuEaqMPZE5T2SVJxrwaaPxp-TXeEK27iy_kauM3oCjteuyagAuMwYmjJWiVKr3j2t9hBf9yXCQDb6Ovh3CS_eecmXlCGqrxTV-gB1zs5otF8f-L3sAzH_RRQ |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Efficient+Low-Rank+Approximation+of+Matrices+Based+on+Randomized+Pivoted+Decomposition&rft.jtitle=IEEE+transactions+on+signal+processing&rft.au=Kaloorazi%2C+Maboud+F.&rft.au=Chen%2C+Jie&rft.date=2020&rft.issn=1053-587X&rft.eissn=1941-0476&rft.volume=68&rft.spage=3575&rft.epage=3589&rft_id=info:doi/10.1109%2FTSP.2020.3001399&rft.externalDBID=n%2Fa&rft.externalDocID=10_1109_TSP_2020_3001399 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1053-587X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1053-587X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1053-587X&client=summon |