{"id":"https://openalex.org/W250209774","doi":"https://doi.org/10.1137/060659624","title":"The Sinkhorn\u2013Knopp Algorithm: Convergence and Applications","display_name":"The Sinkhorn\u2013Knopp Algorithm: Convergence and Applications","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W250209774","doi":"https://doi.org/10.1137/060659624","mag":"250209774"},"language":"en","primary_location":{"id":"doi:10.1137/060659624","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060659624","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://strathprints.strath.ac.uk/19685/1/skapp.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002885020","display_name":"Philip A. Knight","orcid":"https://orcid.org/0000-0001-9511-5692"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Philip A. Knight","raw_affiliation_strings":["pk@maths.strath.ac.uk#TAB#"],"affiliations":[{"raw_affiliation_string":"pk@maths.strath.ac.uk#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5002885020"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.02025764,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"30","issue":"1","first_page":"261","last_page":"275"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6027370095252991},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.599743127822876},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.5733042359352112},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.49849748611450195},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4168429970741272},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.34381598234176636},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3240479826927185},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.07478940486907959}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6027370095252991},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.599743127822876},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.5733042359352112},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.49849748611450195},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4168429970741272},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.34381598234176636},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3240479826927185},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.07478940486907959},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1137/060659624","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060659624","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:strathprints.strath.ac.uk:19685","is_oa":true,"landing_page_url":"https://strathprints.strath.ac.uk/view/author/19922.html>","pdf_url":"https://strathprints.strath.ac.uk/19685/1/skapp.pdf","source":{"id":"https://openalex.org/S4306402226","display_name":"Strathprints: The University of Strathclyde institutional repository (University of Strathclyde)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I181647926","host_organization_name":"University of Strathclyde","host_organization_lineage":["https://openalex.org/I181647926"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"PeerReviewed"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.214.9066","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.214.9066","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://drops.dagstuhl.de/opus/volltexte/2007/1064/pdf/07071.KnightPhilip.Paper.1064.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.95.5830","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.95.5830","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.maths.strath.ac.uk/~caas65/skapp.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.96.3134","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.96.3134","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cerfacs.fr/algor/reports/2006/TR_PA_06_42.pdf","raw_type":"text"},{"id":"pmh:oai:drops-oai.dagstuhl.de:1064","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07071.16","pdf_url":null,"source":{"id":"https://openalex.org/S4377196569","display_name":"DROPS (Schloss Dagstuhl \u2013 Leibniz Center for Informatics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"InProceedings"},{"id":"doi:10.4230/dagsemproc.07071.16","is_oa":true,"landing_page_url":"https://doi.org/10.4230/dagsemproc.07071.16","pdf_url":null,"source":{"id":"https://openalex.org/S7407052059","display_name":"Dagstuhl Research Online Publication Server","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"},{"id":"mag:250209774","is_oa":false,"landing_page_url":"https://drops.dagstuhl.de/opus/volltexte/2007/1064/pdf/07071.KnightPhilip.Paper.1064.pdf/","pdf_url":null,"source":{"id":"https://openalex.org/S4306418148","display_name":"Dagstuhl Seminar Proceedings","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"conference"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":"Dagstuhl Seminar Proceedings","raw_type":null}],"best_oa_location":{"id":"pmh:oai:strathprints.strath.ac.uk:19685","is_oa":true,"landing_page_url":"https://strathprints.strath.ac.uk/view/author/19922.html>","pdf_url":"https://strathprints.strath.ac.uk/19685/1/skapp.pdf","source":{"id":"https://openalex.org/S4306402226","display_name":"Strathprints: The University of Strathclyde institutional repository (University of Strathclyde)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I181647926","host_organization_name":"University of Strathclyde","host_organization_lineage":["https://openalex.org/I181647926"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"PeerReviewed"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320174","display_name":"Carnegie Trust for the Universities of Scotland","ror":"https://ror.org/00dfrzy25"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W250209774.pdf","grobid_xml":"https://content.openalex.org/works/W250209774.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W1500921805","https://openalex.org/W1854214752","https://openalex.org/W1964061814","https://openalex.org/W1964579932","https://openalex.org/W1966824417","https://openalex.org/W1979710858","https://openalex.org/W1992907670","https://openalex.org/W2000399769","https://openalex.org/W2007153785","https://openalex.org/W2013918178","https://openalex.org/W2034025033","https://openalex.org/W2035990937","https://openalex.org/W2045361520","https://openalex.org/W2059949081","https://openalex.org/W2068874164","https://openalex.org/W2073219008","https://openalex.org/W2082205234","https://openalex.org/W2138621811"],"related_works":["https://openalex.org/W2095105219","https://openalex.org/W2774659858","https://openalex.org/W2229285313","https://openalex.org/W1968288349","https://openalex.org/W2090418688","https://openalex.org/W1802019370","https://openalex.org/W2115752294","https://openalex.org/W2102566880","https://openalex.org/W1802310826","https://openalex.org/W2090768360","https://openalex.org/W2290808543","https://openalex.org/W2519706569","https://openalex.org/W2565375212","https://openalex.org/W1970156439","https://openalex.org/W2344933654","https://openalex.org/W2554715437","https://openalex.org/W1919585499","https://openalex.org/W2055070149","https://openalex.org/W2761427269","https://openalex.org/W2059006989"],"abstract_inverted_index":{"As":[0],"long":[1],"as":[2],"a":[3,27,86,114],"square":[4],"nonnegative":[5],"matrix":[6],"A":[7,31],"contains":[8],"sufficient":[9],"nonzero":[10],"elements,":[11],"then":[12],"the":[13,21,40,52,68,94,110,119],"Sinkhorn\u2013Knopp":[14,111],"algorithm":[15,112],"can":[16,81],"be":[17,82],"used":[18,83],"to":[19,25,84],"balance":[20],"matrix,":[22],"that":[23,32,39],"is,":[24],"find":[26],"diagonal":[28],"scaling":[29],"of":[30,54,70,88],"is":[33,37,42,113],"doubly":[34],"stochastic.":[35],"It":[36],"known":[38,99],"convergence":[41,55,71],"linear,":[43],"and":[44],"an":[45,64,107],"upper":[46],"bound":[47],"has":[48],"been":[49],"given":[50],"for":[51,56,67,72,117],"rate":[53,69],"positive":[57],"matrices.":[58,75],"In":[59],"this":[60],"paper":[61],"we":[62],"give":[63,85],"explicit":[65],"expression":[66],"fully":[73],"indecomposable":[74],"We":[76,92,103],"describe":[77],"how":[78],"balancing":[79],"algorithms":[80],"measure":[87,95,120],"web":[89],"page":[90],"significance.":[91],"compare":[93],"with":[96,106],"some":[97],"well":[98],"alternatives,":[100],"including":[101],"PageRank.":[102],"show":[104],"that,":[105],"appropriate":[108],"modification,":[109],"natural":[115],"candidate":[116],"computing":[118],"on":[121],"enormous":[122],"data":[123],"sets.":[124]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2014,"cited_by_count":14},{"year":2012,"cited_by_count":1}],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}