{"id":"https://openalex.org/W2765620689","doi":"https://doi.org/10.1007/s00454-019-00162-2","title":"Impossibility of Dimension Reduction in the Nuclear Norm","display_name":"Impossibility of Dimension Reduction in the Nuclear Norm","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2765620689","doi":"https://doi.org/10.1007/s00454-019-00162-2","mag":"2765620689"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-019-00162-2","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00454-019-00162-2","pdf_url":null,"source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1710.08896","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5052985577","display_name":"Assaf Naor","orcid":null},"institutions":[{"id":"https://openalex.org/I20089843","display_name":"Princeton University","ror":"https://ror.org/00hx57361","country_code":"US","type":"education","lineage":["https://openalex.org/I20089843"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Assaf Naor","raw_affiliation_strings":["Mathematics Department, Princeton University, Princeton, NJ, 08544-1000, USA","Mathematics department, Princeton University, Princeton, USA"],"affiliations":[{"raw_affiliation_string":"Mathematics Department, Princeton University, Princeton, NJ, 08544-1000, USA","institution_ids":["https://openalex.org/I20089843"]},{"raw_affiliation_string":"Mathematics department, Princeton University, Princeton, USA","institution_ids":["https://openalex.org/I20089843"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5035197493","display_name":"Gilles Pisier","orcid":"https://orcid.org/0000-0002-3091-2049"},"institutions":[{"id":"https://openalex.org/I91045830","display_name":"Texas A&M University","ror":"https://ror.org/01f5ytq51","country_code":"US","type":"education","lineage":["https://openalex.org/I91045830"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Gilles Pisier","raw_affiliation_strings":["Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA","Department of Mathematics Texas A&M University College Station USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA","institution_ids":["https://openalex.org/I91045830"]},{"raw_affiliation_string":"Department of Mathematics Texas A&M University College Station USA","institution_ids":["https://openalex.org/I91045830"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023132252","display_name":"Gideon Schechtman","orcid":"https://orcid.org/0000-0002-7981-6402"},"institutions":[{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Gideon Schechtman","raw_affiliation_strings":["Department of Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel","Department of Mathematics, Weizmann Institute of Science Rehovot, Israel"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel","institution_ids":["https://openalex.org/I53964585"]},{"raw_affiliation_string":"Department of Mathematics, Weizmann Institute of Science Rehovot, Israel","institution_ids":["https://openalex.org/I53964585"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5052985577"],"corresponding_institution_ids":["https://openalex.org/I20089843"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":null,"fwci":0.3462,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.49426862,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"63","issue":"2","first_page":"1345","last_page":"1352"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10884","display_name":"Holomorphic and Operator Theory","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10884","display_name":"Holomorphic and Operator Theory","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11545","display_name":"Advanced Banach Space Theory","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/combinatorics","display_name":"Combinatorics","score":0.634059488773346},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5388749241828918},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.49563705921173096},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.48963290452957153},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.30480319261550903}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.634059488773346},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5388749241828918},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.49563705921173096},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.48963290452957153},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.30480319261550903}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1007/s00454-019-00162-2","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00454-019-00162-2","pdf_url":null,"source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},{"id":"doi:10.1137/1.9781611975031.88","is_oa":false,"landing_page_url":"https://doi.org/10.1137/1.9781611975031.88","pdf_url":null,"source":{"id":"https://openalex.org/S4306463922","display_name":"Society for Industrial and Applied Mathematics eBooks","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"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":"ebook platform"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms","raw_type":"book-chapter"},{"id":"pmh:oai:arXiv.org:1710.08896","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1710.08896","pdf_url":"https://arxiv.org/pdf/1710.08896","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:2765620689","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1710.08896","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1710.08896","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1710.08896","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1710.08896","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1710.08896","pdf_url":"https://arxiv.org/pdf/1710.08896","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306079","display_name":"David and Lucile Packard Foundation","ror":"https://ror.org/032atxq54"},{"id":"https://openalex.org/F4320306164","display_name":"Simons Foundation","ror":"https://ror.org/01cmst727"},{"id":"https://openalex.org/F4320320950","display_name":"United States-Israel Binational Science Foundation","ror":"https://ror.org/00j8z2m73"},{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":72,"referenced_works":["https://openalex.org/W4516790","https://openalex.org/W48861618","https://openalex.org/W862539679","https://openalex.org/W1493032731","https://openalex.org/W1516099340","https://openalex.org/W1523004878","https://openalex.org/W1523918473","https://openalex.org/W1525717755","https://openalex.org/W1594954118","https://openalex.org/W1792910226","https://openalex.org/W1963961606","https://openalex.org/W1966053554","https://openalex.org/W1978020111","https://openalex.org/W1997263634","https://openalex.org/W2001085501","https://openalex.org/W2005848700","https://openalex.org/W2010832227","https://openalex.org/W2014279255","https://openalex.org/W2017580271","https://openalex.org/W2024930473","https://openalex.org/W2025029954","https://openalex.org/W2027697499","https://openalex.org/W2028293361","https://openalex.org/W2034889029","https://openalex.org/W2036959865","https://openalex.org/W2045745231","https://openalex.org/W2048695508","https://openalex.org/W2049388783","https://openalex.org/W2054974227","https://openalex.org/W2056532343","https://openalex.org/W2060280149","https://openalex.org/W2060691299","https://openalex.org/W2063491776","https://openalex.org/W2065551446","https://openalex.org/W2079659149","https://openalex.org/W2082287043","https://openalex.org/W2087710599","https://openalex.org/W2090143112","https://openalex.org/W2093480133","https://openalex.org/W2094492505","https://openalex.org/W2105160468","https://openalex.org/W2109521352","https://openalex.org/W2118550318","https://openalex.org/W2132455346","https://openalex.org/W2133711056","https://openalex.org/W2134332047","https://openalex.org/W2137524884","https://openalex.org/W2149463264","https://openalex.org/W2151242030","https://openalex.org/W2151592438","https://openalex.org/W2176446742","https://openalex.org/W2187065282","https://openalex.org/W2188989666","https://openalex.org/W2281727696","https://openalex.org/W2505029951","https://openalex.org/W2571257893","https://openalex.org/W2611328865","https://openalex.org/W2809514611","https://openalex.org/W2952539829","https://openalex.org/W2962854742","https://openalex.org/W2963045194","https://openalex.org/W2963441247","https://openalex.org/W2963766025","https://openalex.org/W2963811477","https://openalex.org/W2971416644","https://openalex.org/W2972929427","https://openalex.org/W2979473749","https://openalex.org/W4230572909","https://openalex.org/W4245085031","https://openalex.org/W4255333822","https://openalex.org/W4298184542","https://openalex.org/W6603939085"],"related_works":["https://openalex.org/W2996404163","https://openalex.org/W3101467415","https://openalex.org/W3148606628","https://openalex.org/W2514130554","https://openalex.org/W2608728403","https://openalex.org/W2809400127","https://openalex.org/W2290861625","https://openalex.org/W1965556777","https://openalex.org/W1991644162","https://openalex.org/W2409767273","https://openalex.org/W2489950765","https://openalex.org/W2470985829","https://openalex.org/W3187894643","https://openalex.org/W2949492003","https://openalex.org/W2967738132","https://openalex.org/W2080883706","https://openalex.org/W3169022418","https://openalex.org/W2951335097","https://openalex.org/W2951169914","https://openalex.org/W2029621817"],"abstract_inverted_index":{"Let":[0],"$${\\mathsf":[1,77,96,99,168,223,243,257,287],"{S}}_1$$":[2,59,100,169,258,288],"(the":[3],"Schatten\u2013von":[4],"Neumann":[5],"trace":[6],"class)":[7],"denote":[8],"the":[9,38,126,142,149,152,172,182,194,204,208,275],"Banach":[10,173],"space":[11,174],"of":[12,41,76,87,90,95,128,144,177,184,219,242,256,261,266,279,286,296],"all":[13],"compact":[14],"linear":[15,74,93,283],"operators":[16,263],"$$T:\\ell":[17],"_2\\rightarrow":[18],"\\ell":[19],"_2$$":[20],"whose":[21],"nuclear":[22],"norm":[23],"$$\\Vert":[24],"T\\Vert":[25],"_{{\\mathsf":[26],"{S}}_1}=\\sum":[27],"_{j=1}^\\infty":[28],"\\upsigma":[29],"_j(T)$$":[30],"is":[31,81,157,227,289],"finite,":[32],"where":[33],"$$\\{\\upsigma":[34],"_j(T)\\}_{j=1}^\\infty":[35],"$$":[36],"are":[37],"singular":[39],"values":[40],"T.":[42],"We":[43,269],"prove":[44],"that":[45,63,151,210,230,259,274],"for":[46,254],"arbitrarily":[47],"large":[48],"$$n\\in":[49],"{\\mathbb":[50],"{N}}$$":[51],"there":[52],"exists":[53],"a":[54,84,104,158,162,217,292],"subset":[55,89],"$${\\mathscr":[56,79,197,231],"{C}}\\subseteq":[57],"{\\mathsf":[58],"with":[60,67,167],"$$|{\\mathscr":[61],"{C}}|=n$$":[62],"cannot":[64,233],"be":[65,200,203,234],"embedded":[66],"bi-Lipschitz":[68],"distortion":[69],"O(1)":[70],"into":[71,154],"any":[72,88,91,280],"$$n^{o(1)}$$-dimensional":[73,92],"subspace":[75,94,241,284],"{S}}_1$$.":[78,97,224],"{C}}$$":[80,198,232],"not":[82,102],"even":[83],"O(1)-Lipschitz":[85],"quotient":[86],"Thus,":[98],"does":[101],"admit":[103],"dimension":[105,146],"reduction":[106,147],"result":[107],"\u00e1":[108],"la":[109],"Johnson":[110],"and":[111,117,134,186,212,247],"Lindenstrauss":[112],"(Conference":[113],"in":[114,222,237],"modern":[115],"analysis":[116],"probability,":[118],"American":[119],"Mathematical":[120],"Society,":[121],"Providence,":[122],"1984),":[123],"which":[124],"complements":[125],"work":[127,183],"Harrow":[129],"et":[130],"al.":[131],"(Automata,":[132],"languages":[133],"programming":[135],"(ICALP":[136],"2011),":[137],"Springer,":[138],"Heidelberg,":[139],"2011)":[140],"on":[141],"limitations":[143],"quantum":[145,159],"under":[148],"assumption":[150],"embedding":[153],"low":[155],"dimensions":[156],"channel.":[160],"Such":[161],"statement":[163],"was":[164],"previously":[165],"known":[166],"replaced":[170],"by":[171,272],"$$\\ell":[175,267],"_1$$":[176],"absolutely":[178],"summable":[179],"sequences":[180],"via":[181],"Brinkman":[185,211,246],"Charikar":[187,213,248],"(J":[188],"ACM":[189],"52(5):766\u2013788,":[190],"2005).":[191],"In":[192],"fact,":[193],"above":[195],"set":[196,206],"can":[199],"taken":[201],"to":[202,228],"same":[205],"as":[207,216],"one":[209],"considered,":[214],"viewed":[215],"collection":[218],"diagonal":[220,262],"matrices":[221],"The":[225],"challenge":[226],"demonstrate":[229],"faithfully":[235],"realized":[236],"an":[238,251],"arbitrary":[239],"low-dimensional":[240],"{S}}_1$$,":[244],"while":[245],"obtained":[249],"such":[250],"assertion":[252],"only":[253],"subspaces":[255,265],"consist":[260],"(i.e.,":[264],"_1$$).":[268],"establish":[270],"this":[271],"proving":[273],"Markov":[276],"2-convexity":[277],"constant":[278,294],"finite":[281],"dimensional":[282],"X":[285],"at":[290],"most":[291],"universal":[293],"multiple":[295],"$$\\sqrt{\\log":[297],"\\dim":[298],"(X)}$$.":[299]},"counts_by_year":[{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}
