{"id":"https://openalex.org/W2624042520","doi":"https://doi.org/10.1137/18m1216134","title":"Clustering with t-SNE, Provably","display_name":"Clustering with t-SNE, Provably","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2624042520","doi":"https://doi.org/10.1137/18m1216134","mag":"2624042520","pmid":"https://pubmed.ncbi.nlm.nih.gov/33073204"},"language":"en","primary_location":{"id":"doi:10.1137/18m1216134","is_oa":true,"landing_page_url":"https://doi.org/10.1137/18m1216134","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"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 Mathematics of Data Science","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite","pubmed"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.1137/18m1216134","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082146323","display_name":"George C. Linderman","orcid":"https://orcid.org/0000-0002-0074-0346"},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]},{"id":"https://openalex.org/I4210131439","display_name":"Applied Mathematics (United States)","ror":"https://ror.org/03seew607","country_code":"US","type":"company","lineage":["https://openalex.org/I4210131439"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"George C. Linderman","raw_affiliation_strings":["Program in Applied Mathematics, Yale University, New Haven, CT 06511, USA","Program in Applied Mathematics, Yale University, New Haven, CT, 06511, USA"],"raw_orcid":"https://orcid.org/0000-0002-0074-0346","affiliations":[{"raw_affiliation_string":"Program in Applied Mathematics, Yale University, New Haven, CT 06511, USA","institution_ids":["https://openalex.org/I32971472","https://openalex.org/I4210131439"]},{"raw_affiliation_string":"Program in Applied Mathematics, Yale University, New Haven, CT, 06511, USA","institution_ids":["https://openalex.org/I32971472","https://openalex.org/I4210131439"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5054778975","display_name":"Stefan Steinerberger","orcid":"https://orcid.org/0000-0002-7745-4217"},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Stefan Steinerberger","raw_affiliation_strings":["Department of Mathematics, Yale University, New Haven, CT 06511, USA","Department of Mathematics, Yale University, New Haven CT 06511, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Yale University, New Haven, CT 06511, USA","institution_ids":["https://openalex.org/I32971472"]},{"raw_affiliation_string":"Department of Mathematics, Yale University, New Haven CT 06511, USA","institution_ids":["https://openalex.org/I32971472"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.415,"has_fulltext":false,"cited_by_count":35,"citation_normalized_percentile":{"value":0.92921104,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"1","issue":"2","first_page":"313","last_page":"332"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9940000176429749,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9940000176429749,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear 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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9860000014305115,"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/T11289","display_name":"Single-cell and spatial transcriptomics","score":0.9677000045776367,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/exaggeration","display_name":"Exaggeration","score":0.8969242572784424},{"id":"https://openalex.org/keywords/cluster-analysis","display_name":"Cluster analysis","score":0.7963613271713257},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.762017548084259},{"id":"https://openalex.org/keywords/connection","display_name":"Connection (principal bundle)","score":0.6254521608352661},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5372306108474731},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.4510502219200134},{"id":"https://openalex.org/keywords/visualization","display_name":"Visualization","score":0.4205549657344818},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.38525158166885376},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.32414138317108154},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.28213614225387573}],"concepts":[{"id":"https://openalex.org/C2777003408","wikidata":"https://www.wikidata.org/wiki/Q5419252","display_name":"Exaggeration","level":2,"score":0.8969242572784424},{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.7963613271713257},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.762017548084259},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.6254521608352661},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5372306108474731},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.4510502219200134},{"id":"https://openalex.org/C36464697","wikidata":"https://www.wikidata.org/wiki/Q451553","display_name":"Visualization","level":2,"score":0.4205549657344818},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.38525158166885376},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32414138317108154},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.28213614225387573},{"id":"https://openalex.org/C118552586","wikidata":"https://www.wikidata.org/wiki/Q7867","display_name":"Psychiatry","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"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":6,"locations":[{"id":"doi:10.1137/18m1216134","is_oa":true,"landing_page_url":"https://doi.org/10.1137/18m1216134","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"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 Mathematics of Data Science","raw_type":"journal-article"},{"id":"pmid:33073204","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33073204","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM journal on mathematics of data science","raw_type":"Journal Article"},{"id":"pmh:oai:arXiv.org:1706.02582","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1706.02582","pdf_url":"https://arxiv.org/pdf/1706.02582","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":null,"raw_type":"text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7561036","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7561036","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM J Math Data Sci","raw_type":"Text"},{"id":"mag:2624042520","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1706.02582.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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.1706.02582","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1706.02582","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"Preprint"}],"best_oa_location":{"id":"doi:10.1137/18m1216134","is_oa":true,"landing_page_url":"https://doi.org/10.1137/18m1216134","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"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 Mathematics of Data Science","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.6700000166893005,"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11"}],"awards":[{"id":"https://openalex.org/G1057984112","display_name":"EFFICIENT SPECTRAL APPROACHES FOR FINDING UNDERLYING STRUCTURES IN BIG DATA","funder_award_id":"1r01hg008383-01a1","funder_id":"https://openalex.org/F4320332161","funder_display_name":"National Institutes of Health"},{"id":"https://openalex.org/G1525194011","display_name":"Medical Scientist Training Program","funder_award_id":"3t32gm007205-44s1","funder_id":"https://openalex.org/F4320332161","funder_display_name":"National Institutes of Health"},{"id":"https://openalex.org/G1914612924","display_name":null,"funder_award_id":"T32-GM-007205","funder_id":"https://openalex.org/F4320337354","funder_display_name":"National Institute of General Medical Sciences"},{"id":"https://openalex.org/G2572482970","display_name":null,"funder_award_id":"R01HG008383","funder_id":"https://openalex.org/F4320337348","funder_display_name":"National Human Genome Research Institute"}],"funders":[{"id":"https://openalex.org/F4320332161","display_name":"National Institutes of Health","ror":"https://ror.org/01cwqze88"},{"id":"https://openalex.org/F4320337348","display_name":"National Human Genome Research Institute","ror":"https://ror.org/00baak391"},{"id":"https://openalex.org/F4320337354","display_name":"National Institute of General Medical Sciences","ror":"https://ror.org/04q48ey07"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W1875842236","https://openalex.org/W2097308346","https://openalex.org/W2102212449"],"related_works":["https://openalex.org/W2187089797","https://openalex.org/W1875842236","https://openalex.org/W2791925274","https://openalex.org/W3210227075","https://openalex.org/W3196433226","https://openalex.org/W3195841004","https://openalex.org/W3194333290","https://openalex.org/W2399487690","https://openalex.org/W2952161373","https://openalex.org/W2970529458","https://openalex.org/W2335861177","https://openalex.org/W2013736751","https://openalex.org/W2938811552","https://openalex.org/W3135509220","https://openalex.org/W2949589636","https://openalex.org/W2165874743","https://openalex.org/W2293425463","https://openalex.org/W2204944655","https://openalex.org/W2078749591","https://openalex.org/W3181145002"],"abstract_inverted_index":{"t-distributed":[0],"stochastic":[1],"neighborhood":[2],"embedding":[3,148],"(t-SNE),":[4],"a":[5,22,26,37,118,160],"clustering":[6,164],"and":[7,15,43,89,100,131],"visualization":[8],"method":[9],"proposed":[10,84],"by":[11,85],"van":[12,86,101],"der":[13,87,102],"Maaten":[14,88,103],"Hinton":[16,90],"in":[17,25,76,143],"2008,":[18],"has":[19],"rapidly":[20],"become":[21],"standard":[23],"tool":[24],"number":[27],"of":[28,40,47,56,140,147,149],"natural":[29],"sciences.":[30],"Despite":[31],"its":[32],"overwhelming":[33],"success,":[34],"there":[35],"is":[36,59,64],"distinct":[38],"lack":[39],"mathematical":[41],"foundations,":[42],"the":[44,48,77,120,127,138,145,153],"inner":[45],"workings":[46],"algorithm":[49],"are":[50],"not":[51],"well":[52],"understood.":[53],"The":[54],"purpose":[55],"this":[57],"paper":[58],"to":[60,66,162],"prove":[61,73],"that":[62,74],"t-SNE":[63,75],"able":[65],"recover":[67],"well-separated":[68],"clusters;":[69],"more":[70],"precisely,":[71],"we":[72],"\u201cearly":[78],"exaggeration\u201d":[79],"phase,":[80],"an":[81],"optimization":[82],"technique":[83],"[":[91,104],"J.":[92,105],"Mach.":[93,106],"Learn.":[94,107],"Res.,":[95,108],"9":[96],"(2008),":[97],"pp.":[98,111],"2579--2605]":[99],"15":[109],"(2014),":[110],"3221--3245],":[112],"can":[113],"be":[114],"rigorously":[115],"analyzed.":[116],"As":[117],"byproduct,":[119],"proof":[121],"suggests":[122],"novel":[123],"ways":[124],"for":[125],"setting":[126],"exaggeration":[128],"parameter":[129],"$\\alpha$":[130],"step":[132],"size":[133],"$h$.":[134],"Numerical":[135],"examples":[136],"illustrate":[137],"effectiveness":[139],"these":[141],"rules:":[142],"particular,":[144],"quality":[146],"topological":[150],"structures":[151],"(e.g.,":[152],"swiss":[154],"roll)":[155],"improves.":[156],"We":[157],"also":[158],"discuss":[159],"connection":[161],"spectral":[163],"methods.":[165]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":10},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":3}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}
