{"id":"https://openalex.org/W2127248062","doi":"https://doi.org/10.1109/sfcs.2003.1238226","title":"Bounded geometries, fractals, and low-distortion embeddings","display_name":"Bounded geometries, fractals, and low-distortion embeddings","publication_year":2004,"publication_date":"2004-03-02","ids":{"openalex":"https://openalex.org/W2127248062","doi":"https://doi.org/10.1109/sfcs.2003.1238226","mag":"2127248062"},"language":"en","primary_location":{"id":"doi:10.1109/sfcs.2003.1238226","is_oa":false,"landing_page_url":"https://doi.org/10.1109/sfcs.2003.1238226","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://figshare.com/articles/Bounded_geometries_fractals_and_low-distortion_embeddings/6603941","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5078381067","display_name":"Anupam Gupta","orcid":"https://orcid.org/0000-0001-5579-3405"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"A. Gupta","raw_affiliation_strings":["Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA","Dept. of Comput. Sci., Carnegie-Mellon Univ., Pittsburgh, PA, USA#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Dept. of Comput. Sci., Carnegie-Mellon Univ., Pittsburgh, PA, USA#TAB#","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037378898","display_name":"Robert Krauthgamer","orcid":"https://orcid.org/0009-0003-8154-3735"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]},{"id":"https://openalex.org/I1297971548","display_name":"International Computer Science Institute","ror":"https://ror.org/01ewh7m12","country_code":"US","type":"nonprofit","lineage":["https://openalex.org/I1297971548"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"R. Krauthgamer","raw_affiliation_strings":["International Computer Science Institute and Computer Science Division, University of California Berkeley, CA, USA","University of California at Berkeley"],"affiliations":[{"raw_affiliation_string":"International Computer Science Institute and Computer Science Division, University of California Berkeley, CA, USA","institution_ids":["https://openalex.org/I1297971548","https://openalex.org/I95457486"]},{"raw_affiliation_string":"University of California at Berkeley","institution_ids":["https://openalex.org/I95457486"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055825419","display_name":"J.R. Lee","orcid":null},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]},{"id":"https://openalex.org/I1290206253","display_name":"Microsoft (United States)","ror":"https://ror.org/00d0nc645","country_code":"US","type":"company","lineage":["https://openalex.org/I1290206253"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J.R. Lee","raw_affiliation_strings":["Microsoft Research and Computer Science Division, University of California Berkeley, USA","University of California at Berkeley"],"affiliations":[{"raw_affiliation_string":"Microsoft Research and Computer Science Division, University of California Berkeley, USA","institution_ids":["https://openalex.org/I1290206253","https://openalex.org/I95457486"]},{"raw_affiliation_string":"University of California at Berkeley","institution_ids":["https://openalex.org/I95457486"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5078381067"],"corresponding_institution_ids":["https://openalex.org/I74973139"],"apc_list":null,"apc_paid":null,"fwci":34.3132,"has_fulltext":true,"cited_by_count":414,"citation_normalized_percentile":{"value":0.99975161,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":100},"biblio":{"volume":null,"issue":null,"first_page":"534","last_page":"543"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9937999844551086,"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"}},"topics":[{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9937999844551086,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9868000149726868,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.98580002784729,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.7685225605964661},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.6627742648124695},{"id":"https://openalex.org/keywords/distortion","display_name":"Distortion (music)","score":0.5769239664077759},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.45448800921440125},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3419661223888397},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3223782777786255},{"id":"https://openalex.org/keywords/telecommunications","display_name":"Telecommunications","score":0.0870867371559143}],"concepts":[{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7685225605964661},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.6627742648124695},{"id":"https://openalex.org/C126780896","wikidata":"https://www.wikidata.org/wiki/Q899871","display_name":"Distortion (music)","level":4,"score":0.5769239664077759},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.45448800921440125},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3419661223888397},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3223782777786255},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0870867371559143},{"id":"https://openalex.org/C2776257435","wikidata":"https://www.wikidata.org/wiki/Q1576430","display_name":"Bandwidth (computing)","level":2,"score":0.0},{"id":"https://openalex.org/C194257627","wikidata":"https://www.wikidata.org/wiki/Q211554","display_name":"Amplifier","level":3,"score":0.0}],"mesh":[],"locations_count":10,"locations":[{"id":"doi:10.1109/sfcs.2003.1238226","is_oa":false,"landing_page_url":"https://doi.org/10.1109/sfcs.2003.1238226","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.","raw_type":"proceedings-article"},{"id":"pmh:oai:figshare.com:article/6603941","is_oa":true,"landing_page_url":"https://figshare.com/articles/journal_contribution/Bounded_geometries_fractals_and_low-distortion_embeddings/6603941","pdf_url":"https://figshare.com/articles/Bounded_geometries_fractals_and_low-distortion_embeddings/6603941","source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"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":"Text"},{"id":"pmh:oai:repository.cmu.edu:compsci-1847","is_oa":false,"landing_page_url":"http://repository.cmu.edu/compsci/848","pdf_url":null,"source":{"id":"https://openalex.org/S4306400668","display_name":"Research Showcase @ Carnegie Mellon University (Carnegie Mellon University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I74973139","host_organization_name":"Carnegie Mellon University","host_organization_lineage":["https://openalex.org/I74973139"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Computer Science Department","raw_type":"text"},{"id":"pmh:doi:10.1184/r1/6603941","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","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":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Journal contribution"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.127.4408","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.4408","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.cmu.edu/~anupamg/papers/focs03b.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.131.5083","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.5083","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.washington.edu/homes/jrl/papers/ass-focs.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.141.6095","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.141.6095","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.cmu.edu/~anupamg/papers/focs03b.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.2.5022","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.2.5022","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.berkeley.edu/~jrl/papers/ass-focs.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.73.8575","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.73.8575","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.wisdom.weizmann.ac.il/~robi/papers/GKL-doubling-FOCS03.ps","raw_type":"text"},{"id":"doi:10.1184/r1/6603941.v1","is_oa":true,"landing_page_url":"https://doi.org/10.1184/r1/6603941.v1","pdf_url":null,"source":{"id":"https://openalex.org/S7407050927","display_name":"KiltHub Repository","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:figshare.com:article/6603941","is_oa":true,"landing_page_url":"https://figshare.com/articles/journal_contribution/Bounded_geometries_fractals_and_low-distortion_embeddings/6603941","pdf_url":"https://figshare.com/articles/Bounded_geometries_fractals_and_low-distortion_embeddings/6603941","source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"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":"Text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2127248062.pdf","grobid_xml":"https://content.openalex.org/works/W2127248062.grobid-xml"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W48861618","https://openalex.org/W100489996","https://openalex.org/W1515282346","https://openalex.org/W1520510766","https://openalex.org/W1523004878","https://openalex.org/W1530239281","https://openalex.org/W1589494766","https://openalex.org/W1594954118","https://openalex.org/W1981853164","https://openalex.org/W1984672232","https://openalex.org/W1987836603","https://openalex.org/W2000876023","https://openalex.org/W2001075173","https://openalex.org/W2001141328","https://openalex.org/W2004322147","https://openalex.org/W2011571064","https://openalex.org/W2017271493","https://openalex.org/W2031212151","https://openalex.org/W2034188144","https://openalex.org/W2036959865","https://openalex.org/W2038964811","https://openalex.org/W2063491776","https://openalex.org/W2066864620","https://openalex.org/W2068871408","https://openalex.org/W2080608361","https://openalex.org/W2083854723","https://openalex.org/W2093171982","https://openalex.org/W2096746450","https://openalex.org/W2097550055","https://openalex.org/W2105160468","https://openalex.org/W2107881198","https://openalex.org/W2149463264","https://openalex.org/W2153319198","https://openalex.org/W2158907148","https://openalex.org/W2167816765","https://openalex.org/W2169036209","https://openalex.org/W2169840105","https://openalex.org/W2176446742","https://openalex.org/W2496163386","https://openalex.org/W2583644876","https://openalex.org/W2962980142","https://openalex.org/W2979473749","https://openalex.org/W3097609957","https://openalex.org/W6630950945","https://openalex.org/W6659084945"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644","https://openalex.org/W2532244109"],"abstract_inverted_index":{"The":[0,32],"doubling":[1,33,55,59,88,98,128],"constant":[2],"of":[3,28,35,51,68,75,133],"a":[4,65,115,134,153],"metric":[5,48,69],"space":[6],"(X,":[7],"d)":[8],"is":[9,37,53,61,64],"the":[10,30,137],"smallest":[11],"value":[12],"/spl":[13,25],"lambda/":[14,26],"such":[15,106],"that":[16,77],"every":[17],"ball":[18],"in":[19,79,136,160],"X":[20,36],"can":[21],"be":[22],"covered":[23],"by":[24],"balls":[27],"half":[29],"radius.":[31],"dimension":[34,60],"then":[38],"defined":[39],"as":[40,100,102,107],"dim":[41],"(X)":[42],"=":[43],"log/sub":[44],"2//spl":[45],"lambda/.":[46],"A":[47],"(or":[49],"sequence":[50],"metrics)":[52],"called":[54],"precisely":[56],"when":[57],"its":[58],"bounded.":[62],"This":[63],"robust":[66],"class":[67],"spaces":[70],"which":[71],"contains":[72],"many":[73],"families":[74,105],"metrics":[76,89],"occur":[78],"applied":[80],"settings.":[81],"We":[82,94],"give":[83],"tight":[84],"bounds":[85],"for":[86,127],"embedding":[87],"into":[90],"(low-dimensional)":[91],"normed":[92],"spaces.":[93],"consider":[95],"both":[96],"general":[97],"metrics,":[99,129],"well":[101],"more":[103],"restricted":[104],"those":[108],"arising":[109],"from":[110,112,119],"trees,":[111],"graphs":[113],"excluding":[114],"fixed":[116],"minor,":[117],"and":[118,130,150],"snowflaked":[120],"metrics.":[121],"Our":[122],"techniques":[123],"include":[124],"decomposition":[125],"theorems":[126],"an":[131],"analysis":[132],"fractal":[135],"plane":[138],"according":[139],"to":[140],"T.":[141],"J.":[142],"Laakso":[143],"(2002).":[144],"Finally,":[145],"we":[146],"discuss":[147],"some":[148],"applications":[149],"point":[151],"out":[152],"central":[154],"open":[155],"question":[156],"regarding":[157],"dimensionality":[158],"reduction":[159],"L/sub":[161],"2/.":[162]},"counts_by_year":[{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":11},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":13},{"year":2021,"cited_by_count":16},{"year":2020,"cited_by_count":10},{"year":2019,"cited_by_count":21},{"year":2018,"cited_by_count":20},{"year":2017,"cited_by_count":14},{"year":2016,"cited_by_count":16},{"year":2015,"cited_by_count":21},{"year":2014,"cited_by_count":26},{"year":2013,"cited_by_count":28},{"year":2012,"cited_by_count":15}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
