{"id":"https://openalex.org/W3095730670","doi":"https://doi.org/10.1109/tit.2022.3148923","title":"Distances Between Probability Distributions of Different Dimensions","display_name":"Distances Between Probability Distributions of Different Dimensions","publication_year":2022,"publication_date":"2022-02-02","ids":{"openalex":"https://openalex.org/W3095730670","doi":"https://doi.org/10.1109/tit.2022.3148923","mag":"3095730670"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2022.3148923","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3148923","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109477179","display_name":"Yuhang Cai","orcid":null},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Yuhang Cai","raw_affiliation_strings":["Department of Statistics, The University of Chicago, Chicago, IL, USA"],"affiliations":[{"raw_affiliation_string":"Department of Statistics, The University of Chicago, Chicago, IL, USA","institution_ids":["https://openalex.org/I40347166"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5050794507","display_name":"Lek\u2010Heng Lim","orcid":"https://orcid.org/0000-0002-9808-0138"},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]},{"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":"Lek-Heng Lim","raw_affiliation_strings":["Computational and Applied Mathematics Initiative, The University of Chicago, Chicago, IL, USA"],"affiliations":[{"raw_affiliation_string":"Computational and Applied Mathematics Initiative, The University of Chicago, Chicago, IL, USA","institution_ids":["https://openalex.org/I40347166","https://openalex.org/I4210131439"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5109477179"],"corresponding_institution_ids":["https://openalex.org/I40347166"],"apc_list":null,"apc_paid":null,"fwci":4.4934,"has_fulltext":false,"cited_by_count":42,"citation_normalized_percentile":{"value":0.95401108,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":93,"max":100},"biblio":{"volume":"68","issue":"6","first_page":"4020","last_page":"4031"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9954000115394592,"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/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9954000115394592,"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9617000222206116,"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"}},{"id":"https://openalex.org/T10136","display_name":"Statistical Methods and Inference","score":0.9520000219345093,"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/notation","display_name":"Notation","score":0.6912586688995361},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.6843805909156799},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6822208762168884},{"id":"https://openalex.org/keywords/divergence","display_name":"Divergence (linguistics)","score":0.6383503675460815},{"id":"https://openalex.org/keywords/probability-distribution","display_name":"Probability distribution","score":0.6363043189048767},{"id":"https://openalex.org/keywords/kullback\u2013leibler-divergence","display_name":"Kullback\u2013Leibler divergence","score":0.5706271529197693},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.513382077217102},{"id":"https://openalex.org/keywords/probability-measure","display_name":"Probability measure","score":0.47263357043266296},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.46686530113220215},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.4240928590297699},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4115995168685913},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.23978665471076965}],"concepts":[{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.6912586688995361},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.6843805909156799},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6822208762168884},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.6383503675460815},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.6363043189048767},{"id":"https://openalex.org/C171752962","wikidata":"https://www.wikidata.org/wiki/Q255166","display_name":"Kullback\u2013Leibler divergence","level":2,"score":0.5706271529197693},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.513382077217102},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.47263357043266296},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46686530113220215},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.4240928590297699},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4115995168685913},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.23978665471076965},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2022.3148923","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3148923","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3845474740","display_name":null,"funder_award_id":"1546413","funder_id":"https://openalex.org/F4320332179","funder_display_name":"Office of the Director"},{"id":"https://openalex.org/G63389949","display_name":null,"funder_award_id":"HR00112190040","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G8468936410","display_name":null,"funder_award_id":"1854831","funder_id":"https://openalex.org/F4320337380","funder_display_name":"Division of Mathematical Sciences"}],"funders":[{"id":"https://openalex.org/F4320332179","display_name":"Office of the Director","ror":"https://ror.org/04nseet23"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":80,"referenced_works":["https://openalex.org/W385466589","https://openalex.org/W404764422","https://openalex.org/W595145996","https://openalex.org/W658020064","https://openalex.org/W1537492533","https://openalex.org/W1549664537","https://openalex.org/W1557758852","https://openalex.org/W1570212550","https://openalex.org/W1597307519","https://openalex.org/W1888321505","https://openalex.org/W1965555277","https://openalex.org/W1980372935","https://openalex.org/W1991410152","https://openalex.org/W1993977015","https://openalex.org/W1995875735","https://openalex.org/W2003243224","https://openalex.org/W2007866395","https://openalex.org/W2009172320","https://openalex.org/W2011181254","https://openalex.org/W2013603106","https://openalex.org/W2014018052","https://openalex.org/W2016370813","https://openalex.org/W2021097403","https://openalex.org/W2021967984","https://openalex.org/W2027842533","https://openalex.org/W2031648200","https://openalex.org/W2036476394","https://openalex.org/W2036476462","https://openalex.org/W2042587503","https://openalex.org/W2043236011","https://openalex.org/W2045656688","https://openalex.org/W2070316439","https://openalex.org/W2094100343","https://openalex.org/W2094689902","https://openalex.org/W2099471712","https://openalex.org/W2101210333","https://openalex.org/W2104144774","https://openalex.org/W2105431676","https://openalex.org/W2109382511","https://openalex.org/W2122882636","https://openalex.org/W2123838014","https://openalex.org/W2125737346","https://openalex.org/W2126163471","https://openalex.org/W2132442278","https://openalex.org/W2134700941","https://openalex.org/W2136525667","https://openalex.org/W2136701143","https://openalex.org/W2143668817","https://openalex.org/W2146950091","https://openalex.org/W2150774511","https://openalex.org/W2158391627","https://openalex.org/W2268884199","https://openalex.org/W2528314500","https://openalex.org/W2547600766","https://openalex.org/W2559515863","https://openalex.org/W2591972302","https://openalex.org/W2615683174","https://openalex.org/W2799055878","https://openalex.org/W2963217065","https://openalex.org/W2963346868","https://openalex.org/W2963426391","https://openalex.org/W2995847508","https://openalex.org/W3099491392","https://openalex.org/W3099494581","https://openalex.org/W3106373805","https://openalex.org/W3110683067","https://openalex.org/W3141207716","https://openalex.org/W3156257261","https://openalex.org/W4233762729","https://openalex.org/W4252979015","https://openalex.org/W4300906944","https://openalex.org/W4302366875","https://openalex.org/W4320013936","https://openalex.org/W6621906925","https://openalex.org/W6659520598","https://openalex.org/W6676210859","https://openalex.org/W6727623532","https://openalex.org/W6779669310","https://openalex.org/W6786768871","https://openalex.org/W6794192968"],"related_works":["https://openalex.org/W2414667732","https://openalex.org/W2183396261","https://openalex.org/W2340232283","https://openalex.org/W4300998569","https://openalex.org/W1600097617","https://openalex.org/W130813085","https://openalex.org/W2317603910","https://openalex.org/W4233290694","https://openalex.org/W2358249721","https://openalex.org/W2188698441"],"abstract_inverted_index":{"Comparing":[0],"probability":[1,24,112,167],"distributions":[2],"is":[3,26,57],"an":[4],"indispensable":[5],"and":[6,12,33,47,64,79,97,137,158,185],"ubiquitous":[7],"task":[8],"in":[9,65,103],"machine":[10],"learning":[11],"statistics.":[13],"The":[14,52],"most":[15,37,54],"common":[16,55,154],"way":[17,56],"to":[18,27,58,162,207,212],"compare":[19],"a":[20,29,60,126,138,209],"pair":[21,110],"of":[22,41,75,85,111,117,156,172],"Borel":[23,166],"measures":[25,113,168],"compute":[28,59],"metric":[30,42,46],"between":[31,62,128,165],"them,":[32,63],"by":[34],"far":[35],"the":[36,44,48,86,109,118,129,133,213],"widely":[38],"used":[39],"notions":[40,155],"are":[43,82,105,114,203],"Wasserstein":[45],"total":[49],"variation":[50],"metric.":[51],"next":[53],"divergence":[61],"this":[66],"case":[67],"almost":[68],"every":[69],"known":[70],"divergences":[71,98,159],"such":[72],"as":[73,206],"those":[74],"Kullback\u2013Leibler,":[76],"Jensen\u2013Shannon,":[77],"R\u00e9nyi,":[78],"many":[80],"more,":[81],"special":[83],"cases":[84],"<inline-formula":[87,142,178,188,196],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[88,143,179,189,197],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">":[89,144,180,190,198],"<tex-math":[90,145,181,191,199],"notation=\"LaTeX\">$f$":[91],"</tex-math></inline-formula>":[92,148,184,194,202],"-divergence.":[93],"Nevertheless":[94],"these":[95,153],"metrics":[96,157],"may":[99],"only":[100,106],"be":[101],"computed,":[102],"fact,":[104],"defined,":[107],"when":[108],"on":[115,132,141,170,177,187],"spaces":[116,171],"same":[119],"dimension.":[120],"How":[121],"would":[122],"one":[123,176],"quantify,":[124],"say,":[125],"KL-divergence":[127],"uniform":[130],"distribution":[131,140],"interval":[134],"[\u22121,":[135],"1]":[136],"Gaussian":[139],"notation=\"LaTeX\">$\\mathbb":[146,182,192],"{R}^{3}$":[147],"?":[149],"We":[150],"show":[151],"that":[152],"give":[160,208],"rise":[161],"natural":[163],"distances":[164],"defined":[169],"different":[173],"dimensions,":[174],"e.g.,":[175],"{R}^{m}$":[183],"another":[186],"{R}^{n}$":[193],"where":[195],"notation=\"LaTeX\">$m,":[200],"n$":[201],"distinct,":[204],"so":[205],"meaningful":[210],"answer":[211],"previous":[214],"question.":[215]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":12},{"year":2024,"cited_by_count":16},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2}],"updated_date":"2026-03-28T08:17:26.163206","created_date":"2025-10-10T00:00:00"}
