{"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":false,"raw_author_name":"Yuhang Cai","raw_affiliation_strings":["Department of Statistics, The University of Chicago, Chicago, IL, USA"],"raw_orcid":null,"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"],"raw_orcid":"https://orcid.org/0000-0002-9808-0138","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":[],"corresponding_institution_ids":[],"apc_list":{"value":2645,"currency":"USD","value_usd":2645},"apc_paid":null,"fwci":4.183,"has_fulltext":false,"cited_by_count":44,"citation_normalized_percentile":{"value":0.95069762,"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":[1,24,105,153],"distributions":[2],"is":[3,26,57],"an":[4],"indispensable":[5],"and":[6,12,33,47,64,79,90,130,144],"ubiquitous":[7],"task":[8],"in":[9,65,96],"machine":[10],"learning":[11],"statistics.":[13],"The":[14,52],"most":[15,37,54],"common":[16,55,140],"way":[17,56],"to":[18,27,58,148,172,177],"compare":[19],"a":[20,29,60,119,131,174],"pair":[21,103],"of":[22,41,75,85,104,110,142,158],"Borel":[23,152],"measures":[25,106,154],"compute":[28,59],"metric":[30,42,46],"between":[31,62,121,151],"them,":[32,63],"by":[34],"far":[35],"the":[36,44,48,102,111,122,126,178],"widely":[38],"used":[39],"notions":[40,141],"are":[43,82,98,107],"Wasserstein":[45],"total":[49],"variation":[50],"metric.":[51],"next":[53],"divergence":[61],"this":[66],"case":[67],"almost":[68],"every":[69],"known":[70],"divergences":[71,91,145],"such":[72],"as":[73,171],"those":[74],"Kullback\u2013Leibler,":[76],"Jensen\u2013Shannon,":[77],"R\u00e9nyi,":[78],"many":[80],"more,":[81],"special":[83],"cases":[84],"the$f$-divergence.":[86],"Nevertheless":[87],"these":[88,139],"metrics":[89,143],"may":[92],"only":[93,99],"be":[94],"computed,":[95],"fact,":[97],"defined,":[100],"when":[101],"on":[108,125,156],"spaces":[109,157],"same":[112],"dimension.":[113],"How":[114],"would":[115],"one":[116,162],"quantify,":[117],"say,":[118],"KL-divergence":[120],"uniform":[123],"distribution":[124,133],"interval":[127],"[\u22121,":[128],"1]":[129],"Gaussian":[132],"on$\\mathbb":[134,163,166],"{R}^{3}$?":[135],"We":[136],"show":[137],"that":[138],"give":[146,173],"rise":[147],"natural":[149],"distances":[150],"defined":[155],"different":[159],"dimensions,":[160],"e.g.,":[161],"{R}^{m}$and":[164],"another":[165],"{R}^{n}$where$m,":[167],"n$are":[168],"distinct,":[169],"so":[170],"meaningful":[175],"answer":[176],"previous":[179],"question.":[180]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":13},{"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":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}