{"id":"https://openalex.org/W4412620466","doi":"https://doi.org/10.1007/s10994-025-06828-8","title":"Computing the distance between unbalanced distributions: the flat metric","display_name":"Computing the distance between unbalanced distributions: the flat metric","publication_year":2025,"publication_date":"2025-07-24","ids":{"openalex":"https://openalex.org/W4412620466","doi":"https://doi.org/10.1007/s10994-025-06828-8","pmid":"https://pubmed.ncbi.nlm.nih.gov/40726635"},"language":"en","primary_location":{"id":"doi:10.1007/s10994-025-06828-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-025-06828-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-025-06828-8.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10994-025-06828-8.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5119060685","display_name":"Henri Schmidt","orcid":"https://orcid.org/0009-0002-2666-2670"},"institutions":[{"id":"https://openalex.org/I223822909","display_name":"Heidelberg University","ror":"https://ror.org/038t36y30","country_code":"DE","type":"education","lineage":["https://openalex.org/I223822909"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Henri Schmidt","raw_affiliation_strings":["Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany","Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany"],"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany","institution_ids":[]},{"raw_affiliation_string":"Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany","institution_ids":["https://openalex.org/I223822909"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078145899","display_name":"Christian D\u00fcll","orcid":"https://orcid.org/0000-0003-4841-9399"},"institutions":[{"id":"https://openalex.org/I223822909","display_name":"Heidelberg University","ror":"https://ror.org/038t36y30","country_code":"DE","type":"education","lineage":["https://openalex.org/I223822909"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Christian D\u00fcll","raw_affiliation_strings":["Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany","Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany"],"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany","institution_ids":[]},{"raw_affiliation_string":"Institute of Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany","institution_ids":["https://openalex.org/I223822909"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5119060685"],"corresponding_institution_ids":["https://openalex.org/I223822909"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.16626832,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"114","issue":"8","first_page":"195","last_page":"195"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.9851999878883362,"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"}},"topics":[{"id":"https://openalex.org/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.9851999878883362,"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"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9775000214576721,"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/T10052","display_name":"Medical Image Segmentation Techniques","score":0.9564999938011169,"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":0.5482276082038879},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5444955825805664},{"id":"https://openalex.org/keywords/normalization","display_name":"Normalization (sociology)","score":0.5415968298912048},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.5271791219711304},{"id":"https://openalex.org/keywords/ground-truth","display_name":"Ground truth","score":0.5073971152305603},{"id":"https://openalex.org/keywords/comparability","display_name":"Comparability","score":0.49256759881973267},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.4871968626976013},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4823998212814331},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.4767533242702484},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4436362087726593},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.42918264865875244},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.29785871505737305},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.15473651885986328},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1537863314151764}],"concepts":[{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5482276082038879},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5444955825805664},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.5415968298912048},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5271791219711304},{"id":"https://openalex.org/C146849305","wikidata":"https://www.wikidata.org/wiki/Q370766","display_name":"Ground truth","level":2,"score":0.5073971152305603},{"id":"https://openalex.org/C197947376","wikidata":"https://www.wikidata.org/wiki/Q5155608","display_name":"Comparability","level":2,"score":0.49256759881973267},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.4871968626976013},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4823998212814331},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.4767533242702484},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4436362087726593},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.42918264865875244},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.29785871505737305},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.15473651885986328},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1537863314151764},{"id":"https://openalex.org/C19165224","wikidata":"https://www.wikidata.org/wiki/Q23404","display_name":"Anthropology","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s10994-025-06828-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-025-06828-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-025-06828-8.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning","raw_type":"journal-article"},{"id":"pmid:40726635","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/40726635","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":"Machine learning","raw_type":null},{"id":"pmh:oai:pubmedcentral.nih.gov:12289810","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/12289810","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mach Learn","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1007/s10994-025-06828-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-025-06828-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-025-06828-8.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5295351072","display_name":null,"funder_award_id":"101071786","funder_id":"https://openalex.org/F4320338453","funder_display_name":"HORIZON EUROPE European Research Council"},{"id":"https://openalex.org/G7527820262","display_name":null,"funder_award_id":"EXC 2181/1 - 390900948","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320320881","display_name":"Universit\u00e4t Heidelberg","ror":"https://ror.org/038t36y30"},{"id":"https://openalex.org/F4320338453","display_name":"HORIZON EUROPE European Research Council","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4412620466.pdf","grobid_xml":"https://content.openalex.org/works/W4412620466.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W1585160083","https://openalex.org/W1990805993","https://openalex.org/W2013176880","https://openalex.org/W2054276177","https://openalex.org/W2079557531","https://openalex.org/W2117616929","https://openalex.org/W2158131535","https://openalex.org/W2163380389","https://openalex.org/W2487202609","https://openalex.org/W2530578841","https://openalex.org/W2618843012","https://openalex.org/W2724892359","https://openalex.org/W2785755647","https://openalex.org/W2898310674","https://openalex.org/W2911279170","https://openalex.org/W2949067670","https://openalex.org/W3046908166","https://openalex.org/W3092241499","https://openalex.org/W3106451119","https://openalex.org/W3135516022","https://openalex.org/W3162855528","https://openalex.org/W3215961257","https://openalex.org/W4206471589","https://openalex.org/W4220762510","https://openalex.org/W4233762729","https://openalex.org/W4382318847","https://openalex.org/W4400957745","https://openalex.org/W6631190155","https://openalex.org/W6676325516","https://openalex.org/W6735913928","https://openalex.org/W6756052093","https://openalex.org/W6787167997"],"related_works":["https://openalex.org/W2365594754","https://openalex.org/W2575292835","https://openalex.org/W3021704418","https://openalex.org/W4287902769","https://openalex.org/W3001140700","https://openalex.org/W2995453361","https://openalex.org/W4406379843","https://openalex.org/W4390295458","https://openalex.org/W1990407237","https://openalex.org/W2092282998"],"abstract_inverted_index":{"We":[0,125],"provide":[1],"an":[2,99],"implementation":[3,43],"to":[4,30,47,53,97],"compute":[5],"the":[6,22,31,34,70,76,89,104,127,130],"flat":[7,13],"metric":[8],"in":[9,132],"any":[10],"dimension.":[11],"The":[12,86],"metric,":[14],"also":[15],"called":[16],"dual":[17],"bounded":[18],"Lipschitz":[19],"distance,":[20],"generalizes":[21],"well-known":[23],"Wasserstein":[24],"distance":[25,105],"<mml:math":[26],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>W</mml:mi>":[27],"<mml:mn>1</mml:mn></mml:msub>":[28],"</mml:math>":[29],"case":[32],"that":[33],"distributions":[35,74],"are":[36],"of":[37,60,72,88,117,129],"unequal":[38],"total":[39],"mass.":[40],"Thus,":[41],"our":[42],"adapts":[44],"very":[45],"well":[46,141],"mass":[48],"differences":[49],"and":[50,68],"uses":[51],"them":[52],"distinguish":[54],"between":[55,106],"different":[56],"distributions.":[57],"This":[58],"is":[59,79,83,91],"particular":[61],"interest":[62],"for":[63,69],"unbalanced":[64],"optimal":[65,100],"transport":[66],"tasks":[67],"analysis":[71],"data":[73],"where":[75,135],"sample":[77],"size":[78],"important":[80],"or":[81],"normalization":[82],"not":[84],"possible.":[85],"core":[87],"method":[90],"based":[92],"on":[93,114],"a":[94],"neural":[95],"network":[96],"determine":[98],"test":[101],"function":[102],"realizing":[103],"two":[107],"given":[108],"measures.":[109],"Special":[110],"focus":[111],"was":[112,138],"put":[113],"achieving":[115],"comparability":[116],"pairwise":[118],"computed":[119],"distances":[120],"from":[121],"independently":[122],"trained":[123],"networks.":[124],"tested":[126],"quality":[128],"output":[131],"several":[133],"experiments":[134],"ground":[136],"truth":[137],"available":[139],"as":[140,142],"with":[143],"simulated":[144],"data.":[145]},"counts_by_year":[],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
