{"id":"https://openalex.org/W3111067318","doi":"https://doi.org/10.1007/s10208-022-09557-9","title":"Continuum Limit of Lipschitz Learning on Graphs","display_name":"Continuum Limit of Lipschitz Learning on Graphs","publication_year":2022,"publication_date":"2022-01-21","ids":{"openalex":"https://openalex.org/W3111067318","doi":"https://doi.org/10.1007/s10208-022-09557-9","mag":"3111067318"},"language":"en","primary_location":{"id":"doi:10.1007/s10208-022-09557-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-022-09557-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-022-09557-9.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"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":"Foundations of Computational Mathematics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10208-022-09557-9.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051931414","display_name":"Tim Roith","orcid":"https://orcid.org/0000-0001-8440-2928"},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Tim Roith","raw_affiliation_strings":["Department Mathematik, Friedrich-Alexander Universit\u00e4t Erlangen-N\u00fcrnberg, Cauerstra\u00dfe 11, 91058, Erlangen, Germany","University of Erlangen Nuremberg"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department Mathematik, Friedrich-Alexander Universit\u00e4t Erlangen-N\u00fcrnberg, Cauerstra\u00dfe 11, 91058, Erlangen, Germany","institution_ids":["https://openalex.org/I181369854"]},{"raw_affiliation_string":"University of Erlangen Nuremberg","institution_ids":["https://openalex.org/I181369854"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035045595","display_name":"Leon Bungert","orcid":"https://orcid.org/0000-0002-6554-9892"},"institutions":[{"id":"https://openalex.org/I135140700","display_name":"University of Bonn","ror":"https://ror.org/041nas322","country_code":"DE","type":"education","lineage":["https://openalex.org/I135140700"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Leon Bungert","raw_affiliation_strings":["Hausdorff Center for Mathematics, Rheinische Friedrich-Wilhelms-Universit\u00e4t Bonn, Endenicher Allee 62, Villa Maria, 53115, Bonn, Germany","University of Bonn >  >  >  >"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hausdorff Center for Mathematics, Rheinische Friedrich-Wilhelms-Universit\u00e4t Bonn, Endenicher Allee 62, Villa Maria, 53115, Bonn, Germany","institution_ids":["https://openalex.org/I135140700"]},{"raw_affiliation_string":"University of Bonn >  >  >  >","institution_ids":["https://openalex.org/I135140700"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.4566,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.63119119,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"23","issue":"2","first_page":"393","last_page":"431"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9995999932289124,"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"}},"topics":[{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9995999932289124,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9882000088691711,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.795896053314209},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7312999963760376},{"id":"https://openalex.org/keywords/hausdorff-space","display_name":"Hausdorff space","score":0.476835161447525},{"id":"https://openalex.org/keywords/limit-of-a-sequence","display_name":"Limit of a sequence","score":0.4665885269641876},{"id":"https://openalex.org/keywords/compact-space","display_name":"Compact space","score":0.4660341143608093},{"id":"https://openalex.org/keywords/uniform-norm","display_name":"Uniform norm","score":0.4250551164150238},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3914395570755005},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38405564427375793},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3200572729110718},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2884874641895294},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.2608630657196045}],"concepts":[{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.795896053314209},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7312999963760376},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.476835161447525},{"id":"https://openalex.org/C71287196","wikidata":"https://www.wikidata.org/wiki/Q847204","display_name":"Limit of a sequence","level":3,"score":0.4665885269641876},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.4660341143608093},{"id":"https://openalex.org/C146324458","wikidata":"https://www.wikidata.org/wiki/Q1202673","display_name":"Uniform norm","level":2,"score":0.4250551164150238},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3914395570755005},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38405564427375793},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3200572729110718},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2884874641895294},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.2608630657196045}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s10208-022-09557-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-022-09557-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-022-09557-9.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"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":"Foundations of Computational Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2012.03772","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2012.03772","pdf_url":"https://arxiv.org/pdf/2012.03772","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"mag:3111067318","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2012.03772.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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.2012.03772","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2012.03772","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"article-journal"}],"best_oa_location":{"id":"doi:10.1007/s10208-022-09557-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-022-09557-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-022-09557-9.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"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":"Foundations of Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1521016025","display_name":"EXC 2047: Hausdorff Center for Mathematics (HCM)","funder_award_id":"390685813","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2621840818","display_name":null,"funder_award_id":"Projekt-ID 390685813","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G6032080779","display_name":null,"funder_award_id":"GZ 2047/1, Projekt-ID 390685813","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G6850111841","display_name":null,"funder_award_id":"05M2020","funder_id":"https://openalex.org/F4320321114","funder_display_name":"Bundesministerium f\u00fcr Bildung und Forschung"},{"id":"https://openalex.org/G8847963564","display_name":null,"funder_award_id":"GZ 2047/1","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/F4320321114","display_name":"Bundesministerium f\u00fcr Bildung und Forschung","ror":"https://ror.org/04pz7b180"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3111067318.pdf","grobid_xml":"https://content.openalex.org/works/W3111067318.grobid-xml"},"referenced_works_count":40,"referenced_works":["https://openalex.org/W308629319","https://openalex.org/W1479807131","https://openalex.org/W1497443639","https://openalex.org/W1545761024","https://openalex.org/W1826824261","https://openalex.org/W1861295442","https://openalex.org/W1887477912","https://openalex.org/W1979463344","https://openalex.org/W1981263972","https://openalex.org/W1990334093","https://openalex.org/W2002276939","https://openalex.org/W2070921008","https://openalex.org/W2086943813","https://openalex.org/W2134086356","https://openalex.org/W2136504847","https://openalex.org/W2139823104","https://openalex.org/W2144280422","https://openalex.org/W2145591097","https://openalex.org/W2167768422","https://openalex.org/W2335062971","https://openalex.org/W2349307120","https://openalex.org/W2504943536","https://openalex.org/W2588998037","https://openalex.org/W2906226118","https://openalex.org/W2952195400","https://openalex.org/W2962701197","https://openalex.org/W2963833097","https://openalex.org/W2964096395","https://openalex.org/W2964179816","https://openalex.org/W2964292172","https://openalex.org/W2992402078","https://openalex.org/W3001332551","https://openalex.org/W3016633500","https://openalex.org/W3034023082","https://openalex.org/W3034398749","https://openalex.org/W3103058618","https://openalex.org/W3144498383","https://openalex.org/W3215062344","https://openalex.org/W4243794903","https://openalex.org/W4253577938"],"related_works":["https://openalex.org/W2925046028","https://openalex.org/W2955614645","https://openalex.org/W3005918773","https://openalex.org/W180867281","https://openalex.org/W2140402456","https://openalex.org/W1972017136","https://openalex.org/W2969083946","https://openalex.org/W3006980704","https://openalex.org/W2890229958","https://openalex.org/W2898507254","https://openalex.org/W2124596865","https://openalex.org/W2904821701","https://openalex.org/W2531697175","https://openalex.org/W2467441529","https://openalex.org/W2070856516","https://openalex.org/W3212553079","https://openalex.org/W3123868121","https://openalex.org/W2767229208","https://openalex.org/W3196514396","https://openalex.org/W1963897537"],"abstract_inverted_index":{"Abstract":[0],"Tackling":[1],"semi-supervised":[2],"learning":[3,135],"problems":[4],"with":[5,240],"graph-based":[6],"methods":[7],"has":[8],"become":[9],"a":[10,25,47,148,160,213,222,253],"trend":[11],"in":[12,172,226],"recent":[13],"years":[14],"since":[15],"graphs":[16],"can":[17],"represent":[18],"all":[19],"kinds":[20],"of":[21,34,58,74,112,123,133,150,159,188,200,206,216,257],"data":[22,218],"and":[23,163],"provide":[24],"suitable":[26],"framework":[27],"for":[28,32],"studying":[29],"continuum":[30,72,110,131],"limits,":[31],"example,":[33],"differential":[35],"operators.":[36],"A":[37],"popular":[38],"strategy":[39],"here":[40],"is":[41,104],"p":[42],"-Laplacian":[43],"learning,":[44,109],"which":[45,103,152,203,219],"poses":[46],"smoothness":[48],"condition":[49],"on":[50,55],"the":[51,56,90,113,121,154,173,185,189,192,201,227],"sought":[52],"inference":[53],"function":[54,162],"set":[57,215,225],"unlabeled":[59],"data.":[60],"For":[61,89],"$$p&lt;\\infty":[62],"$$":[63,83,93,138,166],"<mml:math":[64,84,94,139,167,176,243],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[65,85,95,140,168,177,244],"<mml:mrow>":[66,96],"<mml:mi>p</mml:mi>":[67,97,247],"<mml:mo>&lt;</mml:mo>":[68],"<mml:mi>\u221e</mml:mi>":[69,99,180],"</mml:mrow>":[70,100],"</mml:math>":[71,87,101,142,170,182,249],"limits":[73,111,132],"this":[75,127],"approach":[76],"were":[77,118],"studied":[78,119],"using":[79,120,136],"tools":[80],"from":[81],"$$\\varGamma":[82,137,165],"<mml:mi>\u0393</mml:mi>":[86,141,169],"-convergence.":[88,143],"case":[91],"$$p=\\infty":[92],"<mml:mo>=</mml:mo>":[98],",":[102],"referred":[105],"to":[106,184,221,234,261],"as":[107,191,252],"Lipschitz":[108,134,157],"related":[114],"infinity":[115],"Laplacian":[116],"equation":[117],"concept":[122],"viscosity":[124],"solutions.":[125],"In":[126,144,208],"work,":[128],"we":[129,146,197,211],"prove":[130,164,255],"particular,":[145],"define":[147],"sequence":[149],"functionals":[151,202],"approximate":[153],"largest":[155],"local":[156],"constant":[158],"graph":[161,193,258],"-convergence":[171],"$$L^{\\infty":[174],"}$$":[175],"<mml:msup>":[178,245],"<mml:mi>L</mml:mi>":[179,246],"</mml:msup>":[181,248],"-topology":[183],"supremum":[186],"norm":[187],"gradient":[190],"becomes":[194],"denser.":[195],"Furthermore,":[196],"show":[198],"compactness":[199],"implies":[204],"convergence":[205,256],"minimizers.":[207],"our":[209,232],"analysis":[210],"allow":[212],"varying":[214],"labeled":[217],"converges":[220],"general":[223],"closed":[224],"Hausdorff":[228],"distance.":[229],"We":[230],"apply":[231],"results":[233],"nonlinear":[235],"ground":[236],"states,":[237],"i.e.,":[238],"minimizers":[239],"constrained":[241],"$$L^p$$":[242],"-norm,":[250],"and,":[251],"by-product,":[254],"distance":[259,263],"functions":[260],"geodesic":[262],"functions.":[264]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1}],"updated_date":"2026-06-22T08:00:12.763002","created_date":"2025-10-10T00:00:00"}