{"id":"https://openalex.org/W2963561542","doi":"https://doi.org/10.1109/tpami.2017.2749575","title":"Fixed Points of Belief Propagation\u2014An Analysis via Polynomial Homotopy Continuation","display_name":"Fixed Points of Belief Propagation\u2014An Analysis via Polynomial Homotopy Continuation","publication_year":2017,"publication_date":"2017-09-07","ids":{"openalex":"https://openalex.org/W2963561542","doi":"https://doi.org/10.1109/tpami.2017.2749575","mag":"2963561542","pmid":"https://pubmed.ncbi.nlm.nih.gov/28885150"},"language":"en","primary_location":{"id":"doi:10.1109/tpami.2017.2749575","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tpami.2017.2749575","pdf_url":null,"source":{"id":"https://openalex.org/S199944782","display_name":"IEEE Transactions on Pattern Analysis and Machine Intelligence","issn_l":"0162-8828","issn":["0162-8828","1939-3539","2160-9292"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320439","host_organization_name":"IEEE Computer Society","host_organization_lineage":["https://openalex.org/P4310320439","https://openalex.org/P4310319808"],"host_organization_lineage_names":["IEEE Computer Society","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 Pattern Analysis and Machine Intelligence","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"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/A5009185697","display_name":"Christian Knoll","orcid":"https://orcid.org/0000-0003-3920-1419"},"institutions":[{"id":"https://openalex.org/I4092182","display_name":"Graz University of Technology","ror":"https://ror.org/00d7xrm67","country_code":"AT","type":"education","lineage":["https://openalex.org/I4092182"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Christian Knoll","raw_affiliation_strings":["Signal Processing and Speech Communication Laboratory, Graz University of Technology, Graz, Austria"],"raw_orcid":"https://orcid.org/0000-0003-3920-1419","affiliations":[{"raw_affiliation_string":"Signal Processing and Speech Communication Laboratory, Graz University of Technology, Graz, Austria","institution_ids":["https://openalex.org/I4092182"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056701307","display_name":"Dhagash Mehta","orcid":"https://orcid.org/0000-0002-1040-9032"},"institutions":[{"id":"https://openalex.org/I107639228","display_name":"University of Notre Dame","ror":"https://ror.org/00mkhxb43","country_code":"US","type":"education","lineage":["https://openalex.org/I107639228"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Dhagash Mehta","raw_affiliation_strings":["Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN","institution_ids":["https://openalex.org/I107639228"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101634382","display_name":"Tianran Chen","orcid":"https://orcid.org/0000-0002-3890-617X"},"institutions":[{"id":"https://openalex.org/I902671590","display_name":"Auburn University at Montgomery","ror":"https://ror.org/05eynd241","country_code":"US","type":"education","lineage":["https://openalex.org/I902671590"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Tianran Chen","raw_affiliation_strings":["Department of Mathematics and Computer Science, Auburn University at Montgomery, Montgomery, AL"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, Auburn University at Montgomery, Montgomery, AL","institution_ids":["https://openalex.org/I902671590"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5015798259","display_name":"Franz Pernkopf","orcid":"https://orcid.org/0000-0002-6356-3367"},"institutions":[{"id":"https://openalex.org/I4092182","display_name":"Graz University of Technology","ror":"https://ror.org/00d7xrm67","country_code":"AT","type":"education","lineage":["https://openalex.org/I4092182"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Franz Pernkopf","raw_affiliation_strings":["Signal Processing and Speech Communication Laboratory, Graz University of Technology, Graz, Austria"],"raw_orcid":"https://orcid.org/0000-0002-6356-3367","affiliations":[{"raw_affiliation_string":"Signal Processing and Speech Communication Laboratory, Graz University of Technology, Graz, Austria","institution_ids":["https://openalex.org/I4092182"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2645,"currency":"USD","value_usd":2645},"apc_paid":null,"fwci":1.1407,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.85310924,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"40","issue":"9","first_page":"2124","last_page":"2136"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11303","display_name":"Bayesian Modeling and Causal Inference","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11303","display_name":"Bayesian Modeling and Causal Inference","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11321","display_name":"Error Correcting Code Techniques","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T10621","display_name":"Gene Regulatory Network Analysis","score":0.970300018787384,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.8109340667724609},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7479202747344971},{"id":"https://openalex.org/keywords/least-fixed-point","display_name":"Least fixed point","score":0.529399037361145},{"id":"https://openalex.org/keywords/fixed-point-iteration","display_name":"Fixed-point iteration","score":0.5180802345275879},{"id":"https://openalex.org/keywords/ising-model","display_name":"Ising model","score":0.5103475451469421},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5054144859313965},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.49761512875556946},{"id":"https://openalex.org/keywords/homotopy","display_name":"Homotopy","score":0.4931037127971649},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.463304340839386},{"id":"https://openalex.org/keywords/fixed-point-theorem","display_name":"Fixed-point theorem","score":0.3219001293182373},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3168591260910034},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2369413673877716},{"id":"https://openalex.org/keywords/schauder-fixed-point-theorem","display_name":"Schauder fixed point theorem","score":0.14519265294075012},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.13312658667564392}],"concepts":[{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.8109340667724609},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7479202747344971},{"id":"https://openalex.org/C27333817","wikidata":"https://www.wikidata.org/wiki/Q6134913","display_name":"Least fixed point","level":5,"score":0.529399037361145},{"id":"https://openalex.org/C157449380","wikidata":"https://www.wikidata.org/wiki/Q1030759","display_name":"Fixed-point iteration","level":3,"score":0.5180802345275879},{"id":"https://openalex.org/C51329190","wikidata":"https://www.wikidata.org/wiki/Q1076349","display_name":"Ising model","level":2,"score":0.5103475451469421},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5054144859313965},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.49761512875556946},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.4931037127971649},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.463304340839386},{"id":"https://openalex.org/C45962547","wikidata":"https://www.wikidata.org/wiki/Q1422068","display_name":"Fixed-point theorem","level":2,"score":0.3219001293182373},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3168591260910034},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2369413673877716},{"id":"https://openalex.org/C28642324","wikidata":"https://www.wikidata.org/wiki/Q656176","display_name":"Schauder fixed point theorem","level":4,"score":0.14519265294075012},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.13312658667564392},{"id":"https://openalex.org/C206929604","wikidata":"https://www.wikidata.org/wiki/Q530152","display_name":"Picard\u2013Lindel\u00f6f theorem","level":3,"score":0.0},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tpami.2017.2749575","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tpami.2017.2749575","pdf_url":null,"source":{"id":"https://openalex.org/S199944782","display_name":"IEEE Transactions on Pattern Analysis and Machine Intelligence","issn_l":"0162-8828","issn":["0162-8828","1939-3539","2160-9292"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320439","host_organization_name":"IEEE Computer Society","host_organization_lineage":["https://openalex.org/P4310320439","https://openalex.org/P4310319808"],"host_organization_lineage_names":["IEEE Computer Society","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 Pattern Analysis and Machine Intelligence","raw_type":"journal-article"},{"id":"pmid:28885150","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/28885150","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":"IEEE transactions on pattern analysis and machine intelligence","raw_type":"Journal Article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.4399999976158142}],"awards":[{"id":"https://openalex.org/G8609102460","display_name":"Learning of Bayesian Network Classifiers and Sum-Product Networks","funder_award_id":"P 27803","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":78,"referenced_works":["https://openalex.org/W186521003","https://openalex.org/W254210011","https://openalex.org/W599135353","https://openalex.org/W607590176","https://openalex.org/W1470223494","https://openalex.org/W1483218536","https://openalex.org/W1507653385","https://openalex.org/W1511986666","https://openalex.org/W1521738998","https://openalex.org/W1525888637","https://openalex.org/W1545898854","https://openalex.org/W1560512119","https://openalex.org/W1570420806","https://openalex.org/W1588772321","https://openalex.org/W1593793857","https://openalex.org/W1607198972","https://openalex.org/W1828211540","https://openalex.org/W1842236176","https://openalex.org/W1844626844","https://openalex.org/W1846689784","https://openalex.org/W1985548249","https://openalex.org/W1985813166","https://openalex.org/W1989008275","https://openalex.org/W1989923040","https://openalex.org/W2013504313","https://openalex.org/W2026129786","https://openalex.org/W2069525907","https://openalex.org/W2098571099","https://openalex.org/W2099164281","https://openalex.org/W2118696796","https://openalex.org/W2134700719","https://openalex.org/W2137813581","https://openalex.org/W2142901448","https://openalex.org/W2143023483","https://openalex.org/W2150859678","https://openalex.org/W2159080219","https://openalex.org/W2163364417","https://openalex.org/W2166765763","https://openalex.org/W2167149929","https://openalex.org/W2169415915","https://openalex.org/W2169447051","https://openalex.org/W2170267707","https://openalex.org/W2175160295","https://openalex.org/W2182990316","https://openalex.org/W2301983752","https://openalex.org/W2398757277","https://openalex.org/W2414396530","https://openalex.org/W2566505556","https://openalex.org/W2617837084","https://openalex.org/W2775175104","https://openalex.org/W2807232057","https://openalex.org/W2947000318","https://openalex.org/W2950466990","https://openalex.org/W2950574223","https://openalex.org/W2950654313","https://openalex.org/W2963123876","https://openalex.org/W2963422512","https://openalex.org/W4234426551","https://openalex.org/W4235499294","https://openalex.org/W4238102028","https://openalex.org/W4285719527","https://openalex.org/W4293052541","https://openalex.org/W6631424144","https://openalex.org/W6632679229","https://openalex.org/W6633434192","https://openalex.org/W6635375731","https://openalex.org/W6638725582","https://openalex.org/W6638856916","https://openalex.org/W6667884216","https://openalex.org/W6681488117","https://openalex.org/W6683951949","https://openalex.org/W6684888907","https://openalex.org/W6685097678","https://openalex.org/W6685897926","https://openalex.org/W6687822886","https://openalex.org/W6697871001","https://openalex.org/W6716207201","https://openalex.org/W6752562311"],"related_works":["https://openalex.org/W3196429779","https://openalex.org/W4244119288","https://openalex.org/W2585118952","https://openalex.org/W2846516035","https://openalex.org/W2963131862","https://openalex.org/W1972020955","https://openalex.org/W2506139452","https://openalex.org/W1559063105","https://openalex.org/W2771083884","https://openalex.org/W3035945792"],"abstract_inverted_index":{"Belief":[0],"propagation":[1],"(BP)":[2],"is":[3,22,43,95,193],"an":[4],"iterative":[5],"method":[6,94,192],"to":[7,45,65,151,169,195],"perform":[8],"approximate":[9],"inference":[10],"on":[11,28,83,88],"arbitrary":[12],"graphical":[13],"models.":[14],"Whether":[15],"BP":[16,66,100,157,167],"converges":[17],"and":[18,32,87,121,133,189],"if":[19],"the":[20,30,33,36,60,77,112,116,122,128,131,145,152,190],"solution":[21],"a":[23,55,72,173,186],"unique":[24,174],"fixed":[25,48,67,101,107,119,158,175,183,198],"point":[26,176,184],"depends":[27],"both":[29,115],"structure":[31,113],"parametrization":[34],"of":[35,57,62,97,118,130,156],"model.":[37],"To":[38,69],"understand":[39],"this":[40,51,182,197],"dependence":[41],"it":[42],"interesting":[44],"find":[46,163],"all":[47,99,148],"points.":[49,68,102],"In":[50],"work,":[52],"we":[53,75,109,162,179],"formulate":[54],"set":[56],"polynomial":[58],"equations,":[59],"solutions":[61],"which":[63,166],"correspond":[64],"solve":[70],"such":[71],"nonlinear":[73],"system":[74],"present":[76],"numerical":[78],"polynomial-homotopy-continuation":[79],"(NPHC)":[80],"method.":[81],"Experiments":[82],"binary":[84],"Ising":[85,104],"models":[86,105],"error-correcting":[89],"codes":[90],"show":[91,110,180],"how":[92,111],"our":[93],"capable":[96],"obtaining":[98],"On":[103],"with":[106,139],"parameters":[108],"influences":[114],"number":[117],"points":[120,159],"convergence":[123],"properties.":[124],"We":[125],"further":[126],"asses":[127],"accuracy":[129,146],"marginals":[132,138],"weighted":[134],"combinations":[135],"thereof.":[136],"Weighting":[137],"their":[140],"respective":[141],"partition":[142],"function":[143],"increases":[144],"in":[147],"experiments.":[149],"Contrary":[150],"conjecture":[153],"that":[154,181],"uniqueness":[155],"implies":[160],"convergence,":[161],"graphs":[164],"for":[165],"fails":[168],"converge,":[170],"even":[171],"though":[172],"exists.":[177],"Moreover,":[178],"gives":[185],"good":[187],"approximation,":[188],"NPHC":[191],"able":[194],"obtain":[196],"point.":[199]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":4}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}