{"id":"https://openalex.org/W7152489468","doi":"https://doi.org/10.1109/mci.2026.3655677","title":"Graph Regularization for High Order Neural Networks: Boundedness, Convergence, and Enhanced Performance With Entropy Error Function [Research Frontier]","display_name":"Graph Regularization for High Order Neural Networks: Boundedness, Convergence, and Enhanced Performance With Entropy Error Function [Research Frontier]","publication_year":2026,"publication_date":"2026-04-09","ids":{"openalex":"https://openalex.org/W7152489468","doi":"https://doi.org/10.1109/mci.2026.3655677"},"language":null,"primary_location":{"id":"doi:10.1109/mci.2026.3655677","is_oa":false,"landing_page_url":"https://doi.org/10.1109/mci.2026.3655677","pdf_url":null,"source":{"id":"https://openalex.org/S104797584","display_name":"IEEE Computational Intelligence Magazine","issn_l":"1556-603X","issn":["1556-603X","1556-6048"],"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 Computational Intelligence Magazine","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/A5022385306","display_name":"Qinwei Fan","orcid":"https://orcid.org/0000-0002-1017-3496"},"institutions":[{"id":"https://openalex.org/I37987034","display_name":"Guangzhou University","ror":"https://ror.org/05ar8rn06","country_code":"CN","type":"education","lineage":["https://openalex.org/I37987034"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Qinwei Fan","raw_affiliation_strings":["School of Mathematics and Information Science, Guangzhou University, Guangzhou, China"],"raw_orcid":"https://orcid.org/0000-0002-1017-3496","affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Science, Guangzhou University, Guangzhou, China","institution_ids":["https://openalex.org/I37987034"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025861210","display_name":"Qianru Huang","orcid":null},"institutions":[{"id":"https://openalex.org/I27599042","display_name":"Xi'an Polytechnic University","ror":"https://ror.org/03442p831","country_code":"CN","type":"education","lineage":["https://openalex.org/I27599042"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qianru Huang","raw_affiliation_strings":["School of Science, Xi&#x2019;an Polytechnic University, Xi&#x2019;an, China"],"raw_orcid":"https://orcid.org/0009-0007-8875-2782","affiliations":[{"raw_affiliation_string":"School of Science, Xi&#x2019;an Polytechnic University, Xi&#x2019;an, China","institution_ids":["https://openalex.org/I27599042"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5015925094","display_name":"Jacek M. \u017burada","orcid":"https://orcid.org/0000-0001-6622-534X"},"institutions":[{"id":"https://openalex.org/I142740786","display_name":"University of Louisville","ror":"https://ror.org/01ckdn478","country_code":"US","type":"education","lineage":["https://openalex.org/I142740786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jacek M. Zurada","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of Louisville, Louisville, KY, USA"],"raw_orcid":"https://orcid.org/0000-0001-6622-534X","affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Louisville, Louisville, KY, USA","institution_ids":["https://openalex.org/I142740786"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5122333443","display_name":"Jigen Peng","orcid":null},"institutions":[{"id":"https://openalex.org/I37987034","display_name":"Guangzhou University","ror":"https://ror.org/05ar8rn06","country_code":"CN","type":"education","lineage":["https://openalex.org/I37987034"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jigen Peng","raw_affiliation_strings":["School of Mathematics and Information Science, Guangzhou University, Guangzhou, China"],"raw_orcid":"https://orcid.org/0000-0002-7207-3183","affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Science, Guangzhou University, Guangzhou, China","institution_ids":["https://openalex.org/I37987034"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5133248046","display_name":"Haiyang Li","orcid":"https://orcid.org/0000-0001-7238-2456"},"institutions":[{"id":"https://openalex.org/I37987034","display_name":"Guangzhou University","ror":"https://ror.org/05ar8rn06","country_code":"CN","type":"education","lineage":["https://openalex.org/I37987034"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Haiyang Li","raw_affiliation_strings":["School of Mathematics and Information Science, Guangzhou University, Guangzhou, China"],"raw_orcid":"https://orcid.org/0000-0001-7238-2456","affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Science, Guangzhou University, Guangzhou, China","institution_ids":["https://openalex.org/I37987034"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5022385306"],"corresponding_institution_ids":["https://openalex.org/I37987034"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.74164199,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"21","issue":"2","first_page":"84","last_page":"98"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11273","display_name":"Advanced Graph Neural Networks","score":0.7174999713897705,"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/T11273","display_name":"Advanced Graph Neural Networks","score":0.7174999713897705,"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/T12676","display_name":"Machine Learning and ELM","score":0.10840000212192535,"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/T10057","display_name":"Face and Expression Recognition","score":0.01759999990463257,"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/entropy","display_name":"Entropy (arrow of time)","score":0.5697000026702881},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4625999927520752},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.4593999981880188},{"id":"https://openalex.org/keywords/binary-entropy-function","display_name":"Binary entropy function","score":0.44909998774528503},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4406999945640564},{"id":"https://openalex.org/keywords/pattern-recognition","display_name":"Pattern recognition (psychology)","score":0.3246000111103058}],"concepts":[{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5697000026702881},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5437999963760376},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5103999972343445},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4625999927520752},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.4593999981880188},{"id":"https://openalex.org/C44415725","wikidata":"https://www.wikidata.org/wiki/Q4913893","display_name":"Binary entropy function","level":3,"score":0.44909998774528503},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4406999945640564},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3772999942302704},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.37549999356269836},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.3246000111103058},{"id":"https://openalex.org/C202286095","wikidata":"https://www.wikidata.org/wiki/Q579262","display_name":"Error function","level":2,"score":0.3034999966621399},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.30239999294281006},{"id":"https://openalex.org/C3018824978","wikidata":"https://www.wikidata.org/wiki/Q2894891","display_name":"Error analysis","level":2,"score":0.2980000078678131},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.2838999927043915},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2838999927043915},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.28290000557899475},{"id":"https://openalex.org/C155032097","wikidata":"https://www.wikidata.org/wiki/Q798503","display_name":"Backpropagation","level":3,"score":0.2628999948501587}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/mci.2026.3655677","is_oa":false,"landing_page_url":"https://doi.org/10.1109/mci.2026.3655677","pdf_url":null,"source":{"id":"https://openalex.org/S104797584","display_name":"IEEE Computational Intelligence Magazine","issn_l":"1556-603X","issn":["1556-603X","1556-6048"],"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 Computational Intelligence Magazine","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1623747056","display_name":null,"funder_award_id":"12471482","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2770850836","display_name":null,"funder_award_id":"12271117","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4088427371","display_name":null,"funder_award_id":"12571492","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5606537135","display_name":null,"funder_award_id":"12031003","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":44,"referenced_works":["https://openalex.org/W1601499986","https://openalex.org/W2016782026","https://openalex.org/W2018549926","https://openalex.org/W2068777106","https://openalex.org/W2106858323","https://openalex.org/W2108119513","https://openalex.org/W2114160202","https://openalex.org/W2122825543","https://openalex.org/W2128451237","https://openalex.org/W2138747680","https://openalex.org/W2156354162","https://openalex.org/W2280737108","https://openalex.org/W2294416312","https://openalex.org/W2406273144","https://openalex.org/W2566853890","https://openalex.org/W2726526844","https://openalex.org/W2800373283","https://openalex.org/W2811153427","https://openalex.org/W2909862814","https://openalex.org/W2982049629","https://openalex.org/W3011054152","https://openalex.org/W3017194036","https://openalex.org/W3092436149","https://openalex.org/W3093726209","https://openalex.org/W3095521706","https://openalex.org/W3105617310","https://openalex.org/W3110884700","https://openalex.org/W3155502361","https://openalex.org/W3205731400","https://openalex.org/W4210341073","https://openalex.org/W4225936396","https://openalex.org/W4281728148","https://openalex.org/W4283029029","https://openalex.org/W4285818004","https://openalex.org/W4296916289","https://openalex.org/W4327967887","https://openalex.org/W4379472738","https://openalex.org/W4380520401","https://openalex.org/W4387717436","https://openalex.org/W4391004041","https://openalex.org/W4392796768","https://openalex.org/W4393972734","https://openalex.org/W4399986264","https://openalex.org/W4414856514"],"related_works":[],"abstract_inverted_index":{"As":[0],"a":[1],"kind":[2],"of":[3,17,29,76,127,132,139,148],"high":[4],"order":[5,60],"neural":[6,99],"network,":[7],"Sigma-Pi-Sigma":[8,98],"has":[9],"strong":[10],"nonlinear":[11],"mapping":[12],"ability,":[13],"but":[14],"the":[15,50,56,64,90,97,105,112,118,125,128,130,133,137,140,146,165],"complexity":[16],"its":[18],"network":[19,65,100],"structure":[20,66],"often":[21],"leads":[22],"to":[23,61,71],"low":[24],"application":[25],"efficiency":[26,69],"and":[27,41,52,70,104,136],"lack":[28],"theoretical":[30,74,166],"results.":[31,167],"Although":[32],"traditional":[33],"regularization":[34,92,114],"methods":[35],"are":[36,142,160],"more":[37,85],"effective":[38],"in":[39,59,161],"sparsity":[40],"preventing":[42],"overfitting,":[43],"they":[44],"do":[45],"not":[46],"take":[47],"into":[48,96],"account":[49],"topology":[51],"connection":[53],"information":[54],"between":[55],"data.":[57,156],"Therefore,":[58],"sparsely":[62],"optimize":[63],"for":[67],"improved":[68],"enable":[72],"rigorous":[73],"analysis":[75],"complex":[77],"inter-node":[78],"relationships,":[79],"graph-structured":[80],"data":[81],"must":[82],"be":[83],"processed":[84],"effectively.":[86],"In":[87],"this":[88,149],"paper,":[89],"graph":[91],"component":[93],"is":[94,115,151],"integrated":[95],"(SPSNN)":[101],"learning":[102],"framework,":[103],"SPSNN":[106],"online":[107],"gradient":[108],"algorithm":[109,141],"based":[110],"on":[111],"Turlaplacian":[113],"studied,":[116],"considering":[117],"entropy":[119],"error":[120,134],"measure.":[121],"Under":[122],"reasonable":[123],"conditions,":[124],"boundedness":[126],"weights,":[129],"monotonicity":[131],"function,":[135],"convergence":[138],"strictly":[143],"proved.":[144],"Subsequently,":[145],"effectiveness":[147],"method":[150],"verified":[152],"through":[153],"extensive":[154],"experimental":[155],"The":[157],"simulation":[158],"results":[159],"good":[162],"agreement":[163],"with":[164]},"counts_by_year":[],"updated_date":"2026-05-05T08:41:31.759640","created_date":"2026-04-10T00:00:00"}
