{"id":"https://openalex.org/W4388676597","doi":"https://doi.org/10.1109/tnnls.2023.3330328","title":"NN2Poly: A Polynomial Representation for Deep Feed-Forward Artificial Neural Networks","display_name":"NN2Poly: A Polynomial Representation for Deep Feed-Forward Artificial Neural Networks","publication_year":2023,"publication_date":"2023-11-14","ids":{"openalex":"https://openalex.org/W4388676597","doi":"https://doi.org/10.1109/tnnls.2023.3330328","pmid":"https://pubmed.ncbi.nlm.nih.gov/37962996"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2023.3330328","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2023.3330328","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"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 Neural Networks and Learning Systems","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/A5080001325","display_name":"Pablo Morala","orcid":"https://orcid.org/0000-0002-4109-2330"},"institutions":[{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Pablo Morala","raw_affiliation_strings":["Department of Statistics and the UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe, Spain"],"affiliations":[{"raw_affiliation_string":"Department of Statistics and the UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe, Spain","institution_ids":["https://openalex.org/I50357001"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5077887275","display_name":"Jenny Cifuentes","orcid":"https://orcid.org/0000-0001-7421-291X"},"institutions":[{"id":"https://openalex.org/I96580804","display_name":"Universidad Pontificia Comillas","ror":"https://ror.org/017mdc710","country_code":"ES","type":"education","lineage":["https://openalex.org/I96580804"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Jenny Alexandra Cifuentes","raw_affiliation_strings":["Department of Quantitative Methods, ICADE, Faculty of Economics and Business Administration, and the Institute for Research in Technology (IIT), ICAI School of Engineering, Universidad Pontificia Comillas, Madrid, Spain","ICADE, Department of Quantitative Methods, Faculty of Economics and Business Administration, and the Institute for Research in Technology (IIT), ICAI School of Engineering, Universidad Pontificia Comillas, Madrid, Spain"],"affiliations":[{"raw_affiliation_string":"Department of Quantitative Methods, ICADE, Faculty of Economics and Business Administration, and the Institute for Research in Technology (IIT), ICAI School of Engineering, Universidad Pontificia Comillas, Madrid, Spain","institution_ids":["https://openalex.org/I96580804"]},{"raw_affiliation_string":"ICADE, Department of Quantitative Methods, Faculty of Economics and Business Administration, and the Institute for Research in Technology (IIT), ICAI School of Engineering, Universidad Pontificia Comillas, Madrid, Spain","institution_ids":["https://openalex.org/I96580804"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5090172801","display_name":"Rosa E. Lillo","orcid":"https://orcid.org/0000-0003-0802-4691"},"institutions":[{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Rosa E. Lillo","raw_affiliation_strings":["Department of Statistics and the UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe, Spain"],"affiliations":[{"raw_affiliation_string":"Department of Statistics and the UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe, Spain","institution_ids":["https://openalex.org/I50357001"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072498572","display_name":"I\u00f1aki \u00dacar","orcid":"https://orcid.org/0000-0001-6403-5550"},"institutions":[{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"I\u00f1aki Ucar","raw_affiliation_strings":["Department of Statistics and the UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe, Spain"],"affiliations":[{"raw_affiliation_string":"Department of Statistics and the UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe, Spain","institution_ids":["https://openalex.org/I50357001"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5080001325"],"corresponding_institution_ids":["https://openalex.org/I50357001"],"apc_list":null,"apc_paid":null,"fwci":1.9033,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.88848014,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"36","issue":"1","first_page":"781","last_page":"795"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.9930999875068665,"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/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.9930999875068665,"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/T12535","display_name":"Machine Learning and Data Classification","score":0.9930999875068665,"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/T10320","display_name":"Neural Networks and Applications","score":0.9926999807357788,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/interpretability","display_name":"Interpretability","score":0.8168256878852844},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6831158995628357},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.661122739315033},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6351010799407959},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.616745114326477},{"id":"https://openalex.org/keywords/perceptron","display_name":"Perceptron","score":0.5916029810905457},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.5650013089179993},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.552386462688446},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.5486718416213989},{"id":"https://openalex.org/keywords/multilayer-perceptron","display_name":"Multilayer perceptron","score":0.5395058989524841},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.49959397315979004},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.49784231185913086},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.36084333062171936},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3367695212364197},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.21407079696655273}],"concepts":[{"id":"https://openalex.org/C2781067378","wikidata":"https://www.wikidata.org/wiki/Q17027399","display_name":"Interpretability","level":2,"score":0.8168256878852844},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6831158995628357},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.661122739315033},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6351010799407959},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.616745114326477},{"id":"https://openalex.org/C60908668","wikidata":"https://www.wikidata.org/wiki/Q690207","display_name":"Perceptron","level":3,"score":0.5916029810905457},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.5650013089179993},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.552386462688446},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.5486718416213989},{"id":"https://openalex.org/C179717631","wikidata":"https://www.wikidata.org/wiki/Q2991667","display_name":"Multilayer perceptron","level":3,"score":0.5395058989524841},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.49959397315979004},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.49784231185913086},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.36084333062171936},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3367695212364197},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.21407079696655273},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnnls.2023.3330328","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2023.3330328","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"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 Neural Networks and Learning Systems","raw_type":"journal-article"},{"id":"pmid:37962996","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/37962996","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 neural networks and learning systems","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":45,"referenced_works":["https://openalex.org/W1522301498","https://openalex.org/W1806891645","https://openalex.org/W1978626680","https://openalex.org/W1988115241","https://openalex.org/W2015627422","https://openalex.org/W2029469881","https://openalex.org/W2041143591","https://openalex.org/W2103496339","https://openalex.org/W2116487523","https://openalex.org/W2137983211","https://openalex.org/W2144212877","https://openalex.org/W2171963778","https://openalex.org/W2295739661","https://openalex.org/W2752061190","https://openalex.org/W2766678531","https://openalex.org/W2808246927","https://openalex.org/W2809090039","https://openalex.org/W2919115771","https://openalex.org/W2967826146","https://openalex.org/W2994821360","https://openalex.org/W2996320484","https://openalex.org/W3013992185","https://openalex.org/W3035865126","https://openalex.org/W3127046748","https://openalex.org/W3131431199","https://openalex.org/W3172710683","https://openalex.org/W3202428668","https://openalex.org/W4242989628","https://openalex.org/W4287811112","https://openalex.org/W4297749952","https://openalex.org/W4310574272","https://openalex.org/W4320458252","https://openalex.org/W4388953010","https://openalex.org/W6631190155","https://openalex.org/W6733862737","https://openalex.org/W6745256532","https://openalex.org/W6751980908","https://openalex.org/W6752495264","https://openalex.org/W6767337212","https://openalex.org/W6771465236","https://openalex.org/W6775446939","https://openalex.org/W6776595772","https://openalex.org/W6797480598","https://openalex.org/W6811190969","https://openalex.org/W6849281104"],"related_works":["https://openalex.org/W2977982891","https://openalex.org/W4298111019","https://openalex.org/W4315435472","https://openalex.org/W2103292044","https://openalex.org/W2296440556","https://openalex.org/W2076543106","https://openalex.org/W2170343060","https://openalex.org/W2053362666","https://openalex.org/W2088798842","https://openalex.org/W1482859693"],"abstract_inverted_index":{"Interpretability":[0],"of":[1,14,21,29,52,117,129,163],"neural":[2],"networks":[3],"(NNs)":[4],"and":[5,92,107,132],"their":[6,22],"underlying":[7],"theoretical":[8,39],"behavior":[9],"remain":[10],"an":[11,43,49,53],"open":[12],"field":[13],"study":[15],"even":[16],"after":[17],"the":[18,27,73,101,115,118,125,133,143,161,164],"great":[19],"success":[20],"practical":[23],"applications,":[24],"particularly":[25],"with":[26,85],"emergence":[28],"deep":[30,87],"learning.":[31],"In":[32],"this":[33,130],"work,":[34],"NN2Poly":[35,95],"is":[36,122],"proposed:":[37],"a":[38,68,97],"approach":[40,66],"to":[41,78,83,113,135,153,159],"obtain":[42],"explicit":[44],"polynomial":[45],"model":[46],"that":[47],"provides":[48],"accurate":[50],"representation":[51],"already":[54],"trained":[55],"fully":[56],"connected":[57],"feed-forward":[58],"artificial":[59],"NN":[60],"[a":[61],"multilayer":[62],"perceptron":[63],"(MLP)].":[64],"This":[65],"extends":[67],"previous":[69],"idea":[70],"proposed":[71,165],"in":[72,89],"literature,":[74],"which":[75],"was":[76],"limited":[77],"single":[79],"hidden":[80],"layer":[81],"networks,":[82],"work":[84],"arbitrarily":[86],"MLPs":[88],"both":[90],"regression":[91],"classification":[93],"tasks.":[94],"uses":[96],"Taylor":[98],"expansion":[99],"on":[100,124],"activation":[102],"function,":[103],"at":[104],"each":[105],"layer,":[106],"then":[108],"applies":[109],"several":[110],"combinatorial":[111],"properties":[112],"calculate":[114],"coefficients":[116],"desired":[119],"polynomials.":[120],"Discussion":[121],"presented":[123,158],"main":[126],"computational":[127],"challenges":[128],"method,":[131],"way":[134],"overcome":[136],"them":[137],"by":[138],"imposing":[139],"certain":[140],"constraints":[141],"during":[142],"training":[144],"phase.":[145],"Finally,":[146],"simulation":[147],"experiments":[148],"as":[149,151],"well":[150],"applications":[152],"real":[154],"tabular":[155],"datasets":[156],"are":[157],"demonstrate":[160],"effectiveness":[162],"method.":[166]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":1}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}