{"id":"https://openalex.org/W4391547703","doi":"https://doi.org/10.1109/tnnls.2024.3350786","title":"Achieving Sharp Upper Bounds on the Expressive Power of Neural Networks via Tropical Polynomials","display_name":"Achieving Sharp Upper Bounds on the Expressive Power of Neural Networks via Tropical Polynomials","publication_year":2024,"publication_date":"2024-02-05","ids":{"openalex":"https://openalex.org/W4391547703","doi":"https://doi.org/10.1109/tnnls.2024.3350786","pmid":"https://pubmed.ncbi.nlm.nih.gov/38315593"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2024.3350786","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2024.3350786","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/A5100409103","display_name":"Zhiwei Li","orcid":"https://orcid.org/0000-0003-3642-5994"},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]},{"id":"https://openalex.org/I1327237609","display_name":"Ministry of Education of the People's Republic of China","ror":"https://ror.org/01mv9t934","country_code":"CN","type":"government","lineage":["https://openalex.org/I1327237609","https://openalex.org/I4210127390"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhiwei Li","raw_affiliation_strings":["Department of Computer Science and Technology, Tongji University, Shanghai, China","Key Laboratory of Embedded System and Service Computing, Ministry of Education, Shanghai, China","he Key Laboratory of Industrial Internet of Things and Networked Control, Ministry of Education, Shanghai, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Technology, Tongji University, Shanghai, China","institution_ids":["https://openalex.org/I116953780"]},{"raw_affiliation_string":"Key Laboratory of Embedded System and Service Computing, Ministry of Education, Shanghai, China","institution_ids":["https://openalex.org/I1327237609"]},{"raw_affiliation_string":"he Key Laboratory of Industrial Internet of Things and Networked Control, Ministry of Education, Shanghai, China","institution_ids":["https://openalex.org/I1327237609"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100417007","display_name":"Cheng Wang","orcid":"https://orcid.org/0000-0002-4752-0316"},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]},{"id":"https://openalex.org/I1327237609","display_name":"Ministry of Education of the People's Republic of China","ror":"https://ror.org/01mv9t934","country_code":"CN","type":"government","lineage":["https://openalex.org/I1327237609","https://openalex.org/I4210127390"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Cheng Wang","raw_affiliation_strings":["Department of Computer Science and Technology, Tongji University, Shanghai, China","Key Laboratory of Embedded System and Service Computing, Ministry of Education, Shanghai, China","he Key Laboratory of Industrial Internet of Things and Networked Control, Ministry of Education, Shanghai, China"],"raw_orcid":"https://orcid.org/0000-0002-4752-0316","affiliations":[{"raw_affiliation_string":"Department of Computer Science and Technology, Tongji University, Shanghai, China","institution_ids":["https://openalex.org/I116953780"]},{"raw_affiliation_string":"Key Laboratory of Embedded System and Service Computing, Ministry of Education, Shanghai, China","institution_ids":["https://openalex.org/I1327237609"]},{"raw_affiliation_string":"he Key Laboratory of Industrial Internet of Things and Networked Control, Ministry of Education, Shanghai, China","institution_ids":["https://openalex.org/I1327237609"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.9577,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.7757176,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"36","issue":"2","first_page":"2931","last_page":"2945"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9990000128746033,"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/T10320","display_name":"Neural Networks and Applications","score":0.9990000128746033,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9980000257492065,"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/T12303","display_name":"Tensor decomposition and applications","score":0.9926000237464905,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"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/expressive-power","display_name":"Expressive power","score":0.5845535397529602},{"id":"https://openalex.org/keywords/power","display_name":"Power (physics)","score":0.5433801412582397},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4698920249938965},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.44181716442108154},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.34507274627685547},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3445638418197632},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32680290937423706},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.28465914726257324},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2340959906578064},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1268388032913208}],"concepts":[{"id":"https://openalex.org/C195818886","wikidata":"https://www.wikidata.org/wiki/Q5421724","display_name":"Expressive power","level":2,"score":0.5845535397529602},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.5433801412582397},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4698920249938965},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.44181716442108154},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.34507274627685547},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3445638418197632},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32680290937423706},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.28465914726257324},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2340959906578064},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1268388032913208},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnnls.2024.3350786","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2024.3350786","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:38315593","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/38315593","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":[{"id":"https://openalex.org/G1012616008","display_name":null,"funder_award_id":"62372328","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G1395231520","display_name":null,"funder_award_id":"22XD1423700","funder_id":"https://openalex.org/F4320335796","funder_display_name":"Program of Shanghai Academic Research Leader"},{"id":"https://openalex.org/G5342982222","display_name":null,"funder_award_id":"2022YF-B4501704","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null},{"id":"https://openalex.org/F4320335796","display_name":"Program of Shanghai Academic Research Leader","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":64,"referenced_works":["https://openalex.org/W1677182931","https://openalex.org/W1869625623","https://openalex.org/W2039415399","https://openalex.org/W2089947415","https://openalex.org/W2095705004","https://openalex.org/W2161388792","https://openalex.org/W2169502938","https://openalex.org/W2804128772","https://openalex.org/W2919163648","https://openalex.org/W2955775727","https://openalex.org/W2963657672","https://openalex.org/W2995788622","https://openalex.org/W2998329861","https://openalex.org/W3002181557","https://openalex.org/W3005600385","https://openalex.org/W3035664258","https://openalex.org/W3080273719","https://openalex.org/W3087020912","https://openalex.org/W3089555680","https://openalex.org/W3092190487","https://openalex.org/W3099937123","https://openalex.org/W3115909741","https://openalex.org/W3120358876","https://openalex.org/W3148563252","https://openalex.org/W3152922554","https://openalex.org/W3176554980","https://openalex.org/W3194828218","https://openalex.org/W3195541832","https://openalex.org/W3197364345","https://openalex.org/W3208910075","https://openalex.org/W4200390863","https://openalex.org/W4206129120","https://openalex.org/W4214493665","https://openalex.org/W4226256913","https://openalex.org/W4281719784","https://openalex.org/W4283798762","https://openalex.org/W4287815523","https://openalex.org/W4297813707","https://openalex.org/W4310078553","https://openalex.org/W4313156423","https://openalex.org/W6631943919","https://openalex.org/W6639079861","https://openalex.org/W6645550903","https://openalex.org/W6674330103","https://openalex.org/W6683823733","https://openalex.org/W6691187937","https://openalex.org/W6696761078","https://openalex.org/W6718212895","https://openalex.org/W6729999211","https://openalex.org/W6743923790","https://openalex.org/W6745505307","https://openalex.org/W6751356287","https://openalex.org/W6751475489","https://openalex.org/W6754929296","https://openalex.org/W6758955634","https://openalex.org/W6760544264","https://openalex.org/W6763744544","https://openalex.org/W6771988894","https://openalex.org/W6779024865","https://openalex.org/W6780147038","https://openalex.org/W6790040128","https://openalex.org/W6796080340","https://openalex.org/W6811590017","https://openalex.org/W6845906210"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2980963558","https://openalex.org/W1882246857","https://openalex.org/W3103304230","https://openalex.org/W2185360082","https://openalex.org/W1992365905","https://openalex.org/W1976691234","https://openalex.org/W3212023493","https://openalex.org/W2090715877","https://openalex.org/W2572370460"],"abstract_inverted_index":{"The":[0,15],"expressive":[1,28,45],"power":[2,46],"of":[3,17,27,38,44,123,168,184,187,207,242],"neural":[4,64,82],"networks":[5,83],"describes":[6],"the":[7,21,36,48,52,63,93,101,108,116,121,133,145,162,165,180,185,205,238],"ability":[8],"to":[9,73,85],"represent":[10],"or":[11],"approximate":[12,181],"complex":[13],"functions.":[14],"number":[16,37,122,186,206],"linear":[18,39,81,188,208],"regions":[19,40,189,209],"is":[20,47,171,196],"standard":[22],"and":[23,55,98,131,156,221,231,240,251],"most":[24],"natural":[25],"measure":[26,43],"power.":[29],"However,":[30],"a":[31,42,75,128,175,223],"major":[32],"challenge":[33],"in":[34,115,153,218],"utilizing":[35],"as":[41,62],"exponential":[49,182],"gap":[50],"between":[51,95],"theoretical":[53],"upper":[54,77,225,245],"lower":[56],"bounds,":[57],"which":[58,195],"becomes":[59],"more":[60],"pronounced":[61],"network":[65],"capacity":[66],"increases.":[67],"In":[68,100,144],"this":[69,87,137,243],"article,":[70],"we":[71,90,106,160,203],"aim":[72],"derive":[74,222],"sharp":[76,224,244],"bound":[78,226,246],"on":[79,247],"piecewise":[80],"(PLNNs)":[84],"bridge":[86],"gap.":[88],"Specifically,":[89],"first":[91],"establish":[92],"relationship":[94],"tropical":[96,103,147],"polynomials":[97,104,148],"PLNNs.":[99,228],"unexpanded":[102],"form,":[105,149],"make":[107],"proposition":[109],"that":[110,136,164,178,210],"hyperplanes":[111],"are":[112],"not":[113],"all":[114],"general":[117],"positions,":[118],"thereby":[119],"reducing":[120],"intersecting":[124],"hyperplanes.":[125],"We":[126,173],"propose":[127,174],"rank-based":[129],"approach":[130,138,177],"present":[132],"empirical":[134],"analysis":[135,230],"outperforms":[139],"previous":[140],"Zaslavsky's":[141],"theorem-based":[142],"methods.":[143],"expanded":[146],"accounting":[150],"for":[151,227,237],"limitations":[152],"weight":[154],"initialization":[155],"model":[157],"computational":[158],"precision,":[159],"raise":[161],"concept":[163],"values":[166],"range":[167],"each":[169,215],"term":[170],"bounded.":[172],"precision-based":[176],"transforms":[179],"growth":[183,192],"into":[190],"polynomial":[191],"with":[193],"width,":[194],"effective":[197],"at":[198],"larger":[199],"layer":[200,217],"widths.":[201],"Finally,":[202],"compare":[204],"can":[211],"be":[212],"represented":[213],"by":[214],"hidden":[216],"both":[219,248],"forms":[220],"Empirical":[229],"experimental":[232],"results":[233],"provide":[234],"compelling":[235],"evidence":[236],"efficacy":[239],"feasibility":[241],"simulated":[249],"experiments":[250],"real":[252],"datasets.":[253]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2025-10-10T00:00:00"}