{"id":"https://openalex.org/W4291464279","doi":"https://doi.org/10.3390/e24081117","title":"Learnability of the Boolean Innerproduct in Deep Neural Networks","display_name":"Learnability of the Boolean Innerproduct in Deep Neural Networks","publication_year":2022,"publication_date":"2022-08-13","ids":{"openalex":"https://openalex.org/W4291464279","doi":"https://doi.org/10.3390/e24081117","pmid":"https://pubmed.ncbi.nlm.nih.gov/36010780"},"language":"en","primary_location":{"id":"doi:10.3390/e24081117","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24081117","pdf_url":"https://www.mdpi.com/1099-4300/24/8/1117/pdf?version=1661324332","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/24/8/1117/pdf?version=1661324332","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5055000989","display_name":"Mehmet Erdal","orcid":"https://orcid.org/0000-0002-8001-3703"},"institutions":[{"id":"https://openalex.org/I196349391","display_name":"Universit\u00e4t Ulm","ror":"https://ror.org/032000t02","country_code":"DE","type":"education","lineage":["https://openalex.org/I196349391"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Mehmet Erdal","raw_affiliation_strings":["Institute of Neural Information Processing, Ulm University, 89081 Ulm, Germany"],"affiliations":[{"raw_affiliation_string":"Institute of Neural Information Processing, Ulm University, 89081 Ulm, Germany","institution_ids":["https://openalex.org/I196349391"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078096723","display_name":"Friedhelm Schwenker","orcid":"https://orcid.org/0000-0001-5118-0812"},"institutions":[{"id":"https://openalex.org/I196349391","display_name":"Universit\u00e4t Ulm","ror":"https://ror.org/032000t02","country_code":"DE","type":"education","lineage":["https://openalex.org/I196349391"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Friedhelm Schwenker","raw_affiliation_strings":["Institute of Neural Information Processing, Ulm University, 89081 Ulm, Germany"],"affiliations":[{"raw_affiliation_string":"Institute of Neural Information Processing, Ulm University, 89081 Ulm, Germany","institution_ids":["https://openalex.org/I196349391"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5055000989","https://openalex.org/A5078096723"],"corresponding_institution_ids":["https://openalex.org/I196349391"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.138,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.54033548,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"24","issue":"8","first_page":"1117","last_page":"1117"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9993000030517578,"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":0.9993000030517578,"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":0.9993000030517578,"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":0.998199999332428,"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/learnability","display_name":"Learnability","score":0.9355679154396057},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.7002362012863159},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6002675294876099},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5945577025413513},{"id":"https://openalex.org/keywords/layer","display_name":"Layer (electronics)","score":0.5434958338737488},{"id":"https://openalex.org/keywords/boolean-network","display_name":"Boolean network","score":0.5379676222801208},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5139421820640564},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4878065884113312},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.46516019105911255},{"id":"https://openalex.org/keywords/convolutional-neural-network","display_name":"Convolutional neural network","score":0.4554332196712494},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.4401566982269287},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.4343959391117096},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.42829880118370056},{"id":"https://openalex.org/keywords/boolean-circuit","display_name":"Boolean circuit","score":0.4225316345691681},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.36482951045036316},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3447645902633667},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3428192436695099},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3187832832336426},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.17205917835235596}],"concepts":[{"id":"https://openalex.org/C2777723229","wikidata":"https://www.wikidata.org/wiki/Q4367921","display_name":"Learnability","level":2,"score":0.9355679154396057},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.7002362012863159},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6002675294876099},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5945577025413513},{"id":"https://openalex.org/C2779227376","wikidata":"https://www.wikidata.org/wiki/Q6505497","display_name":"Layer (electronics)","level":2,"score":0.5434958338737488},{"id":"https://openalex.org/C134444547","wikidata":"https://www.wikidata.org/wiki/Q585230","display_name":"Boolean network","level":3,"score":0.5379676222801208},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5139421820640564},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4878065884113312},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.46516019105911255},{"id":"https://openalex.org/C81363708","wikidata":"https://www.wikidata.org/wiki/Q17084460","display_name":"Convolutional neural network","level":2,"score":0.4554332196712494},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.4401566982269287},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.4343959391117096},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.42829880118370056},{"id":"https://openalex.org/C141796577","wikidata":"https://www.wikidata.org/wiki/Q837479","display_name":"Boolean circuit","level":3,"score":0.4225316345691681},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.36482951045036316},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3447645902633667},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3428192436695099},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3187832832336426},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.17205917835235596},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","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}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e24081117","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24081117","pdf_url":"https://www.mdpi.com/1099-4300/24/8/1117/pdf?version=1661324332","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:36010780","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/36010780","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":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:ab1f3449186749b4a6d0e4b936f61f3f","is_oa":true,"landing_page_url":"https://doaj.org/article/ab1f3449186749b4a6d0e4b936f61f3f","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 24, Iss 8, p 1117 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/24/8/1117/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e24081117","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy; Volume 24; Issue 8; Pages: 1117","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:9407482","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/9407482","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e24081117","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24081117","pdf_url":"https://www.mdpi.com/1099-4300/24/8/1117/pdf?version=1661324332","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4291464279.pdf","grobid_xml":"https://content.openalex.org/works/W4291464279.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W104184427","https://openalex.org/W1526741802","https://openalex.org/W1606643095","https://openalex.org/W1829366209","https://openalex.org/W2014850486","https://openalex.org/W2064630666","https://openalex.org/W2097512637","https://openalex.org/W2103496339","https://openalex.org/W2117549320","https://openalex.org/W2122613489","https://openalex.org/W2126229007","https://openalex.org/W2134286587","https://openalex.org/W2140833774","https://openalex.org/W2155258457","https://openalex.org/W2156791867","https://openalex.org/W2161336914","https://openalex.org/W2554646042","https://openalex.org/W2566079294","https://openalex.org/W2604268533","https://openalex.org/W2919115771","https://openalex.org/W2954996726","https://openalex.org/W2963060719","https://openalex.org/W2984306354","https://openalex.org/W2995277268","https://openalex.org/W3022436500","https://openalex.org/W3117450949","https://openalex.org/W3176923149","https://openalex.org/W3198414418","https://openalex.org/W4230719014","https://openalex.org/W4231109964","https://openalex.org/W4246435782","https://openalex.org/W6636077916","https://openalex.org/W6680887930","https://openalex.org/W6691187937","https://openalex.org/W6743923790","https://openalex.org/W6759638512","https://openalex.org/W6773851591"],"related_works":["https://openalex.org/W2728195892","https://openalex.org/W1991822629","https://openalex.org/W2011175857","https://openalex.org/W2246807618","https://openalex.org/W2061512596","https://openalex.org/W2106622045","https://openalex.org/W3141532631","https://openalex.org/W2064457754","https://openalex.org/W2558644562","https://openalex.org/W2131164334"],"abstract_inverted_index":{"In":[0,75],"this":[1,81],"paper,":[2],"we":[3,78],"study":[4],"the":[5,8,20,44,86,97,104,111,114,119,122,138,144,167,180,183,189,203],"learnability":[6,181],"of":[7,19,33,36,58,85,113,121,156,182,188],"Boolean":[9,21,87],"inner":[10,22,88,190],"product":[11,23,89,191],"by":[12,30,50,149,195],"a":[13,51,67,150,198],"systematic":[14],"simulation":[15],"study.":[16],"The":[17],"family":[18],"function":[24],"is":[25,90],"known":[26],"to":[27,92,125,136,171,178],"be":[28,57,64,193],"representable":[29],"neural":[31,72],"networks":[32],"threshold":[34],"neurons":[35],"depth":[37,52,82,98,140,185],"3":[38,83,141,186],"with":[39,118,161],"only":[40],"2n+1":[41],"units":[42,201],"(<i>n</i>":[43],"input":[45,106,123],"dimension)-whereas":[46],"an":[47],"exact":[48],"representation":[49,187],"2":[53,99],"network":[54,73],"cannot":[55],"possibly":[56],"polynomial":[59],"size.":[60],"This":[61],"result":[62],"can":[63,164],"seen":[65],"as":[66],"strong":[68],"argument":[69],"for":[70,103],"deep":[71,115],"architectures.":[74],"our":[76],"study,":[77],"found":[79],"that":[80,131],"architecture":[84,116],"difficult":[91],"train,":[93],"much":[94],"harder":[95],"than":[96],"network,":[100],"at":[101],"least":[102],"small":[105],"size":[107],"scenarios":[108],"n\u226416.":[109],"Nonetheless,":[110],"accuracy":[112,173],"increased":[117],"dimension":[120],"space":[124],"94%":[126],"on":[127],"average,":[128],"which":[129],"means":[130],"multiple":[132],"restarts":[133],"are":[134],"needed":[135],"find":[137],"compact":[139,184],"architecture.":[142],"Replacing":[143],"fully":[145],"connected":[146,152,160],"first":[147,204],"layer":[148,153,158],"partially":[151],"(a":[154],"kind":[155],"convolutional":[157],"sparsely":[159],"weight":[162],"sharing)":[163],"significantly":[165],"improve":[166,179],"learning":[168],"performance":[169],"up":[170],"99%":[172],"in":[174],"simulations.":[175],"Another":[176],"way":[177],"could":[192],"achieved":[194],"adding":[196],"just":[197],"few":[199],"additional":[200],"into":[202],"hidden":[205],"layer.":[206]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}