{"id":"https://openalex.org/W3213601708","doi":"https://doi.org/10.3390/e23111494","title":"Hidden Hypergraphs, Error-Correcting Codes, and Critical Learning in Hopfield Networks","display_name":"Hidden Hypergraphs, Error-Correcting Codes, and Critical Learning in Hopfield Networks","publication_year":2021,"publication_date":"2021-11-11","ids":{"openalex":"https://openalex.org/W3213601708","doi":"https://doi.org/10.3390/e23111494","mag":"3213601708","pmid":"https://pubmed.ncbi.nlm.nih.gov/34828192"},"language":"en","primary_location":{"id":"doi:10.3390/e23111494","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23111494","pdf_url":"https://www.mdpi.com/1099-4300/23/11/1494/pdf?version=1637218499","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/1099-4300/23/11/1494/pdf?version=1637218499","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039959148","display_name":"Christopher J. Hillar","orcid":"https://orcid.org/0000-0001-8575-2663"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Christopher Hillar","raw_affiliation_strings":["Awecom, Inc., San Francisco, CA 94103, USA"],"affiliations":[{"raw_affiliation_string":"Awecom, Inc., San Francisco, CA 94103, USA","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5047137852","display_name":"Tenzin Chan","orcid":"https://orcid.org/0009-0002-8474-616X"},"institutions":[{"id":"https://openalex.org/I152815399","display_name":"Singapore University of Technology and Design","ror":"https://ror.org/05j6fvn87","country_code":"SG","type":"education","lineage":["https://openalex.org/I152815399"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Tenzin Chan","raw_affiliation_strings":["Singapore University of Technology and Design, Singapore 487372, Singapore"],"affiliations":[{"raw_affiliation_string":"Singapore University of Technology and Design, Singapore 487372, Singapore","institution_ids":["https://openalex.org/I152815399"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039921419","display_name":"Rachel Taubman","orcid":"https://orcid.org/0000-0002-7779-5185"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rachel Taubman","raw_affiliation_strings":["Awecom, Inc., San Francisco, CA 94103, USA"],"affiliations":[{"raw_affiliation_string":"Awecom, Inc., San Francisco, CA 94103, USA","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056513353","display_name":"David Rolnick","orcid":"https://orcid.org/0000-0002-2855-393X"},"institutions":[{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"David Rolnick","raw_affiliation_strings":["School of Computer Science, McGill University, Montreal, QC H3A 0G4, Canada"],"affiliations":[{"raw_affiliation_string":"School of Computer Science, McGill University, Montreal, QC H3A 0G4, Canada","institution_ids":["https://openalex.org/I5023651"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5039959148"],"corresponding_institution_ids":[],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.3553,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.56298455,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"23","issue":"11","first_page":"1494","last_page":"1494"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10581","display_name":"Neural dynamics and brain function","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2805","display_name":"Cognitive Neuroscience"},"field":{"id":"https://openalex.org/fields/28","display_name":"Neuroscience"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T10581","display_name":"Neural dynamics and brain function","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2805","display_name":"Cognitive Neuroscience"},"field":{"id":"https://openalex.org/fields/28","display_name":"Neuroscience"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10621","display_name":"Gene Regulatory Network Analysis","score":0.9975000023841858,"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"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.987500011920929,"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/computer-science","display_name":"Computer science","score":0.7103830575942993},{"id":"https://openalex.org/keywords/hopfield-network","display_name":"Hopfield network","score":0.6737576723098755},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.5944156050682068},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.47132137417793274},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.4434017837047577},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4414829909801483},{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.4353983402252197},{"id":"https://openalex.org/keywords/cluster-analysis","display_name":"Cluster analysis","score":0.43034863471984863},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3306649327278137},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.22603380680084229}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.7103830575942993},{"id":"https://openalex.org/C46421273","wikidata":"https://www.wikidata.org/wiki/Q1407668","display_name":"Hopfield network","level":3,"score":0.6737576723098755},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.5944156050682068},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.47132137417793274},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.4434017837047577},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4414829909801483},{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.4353983402252197},{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.43034863471984863},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3306649327278137},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.22603380680084229},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","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}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e23111494","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23111494","pdf_url":"https://www.mdpi.com/1099-4300/23/11/1494/pdf?version=1637218499","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:34828192","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/34828192","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:530fe3130c4548da95e254bcc23a234c","is_oa":true,"landing_page_url":"https://doaj.org/article/530fe3130c4548da95e254bcc23a234c","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 23, Iss 11, p 1494 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/23/11/1494/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e23111494","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 23; Issue 11; Pages: 1494","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:8622935","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/8622935","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/e23111494","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23111494","pdf_url":"https://www.mdpi.com/1099-4300/23/11/1494/pdf?version=1637218499","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":[{"display_name":"Affordable and clean energy","score":0.8600000143051147,"id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3213601708.pdf","grobid_xml":"https://content.openalex.org/works/W3213601708.grobid-xml"},"referenced_works_count":61,"referenced_works":["https://openalex.org/W160989634","https://openalex.org/W1514853588","https://openalex.org/W1535810436","https://openalex.org/W1547224907","https://openalex.org/W1560816886","https://openalex.org/W1597286183","https://openalex.org/W1970391837","https://openalex.org/W1995341919","https://openalex.org/W1996589480","https://openalex.org/W1999678910","https://openalex.org/W2019545912","https://openalex.org/W2020999234","https://openalex.org/W2039122980","https://openalex.org/W2040870580","https://openalex.org/W2046606639","https://openalex.org/W2064344303","https://openalex.org/W2068674173","https://openalex.org/W2086760168","https://openalex.org/W2117489143","https://openalex.org/W2128084896","https://openalex.org/W2129160848","https://openalex.org/W2134797427","https://openalex.org/W2139326808","https://openalex.org/W2141361751","https://openalex.org/W2147238273","https://openalex.org/W2149413876","https://openalex.org/W2152471738","https://openalex.org/W2159793005","https://openalex.org/W2163605009","https://openalex.org/W2168614937","https://openalex.org/W2213869743","https://openalex.org/W2248686023","https://openalex.org/W2569174811","https://openalex.org/W2612591677","https://openalex.org/W2617302127","https://openalex.org/W2796932341","https://openalex.org/W2949512190","https://openalex.org/W2950846759","https://openalex.org/W2963805875","https://openalex.org/W2964346435","https://openalex.org/W2970909495","https://openalex.org/W2978725006","https://openalex.org/W2979357328","https://openalex.org/W2997168740","https://openalex.org/W3102572323","https://openalex.org/W3105441780","https://openalex.org/W3170354226","https://openalex.org/W3172269833","https://openalex.org/W3197886166","https://openalex.org/W4233413206","https://openalex.org/W4238893454","https://openalex.org/W4241400578","https://openalex.org/W4250492671","https://openalex.org/W4250589301","https://openalex.org/W6632924624","https://openalex.org/W6650283448","https://openalex.org/W6678269961","https://openalex.org/W6679909955","https://openalex.org/W6680545848","https://openalex.org/W6767629652","https://openalex.org/W6782959951"],"related_works":["https://openalex.org/W2960434112","https://openalex.org/W3049714198","https://openalex.org/W2036805113","https://openalex.org/W2592279130","https://openalex.org/W2049158118","https://openalex.org/W2123827981","https://openalex.org/W4205994410","https://openalex.org/W1499880030","https://openalex.org/W34461864","https://openalex.org/W2525520510"],"abstract_inverted_index":{"In":[0,133],"1943,":[1],"McCulloch":[2],"and":[3,28,47,77,95,120],"Pitts":[4],"introduced":[5],"a":[6,12,41,59,141],"discrete":[7],"recurrent":[8],"neural":[9],"network":[10,65],"as":[11,23,58,74],"model":[13],"for":[14,63,130],"computation":[15],"in":[16,38,83,108],"brains.":[17],"The":[18],"work":[19],"inspired":[20],"breakthroughs":[21],"such":[22,73],"the":[24,29,84,134],"first":[25],"computer":[26],"design":[27],"theory":[30,85],"of":[31,71,86,127,136,144],"finite":[32],"automata.":[33],"We":[34,67,111],"focus":[35],"on":[36],"learning":[37],"Hopfield":[39,89],"networks,":[40],"special":[42],"case":[43,135],"with":[44,124],"symmetric":[45],"weights":[46],"fixed-point":[48],"attractor":[49],"dynamics.":[50],"Specifically,":[51],"we":[52,138],"explore":[53],"minimum":[54],"energy":[55],"flow":[56],"(MEF)":[57],"scalable":[60],"convex":[61],"objective":[62],"determining":[64],"parameters.":[66],"catalog":[68],"various":[69],"properties":[70],"MEF,":[72],"biological":[75],"plausibility,":[76],"then":[78],"compare":[79],"to":[80,118],"classical":[81],"approaches":[82],"learning.":[87],"Trained":[88],"networks":[90,123,149],"can":[91],"perform":[92],"unsupervised":[93],"clustering":[94],"define":[96],"novel":[97],"error-correcting":[98],"coding":[99],"schemes.":[100],"They":[101],"also":[102,139],"efficiently":[103],"find":[104],"hidden":[105],"structures":[106],"(cliques)":[107],"graph":[109],"theory.":[110],"extend":[112],"this":[113],"known":[114],"connection":[115],"from":[116],"graphs":[117],"hypergraphs":[119],"discover":[121],"<i>n</i>-node":[122],"robust":[125],"storage":[126],"2\u03a9(n1-\u03f5)":[128],"memories":[129],"any":[131],"\u03f5>0.":[132],"graphs,":[137],"determine":[140],"critical":[142],"ratio":[143],"training":[145],"samples":[146],"at":[147],"which":[148],"generalize":[150],"completely.":[151]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}