{"id":"https://openalex.org/W7160398032","doi":"https://doi.org/10.1016/j.neucom.2026.133789","title":"The importance of being empty: Hopfield neural networks with diluted examples","display_name":"The importance of being empty: Hopfield neural networks with diluted examples","publication_year":2026,"publication_date":"2026-05-06","ids":{"openalex":"https://openalex.org/W7160398032","doi":"https://doi.org/10.1016/j.neucom.2026.133789"},"language":"en","primary_location":{"id":"doi:10.1016/j.neucom.2026.133789","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.neucom.2026.133789","pdf_url":null,"source":{"id":"https://openalex.org/S45693802","display_name":"Neurocomputing","issn_l":"0925-2312","issn":["0925-2312","1872-8286"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neurocomputing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1016/j.neucom.2026.133789","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041457930","display_name":"Elena Agliari","orcid":"https://orcid.org/0000-0002-5121-3511"},"institutions":[{"id":"https://openalex.org/I2802148964","display_name":"Istituto Nazionale di Alta Matematica Francesco Severi","ror":"https://ror.org/01vx64p53","country_code":"IT","type":"funder","lineage":["https://openalex.org/I2802148964"]},{"id":"https://openalex.org/I4210130905","display_name":"Unitelma Sapienza University","ror":"https://ror.org/04dfrdm61","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210130905"]},{"id":"https://openalex.org/I861853513","display_name":"Sapienza University of Rome","ror":"https://ror.org/02be6w209","country_code":"IT","type":"education","lineage":["https://openalex.org/I861853513"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Elena Agliari","raw_affiliation_strings":["Department of Mathematics G. Castelnuovo, Sapienza Universit\u00e0 di Roma, Roma, Italy","GNFM-INdAM, Gruppo Nazionale di Fisica Matematica (Istituto Nazionale di Alta Matematica), Rome, Italy"],"raw_orcid":"https://orcid.org/0000-0002-5121-3511","affiliations":[{"raw_affiliation_string":"Department of Mathematics G. Castelnuovo, Sapienza Universit\u00e0 di Roma, Roma, Italy","institution_ids":["https://openalex.org/I4210130905","https://openalex.org/I861853513"]},{"raw_affiliation_string":"GNFM-INdAM, Gruppo Nazionale di Fisica Matematica (Istituto Nazionale di Alta Matematica), Rome, Italy","institution_ids":["https://openalex.org/I2802148964"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5048265125","display_name":"Alberto Fachechi","orcid":"https://orcid.org/0000-0002-8392-1678"},"institutions":[{"id":"https://openalex.org/I2802148964","display_name":"Istituto Nazionale di Alta Matematica Francesco Severi","ror":"https://ror.org/01vx64p53","country_code":"IT","type":"funder","lineage":["https://openalex.org/I2802148964"]},{"id":"https://openalex.org/I4210130905","display_name":"Unitelma Sapienza University","ror":"https://ror.org/04dfrdm61","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210130905"]},{"id":"https://openalex.org/I861853513","display_name":"Sapienza University of Rome","ror":"https://ror.org/02be6w209","country_code":"IT","type":"education","lineage":["https://openalex.org/I861853513"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Alberto Fachechi","raw_affiliation_strings":["Department of Mathematics G. Castelnuovo, Sapienza Universit\u00e0 di Roma, Roma, Italy","GNFM-INdAM, Gruppo Nazionale di Fisica Matematica (Istituto Nazionale di Alta Matematica), Rome, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics G. Castelnuovo, Sapienza Universit\u00e0 di Roma, Roma, Italy","institution_ids":["https://openalex.org/I4210130905","https://openalex.org/I861853513"]},{"raw_affiliation_string":"GNFM-INdAM, Gruppo Nazionale di Fisica Matematica (Istituto Nazionale di Alta Matematica), Rome, Italy","institution_ids":["https://openalex.org/I2802148964"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5093826621","display_name":"Domenico Luongo","orcid":null},"institutions":[{"id":"https://openalex.org/I157210198","display_name":"Scuola Normale Superiore","ror":"https://ror.org/03aydme10","country_code":"IT","type":"education","lineage":["https://openalex.org/I157210198"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Domenico Luongo","raw_affiliation_strings":["Scuola Normale Superiore, Pisa, Italy"],"raw_orcid":"https://orcid.org/0009-0002-8356-8082","affiliations":[{"raw_affiliation_string":"Scuola Normale Superiore, Pisa, Italy","institution_ids":["https://openalex.org/I157210198"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5041457930"],"corresponding_institution_ids":["https://openalex.org/I2802148964","https://openalex.org/I4210130905","https://openalex.org/I861853513"],"apc_list":{"value":2470,"currency":"USD","value_usd":2470},"apc_paid":{"value":2470,"currency":"USD","value_usd":2470},"fwci":7.8105,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.97165998,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"694","issue":null,"first_page":"133789","last_page":"133789"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.7023000121116638,"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":0.7023000121116638,"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/T11347","display_name":"Neural Networks Stability and Synchronization","score":0.049800001084804535,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T13748","display_name":"Advanced Statistical Modeling Techniques","score":0.027499999850988388,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/artificial-neural-network","display_name":"Artificial neural network","score":0.6420999765396118},{"id":"https://openalex.org/keywords/hopfield-network","display_name":"Hopfield network","score":0.4047999978065491},{"id":"https://openalex.org/keywords/pattern-recognition","display_name":"Pattern recognition (psychology)","score":0.3797000050544739},{"id":"https://openalex.org/keywords/deep-neural-networks","display_name":"Deep neural networks","score":0.3116999864578247},{"id":"https://openalex.org/keywords/types-of-artificial-neural-networks","display_name":"Types of artificial neural networks","score":0.29019999504089355}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6858000159263611},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6420999765396118},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.6284000277519226},{"id":"https://openalex.org/C46421273","wikidata":"https://www.wikidata.org/wiki/Q1407668","display_name":"Hopfield network","level":3,"score":0.4047999978065491},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.3797000050544739},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.313400000333786},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.3116999864578247},{"id":"https://openalex.org/C177973122","wikidata":"https://www.wikidata.org/wiki/Q7860946","display_name":"Types of artificial neural networks","level":4,"score":0.29019999504089355},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2515000104904175},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.24799999594688416}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1016/j.neucom.2026.133789","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.neucom.2026.133789","pdf_url":null,"source":{"id":"https://openalex.org/S45693802","display_name":"Neurocomputing","issn_l":"0925-2312","issn":["0925-2312","1872-8286"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neurocomputing","raw_type":"journal-article"},{"id":"pmh:oai:iris.uniroma1.it:11573/1768673","is_oa":true,"landing_page_url":"https://hdl.handle.net/11573/1768673","pdf_url":null,"source":{"id":"https://openalex.org/S4377196107","display_name":"IRIS Research product catalog (Sapienza University of Rome)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1016/j.neucom.2026.133789","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.neucom.2026.133789","pdf_url":null,"source":{"id":"https://openalex.org/S45693802","display_name":"Neurocomputing","issn_l":"0925-2312","issn":["0925-2312","1872-8286"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neurocomputing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3769637172","display_name":null,"funder_award_id":"RM12218169691087","funder_id":"https://openalex.org/F4320322510","funder_display_name":"Sapienza Universit\u00e0 di Roma"},{"id":"https://openalex.org/G7680209173","display_name":null,"funder_award_id":"PE0000013-FAIR","funder_id":"https://openalex.org/F5497039910","funder_display_name":"Ministero dell'Istruzione e del Merito"},{"id":"https://openalex.org/G8022218832","display_name":null,"funder_award_id":"RM12117A8590B3FA","funder_id":"https://openalex.org/F4320322510","funder_display_name":"Sapienza Universit\u00e0 di Roma"}],"funders":[{"id":"https://openalex.org/F4320322510","display_name":"Sapienza Universit\u00e0 di Roma","ror":"https://ror.org/02be6w209"},{"id":"https://openalex.org/F5497039910","display_name":"Ministero dell'Istruzione e del Merito","ror":"https://ror.org/01ehyh486"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W1984431842","https://openalex.org/W2000258229","https://openalex.org/W2029667457","https://openalex.org/W2030358500","https://openalex.org/W2037323183","https://openalex.org/W2080792322","https://openalex.org/W2111406701","https://openalex.org/W2128084896","https://openalex.org/W2141442470","https://openalex.org/W2157418105","https://openalex.org/W2906357013","https://openalex.org/W2911964244","https://openalex.org/W2954996726","https://openalex.org/W2963208784","https://openalex.org/W3143934758","https://openalex.org/W3173484311","https://openalex.org/W4210620801","https://openalex.org/W4289782209","https://openalex.org/W4309866921","https://openalex.org/W4389060138","https://openalex.org/W4391743824","https://openalex.org/W4396920624"],"related_works":[],"abstract_inverted_index":{"We":[0,36],"consider":[1],"Hopfield":[2],"networks,":[3],"where":[4,41],"neurons":[5],"interact":[6],"pair-wise":[7],"by":[8,81,115],"Hebbian":[9,153],"couplings":[10],"built":[11,155],"over":[12,173],".":[13,20,28],"a":[14,21,29,167,174],"set":[15],"of":[16,23,31,51,73,97,122,139,166],"definite":[17],"patterns":[18,107],"(ground-truths),":[19],"sample":[22,30],"labeled":[24],"examples":[25,33,47],"(supervised":[26],"setting),":[27],"unlabeled":[32],"(unsupervised":[34],"setting).":[35],"focus":[37],"on":[38,89,134,156],"the":[39,70,74,78,85,94,98,120,137,152,163],"case":[40],"ground-truths":[42],"are":[43,48],"Rademacher":[44],"vectors":[45],"and":[46,67,83,100],"noisy":[49,109],"versions":[50],"these":[52,140],"ground-truths,":[53],"possibly":[54],"displaying":[55],"some":[56,129],"blank":[57,123],"or":[58,64],"empty":[59],"entries":[60,124],"(e.g.,":[61],"mimicking":[62],"missing":[63],"dropped":[65],"data),":[66],"we":[68,91,149],"determine":[69],"spectral":[71],"distribution":[72],"coupling":[75],"matrices":[76],"in":[77,128,142],"three":[79],"scenarios,":[80],"exploiting":[82],"extending":[84],"Marchenko-Pastur":[86],"theorem.":[87],"Building":[88],"this,":[90],"analytically":[92],"inspect":[93],"generalization":[95],"capabilities":[96],"networks":[99],"examining":[101],"their":[102],"ability":[103],"to":[104],"recover":[105],"ground-truth":[106],"from":[108],"inputs.":[110],"In":[111],"particular,":[112],"as":[113,162],"corroborated":[114],"long-running":[116],"Monte":[117],"Carlo":[118],"simulations,":[119],"presence":[121],"can":[125,159],"be":[126,160],"beneficial":[127],"conditions,":[130],"suggesting":[131],"strategies":[132],"based":[133],"data":[135],"sparsification;":[136],"robustness":[138],"results":[141],"structured":[143],"datasets":[144],"is":[145],"confirmed":[146],"numerically.":[147],"Finally,":[148],"demonstrate":[150],"that":[151],"matrix,":[154],"sparse":[157],"examples,":[158],"recovered":[161],"fixed":[164],"point":[165],"gradient":[168],"descent":[169],"algorithm":[170],"with":[171],"dropout,":[172],"suitable":[175],"loss":[176],"function.":[177]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-08-13T07:04:57.449891","created_date":"2026-05-07T00:00:00"}
