{"id":"https://openalex.org/W2134101287","doi":"https://doi.org/10.1162/089976603765202640","title":"Learning Coefficients of Layered Models When the True Distribution Mismatches the Singularities","display_name":"Learning Coefficients of Layered Models When the True Distribution Mismatches the Singularities","publication_year":2003,"publication_date":"2003-05-01","ids":{"openalex":"https://openalex.org/W2134101287","doi":"https://doi.org/10.1162/089976603765202640","mag":"2134101287"},"language":"en","primary_location":{"id":"doi:10.1162/089976603765202640","is_oa":false,"landing_page_url":"https://doi.org/10.1162/089976603765202640","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"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/A5089768991","display_name":"Sumio Watanabe","orcid":"https://orcid.org/0000-0001-8341-5639"},"institutions":[{"id":"https://openalex.org/I114531698","display_name":"Tokyo Institute of Technology","ror":"https://ror.org/0112mx960","country_code":"JP","type":"education","lineage":["https://openalex.org/I114531698"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Sumio Watanabe","raw_affiliation_strings":["Precision and Intelligence Laboratory, Tokyo Institute of Technology, Midori-ku, Yokohama, 226-8503 Japan,"],"affiliations":[{"raw_affiliation_string":"Precision and Intelligence Laboratory, Tokyo Institute of Technology, Midori-ku, Yokohama, 226-8503 Japan,","institution_ids":["https://openalex.org/I114531698"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055812530","display_name":"\u0428\u0443\u043d-\u0438\u0447\u0438 \u0410\u043c\u0430\u0440\u0438","orcid":"https://orcid.org/0000-0001-8860-8675"},"institutions":[{"id":"https://openalex.org/I2800939219","display_name":"RIKEN Center for Brain Science","ror":"https://ror.org/04j1n1c04","country_code":"JP","type":"facility","lineage":["https://openalex.org/I2800939219","https://openalex.org/I4210110652"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Shun-ichi Amari","raw_affiliation_strings":["Laboratory for Mathematical Neuroscience, RIKEN Brain Science Institute, Wako-shi, Saitama, 351-0198, Japan,","Laboratory for Mathematical Neuroscience, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan#TAB#"],"affiliations":[{"raw_affiliation_string":"Laboratory for Mathematical Neuroscience, RIKEN Brain Science Institute, Wako-shi, Saitama, 351-0198, Japan,","institution_ids":["https://openalex.org/I2800939219"]},{"raw_affiliation_string":"Laboratory for Mathematical Neuroscience, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan#TAB#","institution_ids":["https://openalex.org/I2800939219"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5089768991"],"corresponding_institution_ids":["https://openalex.org/I114531698"],"apc_list":null,"apc_paid":null,"fwci":4.1124,"has_fulltext":false,"cited_by_count":52,"citation_normalized_percentile":{"value":0.94183172,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"15","issue":"5","first_page":"1013","last_page":"1033"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9975000023841858,"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":0.9975000023841858,"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/T10057","display_name":"Face and Expression Recognition","score":0.9939000010490417,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.9876999855041504,"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/generalization","display_name":"Generalization","score":0.7898741960525513},{"id":"https://openalex.org/keywords/gravitational-singularity","display_name":"Gravitational singularity","score":0.734453558921814},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7173536419868469},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.589648962020874},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5174563527107239},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5082106590270996},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4934045374393463},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.4560309946537018},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3501453399658203},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.25325238704681396},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08023419976234436}],"concepts":[{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.7898741960525513},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.734453558921814},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7173536419868469},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.589648962020874},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5174563527107239},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5082106590270996},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4934045374393463},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.4560309946537018},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3501453399658203},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.25325238704681396},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08023419976234436},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1162/089976603765202640","is_oa":false,"landing_page_url":"https://doi.org/10.1162/089976603765202640","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W1594725708","https://openalex.org/W1598266570","https://openalex.org/W1968936725","https://openalex.org/W1974755789","https://openalex.org/W2012644626","https://openalex.org/W2072857774","https://openalex.org/W2120217353","https://openalex.org/W2132474962","https://openalex.org/W2133370198","https://openalex.org/W2138988196","https://openalex.org/W2159626963","https://openalex.org/W2162858358","https://openalex.org/W2168016523","https://openalex.org/W3022903251","https://openalex.org/W3040861372","https://openalex.org/W4238306122","https://openalex.org/W4390463329"],"related_works":["https://openalex.org/W2353039109","https://openalex.org/W4315588385","https://openalex.org/W4245687327","https://openalex.org/W2061564118","https://openalex.org/W3124638673","https://openalex.org/W2066153363","https://openalex.org/W2900100504","https://openalex.org/W3137640173","https://openalex.org/W4321151660","https://openalex.org/W2079849989"],"abstract_inverted_index":{"Hierarchical":[0],"learning":[1,28,56],"machines":[2],"such":[3,125,184],"as":[4],"layered":[5],"neural":[6],"networks":[7],"have":[8],"singularities":[9,53,149],"in":[10,51,61,115,121,151],"their":[11],"parameter":[12,44,77,165],"spaces.":[13],"At":[14],"singularities,":[15],"the":[16,23,26,43,46,52,55,58,73,76,80,86,91,98,106,116,129,134,137,141,145,161,164,186,194,200,216,224],"Fisher":[17],"information":[18],"matrix":[19],"becomes":[20],"degenerate,":[21],"with":[22],"result":[24],"that":[25,41,136,185,192,222],"conventional":[27],"theory":[29],"of":[30,45,54,75,82,95,140,163,181,193,223],"regular":[31,196,226],"statistical":[32],"models":[33],"does":[34],"not":[35,102,112,172],"hold.":[36],"Recently,":[37],"it":[38],"was":[39],"proved":[40],"if":[42,160,199],"true":[47,107,142,182,214],"distribution":[48,108,143,146],"is":[49,64,70,79,101,109,150,171,189,207,219],"contained":[50,114],"machine,":[57],"generalization":[59,99,131,187,217],"error":[60,100,132,188,218],"Bayes":[62,130],"estimation":[63],"asymptotically":[65],"equal":[66],"to\u03bb/n,":[67],"where":[68],"2\u03bb":[69],"smaller":[71,220],"than":[72,174,191,209,221],"dimension":[74,162,201],"andn":[78],"number":[81],"training":[83],"samples.":[84],"However,":[85],"constant\u03bb":[87],"strongly":[88],"depends":[89],"on":[90],"local":[92],"geometrical":[93],"structure":[94],"singularities;":[96],"hence,":[97],"yet":[103],"clarified":[104],"when":[105],"almost":[110],"but":[111],"completely":[113],"singularities.":[117],"In":[118],"this":[119],"article,":[120],"order":[122],"to":[123,153,168,204],"analyze":[124],"cases,":[126],"we":[127],"study":[128],"under":[133],"condition":[135],"Kullback":[138],"distance":[139],"from":[144,166,202],"represented":[147],"by":[148],"proportion":[152],"1/n":[154],"and":[155],"show":[156],"two":[157],"results.":[158],"First,":[159],"inputs":[167,203],"hidden":[169,205],"units":[170,206],"larger":[173,190,208],"three,":[175,210],"then":[176,211],"there":[177],"exists":[178],"a":[179],"region":[180],"parameters":[183],"corresponding":[195,225],"model.":[197,227],"Second,":[198],"for":[212],"arbitrary":[213],"distribution,":[215]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":26},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
