{"id":"https://openalex.org/W7168018504","doi":"https://doi.org/10.1007/s10994-026-07097-9","title":"Deep Linear Discriminant Analysis Revisited","display_name":"Deep Linear Discriminant Analysis Revisited","publication_year":2026,"publication_date":"2026-07-01","ids":{"openalex":"https://openalex.org/W7168018504","doi":"https://doi.org/10.1007/s10994-026-07097-9"},"language":"en","primary_location":{"id":"doi:10.1007/s10994-026-07097-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-026-07097-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-026-07097-9.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10994-026-07097-9.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5007740827","display_name":"Maxat Tezekbayev","orcid":"https://orcid.org/0000-0001-8100-2697"},"institutions":[{"id":"https://openalex.org/I60559429","display_name":"Nazarbayev University","ror":"https://ror.org/052bx8q98","country_code":"KZ","type":"education","lineage":["https://openalex.org/I60559429"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"Maxat Tezekbayev","raw_affiliation_strings":["Department of Mathematics, Nazarbayev University, 53 Kabanbay Batyr ave, Astana, 010000, Kazakhstan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Nazarbayev University, 53 Kabanbay Batyr ave, Astana, 010000, Kazakhstan","institution_ids":["https://openalex.org/I60559429"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029285634","display_name":"Rustem Takhanov","orcid":"https://orcid.org/0000-0001-7405-8254"},"institutions":[{"id":"https://openalex.org/I60559429","display_name":"Nazarbayev University","ror":"https://ror.org/052bx8q98","country_code":"KZ","type":"education","lineage":["https://openalex.org/I60559429"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"Rustem Takhanov","raw_affiliation_strings":["Department of Mathematics, Nazarbayev University, 53 Kabanbay Batyr ave, Astana, 010000, Kazakhstan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Nazarbayev University, 53 Kabanbay Batyr ave, Astana, 010000, Kazakhstan","institution_ids":["https://openalex.org/I60559429"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021173706","display_name":"Arman Bolatov","orcid":"https://orcid.org/0000-0002-4238-487X"},"institutions":[{"id":"https://openalex.org/I4210113480","display_name":"Mohamed bin Zayed University of Artificial Intelligence","ror":"https://ror.org/0258gkt32","country_code":"AE","type":"education","lineage":["https://openalex.org/I4210113480"]},{"id":"https://openalex.org/I4210119247","display_name":"National Center of Space Research and Technology","ror":"https://ror.org/02f0t3e51","country_code":"KZ","type":"government","lineage":["https://openalex.org/I4210119247"]}],"countries":["AE","KZ"],"is_corresponding":false,"raw_author_name":"Arman Bolatov","raw_affiliation_strings":["Machine Learning Department, Mohamed bin Zayed University of Artificial Intelligence, Masdar City, Abu Dhabi, 00000, UAE","Remote Sensing Department, National Center of Space Research and Technology, 15 Shevchenko St., Almaty, 050010, Kazakhstan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Machine Learning Department, Mohamed bin Zayed University of Artificial Intelligence, Masdar City, Abu Dhabi, 00000, UAE","institution_ids":["https://openalex.org/I4210113480"]},{"raw_affiliation_string":"Remote Sensing Department, National Center of Space Research and Technology, 15 Shevchenko St., Almaty, 050010, Kazakhstan","institution_ids":["https://openalex.org/I4210119247"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053731169","display_name":"Zhenisbek Assylbekov","orcid":"https://orcid.org/0000-0003-0095-9409"},"institutions":[{"id":"https://openalex.org/I162817326","display_name":"Indiana University \u2013 Purdue University Fort Wayne","ror":"https://ror.org/01jxzq227","country_code":"US","type":"education","lineage":["https://openalex.org/I162817326"]},{"id":"https://openalex.org/I4210119247","display_name":"National Center of Space Research and Technology","ror":"https://ror.org/02f0t3e51","country_code":"KZ","type":"government","lineage":["https://openalex.org/I4210119247"]},{"id":"https://openalex.org/I4210130184","display_name":"Purdue University Fort Wayne","ror":"https://ror.org/04c4hz115","country_code":"US","type":"education","lineage":["https://openalex.org/I4210130184"]}],"countries":["KZ","US"],"is_corresponding":true,"raw_author_name":"Zhenisbek Assylbekov","raw_affiliation_strings":["Department of Mathematical Sciences, Purdue University Fort Wayne, 2101 East Coliseum Boulevard, Fort Wayne, 46805, IN, USA","Remote Sensing Department, National Center of Space Research and Technology, 15 Shevchenko St., Almaty, 050010, Kazakhstan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Purdue University Fort Wayne, 2101 East Coliseum Boulevard, Fort Wayne, 46805, IN, USA","institution_ids":["https://openalex.org/I162817326","https://openalex.org/I4210130184"]},{"raw_affiliation_string":"Remote Sensing Department, National Center of Space Research and Technology, 15 Shevchenko St., Almaty, 050010, Kazakhstan","institution_ids":["https://openalex.org/I4210119247"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5053731169"],"corresponding_institution_ids":["https://openalex.org/I162817326","https://openalex.org/I4210119247","https://openalex.org/I4210130184"],"apc_list":{"value":2990,"currency":"USD","value_usd":2990},"apc_paid":{"value":2990,"currency":"USD","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.8694808,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"115","issue":"7","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.7678999900817871,"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"}},"topics":[{"id":"https://openalex.org/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.7678999900817871,"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/T11448","display_name":"Face recognition and analysis","score":0.03440000116825104,"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/T12535","display_name":"Machine Learning and Data Classification","score":0.019899999722838402,"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/discriminative-model","display_name":"Discriminative model","score":0.8615999817848206},{"id":"https://openalex.org/keywords/softmax-function","display_name":"Softmax function","score":0.7692000269889832},{"id":"https://openalex.org/keywords/pattern-recognition","display_name":"Pattern recognition (psychology)","score":0.7480999827384949},{"id":"https://openalex.org/keywords/linear-discriminant-analysis","display_name":"Linear discriminant analysis","score":0.694100022315979},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.4487999975681305},{"id":"https://openalex.org/keywords/discriminant","display_name":"Discriminant","score":0.43689998984336853},{"id":"https://openalex.org/keywords/generative-model","display_name":"Generative model","score":0.3765999972820282},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.36980000138282776}],"concepts":[{"id":"https://openalex.org/C97931131","wikidata":"https://www.wikidata.org/wiki/Q5282087","display_name":"Discriminative model","level":2,"score":0.8615999817848206},{"id":"https://openalex.org/C188441871","wikidata":"https://www.wikidata.org/wiki/Q7554146","display_name":"Softmax function","level":3,"score":0.7692000269889832},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.7480999827384949},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.7475000023841858},{"id":"https://openalex.org/C69738355","wikidata":"https://www.wikidata.org/wiki/Q1228929","display_name":"Linear discriminant analysis","level":2,"score":0.694100022315979},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6065000295639038},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.4487999975681305},{"id":"https://openalex.org/C78397625","wikidata":"https://www.wikidata.org/wiki/Q192487","display_name":"Discriminant","level":2,"score":0.43689998984336853},{"id":"https://openalex.org/C167966045","wikidata":"https://www.wikidata.org/wiki/Q5532625","display_name":"Generative model","level":3,"score":0.3765999972820282},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.36980000138282776},{"id":"https://openalex.org/C39890363","wikidata":"https://www.wikidata.org/wiki/Q36108","display_name":"Generative grammar","level":2,"score":0.3693999946117401},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.34709998965263367},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.3395000100135803},{"id":"https://openalex.org/C27438332","wikidata":"https://www.wikidata.org/wiki/Q2873","display_name":"Principal component analysis","level":2,"score":0.3190000057220459},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.313400000333786},{"id":"https://openalex.org/C94124525","wikidata":"https://www.wikidata.org/wiki/Q912550","display_name":"Categorization","level":2,"score":0.2777999937534332},{"id":"https://openalex.org/C51632099","wikidata":"https://www.wikidata.org/wiki/Q3985153","display_name":"Training set","level":2,"score":0.27480000257492065},{"id":"https://openalex.org/C75294576","wikidata":"https://www.wikidata.org/wiki/Q5165192","display_name":"Contextual image classification","level":3,"score":0.27079999446868896},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.25360000133514404},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.2529999911785126}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10994-026-07097-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-026-07097-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-026-07097-9.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10994-026-07097-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-026-07097-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-026-07097-9.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.7733344435691833,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7168018504.pdf","grobid_xml":"https://content.openalex.org/works/W7168018504.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W1614659291","https://openalex.org/W2001619934","https://openalex.org/W2057069782","https://openalex.org/W2150796457","https://openalex.org/W2601450892","https://openalex.org/W2626967530","https://openalex.org/W2750378047","https://openalex.org/W2903284382","https://openalex.org/W2964121744","https://openalex.org/W3047965733","https://openalex.org/W4287865413","https://openalex.org/W4293380554","https://openalex.org/W4313306215"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"We":[1],"show":[2],"that":[3,92],"for":[4],"unconstrained":[5],"Deep":[6,102],"Linear":[7],"Discriminant":[8],"Analysis":[9],"(LDA)":[10],"classifiers,":[11],"maximum-likelihood":[12],"training":[13,34],"admits":[14],"pathological":[15],"solutions":[16],"in":[17],"which":[18,68],"class":[19],"means":[20],"drift":[21],"together,":[22],"covariances":[23],"collapse,":[24],"and":[25,122,126],"the":[26,40,43,62,70,78,113],"learned":[27],"representation":[28],"becomes":[29],"almost":[30],"non-discriminative.":[31],"Conversely,":[32],"cross-entropy":[33],"yields":[35,127],"excellent":[36],"accuracy":[37,115],"but":[38],"decouples":[39],"head":[41],"from":[42],"underlying":[44],"generative":[45,55],"model,":[46],"leading":[47],"to":[48],"highly":[49],"inconsistent":[50],"parameter":[51],"estimates.":[52],"To":[53],"reconcile":[54],"structure":[56],"with":[57,73,105],"discriminative":[58],"performance,":[59],"we":[60],"introduce":[61],"Discriminative":[63],"Negative":[64],"Log-Likelihood":[65],"(DNLL)":[66],"loss,":[67],"augments":[69],"LDA":[71,87,103],"log-likelihood":[72],"a":[74,90],"simple":[75],"penalty":[76],"on":[77,119],"mixture":[79],"density.":[80],"DNLL":[81,106],"can":[82],"be":[83],"interpreted":[84],"as":[85],"standard":[86,123],"NLL":[88],"plus":[89],"term":[91],"explicitly":[93],"discourages":[94],"regions":[95],"where":[96],"several":[97],"classes":[98],"are":[99],"simultaneously":[100],"likely.":[101],"trained":[104],"produces":[107],"clean,":[108],"well-separated":[109],"latent":[110],"spaces,":[111],"matches":[112],"test":[114],"of":[116],"softmax":[117],"classifiers":[118],"synthetic":[120],"data":[121],"image":[124],"benchmarks,":[125],"substantially":[128],"better":[129],"calibrated":[130],"predictive":[131],"probabilities.":[132]},"counts_by_year":[],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2026-07-12T00:00:00"}