{"id":"https://openalex.org/W7162290078","doi":"https://doi.org/10.48550/arxiv.2605.23087","title":"The Implicit Bias of Depth: From Neural Collapse to Softmax Codes","display_name":"The Implicit Bias of Depth: From Neural Collapse to Softmax Codes","publication_year":2026,"publication_date":"2026-05-21","ids":{"openalex":"https://openalex.org/W7162290078","doi":"https://doi.org/10.48550/arxiv.2605.23087"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.23087","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.23087","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.23087","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5095335163","display_name":"Connall Garrod","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Garrod, Connall","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080921125","display_name":"Jonathan P. Keating","orcid":"https://orcid.org/0000-0003-0864-038X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Keating, Jonathan P.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5136933162","display_name":"Christos Thrampoulidis","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thrampoulidis, Christos","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9327999949455261,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9327999949455261,"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/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.006500000134110451,"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/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.005400000140070915,"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/softmax-function","display_name":"Softmax function","score":0.8201000094413757},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5641999840736389},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.5546000003814697},{"id":"https://openalex.org/keywords/deep-neural-networks","display_name":"Deep neural networks","score":0.4909999966621399},{"id":"https://openalex.org/keywords/feature","display_name":"Feature (linguistics)","score":0.44850000739097595},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.4300999939441681}],"concepts":[{"id":"https://openalex.org/C188441871","wikidata":"https://www.wikidata.org/wiki/Q7554146","display_name":"Softmax function","level":3,"score":0.8201000094413757},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5641999840736389},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.5546000003814697},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.517300009727478},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.4909999966621399},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4788999855518341},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.46140000224113464},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.44850000739097595},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.4300999939441681},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.3901999890804291},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.374099999666214},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.3707999885082245},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.3456000089645386},{"id":"https://openalex.org/C180623205","wikidata":"https://www.wikidata.org/wiki/Q1268589","display_name":"Outer product","level":3,"score":0.31060001254081726},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.28859999775886536},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.274399995803833},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26019999384880066}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.23087","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.23087","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.23087","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.23087","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Neural":[0],"collapse":[1],"(NC)":[2],"describes":[3],"the":[4,10,39,138,155],"structured":[5],"geometry":[6],"that":[7,67,117,133],"emerges":[8],"in":[9,24,100,137,164],"features":[11],"and":[12,60,121,158],"weights":[13],"of":[14,128,161],"trained":[15,167],"classifiers.":[16],"Recent":[17],"theory":[18],"suggests":[19],"NC":[20],"can":[21],"be":[22],"suboptimal":[23],"deep":[25,40,47,165],"architectures,":[26],"attributing":[27],"this":[28],"to":[29,45,55,86,93],"an":[30,70,111],"explicit":[31],"low-rank":[32,72,74,84,119],"bias":[33,163],"from":[34],"L2":[35],"regularization.":[36],"We":[37,65],"study":[38],"unconstrained":[41],"feature":[42],"model":[43],"(UFM)-equivalent":[44],"a":[46],"linear":[48],"network":[49],"with":[50,168],"orthogonal":[51],"inputs-trained":[52],"without":[53],"regularization,":[54],"isolate":[56],"how":[57,123],"gradient":[58],"descent":[59],"depth":[61,68,124],"alone":[62],"shape":[63],"NC.":[64,87],"show":[66,132],"induces":[69],"implicit":[71,162],"bias:":[73],"matrices":[75],"propagate":[76],"norm":[77],"more":[78],"efficiently":[79],"through":[80],"successive":[81],"multiplications,":[82],"promoting":[83],"alternatives":[85],"These":[88],"alternatives,":[89],"we":[90,109,131],"argue,":[91],"correspond":[92],"softmax":[94],"codes:":[95],"max-margin":[96],"solutions":[97],"previously":[98],"found":[99],"width-bottlenecked":[101],"networks.":[102],"Analyzing":[103],"training":[104,148],"dynamics":[105],"under":[106],"spectral":[107],"initialization,":[108],"identify":[110],"early-time":[112],"repulsion":[113],"among":[114],"singular":[115],"values":[116],"drives":[118],"emergence,":[120],"characterize":[122],"shrinks":[125],"NC's":[126],"basin":[127],"attraction.":[129],"Finally,":[130],"some":[134],"effects":[135],"act":[136],"opposite":[139],"direction:":[140],"for":[141],"randomly":[142],"initialized":[143],"networks,":[144],"increasing":[145],"width":[146],"biases":[147],"toward":[149],"higher-rank":[150],"solutions.":[151],"Our":[152],"results":[153],"provide":[154],"first":[156],"asymptotic":[157],"dynamic":[159],"characterization":[160],"UFMs":[166],"unregularized":[169],"multiclass":[170],"cross-entropy.":[171]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-05-26T00:00:00"}
