{"id":"https://openalex.org/W7128511071","doi":"https://doi.org/10.48550/arxiv.2602.07697","title":"On the Infinite Width and Depth Limits of Predictive Coding Networks","display_name":"On the Infinite Width and Depth Limits of Predictive Coding Networks","publication_year":2026,"publication_date":"2026-02-07","ids":{"openalex":"https://openalex.org/W7128511071","doi":"https://doi.org/10.48550/arxiv.2602.07697"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2602.07697","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","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":"Article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5125331764","display_name":"Francesco Innocenti","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Innocenti, Francesco","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5125305957","display_name":"El Mehdi Achour","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Achour, El Mehdi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5006746885","display_name":"Rafal Bogacz","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bogacz, Rafal","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/T11289","display_name":"Single-cell and spatial transcriptomics","score":0.2872999906539917,"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"}},"topics":[{"id":"https://openalex.org/T11289","display_name":"Single-cell and spatial transcriptomics","score":0.2872999906539917,"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/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.1006999984383583,"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/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.09000000357627869,"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/scaling","display_name":"Scaling","score":0.5856999754905701},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.5773000121116638},{"id":"https://openalex.org/keywords/backpropagation","display_name":"Backpropagation","score":0.5738999843597412},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.5157999992370605},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5058000087738037},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.490200012922287},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.48260000348091125},{"id":"https://openalex.org/keywords/predictive-coding","display_name":"Predictive coding","score":0.4672999978065491}],"concepts":[{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.5856999754905701},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.5773000121116638},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.57669997215271},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5751000046730042},{"id":"https://openalex.org/C155032097","wikidata":"https://www.wikidata.org/wiki/Q798503","display_name":"Backpropagation","level":3,"score":0.5738999843597412},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.5157999992370605},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5058000087738037},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.490200012922287},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.48260000348091125},{"id":"https://openalex.org/C2778061373","wikidata":"https://www.wikidata.org/wiki/Q1315146","display_name":"Predictive coding","level":3,"score":0.4672999978065491},{"id":"https://openalex.org/C179518139","wikidata":"https://www.wikidata.org/wiki/Q5140297","display_name":"Coding (social sciences)","level":2,"score":0.4442000091075897},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.42989999055862427},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4268999993801117},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.41510000824928284},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.36980000138282776},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.34619998931884766},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.34279999136924744},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3393999934196472},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3386000096797943},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3330000042915344},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.3321000039577484},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.32359999418258667},{"id":"https://openalex.org/C170122806","wikidata":"https://www.wikidata.org/wiki/Q1914828","display_name":"Linear scale","level":2,"score":0.2793999910354614},{"id":"https://openalex.org/C104122410","wikidata":"https://www.wikidata.org/wiki/Q1416406","display_name":"Network model","level":2,"score":0.26919999718666077},{"id":"https://openalex.org/C98856871","wikidata":"https://www.wikidata.org/wiki/Q1588488","display_name":"Radial basis function","level":3,"score":0.2671000063419342},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.262800008058548}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2602.07697","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","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":"Article"},{"id":"doi:10.48550/arxiv.2602.07697","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.07697","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":"pmh:doi:10.48550/arxiv.2602.07697","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","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":"Article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.8860676288604736}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Predictive":[0],"coding":[1],"(PC)":[2],"is":[3,85,114,138],"a":[4,167],"biologically":[5],"plausible":[6],"alternative":[7],"to":[8,19,105,141],"standard":[9],"backpropagation":[10],"(BP)":[11],"that":[12,74,160],"minimises":[13],"an":[14,135],"energy":[15,100],"function":[16],"with":[17,101,163,176],"respect":[18],"network":[20],"activities":[21,103],"before":[22],"updating":[23],"weights.":[24],"Recent":[25],"work":[26,154],"has":[27],"improved":[28],"the":[29,43,60,75,87,98,106,111,118,124,156,187],"training":[30],"stability":[31],"of":[32,49,66,77,95,158],"deep":[33,184],"PC":[34,84,99,122],"networks":[35,149,185],"(PCNs)":[36],"by":[37],"leveraging":[38],"some":[39],"BP-inspired":[40],"reparameterisations.":[41],"However,":[42],"full":[44],"scalability":[45],"and":[46,63,79,150],"theoretical":[47],"basis":[48],"these":[50,96],"methods":[51],"remain":[52],"unclear.":[53],"To":[54],"address":[55],"this":[56,153],"gap,":[57],"we":[58,72],"study":[59],"infinite":[61],"width":[62,113],"depth":[64],"limits":[65],"PCNs.":[67],"For":[68],"linear":[69],"residual":[70],"networks,":[71],"show":[73,130],"set":[76],"width-":[78],"depth-stable":[80],"feature-learning":[81],"parameterisations":[82],"for":[83,90,144],"exactly":[86],"same":[88,125],"as":[89,127,132,134],"BP.":[91,128],"Moreover,":[92],"under":[93],"any":[94],"parameterisations,":[97],"equilibrated":[102],"converges":[104],"quadratic":[107],"BP":[108,142,171],"loss":[109],"when":[110],"model":[112],"much":[115,181],"larger":[116],"than":[117,183],"depth,":[119],"resulting":[120],"in":[121,169,180],"computing":[123],"gradients":[126],"Experiments":[129],"that,":[131],"long":[133],"activity":[136],"equilibrium":[137],"reached,":[139],"convergence":[140],"holds":[143],"nonlinear":[145],"models":[146],"including":[147],"convolutional":[148],"transformers.":[151],"Overall,":[152],"constrains":[155],"types":[157],"parameterisation":[159],"are":[161],"scalable":[162],"PC,":[164],"while":[165],"showing":[166],"way":[168],"which":[170],"can":[172],"be":[173],"effectively":[174],"implemented":[175],"only":[177],"local":[178],"updates":[179],"wider":[182],"like":[186],"brain.":[188]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-02-11T00:00:00"}
