{"id":"https://openalex.org/W3006348122","doi":"https://doi.org/10.1109/tnnls.2020.2968509","title":"Understanding Convolutional Neural Networks With Information Theory: An Initial Exploration","display_name":"Understanding Convolutional Neural Networks With Information Theory: An Initial Exploration","publication_year":2020,"publication_date":"2020-02-13","ids":{"openalex":"https://openalex.org/W3006348122","doi":"https://doi.org/10.1109/tnnls.2020.2968509","mag":"3006348122","pmid":"https://pubmed.ncbi.nlm.nih.gov/32071010"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2020.2968509","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2020.2968509","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks and Learning Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"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/A5041358503","display_name":"Shujian Yu","orcid":"https://orcid.org/0000-0002-6385-1705"},"institutions":[{"id":"https://openalex.org/I33213144","display_name":"University of Florida","ror":"https://ror.org/02y3ad647","country_code":"US","type":"education","lineage":["https://openalex.org/I33213144"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Shujian Yu","raw_affiliation_strings":["University of Florida, Gainesville, FL, USA"],"affiliations":[{"raw_affiliation_string":"University of Florida, Gainesville, FL, USA","institution_ids":["https://openalex.org/I33213144"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031748191","display_name":"Kristoffer Wickstr\u00f8m","orcid":"https://orcid.org/0000-0003-1395-7154"},"institutions":[{"id":"https://openalex.org/I78037679","display_name":"UiT The Arctic University of Norway","ror":"https://ror.org/00wge5k78","country_code":"NO","type":"education","lineage":["https://openalex.org/I78037679"]}],"countries":["NO"],"is_corresponding":false,"raw_author_name":"Kristoffer Wickstrom","raw_affiliation_strings":["UiT The Arctic University of Norway, Troms\u00f8, Norway"],"affiliations":[{"raw_affiliation_string":"UiT The Arctic University of Norway, Troms\u00f8, Norway","institution_ids":["https://openalex.org/I78037679"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013414039","display_name":"Robert Jenssen","orcid":"https://orcid.org/0000-0002-7496-8474"},"institutions":[{"id":"https://openalex.org/I78037679","display_name":"UiT The Arctic University of Norway","ror":"https://ror.org/00wge5k78","country_code":"NO","type":"education","lineage":["https://openalex.org/I78037679"]}],"countries":["NO"],"is_corresponding":false,"raw_author_name":"Robert Jenssen","raw_affiliation_strings":["UiT The Arctic University of Norway, Troms\u00f8, Norway"],"affiliations":[{"raw_affiliation_string":"UiT The Arctic University of Norway, Troms\u00f8, Norway","institution_ids":["https://openalex.org/I78037679"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019504861","display_name":"Jos\u00e9 C. Pr\u0131\u0301ncipe","orcid":"https://orcid.org/0000-0002-3449-3531"},"institutions":[{"id":"https://openalex.org/I33213144","display_name":"University of Florida","ror":"https://ror.org/02y3ad647","country_code":"US","type":"education","lineage":["https://openalex.org/I33213144"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jose Principe","raw_affiliation_strings":["University of Florida, Gainesville, FL, USA"],"affiliations":[{"raw_affiliation_string":"University of Florida, Gainesville, FL, USA","institution_ids":["https://openalex.org/I33213144"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5041358503"],"corresponding_institution_ids":["https://openalex.org/I33213144"],"apc_list":null,"apc_paid":null,"fwci":7.8639,"has_fulltext":false,"cited_by_count":81,"citation_normalized_percentile":{"value":0.98154271,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":100},"biblio":{"volume":"32","issue":"1","first_page":"435","last_page":"442"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9976000189781189,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.7027326822280884},{"id":"https://openalex.org/keywords/convolutional-neural-network","display_name":"Convolutional neural network","score":0.6707542538642883},{"id":"https://openalex.org/keywords/reproducing-kernel-hilbert-space","display_name":"Reproducing kernel Hilbert space","score":0.6463472843170166},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6165972352027893},{"id":"https://openalex.org/keywords/hermitian-matrix","display_name":"Hermitian matrix","score":0.5947855114936829},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.517479658126831},{"id":"https://openalex.org/keywords/information-theory","display_name":"Information theory","score":0.4974980652332306},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.44921237230300903},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.41930797696113586},{"id":"https://openalex.org/keywords/redundancy","display_name":"Redundancy (engineering)","score":0.41917213797569275},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.4168553054332733},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.40990713238716125},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.340730220079422},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3340079188346863},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.1333719789981842},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.11719080805778503},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1149643063545227}],"concepts":[{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.7027326822280884},{"id":"https://openalex.org/C81363708","wikidata":"https://www.wikidata.org/wiki/Q17084460","display_name":"Convolutional neural network","level":2,"score":0.6707542538642883},{"id":"https://openalex.org/C80884492","wikidata":"https://www.wikidata.org/wiki/Q3345678","display_name":"Reproducing kernel Hilbert space","level":3,"score":0.6463472843170166},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6165972352027893},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.5947855114936829},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.517479658126831},{"id":"https://openalex.org/C52622258","wikidata":"https://www.wikidata.org/wiki/Q131222","display_name":"Information theory","level":2,"score":0.4974980652332306},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.44921237230300903},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.41930797696113586},{"id":"https://openalex.org/C152124472","wikidata":"https://www.wikidata.org/wiki/Q1204361","display_name":"Redundancy (engineering)","level":2,"score":0.41917213797569275},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4168553054332733},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.40990713238716125},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.340730220079422},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3340079188346863},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.1333719789981842},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.11719080805778503},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1149643063545227},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnnls.2020.2968509","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2020.2968509","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks and Learning Systems","raw_type":"journal-article"},{"id":"pmid:32071010","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/32071010","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE transactions on neural networks and learning systems","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7420929496","display_name":null,"funder_award_id":"N00014-18-1-2306","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G7703754090","display_name":null,"funder_award_id":"FA9453-18-1-0039","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"}],"funders":[{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":85,"referenced_works":["https://openalex.org/W110813552","https://openalex.org/W1556911442","https://openalex.org/W1686810756","https://openalex.org/W1686946872","https://openalex.org/W1770458422","https://openalex.org/W1866114900","https://openalex.org/W1932088538","https://openalex.org/W2027665171","https://openalex.org/W2045739538","https://openalex.org/W2080722576","https://openalex.org/W2081988641","https://openalex.org/W2092939357","https://openalex.org/W2095841525","https://openalex.org/W2112796928","https://openalex.org/W2114771311","https://openalex.org/W2121947440","https://openalex.org/W2122166994","https://openalex.org/W2122882636","https://openalex.org/W2129000925","https://openalex.org/W2129905273","https://openalex.org/W2142357860","https://openalex.org/W2146705998","https://openalex.org/W2149803014","https://openalex.org/W2156504490","https://openalex.org/W2163605009","https://openalex.org/W2194775991","https://openalex.org/W2339172597","https://openalex.org/W2467203645","https://openalex.org/W2467760016","https://openalex.org/W2495425901","https://openalex.org/W2583538381","https://openalex.org/W2593634001","https://openalex.org/W2622719862","https://openalex.org/W2750384547","https://openalex.org/W2753338388","https://openalex.org/W2785885194","https://openalex.org/W2796664995","https://openalex.org/W2902227449","https://openalex.org/W2902553997","https://openalex.org/W2950850913","https://openalex.org/W2962679290","https://openalex.org/W2962719046","https://openalex.org/W2962779416","https://openalex.org/W2962783375","https://openalex.org/W2962807446","https://openalex.org/W2962835968","https://openalex.org/W2962965870","https://openalex.org/W2963386266","https://openalex.org/W2963912358","https://openalex.org/W2964080999","https://openalex.org/W2964184826","https://openalex.org/W2964233199","https://openalex.org/W2966030703","https://openalex.org/W2996320484","https://openalex.org/W3098857864","https://openalex.org/W3099855006","https://openalex.org/W3105841952","https://openalex.org/W3106022083","https://openalex.org/W3118608800","https://openalex.org/W4230367971","https://openalex.org/W4240807122","https://openalex.org/W4248721357","https://openalex.org/W4297749952","https://openalex.org/W4300972832","https://openalex.org/W4362508137","https://openalex.org/W6604542330","https://openalex.org/W6637108112","https://openalex.org/W6637373629","https://openalex.org/W6637886330","https://openalex.org/W6640571472","https://openalex.org/W6670682178","https://openalex.org/W6680940449","https://openalex.org/W6682686508","https://openalex.org/W6684191040","https://openalex.org/W6703720303","https://openalex.org/W6723181079","https://openalex.org/W6726275242","https://openalex.org/W6732990564","https://openalex.org/W6743688258","https://openalex.org/W6748205129","https://openalex.org/W6752757957","https://openalex.org/W6754984521","https://openalex.org/W6757512965","https://openalex.org/W6787972765","https://openalex.org/W7053133691"],"related_works":["https://openalex.org/W2018264451","https://openalex.org/W3196977684","https://openalex.org/W2591253179","https://openalex.org/W3104367283","https://openalex.org/W4317464059","https://openalex.org/W4378757502","https://openalex.org/W2994793674","https://openalex.org/W4234882900","https://openalex.org/W4387349426","https://openalex.org/W2940361845"],"abstract_inverted_index":{"A":[0],"novel":[1],"functional":[2],"estimator":[3,62],"for":[4],"R\u00e9nyi's":[5],"\u03b1":[6],"-entropy":[7],"and":[8,39,49,87,95,109],"its":[9],"multivariate":[10],"extension":[11],"was":[12],"recently":[13],"proposed":[14],"in":[15,29,69,97],"terms":[16],"of":[17,21,25,42,66],"the":[18,26,37,81,93],"normalized":[19],"eigenspectrum":[20],"a":[22,30],"Hermitian":[23],"matrix":[24],"projected":[27],"data":[28,106],"reproducing":[31],"kernel":[32],"Hilbert":[33],"space":[34],"(RKHS).":[35],"However,":[36],"utility":[38],"possible":[40],"applications":[41],"these":[43],"new":[44,48],"estimators":[45],"are":[46],"rather":[47],"mostly":[50],"unknown":[51],"to":[52,91],"practitioners.":[53],"In":[54],"this":[55,61],"brief,":[56],"we":[57,79],"first":[58],"show":[59],"that":[60],"enables":[63],"straightforward":[64],"measurement":[65],"information":[67,83],"flow":[68],"realistic":[70],"convolutional":[71,98],"neural":[72],"networks":[73],"(CNNs)":[74],"without":[75],"any":[76],"approximation.":[77],"Then,":[78],"introduce":[80],"partial":[82],"decomposition":[84],"(PID)":[85],"framework":[86],"develop":[88],"three":[89],"quantities":[90],"analyze":[92],"synergy":[94],"redundancy":[96],"layer":[99],"representations.":[100],"Our":[101],"results":[102],"validate":[103],"two":[104],"fundamental":[105],"processing":[107],"inequalities":[108],"reveal":[110],"more":[111],"inner":[112],"properties":[113],"concerning":[114],"CNN":[115],"training.":[116]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":11},{"year":2022,"cited_by_count":15},{"year":2021,"cited_by_count":28},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":5}],"updated_date":"2026-04-13T07:58:08.660418","created_date":"2025-10-10T00:00:00"}