{"id":"https://openalex.org/W7117776993","doi":"https://doi.org/10.3390/e28010057","title":"Mitigating the Vanishing Gradient Problem Using a Pseudo-Normalizing Method","display_name":"Mitigating the Vanishing Gradient Problem Using a Pseudo-Normalizing Method","publication_year":2025,"publication_date":"2025-12-31","ids":{"openalex":"https://openalex.org/W7117776993","doi":"https://doi.org/10.3390/e28010057","pmid":"https://pubmed.ncbi.nlm.nih.gov/41593963"},"language":"en","primary_location":{"id":"doi:10.3390/e28010057","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28010057","pdf_url":"https://www.mdpi.com/1099-4300/28/1/57/pdf?version=1767189622","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/28/1/57/pdf?version=1767189622","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088867739","display_name":"Yun Bu","orcid":"https://orcid.org/0000-0003-0405-9512"},"institutions":[{"id":"https://openalex.org/I102345215","display_name":"Xihua University","ror":"https://ror.org/04gwtvf26","country_code":"CN","type":"education","lineage":["https://openalex.org/I102345215"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yun Bu","raw_affiliation_strings":["School of Electrical Engineering and Electronic Information, Xihua University, Chengdu 610039, China"],"affiliations":[{"raw_affiliation_string":"School of Electrical Engineering and Electronic Information, Xihua University, Chengdu 610039, China","institution_ids":["https://openalex.org/I102345215"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033430212","display_name":"Wenbo Jiang","orcid":null},"institutions":[{"id":"https://openalex.org/I102345215","display_name":"Xihua University","ror":"https://ror.org/04gwtvf26","country_code":"CN","type":"education","lineage":["https://openalex.org/I102345215"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wenbo Jiang","raw_affiliation_strings":["School of Electrical Engineering and Electronic Information, Xihua University, Chengdu 610039, China"],"affiliations":[{"raw_affiliation_string":"School of Electrical Engineering and Electronic Information, Xihua University, Chengdu 610039, China","institution_ids":["https://openalex.org/I102345215"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036178036","display_name":"Gang Lu","orcid":"https://orcid.org/0000-0002-7406-8695"},"institutions":[{"id":"https://openalex.org/I102345215","display_name":"Xihua University","ror":"https://ror.org/04gwtvf26","country_code":"CN","type":"education","lineage":["https://openalex.org/I102345215"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Gang Lu","raw_affiliation_strings":["School of Electrical Engineering and Electronic Information, Xihua University, Chengdu 610039, China"],"affiliations":[{"raw_affiliation_string":"School of Electrical Engineering and Electronic Information, Xihua University, Chengdu 610039, China","institution_ids":["https://openalex.org/I102345215"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066303903","display_name":"Q. Zhang","orcid":null},"institutions":[{"id":"https://openalex.org/I165745306","display_name":"Southwest Petroleum University","ror":"https://ror.org/03h17x602","country_code":"CN","type":"education","lineage":["https://openalex.org/I165745306"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qiang Zhang","raw_affiliation_strings":["School of Sciences, Southwest Petroleum University, Chengdu 610500, China"],"affiliations":[{"raw_affiliation_string":"School of Sciences, Southwest Petroleum University, Chengdu 610500, China","institution_ids":["https://openalex.org/I165745306"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5088867739"],"corresponding_institution_ids":["https://openalex.org/I102345215"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.68182426,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"28","issue":"1","first_page":"57","last_page":"57"},"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":0.4223000109195709,"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":0.4223000109195709,"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":0.13169999420642853,"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/T12417","display_name":"Morphological variations and asymmetry","score":0.02280000038444996,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/interpretability","display_name":"Interpretability","score":0.8878999948501587},{"id":"https://openalex.org/keywords/tangent","display_name":"Tangent","score":0.7224000096321106},{"id":"https://openalex.org/keywords/hyperbolic-function","display_name":"Hyperbolic function","score":0.6306999921798706},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.6200000047683716},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.6061000227928162},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5795000195503235},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.5540000200271606},{"id":"https://openalex.org/keywords/amplitude","display_name":"Amplitude","score":0.5317999720573425}],"concepts":[{"id":"https://openalex.org/C2781067378","wikidata":"https://www.wikidata.org/wiki/Q17027399","display_name":"Interpretability","level":2,"score":0.8878999948501587},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.7224000096321106},{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.6306999921798706},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.6200000047683716},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.6061000227928162},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5795000195503235},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.5540000200271606},{"id":"https://openalex.org/C180205008","wikidata":"https://www.wikidata.org/wiki/Q159190","display_name":"Amplitude","level":2,"score":0.5317999720573425},{"id":"https://openalex.org/C2011187","wikidata":"https://www.wikidata.org/wiki/Q383851","display_name":"Directional derivative","level":2,"score":0.4966000020503998},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.48010000586509705},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4787999987602234},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4578999876976013},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.44200000166893005},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.39890000224113464},{"id":"https://openalex.org/C126780896","wikidata":"https://www.wikidata.org/wiki/Q899871","display_name":"Distortion (music)","level":4,"score":0.3691999912261963},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.362199991941452},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.3544999957084656},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32179999351501465},{"id":"https://openalex.org/C33257320","wikidata":"https://www.wikidata.org/wiki/Q1932744","display_name":"Second derivative","level":2,"score":0.3124000132083893},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3009999990463257},{"id":"https://openalex.org/C17685861","wikidata":"https://www.wikidata.org/wiki/Q6960532","display_name":"Time derivative","level":2,"score":0.289000004529953},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C91873725","wikidata":"https://www.wikidata.org/wiki/Q3445816","display_name":"Function approximation","level":3,"score":0.26910001039505005},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.26739999651908875},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.25870001316070557}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e28010057","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28010057","pdf_url":"https://www.mdpi.com/1099-4300/28/1/57/pdf?version=1767189622","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:41593963","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/41593963","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":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:bdd28894a4674d76b18fe64fa0770a06","is_oa":true,"landing_page_url":"https://doaj.org/article/bdd28894a4674d76b18fe64fa0770a06","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 28, Iss 1, p 57 (2025)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:12839799","is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC12839799/","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e28010057","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28010057","pdf_url":"https://www.mdpi.com/1099-4300/28/1/57/pdf?version=1767189622","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7117776993.pdf","grobid_xml":"https://content.openalex.org/works/W7117776993.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"When":[0],"training":[1],"a":[2,18,48,61,132],"neural":[3],"network,":[4],"the":[5,23,26,33,58,70,82,96,104,136,144,160,164],"choice":[6],"of":[7,25,51,60,72,106,135,167],"activation":[8,45,126],"function":[9,16,46],"can":[10,55,176],"greatly":[11],"impact":[12],"its":[13],"performance.":[14],"A":[15],"with":[17,47,123],"larger":[19,99],"derivative":[20,49],"may":[21],"cause":[22],"coefficients":[24],"latter":[27],"layers":[28,92],"to":[29,41,74,93,118,142,153,181],"deviate":[30],"further":[31],"from":[32],"calculated":[34],"direction,":[35],"making":[36],"deep":[37,120],"learning":[38,121],"more":[39],"difficult":[40],"train.":[42],"However,":[43],"an":[44],"amplitude":[50],"less":[52],"than":[53,100],"one":[54],"result":[56],"in":[57,151,179],"problem":[59],"vanishing":[62,107],"gradient.":[63],"To":[64,130],"overcome":[65],"this":[66,116],"drawback,":[67],"we":[68,138],"propose":[69],"application":[71],"pseudo-normalization":[73],"enlarge":[75],"some":[76],"gradients":[77],"by":[78,81],"dividing":[79],"them":[80],"root":[83],"mean":[84],"square.":[85],"This":[86,171],"amplification":[87],"is":[88],"performed":[89],"every":[90],"few":[91],"ensure":[94],"that":[95,156,173],"amplitudes":[97],"are":[98],"one,":[101],"thus":[102],"avoiding":[103],"condition":[105],"gradient":[108,111],"and":[109],"preventing":[110],"explosion.":[112],"We":[113,148],"successfully":[114],"applied":[115],"approach":[117],"several":[119],"networks":[122,155,161],"hyperbolic":[124],"tangent":[125],"for":[127,169],"image":[128],"classifications.":[129],"gain":[131],"deeper":[133],"understanding":[134],"algorithm,":[137],"employed":[139],"interpretability":[140],"techniques":[141],"examine":[143],"network's":[145],"prediction":[146],"outcomes.":[147],"discovered":[149],"that,":[150],"contrast":[152],"popular":[154],"learn":[157],"picture":[158],"characteristics,":[159],"primarily":[162],"employ":[163],"contour":[165],"information":[166],"images":[168],"categorization.":[170],"suggests":[172],"our":[174],"technique":[175],"be":[177],"utilized":[178],"addition":[180],"other":[182],"widely":[183],"used":[184],"algorithms.":[185]},"counts_by_year":[],"updated_date":"2026-03-12T06:13:28.667946","created_date":"2025-12-31T00:00:00"}