{"id":"https://openalex.org/W4313245158","doi":"https://doi.org/10.3390/computers12010004","title":"Batch Gradient Learning Algorithm with Smoothing L1 Regularization for Feedforward Neural Networks","display_name":"Batch Gradient Learning Algorithm with Smoothing L1 Regularization for Feedforward Neural Networks","publication_year":2022,"publication_date":"2022-12-23","ids":{"openalex":"https://openalex.org/W4313245158","doi":"https://doi.org/10.3390/computers12010004"},"language":"en","primary_location":{"id":"doi:10.3390/computers12010004","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computers12010004","pdf_url":"https://www.mdpi.com/2073-431X/12/1/4/pdf?version=1672296495","source":{"id":"https://openalex.org/S4210228075","display_name":"Computers","issn_l":"2073-431X","issn":["2073-431X"],"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":"Computers","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-431X/12/1/4/pdf?version=1672296495","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5075390658","display_name":"Khidir Shaib Mohamed","orcid":"https://orcid.org/0000-0001-8309-6121"},"institutions":[{"id":"https://openalex.org/I156216236","display_name":"Qassim University","ror":"https://ror.org/01wsfe280","country_code":"SA","type":"education","lineage":["https://openalex.org/I156216236"]},{"id":"https://openalex.org/I4210113743","display_name":"Dalanj University","ror":"https://ror.org/02ayk5126","country_code":"SD","type":"education","lineage":["https://openalex.org/I4210113743"]}],"countries":["SA","SD"],"is_corresponding":true,"raw_author_name":"Khidir Shaib Mohamed","raw_affiliation_strings":["Department of Mathematics and Computer, College of Science, Dalanj University, Dilling P.O. Box 14, Sudan","Department of Mathematics, College of Sciences and Arts in Uglat Asugour, Qassim University, Buraydah 51452, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-8309-6121","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer, College of Science, Dalanj University, Dilling P.O. Box 14, Sudan","institution_ids":["https://openalex.org/I4210113743"]},{"raw_affiliation_string":"Department of Mathematics, College of Sciences and Arts in Uglat Asugour, Qassim University, Buraydah 51452, Saudi Arabia","institution_ids":["https://openalex.org/I156216236"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5075390658"],"corresponding_institution_ids":["https://openalex.org/I156216236","https://openalex.org/I4210113743"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1732},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1732},"fwci":1.3873,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.84680112,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"12","issue":"1","first_page":"4","last_page":"4"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12676","display_name":"Machine Learning and ELM","score":0.9998999834060669,"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/T12676","display_name":"Machine Learning and ELM","score":0.9998999834060669,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10057","display_name":"Face and Expression Recognition","score":0.996399998664856,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.683838963508606},{"id":"https://openalex.org/keywords/smoothing","display_name":"Smoothing","score":0.6553592681884766},{"id":"https://openalex.org/keywords/proximal-gradient-methods-for-learning","display_name":"Proximal gradient methods for learning","score":0.5965387225151062},{"id":"https://openalex.org/keywords/regularization-perspectives-on-support-vector-machines","display_name":"Regularization perspectives on support vector machines","score":0.5914492607116699},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5357460379600525},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5310423970222473},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5264576077461243},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5058848261833191},{"id":"https://openalex.org/keywords/overfitting","display_name":"Overfitting","score":0.5034796595573425},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4812428951263428},{"id":"https://openalex.org/keywords/early-stopping","display_name":"Early stopping","score":0.4533804953098297},{"id":"https://openalex.org/keywords/feedforward-neural-network","display_name":"Feedforward neural network","score":0.42999914288520813},{"id":"https://openalex.org/keywords/backus\u2013gilbert-method","display_name":"Backus\u2013Gilbert method","score":0.4212341904640198},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.41056495904922485},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.38412514328956604},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3326685428619385},{"id":"https://openalex.org/keywords/tikhonov-regularization","display_name":"Tikhonov regularization","score":0.16692444682121277},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.11880818009376526}],"concepts":[{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.683838963508606},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.6553592681884766},{"id":"https://openalex.org/C79248915","wikidata":"https://www.wikidata.org/wiki/Q17086776","display_name":"Proximal gradient methods for learning","level":5,"score":0.5965387225151062},{"id":"https://openalex.org/C141718189","wikidata":"https://www.wikidata.org/wiki/Q7309628","display_name":"Regularization perspectives on support vector machines","level":4,"score":0.5914492607116699},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5357460379600525},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5310423970222473},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5264576077461243},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5058848261833191},{"id":"https://openalex.org/C22019652","wikidata":"https://www.wikidata.org/wiki/Q331309","display_name":"Overfitting","level":3,"score":0.5034796595573425},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4812428951263428},{"id":"https://openalex.org/C5465570","wikidata":"https://www.wikidata.org/wiki/Q5326898","display_name":"Early stopping","level":3,"score":0.4533804953098297},{"id":"https://openalex.org/C47702885","wikidata":"https://www.wikidata.org/wiki/Q5441227","display_name":"Feedforward neural network","level":3,"score":0.42999914288520813},{"id":"https://openalex.org/C27872270","wikidata":"https://www.wikidata.org/wiki/Q4839810","display_name":"Backus\u2013Gilbert method","level":5,"score":0.4212341904640198},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.41056495904922485},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.38412514328956604},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3326685428619385},{"id":"https://openalex.org/C152442038","wikidata":"https://www.wikidata.org/wiki/Q2778212","display_name":"Tikhonov regularization","level":3,"score":0.16692444682121277},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.11880818009376526},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/computers12010004","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computers12010004","pdf_url":"https://www.mdpi.com/2073-431X/12/1/4/pdf?version=1672296495","source":{"id":"https://openalex.org/S4210228075","display_name":"Computers","issn_l":"2073-431X","issn":["2073-431X"],"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":"Computers","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:a653afa5d78b402097e464de3f641831","is_oa":false,"landing_page_url":"https://doaj.org/article/a653afa5d78b402097e464de3f641831","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Computers, Vol 12, Iss 1, p 4 (2022)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/computers12010004","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computers12010004","pdf_url":"https://www.mdpi.com/2073-431X/12/1/4/pdf?version=1672296495","source":{"id":"https://openalex.org/S4210228075","display_name":"Computers","issn_l":"2073-431X","issn":["2073-431X"],"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":"Computers","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4313245158.pdf"},"referenced_works_count":43,"referenced_works":["https://openalex.org/W115006267","https://openalex.org/W2001172396","https://openalex.org/W2013640190","https://openalex.org/W2030302676","https://openalex.org/W2034978228","https://openalex.org/W2041891835","https://openalex.org/W2073730966","https://openalex.org/W2080210651","https://openalex.org/W2088350412","https://openalex.org/W2095705004","https://openalex.org/W2111719156","https://openalex.org/W2129129079","https://openalex.org/W2129131372","https://openalex.org/W2135046866","https://openalex.org/W2145085734","https://openalex.org/W2165780329","https://openalex.org/W2168728319","https://openalex.org/W2188581537","https://openalex.org/W2213612645","https://openalex.org/W2316144570","https://openalex.org/W2334522350","https://openalex.org/W2546700349","https://openalex.org/W2726526844","https://openalex.org/W2730619644","https://openalex.org/W2766826419","https://openalex.org/W2790029018","https://openalex.org/W2897028039","https://openalex.org/W2928762566","https://openalex.org/W2955823899","https://openalex.org/W2969534524","https://openalex.org/W3005798491","https://openalex.org/W3093743744","https://openalex.org/W3148590207","https://openalex.org/W3159243351","https://openalex.org/W3172971963","https://openalex.org/W4206569030","https://openalex.org/W4243367687","https://openalex.org/W4256530643","https://openalex.org/W4283703912","https://openalex.org/W6604730130","https://openalex.org/W6674330103","https://openalex.org/W6761286452","https://openalex.org/W6837694383"],"related_works":["https://openalex.org/W1977073705","https://openalex.org/W2807240017","https://openalex.org/W2183525982","https://openalex.org/W2366023887","https://openalex.org/W2059130945","https://openalex.org/W2805921369","https://openalex.org/W2020951562","https://openalex.org/W2155619582","https://openalex.org/W2896323223","https://openalex.org/W1481686068"],"abstract_inverted_index":{"Regularization":[0],"techniques":[1],"are":[2,17,145],"critical":[3],"in":[4,46,110,156],"the":[5,27,33,52,55,58,114,120,123,128,139,163,186,193,196,199,206,210,213],"development":[6],"of":[7,54,116,195,198,212],"machine":[8],"learning":[9,81,89],"models.":[10],"Complex":[11],"models,":[12],"such":[13],"as":[14],"neural":[15,94],"networks,":[16],"particularly":[18],"prone":[19],"to":[20,23,37,51,112],"overfitting":[21],"and":[22,65,90,132,141,152,168,178,191,209],"performing":[24],"poorly":[25],"on":[26],"training":[28],"data.":[29],"L1":[30,59,85,117],"regularization":[31,60,86,118],"is":[32],"most":[34],"extreme":[35],"way":[36],"enforce":[38],"sparsity,":[39],"but,":[40],"regrettably,":[41],"it":[42],"does":[43],"not":[44],"result":[45],"an":[47],"NP-hard":[48],"problem":[49],"due":[50],"non-differentiability":[53,115],"1-norm.":[56],"However,":[57],"term":[61],"achieved":[62],"convergence":[63,124,143,166],"speed":[64,125],"efficiency":[66],"optimization":[67],"solution":[68],"through":[69],"a":[70,78,92,106,181],"proximal":[71],"method.":[72],"In":[73],"this":[74,137],"paper,":[75],"we":[76,104,183],"propose":[77,105],"batch":[79],"gradient":[80,197],"algorithm":[82],"with":[83,96,174],"smoothing":[84,107],"(BGSL1)":[87],"for":[88],"pruning":[91],"feedforward":[93],"network":[95,129],"hidden":[97],"nodes.":[98],"To":[99],"achieve":[100],"our":[101,157],"study":[102],"purpose,":[103],"(differentiable)":[108],"function":[109,153,188,201],"order":[111],"address":[113],"at":[119],"origin,":[121],"make":[122],"faster,":[126],"improve":[127],"structure":[130],"ability,":[131],"build":[133],"stronger":[134],"mapping.":[135],"Under":[136],"condition,":[138],"strong":[140],"weak":[142],"theorems":[144],"provided.":[146],"We":[147],"used":[148],"N-dimensional":[149],"parity":[150],"problems":[151,155],"approximation":[154],"experiments.":[158],"Preliminary":[159],"findings":[160],"indicate":[161],"that":[162,185,192],"BGSL1":[164],"has":[165],"faster":[167],"good":[169],"generalization":[170],"abilities":[171],"when":[172],"compared":[173],"BGL1/2,":[175],"BGL1,":[176],"BGL2,":[177],"BGSL1/2.":[179],"As":[180],"result,":[182],"demonstrate":[184],"error":[187,200],"decreases":[189],"monotonically":[190],"norm":[194],"approaches":[202],"zero,":[203],"thereby":[204],"validating":[205],"theoretical":[207],"finding":[208],"supremacy":[211],"suggested":[214],"technique.":[215]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":5}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2023-01-06T00:00:00"}
