{"id":"https://openalex.org/W4403400928","doi":"https://doi.org/10.3390/sym16101363","title":"Enhancing Adversarial Robustness through Stable Adversarial Training","display_name":"Enhancing Adversarial Robustness through Stable Adversarial Training","publication_year":2024,"publication_date":"2024-10-14","ids":{"openalex":"https://openalex.org/W4403400928","doi":"https://doi.org/10.3390/sym16101363"},"language":"en","primary_location":{"id":"doi:10.3390/sym16101363","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16101363","pdf_url":"https://www.mdpi.com/2073-8994/16/10/1363/pdf?version=1728898013","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/16/10/1363/pdf?version=1728898013","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100615077","display_name":"Kun Yan","orcid":"https://orcid.org/0000-0003-1532-5935"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Kun Yan","raw_affiliation_strings":["Beijing Institute of Technology, Beijing 100081, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Beijing Institute of Technology, Beijing 100081, China","institution_ids":["https://openalex.org/I125839683"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017857828","display_name":"Luyi Yang","orcid":"https://orcid.org/0000-0003-2533-7470"},"institutions":[{"id":"https://openalex.org/I169689159","display_name":"PLA Information Engineering University","ror":"https://ror.org/00mm1qk40","country_code":"CN","type":"education","lineage":["https://openalex.org/I169689159"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Luyi Yang","raw_affiliation_strings":["Foreign Language Department, Information and Engineering University, Zhengzhou 450000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Foreign Language Department, Information and Engineering University, Zhengzhou 450000, China","institution_ids":["https://openalex.org/I169689159"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103281593","display_name":"Zhanpeng Yang","orcid":null},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210137199","display_name":"Aerospace Information Research Institute","ror":"https://ror.org/0419fj215","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210137199"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhanpeng Yang","raw_affiliation_strings":["The Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100190, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"The Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100190, China","institution_ids":["https://openalex.org/I4210137199","https://openalex.org/I19820366"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039672587","display_name":"Wenjuan Ren","orcid":"https://orcid.org/0009-0009-2950-1224"},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210137199","display_name":"Aerospace Information Research Institute","ror":"https://ror.org/0419fj215","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210137199"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Wenjuan Ren","raw_affiliation_strings":["The Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100190, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"The Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100190, China","institution_ids":["https://openalex.org/I4210137199","https://openalex.org/I19820366"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5039672587","https://openalex.org/A5103281593"],"corresponding_institution_ids":["https://openalex.org/I19820366","https://openalex.org/I4210137199"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.8335,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.77926868,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"16","issue":"10","first_page":"1363","last_page":"1363"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9998000264167786,"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/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9998000264167786,"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/T11242","display_name":"Nuclear Materials and Properties","score":0.9458000063896179,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.9390000104904175,"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/overfitting","display_name":"Overfitting","score":0.8466231822967529},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6368311643600464},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.6326776146888733},{"id":"https://openalex.org/keywords/adversarial-system","display_name":"Adversarial system","score":0.5779473781585693},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.5777885317802429},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.573797345161438},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5447716116905212},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5220860838890076},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.509933590888977},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.4502045214176178},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.33665746450424194},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.31983691453933716}],"concepts":[{"id":"https://openalex.org/C22019652","wikidata":"https://www.wikidata.org/wiki/Q331309","display_name":"Overfitting","level":3,"score":0.8466231822967529},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6368311643600464},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.6326776146888733},{"id":"https://openalex.org/C37736160","wikidata":"https://www.wikidata.org/wiki/Q1801315","display_name":"Adversarial system","level":2,"score":0.5779473781585693},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.5777885317802429},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.573797345161438},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5447716116905212},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5220860838890076},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.509933590888977},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.4502045214176178},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.33665746450424194},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.31983691453933716},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym16101363","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16101363","pdf_url":"https://www.mdpi.com/2073-8994/16/10/1363/pdf?version=1728898013","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:4648795261e94184a1436bc1b24dcaf1","is_oa":false,"landing_page_url":"https://doaj.org/article/4648795261e94184a1436bc1b24dcaf1","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":"Symmetry, Vol 16, Iss 10, p 1363 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym16101363","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16101363","pdf_url":"https://www.mdpi.com/2073-8994/16/10/1363/pdf?version=1728898013","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5783530182","display_name":null,"funder_award_id":"62131019","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320335892","display_name":"Youth Innovation Promotion Association","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4403400928.pdf","grobid_xml":"https://content.openalex.org/works/W4403400928.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W1505829478","https://openalex.org/W2003874621","https://openalex.org/W2037920050","https://openalex.org/W2146489275","https://openalex.org/W2972524765","https://openalex.org/W3038319179","https://openalex.org/W3101496892","https://openalex.org/W3118608800","https://openalex.org/W3135151133","https://openalex.org/W3191453585","https://openalex.org/W4200633541","https://openalex.org/W4221165940","https://openalex.org/W4225796928","https://openalex.org/W4226110825","https://openalex.org/W4285602371","https://openalex.org/W4303980686","https://openalex.org/W4306786696","https://openalex.org/W4317929775","https://openalex.org/W4366987820","https://openalex.org/W4386065396","https://openalex.org/W4386066141","https://openalex.org/W4402703010","https://openalex.org/W4402716031","https://openalex.org/W4402727757","https://openalex.org/W4402753786","https://openalex.org/W4402753982","https://openalex.org/W6767248609","https://openalex.org/W6774681163"],"related_works":["https://openalex.org/W4362597605","https://openalex.org/W1574414179","https://openalex.org/W4297676672","https://openalex.org/W3009056573","https://openalex.org/W2922073769","https://openalex.org/W4281702477","https://openalex.org/W2490526372","https://openalex.org/W4376166922","https://openalex.org/W4378510483","https://openalex.org/W3099765033"],"abstract_inverted_index":{"Deep":[0],"neural":[1,80,96,134,169],"network":[2,170],"models":[3],"are":[4],"vulnerable":[5],"to":[6,29,107,197,253],"attacks":[7,37],"from":[8,75],"adversarial":[9,33,40,45,210,216],"methods,":[10],"such":[11,83,266],"as":[12,84,110,267],"gradient":[13,36],"attacks.":[14],"Evening":[15],"small":[16],"perturbations":[17],"can":[18,174,221,257,277],"cause":[19],"significant":[20],"differences":[21],"in":[22,120],"their":[23,111],"predictions.":[24],"Adversarial":[25],"training":[26,46,100,112,201,211,217,246,284],"(AT)":[27],"aims":[28],"improve":[30,241,279],"the":[31,44,50,72,76,85,127,141,144,158,232,235,242,280],"model\u2019s":[32,243],"robustness":[34,244],"against":[35],"by":[38,180],"generating":[39],"samples":[41],"and":[42,62,65,116,160,178,186,227,239,245,251,255,269,283],"optimizing":[43],"objective":[47],"function":[48,182],"of":[49,78,126,129,143,166,234,286],"model.":[51],"Existing":[52],"methods":[53,276],"mainly":[54],"focus":[55],"on":[56,204],"improving":[57],"robust":[58,63,67,281],"accuracy,":[59],"balancing":[60],"natural":[61],"accuracy":[64,282],"suppressing":[66],"overfitting.":[68],"They":[69],"rarely":[70],"consider":[71],"AT":[73,229,264],"problem":[74],"characteristics":[77],"deep":[79,95,133,168],"networks":[81,97],"themselves,":[82],"stability":[86,131,137,173,215,225,285],"properties":[87,185],"under":[88],"certain":[89],"conditions.":[90],"From":[91],"a":[92,104,124,167,181,187,198],"mathematical":[93],"perspective,":[94],"with":[98,171,183,261],"stable":[99,200],"processes":[101],"may":[102],"have":[103],"better":[105],"ability":[106],"suppress":[108],"overfitting,":[109],"process":[113],"is":[114,249],"smoother":[115],"avoids":[117],"sudden":[118],"drops":[119],"performance.":[121],"We":[122],"provide":[123],"proof":[125],"existence":[128,142],"Ulam":[130,136,172,214,224],"for":[132,146],"networks.":[135],"not":[138],"only":[139],"determines":[140],"solution":[145],"an":[147,154,209],"operator":[148],"inequality,":[149],"but":[150],"it":[151,256],"also":[152],"provides":[153],"error":[155,189],"bound":[156],"between":[157],"exact":[159],"approximate":[161],"solutions.":[162],"The":[163],"feature":[164,194,237],"subspace":[165,195],"be":[175,258],"accurately":[176],"characterized":[177],"constrained":[179],"special":[184],"controlled":[188],"boundary":[190],"constant.":[191],"This":[192,219],"restricted":[193],"leads":[196],"more":[199],"process.":[202],"Based":[203],"these":[205],"properties,":[206],"we":[207],"propose":[208],"framework":[212,220],"called":[213],"(US-AT).":[218],"incorporate":[222],"different":[223],"conditions":[226],"benchmark":[228,287],"models,":[230,265],"optimize":[231],"construction":[233],"optimal":[236],"subspace,":[238],"consistently":[240,278],"stability.":[247],"US-AT":[248,275],"simple":[250],"easy":[252],"use,":[254],"easily":[259],"integrated":[260],"existing":[262],"multi-class":[263],"GradAlign":[268],"TRADES.":[270],"Experimental":[271],"results":[272],"show":[273],"that":[274],"models.":[288]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}