{"id":"https://openalex.org/W3198519141","doi":"https://doi.org/10.1007/s00165-021-00553-4","title":"Verification of piecewise deep neural networks: a star set approach with zonotope pre-filter","display_name":"Verification of piecewise deep neural networks: a star set approach with zonotope pre-filter","publication_year":2021,"publication_date":"2021-08-01","ids":{"openalex":"https://openalex.org/W3198519141","doi":"https://doi.org/10.1007/s00165-021-00553-4","mag":"3198519141"},"language":"en","primary_location":{"id":"doi:10.1007/s00165-021-00553-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00165-021-00553-4","pdf_url":"https://dl.acm.org/doi/pdf/10.1007/s00165-021-00553-4","source":{"id":"https://openalex.org/S128250343","display_name":"Formal Aspects of Computing","issn_l":"0934-5043","issn":["0934-5043","1433-299X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formal Aspects of Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://dl.acm.org/doi/pdf/10.1007/s00165-021-00553-4","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023289374","display_name":"Hoang-Dung Tran","orcid":"https://orcid.org/0000-0001-6946-9526"},"institutions":[{"id":"https://openalex.org/I114395901","display_name":"University of Nebraska\u2013Lincoln","ror":"https://ror.org/043mer456","country_code":"US","type":"education","lineage":["https://openalex.org/I114395901"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Hoang-Dung Tran","raw_affiliation_strings":["University of Nebraska Lincoln, Lincoln, NE, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Nebraska Lincoln, Lincoln, NE, USA","institution_ids":["https://openalex.org/I114395901"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005381976","display_name":"Neelanjana Pal","orcid":"https://orcid.org/0000-0002-5978-8168"},"institutions":[{"id":"https://openalex.org/I200719446","display_name":"Vanderbilt University","ror":"https://ror.org/02vm5rt34","country_code":"US","type":"education","lineage":["https://openalex.org/I200719446"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Neelanjana Pal","raw_affiliation_strings":["Vanderbilt University, Nashville, TN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Vanderbilt University, Nashville, TN, USA","institution_ids":["https://openalex.org/I200719446"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021939745","display_name":"Diego Manzanas Lopez","orcid":"https://orcid.org/0000-0003-0721-1241"},"institutions":[{"id":"https://openalex.org/I200719446","display_name":"Vanderbilt University","ror":"https://ror.org/02vm5rt34","country_code":"US","type":"education","lineage":["https://openalex.org/I200719446"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Diego Manzanas Lopez","raw_affiliation_strings":["Vanderbilt University, Nashville, TN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Vanderbilt University, Nashville, TN, USA","institution_ids":["https://openalex.org/I200719446"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056925300","display_name":"Patrick Musau","orcid":null},"institutions":[{"id":"https://openalex.org/I200719446","display_name":"Vanderbilt University","ror":"https://ror.org/02vm5rt34","country_code":"US","type":"education","lineage":["https://openalex.org/I200719446"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Patrick Musau","raw_affiliation_strings":["Vanderbilt University, Nashville, TN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Vanderbilt University, Nashville, TN, USA","institution_ids":["https://openalex.org/I200719446"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005299838","display_name":"Xiaodong Yang","orcid":"https://orcid.org/0000-0003-1310-2615"},"institutions":[{"id":"https://openalex.org/I200719446","display_name":"Vanderbilt University","ror":"https://ror.org/02vm5rt34","country_code":"US","type":"education","lineage":["https://openalex.org/I200719446"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xiaodong Yang","raw_affiliation_strings":["Vanderbilt University, Nashville, TN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Vanderbilt University, Nashville, TN, USA","institution_ids":["https://openalex.org/I200719446"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051151589","display_name":"Luan Viet Nguyen","orcid":"https://orcid.org/0000-0001-5516-2443"},"institutions":[{"id":"https://openalex.org/I127591826","display_name":"University of Dayton","ror":"https://ror.org/021v3qy27","country_code":"US","type":"education","lineage":["https://openalex.org/I127591826"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Luan Viet Nguyen","raw_affiliation_strings":["University of Dayton, Dayton, OH, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Dayton, Dayton, OH, USA","institution_ids":["https://openalex.org/I127591826"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5090573416","display_name":"Weiming Xiang","orcid":"https://orcid.org/0000-0001-9065-8428"},"institutions":[{"id":"https://openalex.org/I25041050","display_name":"Augusta University","ror":"https://ror.org/012mef835","country_code":"US","type":"education","lineage":["https://openalex.org/I25041050"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Weiming Xiang","raw_affiliation_strings":["Augusta University, Augusta, GA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Augusta University, Augusta, GA, USA","institution_ids":["https://openalex.org/I25041050"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033330108","display_name":"Stanley Bak","orcid":"https://orcid.org/0000-0003-4947-9553"},"institutions":[{"id":"https://openalex.org/I59553526","display_name":"Stony Brook University","ror":"https://ror.org/05qghxh33","country_code":"US","type":"education","lineage":["https://openalex.org/I59553526"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Stanley Bak","raw_affiliation_strings":["Stony Brook University, Stony Brook, NY, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Stony Brook University, Stony Brook, NY, USA","institution_ids":["https://openalex.org/I59553526"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067901159","display_name":"Taylor T. Johnson","orcid":"https://orcid.org/0000-0001-8021-9923"},"institutions":[{"id":"https://openalex.org/I200719446","display_name":"Vanderbilt University","ror":"https://ror.org/02vm5rt34","country_code":"US","type":"education","lineage":["https://openalex.org/I200719446"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Taylor T. Johnson","raw_affiliation_strings":["Vanderbilt University, Nashville, TN, USA"],"raw_orcid":"https://orcid.org/0000-0001-8021-9923","affiliations":[{"raw_affiliation_string":"Vanderbilt University, Nashville, TN, USA","institution_ids":["https://openalex.org/I200719446"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":9,"corresponding_author_ids":["https://openalex.org/A5023289374"],"corresponding_institution_ids":["https://openalex.org/I114395901"],"apc_list":null,"apc_paid":null,"fwci":1.5391,"has_fulltext":true,"cited_by_count":17,"citation_normalized_percentile":{"value":0.86165743,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"33","issue":"4-5","first_page":"519","last_page":"545"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11689","display_name":"Adversarial Robustness in Machine Learning","score":1.0,"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":1.0,"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/T11005","display_name":"Radiation Effects in Electronics","score":0.9898999929428101,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T12122","display_name":"Physical Unclonable Functions (PUFs) and Hardware Security","score":0.9882000088691711,"subfield":{"id":"https://openalex.org/subfields/1708","display_name":"Hardware and Architecture"},"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/reachability","display_name":"Reachability","score":0.7262654304504395},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6769150495529175},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.5676814913749695},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.5553511381149292},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5251302719116211},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5075901746749878},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.35997849702835083},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.215508371591568}],"concepts":[{"id":"https://openalex.org/C136643341","wikidata":"https://www.wikidata.org/wiki/Q1361526","display_name":"Reachability","level":2,"score":0.7262654304504395},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6769150495529175},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.5676814913749695},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.5553511381149292},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5251302719116211},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5075901746749878},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.35997849702835083},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.215508371591568},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","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":1,"locations":[{"id":"doi:10.1007/s00165-021-00553-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00165-021-00553-4","pdf_url":"https://dl.acm.org/doi/pdf/10.1007/s00165-021-00553-4","source":{"id":"https://openalex.org/S128250343","display_name":"Formal Aspects of Computing","issn_l":"0934-5043","issn":["0934-5043","1433-299X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formal Aspects of Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00165-021-00553-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00165-021-00553-4","pdf_url":"https://dl.acm.org/doi/pdf/10.1007/s00165-021-00553-4","source":{"id":"https://openalex.org/S128250343","display_name":"Formal Aspects of Computing","issn_l":"0934-5043","issn":["0934-5043","1433-299X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formal Aspects of Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1470115129","display_name":null,"funder_award_id":"FA8750-18-C-0089","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G1523888516","display_name":null,"funder_award_id":"FA9550-","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G36914211","display_name":null,"funder_award_id":"FA9550-18-1-0122","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G4713059963","display_name":null,"funder_award_id":"FA8750","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G483564692","display_name":null,"funder_award_id":"FA9550-19-1-0288","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G5809100787","display_name":null,"funder_award_id":"FA9550","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G7377016339","display_name":null,"funder_award_id":"1910017","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320332815","display_name":"Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3198519141.pdf","grobid_xml":"https://content.openalex.org/works/W3198519141.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W1511213320","https://openalex.org/W2160815625","https://openalex.org/W2165073069","https://openalex.org/W2243397390","https://openalex.org/W2543296129","https://openalex.org/W2565516711","https://openalex.org/W2592929672","https://openalex.org/W2594877703","https://openalex.org/W2721006554","https://openalex.org/W2734705105","https://openalex.org/W2791251367","https://openalex.org/W2794609696","https://openalex.org/W2895120768","https://openalex.org/W2896500468","https://openalex.org/W2899156512","https://openalex.org/W2900153411","https://openalex.org/W2902630996","https://openalex.org/W2905299363","https://openalex.org/W2955364893","https://openalex.org/W2957311447","https://openalex.org/W2962692913","https://openalex.org/W2963054787","https://openalex.org/W2963207607","https://openalex.org/W2974147052","https://openalex.org/W3042471904","https://openalex.org/W3043655933","https://openalex.org/W3046272654","https://openalex.org/W3046662910","https://openalex.org/W3103148066","https://openalex.org/W3186627997"],"related_works":["https://openalex.org/W381935118","https://openalex.org/W2573691487","https://openalex.org/W2368796975","https://openalex.org/W2364921833","https://openalex.org/W1486100501","https://openalex.org/W2302028273","https://openalex.org/W4299419634","https://openalex.org/W1525643724","https://openalex.org/W1534259281","https://openalex.org/W2067938758"],"abstract_inverted_index":{"Abstract":[0],"Verification":[1],"has":[2],"emerged":[3],"as":[4,224],"a":[5,26,49,77,80,90,119,147,181,201,247],"means":[6],"to":[7,134,150,180,214,241],"provide":[8],"formal":[9],"guarantees":[10],"on":[11],"learning-based":[12],"systems":[13],"incorporating":[14],"neural":[15,32,91,110,206],"network":[16,78,92,207],"before":[17,84],"using":[18,40,54,85],"them":[19,86],"in":[20,175,183,200],"safety-critical":[21],"applications.":[22],"This":[23,158],"paper":[24],"proposes":[25],"new":[27],"verification":[28,208,228],"approach":[29,47,99,236,255],"for":[30,87],"deep":[31],"networks":[33],"(DNNs)":[34],"with":[35,79,93,124],"piecewise":[36,94],"linear":[37,95,167],"activation":[38,96],"functions":[39],"reachability":[41,52,69,196],"analysis.":[42],"The":[43,67],"core":[44],"of":[45,51,64,76,108,116,130,142,166,190,219,229],"our":[46,98,117,235],"is":[48,122,256],"collection":[50],"algorithms":[53,70,197],"star":[55,120,132,192,253],"sets":[56,75,107],"(or":[57],"shortly,":[58],"stars),":[59],"an":[60,125,143],"effective":[61],"symbolic":[62],"representation":[63],"high-dimensional":[65],"polytopes.":[66],"star-based":[68],"compute":[71],"the":[72,109,114,131,137,164,176,188,191,205,217],"output":[73],"reachable":[74,106],"given":[81],"input":[82,144],"set":[83,121,145,193,254],"verification.":[88],"For":[89],"functions,":[97],"can":[100,211,237],"construct":[101],"both":[102],"exact":[103],"and":[104,139,178,210,226,264],"over-approximate":[105],"network.":[111],"To":[112],"enhance":[113],"scalability":[115,189],"approach,":[118],"equipped":[123],"outer-zonotope":[126],"(a":[127],"zonotope":[128,159,263],"over-approximation":[129],"set)":[133],"quickly":[135],"estimate":[136],"lower":[138],"upper":[140],"bounds":[141],"at":[146,155],"specific":[148],"neuron":[149],"determine":[151],"if":[152],"splitting":[153],"occurs":[154],"that":[156,171,234],"neuron.":[157],"pre-filtering":[160],"step":[161],"reduces":[162],"significantly":[163],"number":[165],"programming":[168],"optimization":[169],"problems":[170,215],"must":[172],"be":[173,212],"solved":[174],"analysis,":[177],"leads":[179],"reduction":[182],"computation":[184],"time,":[185],"which":[186],"enhances":[187],"approach.":[194,251],"Our":[195,231,252],"are":[198],"implemented":[199],"software":[202],"prototype":[203],"called":[204],"tool,":[209],"applied":[213],"analyzing":[216],"robustness":[218,227],"machine":[220],"learning":[221],"methods,":[222],"such":[223],"safety":[225],"DNNs.":[230],"experiments":[232],"show":[233],"achieve":[238],"runtimes":[239],"twenty":[240],"1400":[242],"times":[243],"faster":[244],"than":[245,260],"Reluplex,":[246],"satisfiability":[248],"modulo":[249],"theory-based":[250],"also":[257],"less":[258],"conservative":[259],"other":[261],"recent":[262],"abstract":[265],"domain":[266],"approaches.":[267]},"counts_by_year":[{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":3}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
