{"id":"https://openalex.org/W4312838810","doi":"https://doi.org/10.1109/tr.2022.3209081","title":"Output Range Analysis for Feed-Forward Deep Neural Networks via Linear Programming","display_name":"Output Range Analysis for Feed-Forward Deep Neural Networks via Linear Programming","publication_year":2022,"publication_date":"2022-10-04","ids":{"openalex":"https://openalex.org/W4312838810","doi":"https://doi.org/10.1109/tr.2022.3209081"},"language":"en","primary_location":{"id":"doi:10.1109/tr.2022.3209081","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tr.2022.3209081","pdf_url":null,"source":{"id":"https://openalex.org/S87725633","display_name":"IEEE Transactions on Reliability","issn_l":"0018-9529","issn":["0018-9529","1558-1721"],"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 Reliability","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"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/A5089943764","display_name":"Zhiwu Xu","orcid":"https://orcid.org/0000-0001-6727-440X"},"institutions":[{"id":"https://openalex.org/I180726961","display_name":"Shenzhen University","ror":"https://ror.org/01vy4gh70","country_code":"CN","type":"education","lineage":["https://openalex.org/I180726961"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhiwu Xu","raw_affiliation_strings":["College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China"],"affiliations":[{"raw_affiliation_string":"College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China","institution_ids":["https://openalex.org/I180726961"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103097958","display_name":"Yazheng Liu","orcid":"https://orcid.org/0009-0005-5573-6861"},"institutions":[{"id":"https://openalex.org/I180726961","display_name":"Shenzhen University","ror":"https://ror.org/01vy4gh70","country_code":"CN","type":"education","lineage":["https://openalex.org/I180726961"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yazheng Liu","raw_affiliation_strings":["College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China"],"affiliations":[{"raw_affiliation_string":"College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China","institution_ids":["https://openalex.org/I180726961"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042013137","display_name":"Shengchao Qin","orcid":"https://orcid.org/0000-0003-3028-8191"},"institutions":[{"id":"https://openalex.org/I180726961","display_name":"Shenzhen University","ror":"https://ror.org/01vy4gh70","country_code":"CN","type":"education","lineage":["https://openalex.org/I180726961"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shengchao Qin","raw_affiliation_strings":["College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China"],"affiliations":[{"raw_affiliation_string":"College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China","institution_ids":["https://openalex.org/I180726961"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100633973","display_name":"Zhong Ming","orcid":"https://orcid.org/0000-0001-9310-3460"},"institutions":[{"id":"https://openalex.org/I180726961","display_name":"Shenzhen University","ror":"https://ror.org/01vy4gh70","country_code":"CN","type":"education","lineage":["https://openalex.org/I180726961"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhong Ming","raw_affiliation_strings":["College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China"],"affiliations":[{"raw_affiliation_string":"College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China","institution_ids":["https://openalex.org/I180726961"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5089943764"],"corresponding_institution_ids":["https://openalex.org/I180726961"],"apc_list":null,"apc_paid":null,"fwci":0.5516,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.72931744,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"72","issue":"3","first_page":"1191","last_page":"1205"},"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/T10036","display_name":"Advanced Neural Network Applications","score":0.9926999807357788,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9868999719619751,"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/artificial-neural-network","display_name":"Artificial neural network","score":0.5641193389892578},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.5507241487503052},{"id":"https://openalex.org/keywords/range","display_name":"Range (aeronautics)","score":0.5040313601493835},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4394882023334503},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42369717359542847},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.41475093364715576},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4066261053085327},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.38371413946151733},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.34355658292770386},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.33397188782691956},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.24244201183319092},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.24008405208587646}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5641193389892578},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.5507241487503052},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.5040313601493835},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4394882023334503},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42369717359542847},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.41475093364715576},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4066261053085327},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.38371413946151733},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34355658292770386},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.33397188782691956},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.24244201183319092},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.24008405208587646},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tr.2022.3209081","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tr.2022.3209081","pdf_url":null,"source":{"id":"https://openalex.org/S87725633","display_name":"IEEE Transactions on Reliability","issn_l":"0018-9529","issn":["0018-9529","1558-1721"],"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 Reliability","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2037522653","display_name":null,"funder_award_id":"61972260","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7097245526","display_name":null,"funder_award_id":"61836005","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7219935079","display_name":null,"funder_award_id":"2019A1515011577","funder_id":"https://openalex.org/F4320337111","funder_display_name":"Basic and Applied Basic Research Foundation of Guangdong Province"},{"id":"https://openalex.org/G7803258927","display_name":null,"funder_award_id":"61772347","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/F4320337111","display_name":"Basic and Applied Basic Research Foundation of Guangdong Province","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":66,"referenced_works":["https://openalex.org/W1673923490","https://openalex.org/W1918816242","https://openalex.org/W2118023920","https://openalex.org/W2130942839","https://openalex.org/W2165073069","https://openalex.org/W2296673577","https://openalex.org/W2543296129","https://openalex.org/W2594877703","https://openalex.org/W2618099328","https://openalex.org/W2721006554","https://openalex.org/W2786163515","https://openalex.org/W2789524546","https://openalex.org/W2791251367","https://openalex.org/W2794609696","https://openalex.org/W2798356176","https://openalex.org/W2800473699","https://openalex.org/W2889490035","https://openalex.org/W2890472662","https://openalex.org/W2891698621","https://openalex.org/W2895020057","https://openalex.org/W2896500468","https://openalex.org/W2900153411","https://openalex.org/W2917875722","https://openalex.org/W2919234322","https://openalex.org/W2921786109","https://openalex.org/W2950029838","https://openalex.org/W2950183737","https://openalex.org/W2962692913","https://openalex.org/W2963054787","https://openalex.org/W2963285578","https://openalex.org/W2963334980","https://openalex.org/W2963440492","https://openalex.org/W2963496101","https://openalex.org/W2963998105","https://openalex.org/W2964130946","https://openalex.org/W2980264580","https://openalex.org/W2986305485","https://openalex.org/W2998709064","https://openalex.org/W3042779836","https://openalex.org/W3090972680","https://openalex.org/W3103148066","https://openalex.org/W3105558219","https://openalex.org/W3176946506","https://openalex.org/W4205950066","https://openalex.org/W4206095852","https://openalex.org/W4294349862","https://openalex.org/W4301187031","https://openalex.org/W6637162671","https://openalex.org/W6679436768","https://openalex.org/W6685562342","https://openalex.org/W6711870810","https://openalex.org/W6740406300","https://openalex.org/W6748277150","https://openalex.org/W6750745550","https://openalex.org/W6751044371","https://openalex.org/W6751161574","https://openalex.org/W6751834733","https://openalex.org/W6754543342","https://openalex.org/W6754619240","https://openalex.org/W6754681091","https://openalex.org/W6754899549","https://openalex.org/W6759698669","https://openalex.org/W6760457120","https://openalex.org/W6765694979","https://openalex.org/W6767570850","https://openalex.org/W6783637622"],"related_works":["https://openalex.org/W3110577345","https://openalex.org/W2504004674","https://openalex.org/W2946235959","https://openalex.org/W2902935192","https://openalex.org/W3134817226","https://openalex.org/W3198895633","https://openalex.org/W3215912596","https://openalex.org/W3034021666","https://openalex.org/W2898900925","https://openalex.org/W3170224572"],"abstract_inverted_index":{"The":[0,97,161],"success":[1],"of":[2,20,31,123,132],"deep":[3,21,91],"neural":[4,22,32,92],"networks":[5,33,93,179,203,228,243],"and":[6,112,129,136,180,211,217,221,263],"their":[7],"potential":[8],"use":[9],"in":[10,121,144],"many":[11],"safety-critical":[12],"applications":[13],"has":[14],"motivated":[15],"research":[16],"on":[17,41,47,140,174,198,223],"formal":[18,29],"verification":[19],"networks.":[23,185,291],"A":[24,148],"fundamental":[25],"primitive":[26],"enabling":[27],"the":[28,35,86,103,124,127,133,137,145,152,270,284],"analysis":[30,44],"is":[34,100,159],"output":[36,42,87],"range":[37,43,88,135,158],"analysis.":[38],"Existing":[39],"approaches":[40],"either":[45],"focus":[46],"some":[48,64,141],"simple":[49],"activation":[50,65,104],"functions,":[51,66,105],"such":[52,67,106],"as":[53,68,107,118,279,281],"<inline-formula":[54,72,108,113,175,181,199,205,212,224,230,239,245,287],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[55,73,109,114,176,182,200,206,213,225,231,240,246,288],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><tex-math":[56,74,110,115,177,183,201,207,214,226,232,241,247,289],"notation=\"LaTeX\">$\\text{relu,}$</tex-math></inline-formula>":[57],"or":[58,236],"compute":[59,85],"a":[60,156,169,276],"relaxed":[61],"result":[62,171,278],"for":[63,89,286],"exponential":[69],"linear":[70,95,119,193],"unit":[71],"notation=\"LaTeX\">$\\text{({elu}}$</tex-math></inline-formula>":[75],").":[76],"In":[77],"this":[78],"article,":[79],"we":[80,266],"propose":[81],"an":[82],"approach":[83,167,188,253],"to":[84,101,150,154,260,274,282],"feed-forward":[90],"via":[94],"programming.":[96],"key":[98],"idea":[99],"encode":[102],"notation=\"LaTeX\">$\\text{{elu}}$</tex-math></inline-formula>":[111,178,202,227,242,290],"notation=\"LaTeX\">$\\text{sigmoid}$</tex-math></inline-formula>":[116,184,215],",":[117,251],"constraints":[120],"term":[122],"line":[125],"between":[126],"left":[128],"right":[130],"end-points":[131],"input":[134,146],"tangent":[138],"lines":[139],"special":[142],"points":[143],"range.":[147],"strategy":[149],"partition":[151,272],"network":[153,271],"get":[155],"tighter":[157,170,277],"presented.":[160],"experimental":[162],"results":[163,256],"show":[164],"that":[165,257,269],"our":[166,187,252],"gets":[168],"than":[172,191,219],"RobustVerifier":[173],"Moreover,":[186],"performs":[189],"better":[190,218],"(the":[192],"encodings":[194],"implemented":[195],"in)":[196],"Crown":[197],"with":[204,229,244],"notation=\"LaTeX\">$\\alpha":[208,233,248],"=0.5,":[209],"1.0$</tex-math></inline-formula>":[210],"networks,":[216],"CNN-Cert":[220],"DeepCert":[222],"=":[234,249],"0.5$</tex-math></inline-formula>":[235],"1.0.":[237],"For":[238],"2.0$</tex-math></inline-formula>":[250],"can":[254],"achieve":[255,275],"are":[258],"closed":[259],"Crown,":[261],"CNN-Cert,":[262],"DeepCert.":[264],"Finally,":[265],"also":[267],"found":[268],"helps":[273],"well":[280],"improve":[283],"efficiency":[285]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":2}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}