{"id":"https://openalex.org/W7204251246","doi":"https://doi.org/10.48550/arxiv.2608.24865","title":"Parameterized Complexity of $L_p$-Lipschitz Constants for Input Convex Neural Networks and $L_p$-Norm Maximization over Zonotopes","display_name":"Parameterized Complexity of $L_p$-Lipschitz Constants for Input Convex Neural Networks and $L_p$-Norm Maximization over Zonotopes","publication_year":2026,"publication_date":"2026-08-25","ids":{"openalex":"https://openalex.org/W7204251246","doi":"https://doi.org/10.48550/arxiv.2608.24865"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.24865","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24865","pdf_url":"https://arxiv.org/pdf/2608.24865","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.24865","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5126822932","display_name":"Aritra Das","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I347237974","display_name":"Ashoka University","ror":"https://ror.org/02j1xr113","country_code":"IN","type":"education","lineage":["https://openalex.org/I347237974"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Das, Aritra","raw_affiliation_strings":["Ashoka University , Truth Audit Labs {aritra. das,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ashoka University , Truth Audit Labs {aritra. das,","institution_ids":["https://openalex.org/I347237974"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042414939","display_name":"Vincent Froese","orcid":"https://orcid.org/0000-0002-8499-0130","observed_orcids":["https://orcid.org/0000-0002-8499-0130"]},"institutions":[{"id":"https://openalex.org/I4577782","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40","country_code":"DE","type":"education","lineage":["https://openalex.org/I4577782"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Froese, Vincent","raw_affiliation_strings":["Technische Universit\u00e4t Berlin ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Technische Universit\u00e4t Berlin ,","institution_ids":["https://openalex.org/I4577782"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5114482385","display_name":"Moritz Grillo","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210119712","display_name":"Max Planck Institute for Mathematics","ror":"https://ror.org/02dh8ja68","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210119712"]},{"id":"https://openalex.org/I4210142348","display_name":"Max Planck Institute for the History of Science","ror":"https://ror.org/0492sjc74","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210142348"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Grillo, Moritz","raw_affiliation_strings":["Max Planck Institute for Mathematics in the Sciences ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences ,","institution_ids":["https://openalex.org/I4210119712","https://openalex.org/I4210142348"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073351462","display_name":"Debayan Gupta","orcid":"https://orcid.org/0000-0002-4457-1556","observed_orcids":["https://orcid.org/0000-0002-4457-1556"]},"institutions":[{"id":"https://openalex.org/I347237974","display_name":"Ashoka University","ror":"https://ror.org/02j1xr113","country_code":"IN","type":"education","lineage":["https://openalex.org/I347237974"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Gupta, Debayan","raw_affiliation_strings":["Ashoka University , Truth Audit Labs {aritra. das,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ashoka University , Truth Audit Labs {aritra. das,","institution_ids":["https://openalex.org/I347237974"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066485339","display_name":"Christoph Hertrich","orcid":"https://orcid.org/0000-0001-5646-8567","observed_orcids":["https://orcid.org/0000-0001-5646-8567"]},"institutions":[{"id":"https://openalex.org/I4387153582","display_name":"University of Technology Nuremberg","ror":"https://ror.org/02sv65x64","country_code":"DE","type":"education","lineage":["https://openalex.org/I4387153582"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Hertrich, Christoph","raw_affiliation_strings":["University of Technology Nuremberg , {christoph.hertrich ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Technology Nuremberg , {christoph.hertrich ,","institution_ids":["https://openalex.org/I4387153582"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5148269261","display_name":"Tharrshann Jayan Logarajah","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Logarajah, Tharrshann Jayan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5001022516","display_name":"Georg Loho","orcid":"https://orcid.org/0000-0001-6500-385X","observed_orcids":["https://orcid.org/0000-0001-6500-385X"]},"institutions":[{"id":"https://openalex.org/I75951250","display_name":"Freie Universit\u00e4t Berlin","ror":"https://ror.org/046ak2485","country_code":"DE","type":"education","lineage":["https://openalex.org/I75951250"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Loho, Georg","raw_affiliation_strings":["Freie Universit\u00e4t Berlin ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Freie Universit\u00e4t Berlin ,","institution_ids":["https://openalex.org/I75951250"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5126790215","display_name":"Mihir More","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I347237974","display_name":"Ashoka University","ror":"https://ror.org/02j1xr113","country_code":"IN","type":"education","lineage":["https://openalex.org/I347237974"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"More, Mihir","raw_affiliation_strings":["Ashoka University , Truth Audit Labs {aritra. das,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ashoka University , Truth Audit Labs {aritra. das,","institution_ids":["https://openalex.org/I347237974"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5120056358","display_name":"Moritz Stargalla","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4387153582","display_name":"University of Technology Nuremberg","ror":"https://ror.org/02sv65x64","country_code":"DE","type":"education","lineage":["https://openalex.org/I4387153582"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Stargalla, Moritz","raw_affiliation_strings":["University of Technology Nuremberg , {christoph.hertrich ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Technology Nuremberg , {christoph.hertrich ,","institution_ids":["https://openalex.org/I4387153582"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.4884999990463257,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.4884999990463257,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.11469999700784683,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10320","display_name":"Neural Networks and Applications","score":0.07150000333786011,"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/lipschitz-constant","display_name":"Lipschitz constant","score":0.8790000081062317},{"id":"https://openalex.org/keywords/input-convex-neural-networks","display_name":"input convex neural networks","score":0.9300000071525574},{"id":"https://openalex.org/keywords/zonotopes","display_name":"zonotopes","score":0.8730000257492065},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"parameterized complexity","score":0.9300000071525574},{"id":"https://openalex.org/keywords/w-1-hardness","display_name":"W[1]-hardness","score":0.9819999933242798},{"id":"https://openalex.org/keywords/exponential-time-hypothesis","display_name":"Exponential Time Hypothesis","score":0.8730000257492065},{"id":"https://openalex.org/keywords/relu-neural-networks","display_name":"ReLU networks","score":0.7210000157356262},{"id":"https://openalex.org/keywords/dual-norm","display_name":"dual norm","score":0.4970000088214874},{"id":"https://openalex.org/keywords/taylor-expansion","display_name":"Taylor series expansion","score":0.5479999780654907},{"id":"https://openalex.org/keywords/neural-network-robustness","display_name":"neural network robustness","score":0.6100000143051147}],"concepts":[{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.8414000272750854},{"id":"https://openalex.org/C2776330181","wikidata":"https://www.wikidata.org/wiki/Q18358244","display_name":"Maximization","level":2,"score":0.6732000112533569},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6298999786376953},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5044000148773193},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4936999976634979},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4909999966621399},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4323999881744385},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.4165000021457672},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4163999855518341},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.41620001196861267},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3702999949455261},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.36629998683929443},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3637999892234802},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.36000001430511475},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.3546000123023987},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3476000130176544},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.3449999988079071},{"id":"https://openalex.org/C127233936","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Entropy maximization","level":3,"score":0.3248000144958496},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.30559998750686646},{"id":"https://openalex.org/C21200559","wikidata":"https://www.wikidata.org/wiki/Q7451068","display_name":"Sensitivity (control systems)","level":2,"score":0.2957000136375427},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.2867000102996826},{"id":"https://openalex.org/C71017364","wikidata":"https://www.wikidata.org/wiki/Q5421532","display_name":"Exponential time hypothesis","level":3,"score":0.2639000117778778},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.2581999897956848},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.2556999921798706}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.24865","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24865","pdf_url":"https://arxiv.org/pdf/2608.24865","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.24865","doi":"https://doi.org/10.48550/arxiv.2608.24865","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.24865","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.24865","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24865","pdf_url":"https://arxiv.org/pdf/2608.24865","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204251246.pdf","grobid_xml":"https://content.openalex.org/works/W7204251246.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4415331449","https://openalex.org/W7203997958","https://openalex.org/W7203700339","https://openalex.org/W4415287333","https://openalex.org/W3160218486","https://openalex.org/W6910704879","https://openalex.org/W2995623499","https://openalex.org/W7114909429","https://openalex.org/W7130315962","https://openalex.org/W3035270717"],"abstract_inverted_index":{"Lipschitz":[0,188],"constants":[1],"are":[2,119,200],"a":[3,37,61,98,166,212],"standard":[4],"way":[5],"to":[6,13,55,106,160],"quantify":[7],"the":[8,47,57,76,80,95,107,126,132,139,150,153,158,173,177,206,216,223,233],"sensitivity":[9],"of":[10,79,142,215,235],"neural":[11,34],"networks":[12,35,52],"small":[14],"input":[15],"perturbations,":[16],"but":[17],"computing":[18,138],"them":[19],"is":[20,53,102],"difficult":[21],"even":[22],"for":[23,31,50,88,122,137,152,181],"shallow":[24],"ReLU":[25,144],"networks.":[26],"We":[27,85],"study":[28],"this":[29,123],"problem":[30,124,195],"two-layer":[32,143,186],"input-convex":[33],"(ICNNs),":[36],"restricted":[38],"architecture":[39],"where":[40],"nonnegative":[41],"output":[42],"weights":[43],"enforce":[44],"convexity.":[45],"Computing":[46],"$L_p$-Lipschitz":[48,140],"constant":[49,141],"these":[51],"equivalent":[54],"maximizing":[56,94],"dual":[58],"norm":[59,183],"over":[60,97],"zonotope.":[62],"While":[63],"$L_1$-":[64],"and":[65,72,155,185,219],"$L_\\infty$-norm":[66],"maximization":[67,184],"on":[68],"zonotopes":[69],"admit":[70],"fixed-parameter":[71],"polynomial-time":[73],"algorithms,":[74],"respectively,":[75],"parameterized":[77,178],"complexity":[78,179],"remaining":[81],"$L_p$-norms":[82],"was":[83],"open.":[84],"prove":[86],"that,":[87],"every":[89],"fixed":[90,162],"$p\\in":[91,163],"(1,\\infty)\\cap":[92],"\\mathbb{Q}$,":[93],"$L_p$-norm":[96],"zonotope":[99,182],"in":[100],"$\\mathbb{R}^d$":[101],"W[1]-hard":[103],"with":[104],"respect":[105],"dimension":[108],"$d$.":[109],"Moreover,":[110],"our":[111,229],"hardness":[112,134],"results":[113,135,171],"imply":[114],"that":[115],"brute-force":[116],"enumeration":[117],"algorithms":[118],"essentially":[120],"optimal":[121],"under":[125],"Exponential":[127],"Time":[128],"Hypothesis.":[129],"By":[130],"duality,":[131],"same":[133,207],"hold":[136],"ICNNs.":[145],"Our":[146,190,209],"proof":[147],"first":[148],"establishes":[149],"result":[151],"$L_2$-norm":[154],"then":[156],"transfers":[157],"construction":[159],"arbitrary":[161],"(1,\\infty)\\cap\\mathbb{Q}$":[164],"using":[165],"suitable":[167],"Taylor":[168],"approximation.":[169],"These":[170],"resolve":[172],"corresponding":[174],"questions":[175],"regarding":[176],"status":[180],"ICNN":[187],"constants.":[189],"paper":[191,210],"resolves":[192],"an":[193],"open":[194],"posted":[196],"at":[197],"COLT'25.":[198],"There":[199],"several":[201],"independent":[202],"concurrent":[203],"papers":[204],"resolving":[205],"problem.":[208],"prioritizes":[211],"clear":[213],"exposition":[214],"underlying":[217],"mathematics":[218],"conceptual":[220],"intuitions":[221],"behind":[222],"proof.":[224],"Additionally,":[225],"we":[226],"explicitly":[227],"describe":[228],"research":[230],"process":[231],"including":[232],"use":[234],"LLMs.":[236]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-08-27T00:00:00"}