{"id":"https://openalex.org/W7220701293","doi":"https://doi.org/10.48550/arxiv.2610.06531","title":"One Extra Dimension Suffices for the Complete-Graph Squared-Stress","display_name":"One Extra Dimension Suffices for the Complete-Graph Squared-Stress","publication_year":2026,"publication_date":"2026-10-05","ids":{"openalex":"https://openalex.org/W7220701293","doi":"https://doi.org/10.48550/arxiv.2610.06531"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.06531","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06531","pdf_url":"https://arxiv.org/pdf/2610.06531","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.06531","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151240881","display_name":"Lilin Yan","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I17145004","display_name":"Northwestern Polytechnical University","ror":"https://ror.org/01y0j0j86","country_code":"CN","type":"education","lineage":["https://openalex.org/I17145004"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yan, Lilin","raw_affiliation_strings":["School of Electronics and Information , Northwestern Polytechnical University , Xi'an 710072 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electronics and Information , Northwestern Polytechnical University , Xi'an 710072 , China","institution_ids":["https://openalex.org/I17145004"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5147582384","display_name":"Hongwei Zhao","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I17145004","display_name":"Northwestern Polytechnical University","ror":"https://ror.org/01y0j0j86","country_code":"CN","type":"education","lineage":["https://openalex.org/I17145004"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhao, Hongwei","raw_affiliation_strings":["School of Electronics and Information , Northwestern Polytechnical University , Xi'an 710072 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electronics and Information , Northwestern Polytechnical University , Xi'an 710072 , China","institution_ids":["https://openalex.org/I17145004"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I17145004"],"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.1574999988079071,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.1574999988079071,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.10459999740123749,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.08669999986886978,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/complete-graphs","display_name":"complete graphs","score":0.7950000166893005},{"id":"https://openalex.org/keywords/dimensionality-reduction","display_name":"dimensionality reduction","score":0.4749999940395355},{"id":"https://openalex.org/keywords/second-order-stationary-points","display_name":"second-order stationary points","score":0.7099999785423279},{"id":"https://openalex.org/keywords/residual-stress","display_name":"residual stress","score":0.7160000205039978},{"id":"https://openalex.org/keywords/rank-nullity-theorem","display_name":"rank-nullity theorem","score":0.8830000162124634},{"id":"https://openalex.org/keywords/formal-verification","display_name":"formal verification","score":0.7179999947547913},{"id":"https://openalex.org/keywords/lean-4","display_name":"Lean 4","score":0.9620000123977661}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8382999897003174},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5281000137329102},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.513700008392334},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.5116999745368958},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5030999779701233},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.4781000018119812},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.4510999917984009},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.42080000042915344},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.3955000042915344},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39399999380111694},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.39070001244544983},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.37619999051094055},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.3427000045776367},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.3400999903678894},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.3343999981880188},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.3246999979019165},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.32429999113082886},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3075000047683716},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.30410000681877136},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.3028999865055084},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.3010999858379364},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.2897000014781952},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.27410000562667847},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2615000009536743},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.25519999861717224}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.06531","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06531","pdf_url":"https://arxiv.org/pdf/2610.06531","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.06531","doi":"https://doi.org/10.48550/arxiv.2610.06531","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.06531","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.06531","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06531","pdf_url":"https://arxiv.org/pdf/2610.06531","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320336350","display_name":"Key Research and Development Projects of Shaanxi Province","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220701293.pdf","grobid_xml":"https://content.openalex.org/works/W7220701293.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4413389464","https://openalex.org/W4417493456","https://openalex.org/W7203691510"],"related_works":["https://openalex.org/W7203691510","https://openalex.org/W7220710783","https://openalex.org/W2524442950","https://openalex.org/W7213787246","https://openalex.org/W2170055001","https://openalex.org/W7221618878","https://openalex.org/W7098162991","https://openalex.org/W3015918970","https://openalex.org/W2114752924","https://openalex.org/W7098741341"],"abstract_inverted_index":{"Let":[0],"all":[1],"pairwise":[2],"distances":[3],"of":[4,8,22,44,72,78,88,102],"an":[5],"arbitrary":[6],"configuration":[7],"points":[9,53],"in":[10,29,114],"$\\mathbb{R}^{\\ell}$":[11],"be":[12],"known":[13],"exactly.":[14],"We":[15],"prove":[16],"that":[17],"every":[18,42],"second-order":[19],"stationary":[20],"point":[21],"the":[23,60,73,79,92,98,103],"complete-graph":[24],"squared-stress":[25],"objective":[26,106],"over":[27],"configurations":[28],"$\\mathbb{R}^{k}$":[30],"has":[31,111],"zero":[32,89,126],"residual":[33,74],"whenever":[34],"$k":[35],">":[36],"\\ell$.":[37],"The":[38,68,95,117],"result":[39],"holds":[40],"for":[41,63],"number":[43],"points,":[45],"without":[46],"a":[47,82,122,130],"genericity":[48],"assumption,":[49],"and":[50,54,86,129],"allows":[51],"repeated":[52],"degenerate":[55],"ground-truth":[56],"configurations.":[57],"It":[58],"resolves":[59],"one-extra-dimension":[61],"conjecture":[62],"complete,":[64],"uniformly":[65],"weighted":[66],"data.":[67],"proof":[69],"combines":[70],"positivity":[71],"stress,":[75],"polar":[76],"normalization":[77],"candidate":[80],"configuration,":[81],"row-wise":[83],"rank-nullity":[84],"argument,":[85],"propagation":[87],"curvature":[90],"through":[91],"full":[93],"Hessian.":[94],"theorem,":[96],"including":[97],"passage":[99],"from":[100],"derivatives":[101],"original":[104],"quartic":[105],"to":[107],"its":[108],"matrix":[109],"formulation,":[110],"been":[112],"formalized":[113],"Lean":[115],"4.":[116],"accompanying":[118],"verification":[119],"records":[120],"include":[121],"kernel":[123],"check":[124],"with":[125],"trust":[127],"level":[128],"transitive":[131],"axiom":[132],"audit.":[133]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}
