{"id":"https://openalex.org/W7222997938","doi":"https://doi.org/10.48550/arxiv.2610.24191","title":"Certifiably Correct Fr\u00e9chet Means in Orbit Spaces","display_name":"Certifiably Correct Fr\u00e9chet Means in Orbit Spaces","publication_year":2026,"publication_date":"2026-10-16","ids":{"openalex":"https://openalex.org/W7222997938","doi":"https://doi.org/10.48550/arxiv.2610.24191"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.24191","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24191","pdf_url":"https://arxiv.org/pdf/2610.24191","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/2610.24191","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015279113","display_name":"Dustin G. Mixon","orcid":"https://orcid.org/0000-0003-2743-7010","observed_orcids":["https://orcid.org/0000-0003-2743-7010"]},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mixon, Dustin G.","raw_affiliation_strings":["Department of Mathematics , The Ohio State University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , The Ohio State University","institution_ids":["https://openalex.org/I52357470"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073878869","display_name":"Tom Needham","orcid":"https://orcid.org/0000-0001-6165-3433","observed_orcids":["https://orcid.org/0000-0001-6165-3433"]},"institutions":[{"id":"https://openalex.org/I103163165","display_name":"Florida State University","ror":"https://ror.org/05g3dte14","country_code":"US","type":"education","lineage":["https://openalex.org/I103163165"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Needham, Tom","raw_affiliation_strings":["Department of Mathematics , Florida State University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Florida State University","institution_ids":["https://openalex.org/I103163165"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032361277","display_name":"Nathan Willey","orcid":"https://orcid.org/0009-0001-1196-9330","observed_orcids":["https://orcid.org/0009-0001-1196-9330"]},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Willey, Nathan","raw_affiliation_strings":["Department of Mathematics , The Ohio State University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , The Ohio State University","institution_ids":["https://openalex.org/I52357470"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.23399999737739563,"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"}},"topics":[{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.23399999737739563,"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"}},{"id":"https://openalex.org/T10637","display_name":"Advanced Clustering Algorithms Research","score":0.23340000212192535,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.19629999995231628,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/frechet-mean","display_name":"Fr\u00e9chet mean","score":0.8690000176429749},{"id":"https://openalex.org/keywords/orbit-spaces","display_name":"orbit spaces","score":0.9929999709129333},{"id":"https://openalex.org/keywords/collective-synchronization","display_name":"collective synchronization","score":0.925000011920929},{"id":"https://openalex.org/keywords/semidefinite-relaxation","display_name":"semidefinite relaxation","score":0.8949999809265137},{"id":"https://openalex.org/keywords/spectral-relaxation","display_name":"spectral relaxation","score":0.8080000281333923},{"id":"https://openalex.org/keywords/pairwise-sequence-alignment","display_name":"pairwise sequence alignment","score":0.7870000004768372},{"id":"https://openalex.org/keywords/np-hardness","display_name":"NP-hardness","score":0.6800000071525574},{"id":"https://openalex.org/keywords/time-series-clustering","display_name":"time series clustering","score":0.871999979019165}],"concepts":[{"id":"https://openalex.org/C101901036","wikidata":"https://www.wikidata.org/wiki/Q2269096","display_name":"Semidefinite programming","level":2,"score":0.6312000155448914},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.5985000133514404},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.5418000221252441},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.5371000170707703},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5009999871253967},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4884999990463257},{"id":"https://openalex.org/C96865113","wikidata":"https://www.wikidata.org/wiki/Q2946816","display_name":"Certificate","level":2,"score":0.4878999888896942},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.45969998836517334},{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.4361000061035156},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42239999771118164},{"id":"https://openalex.org/C2778562939","wikidata":"https://www.wikidata.org/wiki/Q1298791","display_name":"Synchronization (alternating current)","level":3,"score":0.4018000066280365},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.39640000462532043},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.35269999504089355},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.3447999954223633},{"id":"https://openalex.org/C105611402","wikidata":"https://www.wikidata.org/wiki/Q2976589","display_name":"Spectral clustering","level":3,"score":0.34139999747276306},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.3407999873161316},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.319599986076355},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.31040000915527344},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.30730000138282776},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.298799991607666},{"id":"https://openalex.org/C183448482","wikidata":"https://www.wikidata.org/wiki/Q1799665","display_name":"Modulo operation","level":2,"score":0.29660001397132874},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.2897999882698059},{"id":"https://openalex.org/C2781215313","wikidata":"https://www.wikidata.org/wiki/Q3493345","display_name":"SPARK (programming language)","level":2,"score":0.2712000012397766},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2606000006198883},{"id":"https://openalex.org/C123403432","wikidata":"https://www.wikidata.org/wiki/Q654068","display_name":"Visibility","level":2,"score":0.25529998540878296}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.24191","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24191","pdf_url":"https://arxiv.org/pdf/2610.24191","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.2610.24191","doi":"https://doi.org/10.48550/arxiv.2610.24191","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.24191","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:2610.24191","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24191","pdf_url":"https://arxiv.org/pdf/2610.24191","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":[{"id":"https://openalex.org/G8904870570","display_name":"Collaborative Research: ATD: Principled machine learning and packing subspaces for improved spatiotemporal data processing","funder_award_id":"2220304","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/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/W7222997938.pdf","grobid_xml":"https://content.openalex.org/works/W7222997938.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4417070636","https://openalex.org/W6967233593"],"related_works":["https://openalex.org/W4416025495","https://openalex.org/W3029695162","https://openalex.org/W3205864334","https://openalex.org/W4313650234","https://openalex.org/W4409737394","https://openalex.org/W7096951888","https://openalex.org/W4406090494","https://openalex.org/W2762408281","https://openalex.org/W2260440663","https://openalex.org/W4367840394"],"abstract_inverted_index":{"We":[0],"study":[1],"Fr\u00e9chet":[2],"means":[3,15],"in":[4,18,61,73],"orbit":[5],"spaces":[6],"of":[7,79],"finite":[8],"orthogonal":[9],"group":[10,24],"actions.":[11],"Since":[12],"computing":[13],"these":[14,71],"is":[16],"NP-hard":[17],"general,":[19],"we":[20,39,55],"take":[21],"inspiration":[22],"from":[23],"synchronization":[25],"to":[26],"develop":[27,40],"tractable":[28],"spectral":[29],"and":[30,38,82],"semidefinite":[31,42],"relaxations.":[32],"Pairwise":[33],"alignment":[34,66],"certifies":[35],"global":[36],"optimality":[37],"a":[41,62],"certificate":[43,59],"that":[44,57],"can":[45],"extend":[46],"beyond":[47],"this":[48],"condition.":[49],"For":[50],"Gaussian":[51],"data":[52],"modulo":[53],"sign,":[54],"prove":[56],"our":[58],"succeeds":[60],"high-noise":[63],"regime":[64],"where":[65],"fails.":[67],"Numerical":[68],"experiments":[69],"evaluate":[70],"methods":[72],"various":[74],"settings,":[75],"including":[76],"summarizing":[77],"ensembles":[78],"districting":[80],"plans":[81],"clustering":[83],"time":[84],"series":[85]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-21T00:00:00"}
