{"id":"https://openalex.org/W7212289682","doi":"https://doi.org/10.48550/arxiv.2609.11016","title":"A Scale Invariance Property of PCA","display_name":"A Scale Invariance Property of PCA","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7212289682","doi":"https://doi.org/10.48550/arxiv.2609.11016"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.11016","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11016","pdf_url":"https://arxiv.org/pdf/2609.11016","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.11016","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034061744","display_name":"Jonathan Landy","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Landy, Jonathan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5034061744"],"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/T13141","display_name":"Statistical Methods and Applications","score":0.9916999936103821,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T13141","display_name":"Statistical Methods and Applications","score":0.9916999936103821,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.0035000001080334187,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T13487","display_name":"Statistical and numerical algorithms","score":0.0012000000569969416,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/principal-component-analysis","display_name":"principal component analysis","score":0.996999979019165},{"id":"https://openalex.org/keywords/scale-invariance","display_name":"scale invariance","score":0.972000002861023},{"id":"https://openalex.org/keywords/eigenvalue-crossing","display_name":"eigenvalue crossing","score":0.9039999842643738}],"concepts":[{"id":"https://openalex.org/C27438332","wikidata":"https://www.wikidata.org/wiki/Q2873","display_name":"Principal component analysis","level":2,"score":0.8169000148773193},{"id":"https://openalex.org/C99821215","wikidata":"https://www.wikidata.org/wiki/Q1136583","display_name":"Swap (finance)","level":2,"score":0.6402999758720398},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.6359999775886536},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6207000017166138},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.6057999730110168},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.5946000218391418},{"id":"https://openalex.org/C135593079","wikidata":"https://www.wikidata.org/wiki/Q1750766","display_name":"Scale invariance","level":2,"score":0.5522000193595886},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.550599992275238},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.366100013256073},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.361299991607666},{"id":"https://openalex.org/C144559511","wikidata":"https://www.wikidata.org/wiki/Q2986279","display_name":"Principal (computer security)","level":2,"score":0.35600000619888306},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3319999873638153},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.31690001487731934},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3082999885082245},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.2921999990940094},{"id":"https://openalex.org/C161326058","wikidata":"https://www.wikidata.org/wiki/Q7245073","display_name":"Principal axis theorem","level":2,"score":0.27799999713897705},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.259799987077713},{"id":"https://openalex.org/C19619285","wikidata":"https://www.wikidata.org/wiki/Q196372","display_name":"Observational error","level":2,"score":0.2567000091075897},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25189998745918274}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.11016","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11016","pdf_url":"https://arxiv.org/pdf/2609.11016","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.2609.11016","doi":"https://doi.org/10.48550/arxiv.2609.11016","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.11016","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:2609.11016","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11016","pdf_url":"https://arxiv.org/pdf/2609.11016","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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212289682.pdf","grobid_xml":"https://content.openalex.org/works/W7212289682.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7095729316","https://openalex.org/W4313447089","https://openalex.org/W6885093514","https://openalex.org/W4321649370","https://openalex.org/W4399100602","https://openalex.org/W1598971805","https://openalex.org/W2063692987","https://openalex.org/W6922612630","https://openalex.org/W3133668574","https://openalex.org/W2547192247"],"abstract_inverted_index":{"The":[0],"PCA":[1,86],"algorithm":[2],"is":[3,34],"sensitive":[4],"to":[5,91,107],"changes":[6,92],"in":[7,16,93,135],"measurement":[8,94],"scale.":[9,95],"Measuring":[10],"one":[11],"variable":[12],"of":[13,85],"a":[14,44,49],"system":[15],"inches":[17],"rather":[18],"than":[19],"centimeters,":[20],"say,":[21],"alters":[22],"both":[23],"its":[24],"principal":[25,28,58,71],"axes":[26],"and":[27],"eigenvalues.":[29],"Although":[30],"this":[31,77,136],"scale":[32,51],"dependence":[33],"generally":[35],"complicated,":[36],"we":[37,79,117,120],"show":[38,118],"here":[39],"that":[40,82,104,119],"it":[41],"nevertheless":[42],"obeys":[43],"strict":[45],"invariance":[46],"property:":[47],"under":[48],"continuous":[50],"adjustment,":[52],"the":[53,66,83,129,139],"initial":[54],"state's":[55,68],"$k$-th":[56,69],"largest":[57,70],"component":[59],"(ordered":[60],"by":[61],"eigenvalue)":[62],"continuously":[63],"evolves":[64],"into":[65],"final":[67],"component,":[72],"for":[73],"each":[74],"$k$.":[75],"In":[76],"sense,":[78],"can":[80,114,121],"say":[81],"modes":[84,130],"are":[87,105],"\"order-stable\"":[88],"with":[89],"respect":[90],"A":[96],"special":[97],"case":[98],"occurs":[99],"when":[100],"scaling":[101],"along":[102],"directions":[103],"orthogonal":[106],"some":[108],"modes.":[109],"Here,":[110],"apparent":[111,124],"eigenvalue":[112],"crossings":[113,125],"occur.":[115],"However,":[116],"interpret":[122],"these":[123],"as":[126],"cases":[127],"where":[128],"instantaneously":[131],"swap":[132],"their":[133],"orientation,":[134],"way":[137],"maintaining":[138],"required":[140],"order":[141],"stability.":[142]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-12T00:00:00"}