{"id":"https://openalex.org/W7208705381","doi":"https://doi.org/10.48550/arxiv.2609.03042","title":"Equivariance, Curvature and Symmetry in Functional Covariance Estimation","display_name":"Equivariance, Curvature and Symmetry in Functional Covariance Estimation","publication_year":2026,"publication_date":"2026-09-02","ids":{"openalex":"https://openalex.org/W7208705381","doi":"https://doi.org/10.48550/arxiv.2609.03042"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.03042","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03042","pdf_url":"https://arxiv.org/pdf/2609.03042","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/2609.03042","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071373642","display_name":"Jocelyn Nemb\u00e9","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Nembe, Jocelyn","raw_affiliation_strings":["Laboratory of Engineering Applied to Business Management P.O Box 190 , Libreville , Gabon"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratory of Engineering Applied to Business Management P.O Box 190 , Libreville , Gabon","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5071373642"],"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9937000274658203,"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9937000274658203,"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/T10136","display_name":"Statistical Methods and Inference","score":0.0010000000474974513,"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/T13141","display_name":"Statistical Methods and Applications","score":0.000699999975040555,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/equivariance","display_name":"equivariance","score":0.746999979019165},{"id":"https://openalex.org/keywords/curvature","display_name":"curvature","score":0.7059999704360962},{"id":"https://openalex.org/keywords/covariance-operator","display_name":"covariance operator","score":0.45899999141693115},{"id":"https://openalex.org/keywords/coordinate-invariance","display_name":"coordinate invariance","score":0.8709999918937683}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7663000226020813},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.7516999840736389},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.6848000288009644},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5906000137329102},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.5584999918937683},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.5130000114440918},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.41040000319480896},{"id":"https://openalex.org/C134261354","wikidata":"https://www.wikidata.org/wiki/Q938438","display_name":"Statistical inference","level":2,"score":0.3959999978542328},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.39570000767707825},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.3921000063419342},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.3846000134944916},{"id":"https://openalex.org/C51820054","wikidata":"https://www.wikidata.org/wiki/Q5508814","display_name":"Functional data analysis","level":2,"score":0.35760000348091125},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.34119999408721924},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.33070001006126404},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32820001244544983},{"id":"https://openalex.org/C2778300598","wikidata":"https://www.wikidata.org/wiki/Q2638561","display_name":"General covariance","level":3,"score":0.321399986743927},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.31779998540878296},{"id":"https://openalex.org/C167928553","wikidata":"https://www.wikidata.org/wiki/Q1376021","display_name":"Estimation theory","level":2,"score":0.3133000135421753},{"id":"https://openalex.org/C44306375","wikidata":"https://www.wikidata.org/wiki/Q902019","display_name":"Symmetry group","level":2,"score":0.30809998512268066},{"id":"https://openalex.org/C180877172","wikidata":"https://www.wikidata.org/wiki/Q5401390","display_name":"Estimation of covariance matrices","level":3,"score":0.3061000108718872},{"id":"https://openalex.org/C83042196","wikidata":"https://www.wikidata.org/wiki/Q5178898","display_name":"Covariance intersection","level":4,"score":0.28139999508857727},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.27970001101493835},{"id":"https://openalex.org/C16977076","wikidata":"https://www.wikidata.org/wiki/Q1589551","display_name":"Principal curvature","level":4,"score":0.2759999930858612},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2736999988555908},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.2702000141143799},{"id":"https://openalex.org/C80551277","wikidata":"https://www.wikidata.org/wiki/Q11210","display_name":"Coordinate system","level":2,"score":0.2624000012874603},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.2621999979019165},{"id":"https://openalex.org/C175017881","wikidata":"https://www.wikidata.org/wiki/Q1318998","display_name":"Mean curvature","level":3,"score":0.26190000772476196},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.25619998574256897},{"id":"https://openalex.org/C156387681","wikidata":"https://www.wikidata.org/wiki/Q6059484","display_name":"Invariance principle","level":2,"score":0.2547999918460846},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.2535000145435333},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.2502000033855438}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.03042","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03042","pdf_url":"https://arxiv.org/pdf/2609.03042","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.2609.03042","doi":"https://doi.org/10.48550/arxiv.2609.03042","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.03042","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:2609.03042","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03042","pdf_url":"https://arxiv.org/pdf/2609.03042","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/W7208705381.pdf","grobid_xml":"https://content.openalex.org/works/W7208705381.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2110849403","https://openalex.org/W4242939741","https://openalex.org/W4415025678","https://openalex.org/W2042700655","https://openalex.org/W1995721524","https://openalex.org/W4407382137","https://openalex.org/W2302261895","https://openalex.org/W7119401416","https://openalex.org/W4286900748","https://openalex.org/W6931337017"],"abstract_inverted_index":{"Statistical":[0],"procedures":[1],"for":[2,36],"functional":[3,43],"data":[4],"are":[5,51],"routinely":[6],"applied":[7],"after":[8],"changes":[9],"of":[10,28,105],"time":[11],"scale,":[12],"registration,":[13],"or":[14],"other":[15],"reparametrisations,":[16],"although":[17],"it":[18],"is":[19,26,72,82,101,122],"generally":[20],"unclear":[21],"when":[22],"the":[23,29,46,59,70,85,103,124,129,137,150,168,172],"resulting":[24],"inference":[25],"independent":[27],"chosen":[30],"coordinates.":[31],"We":[32,108],"characterize":[33],"this":[34],"equivariance":[35,81],"local-linear":[37,90],"covariance":[38,49],"estimation":[39,60,126],"from":[40,80,171],"sparsely":[41],"observed":[42],"data.":[44],"At":[45,58],"population":[47,169],"level,":[48,61],"operators":[50],"unitarily":[52],"conjugate":[53],"under":[54],"every":[55],"diffeomorphic":[56],"reparametrisation.":[57],"exact":[62],"commutation":[63],"holds":[64],"universally":[65],"if":[66,69],"and":[67,141,158],"only":[68],"reparametrisation":[71],"affine.":[73],"For":[74],"a":[75],"general":[76],"\\(C^{2,1}\\)":[77],"diffeomorphism,":[78],"departure":[79],"controlled":[83],"by":[84,176],"normalized":[86],"curvature":[87,100],"\\(\u03ba_\u03c8=\\|\u03c8''/\u03c8'\\|_\\infty\\),":[88],"with":[89],"defect":[91],"\\[":[92],"O_P\\!\\left\\{\u03ba_\u03c8\\left(h^2+h":[93],"n_{\\mathrm{loc}}^{-1/2}":[94],"+h_0^2+h_0":[95],"n_{\\mathrm{loc},1}^{-1/2}\\right)\\right\\}.":[96],"\\]":[97],"Thus":[98],"zero":[99],"exactly":[102,123],"boundary":[104],"statistical":[106,177],"equivariance.":[107],"then":[109],"show":[110],"that":[111],"finite-group":[112],"symmetry":[113,131],"acts":[114],"as":[115],"an":[116],"orthogonal-projection":[117],"regularizer:":[118],"its":[119],"risk":[120],"gain":[121],"anti-invariant":[125],"error":[127],"minus":[128],"squared":[130],"misspecification.":[132],"An":[133],"orbit-covariance":[134],"identity":[135],"quantifies":[136],"attainable":[138],"variance":[139],"reduction":[140],"shows":[142],"why":[143],"group":[144],"size":[145],"alone":[146],"does":[147],"not":[148],"determine":[149],"gain.":[151],"These":[152],"principles":[153],"propagate":[154],"to":[155],"eigenvalues,":[156],"eigenspaces":[157],"truncated":[159],"PACE":[160],"prediction.":[161],"The":[162],"results":[163],"separate":[164],"coordinate":[165],"invariance":[166],"at":[167],"level":[170],"geometric":[173],"obstructions":[174],"introduced":[175],"smoothing.":[178]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-05T00:00:00"}