{"id":"https://openalex.org/W7208707677","doi":"https://doi.org/10.48550/arxiv.2609.03361","title":"Grassmann--Pl\u00fccker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings","display_name":"Grassmann--Pl\u00fccker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings","publication_year":2026,"publication_date":"2026-09-03","ids":{"openalex":"https://openalex.org/W7208707677","doi":"https://doi.org/10.48550/arxiv.2609.03361"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.03361","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03361","pdf_url":"https://arxiv.org/pdf/2609.03361","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","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.03361","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5150055258","display_name":"Hongyu Yuan","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuan, Hongyu","raw_affiliation_strings":["Zhili College , Tsinghua University , Beijing , 100084 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Zhili College , Tsinghua University , Beijing , 100084 , P. R. China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5149990779","display_name":"Huaiqing Zuo","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zuo, Huaiqing","raw_affiliation_strings":["Department of Mathematical Sciences , Tsinghua University , Beijing , 100084 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences , Tsinghua University , Beijing , 100084 , P. R. China","institution_ids":["https://openalex.org/I99065089"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I99065089"],"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/T10320","display_name":"Neural Networks and Applications","score":0.12399999797344208,"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"}},"topics":[{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.12399999797344208,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.12020000070333481,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.11020000278949738,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/convolutional-filters","display_name":"convolutional filters","score":0.7749999761581421},{"id":"https://openalex.org/keywords/plucker-embedding","display_name":"Pl\u00fccker embedding","score":0.9620000123977661},{"id":"https://openalex.org/keywords/closed-embeddings","display_name":"closed embeddings","score":0.7129999995231628},{"id":"https://openalex.org/keywords/filter-redundancy","display_name":"filter redundancy","score":0.6970000267028809}],"concepts":[{"id":"https://openalex.org/C162929932","wikidata":"https://www.wikidata.org/wiki/Q129638","display_name":"Grassmannian","level":2,"score":0.7775999903678894},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7773000001907349},{"id":"https://openalex.org/C202887219","wikidata":"https://www.wikidata.org/wiki/Q3895221","display_name":"Parametrization (atmospheric modeling)","level":3,"score":0.6434000134468079},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.6126000285148621},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.49149999022483826},{"id":"https://openalex.org/C106131492","wikidata":"https://www.wikidata.org/wiki/Q3072260","display_name":"Filter (signal processing)","level":2,"score":0.4163999855518341},{"id":"https://openalex.org/C75280867","wikidata":"https://www.wikidata.org/wiki/Q877775","display_name":"Projective space","level":3,"score":0.4147000014781952},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.4097000062465668},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4056999981403351},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.3896999955177307},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3296000063419342},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.32420000433921814},{"id":"https://openalex.org/C134949993","wikidata":"https://www.wikidata.org/wiki/Q2068617","display_name":"Pencil (optics)","level":2,"score":0.32260000705718994},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.313400000333786},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.30630001425743103},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.29809999465942383},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.2944999933242798},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2922999858856201},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.28279998898506165},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.2825999855995178},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.2793999910354614},{"id":"https://openalex.org/C62139920","wikidata":"https://www.wikidata.org/wiki/Q3554818","display_name":"Projective variety","level":2,"score":0.27730000019073486},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C28903235","wikidata":"https://www.wikidata.org/wiki/Q746550","display_name":"Tangent bundle","level":3,"score":0.2734000086784363},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C191961195","wikidata":"https://www.wikidata.org/wiki/Q5477902","display_name":"Frame bundle","level":4,"score":0.2678000032901764},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.2581000030040741},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.25529998540878296}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.03361","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03361","pdf_url":"https://arxiv.org/pdf/2609.03361","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.2609.03361","doi":"https://doi.org/10.48550/arxiv.2609.03361","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.03361","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:2609.03361","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03361","pdf_url":"https://arxiv.org/pdf/2609.03361","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/W7208707677.pdf","grobid_xml":"https://content.openalex.org/works/W7208707677.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W1996901117","https://openalex.org/W2001689396","https://openalex.org/W2045512849","https://openalex.org/W2551687757","https://openalex.org/W3191453487"],"related_works":["https://openalex.org/W1986551561","https://openalex.org/W2951649848","https://openalex.org/W1965898708","https://openalex.org/W7097012893","https://openalex.org/W2032540051","https://openalex.org/W2488750511","https://openalex.org/W2952420656","https://openalex.org/W4413142809","https://openalex.org/W2968266025","https://openalex.org/W1981216200"],"abstract_inverted_index":{"We":[0,93],"propose":[1],"a":[2,9,25,65,107,114,122,147],"geometric":[3,203],"parametrization":[4,67,136],"of":[5,21,28,76,86],"the":[6,13,29,36,61,74,77,84,96,126,135,150,167,185,201],"filters":[7],"in":[8,51,56],"single":[10],"convolutional":[11],"layer:":[12],"parameter":[14,127],"is":[15,88,113,121,129,137,146,155],"no":[16],"longer":[17],"an":[18,40],"ordered":[19],"family":[20],"filter":[22,30,49,194],"vectors,":[23],"but":[24],"fixed-dimensional":[26],"subspace":[27],"space.":[31],"For":[32,157],"one-dimensional":[33],"finite-stride":[34],"convolution,":[35,198],"filter-to-convolution-operator":[37],"correspondence":[38],"gives":[39],"injective":[41,89],"linear":[42],"map":[43,47,80],"$\\mathcal{C}:\\mathcal{K}\\to":[44],"H$.":[45],"This":[46,179],"sends":[48],"subspaces":[50,55],"$\\mathrm{Gr}(q,\\mathcal{K})$":[52],"to":[53,109,131,165],"operator":[54],"$\\mathrm{Gr}(q,H)$;":[57],"composing":[58],"it":[59],"with":[60,193],"Pl\u00fccker":[62,100],"embedding":[63],"yields":[64],"projective":[66,133,152],"$\u03a6:\\mathrm{Gr}(q,\\mathcal{K})\\to\\mathbb{P}(\\bigwedge^q":[68],"H)$.":[69],"Using":[70],"$T_U\\mathrm{Gr}(q,\\mathcal{K})\\cong\\mathrm{Hom}(U,\\mathcal{K}/U)$,":[71],"we":[72,161,189],"compute":[73],"differential":[75,85],"induced":[78],"Grassmannian":[79,108],"and":[81,102,117,139,149,159,170,177,196],"show":[82],"that":[83,111,119],"$\u03a6$":[87,120],"at":[90],"every":[91,144],"point.":[92],"then":[94],"use":[95,163],"vanishing":[97],"equations":[98],"for":[99],"coordinates":[101,105],"standard":[103],"affine":[104],"on":[106],"prove":[110],"$\\mathrm{Gr}(q,\\mathcal{C}(\\mathcal{K}))\\hookrightarrow\\mathrm{Gr}(q,H)$":[112],"closed":[115,123],"embedding,":[116],"hence":[118],"embedding.":[124],"Consequently,":[125],"space":[128],"isomorphic":[130],"its":[132,142,172],"image,":[134,143],"finite":[138],"birational":[140],"onto":[141],"fiber":[145],"singleton,":[148],"resulting":[151],"neural":[153],"variety":[154],"smooth.":[156],"$k=4$":[158],"$q=2$,":[160],"also":[162],"Singular":[164],"recover":[166],"image":[168],"ideal":[169],"check":[171],"dimension,":[173],"degree,":[174],"chart":[175],"rank,":[176],"smoothness.":[178],"computation":[180],"illustrates,":[181],"rather":[182],"than":[183],"replaces,":[184],"general":[186],"proof.":[187],"Finally,":[188],"discuss":[190],"possible":[191],"connections":[192],"redundancy":[195],"low-rank":[197],"while":[199],"distinguishing":[200],"proved":[202],"results":[204],"from":[205],"application":[206],"proposals":[207],"requiring":[208],"numerical":[209],"validation.":[210]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-05T00:00:00"}