{"id":"https://openalex.org/W3191453487","doi":"https://doi.org/10.1137/21m1441183","title":"Geometry of Linear Convolutional Networks","display_name":"Geometry of Linear Convolutional Networks","publication_year":2022,"publication_date":"2022-08-02","ids":{"openalex":"https://openalex.org/W3191453487","doi":"https://doi.org/10.1137/21m1441183","mag":"3191453487"},"language":"en","primary_location":{"id":"doi:10.1137/21m1441183","is_oa":true,"landing_page_url":"https://doi.org/10.1137/21m1441183","pdf_url":null,"source":{"id":"https://openalex.org/S2996807011","display_name":"SIAM Journal on Applied Algebra and Geometry","issn_l":"2470-6566","issn":["2470-6566"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Algebra and Geometry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1137/21m1441183","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003092978","display_name":"Kathl\u00e9n Kohn","orcid":"https://orcid.org/0000-0002-4627-8812"},"institutions":[{"id":"https://openalex.org/I86987016","display_name":"KTH Royal Institute of Technology","ror":"https://ror.org/026vcq606","country_code":"SE","type":"education","lineage":["https://openalex.org/I86987016"]}],"countries":["SE"],"is_corresponding":true,"raw_author_name":"Kathl\u00e9n Kohn","raw_affiliation_strings":["Royal Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Royal Institute of Technology","institution_ids":["https://openalex.org/I86987016"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021863269","display_name":"Thomas Merkh","orcid":"https://orcid.org/0000-0002-1169-3960"},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Thomas Merkh","raw_affiliation_strings":["University of California-Los Angeles"],"affiliations":[{"raw_affiliation_string":"University of California-Los Angeles","institution_ids":["https://openalex.org/I161318765"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072280192","display_name":"Guido Mont\u00fafar","orcid":"https://orcid.org/0000-0002-0131-2669"},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Guido Mont\u00fafar","raw_affiliation_strings":["University of California-Los Angeles"],"affiliations":[{"raw_affiliation_string":"University of California-Los Angeles","institution_ids":["https://openalex.org/I161318765"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030073760","display_name":"Matthew Trager","orcid":"https://orcid.org/0000-0001-9304-6427"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Matthew Trager","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5003092978"],"corresponding_institution_ids":["https://openalex.org/I86987016"],"apc_list":null,"apc_paid":null,"fwci":1.6069,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.84648759,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"6","issue":"3","first_page":"368","last_page":"406"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9995999932289124,"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":0.9995999932289124,"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/T11304","display_name":"Advanced Neuroimaging Techniques and Applications","score":0.9904999732971191,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9722999930381775,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5948542356491089},{"id":"https://openalex.org/keywords/linear-space","display_name":"Linear space","score":0.5381985306739807},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5320250391960144},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5236063599586487},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5176542401313782},{"id":"https://openalex.org/keywords/algebraic-geometry","display_name":"Algebraic geometry","score":0.4933268129825592},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.48420003056526184},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.39644598960876465},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3893741965293884},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.36977481842041016},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.340162992477417},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3331213593482971},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2870925962924957},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.2830107808113098},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.13055461645126343}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5948542356491089},{"id":"https://openalex.org/C176370821","wikidata":"https://www.wikidata.org/wiki/Q1826459","display_name":"Linear space","level":2,"score":0.5381985306739807},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5320250391960144},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5236063599586487},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5176542401313782},{"id":"https://openalex.org/C68363185","wikidata":"https://www.wikidata.org/wiki/Q180969","display_name":"Algebraic geometry","level":2,"score":0.4933268129825592},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.48420003056526184},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.39644598960876465},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3893741965293884},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36977481842041016},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.340162992477417},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3331213593482971},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2870925962924957},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2830107808113098},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.13055461645126343},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/21m1441183","is_oa":true,"landing_page_url":"https://doi.org/10.1137/21m1441183","pdf_url":null,"source":{"id":"https://openalex.org/S2996807011","display_name":"SIAM Journal on Applied Algebra and Geometry","issn_l":"2470-6566","issn":["2470-6566"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Algebra and Geometry","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1137/21m1441183","is_oa":true,"landing_page_url":"https://doi.org/10.1137/21m1441183","pdf_url":null,"source":{"id":"https://openalex.org/S2996807011","display_name":"SIAM Journal on Applied Algebra and Geometry","issn_l":"2470-6566","issn":["2470-6566"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Algebra and Geometry","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/11","display_name":"Sustainable cities and communities","score":0.6899999976158142}],"awards":[{"id":"https://openalex.org/G7472961863","display_name":null,"funder_award_id":"757983","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"}],"funders":[{"id":"https://openalex.org/F4320322327","display_name":"Knut och Alice Wallenbergs Stiftelse","ror":"https://ror.org/004hzzk67"},{"id":"https://openalex.org/F4320332999","display_name":"Horizon 2020 Framework Programme","ror":"https://ror.org/00k4n6c32"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":44,"referenced_works":["https://openalex.org/W1595559231","https://openalex.org/W1994530392","https://openalex.org/W2064206674","https://openalex.org/W2074940496","https://openalex.org/W2078626246","https://openalex.org/W2124706092","https://openalex.org/W2134022723","https://openalex.org/W2137902180","https://openalex.org/W2171902495","https://openalex.org/W2218145469","https://openalex.org/W2263126354","https://openalex.org/W2293909219","https://openalex.org/W2559655401","https://openalex.org/W2565538933","https://openalex.org/W2593380010","https://openalex.org/W2788800397","https://openalex.org/W2809090039","https://openalex.org/W2894604724","https://openalex.org/W2899748887","https://openalex.org/W2911742574","https://openalex.org/W2951520791","https://openalex.org/W2962726660","https://openalex.org/W2962782051","https://openalex.org/W2963093364","https://openalex.org/W2963446085","https://openalex.org/W2963504252","https://openalex.org/W2963524516","https://openalex.org/W2963651774","https://openalex.org/W2963669332","https://openalex.org/W2963826371","https://openalex.org/W2964072429","https://openalex.org/W2969215180","https://openalex.org/W2979900602","https://openalex.org/W2989989463","https://openalex.org/W2990138404","https://openalex.org/W2996284974","https://openalex.org/W3013909493","https://openalex.org/W3023563279","https://openalex.org/W3035635935","https://openalex.org/W3094491077","https://openalex.org/W3127516423","https://openalex.org/W3149313975","https://openalex.org/W4255213956","https://openalex.org/W4256619042"],"related_works":["https://openalex.org/W2052497030","https://openalex.org/W2156697727","https://openalex.org/W2038106543","https://openalex.org/W2885964212","https://openalex.org/W2337746918","https://openalex.org/W4311725933","https://openalex.org/W2013589960","https://openalex.org/W2356702416","https://openalex.org/W2963728095","https://openalex.org/W2949253011"],"abstract_inverted_index":{"We":[0,48,86,143],"study":[1,88],"the":[2,22,35,51,72,75,79,82,89,120,160],"family":[3],"of":[4,21,24,37,74,81,91,123,159],"functions":[5,16,38,52],"that":[6,50,61,119,136,150],"are":[7],"represented":[8,39,53],"by":[9,40,54],"a":[10,18],"linear":[11,25,43],"convolutional":[12],"network":[13],"(LCN).":[14],"These":[15],"form":[17,45],"semi-algebraic":[19],"subset":[20],"set":[23],"maps":[26],"from":[27],"input":[28],"space":[29,103,107],"to":[30,70,129],"output":[31],"space.":[32,85],"In":[33],"contrast,":[34],"families":[36],"fully":[41],"connected":[42],"networks":[44],"algebraic":[46],"sets.":[47],"observe":[49],"LCNs":[55],"can":[56,137],"be":[57,138],"identified":[58],"with":[59],"polynomials":[60],"admit":[62],"certain":[63],"factorizations,":[64],"and":[65,104,108,147,154],"we":[66],"use":[67],"this":[68],"perspective":[69],"describe":[71],"impact":[73],"network's":[76],"architecture":[77],"on":[78],"geometry":[80],"resulting":[83],"function":[84,94,102],"further":[87],"optimization":[90],"an":[92,96,124,156],"objective":[93],"over":[95],"LCN,":[97],"analyzing":[98],"critical":[99],"points":[100],"in":[101,105],"parameter":[106],"describing":[109],"dynamical":[110],"invariants":[111],"for":[112,162],"gradient":[113],"descent.":[114],"Overall,":[115],"our":[116,152],"theory":[117],"predicts":[118],"optimized":[121],"parameters":[122],"LCN":[125],"will":[126],"often":[127],"correspond":[128],"repeated":[130,141],"filters":[131,135],"across":[132],"layers,":[133],"or":[134],"decomposed":[139],"as":[140],"filters.":[142],"also":[144],"conduct":[145],"numerical":[146],"symbolic":[148],"experiments":[149],"illustrate":[151],"results":[153],"present":[155],"in-depth":[157],"analysis":[158],"landscape":[161],"small":[163],"architectures.":[164]},"counts_by_year":[{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
