{"id":"https://openalex.org/W2809871795","doi":"https://doi.org/10.1137/18m119834x","title":"Multilinear Compressive Sensing and an Application to Convolutional Linear Networks","display_name":"Multilinear Compressive Sensing and an Application to Convolutional Linear Networks","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2809871795","doi":"https://doi.org/10.1137/18m119834x","mag":"2809871795"},"language":"en","primary_location":{"id":"doi:10.1137/18m119834x","is_oa":true,"landing_page_url":"https://doi.org/10.1137/18m119834x","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/18M119834X","source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://epubs.siam.org/doi/pdf/10.1137/18M119834X","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016303776","display_name":"Fran\u00e7ois Malgouyres","orcid":"https://orcid.org/0000-0002-1213-858X"},"institutions":[{"id":"https://openalex.org/I134560555","display_name":"Universit\u00e9 Toulouse III - Paul Sabatier","ror":"https://ror.org/02v6kpv12","country_code":"FR","type":"education","lineage":["https://openalex.org/I134560555"]},{"id":"https://openalex.org/I196454796","display_name":"Institut National des Sciences Appliqu\u00e9es de Toulouse","ror":"https://ror.org/01h8pf755","country_code":"FR","type":"education","lineage":["https://openalex.org/I196454796","https://openalex.org/I4405258862"]},{"id":"https://openalex.org/I84500057","display_name":"Institut de Math\u00e9matiques de Toulouse","ror":"https://ror.org/014vp6c30","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I196454796","https://openalex.org/I4210141950","https://openalex.org/I4210152422","https://openalex.org/I4405258862","https://openalex.org/I4405258862","https://openalex.org/I4405258862","https://openalex.org/I4405259414","https://openalex.org/I84500057"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Fran\u00e7ois Malgouyres","raw_affiliation_strings":["IMT - Institut de Math\u00e9matiques de Toulouse UMR5219 (UPS IMT, F-31062 Toulouse Cedex 9, \r\nINSA Toulouse, F-31077 Toulouse,\r\nFrance UT1, F-31042 Toulouse, \r\nFrance UT2, F-31058 Toulouse,\r\n\r\nT\u00e9l\u00e9phone : 05.61.55.67.90 - France)"],"affiliations":[{"raw_affiliation_string":"IMT - Institut de Math\u00e9matiques de Toulouse UMR5219 (UPS IMT, F-31062 Toulouse Cedex 9, \r\nINSA Toulouse, F-31077 Toulouse,\r\nFrance UT1, F-31042 Toulouse, \r\nFrance UT2, F-31058 Toulouse,\r\n\r\nT\u00e9l\u00e9phone : 05.61.55.67.90 - France)","institution_ids":["https://openalex.org/I84500057","https://openalex.org/I134560555","https://openalex.org/I196454796"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5068623131","display_name":"J. M. Landsberg","orcid":null},"institutions":[{"id":"https://openalex.org/I91045830","display_name":"Texas A&M University","ror":"https://ror.org/01f5ytq51","country_code":"US","type":"education","lineage":["https://openalex.org/I91045830"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Joseph Landsberg","raw_affiliation_strings":["TAMU - Department of Mathematics [Texas] (Mailstop 3368 - Texas A&M University - College Station, TX 77843-3368 - United States)"],"affiliations":[{"raw_affiliation_string":"TAMU - Department of Mathematics [Texas] (Mailstop 3368 - Texas A&M University - College Station, TX 77843-3368 - United States)","institution_ids":["https://openalex.org/I91045830"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5016303776"],"corresponding_institution_ids":["https://openalex.org/I134560555","https://openalex.org/I196454796","https://openalex.org/I84500057"],"apc_list":null,"apc_paid":null,"fwci":0.5823,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.63152653,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"1","issue":"3","first_page":"446","last_page":"475"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11447","display_name":"Blind Source Separation Techniques","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/T11739","display_name":"Microwave Imaging and Scattering Analysis","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/restricted-isometry-property","display_name":"Restricted isometry property","score":0.6278883814811707},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5517497062683105},{"id":"https://openalex.org/keywords/identifiability","display_name":"Identifiability","score":0.4724612236022949},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.4663476347923279},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.4475480020046234},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.43978777527809143},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.42350903153419495},{"id":"https://openalex.org/keywords/compressed-sensing","display_name":"Compressed sensing","score":0.38318049907684326},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.37720635533332825},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.36400383710861206},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.35454678535461426},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3451322317123413},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.15187236666679382},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.1507643461227417},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.13653439283370972}],"concepts":[{"id":"https://openalex.org/C17902559","wikidata":"https://www.wikidata.org/wiki/Q17099734","display_name":"Restricted isometry property","level":3,"score":0.6278883814811707},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5517497062683105},{"id":"https://openalex.org/C122770356","wikidata":"https://www.wikidata.org/wiki/Q1656753","display_name":"Identifiability","level":2,"score":0.4724612236022949},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.4663476347923279},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.4475480020046234},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.43978777527809143},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.42350903153419495},{"id":"https://openalex.org/C124851039","wikidata":"https://www.wikidata.org/wiki/Q2665459","display_name":"Compressed sensing","level":2,"score":0.38318049907684326},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.37720635533332825},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36400383710861206},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.35454678535461426},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3451322317123413},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.15187236666679382},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.1507643461227417},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.13653439283370972},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/18m119834x","is_oa":true,"landing_page_url":"https://doi.org/10.1137/18m119834x","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/18M119834X","source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1703.08044","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1703.08044","pdf_url":"https://arxiv.org/pdf/1703.08044","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"text"},{"id":"pmh:oai:HAL:hal-01494267v4","is_oa":true,"landing_page_url":"https://hal.science/hal-01494267","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Mathematics of Data Science, 2019, 1 (3), pp.446-475. ⟨10.1137/18M119834X⟩","raw_type":"Journal articles"}],"best_oa_location":{"id":"doi:10.1137/18m119834x","is_oa":true,"landing_page_url":"https://doi.org/10.1137/18m119834x","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/18M119834X","source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3608357509","display_name":null,"funder_award_id":"DMS-1405348","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G591295936","display_name":null,"funder_award_id":"AF1814254","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"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2809871795.pdf","grobid_xml":"https://content.openalex.org/works/W2809871795.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W1787224781","https://openalex.org/W1902027874","https://openalex.org/W1953819449","https://openalex.org/W1970377488","https://openalex.org/W2020390700","https://openalex.org/W2024254345","https://openalex.org/W2034683677","https://openalex.org/W2048687561","https://openalex.org/W2068943743","https://openalex.org/W2078397124","https://openalex.org/W2078626246","https://openalex.org/W2095168618","https://openalex.org/W2102019642","https://openalex.org/W2118550318","https://openalex.org/W2124172487","https://openalex.org/W2127114523","https://openalex.org/W2134332047","https://openalex.org/W2137902180","https://openalex.org/W2140089781","https://openalex.org/W2140867429","https://openalex.org/W2145096794","https://openalex.org/W2150871663","https://openalex.org/W2166116275","https://openalex.org/W2195388612","https://openalex.org/W2274086169","https://openalex.org/W2559655401","https://openalex.org/W2604272474","https://openalex.org/W2611328865","https://openalex.org/W2612690371","https://openalex.org/W2612972698","https://openalex.org/W2963494673","https://openalex.org/W2963819344","https://openalex.org/W2963855280","https://openalex.org/W3099840502","https://openalex.org/W3103287012","https://openalex.org/W3104684837","https://openalex.org/W4250955649"],"related_works":["https://openalex.org/W1595179898","https://openalex.org/W2949399405","https://openalex.org/W4301619540","https://openalex.org/W4297570801","https://openalex.org/W1639246335","https://openalex.org/W116214155","https://openalex.org/W2568623927","https://openalex.org/W2952050240","https://openalex.org/W2085436476","https://openalex.org/W2025666718"],"abstract_inverted_index":{"We":[0,93,219,254,273],"study":[1],"a":[2,41,46,51,105,183,203,216,222,259,269,275],"deep":[3,175,264],"linear":[4,43,177,266,271],"network":[5,101,138,178,267,294],"endowed":[6],"with":[7,125,236,258],"the":[8,25,28,70,73,76,82,87,90,100,114,117,120,126,130,137,162,189,194,198,209,212,228,234,237,243,250,256,263,278,281,293,302],"following":[9],"structure:":[10],"A":[11,289],"matrix":[12],"$X$":[13,62,80],"is":[14,37,57,63,85,193,268,284],"obtained":[15,38],"by":[16,39],"multiplying":[17],"$K$":[18],"matrices":[19],"(called":[20],"factors":[21,66,78,122,213],"and":[22,65,79,96,165,205,224],"corresponding":[23],"to":[24,45,145,155,215,299],"action":[26,31],"of":[27,32,48,55,75,211,233,280],"layers).":[29],"The":[30,53,109],"each":[33],"layer":[34],"(i.e.,":[35,81,129],"factor)":[36],"applying":[40],"fixed":[42],"operator":[44],"vector":[47],"parameters":[49,118],"satisfying":[50],"constraint.":[52],"number":[54],"layers":[56],"not":[58,287],"limited.":[59],"Assuming":[60],"that":[61,113,149,173,187,249],"given":[64],"have":[67],"been":[68],"estimated,":[69],"error":[71,115,128],"between":[72],"product":[74],"estimated":[77],"reconstruction":[83,127],"error)":[84],"either":[86],"statistical":[88],"or":[89],"empirical":[91],"risk.":[92],"provide":[94,202,221],"necessary":[95,204,223],"sufficient":[97,206,225],"conditions":[98,135],"on":[99,116,136,277,292],"topology":[102,295],"under":[103,133],"which":[104,247,283],"stability":[106,110,251],"property":[107,111,252],"holds.":[108,253,304],"requires":[112],"defining":[119],"near-optimal":[121],"scales":[123],"linearly":[124],"risk).":[131],"Therefore,":[132],"these":[134],"topology,":[139],"any":[140,174],"successful":[141],"learning":[142],"task":[143],"leads":[144],"stably":[146],"defined":[147],"features":[148],"can":[150,179,296],"be":[151,180,297],"interpreted.":[152],"In":[153],"order":[154],"do":[156],"so,":[157],"we":[158,171,201],"first":[159],"evaluate":[160],"how":[161],"Segre":[163,190],"embedding":[164],"its":[166],"inverse":[167],"distort":[168],"distances.":[169],"Then":[170],"show":[172],"structured":[176,265],"cast":[181],"as":[182],"generic":[184],"multilinear":[185],"problem":[186],"uses":[188],"embedding.":[191],"This":[192],"tensorial":[195,199],"lifting.":[196],"Using":[197],"lifting,":[200],"condition":[207,226,276,303],"for":[208],"identifiability":[210],"up":[214],"scale":[217],"rearrangement.":[218],"finally":[220],"called":[227],"deep-Null":[229],"Space":[230,240],"Property":[231,241],"(because":[232],"analogy":[235],"usual":[238],"Null":[239],"in":[242],"compressed":[244],"sensing":[245],"framework)":[246],"guarantees":[248],"illustrate":[255],"theory":[257],"practical":[260],"example":[261],"where":[262],"convolutional":[270],"network.":[272],"obtain":[274],"scattering":[279],"supports":[282],"strong":[285],"but":[286],"empty.":[288],"simple":[290],"test":[291,300],"implemented":[298],"whether":[301]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-15T08:11:43.952461","created_date":"2025-10-10T00:00:00"}