{"id":"https://openalex.org/W2782090365","doi":"https://doi.org/10.1137/17m1141503","title":"A Probabilistic Subspace Bound with Application to Active Subspaces","display_name":"A Probabilistic Subspace Bound with Application to Active Subspaces","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2782090365","doi":"https://doi.org/10.1137/17m1141503","mag":"2782090365"},"language":"en","primary_location":{"id":"doi:10.1137/17m1141503","is_oa":false,"landing_page_url":"https://doi.org/10.1137/17m1141503","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1801.00682","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086932068","display_name":"John T. Holodnak","orcid":"https://orcid.org/0000-0002-6603-3046"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"John T. Holodnak","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036959652","display_name":"Ilse C. F. Ipsen","orcid":"https://orcid.org/0000-0001-5645-5854"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ilse C. F. Ipsen","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0001-5645-5854","affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5040088245","display_name":"Ralph C. Smith","orcid":"https://orcid.org/0000-0001-7434-5712"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ralph C. Smith","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5338,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.57192304,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"39","issue":"3","first_page":"1208","last_page":"1220"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9994000196456909,"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":0.9994000196456909,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9990000128746033,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.998199999332428,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.7975085973739624},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6301823258399963},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5871503949165344},{"id":"https://openalex.org/keywords/monte-carlo-method","display_name":"Monte Carlo method","score":0.5852708220481873},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5544961094856262},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.5339931845664978},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5092326402664185},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5081729888916016},{"id":"https://openalex.org/keywords/sampling","display_name":"Sampling (signal processing)","score":0.4826686680316925},{"id":"https://openalex.org/keywords/intrinsic-dimension","display_name":"Intrinsic dimension","score":0.44982004165649414},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.44972822070121765},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.43403440713882446},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.41578954458236694},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38620153069496155},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.35280758142471313},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2287571132183075},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.15883579850196838},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.09193110466003418},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08545157313346863}],"concepts":[{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.7975085973739624},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6301823258399963},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5871503949165344},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.5852708220481873},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5544961094856262},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.5339931845664978},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5092326402664185},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5081729888916016},{"id":"https://openalex.org/C140779682","wikidata":"https://www.wikidata.org/wiki/Q210868","display_name":"Sampling (signal processing)","level":3,"score":0.4826686680316925},{"id":"https://openalex.org/C30732413","wikidata":"https://www.wikidata.org/wiki/Q17092636","display_name":"Intrinsic dimension","level":3,"score":0.44982004165649414},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.44972822070121765},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.43403440713882446},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.41578954458236694},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38620153069496155},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.35280758142471313},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2287571132183075},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.15883579850196838},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.09193110466003418},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08545157313346863},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C106131492","wikidata":"https://www.wikidata.org/wiki/Q3072260","display_name":"Filter (signal processing)","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/17m1141503","is_oa":false,"landing_page_url":"https://doi.org/10.1137/17m1141503","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1801.00682","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1801.00682","pdf_url":"https://arxiv.org/pdf/1801.00682","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,"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":"mag:2782090365","is_oa":true,"landing_page_url":"http://arxiv.org/pdf/1801.00682.pdf","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,"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1801.00682","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1801.00682","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,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1801.00682","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1801.00682","pdf_url":"https://arxiv.org/pdf/1801.00682","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,"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":[],"awards":[{"id":"https://openalex.org/G1645119126","display_name":null,"funder_award_id":"AC05-00OR22725","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G203625172","display_name":null,"funder_award_id":"DE--AC05--00OR22725","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G2411683797","display_name":null,"funder_award_id":"DEAC05- 00OR22725","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2988778103","display_name":null,"funder_award_id":"DE-AC05?","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G3945850812","display_name":null,"funder_award_id":"00OR22725","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4005865737","display_name":null,"funder_award_id":"FA9550-15-1- 0299","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G4096326766","display_name":null,"funder_award_id":"FA8750-12-C0323","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G454454283","display_name":null,"funder_award_id":"DE-AC05-00OR22725","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G4713059963","display_name":null,"funder_award_id":"FA8750","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G773731761","display_name":null,"funder_award_id":"FA8750-12-C-0323","funder_id":"https://openalex.org/F4320338294","funder_display_name":"Air Force Research Laboratory"},{"id":"https://openalex.org/G8573235543","display_name":"EAGER: Numerical Accuracy of Randomized Algorithms for Matrix Multiplication and Least Squares","funder_award_id":"1145383","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8906985441","display_name":null,"funder_award_id":"00OR22725","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320332815","display_name":"Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"},{"id":"https://openalex.org/F4320338294","display_name":"Air Force Research Laboratory","ror":"https://ror.org/02e2egq70"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2782090365.pdf","grobid_xml":"https://content.openalex.org/works/W2782090365.grobid-xml"},"referenced_works_count":27,"referenced_works":["https://openalex.org/W575374134","https://openalex.org/W634109894","https://openalex.org/W658559791","https://openalex.org/W781784501","https://openalex.org/W1493199243","https://openalex.org/W1804110266","https://openalex.org/W1836047100","https://openalex.org/W1977824907","https://openalex.org/W2014042806","https://openalex.org/W2035363534","https://openalex.org/W2053469438","https://openalex.org/W2059586807","https://openalex.org/W2117756735","https://openalex.org/W2168016228","https://openalex.org/W2278156475","https://openalex.org/W2322299560","https://openalex.org/W2516218596","https://openalex.org/W2610857016","https://openalex.org/W2949583171","https://openalex.org/W2963441460","https://openalex.org/W2963459001","https://openalex.org/W2964089577","https://openalex.org/W2964214436","https://openalex.org/W3103869760","https://openalex.org/W3125756894","https://openalex.org/W4206039841","https://openalex.org/W4245636761"],"related_works":["https://openalex.org/W1979750072","https://openalex.org/W2270821680","https://openalex.org/W2954018354","https://openalex.org/W2086486316","https://openalex.org/W1499670810","https://openalex.org/W2951791000","https://openalex.org/W2727527704","https://openalex.org/W2921360821","https://openalex.org/W2588219153","https://openalex.org/W3016868194","https://openalex.org/W1657368493","https://openalex.org/W2964265363","https://openalex.org/W2489120772","https://openalex.org/W2964013305","https://openalex.org/W2905635580","https://openalex.org/W2804080355","https://openalex.org/W2247462232","https://openalex.org/W2974353649","https://openalex.org/W3203654258","https://openalex.org/W2965741480"],"abstract_inverted_index":{"Given":[0],"a":[1,14,92,117,122,131],"real":[2],"symmetric":[3],"positive":[4],"semidefinite":[5],"matrix":[6,41],"$E$,":[7],"and":[8,112,133,142],"an":[9],"approximation":[10,58],"$S$":[11,30,83,113],"that":[12,100,155],"is":[13,84,114,121,130,175],"sum":[15],"of":[16,51,59,75,97,110,165],"$n$":[17],"independent":[18],"matrix-valued":[19],"random":[20,63],"variables,":[21],"we":[22,90],"present":[23,91],"bounds":[24,35],"on":[25,39,45,79,94,137],"the":[26,40,46,56,73,95,104,107,138,145,149,163,171],"relative":[27],"error":[28],"in":[29,162],"due":[31],"to":[32,147],"randomization.":[33],"The":[34],"do":[36],"not":[37],"depend":[38],"dimensions":[42],"but":[43,65],"only":[44],"numerical":[47],"rank":[48],"(intrinsic":[49],"dimension)":[50],"$E$.":[52],"Our":[53],"approach":[54],"resembles":[55],"low-rank":[57],"kernel":[60],"matrices":[61],"from":[62],"features,":[64],"our":[66],"accuracy":[67],"measures":[68],"are":[69],"more":[70],"stringent.":[71],"In":[72],"context":[74],"parameter":[76],"selection":[77],"based":[78],"active":[80],"subspaces,":[81],"where":[82],"computed":[85],"via":[86],"Monte":[87,156],"Carlo":[88,157],"sampling,":[89],"bound":[93,136],"number":[96],"samples":[98],"so":[99],"with":[101],"high":[102],"probability":[103],"angle":[105],"between":[106],"dominant":[108],"subspaces":[109],"$E$":[111],"less":[115],"than":[116],"user-specified":[118],"tolerance.":[119],"This":[120],"substantial":[123],"improvement":[124],"over":[125],"existing":[126],"work,":[127],"as":[128,168,170],"it":[129,143],"nonasymptotic":[132],"fully":[134],"explicit":[135],"sampling":[139,158],"amount":[140],"$n$,":[141],"allows":[144],"user":[146],"tune":[148],"success":[150],"probability.":[151],"It":[152],"also":[153],"suggests":[154],"can":[159],"be":[160],"efficient":[161],"presence":[164],"many":[166],"parameters,":[167],"long":[169],"underlying":[172],"function":[173],"$f$":[174],"sufficiently":[176],"smooth.":[177]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2025-10-10T00:00:00"}