{"id":"https://openalex.org/W2963669613","doi":"https://doi.org/10.1137/130935434","title":"Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem","display_name":"Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2963669613","doi":"https://doi.org/10.1137/130935434","mag":"2963669613","pmid":"https://pubmed.ncbi.nlm.nih.gov/25699132"},"language":"en","primary_location":{"id":"doi:10.1137/130935434","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130935434","pdf_url":null,"source":{"id":"https://openalex.org/S152600803","display_name":"SIAM Journal on Imaging Sciences","issn_l":"1936-4954","issn":["1936-4954"],"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 Imaging Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://stars.library.ucf.edu/facultybib2010/6621","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020452709","display_name":"Eugene Katsevich","orcid":"https://orcid.org/0000-0003-0598-2050"},"institutions":[{"id":"https://openalex.org/I20089843","display_name":"Princeton University","ror":"https://ror.org/00hx57361","country_code":"US","type":"education","lineage":["https://openalex.org/I20089843"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"E. Katsevich","raw_affiliation_strings":["Department of Mathematics, Princeton University, Princeton, NJ 08544"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Princeton University, Princeton, NJ 08544","institution_ids":["https://openalex.org/I20089843"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034138273","display_name":"Alexander Katsevich","orcid":"https://orcid.org/0000-0002-6586-1210"},"institutions":[{"id":"https://openalex.org/I106165777","display_name":"University of Central Florida","ror":"https://ror.org/036nfer12","country_code":"US","type":"education","lineage":["https://openalex.org/I106165777"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"A. Katsevich","raw_affiliation_strings":["Department of Mathematics, University of Central Florida, Orlando, FL 32816"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Central Florida, Orlando, FL 32816","institution_ids":["https://openalex.org/I106165777"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062844820","display_name":"Amit Singer","orcid":"https://orcid.org/0000-0002-6975-7955"},"institutions":[{"id":"https://openalex.org/I20089843","display_name":"Princeton University","ror":"https://ror.org/00hx57361","country_code":"US","type":"education","lineage":["https://openalex.org/I20089843"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"A. Singer","raw_affiliation_strings":["Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544-1000"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544-1000","institution_ids":["https://openalex.org/I20089843"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5020452709"],"corresponding_institution_ids":["https://openalex.org/I20089843"],"apc_list":null,"apc_paid":null,"fwci":9.5495,"has_fulltext":false,"cited_by_count":69,"citation_normalized_percentile":{"value":0.97264875,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"8","issue":"1","first_page":"126","last_page":"185"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10857","display_name":"Advanced Electron Microscopy Techniques and Applications","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/1315","display_name":"Structural Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T10857","display_name":"Advanced Electron Microscopy Techniques and Applications","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/1315","display_name":"Structural Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9959999918937683,"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/T12039","display_name":"Electron and X-Ray Spectroscopy Techniques","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/2508","display_name":"Surfaces, Coatings and Films"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/covariance-matrix","display_name":"Covariance matrix","score":0.6784392595291138},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.6751986742019653},{"id":"https://openalex.org/keywords/estimation-of-covariance-matrices","display_name":"Estimation of covariance matrices","score":0.6385685205459595},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.587992787361145},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.5187609195709229},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.5056300163269043},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5018055438995361},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.47705212235450745},{"id":"https://openalex.org/keywords/principal-component-analysis","display_name":"Principal component analysis","score":0.46263545751571655},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.45202115178108215},{"id":"https://openalex.org/keywords/filter","display_name":"Filter (signal processing)","score":0.4143102467060089},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3247080445289612},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.30446866154670715},{"id":"https://openalex.org/keywords/computer-vision","display_name":"Computer vision","score":0.20928528904914856},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.15538761019706726},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11841842532157898}],"concepts":[{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.6784392595291138},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.6751986742019653},{"id":"https://openalex.org/C180877172","wikidata":"https://www.wikidata.org/wiki/Q5401390","display_name":"Estimation of covariance matrices","level":3,"score":0.6385685205459595},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.587992787361145},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.5187609195709229},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5056300163269043},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5018055438995361},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.47705212235450745},{"id":"https://openalex.org/C27438332","wikidata":"https://www.wikidata.org/wiki/Q2873","display_name":"Principal component analysis","level":2,"score":0.46263545751571655},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.45202115178108215},{"id":"https://openalex.org/C106131492","wikidata":"https://www.wikidata.org/wiki/Q3072260","display_name":"Filter (signal processing)","level":2,"score":0.4143102467060089},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3247080445289612},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.30446866154670715},{"id":"https://openalex.org/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.20928528904914856},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.15538761019706726},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11841842532157898},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/130935434","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130935434","pdf_url":null,"source":{"id":"https://openalex.org/S152600803","display_name":"SIAM Journal on Imaging Sciences","issn_l":"1936-4954","issn":["1936-4954"],"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 Imaging Sciences","raw_type":"journal-article"},{"id":"pmid:25699132","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/25699132","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM journal on imaging sciences","raw_type":null},{"id":"pmh:oai:stars.library.ucf.edu:facultybib2010-7620","is_oa":true,"landing_page_url":"https://stars.library.ucf.edu/facultybib2010/6621","pdf_url":null,"source":{"id":"https://openalex.org/S4210172555","display_name":"Journal of International Crisis and Risk Communication Research","issn_l":"2576-0017","issn":["2576-0017","2576-0025"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Faculty Bibliography 2010s","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.755.1224","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.755.1224","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1309.1737.pdf","raw_type":"text"},{"id":"pmh:oai:europepmc.org:3296218","is_oa":false,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/4331039","pdf_url":null,"source":{"id":"https://openalex.org/S4306400806","display_name":"Europe PMC (PubMed Central)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1303153112","host_organization_name":"European Bioinformatics Institute","host_organization_lineage":["https://openalex.org/I1303153112"],"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"}],"best_oa_location":{"id":"pmh:oai:stars.library.ucf.edu:facultybib2010-7620","is_oa":true,"landing_page_url":"https://stars.library.ucf.edu/facultybib2010/6621","pdf_url":null,"source":{"id":"https://openalex.org/S4210172555","display_name":"Journal of International Crisis and Risk Communication Research","issn_l":"2576-0017","issn":["2576-0017","2576-0025"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Faculty Bibliography 2010s","raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":72,"referenced_works":["https://openalex.org/W29265268","https://openalex.org/W636061944","https://openalex.org/W1520752838","https://openalex.org/W1548811283","https://openalex.org/W1559907478","https://openalex.org/W1576347883","https://openalex.org/W1577655666","https://openalex.org/W1663973292","https://openalex.org/W1736339626","https://openalex.org/W1920010643","https://openalex.org/W1967134148","https://openalex.org/W1978702162","https://openalex.org/W2001152940","https://openalex.org/W2006025109","https://openalex.org/W2019549276","https://openalex.org/W2024183250","https://openalex.org/W2028660771","https://openalex.org/W2029094924","https://openalex.org/W2029626785","https://openalex.org/W2032347855","https://openalex.org/W2036823849","https://openalex.org/W2039953809","https://openalex.org/W2041969899","https://openalex.org/W2044098317","https://openalex.org/W2044758663","https://openalex.org/W2047071281","https://openalex.org/W2049633694","https://openalex.org/W2051925784","https://openalex.org/W2057197276","https://openalex.org/W2058583833","https://openalex.org/W2060581589","https://openalex.org/W2066459155","https://openalex.org/W2067145564","https://openalex.org/W2072951627","https://openalex.org/W2078096876","https://openalex.org/W2087995885","https://openalex.org/W2092709807","https://openalex.org/W2097714737","https://openalex.org/W2098056745","https://openalex.org/W2099210013","https://openalex.org/W2100758571","https://openalex.org/W2104234755","https://openalex.org/W2106005123","https://openalex.org/W2106084579","https://openalex.org/W2107411554","https://openalex.org/W2118550318","https://openalex.org/W2134189568","https://openalex.org/W2139916860","https://openalex.org/W2148575499","https://openalex.org/W2149360962","https://openalex.org/W2150790075","https://openalex.org/W2150938882","https://openalex.org/W2155688047","https://openalex.org/W2160341948","https://openalex.org/W2160559764","https://openalex.org/W2160880244","https://openalex.org/W2164659462","https://openalex.org/W2165593770","https://openalex.org/W2167034439","https://openalex.org/W2174160981","https://openalex.org/W2294798173","https://openalex.org/W2323284653","https://openalex.org/W2340736170","https://openalex.org/W2341535507","https://openalex.org/W2963893933","https://openalex.org/W2963927955","https://openalex.org/W2965497096","https://openalex.org/W3098045837","https://openalex.org/W3099609308","https://openalex.org/W3103380172","https://openalex.org/W3105364218","https://openalex.org/W4235941905"],"related_works":["https://openalex.org/W3000541886","https://openalex.org/W4297802450","https://openalex.org/W2921280830","https://openalex.org/W2126916073","https://openalex.org/W2887132723","https://openalex.org/W2572601863","https://openalex.org/W2038723318","https://openalex.org/W1679731869","https://openalex.org/W2886934452","https://openalex.org/W2024369332"],"abstract_inverted_index":{"In":[0,53,130,200],"cryo-electron":[1],"microscopy":[2],"(cryo-EM),":[3],"a":[4,7,11,18,65,84,135,196,234,245],"microscope":[5],"generates":[6],"top":[8],"view":[9],"of":[10,13,17,22,35,50,76,80,83,95,99,120,138,144,177,185,274,280],"sample":[12],"randomly":[14],"oriented":[15],"copies":[16],"molecule.":[19,52,85],"The":[20,70,187],"problem":[21,72,137,147],"single":[23],"particle":[24],"reconstruction":[25],"(SPR)":[26],"from":[27,141],"cryo-EM":[28,202],"is":[29,73,115,217,252],"to":[30,44,106,195,207,237,277],"use":[31],"the":[32,46,51,56,74,78,92,96,100,108,112,121,127,175,178,183,193,201,204,213,238,272],"resulting":[33],"set":[34],"noisy":[36,142],"two-dimensional":[37],"projection":[38,214],"images":[39],"taken":[40],"at":[41],"unknown":[42],"directions":[43],"reconstruct":[45],"three-dimensional":[47],"(3D)":[48],"structure":[49],"some":[54],"situations,":[55],"molecule":[57],"under":[58],"examination":[59],"exhibits":[60],"structural":[61,228],"variability,":[62],"which":[63,210,248],"poses":[64],"fundamental":[66],"challenge":[67],"in":[68,221,241,247],"SPR.":[69],"heterogeneity":[71,109,173],"task":[75],"mapping":[77],"space":[79],"conformational":[81],"states":[82],"It":[86],"has":[87,148],"been":[88],"previously":[89],"suggested":[90],"that":[91],"leading":[93],"eigenvectors":[94],"covariance":[97,113,139,180,215,222],"matrix":[98,114,152,181],"3D":[101],"molecules":[102,122,128],"can":[103,189,256],"be":[104,190,208,257],"used":[105],"solve":[107],"problem.":[110],"Estimating":[111],"challenging,":[116],"since":[117],"only":[118],"projections":[119,143],"are":[123,170],"observed,":[124],"but":[125],"not":[126],"themselves.":[129],"this":[131,249],"paper,":[132],"we":[133,211],"formulate":[134],"general":[136],"estimation":[140,223],"samples.":[145],"This":[146],"intimate":[149],"connections":[150],"with":[151],"completion":[153],"problems":[154,226],"and":[155,164,254],"high-dimensional":[156],"principal":[157],"component":[158],"analysis.":[159],"We":[160,243,264],"propose":[161],"an":[162,218],"estimator":[163,188],"prove":[165],"its":[166,261],"consistency.":[167],"When":[168],"there":[169],"finitely":[171],"many":[172],"classes,":[174],"spectrum":[176],"estimated":[179],"reveals":[182],"number":[184],"classes.":[186],"found":[191],"as":[192],"solution":[194],"certain":[197],"linear":[198,205,250],"system.":[199],"case,":[203],"operator":[206,251],"inverted,":[209],"term":[212],"transform,":[216],"important":[219],"object":[220],"for":[224],"tomographic":[225],"involving":[227],"variation.":[229],"Inverting":[230],"it":[231],"involves":[232],"applying":[233],"filter":[235,240],"akin":[236],"ramp":[239],"tomography.":[242],"design":[244],"basis":[246],"sparse":[253],"thus":[255],"tractably":[258],"inverted":[259],"despite":[260],"large":[262],"size.":[263],"demonstrate":[265],"via":[266],"numerical":[267],"experiments":[268],"on":[269],"synthetic":[270],"datasets":[271],"robustness":[273],"our":[275],"algorithm":[276],"high":[278],"levels":[279],"noise.":[281]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":14},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":7},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}