{"id":"https://openalex.org/W2798620024","doi":"https://doi.org/10.1109/acssc.2017.8335623","title":"Hilbert space geometry of quadratic covariance bounds","display_name":"Hilbert space geometry of quadratic covariance bounds","publication_year":2017,"publication_date":"2017-10-01","ids":{"openalex":"https://openalex.org/W2798620024","doi":"https://doi.org/10.1109/acssc.2017.8335623","mag":"2798620024"},"language":"en","primary_location":{"id":"doi:10.1109/acssc.2017.8335623","is_oa":false,"landing_page_url":"https://doi.org/10.1109/acssc.2017.8335623","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2017 51st Asilomar Conference on Signals, Systems, and Computers","raw_type":"proceedings-article"},"type":"conference-paper","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Stephen D. Howard","orcid":null},"institutions":[{"id":"https://openalex.org/I1303474014","display_name":"Defence Science and Technology Group","ror":"https://ror.org/05ddrvt52","country_code":"AU","type":"government","lineage":["https://openalex.org/I1303474014","https://openalex.org/I2801453606","https://openalex.org/I3139952251"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Stephen D. Howard","raw_affiliation_strings":["Defence Science and Technology Group, Edinburgh, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Defence Science and Technology Group, Edinburgh, Australia","institution_ids":["https://openalex.org/I1303474014"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030306670","display_name":"William Moran","orcid":"https://orcid.org/0000-0001-6219-2341"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"William Moran","raw_affiliation_strings":["School of Engineering, University of Melbourne, Melbourne, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Engineering, University of Melbourne, Melbourne, Australia","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031972705","display_name":"Pooria Pakrooh","orcid":"https://orcid.org/0000-0002-6468-3133"},"institutions":[{"id":"https://openalex.org/I92446798","display_name":"Colorado State University","ror":"https://ror.org/03k1gpj17","country_code":"US","type":"education","lineage":["https://openalex.org/I92446798"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Pooria Pakrooh","raw_affiliation_strings":["Department of Mathematics, Colorado state University, Fort Collins, CO, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Colorado state University, Fort Collins, CO, USA","institution_ids":["https://openalex.org/I92446798"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080469112","display_name":"Louis L. Scharf","orcid":"https://orcid.org/0000-0003-1764-9335"},"institutions":[{"id":"https://openalex.org/I92446798","display_name":"Colorado State University","ror":"https://ror.org/03k1gpj17","country_code":"US","type":"education","lineage":["https://openalex.org/I92446798"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Louis L. Scharf","raw_affiliation_strings":["Department of Mathematics, Colorado state University, Fort Collins, CO, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Colorado state University, Fort Collins, CO, USA","institution_ids":["https://openalex.org/I92446798"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":"9","issue":null,"first_page":"1578","last_page":"1581"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10931","display_name":"Direction-of-Arrival Estimation Techniques","score":0.982699990272522,"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"}},"topics":[{"id":"https://openalex.org/T10931","display_name":"Direction-of-Arrival Estimation Techniques","score":0.982699990272522,"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9807000160217285,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10243","display_name":"Statistical Methods and Bayesian Inference","score":0.9793999791145325,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/covariance","display_name":"Covariance","score":0.8133995532989502},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8033648729324341},{"id":"https://openalex.org/keywords/rational-quadratic-covariance-function","display_name":"Rational quadratic covariance function","score":0.7526810169219971},{"id":"https://openalex.org/keywords/covariance-intersection","display_name":"Covariance intersection","score":0.5508079528808594},{"id":"https://openalex.org/keywords/covariance-function","display_name":"Covariance function","score":0.5489169359207153},{"id":"https://openalex.org/keywords/mat\u00e9rn-covariance-function","display_name":"Mat\u00e9rn covariance function","score":0.5308524370193481},{"id":"https://openalex.org/keywords/estimation-of-covariance-matrices","display_name":"Estimation of covariance matrices","score":0.48058319091796875},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.441921591758728},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4243202209472656},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.4189390540122986},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4140388071537018},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.4132426679134369},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3251078128814697},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2402796745300293},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.23995497822761536}],"concepts":[{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.8133995532989502},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8033648729324341},{"id":"https://openalex.org/C148893098","wikidata":"https://www.wikidata.org/wiki/Q7295778","display_name":"Rational quadratic covariance function","level":5,"score":0.7526810169219971},{"id":"https://openalex.org/C83042196","wikidata":"https://www.wikidata.org/wiki/Q5178898","display_name":"Covariance intersection","level":4,"score":0.5508079528808594},{"id":"https://openalex.org/C137250428","wikidata":"https://www.wikidata.org/wiki/Q5178897","display_name":"Covariance function","level":3,"score":0.5489169359207153},{"id":"https://openalex.org/C118006245","wikidata":"https://www.wikidata.org/wiki/Q6792079","display_name":"Mat\u00e9rn covariance function","level":5,"score":0.5308524370193481},{"id":"https://openalex.org/C180877172","wikidata":"https://www.wikidata.org/wiki/Q5401390","display_name":"Estimation of covariance matrices","level":3,"score":0.48058319091796875},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.441921591758728},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4243202209472656},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.4189390540122986},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4140388071537018},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.4132426679134369},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3251078128814697},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2402796745300293},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.23995497822761536}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/acssc.2017.8335623","is_oa":false,"landing_page_url":"https://doi.org/10.1109/acssc.2017.8335623","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2017 51st Asilomar Conference on Signals, Systems, and Computers","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W639989723","https://openalex.org/W1479979375","https://openalex.org/W1887457978","https://openalex.org/W2040832208","https://openalex.org/W2079640253","https://openalex.org/W2167933251","https://openalex.org/W2524449277","https://openalex.org/W4211251910","https://openalex.org/W6628836592"],"related_works":["https://openalex.org/W2022823194","https://openalex.org/W2126916073","https://openalex.org/W3211883524","https://openalex.org/W2109377650","https://openalex.org/W1987404909","https://openalex.org/W4386993326","https://openalex.org/W1627656821","https://openalex.org/W3025277714","https://openalex.org/W2018001152","https://openalex.org/W2050026105"],"abstract_inverted_index":{"In":[0,138],"this":[1,168],"paper,":[2],"we":[3,141],"study":[4],"the":[5,12,32,39,42,48,52,86,111,116,119,127,131,135,161],"geometry":[6,165],"of":[7,22,41,68,81,105,118,121],"quadratic":[8,82],"covariance":[9,34,83,113],"bounds":[10,84],"on":[11,31],"estimation":[13],"error":[14,33,43],"covariance,":[15,153],"in":[16,64,167],"a":[17,28,65,78],"properly":[18],"defined":[19,58],"Hilbert":[20,66,163],"space":[21,67,158,164],"random":[23,71],"variables.":[24,72],"We":[25],"show":[26,142],"that":[27,143],"lower":[29],"bound":[30,94,114],"may":[35],"be":[36],"represented":[37],"by":[38,51,97,134],"Grammian":[40,56,120],"score":[44],"after":[45,124],"projection":[46,125],"onto":[47,126],"subspace":[49,128,132],"spanned":[50,133],"measurement":[53,100],"scores.":[54,101,137],"The":[55],"is":[57,95,115],"with":[59,149],"respect":[60],"to":[61,130],"inner":[62],"products":[63],"second":[69],"order":[70],"This":[73],"geometric":[74],"result":[75],"holds":[76],"for":[77,144,160],"large":[79],"class":[80],"including":[85],"Barankin,":[87],"Cram\u00e9r-Rao,":[88],"and":[89,108],"Bhattacharyya":[90],"bounds,":[91],"where":[92],"each":[93],"characterized":[96],"its":[98],"corresponding":[99],"When":[102],"parameters":[103,107],"consist":[104],"essential":[106,122],"nuisance":[109,136],"parameters,":[110],"Cram\u00e9r-Rao":[112],"inverse":[117],"scores":[123],"orthogonal":[129],"two":[139],"examples,":[140],"complex":[145],"multivariate":[146],"normal":[147],"measurements":[148],"parameterized":[150],"mean":[151],"or":[152],"there":[154],"exist":[155],"well-known":[156],"Euclidean":[157],"geometries":[159],"general":[162],"derived":[166],"paper.":[169]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-07-14T23:27:15.235271","created_date":"2025-10-10T00:00:00"}