{"id":"https://openalex.org/W2014509644","doi":"https://doi.org/10.1109/tsp.2014.2360826","title":"Regularized &lt;formula formulatype=\"inline\"&gt;&lt;tex Notation=\"TeX\"&gt;$M$&lt;/tex&gt; &lt;/formula&gt;-Estimators of Scatter Matrix","display_name":"Regularized &lt;formula formulatype=\"inline\"&gt;&lt;tex Notation=\"TeX\"&gt;$M$&lt;/tex&gt; &lt;/formula&gt;-Estimators of Scatter Matrix","publication_year":2014,"publication_date":"2014-09-29","ids":{"openalex":"https://openalex.org/W2014509644","doi":"https://doi.org/10.1109/tsp.2014.2360826","mag":"2014509644"},"language":"en","primary_location":{"id":"doi:10.1109/tsp.2014.2360826","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2014.2360826","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Signal Processing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1405.2528","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031785324","display_name":"Esa Ollila","orcid":"https://orcid.org/0000-0002-0898-5313"},"institutions":[{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]},{"id":"https://openalex.org/I4210121626","display_name":"Signal Processing (United States)","ror":"https://ror.org/021gzyw51","country_code":"US","type":"company","lineage":["https://openalex.org/I4210121626"]}],"countries":["FI","US"],"is_corresponding":true,"raw_author_name":"Esa Ollila","raw_affiliation_strings":["Department of Signal Processing and Acoustics, Aalto University, FINLAND","Department of Signal Processing and Acoustics, Aalto University, Finland#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Signal Processing and Acoustics, Aalto University, FINLAND","institution_ids":["https://openalex.org/I9927081","https://openalex.org/I4210121626"]},{"raw_affiliation_string":"Department of Signal Processing and Acoustics, Aalto University, Finland#TAB#","institution_ids":["https://openalex.org/I9927081"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5057170468","display_name":"David E. Tyler","orcid":"https://orcid.org/0000-0003-2040-6318"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David E. Tyler","raw_affiliation_strings":["Department of Statistics & Biostatistics, The State University of New Jersey, Piscataway, NJ","[Department of Statistics & Biostatistics, The State University of New Jersey, Piscataway, NJ]"],"affiliations":[{"raw_affiliation_string":"Department of Statistics & Biostatistics, The State University of New Jersey, Piscataway, NJ","institution_ids":["https://openalex.org/I102322142"]},{"raw_affiliation_string":"[Department of Statistics & Biostatistics, The State University of New Jersey, Piscataway, NJ]","institution_ids":["https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5031785324"],"corresponding_institution_ids":["https://openalex.org/I4210121626","https://openalex.org/I9927081"],"apc_list":null,"apc_paid":null,"fwci":186.5917,"has_fulltext":false,"cited_by_count":115,"citation_normalized_percentile":{"value":0.99942922,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"62","issue":"22","first_page":"6059","last_page":"6070"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10891","display_name":"Radar Systems and Signal Processing","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"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/T10891","display_name":"Radar Systems and Signal Processing","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"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/T11038","display_name":"Advanced SAR Imaging Techniques","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.995199978351593,"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/mathematics","display_name":"Mathematics","score":0.7924413084983826},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.6629051566123962},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5026004314422607},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.48895561695098877},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.4665068984031677},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.39530354738235474},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3409423530101776},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.2037748396396637},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.18318268656730652},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13677945733070374},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.12986430525779724}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7924413084983826},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.6629051566123962},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5026004314422607},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.48895561695098877},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.4665068984031677},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.39530354738235474},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3409423530101776},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2037748396396637},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.18318268656730652},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13677945733070374},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.12986430525779724}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tsp.2014.2360826","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2014.2360826","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Signal Processing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1405.2528","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1405.2528","pdf_url":"https://arxiv.org/pdf/1405.2528","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":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1405.2528","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1405.2528","pdf_url":"https://arxiv.org/pdf/1405.2528","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":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W1965928820","https://openalex.org/W1973176871","https://openalex.org/W1974640819","https://openalex.org/W1979573625","https://openalex.org/W1984614739","https://openalex.org/W1992511687","https://openalex.org/W1994065850","https://openalex.org/W2008330147","https://openalex.org/W2008418230","https://openalex.org/W2014937354","https://openalex.org/W2021553162","https://openalex.org/W2025907885","https://openalex.org/W2026239436","https://openalex.org/W2037062671","https://openalex.org/W2051108596","https://openalex.org/W2053583253","https://openalex.org/W2062125287","https://openalex.org/W2063926307","https://openalex.org/W2085653682","https://openalex.org/W2093608496","https://openalex.org/W2107820823","https://openalex.org/W2109363365","https://openalex.org/W2115972246","https://openalex.org/W2119933098","https://openalex.org/W2124759134","https://openalex.org/W2125341398","https://openalex.org/W2132555912","https://openalex.org/W2147116730","https://openalex.org/W2160537546","https://openalex.org/W2161503240","https://openalex.org/W2170257405","https://openalex.org/W2489822048","https://openalex.org/W3004366399","https://openalex.org/W4238299808","https://openalex.org/W6685397067"],"related_works":["https://openalex.org/W571281153","https://openalex.org/W2949403936","https://openalex.org/W4301247218","https://openalex.org/W4245953108","https://openalex.org/W1977619534","https://openalex.org/W2236763869","https://openalex.org/W2097626286","https://openalex.org/W4293568347","https://openalex.org/W2808230936","https://openalex.org/W2023659367"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"a":[3,34,48,51,156],"general":[4,88],"class":[5,32],"of":[6,9,37,39,59,68,72,79,94,116,135,185,190,204],"regularized":[7,70,97,118,133,150,188,220],"M-estimators":[8,38,71,93,189,221],"scatter":[10,40,73,191],"matrix":[11,41,171],"are":[12,15,45,99,145,222],"proposed":[13,187],"that":[14,213],"suitable":[16],"also":[17,122,154],"for":[18,103,127,142,163],"low":[19,198],"or":[20],"insufficient":[21],"sample":[22,199],"support":[23,200],"(small":[24],"n":[25],"and":[26,44,66,74,77,139,160],"large":[27],"p)":[28],"problems.":[29,201],"The":[30],"considered":[31],"constitutes":[33],"natural":[35],"generalization":[36],"(Maronna,":[42],"1976)":[43],"defined":[46],"as":[47],"solution":[49,81,115,162],"to":[50,82,101,113,193,224],"penalized":[52],"M-estimation":[53],"cost":[54],"function.":[55],"Using":[56],"the":[57,64,69,75,80,83,91,96,114,117,125,132,149,165,174,182,186,227],"concept":[58],"geodesic":[60],"convexity,":[61],"we":[62,153],"prove":[63],"existence":[65,76],"uniqueness":[67,78,128,144],"corresponding":[84],"M-estimating":[85,119],"equations":[86],"under":[87],"conditions.":[89],"Unlike":[90],"non-regularized":[92,195],"scatter,":[95],"estimators":[98],"shown":[100],"exist":[102],"any":[104],"data":[105],"configuration.":[106],"An":[107,202],"iterative":[108],"algorithm":[109],"with":[110],"proven":[111],"convergence":[112],"equation":[120],"is":[121],"given.":[123],"Since":[124],"conditions":[126,141],"do":[129],"not":[130],"include":[131],"versions":[134],"Tyler's":[136,151],"M-estimator,":[137],"necessary":[138],"sufficient":[140],"their":[143,194],"established":[146],"separately.":[147],"For":[148],"M-estimators,":[152],"derive":[155],"simple,":[157],"closed":[158],"form,":[159],"data-dependent":[161],"choosing":[164],"regularization":[166],"parameter":[167],"based":[168,218],"on":[169,219],"shape":[170],"matching":[172],"in":[173,197],"mean-squared":[175],"sense.":[176],"Finally,":[177],"some":[178],"simulations":[179],"studies":[180],"illustrate":[181,212],"improved":[183],"accuracy":[184],"compared":[192],"counterparts":[196],"example":[203],"radar":[205],"detection":[206],"using":[207],"normalized":[208],"matched":[209],"filter":[210],"(NMF)":[211],"an":[214],"adaptive":[215],"NMF":[216],"detector":[217],"able":[223],"maintain":[225],"accurately":[226],"preset":[228],"CFAR":[229],"level.":[230]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":10},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":17},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":8},{"year":2018,"cited_by_count":23},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":16},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":2}],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2016-06-24T00:00:00"}
