{"id":"https://openalex.org/W2090458652","doi":"https://doi.org/10.1109/allerton.2012.6483330","title":"On a relation between the minimax risk and the phase transitions of compressed recovery","display_name":"On a relation between the minimax risk and the phase transitions of compressed recovery","publication_year":2012,"publication_date":"2012-10-01","ids":{"openalex":"https://openalex.org/W2090458652","doi":"https://doi.org/10.1109/allerton.2012.6483330","mag":"2090458652"},"language":"en","primary_location":{"id":"doi:10.1109/allerton.2012.6483330","is_oa":false,"landing_page_url":"https://doi.org/10.1109/allerton.2012.6483330","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)","raw_type":"proceedings-article"},"type":"article","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":"https://openalex.org/A5050547472","display_name":"Samet Oymak","orcid":"https://orcid.org/0000-0001-5203-0752"},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Samet Oymak","raw_affiliation_strings":["Department of Electrical Engineering, Caltech, Pasadena, USA","Department of Electrical Engineering Caltech, Pasadena - 91125"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Caltech, Pasadena, USA","institution_ids":["https://openalex.org/I122411786"]},{"raw_affiliation_string":"Department of Electrical Engineering Caltech, Pasadena - 91125","institution_ids":["https://openalex.org/I122411786"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5002430773","display_name":"Babak Hassibi","orcid":"https://orcid.org/0000-0002-1375-5838"},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Babak Hassibi","raw_affiliation_strings":["Department of Electrical Engineering, Caltech, Pasadena, USA","Department of Electrical Engineering Caltech, Pasadena - 91125"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Caltech, Pasadena, USA","institution_ids":["https://openalex.org/I122411786"]},{"raw_affiliation_string":"Department of Electrical Engineering Caltech, Pasadena - 91125","institution_ids":["https://openalex.org/I122411786"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5050547472"],"corresponding_institution_ids":["https://openalex.org/I122411786"],"apc_list":null,"apc_paid":null,"fwci":3.15283359,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.91233076,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":null,"issue":null,"first_page":"1018","last_page":"1025"},"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/T10688","display_name":"Image and Signal Denoising Methods","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11447","display_name":"Blind Source Separation Techniques","score":0.9930999875068665,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/minimax","display_name":"Minimax","score":0.6135172247886658},{"id":"https://openalex.org/keywords/compressed-sensing","display_name":"Compressed sensing","score":0.5853759050369263},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.48184287548065186},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.45449721813201904},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4245436191558838},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4025109112262726},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.35144364833831787},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.32500821352005005},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32074588537216187},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.19409304857254028}],"concepts":[{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.6135172247886658},{"id":"https://openalex.org/C124851039","wikidata":"https://www.wikidata.org/wiki/Q2665459","display_name":"Compressed sensing","level":2,"score":0.5853759050369263},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.48184287548065186},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.45449721813201904},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4245436191558838},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4025109112262726},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.35144364833831787},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.32500821352005005},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32074588537216187},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.19409304857254028},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/allerton.2012.6483330","is_oa":false,"landing_page_url":"https://doi.org/10.1109/allerton.2012.6483330","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W74946117","https://openalex.org/W1481321471","https://openalex.org/W1485629353","https://openalex.org/W1530702808","https://openalex.org/W1630108645","https://openalex.org/W2078204800","https://openalex.org/W2079724595","https://openalex.org/W2082029531","https://openalex.org/W2090458652","https://openalex.org/W2098996169","https://openalex.org/W2103539935","https://openalex.org/W2123202508","https://openalex.org/W2124541940","https://openalex.org/W2124608575","https://openalex.org/W2134074206","https://openalex.org/W2145096794","https://openalex.org/W2145962650","https://openalex.org/W2146678402","https://openalex.org/W2146842127","https://openalex.org/W2168946004","https://openalex.org/W2170929819","https://openalex.org/W2296319761","https://openalex.org/W2296616510","https://openalex.org/W2611328865","https://openalex.org/W2949066786","https://openalex.org/W2949789221","https://openalex.org/W2951457851","https://openalex.org/W3124617746","https://openalex.org/W3141595720","https://openalex.org/W4250589301","https://openalex.org/W4250955649","https://openalex.org/W4255521522","https://openalex.org/W4298371340"],"related_works":["https://openalex.org/W2016058626","https://openalex.org/W2474724840","https://openalex.org/W2158224665","https://openalex.org/W2895916002","https://openalex.org/W2379589510","https://openalex.org/W1814049089","https://openalex.org/W4300044672","https://openalex.org/W2810730439","https://openalex.org/W1977348009","https://openalex.org/W2369683208"],"abstract_inverted_index":{"This":[0,230],"paper":[1],"provides":[2],"a":[3,14,39,52,67,117,180,215,241],"sharp":[4],"analysis":[5],"of":[6,65,211,225],"the":[7,17,58,63,84,110,121,166,175,185,192,196,208,222,226],"optimally":[8],"tuned":[9],"denoising":[10,61,168,197],"problem":[11,64,169,198],"and":[12,22,31,107,120,136,149,153,170,191],"establishes":[13],"relation":[15,231],"between":[16],"estimation":[18,228],"error":[19],"(minimax":[20],"risk)":[21],"phase":[23,176],"transition":[24,177],"for":[25,147,156],"compressed":[26,43,189,216],"sensing":[27],"recovery":[28,190],"using":[29,83],"convex":[30,54],"continuous":[32],"functions.":[33],"Phase":[34],"transitions":[35],"deal":[36],"with":[37],"recovering":[38],"signal":[40,68,111,217],"xo":[41],"from":[42,73],"linear":[44],"observations":[45,75,212],"Ax":[46],"<sub":[47,70,79,87,93,113,133,138,143],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[48,71,80,88,94,97,114,134,139,144],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">0</sub>":[49,72,81,115],"by":[50],"minimizing":[51],"certain":[53,118],"function":[55,122],"f(\u00b7).":[56],"On":[57],"other":[59],"hand,":[60],"is":[62,124,218],"estimating":[66],"x":[69,78,112],"noisy":[74],"y":[76],"=":[77],"+z":[82],"regularization":[85],"min":[86],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x</sub>":[89],"\u03bb/f(x)":[90],"+":[91],"1/2\u2225y-x\u2225":[92],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>":[95,145],"<sup":[96],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sup>":[98],".":[99],"In":[100,160],"general,":[101],"these":[102],"problems":[103],"are":[104,199],"more":[105],"meaningful":[106],"useful":[108],"when":[109],"has":[116],"structure":[119],"f(\u00b7)":[123],"chosen":[125],"to":[126,174,213,221],"exploit":[127],"this":[128,161],"structure.":[129],"Examples":[130],"include,":[131],"l":[132,137,142],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">1</sub>":[135,140],"-":[141],"norms":[146],"sparse":[148,151],"block":[150],"vectors":[152],"nuclear":[154],"norm":[155],"low":[157],"rank":[158],"matrices.":[159],"work,":[162],"we":[163,239],"carefully":[164],"analyze":[165],"minimax":[167],"relate":[171],"our":[172],"results":[173,205],"performance":[178],"under":[179],"considerably":[181],"general":[182],"setting":[183],"where":[184],"measurement":[186],"A":[187],"in":[188,195,236],"noise":[193],"z":[194],"iid":[200],"Gaussian":[201],"random":[202],"variables.":[203],"Our":[204],"suggest":[206],"that":[207],"required":[209],"number":[210],"recover":[214],"closely":[219],"related":[220],"asymptotic":[223],"variance":[224],"optimal":[227],"error.":[229],"was":[232],"first":[233],"empirically":[234],"noted":[235],"[9].":[237],"Here":[238],"provide":[240],"rigorous":[242],"foundation.":[243]},"counts_by_year":[{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
