{"id":"https://openalex.org/W2009006054","doi":"https://doi.org/10.1007/s10107-010-0422-2","title":"Null space conditions and thresholds for rank minimization","display_name":"Null space conditions and thresholds for rank minimization","publication_year":2010,"publication_date":"2010-10-12","ids":{"openalex":"https://openalex.org/W2009006054","doi":"https://doi.org/10.1007/s10107-010-0422-2","mag":"2009006054"},"language":"en","primary_location":{"id":"doi:10.1007/s10107-010-0422-2","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10107-010-0422-2","pdf_url":null,"source":{"id":"https://openalex.org/S193920097","display_name":"Mathematical Programming","issn_l":"0025-5610","issn":["0025-5610","1436-4646"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Programming","raw_type":"journal-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/A5012870568","display_name":"Benjamin Recht","orcid":"https://orcid.org/0000-0002-0293-593X"},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Benjamin Recht","raw_affiliation_strings":["Department of Computer Sciences, University of Wisconsin, 1210 W Dayton St., Madison, WI, 53703, USA","University of Wisconsin, Department of Computer Sciences, 1210 W Dayton St., 53703, Madison, WI, USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Sciences, University of Wisconsin, 1210 W Dayton St., Madison, WI, 53703, USA","institution_ids":["https://openalex.org/I135310074"]},{"raw_affiliation_string":"University of Wisconsin, Department of Computer Sciences, 1210 W Dayton St., 53703, Madison, WI, USA#TAB#","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101767475","display_name":"Weiyu Xu","orcid":"https://orcid.org/0000-0003-4256-6072"},"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":"Weiyu Xu","raw_affiliation_strings":["Electrical Engineering, California Institute of Technology, Pasadena, CA, USA","California Institute of Technology, Electrical Engineering, Pasadena, CA, USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Electrical Engineering, California Institute of Technology, Pasadena, CA, USA","institution_ids":["https://openalex.org/I122411786"]},{"raw_affiliation_string":"California Institute of Technology, Electrical Engineering, Pasadena, CA, USA#TAB#","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":["Electrical Engineering, California Institute of Technology, Pasadena, CA, USA","California Institute of Technology, Electrical Engineering, Pasadena, CA, USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Electrical Engineering, California Institute of Technology, Pasadena, CA, USA","institution_ids":["https://openalex.org/I122411786"]},{"raw_affiliation_string":"California Institute of Technology, Electrical Engineering, Pasadena, CA, USA#TAB#","institution_ids":["https://openalex.org/I122411786"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5012870568"],"corresponding_institution_ids":["https://openalex.org/I135310074"],"apc_list":{"value":2190,"currency":"EUR","value_usd":2890},"apc_paid":null,"fwci":19.7698,"has_fulltext":false,"cited_by_count":157,"citation_normalized_percentile":{"value":0.99649991,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"127","issue":"1","first_page":"175","last_page":"202"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998999834060669,"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":0.9998999834060669,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9811999797821045,"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/mathematics","display_name":"Mathematics","score":0.7641165256500244},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.6358440518379211},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.617840051651001},{"id":"https://openalex.org/keywords/heuristic","display_name":"Heuristic","score":0.5740991234779358},{"id":"https://openalex.org/keywords/low-rank-approximation","display_name":"Low-rank approximation","score":0.5592209100723267},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4940796196460724},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.4815566837787628},{"id":"https://openalex.org/keywords/null","display_name":"Null (SQL)","score":0.441182017326355},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4344322383403778},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.41260620951652527},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.19037014245986938},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.17401790618896484},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16716209053993225}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7641165256500244},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6358440518379211},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.617840051651001},{"id":"https://openalex.org/C173801870","wikidata":"https://www.wikidata.org/wiki/Q201413","display_name":"Heuristic","level":2,"score":0.5740991234779358},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.5592209100723267},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4940796196460724},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.4815566837787628},{"id":"https://openalex.org/C203763787","wikidata":"https://www.wikidata.org/wiki/Q371029","display_name":"Null (SQL)","level":2,"score":0.441182017326355},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4344322383403778},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.41260620951652527},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.19037014245986938},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.17401790618896484},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16716209053993225},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","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},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C25023664","wikidata":"https://www.wikidata.org/wiki/Q1575637","display_name":"Hankel matrix","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s10107-010-0422-2","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10107-010-0422-2","pdf_url":null,"source":{"id":"https://openalex.org/S193920097","display_name":"Mathematical Programming","issn_l":"0025-5610","issn":["0025-5610","1436-4646"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Programming","raw_type":"journal-article"},{"id":"pmh:oai:authors.library.caltech.edu:23202","is_oa":false,"landing_page_url":"https://resolver.caltech.edu/CaltechAUTHORS:20110401-112500555","pdf_url":null,"source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.150.6296","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.6296","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ist.caltech.edu/~brecht/papers/08.RecXuHas.Thresholds.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.7799999713897705,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306079","display_name":"David and Lucile Packard Foundation","ror":"https://ror.org/032atxq54"},{"id":"https://openalex.org/F4320309398","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055"},{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":41,"referenced_works":["https://openalex.org/W188867022","https://openalex.org/W1488435683","https://openalex.org/W1495302720","https://openalex.org/W1560133479","https://openalex.org/W1966096622","https://openalex.org/W1967073510","https://openalex.org/W1976618413","https://openalex.org/W1980052309","https://openalex.org/W2010669260","https://openalex.org/W2030449718","https://openalex.org/W2040745222","https://openalex.org/W2054443446","https://openalex.org/W2056042427","https://openalex.org/W2059283452","https://openalex.org/W2060581589","https://openalex.org/W2065180801","https://openalex.org/W2069754508","https://openalex.org/W2099641086","https://openalex.org/W2101593431","https://openalex.org/W2103699341","https://openalex.org/W2103972604","https://openalex.org/W2116616571","https://openalex.org/W2118408802","https://openalex.org/W2118550318","https://openalex.org/W2129131372","https://openalex.org/W2132164018","https://openalex.org/W2135127969","https://openalex.org/W2145096794","https://openalex.org/W2149863017","https://openalex.org/W2151211319","https://openalex.org/W2163916252","https://openalex.org/W2170929819","https://openalex.org/W2176446742","https://openalex.org/W2202343345","https://openalex.org/W2611328865","https://openalex.org/W2949351217","https://openalex.org/W3013880646","https://openalex.org/W3102105495","https://openalex.org/W3140158754","https://openalex.org/W4285719527","https://openalex.org/W4301425558"],"related_works":["https://openalex.org/W4221157149","https://openalex.org/W3174042135","https://openalex.org/W4283122425","https://openalex.org/W2057590040","https://openalex.org/W4297415474","https://openalex.org/W3106318770","https://openalex.org/W3135513863","https://openalex.org/W1991825408","https://openalex.org/W2969889343","https://openalex.org/W4306936641"],"abstract_inverted_index":null,"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":23},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":13},{"year":2015,"cited_by_count":14},{"year":2014,"cited_by_count":15},{"year":2013,"cited_by_count":18},{"year":2012,"cited_by_count":17}],"updated_date":"2026-06-19T15:47:20.252518","created_date":"2025-10-10T00:00:00"}