{"id":"https://openalex.org/W3201021867","doi":"https://doi.org/10.1109/tit.2022.3159100","title":"An Open Problem on Sparse Representations in Unions of Bases","display_name":"An Open Problem on Sparse Representations in Unions of Bases","publication_year":2022,"publication_date":"2022-03-16","ids":{"openalex":"https://openalex.org/W3201021867","doi":"https://doi.org/10.1109/tit.2022.3159100","mag":"3201021867"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2022.3159100","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3159100","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","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/A5089903944","display_name":"Yi Shen","orcid":"https://orcid.org/0000-0002-2554-9812"},"institutions":[{"id":"https://openalex.org/I1328775524","display_name":"Zhejiang Sci-Tech University","ror":"https://ror.org/03893we55","country_code":"CN","type":"education","lineage":["https://openalex.org/I1328775524"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yi Shen","raw_affiliation_strings":["Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, China","institution_ids":["https://openalex.org/I1328775524"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103021093","display_name":"Chenyun Yu","orcid":"https://orcid.org/0000-0001-7626-1256"},"institutions":[{"id":"https://openalex.org/I1328775524","display_name":"Zhejiang Sci-Tech University","ror":"https://ror.org/03893we55","country_code":"CN","type":"education","lineage":["https://openalex.org/I1328775524"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chenyun Yu","raw_affiliation_strings":["Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, China","institution_ids":["https://openalex.org/I1328775524"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052022779","display_name":"Yuan Shen","orcid":"https://orcid.org/0000-0002-9396-1964"},"institutions":[{"id":"https://openalex.org/I1328775524","display_name":"Zhejiang Sci-Tech University","ror":"https://ror.org/03893we55","country_code":"CN","type":"education","lineage":["https://openalex.org/I1328775524"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuan Shen","raw_affiliation_strings":["Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, China","institution_ids":["https://openalex.org/I1328775524"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100448207","display_name":"Song Li","orcid":"https://orcid.org/0000-0002-7307-3935"},"institutions":[{"id":"https://openalex.org/I76130692","display_name":"Zhejiang University","ror":"https://ror.org/00a2xv884","country_code":"CN","type":"education","lineage":["https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Song Li","raw_affiliation_strings":["School of Mathematical Science, Zhejiang University, Hangzhou, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Science, Zhejiang University, Hangzhou, China","institution_ids":["https://openalex.org/I76130692"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5089903944"],"corresponding_institution_ids":["https://openalex.org/I1328775524"],"apc_list":null,"apc_paid":null,"fwci":1.7805,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.82124659,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":96},"biblio":{"volume":"68","issue":"7","first_page":"4230","last_page":"4243"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11447","display_name":"Blind Source Separation Techniques","score":0.9997000098228455,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991000294685364,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/orthonormal-basis","display_name":"Orthonormal basis","score":0.9225651025772095},{"id":"https://openalex.org/keywords/mutual-coherence","display_name":"Mutual coherence","score":0.8664488196372986},{"id":"https://openalex.org/keywords/constructive","display_name":"Constructive","score":0.7525115013122559},{"id":"https://openalex.org/keywords/coherence","display_name":"Coherence (philosophical gambling strategy)","score":0.6471205949783325},{"id":"https://openalex.org/keywords/spark","display_name":"SPARK (programming language)","score":0.6039477586746216},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.550502598285675},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5330808162689209},{"id":"https://openalex.org/keywords/mutually-unbiased-bases","display_name":"Mutually unbiased bases","score":0.5324476957321167},{"id":"https://openalex.org/keywords/sparse-approximation","display_name":"Sparse approximation","score":0.4631125032901764},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.43762776255607605},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.38287782669067383},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.32846736907958984},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2099887728691101},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.19349613785743713},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.09165579080581665}],"concepts":[{"id":"https://openalex.org/C5806529","wikidata":"https://www.wikidata.org/wiki/Q2365325","display_name":"Orthonormal basis","level":2,"score":0.9225651025772095},{"id":"https://openalex.org/C45900066","wikidata":"https://www.wikidata.org/wiki/Q6944191","display_name":"Mutual coherence","level":3,"score":0.8664488196372986},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.7525115013122559},{"id":"https://openalex.org/C2781181686","wikidata":"https://www.wikidata.org/wiki/Q4226068","display_name":"Coherence (philosophical gambling strategy)","level":2,"score":0.6471205949783325},{"id":"https://openalex.org/C2781215313","wikidata":"https://www.wikidata.org/wiki/Q3493345","display_name":"SPARK (programming language)","level":2,"score":0.6039477586746216},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.550502598285675},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5330808162689209},{"id":"https://openalex.org/C158587675","wikidata":"https://www.wikidata.org/wiki/Q17160413","display_name":"Mutually unbiased bases","level":3,"score":0.5324476957321167},{"id":"https://openalex.org/C124066611","wikidata":"https://www.wikidata.org/wiki/Q28684319","display_name":"Sparse approximation","level":2,"score":0.4631125032901764},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.43762776255607605},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.38287782669067383},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.32846736907958984},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2099887728691101},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.19349613785743713},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.09165579080581665},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"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/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2022.3159100","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3159100","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Quality Education","score":0.5899999737739563,"id":"https://metadata.un.org/sdg/4"}],"awards":[{"id":"https://openalex.org/G2207679734","display_name":null,"funder_award_id":"12071426","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2996493410","display_name":null,"funder_award_id":"LR19A010001","funder_id":"https://openalex.org/F4320338464","funder_display_name":"Natural Science Foundation of Zhejiang Province"},{"id":"https://openalex.org/G3642703624","display_name":null,"funder_award_id":"11671358","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G420315194","display_name":null,"funder_award_id":"U21A20426","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8219867813","display_name":null,"funder_award_id":"12022112","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320338464","display_name":"Natural Science Foundation of Zhejiang Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W143004564","https://openalex.org/W401986301","https://openalex.org/W761890699","https://openalex.org/W1624182979","https://openalex.org/W1954214817","https://openalex.org/W1965142549","https://openalex.org/W1976709621","https://openalex.org/W2019741018","https://openalex.org/W2024165284","https://openalex.org/W2035816523","https://openalex.org/W2069776505","https://openalex.org/W2086869478","https://openalex.org/W2094235750","https://openalex.org/W2098704159","https://openalex.org/W2099641086","https://openalex.org/W2122315118","https://openalex.org/W2122340891","https://openalex.org/W2125455772","https://openalex.org/W2136235822","https://openalex.org/W2143130349","https://openalex.org/W2143989619","https://openalex.org/W2145096794","https://openalex.org/W2154332973","https://openalex.org/W2156043924","https://openalex.org/W2159530242","https://openalex.org/W2160396672","https://openalex.org/W2167839759","https://openalex.org/W2900661994","https://openalex.org/W2950978955","https://openalex.org/W2963680357","https://openalex.org/W2963812225","https://openalex.org/W4254448470","https://openalex.org/W4300263211"],"related_works":["https://openalex.org/W2031104901","https://openalex.org/W2644857425","https://openalex.org/W4298326036","https://openalex.org/W2476820244","https://openalex.org/W2566126253","https://openalex.org/W1930116263","https://openalex.org/W2561877285","https://openalex.org/W1568625050","https://openalex.org/W3201021867","https://openalex.org/W2480957843"],"abstract_inverted_index":{"We":[0,59],"consider":[1],"sparse":[2,28,81],"representations":[3,82],"of":[4,12,33,40,64,85,94],"signals":[5],"from":[6],"redundant":[7],"dictionaries":[8],"which":[9,98],"are":[10,35,66],"unions":[11,39,84],"several":[13,41],"orthonormal":[14,42,86],"bases.":[15,87],"The":[16],"spark":[17,49],"introduced":[18],"by":[19],"Donoho":[20],"and":[21,76],"Elad":[22],"plays":[23],"an":[24],"important":[25],"role":[26],"in":[27,56,83,101],"representations.":[29],"However,":[30],"numerical":[31],"computations":[32],"sparks":[34],"generally":[36],"combinatorial.":[37],"For":[38],"bases,":[43],"two":[44],"lower":[45],"bounds":[46],"on":[47,80,91],"the":[48,51],"via":[50],"mutual":[52],"coherence":[53],"were":[54],"established":[55],"previous":[57],"work.":[58],"constructively":[60],"prove":[61],"that":[62],"both":[63],"them":[65],"tight.":[67],"Our":[68],"main":[69],"results":[70],"give":[71],"positive":[72],"answers":[73],"to":[74],"Gribonval":[75],"Nielsen\u2019s":[77],"open":[78],"problem":[79],"Constructive":[88],"proofs":[89],"rely":[90],"a":[92],"family":[93],"mutually":[95],"unbiased":[96],"bases":[97],"first":[99],"appears":[100],"quantum":[102],"information":[103],"theory.":[104]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}