{"id":"https://openalex.org/W4312991203","doi":"https://doi.org/10.1109/tit.2022.3229058","title":"Some Notes on Variational Principle for Metric Mean Dimension","display_name":"Some Notes on Variational Principle for Metric Mean Dimension","publication_year":2022,"publication_date":"2022-12-14","ids":{"openalex":"https://openalex.org/W4312991203","doi":"https://doi.org/10.1109/tit.2022.3229058"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2022.3229058","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3229058","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/A5100614867","display_name":"Rui Yang","orcid":"https://orcid.org/0000-0003-4486-7696"},"institutions":[{"id":"https://openalex.org/I152031979","display_name":"Nanjing Normal University","ror":"https://ror.org/036trcv74","country_code":"CN","type":"education","lineage":["https://openalex.org/I152031979"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Rui Yang","raw_affiliation_strings":["School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, China"],"raw_orcid":"https://orcid.org/0000-0003-4486-7696","affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, China","institution_ids":["https://openalex.org/I152031979"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052807376","display_name":"Ercai Chen","orcid":"https://orcid.org/0000-0002-2600-7548"},"institutions":[{"id":"https://openalex.org/I152031979","display_name":"Nanjing Normal University","ror":"https://ror.org/036trcv74","country_code":"CN","type":"education","lineage":["https://openalex.org/I152031979"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ercai Chen","raw_affiliation_strings":["School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, China","institution_ids":["https://openalex.org/I152031979"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046835562","display_name":"Xiaoyao Zhou","orcid":"https://orcid.org/0000-0003-2798-6271"},"institutions":[{"id":"https://openalex.org/I152031979","display_name":"Nanjing Normal University","ror":"https://ror.org/036trcv74","country_code":"CN","type":"education","lineage":["https://openalex.org/I152031979"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaoyao Zhou","raw_affiliation_strings":["School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, China"],"raw_orcid":"https://orcid.org/0000-0003-2798-6271","affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, China","institution_ids":["https://openalex.org/I152031979"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I152031979"],"apc_list":null,"apc_paid":null,"fwci":1.1145,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.74876485,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"69","issue":"5","first_page":"2796","last_page":"2800"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9955000281333923,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.983299970626831,"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/dimension","display_name":"Dimension (graph theory)","score":0.7552684545516968},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7292894124984741},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.6975698471069336},{"id":"https://openalex.org/keywords/variational-principle","display_name":"Variational principle","score":0.6025364995002747},{"id":"https://openalex.org/keywords/infimum-and-supremum","display_name":"Infimum and supremum","score":0.5659880638122559},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.43491992354393005},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3958353102207184},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32105183601379395},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2655022144317627}],"concepts":[{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.7552684545516968},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7292894124984741},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.6975698471069336},{"id":"https://openalex.org/C9259290","wikidata":"https://www.wikidata.org/wiki/Q745215","display_name":"Variational principle","level":2,"score":0.6025364995002747},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.5659880638122559},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.43491992354393005},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3958353102207184},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32105183601379395},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2655022144317627},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2022.3229058","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3229058","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":[{"id":"https://metadata.un.org/sdg/10","score":0.4300000071525574,"display_name":"Reduced inequalities"}],"awards":[{"id":"https://openalex.org/G2154353796","display_name":null,"funder_award_id":"11826102","funder_id":"https://openalex.org/F4320322990","funder_display_name":"Sichuan University"},{"id":"https://openalex.org/G2587349982","display_name":null,"funder_award_id":"2016M591873","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"},{"id":"https://openalex.org/G3370888981","display_name":null,"funder_award_id":"2017T100384","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"},{"id":"https://openalex.org/G4108726380","display_name":"\u62d3\u6251\u52a8\u529b\u7cfb\u7edf\u4e2d\u7684\u71b5\u4e0e\u6df7\u6c8c\u7684\u7814\u7a76","funder_award_id":"11971236","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6754888378","display_name":null,"funder_award_id":"12071222","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/F4320321543","display_name":"China Postdoctoral Science Foundation","ror":"https://ror.org/0426zh255"},{"id":"https://openalex.org/F4320322720","display_name":"Southwest Jiaotong University","ror":"https://ror.org/00hn7w693"},{"id":"https://openalex.org/F4320322990","display_name":"Sichuan University","ror":"https://ror.org/011ashp19"},{"id":"https://openalex.org/F4320327518","display_name":"Priority Academic Program Development of Jiangsu Higher Education Institutions","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W149658591","https://openalex.org/W187295716","https://openalex.org/W1485816232","https://openalex.org/W1496462336","https://openalex.org/W2040635624","https://openalex.org/W2040840049","https://openalex.org/W2046045869","https://openalex.org/W2065979686","https://openalex.org/W2070509948","https://openalex.org/W2117263186","https://openalex.org/W2165253422","https://openalex.org/W2263898306","https://openalex.org/W2478708596","https://openalex.org/W2586913934","https://openalex.org/W2591072568","https://openalex.org/W2903469463","https://openalex.org/W2962940475","https://openalex.org/W2963651978","https://openalex.org/W2963695017","https://openalex.org/W2996272230","https://openalex.org/W2998243751","https://openalex.org/W3099349146","https://openalex.org/W3100329389","https://openalex.org/W3119565752","https://openalex.org/W3137487347","https://openalex.org/W3176887306","https://openalex.org/W4231009334","https://openalex.org/W4295679940","https://openalex.org/W6741882415"],"related_works":["https://openalex.org/W2953175190","https://openalex.org/W4288318511","https://openalex.org/W2257286100","https://openalex.org/W2784344684","https://openalex.org/W2562705109","https://openalex.org/W4283118620","https://openalex.org/W2098184873","https://openalex.org/W427674437","https://openalex.org/W2012159417","https://openalex.org/W2773544026"],"abstract_inverted_index":{"In":[0],"this":[1],"note,":[2],"we":[3,94],"establish":[4,22,66],"variational":[5,24,53,69,105],"principles":[6],"for":[7,26,55,71,81,107,126],"mean":[8,12,28,57,72,77,109],"dimension":[9,29,73,80,110,116],"and":[10,48,101,120],"metric":[11,27,56,108],"dimension.":[13],"Our":[14],"main":[15],"results":[16],"are":[17],"as":[18],"follows.":[19],"(1)":[20],"We":[21,65],"a":[23,43,67],"principle":[25,54,70,106],"in":[30,50,74,111],"terms":[31,75,112],"of$L^{\\infty":[32],"}$rate-distortion":[33],"function":[34],"with":[35],"supremum":[36],"over":[37],"all":[38],"ergodic":[39],"measures,":[40],"which":[41],"answers":[42],"question":[44],"posed":[45],"by":[46,118],"Gutman":[47,119],"\u015apiewak":[49,121],"(Around":[51],"the":[52,82,89,97,104],"dimension,":[58],"Studia":[59],"Math.":[60],"(2021)":[61,122],"261":[62],"345-60).":[63],"(2)":[64],"double":[68],"of":[76,99,103,113],"R\u00e9nyi":[78,114],"information":[79,115],"systems":[83],"admitting":[84],"marker":[85,92],"property.":[86],"(3)":[87],"If":[88],"system":[90],"admits":[91],"property,":[93],"show":[95],"that":[96],"order":[98],"sup":[100],"limsup":[102],"introduced":[117],"can":[123],"be":[124],"exchanged":[125],"some":[127],"\u201cnice\u201d":[128],"metrics.":[129]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2025-10-10T00:00:00"}
