{"id":"https://openalex.org/W2057946045","doi":"https://doi.org/10.1109/tit.1984.1056873","title":"Monotony of Lloyd's method II for log-concave density and convex error weighting function (Corresp.)","display_name":"Monotony of Lloyd's method II for log-concave density and convex error weighting function (Corresp.)","publication_year":1984,"publication_date":"1984-03-01","ids":{"openalex":"https://openalex.org/W2057946045","doi":"https://doi.org/10.1109/tit.1984.1056873","mag":"2057946045"},"language":"en","primary_location":{"id":"doi:10.1109/tit.1984.1056873","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1984.1056873","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/A5043655298","display_name":"A. Trushkin","orcid":null},"institutions":[{"id":"https://openalex.org/I4210107660","display_name":"Institute for Information Transmission Problems","ror":"https://ror.org/013w2d378","country_code":"RU","type":"facility","lineage":["https://openalex.org/I4210107660"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"A. Trushkin","raw_affiliation_strings":["Institute for Information Transmission Problems, USSR Academy of Science, Moscow, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Information Transmission Problems, USSR Academy of Science, Moscow, Russia","institution_ids":["https://openalex.org/I4210107660"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5043655298"],"corresponding_institution_ids":["https://openalex.org/I4210107660"],"apc_list":{"value":2645,"currency":"USD","value_usd":2645},"apc_paid":null,"fwci":0.6421,"has_fulltext":false,"cited_by_count":19,"citation_normalized_percentile":{"value":0.72030361,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"30","issue":"2","first_page":"380","last_page":"383"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10901","display_name":"Advanced Data Compression Techniques","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T10901","display_name":"Advanced Data Compression Techniques","score":0.9997000098228455,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9994999766349792,"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":0.9939000010490417,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/weighting","display_name":"Weighting","score":0.7129587531089783},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.668861448764801},{"id":"https://openalex.org/keywords/probability-density-function","display_name":"Probability density function","score":0.6016954779624939},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.590796709060669},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.5649640560150146},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5072633624076843},{"id":"https://openalex.org/keywords/concave-function","display_name":"Concave function","score":0.49576982855796814},{"id":"https://openalex.org/keywords/quantization","display_name":"Quantization (signal processing)","score":0.48351624608039856},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.47466638684272766},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4672059416770935},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.44507136940956116},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.41258031129837036},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38350874185562134},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.32545921206474304},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.24783185124397278},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22499528527259827}],"concepts":[{"id":"https://openalex.org/C183115368","wikidata":"https://www.wikidata.org/wiki/Q856577","display_name":"Weighting","level":2,"score":0.7129587531089783},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.668861448764801},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.6016954779624939},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.590796709060669},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.5649640560150146},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5072633624076843},{"id":"https://openalex.org/C66690126","wikidata":"https://www.wikidata.org/wiki/Q2914302","display_name":"Concave function","level":3,"score":0.49576982855796814},{"id":"https://openalex.org/C28855332","wikidata":"https://www.wikidata.org/wiki/Q198099","display_name":"Quantization (signal processing)","level":2,"score":0.48351624608039856},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47466638684272766},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4672059416770935},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.44507136940956116},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.41258031129837036},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38350874185562134},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32545921206474304},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.24783185124397278},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22499528527259827},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","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},{"id":"https://openalex.org/C126838900","wikidata":"https://www.wikidata.org/wiki/Q77604","display_name":"Radiology","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},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.1984.1056873","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1984.1056873","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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W2079232128","https://openalex.org/W2140942991","https://openalex.org/W2150593711"],"related_works":["https://openalex.org/W4285277090","https://openalex.org/W4385382439","https://openalex.org/W4295356201","https://openalex.org/W4310854535","https://openalex.org/W4306754731","https://openalex.org/W4297903463","https://openalex.org/W2909546413","https://openalex.org/W2951835737","https://openalex.org/W996039869","https://openalex.org/W2889827900"],"abstract_inverted_index":{"Kieffer":[0],"recently":[1],"showed":[2],"that":[3,42],"for":[4],"a":[5,11,16,65],"log-concave":[6],"probability":[7],"density":[8],"there":[9],"is":[10,15,40,62],"uniqueN-level":[12],"quantizer":[13],"which":[14,61],"stationary":[17],"point":[18],"of":[19,23,45,54,60,67],"the":[20,35,43,52,58,71],"expected":[21],"value":[22],"an":[24],"increasing,":[25],"convex,":[26],"and":[27],"continuously":[28],"differentiable":[29],"error":[30],"weighting":[31],"function,":[32],"dependent":[33],"on":[34,64],"absolute":[36],"quantization":[37],"error.":[38],"It":[39],"shown":[41],"requirement":[44],"continuous":[46],"differentiability":[47],"can":[48],"be":[49],"dropped,":[50],"using":[51],"monotony":[53],"Lloyd's":[55],"Method":[56],"II,":[57],"proof":[59],"based":[63],"generalization":[66],"previous":[68],"work":[69],"by":[70],"author.":[72]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}
