{"id":"https://openalex.org/W2129623910","doi":"https://doi.org/10.1109/tit.2012.2235172","title":"Covering Numbers for Convex Functions","display_name":"Covering Numbers for Convex Functions","publication_year":2012,"publication_date":"2012-12-21","ids":{"openalex":"https://openalex.org/W2129623910","doi":"https://doi.org/10.1109/tit.2012.2235172","mag":"2129623910"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2012.2235172","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2012.2235172","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/A5084461752","display_name":"Adityanand Guntuboyina","orcid":"https://orcid.org/0009-0000-2674-4065"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Adityanand Guntuboyina","raw_affiliation_strings":["Department of Statistics, University of California, Berkeley, CA, USA","Department of Statistics, University of California, Berkeley, CA, USA#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Statistics, University of California, Berkeley, CA, USA","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"Department of Statistics, University of California, Berkeley, CA, USA#TAB#","institution_ids":["https://openalex.org/I95457486"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5033475436","display_name":"Bodhisattva Sen","orcid":"https://orcid.org/0000-0002-4667-3150"},"institutions":[{"id":"https://openalex.org/I78577930","display_name":"Columbia University","ror":"https://ror.org/00hj8s172","country_code":"US","type":"education","lineage":["https://openalex.org/I78577930"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Bodhisattva Sen","raw_affiliation_strings":["Department of Statistics, Columbia University, New York, NY, USA","Department of Statistics Columbia University New York, NY USA"],"affiliations":[{"raw_affiliation_string":"Department of Statistics, Columbia University, New York, NY, USA","institution_ids":["https://openalex.org/I78577930"]},{"raw_affiliation_string":"Department of Statistics Columbia University New York, NY USA","institution_ids":["https://openalex.org/I78577930"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5084461752"],"corresponding_institution_ids":["https://openalex.org/I95457486"],"apc_list":null,"apc_paid":null,"fwci":5.1551,"has_fulltext":false,"cited_by_count":77,"citation_normalized_percentile":{"value":0.95820162,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"59","issue":"4","first_page":"1957","last_page":"1965"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9921000003814697,"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":0.9921000003814697,"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/T10545","display_name":"Optimization and Variational Analysis","score":0.9782999753952026,"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"}},{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.9663000106811523,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/convexity","display_name":"Convexity","score":0.7925987839698792},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6825981140136719},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5758106708526611},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5416825413703918},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.521605372428894},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.5022096633911133},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.47981786727905273},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4715970456600189},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.44234901666641235},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.41777825355529785},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4108278453350067},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08396625518798828}],"concepts":[{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.7925987839698792},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6825981140136719},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5758106708526611},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5416825413703918},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.521605372428894},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5022096633911133},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47981786727905273},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4715970456600189},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.44234901666641235},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.41777825355529785},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4108278453350067},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08396625518798828},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/tit.2012.2235172","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2012.2235172","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.706.6676","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.706.6676","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.stat.berkeley.edu/%7Eaditya/resources/IEEEconco.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.746.9949","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.746.9949","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1204.0147.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W156419107","https://openalex.org/W1497683195","https://openalex.org/W1507545330","https://openalex.org/W1524622012","https://openalex.org/W1666223003","https://openalex.org/W1675105563","https://openalex.org/W1854990296","https://openalex.org/W1963490725","https://openalex.org/W1969705577","https://openalex.org/W1973513227","https://openalex.org/W1976101219","https://openalex.org/W1998643067","https://openalex.org/W2004060143","https://openalex.org/W2037134555","https://openalex.org/W2086117410","https://openalex.org/W2101210333","https://openalex.org/W2101626984","https://openalex.org/W2130315424","https://openalex.org/W2146433171","https://openalex.org/W3100270455","https://openalex.org/W3105936967","https://openalex.org/W3105994413","https://openalex.org/W4239022279","https://openalex.org/W4248484045"],"related_works":["https://openalex.org/W4288573727","https://openalex.org/W2912909874","https://openalex.org/W2981448761","https://openalex.org/W2768754015","https://openalex.org/W4376864979","https://openalex.org/W2494876756","https://openalex.org/W2393350255","https://openalex.org/W257189624","https://openalex.org/W2356369451","https://openalex.org/W4297903463"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"study":[4,117],"the":[5,9,27,39,53,77,116,133],"covering":[6,96],"numbers":[7,97],"of":[8,11,30,52,79,106,118,120,122],"space":[10],"convex":[12,81,99],"and":[13,23,68,101],"uniformly":[14,86],"bounded":[15,87],"functions":[16,82,100],"in":[17,38,50,115,132],"multidimension.":[18],"We":[19,90],"find":[20],"optimal":[21,129],"upper":[22],"lower":[24],"bounds":[25],"for":[26,98],"\u03b5-covering":[28],"number":[29],"C([a,":[31,69],"b]":[32,70],"<sup":[33,71],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[34,42,72],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">d</sup>":[35,73],",":[36,74],"B),":[37],"L":[40],"<sub":[41],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">p</sub>":[43],"-metric,":[44],"1":[45],"\u2264":[46],"p":[47],"<;":[48,61],"\u221e,":[49],"terms":[51],"relevant":[54],"constants,":[55],"where":[56],"d":[57],"\u2265":[58],"1,":[59],"a":[60],"b":[62],"\u2208":[63],"\u211d,":[64],"B":[65],">":[66],"0,":[67],"B)":[75],"denotes":[76],"set":[78],"all":[80],"on":[83,95],"that":[84],"are":[85],"by":[88],"B.":[89],"summarize":[91],"previously":[92],"known":[93,108],"results":[94,111],"also":[102],"provide":[103],"alternate":[104],"proofs":[105],"some":[107],"results.":[109],"Our":[110],"have":[112],"direct":[113],"implications":[114],"rates":[119,131],"convergence":[121,130],"empirical":[123],"minimization":[124],"procedures":[125],"as":[126,128],"well":[127],"numerous":[134],"convexity":[135],"constrained":[136],"function":[137],"estimation":[138],"problems.":[139]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":23},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":10},{"year":2015,"cited_by_count":8},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2026-03-01T08:55:55.761014","created_date":"2025-10-10T00:00:00"}
