{"id":"https://openalex.org/W2167349346","doi":"https://doi.org/10.1109/tit.2010.2069250","title":"$K$-Dimensional Coding Schemes in Hilbert Spaces","display_name":"$K$-Dimensional Coding Schemes in Hilbert Spaces","publication_year":2010,"publication_date":"2010-10-20","ids":{"openalex":"https://openalex.org/W2167349346","doi":"https://doi.org/10.1109/tit.2010.2069250","mag":"2167349346"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2010.2069250","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2010.2069250","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/A5008737035","display_name":"Andreas Maurer","orcid":"https://orcid.org/0000-0002-2916-7475"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Andreas Maurer","raw_affiliation_strings":["Munich, Germany"],"affiliations":[{"raw_affiliation_string":"Munich, Germany","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5034260726","display_name":"Massimiliano Pontil","orcid":"https://orcid.org/0000-0001-9415-098X"},"institutions":[{"id":"https://openalex.org/I45129253","display_name":"University College London","ror":"https://ror.org/02jx3x895","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I45129253"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Massimiliano Pontil","raw_affiliation_strings":["Department of Computer Science, University College London, London, UK"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University College London, London, UK","institution_ids":["https://openalex.org/I45129253"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5008737035"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.4915,"has_fulltext":false,"cited_by_count":92,"citation_normalized_percentile":{"value":0.96327887,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":100},"biblio":{"volume":"56","issue":"11","first_page":"5839","last_page":"5846"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991999864578247,"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":0.9991999864578247,"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/T10052","display_name":"Medical Image Segmentation Techniques","score":0.9965999722480774,"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/T10057","display_name":"Face and Expression Recognition","score":0.9943000078201294,"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/codebook","display_name":"Codebook","score":0.7823414206504822},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.5928052067756653},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5752527117729187},{"id":"https://openalex.org/keywords/variable-length-code","display_name":"Variable-length code","score":0.5626137256622314},{"id":"https://openalex.org/keywords/non-negative-matrix-factorization","display_name":"Non-negative matrix factorization","score":0.47428232431411743},{"id":"https://openalex.org/keywords/shannon\u2013fano-coding","display_name":"Shannon\u2013Fano coding","score":0.4704510271549225},{"id":"https://openalex.org/keywords/coding","display_name":"Coding (social sciences)","score":0.467564195394516},{"id":"https://openalex.org/keywords/cluster-analysis","display_name":"Cluster analysis","score":0.4545804560184479},{"id":"https://openalex.org/keywords/matrix-decomposition","display_name":"Matrix decomposition","score":0.4511623978614807},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4479522109031677},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.3859216868877411},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3856954574584961},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38333985209465027},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.2549395263195038},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.25139689445495605},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2174186408519745},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.14604750275611877}],"concepts":[{"id":"https://openalex.org/C127759330","wikidata":"https://www.wikidata.org/wiki/Q637416","display_name":"Codebook","level":2,"score":0.7823414206504822},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5928052067756653},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5752527117729187},{"id":"https://openalex.org/C60603091","wikidata":"https://www.wikidata.org/wiki/Q2981616","display_name":"Variable-length code","level":3,"score":0.5626137256622314},{"id":"https://openalex.org/C152671427","wikidata":"https://www.wikidata.org/wiki/Q10843505","display_name":"Non-negative matrix factorization","level":4,"score":0.47428232431411743},{"id":"https://openalex.org/C130811719","wikidata":"https://www.wikidata.org/wiki/Q2645","display_name":"Shannon\u2013Fano coding","level":4,"score":0.4704510271549225},{"id":"https://openalex.org/C179518139","wikidata":"https://www.wikidata.org/wiki/Q5140297","display_name":"Coding (social sciences)","level":2,"score":0.467564195394516},{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.4545804560184479},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.4511623978614807},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4479522109031677},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3859216868877411},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3856954574584961},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38333985209465027},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.2549395263195038},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.25139689445495605},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2174186408519745},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.14604750275611877},{"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/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/tit.2010.2069250","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2010.2069250","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:10.1.1.668.5083","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.668.5083","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.andreas-maurer.eu/SpaCodeIEEE.pdf","raw_type":"text"},{"id":"pmh:oai:eprints.ucl.ac.uk.OAI2:1350086","is_oa":false,"landing_page_url":"http://discovery.ucl.ac.uk/1350086/","pdf_url":null,"source":{"id":"https://openalex.org/S4306400024","display_name":"UCL Discovery (University College London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I45129253","host_organization_name":"University College London","host_organization_lineage":["https://openalex.org/I45129253"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"   IEEE TRANSACTIONS ON INFORMATION THEORY , 56  (11)   pp. 5839-5846.   (2010)      ","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3951961987","display_name":null,"funder_award_id":"EP/D071542/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W6713289","https://openalex.org/W188867022","https://openalex.org/W1533128638","https://openalex.org/W1608733719","https://openalex.org/W1902027874","https://openalex.org/W2010353172","https://openalex.org/W2047278710","https://openalex.org/W2069754508","https://openalex.org/W2103012681","https://openalex.org/W2118718620","https://openalex.org/W2124623336","https://openalex.org/W2131680270","https://openalex.org/W2136171036","https://openalex.org/W2137216203","https://openalex.org/W2139095575","https://openalex.org/W2142766915","https://openalex.org/W2145889472","https://openalex.org/W2579923771","https://openalex.org/W4233413206","https://openalex.org/W4241684091","https://openalex.org/W4253712714","https://openalex.org/W4301425558","https://openalex.org/W6635206662","https://openalex.org/W6677759377","https://openalex.org/W6732517885"],"related_works":["https://openalex.org/W2127243424","https://openalex.org/W2037504162","https://openalex.org/W4390394189","https://openalex.org/W2792706544","https://openalex.org/W1568451138","https://openalex.org/W2539013788","https://openalex.org/W2156699640","https://openalex.org/W2045265907","https://openalex.org/W2972997031","https://openalex.org/W4312628177"],"abstract_inverted_index":{"This":[0],"paper":[1],"presents":[2],"a":[3,10,29],"general":[4],"coding":[5,18,40,91],"method":[6,21,55],"where":[7],"data":[8],"in":[9],"Hilbert":[11,44],"space":[12],"are":[13,56,74],"represented":[14],"by":[15,63],"finite":[16],"dimensional":[17],"vectors.":[19],"The":[20,72],"is":[22],"based":[23],"on":[24],"empirical":[25],"risk":[26],"minimization":[27],"within":[28],"certain":[30],"class":[31,68],"of":[32,39,53,69,79],"linear":[33,70],"operators,":[34],"which":[35,58],"map":[36],"the":[37,43,49,54,60,64,67],"set":[38],"vectors":[41],"to":[42,76],"space.":[45],"Two":[46],"results":[47,73],"bounding":[48],"expected":[50],"reconstruction":[51],"error":[52],"derived,":[57],"highlight":[59],"role":[61],"played":[62],"codebook":[65],"and":[66,88],"operators.":[71],"specialized":[75],"some":[77],"cases":[78],"practical":[80],"importance,":[81],"including":[82],"K-means":[83],"clustering,":[84],"nonnegative":[85],"matrix":[86],"factorization":[87],"other":[89],"sparse":[90],"methods.":[92]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":12},{"year":2015,"cited_by_count":16},{"year":2014,"cited_by_count":14},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":7}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
