{"id":"https://openalex.org/W2094726556","doi":"https://doi.org/10.1109/isit.2007.4557483","title":"Unequal dimensional small balls and quantization on Grassmann Manifolds","display_name":"Unequal dimensional small balls and quantization on Grassmann Manifolds","publication_year":2007,"publication_date":"2007-06-01","ids":{"openalex":"https://openalex.org/W2094726556","doi":"https://doi.org/10.1109/isit.2007.4557483","mag":"2094726556"},"language":"en","primary_location":{"id":"doi:10.1109/isit.2007.4557483","is_oa":false,"landing_page_url":"https://doi.org/10.1109/isit.2007.4557483","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2007 IEEE International Symposium on Information Theory","raw_type":"proceedings-article"},"type":"preprint","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/0705.2278","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100752236","display_name":"Wei Dai","orcid":"https://orcid.org/0000-0002-4781-3485"},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wei Dai","raw_affiliation_strings":["Department of Electrical and Computer Engineering","Department of Electrical and Computer Engineering, University of Colorado Boulder, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering","institution_ids":[]},{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Colorado Boulder, USA","institution_ids":["https://openalex.org/I188538660"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017012202","display_name":"Brian Rider","orcid":"https://orcid.org/0000-0002-0538-8441"},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Brian Rider","raw_affiliation_strings":["Department of Mathematics, University of Colorado at Boulder"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Colorado at Boulder","institution_ids":["https://openalex.org/I188538660"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084454849","display_name":"Youjian Liu","orcid":"https://orcid.org/0000-0003-4681-0546"},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Youjian Liu","raw_affiliation_strings":["Department of Electrical and Computer Engineering","Department of Electrical and Computer Engineering, University of Colorado Boulder, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering","institution_ids":[]},{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Colorado Boulder, USA","institution_ids":["https://openalex.org/I188538660"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I188538660"],"apc_list":null,"apc_paid":null,"fwci":0.2665,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.59072111,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"1806","last_page":"1810"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12287","display_name":"Fibroblast Growth Factor Research","score":0.9860000014305115,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T12287","display_name":"Fibroblast Growth Factor Research","score":0.9860000014305115,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10901","display_name":"Advanced Data Compression Techniques","score":0.9581000208854675,"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/T11157","display_name":"Chronic Lymphocytic Leukemia Research","score":0.9380000233650208,"subfield":{"id":"https://openalex.org/subfields/2716","display_name":"Genetics"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/ball","display_name":"Ball (mathematics)","score":0.6320036053657532},{"id":"https://openalex.org/keywords/quantization","display_name":"Quantization (signal processing)","score":0.496282160282135},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4839625656604767},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.44155943393707275},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.43541547656059265},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2969309091567993},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.26138395071029663},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.18752917647361755}],"concepts":[{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.6320036053657532},{"id":"https://openalex.org/C28855332","wikidata":"https://www.wikidata.org/wiki/Q198099","display_name":"Quantization (signal processing)","level":2,"score":0.496282160282135},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4839625656604767},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.44155943393707275},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.43541547656059265},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2969309091567993},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.26138395071029663},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.18752917647361755},{"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}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1109/isit.2007.4557483","is_oa":false,"landing_page_url":"https://doi.org/10.1109/isit.2007.4557483","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2007 IEEE International Symposium on Information Theory","raw_type":"proceedings-article"},{"id":"pmh:oai:arXiv.org:0705.2278","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0705.2278","pdf_url":"https://arxiv.org/pdf/0705.2278","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.142.5419","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.142.5419","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://ece.colorado.edu/~liue/publications/papers/Liu_ISIT07_Unequal_dimension_small_ball.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.246.2945","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.246.2945","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/0705.2278v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.93.5065","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.93.5065","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://math.colorado.edu/~brider/isit2007.pdf","raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:0705.2278","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0705.2278","pdf_url":"https://arxiv.org/pdf/0705.2278","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W1563889663","https://openalex.org/W2009188663","https://openalex.org/W2035469609","https://openalex.org/W2052775030","https://openalex.org/W2074863717","https://openalex.org/W2075416888","https://openalex.org/W2091497247","https://openalex.org/W2117616158","https://openalex.org/W2119582213","https://openalex.org/W2129972516","https://openalex.org/W2130256523","https://openalex.org/W2130521103","https://openalex.org/W2133214726","https://openalex.org/W2134765312","https://openalex.org/W2142361128","https://openalex.org/W2147643489","https://openalex.org/W2158148081","https://openalex.org/W2162876202","https://openalex.org/W2951632485","https://openalex.org/W2952817145","https://openalex.org/W2972424502","https://openalex.org/W4246203530","https://openalex.org/W4300546246","https://openalex.org/W6633653771","https://openalex.org/W6680229276","https://openalex.org/W6681401921","https://openalex.org/W6844905184"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2980963558","https://openalex.org/W1882246857","https://openalex.org/W3103304230","https://openalex.org/W2185360082","https://openalex.org/W1992365905","https://openalex.org/W1976691234","https://openalex.org/W4287664878","https://openalex.org/W3212023493","https://openalex.org/W2268755818"],"abstract_inverted_index":{"The":[0,84,178],"Grassmann":[1,139,248],"manifold":[2,249],"G":[3,47,58,100,109],"<sub":[4,48,59,101,110],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[5,25,49,60,102,111],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n,p</sub>":[6,50,103],"(L)":[7,51,104],"is":[8,30,52,73,90,107,117,131,175,250],"the":[9,16,19,70,93,122,129,167,172,184,195,210,215,225,237,247],"set":[10],"of":[11,95,124,236,255],"all":[12],"p-dimensional":[13],"planes":[14],"(through":[15],"origin)":[17],"in":[18,42,46,57,78,99,108,209,246],"n-dimensional":[20],"Euclidean":[21],"space":[22],"L":[23,29],"<sup":[24],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</sup>":[26],",":[27],"where":[28,63,80],"either":[31],"R":[32],"or":[33],"C.":[34],"This":[35,134],"paper":[36],"considers":[37],"an":[38],"unequal":[39,87],"dimensional":[40,88],"quantization":[41,89,173],"which":[43],"a":[44,55,96,118,125],"source":[45],"quantized":[53],"through":[54],"code":[56,226],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n,q</sub>":[61,112],"(L),":[62],"p":[64,81],"and":[65,146,180,191,224],"q":[66,145],"are":[67,164,188,206],"not":[68],"necessarily":[69],"same.":[71],"It":[72],"different":[74],"from":[75],"most":[76],"works":[77],"literature":[79],"\u2261":[82],"q.":[83],"analysis":[85],"for":[86,121,138,166,252],"based":[91],"on":[92,158,183],"volume":[94,123,135,160],"metric":[97,126],"ball":[98,127],"whose":[105],"center":[106],".":[113],"Our":[114],"chief":[115],"result":[116],"closed-form":[119],"formula":[120,136],"when":[128],"radius":[130],"sufficiently":[132,176],"small.":[133],"holds":[137],"manifolds":[140],"with":[141,219],"arbitrary":[142],"n,":[143],"p,":[144],"L,":[147],"while":[148],"previous":[149],"results":[150,239],"pertained":[151],"only":[152],"to":[153,240],"some":[154,234],"special":[155],"cases.":[156],"Based":[157],"this":[159],"formula,":[161],"several":[162],"bounds":[163,182],"derived":[165,238],"rate":[168,174,186,197,227],"distortion":[169,185,198,218],"tradeoff":[170],"assuming":[171],"high.":[177],"lower":[179],"upper":[181],"function":[187],"asymptotically":[189,207],"identical,":[190],"so":[192],"precisely":[193],"quantify":[194],"asymptotic":[196],"tradeoff.":[199],"We":[200],"also":[201],"show":[202],"that":[203,212],"random":[204],"codes":[205],"optimal":[208],"sense":[211],"they":[213],"achieve":[214],"minimum":[216],"achievable":[217],"probability":[220],"one":[221],"as":[222],"n":[223],"approach":[228],"infinity":[229],"linearly.":[230],"Finally,":[231],"we":[232],"discuss":[233],"applications":[235],"communication":[241],"theory.":[242],"A":[243],"geometric":[244],"interpretation":[245],"developed":[251],"capacity":[253],"calculation":[254],"additive":[256],"white":[257],"Gaussian":[258],"noise":[259],"channel.":[260]},"counts_by_year":[],"updated_date":"2026-07-03T08:13:44.112507","created_date":"2025-10-10T00:00:00"}
