{"id":"https://openalex.org/W3021639139","doi":"https://doi.org/10.3390/e22101105","title":"Fractional Norms and Quasinorms Do Not Help to Overcome the Curse of Dimensionality","display_name":"Fractional Norms and Quasinorms Do Not Help to Overcome the Curse of Dimensionality","publication_year":2020,"publication_date":"2020-09-30","ids":{"openalex":"https://openalex.org/W3021639139","doi":"https://doi.org/10.3390/e22101105","mag":"3021639139","pmid":"https://pubmed.ncbi.nlm.nih.gov/33286874"},"language":"en","primary_location":{"id":"doi:10.3390/e22101105","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22101105","pdf_url":"https://www.mdpi.com/1099-4300/22/10/1105/pdf?version=1603175755","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/22/10/1105/pdf?version=1603175755","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Evgeny M. Mirkes","orcid":"https://orcid.org/0000-0003-1474-1734"},"institutions":[{"id":"https://openalex.org/I153648349","display_name":"University of Leicester","ror":"https://ror.org/04h699437","country_code":"GB","type":"education","lineage":["https://openalex.org/I153648349"]},{"id":"https://openalex.org/I79563041","display_name":"N. I. Lobachevsky State University of Nizhny Novgorod","ror":"https://ror.org/01bb1zm18","country_code":"RU","type":"education","lineage":["https://openalex.org/I79563041"]}],"countries":["GB","RU"],"is_corresponding":true,"raw_author_name":"Evgeny M. Mirkes","raw_affiliation_strings":["Laboratory of Advanced Methods for High-Dimensional Data Analysis, Lobachevsky State University, 603105 Nizhny Novgorod, Russia","School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7HR, UK"],"raw_orcid":"https://orcid.org/0000-0003-1474-1734","affiliations":[{"raw_affiliation_string":"Laboratory of Advanced Methods for High-Dimensional Data Analysis, Lobachevsky State University, 603105 Nizhny Novgorod, Russia","institution_ids":["https://openalex.org/I79563041"]},{"raw_affiliation_string":"School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7HR, UK","institution_ids":["https://openalex.org/I153648349"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Jeza Allohibi","orcid":"https://orcid.org/0000-0002-3439-1500"},"institutions":[{"id":"https://openalex.org/I153648349","display_name":"University of Leicester","ror":"https://ror.org/04h699437","country_code":"GB","type":"education","lineage":["https://openalex.org/I153648349"]},{"id":"https://openalex.org/I23075662","display_name":"Taibah University","ror":"https://ror.org/01xv1nn60","country_code":"SA","type":"education","lineage":["https://openalex.org/I23075662"]}],"countries":["GB","SA"],"is_corresponding":false,"raw_author_name":"Jeza Allohibi","raw_affiliation_strings":["Department of Mathematics, Taibah University, Janadah Bin Umayyah Road, Tayba, Medina 42353, Saudi Arabia","School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7HR, UK"],"raw_orcid":"https://orcid.org/0000-0002-3439-1500","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Taibah University, Janadah Bin Umayyah Road, Tayba, Medina 42353, Saudi Arabia","institution_ids":["https://openalex.org/I23075662"]},{"raw_affiliation_string":"School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7HR, UK","institution_ids":["https://openalex.org/I153648349"]}]},{"author_position":"last","author":{"id":null,"display_name":"Alexander Gorban","orcid":"https://orcid.org/0000-0001-6224-1430"},"institutions":[{"id":"https://openalex.org/I153648349","display_name":"University of Leicester","ror":"https://ror.org/04h699437","country_code":"GB","type":"education","lineage":["https://openalex.org/I153648349"]},{"id":"https://openalex.org/I79563041","display_name":"N. I. Lobachevsky State University of Nizhny Novgorod","ror":"https://ror.org/01bb1zm18","country_code":"RU","type":"education","lineage":["https://openalex.org/I79563041"]}],"countries":["GB","RU"],"is_corresponding":false,"raw_author_name":"Alexander Gorban","raw_affiliation_strings":["Laboratory of Advanced Methods for High-Dimensional Data Analysis, Lobachevsky State University, 603105 Nizhny Novgorod, Russia","School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7HR, UK"],"raw_orcid":"https://orcid.org/0000-0001-6224-1430","affiliations":[{"raw_affiliation_string":"Laboratory of Advanced Methods for High-Dimensional Data Analysis, Lobachevsky State University, 603105 Nizhny Novgorod, Russia","institution_ids":["https://openalex.org/I79563041"]},{"raw_affiliation_string":"School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7HR, UK","institution_ids":["https://openalex.org/I153648349"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I153648349","https://openalex.org/I79563041"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.7202,"has_fulltext":false,"cited_by_count":43,"citation_normalized_percentile":{"value":0.92080004,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"22","issue":"10","first_page":"1105","last_page":"1105"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12535","display_name":"Machine Learning and Data Classification","score":0.25290000438690186,"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"}},"topics":[{"id":"https://openalex.org/T12535","display_name":"Machine Learning and Data Classification","score":0.25290000438690186,"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"}},{"id":"https://openalex.org/T10057","display_name":"Face and Expression Recognition","score":0.20110000669956207,"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/T11652","display_name":"Imbalanced Data Classification Techniques","score":0.09269999712705612,"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/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.9429000020027161},{"id":"https://openalex.org/keywords/curse","display_name":"Curse","score":0.728600025177002},{"id":"https://openalex.org/keywords/contrast","display_name":"Contrast (vision)","score":0.7014999985694885},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.555400013923645},{"id":"https://openalex.org/keywords/blessing","display_name":"Blessing","score":0.49070000648498535},{"id":"https://openalex.org/keywords/euclidean-distance","display_name":"Euclidean distance","score":0.45260000228881836},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.4318999946117401}],"concepts":[{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.9429000020027161},{"id":"https://openalex.org/C2780273121","wikidata":"https://www.wikidata.org/wiki/Q109411","display_name":"Curse","level":2,"score":0.728600025177002},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.7014999985694885},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6218000054359436},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.555400013923645},{"id":"https://openalex.org/C2776195157","wikidata":"https://www.wikidata.org/wiki/Q626510","display_name":"Blessing","level":2,"score":0.49070000648498535},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.45260000228881836},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4318999946117401},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.42719998955726624},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3750999867916107},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.35929998755455017},{"id":"https://openalex.org/C70518039","wikidata":"https://www.wikidata.org/wiki/Q16000077","display_name":"Dimensionality reduction","level":2,"score":0.35179999470710754},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.335999995470047},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3287999927997589},{"id":"https://openalex.org/C3018023364","wikidata":"https://www.wikidata.org/wiki/Q425265","display_name":"Significant difference","level":2,"score":0.3212999999523163},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3158000111579895},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.27160000801086426},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.26750001311302185}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.3390/e22101105","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22101105","pdf_url":"https://www.mdpi.com/1099-4300/22/10/1105/pdf?version=1603175755","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:33286874","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33286874","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:arXiv.org:2004.14230","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2004.14230","pdf_url":"https://arxiv.org/pdf/2004.14230","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:doaj.org/article:4ab622c5196541fea3e463d29b049b7a","is_oa":true,"landing_page_url":"https://doaj.org/article/4ab622c5196541fea3e463d29b049b7a","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 22, Iss 10, p 1105 (2020)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/22/10/1105/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e22101105","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy; Volume 22; Issue 10; Pages: 1105","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7597215","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7597215","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e22101105","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22101105","pdf_url":"https://www.mdpi.com/1099-4300/22/10/1105/pdf?version=1603175755","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8931689542","display_name":null,"funder_award_id":"14.Y26.31.0022","funder_id":"https://openalex.org/F4320327494","funder_display_name":"Ministry of Science and Higher Education of the Russian Federation"}],"funders":[{"id":"https://openalex.org/F4320327494","display_name":"Ministry of Science and Higher Education of the Russian Federation","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":80,"referenced_works":["https://openalex.org/W8060374","https://openalex.org/W93922237","https://openalex.org/W231970315","https://openalex.org/W568124517","https://openalex.org/W1565746575","https://openalex.org/W1595303882","https://openalex.org/W1612304093","https://openalex.org/W1672197616","https://openalex.org/W1680797894","https://openalex.org/W1908619978","https://openalex.org/W1973286131","https://openalex.org/W1987698418","https://openalex.org/W1991566301","https://openalex.org/W2000432226","https://openalex.org/W2016944307","https://openalex.org/W2028569884","https://openalex.org/W2030644212","https://openalex.org/W2057074619","https://openalex.org/W2061122559","https://openalex.org/W2061128187","https://openalex.org/W2068458829","https://openalex.org/W2074166357","https://openalex.org/W2078894942","https://openalex.org/W2088422930","https://openalex.org/W2104027187","https://openalex.org/W2110847962","https://openalex.org/W2114588272","https://openalex.org/W2115990376","https://openalex.org/W2131954086","https://openalex.org/W2135046866","https://openalex.org/W2144407188","https://openalex.org/W2146453955","https://openalex.org/W2146801187","https://openalex.org/W2148654450","https://openalex.org/W2152035667","https://openalex.org/W2152799677","https://openalex.org/W2157133710","https://openalex.org/W2160172778","https://openalex.org/W2162006472","https://openalex.org/W2165533158","https://openalex.org/W2168532736","https://openalex.org/W2170702893","https://openalex.org/W2294798173","https://openalex.org/W2298860367","https://openalex.org/W2328142025","https://openalex.org/W2344260814","https://openalex.org/W2510620627","https://openalex.org/W2567830010","https://openalex.org/W2579703217","https://openalex.org/W2593455419","https://openalex.org/W2606759007","https://openalex.org/W2612932006","https://openalex.org/W2784010521","https://openalex.org/W2787894218","https://openalex.org/W2884716758","https://openalex.org/W2892053535","https://openalex.org/W2898060137","https://openalex.org/W2914580252","https://openalex.org/W2963386266","https://openalex.org/W2968231406","https://openalex.org/W2977886961","https://openalex.org/W2979173662","https://openalex.org/W2981496185","https://openalex.org/W2999763174","https://openalex.org/W3000084948","https://openalex.org/W3015598826","https://openalex.org/W3046140350","https://openalex.org/W3148116038","https://openalex.org/W4237171445","https://openalex.org/W4241727697","https://openalex.org/W4250954493","https://openalex.org/W4252684946","https://openalex.org/W4388297464","https://openalex.org/W6676352880","https://openalex.org/W6681758375","https://openalex.org/W6684836034","https://openalex.org/W6685529966","https://openalex.org/W6765970415","https://openalex.org/W6771030649","https://openalex.org/W6864722823"],"related_works":[],"abstract_inverted_index":{"The":[0],"curse":[1,39,164],"of":[2,40,66,92,147,163,167,172,181,189],"dimensionality":[3,41,168,174],"causes":[4],"the":[5,21,38,69,90,96,127,142,145,179,190],"well-known":[6],"and":[7,24,64,103,157,165],"widely":[8],"discussed":[9],"problems":[10],"for":[11,99,124,150],"machine":[12],"learning":[13],"methods.":[14],"There":[15],"is":[16,54,75,106,159],"a":[17,60,109,116],"hypothesis":[18],"that":[19,56,77,89,108,123,141,175],"using":[20],"Manhattan":[22],"distance":[23],"even":[25],"fractional":[26,57],"lp":[27,151],"quasinorms":[28,58],"(for":[29],"p":[30,153],"less":[31],"than":[32,68],"1)":[33],"can":[34],"help":[35],"to":[36],"overcome":[37],"in":[42,144],"classification":[43,118],"problems.":[44],"In":[45],"this":[46,51,78],"study,":[47],"we":[48],"systematically":[49,184],"test":[50],"hypothesis.":[52],"It":[53,85,105,120],"illustrated":[55],"have":[59],"greater":[61,110],"relative":[62,111],"contrast":[63,112],"coefficient":[65],"variation":[67],"Euclidean":[70],"norm":[71],"l2,":[72],"but":[73],"it":[74],"shown":[76,107],"difference":[79,143],"decays":[80],"with":[81,178],"increasing":[82],"space":[83],"dimension.":[84],"has":[86],"been":[87],"demonstrated":[88],"concentration":[91],"distances":[93],"shows":[94,140],"qualitatively":[95],"same":[97],"behaviour":[98],"all":[100],"tested":[101],"norms":[102,135],"quasinorms.":[104],"does":[113],"not":[114],"mean":[115],"better":[117],"quality.":[119],"was":[121,131],"revealed":[122],"different":[125,134],"databases":[126],"best":[128],"(worst)":[129],"performance":[130,146],"achieved":[132],"under":[133],"(quasinorms).":[136],"A":[137],"systematic":[138],"comparison":[139],"kNN":[148],"classifiers":[149],"at":[152],"=":[154],"0.5,":[155],"1,":[156],"2":[158],"statistically":[160],"insignificant.":[161],"Analysis":[162],"blessing":[166],"requires":[169],"careful":[170],"definition":[171],"data":[173],"rarely":[176],"coincides":[177],"number":[180],"attributes.":[182],"We":[183],"examined":[185],"several":[186],"intrinsic":[187],"dimensions":[188],"data.":[191]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":7},{"year":2024,"cited_by_count":13},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":6}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2020-05-13T00:00:00"}