{"id":"https://openalex.org/W2163696743","doi":"https://doi.org/10.1162/089976699300016403","title":"On the Approximation Rate of Hierarchical Mixtures-of-Experts for Generalized Linear Models","display_name":"On the Approximation Rate of Hierarchical Mixtures-of-Experts for Generalized Linear Models","publication_year":1999,"publication_date":"1999-07-01","ids":{"openalex":"https://openalex.org/W2163696743","doi":"https://doi.org/10.1162/089976699300016403","mag":"2163696743","pmid":"https://pubmed.ncbi.nlm.nih.gov/10418162"},"language":"en","primary_location":{"id":"doi:10.1162/089976699300016403","is_oa":false,"landing_page_url":"https://doi.org/10.1162/089976699300016403","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"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/A5101578550","display_name":"Wenxin Jiang","orcid":"https://orcid.org/0000-0003-0868-0402"},"institutions":[{"id":"https://openalex.org/I111979921","display_name":"Northwestern University","ror":"https://ror.org/000e0be47","country_code":"US","type":"education","lineage":["https://openalex.org/I111979921"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Wenxin Jiang","raw_affiliation_strings":["Department of Statistics, Northwestern University, Evanston, IL 60208, U.S.A"],"affiliations":[{"raw_affiliation_string":"Department of Statistics, Northwestern University, Evanston, IL 60208, U.S.A","institution_ids":["https://openalex.org/I111979921"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5045302193","display_name":"Martin A. Tanner","orcid":"https://orcid.org/0000-0002-9536-5373"},"institutions":[{"id":"https://openalex.org/I111979921","display_name":"Northwestern University","ror":"https://ror.org/000e0be47","country_code":"US","type":"education","lineage":["https://openalex.org/I111979921"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Martin A. Tanner","raw_affiliation_strings":["Department of Statistics, Northwestern University, Evanston, IL 60208, U.S.A"],"affiliations":[{"raw_affiliation_string":"Department of Statistics, Northwestern University, Evanston, IL 60208, U.S.A","institution_ids":["https://openalex.org/I111979921"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5101578550"],"corresponding_institution_ids":["https://openalex.org/I111979921"],"apc_list":null,"apc_paid":null,"fwci":3.2825,"has_fulltext":false,"cited_by_count":52,"citation_normalized_percentile":{"value":0.9324639,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"11","issue":"5","first_page":"1183","last_page":"1198"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6530486941337585},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.593303918838501},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5273340940475464},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5113930106163025},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4791550040245056},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.4770069420337677},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4700416922569275},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.46902480721473694},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.45643770694732666},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4523686468601227},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4522002041339874},{"id":"https://openalex.org/keywords/linear-approximation","display_name":"Linear approximation","score":0.4410780966281891},{"id":"https://openalex.org/keywords/generalized-linear-model","display_name":"Generalized linear model","score":0.42765557765960693},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.38302722573280334},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3292543888092041},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2686406075954437},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13233518600463867},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11395424604415894},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.0690663754940033},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.06819945573806763}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6530486941337585},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.593303918838501},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5273340940475464},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5113930106163025},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4791550040245056},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.4770069420337677},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4700416922569275},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.46902480721473694},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.45643770694732666},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4523686468601227},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4522002041339874},{"id":"https://openalex.org/C160824197","wikidata":"https://www.wikidata.org/wiki/Q2071054","display_name":"Linear approximation","level":3,"score":0.4410780966281891},{"id":"https://openalex.org/C41587187","wikidata":"https://www.wikidata.org/wiki/Q1501882","display_name":"Generalized linear model","level":2,"score":0.42765557765960693},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.38302722573280334},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3292543888092041},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2686406075954437},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13233518600463867},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11395424604415894},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0690663754940033},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.06819945573806763},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","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}],"mesh":[{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D009415","descriptor_name":"Nerve Net","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D009415","descriptor_name":"Nerve Net","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D009415","descriptor_name":"Nerve Net","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true}],"locations_count":3,"locations":[{"id":"doi:10.1162/089976699300016403","is_oa":false,"landing_page_url":"https://doi.org/10.1162/089976699300016403","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},{"id":"pmid:10418162","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/10418162","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":"Neural computation","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.26.8402","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.8402","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://neyman.stats.northwestern.edu/jiang/hme_dens.ps","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W1528905581","https://openalex.org/W2011277999","https://openalex.org/W2025653905","https://openalex.org/W2027628336","https://openalex.org/W2079224763","https://openalex.org/W2130136332","https://openalex.org/W2135879592","https://openalex.org/W2147503863","https://openalex.org/W2148845857","https://openalex.org/W2150884987","https://openalex.org/W3015463134"],"related_works":["https://openalex.org/W4382469906","https://openalex.org/W2087968352","https://openalex.org/W2105325102","https://openalex.org/W4385890208","https://openalex.org/W1999955141","https://openalex.org/W2351859806","https://openalex.org/W2048235058","https://openalex.org/W4283652896","https://openalex.org/W2011212954","https://openalex.org/W4360977145"],"abstract_inverted_index":{"We":[0],"investigate":[1],"a":[2,46],"class":[3,47,63],"of":[4,17,40,48,51,70,80,99,112],"hierarchical":[5],"mixtures-of-experts":[6],"(HME)":[7],"models":[8,12],"where":[9,56,107],"generalized":[10],"linear":[11],"with":[13,102],"nonlinear":[14],"mean":[15,42],"functions":[16,43,50],"the":[18,31,52,68,74,81,97,110,113],"form":[19,53],"psi":[20,28,54],"(alpha":[21],"+":[22],"xT":[23],"beta)":[24],"are":[25],"mixed.":[26],"Here":[27],"(.)":[29],"is":[30,36,84,109],"inverse":[32],"link":[33],"function.":[34],"It":[35],"shown":[37],"that":[38],"mixtures":[39],"such":[41],"can":[44,93],"approximate":[45],"smooth":[49],"(h(x)),":[55],"h(.)":[57],"epsilon":[58],"W2;K":[59],"infinity":[60],"(a":[61],"Sobolev":[62],"over":[64],"[0,":[65],"1]s),":[66],"as":[67,86],"number":[69],"experts":[71],"m":[72],"in":[73,88],"network":[75],"increases.":[76],"An":[77],"upper":[78],"bound":[79],"approximation":[82],"rate":[83,92],"given":[85],"O(m-2/s)":[87],"Lp":[89],"norm.":[90],"This":[91],"be":[94],"achieved":[95],"within":[96],"family":[98],"HME":[100],"structures":[101],"no":[103],"more":[104],"than":[105],"s-layers,":[106],"s":[108],"dimension":[111],"predictor":[114],"x.":[115]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}