{"id":"https://openalex.org/W2805387616","doi":"https://doi.org/10.1109/lsp.2018.2860238","title":"Data-Free/Data-Sparse Softmax Parameter Estimation With Structured Class Geometries","display_name":"Data-Free/Data-Sparse Softmax Parameter Estimation With Structured Class Geometries","publication_year":2018,"publication_date":"2018-07-26","ids":{"openalex":"https://openalex.org/W2805387616","doi":"https://doi.org/10.1109/lsp.2018.2860238","mag":"2805387616"},"language":"en","primary_location":{"id":"doi:10.1109/lsp.2018.2860238","is_oa":false,"landing_page_url":"https://doi.org/10.1109/lsp.2018.2860238","pdf_url":null,"source":{"id":"https://openalex.org/S120629676","display_name":"IEEE Signal Processing Letters","issn_l":"1070-9908","issn":["1070-9908","1558-2361"],"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 Signal Processing Letters","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1806.00728","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100664008","display_name":"Nisar Ahmed","orcid":"https://orcid.org/0000-0002-7555-5671"},"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":true,"raw_author_name":"Nisar Ahmed","raw_affiliation_strings":["Ann and H. J. Smead Aerospace Engineering Sciences, University of Colorado, Boulder, USA"],"affiliations":[{"raw_affiliation_string":"Ann and H. J. Smead Aerospace Engineering Sciences, University of Colorado, Boulder, USA","institution_ids":["https://openalex.org/I188538660"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5100664008"],"corresponding_institution_ids":["https://openalex.org/I188538660"],"apc_list":null,"apc_paid":null,"fwci":2.1999,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.90539068,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"25","issue":"9","first_page":"1408","last_page":"1412"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.998199999332428,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.998199999332428,"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/T12535","display_name":"Machine Learning and Data Classification","score":0.9969000220298767,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9961000084877014,"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/softmax-function","display_name":"Softmax function","score":0.9436312913894653},{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.672601580619812},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5630424618721008},{"id":"https://openalex.org/keywords/maximum-a-posteriori-estimation","display_name":"Maximum a posteriori estimation","score":0.5175156593322754},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5091950297355652},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42842257022857666},{"id":"https://openalex.org/keywords/parameter-space","display_name":"Parameter space","score":0.4223717749118805},{"id":"https://openalex.org/keywords/feature-vector","display_name":"Feature vector","score":0.4112856686115265},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3545694947242737},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3286653459072113},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2988772988319397},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.23933202028274536},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23253914713859558},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.21073505282402039},{"id":"https://openalex.org/keywords/maximum-likelihood","display_name":"Maximum likelihood","score":0.13007885217666626},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.12621694803237915}],"concepts":[{"id":"https://openalex.org/C188441871","wikidata":"https://www.wikidata.org/wiki/Q7554146","display_name":"Softmax function","level":3,"score":0.9436312913894653},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.672601580619812},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5630424618721008},{"id":"https://openalex.org/C9810830","wikidata":"https://www.wikidata.org/wiki/Q635384","display_name":"Maximum a posteriori estimation","level":3,"score":0.5175156593322754},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5091950297355652},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42842257022857666},{"id":"https://openalex.org/C73586568","wikidata":"https://www.wikidata.org/wiki/Q2600211","display_name":"Parameter space","level":2,"score":0.4223717749118805},{"id":"https://openalex.org/C83665646","wikidata":"https://www.wikidata.org/wiki/Q42139305","display_name":"Feature vector","level":2,"score":0.4112856686115265},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3545694947242737},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3286653459072113},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2988772988319397},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.23933202028274536},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23253914713859558},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.21073505282402039},{"id":"https://openalex.org/C49781872","wikidata":"https://www.wikidata.org/wiki/Q1045555","display_name":"Maximum likelihood","level":2,"score":0.13007885217666626},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.12621694803237915}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/lsp.2018.2860238","is_oa":false,"landing_page_url":"https://doi.org/10.1109/lsp.2018.2860238","pdf_url":null,"source":{"id":"https://openalex.org/S120629676","display_name":"IEEE Signal Processing Letters","issn_l":"1070-9908","issn":["1070-9908","1558-2361"],"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 Signal Processing Letters","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1806.00728","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1806.00728","pdf_url":"https://arxiv.org/pdf/1806.00728","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":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1806.00728","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1806.00728","pdf_url":"https://arxiv.org/pdf/1806.00728","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":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W334520530","https://openalex.org/W1550812296","https://openalex.org/W1561907491","https://openalex.org/W1599010327","https://openalex.org/W1663973292","https://openalex.org/W1821462560","https://openalex.org/W1999244880","https://openalex.org/W2009086942","https://openalex.org/W2014158652","https://openalex.org/W2062122868","https://openalex.org/W2097148013","https://openalex.org/W2117063635","https://openalex.org/W2119129950","https://openalex.org/W2129377018","https://openalex.org/W2150579376","https://openalex.org/W2160409433","https://openalex.org/W2164820644","https://openalex.org/W2505439654","https://openalex.org/W2508271142","https://openalex.org/W2784163702","https://openalex.org/W2963547728","https://openalex.org/W2963656735","https://openalex.org/W3103152812","https://openalex.org/W3105988882","https://openalex.org/W4285719527","https://openalex.org/W6611461683","https://openalex.org/W6633117799","https://openalex.org/W6635795003","https://openalex.org/W6638523607","https://openalex.org/W6666091668","https://openalex.org/W6735013348"],"related_works":["https://openalex.org/W3107204728","https://openalex.org/W4287591324","https://openalex.org/W3108503355","https://openalex.org/W2980176872","https://openalex.org/W4226420367","https://openalex.org/W2962876041","https://openalex.org/W3090555870","https://openalex.org/W2249953602","https://openalex.org/W2912971006","https://openalex.org/W3022820045"],"abstract_inverted_index":{"This":[0,79],"note":[1],"considers":[2],"softmax":[3,33,69,81,155],"parameter":[4,43,70,82,121],"estimation":[5,107],"when":[6],"little/no":[7],"labeled":[8],"training":[9],"data":[10,91],"is":[11,26,29,129],"available,":[12],"but":[13],"a":[14,39],"priori":[15],"information":[16],"about":[17],"the":[18,58,64,119,146],"relative":[19],"geometry":[20],"of":[21,42],"class":[22,48,74],"label":[23],"log-odds":[24,49,147],"boundaries":[25,50,148],"available.":[27],"It":[28],"shown":[30,132],"that":[31,56,133],"\u201cdata-free\u201d":[32],"model":[34],"synthesis":[35],"corresponds":[36],"to":[37,84,103,116],"solving":[38],"linear":[40,65,97,120],"system":[41],"equations,":[44],"wherein":[45],"desired":[46],"dominant":[47],"are":[51],"encoded":[52],"via":[53],"convex":[54],"polytopes":[55],"decompose":[57],"input":[59],"feature":[60],"space.":[61],"When":[62],"solvable,":[63],"equations":[66,98,122],"yield":[67],"closed-form":[68],"solution":[71],"families":[72],"using":[73,152],"boundary":[75],"polytope":[76,126],"specifications":[77],"only.":[78],"allows":[80],"learning":[83],"be":[85,101,150],"implemented":[86],"without":[87],"expensive":[88],"brute":[89],"force":[90],"sampling":[92],"and":[93],"numerical":[94],"optimization.":[95],"The":[96],"can":[99],"also":[100,114,131],"adapted":[102],"constrained":[104],"maximum":[105],"likelihood":[106],"in":[108],"data-sparse":[109],"settings.":[110],"Since":[111],"solutions":[112],"may":[113],"fail":[115],"exist":[117,135],"for":[118,144],"derived":[123],"from":[124],"certain":[125],"specifications,":[127],"it":[128],"thus":[130],"there":[134],"probabilistic":[136],"classification":[137],"problems":[138],"over":[139],"m":[140],"convexly":[141],"separable":[142],"classes":[143],"which":[145],"cannot":[149],"learned":[151],"an":[153],"m-class":[154],"model.":[156]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":2}],"updated_date":"2026-03-27T14:29:43.386196","created_date":"2025-10-10T00:00:00"}