{"id":"https://openalex.org/W4413916360","doi":"https://doi.org/10.3390/make7030093","title":"Geometric Reasoning in the Embedding Space","display_name":"Geometric Reasoning in the Embedding Space","publication_year":2025,"publication_date":"2025-09-02","ids":{"openalex":"https://openalex.org/W4413916360","doi":"https://doi.org/10.3390/make7030093"},"language":"en","primary_location":{"id":"doi:10.3390/make7030093","is_oa":true,"landing_page_url":"https://doi.org/10.3390/make7030093","pdf_url":"https://www.mdpi.com/2504-4990/7/3/93/pdf?version=1756816109","source":{"id":"https://openalex.org/S4210213891","display_name":"Machine Learning and Knowledge Extraction","issn_l":"2504-4990","issn":["2504-4990"],"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":"Machine Learning and Knowledge Extraction","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2504-4990/7/3/93/pdf?version=1756816109","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5065303754","display_name":"David Moj\u017e\u00ed\u0161ek","orcid":"https://orcid.org/0000-0002-3867-644X"},"institutions":[{"id":"https://openalex.org/I187293425","display_name":"University of Ostrava","ror":"https://ror.org/00pyqav47","country_code":"CZ","type":"education","lineage":["https://openalex.org/I187293425"]}],"countries":["CZ"],"is_corresponding":true,"raw_author_name":"David Moj\u017e\u00ed\u0161ek","raw_affiliation_strings":["Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 702 00 Ostrava, Czech Republic"],"affiliations":[{"raw_affiliation_string":"Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 702 00 Ostrava, Czech Republic","institution_ids":["https://openalex.org/I187293425"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076891199","display_name":"Jan H\u016fla","orcid":"https://orcid.org/0000-0001-7639-864X"},"institutions":[{"id":"https://openalex.org/I44504214","display_name":"Czech Technical University in Prague","ror":"https://ror.org/03kqpb082","country_code":"CZ","type":"education","lineage":["https://openalex.org/I44504214"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Jan H\u016fla","raw_affiliation_strings":["Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague, 160 00 Prague, Czech Republic"],"affiliations":[{"raw_affiliation_string":"Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague, 160 00 Prague, Czech Republic","institution_ids":["https://openalex.org/I44504214"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5058168742","display_name":"Ji\u0159\u0131\u0301 Jane\u010dek","orcid":"https://orcid.org/0000-0003-4397-9004"},"institutions":[{"id":"https://openalex.org/I187293425","display_name":"University of Ostrava","ror":"https://ror.org/00pyqav47","country_code":"CZ","type":"education","lineage":["https://openalex.org/I187293425"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Ji\u0159\u00ed Jane\u010dek","raw_affiliation_strings":["Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 702 00 Ostrava, Czech Republic"],"affiliations":[{"raw_affiliation_string":"Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 702 00 Ostrava, Czech Republic","institution_ids":["https://openalex.org/I187293425"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5022071541","display_name":"David Herel","orcid":"https://orcid.org/0009-0000-1861-3778"},"institutions":[{"id":"https://openalex.org/I44504214","display_name":"Czech Technical University in Prague","ror":"https://ror.org/03kqpb082","country_code":"CZ","type":"education","lineage":["https://openalex.org/I44504214"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"David Herel","raw_affiliation_strings":["Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague, 160 00 Prague, Czech Republic"],"affiliations":[{"raw_affiliation_string":"Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague, 160 00 Prague, Czech Republic","institution_ids":["https://openalex.org/I44504214"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083748570","display_name":"Mikol\u00e1\u0161 Janota","orcid":"https://orcid.org/0000-0003-3487-784X"},"institutions":[{"id":"https://openalex.org/I44504214","display_name":"Czech Technical University in Prague","ror":"https://ror.org/03kqpb082","country_code":"CZ","type":"education","lineage":["https://openalex.org/I44504214"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Mikol\u00e1\u0161 Janota","raw_affiliation_strings":["Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague, 160 00 Prague, Czech Republic"],"affiliations":[{"raw_affiliation_string":"Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague, 160 00 Prague, Czech Republic","institution_ids":["https://openalex.org/I44504214"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5065303754"],"corresponding_institution_ids":["https://openalex.org/I187293425"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.23764609,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"7","issue":"3","first_page":"93","last_page":"93"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11450","display_name":"Model-Driven Software Engineering Techniques","score":0.9598000049591064,"subfield":{"id":"https://openalex.org/subfields/1712","display_name":"Software"},"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/T11450","display_name":"Model-Driven Software Engineering Techniques","score":0.9598000049591064,"subfield":{"id":"https://openalex.org/subfields/1712","display_name":"Software"},"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/T11902","display_name":"Intelligent Tutoring Systems and Adaptive Learning","score":0.9387000203132629,"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/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.9302999973297119,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/embedding","display_name":"Embedding","score":0.6912661790847778},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5363221764564514},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4194066524505615},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.34475836157798767},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3125429153442383}],"concepts":[{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.6912661790847778},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5363221764564514},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4194066524505615},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.34475836157798767},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3125429153442383},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/make7030093","is_oa":true,"landing_page_url":"https://doi.org/10.3390/make7030093","pdf_url":"https://www.mdpi.com/2504-4990/7/3/93/pdf?version=1756816109","source":{"id":"https://openalex.org/S4210213891","display_name":"Machine Learning and Knowledge Extraction","issn_l":"2504-4990","issn":["2504-4990"],"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":"Machine Learning and Knowledge Extraction","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:123d3188726d4887a81d5f62d86e9da1","is_oa":true,"landing_page_url":"https://doaj.org/article/123d3188726d4887a81d5f62d86e9da1","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Machine Learning and Knowledge Extraction, Vol 7, Iss 3, p 93 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/make7030093","is_oa":true,"landing_page_url":"https://doi.org/10.3390/make7030093","pdf_url":"https://www.mdpi.com/2504-4990/7/3/93/pdf?version=1756816109","source":{"id":"https://openalex.org/S4210213891","display_name":"Machine Learning and Knowledge Extraction","issn_l":"2504-4990","issn":["2504-4990"],"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":"Machine Learning and Knowledge Extraction","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G112504165","display_name":null,"funder_award_id":"CZ.02.01.01/00/22_008/0004590","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G1408456483","display_name":null,"funder_award_id":"CZ.10.03.01/00/22_003/0000048","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G2678520069","display_name":null,"funder_award_id":"CZ.02.01","funder_id":"https://openalex.org/F4320321006","funder_display_name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky"},{"id":"https://openalex.org/G3771185154","display_name":null,"funder_award_id":"CZ.10.03","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G4273056302","display_name":null,"funder_award_id":".10.03.01/00/22_003/0000048","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G4821928915","display_name":null,"funder_award_id":"CZ.10.03.01/00/22_003/000004","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G4956428346","display_name":null,"funder_award_id":"Horizon 2020 research and innovatio","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G6837089890","display_name":null,"funder_award_id":"ROBOPROX","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8318064016","display_name":null,"funder_award_id":"Horizon","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8343064509","display_name":null,"funder_award_id":"857306","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320321006","display_name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","ror":"https://ror.org/01pv73b02"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4413916360.pdf","grobid_xml":"https://content.openalex.org/works/W4413916360.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W2064675550","https://openalex.org/W2077378514","https://openalex.org/W2734458248","https://openalex.org/W3042993977","https://openalex.org/W3185710702","https://openalex.org/W4292732060","https://openalex.org/W4362028962","https://openalex.org/W4388979610","https://openalex.org/W4389520071","https://openalex.org/W4390940921","https://openalex.org/W4403582418"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027"],"abstract_inverted_index":{"While":[0],"neural":[1,34,169],"networks":[2,35,170],"can":[3,171],"solve":[4,124],"complex":[5],"geometric":[6,57,75,104],"problems,":[7],"as":[8],"demonstrated":[9],"by":[10,40],"systems":[11],"like":[12],"AlphaGeometry,":[13],"we":[14,31,83],"have":[15],"limited":[16],"understanding":[17,39,174],"of":[18,77,151],"how":[19,33,113,168],"they":[20,80],"internally":[21],"represent":[22],"and":[23,45,95,120,155,175],"reason":[24],"about":[25],"spatial":[26,38],"relationships.":[27],"In":[28],"this":[29,149],"work,":[30],"investigate":[32],"develop":[36,68,172],"internal":[37,70],"training":[41],"Graph":[42,140],"Neural":[43,141],"Networks":[44,142],"Transformers":[46,147],"to":[47,159,177],"predict":[48],"point":[49,86],"positions":[50],"on":[51],"a":[52],"discrete":[53],"2D":[54,90],"grid":[55,91],"from":[56],"constraints":[58,125],"that":[59,65,72,85,122,139],"describe":[60],"hidden":[61,103],"figures.":[62],"We":[63,136],"show":[64],"both":[66],"models":[67,99,123],"interpretable":[69],"representations":[71],"mirror":[73],"the":[74,78,98,102],"structure":[76],"problems":[79],"solve.":[81],"Specifically,":[82],"observe":[84],"embeddings":[87],"self-organize":[88],"into":[89,167],"structures":[92],"during":[93,96],"training,":[94],"inference,":[97],"iteratively":[100],"construct":[101],"figures":[105],"within":[106],"their":[107,178],"embedding":[108],"spaces.":[109],"Our":[110],"analysis":[111],"reveals":[112],"reasoning":[114,154],"complexity":[115],"correlates":[116],"with":[117],"prediction":[118],"accuracy,":[119],"shows":[121],"through":[126],"an":[127],"iterative":[128],"refinement":[129],"process,":[130],"which":[131],"might":[132],"resemble":[133],"continuous":[134],"optimization.":[135],"also":[137],"find":[138],"prove":[143],"more":[144,157],"suitable":[145],"than":[146],"for":[148],"type":[150],"structured":[152,173],"constraint":[153],"scale":[156],"effectively":[158],"larger":[160],"problems.":[161],"These":[162],"findings":[163],"provide":[164],"initial":[165],"insights":[166],"contribute":[176],"interpretability.":[179]},"counts_by_year":[],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}
