{"id":"https://openalex.org/W4415208680","doi":"https://doi.org/10.1007/978-3-032-08324-1_13","title":"Mathematical Foundation of Interpretable Equivariant Surrogate Models","display_name":"Mathematical Foundation of Interpretable Equivariant Surrogate Models","publication_year":2025,"publication_date":"2025-10-15","ids":{"openalex":"https://openalex.org/W4415208680","doi":"https://doi.org/10.1007/978-3-032-08324-1_13"},"language":"en","primary_location":{"id":"doi:10.1007/978-3-032-08324-1_13","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-032-08324-1_13","pdf_url":"https://link.springer.com/content/pdf/10.1007/978-3-032-08324-1_13.pdf","source":{"id":"https://openalex.org/S2764900261","display_name":"Communications in computer and information science","issn_l":"1865-0929","issn":["1865-0929","1865-0937"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Communications in Computer and Information Science","raw_type":"book-chapter"},"type":"conference-paper","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/978-3-032-08324-1_13.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120008744","display_name":"Jacopo Joy Colombini","orcid":null},"institutions":[{"id":"https://openalex.org/I157210198","display_name":"Scuola Normale Superiore","ror":"https://ror.org/03aydme10","country_code":"IT","type":"education","lineage":["https://openalex.org/I157210198"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Jacopo Joy Colombini","raw_affiliation_strings":["Scuola Normale Superiore, Pisa, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Scuola Normale Superiore, Pisa, Italy","institution_ids":["https://openalex.org/I157210198"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004609161","display_name":"Filippo Bonchi","orcid":"https://orcid.org/0000-0002-3433-723X"},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Filippo Bonchi","raw_affiliation_strings":["Universit\u00e0 di Pisa, Pisa, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e0 di Pisa, Pisa, Italy","institution_ids":["https://openalex.org/I108290504"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5058772124","display_name":"Francesco Giannini","orcid":"https://orcid.org/0000-0001-8492-8110"},"institutions":[{"id":"https://openalex.org/I157210198","display_name":"Scuola Normale Superiore","ror":"https://ror.org/03aydme10","country_code":"IT","type":"education","lineage":["https://openalex.org/I157210198"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Francesco Giannini","raw_affiliation_strings":["Scuola Normale Superiore, Pisa, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Scuola Normale Superiore, Pisa, Italy","institution_ids":["https://openalex.org/I157210198"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024505700","display_name":"Fosca Giannotti","orcid":"https://orcid.org/0000-0003-3099-3835"},"institutions":[{"id":"https://openalex.org/I157210198","display_name":"Scuola Normale Superiore","ror":"https://ror.org/03aydme10","country_code":"IT","type":"education","lineage":["https://openalex.org/I157210198"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Fosca Giannotti","raw_affiliation_strings":["Scuola Normale Superiore, Pisa, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Scuola Normale Superiore, Pisa, Italy","institution_ids":["https://openalex.org/I157210198"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5012682000","display_name":"Roberto Pellungrini","orcid":"https://orcid.org/0000-0003-3268-9271"},"institutions":[{"id":"https://openalex.org/I157210198","display_name":"Scuola Normale Superiore","ror":"https://ror.org/03aydme10","country_code":"IT","type":"education","lineage":["https://openalex.org/I157210198"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Roberto Pellungrini","raw_affiliation_strings":["Scuola Normale Superiore, Pisa, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Scuola Normale Superiore, Pisa, Italy","institution_ids":["https://openalex.org/I157210198"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5045329472","display_name":"Patrizio Frosini","orcid":"https://orcid.org/0000-0001-6711-1926"},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Patrizio Frosini","raw_affiliation_strings":["Universit\u00e0 di Pisa, Pisa, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e0 di Pisa, Pisa, Italy","institution_ids":["https://openalex.org/I108290504"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5120008744"],"corresponding_institution_ids":["https://openalex.org/I157210198"],"apc_list":null,"apc_paid":null,"fwci":1.3985,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.83352296,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":94},"biblio":{"volume":null,"issue":null,"first_page":"294","last_page":"318"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.9977999925613403,"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/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.9977999925613403,"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/T14351","display_name":"Statistical and Computational Modeling","score":0.992900013923645,"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/T10320","display_name":"Neural Networks and Applications","score":0.9769999980926514,"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/interpretability","display_name":"Interpretability","score":0.9445000290870667},{"id":"https://openalex.org/keywords/equivariant-map","display_name":"Equivariant map","score":0.9235000014305115},{"id":"https://openalex.org/keywords/convolutional-neural-network","display_name":"Convolutional neural network","score":0.5461000204086304},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.4690999984741211},{"id":"https://openalex.org/keywords/foundation","display_name":"Foundation (evidence)","score":0.3968999981880188},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.37220001220703125}],"concepts":[{"id":"https://openalex.org/C2781067378","wikidata":"https://www.wikidata.org/wiki/Q17027399","display_name":"Interpretability","level":2,"score":0.9445000290870667},{"id":"https://openalex.org/C171036898","wikidata":"https://www.wikidata.org/wiki/Q256355","display_name":"Equivariant map","level":2,"score":0.9235000014305115},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.5601000189781189},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5548999905586243},{"id":"https://openalex.org/C81363708","wikidata":"https://www.wikidata.org/wiki/Q17084460","display_name":"Convolutional neural network","level":2,"score":0.5461000204086304},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4690999984741211},{"id":"https://openalex.org/C2780966255","wikidata":"https://www.wikidata.org/wiki/Q5474306","display_name":"Foundation (evidence)","level":2,"score":0.3968999981880188},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.37220001220703125},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.36010000109672546},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.34700000286102295},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.32280001044273376},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.3176000118255615},{"id":"https://openalex.org/C130327152","wikidata":"https://www.wikidata.org/wiki/Q748349","display_name":"Mathematical structure","level":2,"score":0.313400000333786},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.25940001010894775},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.25360000133514404}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/978-3-032-08324-1_13","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-032-08324-1_13","pdf_url":"https://link.springer.com/content/pdf/10.1007/978-3-032-08324-1_13.pdf","source":{"id":"https://openalex.org/S2764900261","display_name":"Communications in computer and information science","issn_l":"1865-0929","issn":["1865-0929","1865-0937"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Communications in Computer and Information Science","raw_type":"book-chapter"},{"id":"pmh:oai:ricerca.sns.it:11384/162004","is_oa":true,"landing_page_url":"https://hdl.handle.net/11384/162004","pdf_url":"https://ricerca.sns.it/bitstream/11384/162004/1/XAI%20-%20GENEO.pdf","source":{"id":"https://openalex.org/S7407050981","display_name":"Scuola Normale Superiore di Pisa","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/conferenceObject"},{"id":"pmh:oai:arpi.unipi.it:11568/1328567","is_oa":true,"landing_page_url":"https://link.springer.com/chapter/10.1007/978-3-032-08324-1_13","pdf_url":null,"source":{"id":"https://openalex.org/S4377196265","display_name":"CINECA IRIS Institutial research information system (University of Pisa)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I108290504","host_organization_name":"University of Pisa","host_organization_lineage":["https://openalex.org/I108290504"],"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":null,"raw_type":"info:eu-repo/semantics/conferenceObject"}],"best_oa_location":{"id":"doi:10.1007/978-3-032-08324-1_13","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-032-08324-1_13","pdf_url":"https://link.springer.com/content/pdf/10.1007/978-3-032-08324-1_13.pdf","source":{"id":"https://openalex.org/S2764900261","display_name":"Communications in computer and information science","issn_l":"1865-0929","issn":["1865-0929","1865-0937"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Communications in Computer and Information Science","raw_type":"book-chapter"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2070155107","display_name":"It takes two to tango: a synergistic approach to human-machine decision making","funder_award_id":"101120763","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G4508289328","display_name":null,"funder_award_id":"PE00000013","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G5245466801","display_name":null,"funder_award_id":"PRIN 2022","funder_id":"https://openalex.org/F4320311030","funder_display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\""},{"id":"https://openalex.org/G5796505130","display_name":null,"funder_award_id":"PRIN 2022 PNRR","funder_id":"https://openalex.org/F4320311030","funder_display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\""},{"id":"https://openalex.org/G7318860614","display_name":null,"funder_award_id":"834756","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8409961469","display_name":null,"funder_award_id":"Spoke 1","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8657926182","display_name":null,"funder_award_id":"ERC-2018-ADG G.A","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8951614284","display_name":null,"funder_award_id":"I53C22000690001","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G972786147","display_name":null,"funder_award_id":"PRIN-2022","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320334045","display_name":"Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4415208680.pdf","grobid_xml":"https://content.openalex.org/works/W4415208680.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1635792279","https://openalex.org/W1644475622","https://openalex.org/W1756438045","https://openalex.org/W2103496339","https://openalex.org/W2163922914","https://openalex.org/W2576915720","https://openalex.org/W2618132268","https://openalex.org/W2745777039","https://openalex.org/W2888405502","https://openalex.org/W2910705748","https://openalex.org/W2949602009","https://openalex.org/W2962772482","https://openalex.org/W2972182854","https://openalex.org/W2982560934","https://openalex.org/W2998142440","https://openalex.org/W3105655952","https://openalex.org/W3130450512","https://openalex.org/W3164401570","https://openalex.org/W3188740654","https://openalex.org/W3215284968","https://openalex.org/W4210938108","https://openalex.org/W4212763939","https://openalex.org/W4283799240","https://openalex.org/W4321786089","https://openalex.org/W4324155120","https://openalex.org/W4361982725","https://openalex.org/W4381331127","https://openalex.org/W4385732922","https://openalex.org/W4386032032","https://openalex.org/W4388928904","https://openalex.org/W4389538060","https://openalex.org/W4396839053","https://openalex.org/W4400439113","https://openalex.org/W4407791595"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"This":[1],"paper":[2,52],"introduces":[3],"a":[4,54,62],"rigorous":[5],"mathematical":[6],"framework":[7,82],"for":[8,15],"neural":[9,100],"network":[10],"explainability,":[11],"and":[12,83],"more":[13],"broadly":[14],"the":[16,39,45,51,77],"explainability":[17],"of":[18,47,56,58,80],"equivariant":[19],"operators":[20],"called":[21],"Group":[22,28],"Equivariant":[23,29],"Operators":[24,31],"(GEOs),":[25],"based":[26],"on":[27],"Non-Expansive":[30],"(GENEOs)":[32],"transformations.":[33],"The":[34],"central":[35],"concept":[36],"involves":[37],"quantifying":[38],"distance":[40],"between":[41],"GEOs":[42,59],"by":[43],"measuring":[44],"non-commutativity":[46],"specific":[48],"diagrams.":[49],"Additionally,":[50],"proposes":[53],"definition":[55],"interpretability":[57],"according":[60,69],"to":[61,70],"complexity":[63],"measure":[64],"that":[65],"can":[66,87],"be":[67,88],"defined":[68],"each":[71],"user\u2019s":[72],"preferences.":[73],"Moreover,":[74],"we":[75],"explore":[76],"formal":[78],"properties":[79],"this":[81],"show":[84],"how":[85],"it":[86],"applied":[89],"in":[90],"classical":[91],"machine":[92],"learning":[93],"scenarios,":[94],"like":[95],"image":[96],"classification":[97],"with":[98],"convolutional":[99],"networks.":[101]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}