{"id":"https://openalex.org/W7110810877","doi":"https://doi.org/10.48550/arxiv.2512.07355","title":"A Geometric Unification of Concept Learning with Concept Cones","display_name":"A Geometric Unification of Concept Learning with Concept Cones","publication_year":2025,"publication_date":"2025-12-08","ids":{"openalex":"https://openalex.org/W7110810877","doi":"https://doi.org/10.48550/arxiv.2512.07355"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2512.07355","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2512.07355","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2512.07355","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Rocchi, Alexandre","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rocchi, Alexandre","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Fel, Thomas","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fel, Thomas","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Franchi, Gianni","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Franchi, Gianni","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.8888000249862671,"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":0.8888000249862671,"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/T11303","display_name":"Bayesian Modeling and Causal Inference","score":0.012799999676644802,"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/T11273","display_name":"Advanced Graph Neural Networks","score":0.012199999764561653,"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":0.7139999866485596},{"id":"https://openalex.org/keywords/unification","display_name":"Unification","score":0.6402999758720398},{"id":"https://openalex.org/keywords/terminology","display_name":"Terminology","score":0.46860000491142273},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4018000066280365},{"id":"https://openalex.org/keywords/supervised-learning","display_name":"Supervised learning","score":0.3937999904155731},{"id":"https://openalex.org/keywords/unsupervised-learning","display_name":"Unsupervised learning","score":0.3853999972343445},{"id":"https://openalex.org/keywords/inductive-bias","display_name":"Inductive bias","score":0.37400001287460327},{"id":"https://openalex.org/keywords/bottleneck","display_name":"Bottleneck","score":0.36230000853538513},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.35510000586509705}],"concepts":[{"id":"https://openalex.org/C2781067378","wikidata":"https://www.wikidata.org/wiki/Q17027399","display_name":"Interpretability","level":2,"score":0.7139999866485596},{"id":"https://openalex.org/C96146094","wikidata":"https://www.wikidata.org/wiki/Q609057","display_name":"Unification","level":2,"score":0.6402999758720398},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6276000142097473},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.6183000206947327},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.4745999872684479},{"id":"https://openalex.org/C547195049","wikidata":"https://www.wikidata.org/wiki/Q1725664","display_name":"Terminology","level":2,"score":0.46860000491142273},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4018000066280365},{"id":"https://openalex.org/C136389625","wikidata":"https://www.wikidata.org/wiki/Q334384","display_name":"Supervised learning","level":3,"score":0.3937999904155731},{"id":"https://openalex.org/C8038995","wikidata":"https://www.wikidata.org/wiki/Q1152135","display_name":"Unsupervised learning","level":2,"score":0.3853999972343445},{"id":"https://openalex.org/C197352929","wikidata":"https://www.wikidata.org/wiki/Q1074074","display_name":"Inductive bias","level":4,"score":0.37400001287460327},{"id":"https://openalex.org/C2780513914","wikidata":"https://www.wikidata.org/wiki/Q18210350","display_name":"Bottleneck","level":2,"score":0.36230000853538513},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.35510000586509705},{"id":"https://openalex.org/C48164120","wikidata":"https://www.wikidata.org/wiki/Q4491893","display_name":"Concept learning","level":2,"score":0.3361999988555908},{"id":"https://openalex.org/C132010649","wikidata":"https://www.wikidata.org/wiki/Q189222","display_name":"Intuition","level":2,"score":0.3239000141620636},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.31520000100135803},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.31290000677108765},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.3084999918937683},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.296099990606308},{"id":"https://openalex.org/C207609745","wikidata":"https://www.wikidata.org/wiki/Q4944086","display_name":"Bootstrapping (finance)","level":2,"score":0.2856999933719635},{"id":"https://openalex.org/C77637269","wikidata":"https://www.wikidata.org/wiki/Q7002051","display_name":"Neural coding","level":2,"score":0.2815999984741211},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.2806999981403351},{"id":"https://openalex.org/C204321447","wikidata":"https://www.wikidata.org/wiki/Q30642","display_name":"Natural language processing","level":1,"score":0.27889999747276306},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.2727000117301941},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.26910001039505005},{"id":"https://openalex.org/C2779808786","wikidata":"https://www.wikidata.org/wiki/Q6664603","display_name":"Locality","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C2777601683","wikidata":"https://www.wikidata.org/wiki/Q6499736","display_name":"Vocabulary","level":2,"score":0.2660999894142151},{"id":"https://openalex.org/C207685749","wikidata":"https://www.wikidata.org/wiki/Q2088941","display_name":"Domain knowledge","level":2,"score":0.2605000138282776},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.2533999979496002},{"id":"https://openalex.org/C152124472","wikidata":"https://www.wikidata.org/wiki/Q1204361","display_name":"Redundancy (engineering)","level":2,"score":0.25119999051094055}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2512.07355","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2512.07355","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2512.07355","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2512.07355","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Two":[0],"traditions":[1],"of":[2,68,130,153,158],"interpretability":[3],"have":[4],"evolved":[5],"side":[6,8],"by":[7,120,183,189],"but":[9,90],"seldom":[10],"spoken":[11],"to":[12,39,50,151,196,248],"each":[13,64],"other:":[14],"Concept":[15],"Bottleneck":[16],"Models":[17],"(CBMs),":[18],"which":[19,30],"prescribe":[20],"what":[21,32],"a":[22,66,78,212,241],"concept":[23,79,238,257],"should":[24],"be,":[25],"and":[26,82,169,176,218,225,236,252],"Sparse":[27],"Autoencoders":[28],"(SAEs),":[29],"discover":[31],"concepts":[33,180],"emerge.":[34],"While":[35],"CBMs":[36,110],"use":[37],"supervision":[38],"align":[40,258],"activations":[41],"with":[42,173,228,259],"human-labeled":[43],"concepts,":[44],"SAEs":[45,116],"rely":[46],"on":[47,98],"sparse":[48],"coding":[49],"uncover":[51,211],"emergent":[52],"ones.":[53],"We":[54],"show":[55],"that":[56,201,221],"both":[57,216,223],"paradigms":[58],"instantiate":[59],"the":[60,107,156,197,203],"same":[61],"geometric":[62,224,243],"structure:":[63],"learns":[65],"set":[67],"linear":[69],"directions":[70],"in":[71,88,91,215],"activation":[72],"space":[73],"whose":[74],"nonnegative":[75],"combinations":[76],"form":[77],"cone.":[80,96],"Supervised":[81],"unsupervised":[83,237],"methods":[84],"thus":[85],"differ":[86],"not":[87,193],"kind":[89],"how":[92,121,254],"they":[93],"select":[94],"this":[95,99],"Building":[97],"view,":[100],"we":[101,210],"propose":[102],"an":[103],"operational":[104],"bridge":[105],"between":[106,162],"two":[108],"paradigms.":[109],"provide":[111],"human-defined":[112],"reference":[113],"geometries,":[114],"while":[115],"can":[117],"be":[118],"evaluated":[119],"well":[122,255],"their":[123],"learned":[124],"cones":[125],"approximate":[126],"or":[127,147,187],"contain":[128],"those":[129],"CBMs.":[131],"This":[132],"containment":[133],"framework":[134],"yields":[135],"quantitative":[136],"metrics":[137,247],"linking":[138],"inductive":[139],"biases":[140],"--":[141,150,185,191],"such":[142],"as":[143],"SAE":[144,250],"type,":[145],"sparsity,":[146],"expansion":[148,219],"ratio":[149],"emergence":[152],"plausible\\footnote{We":[154],"adopt":[155],"terminology":[157],"\\citet{jacovi2020towards},":[159],"who":[160],"distinguish":[161],"faithful":[163,195],"explanations":[164,171],"(accurately":[165],"reflecting":[166],"model":[167],"computations)":[168],"plausible":[170,182,260],"(aligning":[172],"human":[174,261],"intuition":[175],"domain":[177],"knowledge).":[178],"CBM":[179,229],"are":[181],"construction":[184],"selected":[186],"annotated":[188],"humans":[190],"though":[192],"necessarily":[194],"true":[198],"latent":[199],"factors":[200],"organise":[202],"data":[204],"manifold.}":[205],"concepts.":[206,230,262],"Using":[207],"these":[208],"metrics,":[209],"``sweet":[213],"spot''":[214],"sparsity":[217],"factor":[220],"maximizes":[222],"semantic":[226],"alignment":[227],"Overall,":[231],"our":[232],"work":[233],"unifies":[234],"supervised":[235],"discovery":[239],"through":[240],"shared":[242],"framework,":[244],"providing":[245],"principled":[246],"measure":[249],"progress":[251],"assess":[253],"discovered":[256]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2025-12-10T00:00:00"}