{"id":"https://openalex.org/W4408919020","doi":"https://doi.org/10.3390/e27040344","title":"The Geometry of Concepts: Sparse Autoencoder Feature Structure","display_name":"The Geometry of Concepts: Sparse Autoencoder Feature Structure","publication_year":2025,"publication_date":"2025-03-27","ids":{"openalex":"https://openalex.org/W4408919020","doi":"https://doi.org/10.3390/e27040344","pmid":"https://pubmed.ncbi.nlm.nih.gov/40282579"},"language":"en","primary_location":{"id":"doi:10.3390/e27040344","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27040344","pdf_url":"https://www.mdpi.com/1099-4300/27/4/344/pdf?version=1743052584","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/27/4/344/pdf?version=1743052584","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044386909","display_name":"Yuxiao Li","orcid":"https://orcid.org/0000-0002-6496-9991"},"institutions":[{"id":"https://openalex.org/I4210115294","display_name":"Beneficial Designs (United States)","ror":"https://ror.org/0221y9f82","country_code":"US","type":"company","lineage":["https://openalex.org/I4210115294"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yuxiao Li","raw_affiliation_strings":["Beneficial AI Foundation (BAIF), Cambridge, MA 02139, USA"],"raw_orcid":"https://orcid.org/0000-0002-6496-9991","affiliations":[{"raw_affiliation_string":"Beneficial AI Foundation (BAIF), Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I4210115294"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071048251","display_name":"Eric J. Michaud","orcid":"https://orcid.org/0000-0001-7912-1953"},"institutions":[{"id":"https://openalex.org/I4210160315","display_name":"The NSF AI Institute for Artificial Intelligence and Fundamental Interactions","ror":"https://ror.org/04pvzz946","country_code":"US","type":"facility","lineage":["https://openalex.org/I121934306","https://openalex.org/I12912129","https://openalex.org/I136199984","https://openalex.org/I4210160315","https://openalex.org/I63966007"]},{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Eric J. Michaud","raw_affiliation_strings":["Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","Institute for Artificial Intelligence and Fundamental Interaction, Cambridge, MA 02139, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Institute for Artificial Intelligence and Fundamental Interaction, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I4210160315"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5093902877","display_name":"David D. Baek","orcid":null},"institutions":[{"id":"https://openalex.org/I4210160315","display_name":"The NSF AI Institute for Artificial Intelligence and Fundamental Interactions","ror":"https://ror.org/04pvzz946","country_code":"US","type":"facility","lineage":["https://openalex.org/I121934306","https://openalex.org/I12912129","https://openalex.org/I136199984","https://openalex.org/I4210160315","https://openalex.org/I63966007"]},{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David D. Baek","raw_affiliation_strings":["Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","Institute for Artificial Intelligence and Fundamental Interaction, Cambridge, MA 02139, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Institute for Artificial Intelligence and Fundamental Interaction, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I4210160315"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5104313316","display_name":"Joshua Engels","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Joshua Engels","raw_affiliation_strings":["Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101866940","display_name":"Xiaoqing Sun","orcid":"https://orcid.org/0000-0002-6263-9569"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xiaoqing Sun","raw_affiliation_strings":["Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5091601455","display_name":"Max Tegmark","orcid":"https://orcid.org/0000-0001-7670-7190"},"institutions":[{"id":"https://openalex.org/I4210160315","display_name":"The NSF AI Institute for Artificial Intelligence and Fundamental Interactions","ror":"https://ror.org/04pvzz946","country_code":"US","type":"facility","lineage":["https://openalex.org/I121934306","https://openalex.org/I12912129","https://openalex.org/I136199984","https://openalex.org/I4210160315","https://openalex.org/I63966007"]},{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Max Tegmark","raw_affiliation_strings":["Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","Institute for Artificial Intelligence and Fundamental Interaction, Cambridge, MA 02139, USA"],"raw_orcid":"https://orcid.org/0000-0001-7670-7190","affiliations":[{"raw_affiliation_string":"Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Institute for Artificial Intelligence and Fundamental Interaction, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I4210160315"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5091601455"],"corresponding_institution_ids":["https://openalex.org/I4210160315","https://openalex.org/I63966007"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":16.6207,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.9884923,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"27","issue":"4","first_page":"344","last_page":"344"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9958999752998352,"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/T10320","display_name":"Neural Networks and Applications","score":0.9958999752998352,"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/T12254","display_name":"Machine Learning in Bioinformatics","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T12946","display_name":"Fractal and DNA sequence analysis","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/pattern-recognition","display_name":"Pattern recognition (psychology)","score":0.5130082964897156},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4997560977935791},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.49214380979537964},{"id":"https://openalex.org/keywords/autoencoder","display_name":"Autoencoder","score":0.4783598780632019},{"id":"https://openalex.org/keywords/feature","display_name":"Feature (linguistics)","score":0.45440807938575745},{"id":"https://openalex.org/keywords/parallelogram","display_name":"Parallelogram","score":0.4304611086845398},{"id":"https://openalex.org/keywords/geodesic","display_name":"Geodesic","score":0.41027766466140747},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.39104390144348145},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.3746197819709778},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.33899587392807007},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.3253708779811859},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.2479502558708191},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.14101651310920715}],"concepts":[{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.5130082964897156},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4997560977935791},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.49214380979537964},{"id":"https://openalex.org/C101738243","wikidata":"https://www.wikidata.org/wiki/Q786435","display_name":"Autoencoder","level":3,"score":0.4783598780632019},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.45440807938575745},{"id":"https://openalex.org/C53118041","wikidata":"https://www.wikidata.org/wiki/Q45867","display_name":"Parallelogram","level":3,"score":0.4304611086845398},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.41027766466140747},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.39104390144348145},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3746197819709778},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.33899587392807007},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3253708779811859},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.2479502558708191},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.14101651310920715},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C90509273","wikidata":"https://www.wikidata.org/wiki/Q11012","display_name":"Robot","level":2,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e27040344","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27040344","pdf_url":"https://www.mdpi.com/1099-4300/27/4/344/pdf?version=1743052584","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},{"id":"pmid:40282579","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/40282579","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":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:14fa34bd5cc647d7be0047e5c168656f","is_oa":true,"landing_page_url":"https://doaj.org/article/14fa34bd5cc647d7be0047e5c168656f","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 27, Iss 4, p 344 (2025)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:12025678","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/12025678","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e27040344","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27040344","pdf_url":"https://www.mdpi.com/1099-4300/27/4/344/pdf?version=1743052584","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.6299999952316284}],"awards":[{"id":"https://openalex.org/G2004883221","display_name":null,"funder_award_id":"PHY-2019786","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G539115378","display_name":null,"funder_award_id":"2141064","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4408919020.pdf"},"referenced_works_count":41,"referenced_works":["https://openalex.org/W1973335304","https://openalex.org/W1987869189","https://openalex.org/W2037243854","https://openalex.org/W2060581589","https://openalex.org/W2101234009","https://openalex.org/W2105464873","https://openalex.org/W2113054345","https://openalex.org/W2131522386","https://openalex.org/W2140610559","https://openalex.org/W2145889472","https://openalex.org/W2160448328","https://openalex.org/W2187089797","https://openalex.org/W2202042777","https://openalex.org/W2250539671","https://openalex.org/W2292751606","https://openalex.org/W2572487292","https://openalex.org/W2800559265","https://openalex.org/W2946857231","https://openalex.org/W3021792233","https://openalex.org/W3048804154","https://openalex.org/W3130880317","https://openalex.org/W3138771162","https://openalex.org/W3170470779","https://openalex.org/W4242100153","https://openalex.org/W4247359480","https://openalex.org/W4281657280","https://openalex.org/W4367628394","https://openalex.org/W4386839891","https://openalex.org/W4389518382","https://openalex.org/W4389518997","https://openalex.org/W4396796749","https://openalex.org/W4402671141","https://openalex.org/W4404575139","https://openalex.org/W4404783416","https://openalex.org/W4404928080","https://openalex.org/W6675354045","https://openalex.org/W6762865300","https://openalex.org/W6792221166","https://openalex.org/W6839328737","https://openalex.org/W6843138171","https://openalex.org/W6852578844"],"related_works":["https://openalex.org/W4298181155","https://openalex.org/W4210985719","https://openalex.org/W2127261674","https://openalex.org/W78786167","https://openalex.org/W2949507605","https://openalex.org/W4249323188","https://openalex.org/W2387120636","https://openalex.org/W2055676168","https://openalex.org/W2350170215","https://openalex.org/W2026896861"],"abstract_inverted_index":{").":[0],"We":[1,60,124],"find":[2,72],"that":[3,73],"the":[4,62,103,128,133],"quality":[5],"of":[6,65,75,102,116],"such":[7,21],"parallelograms":[8],"and":[9,45,71],"associated":[10],"function":[11],"vectors":[12],"improves":[13],"greatly":[14],"when":[15],"projecting":[16],"out":[17],"global":[18],"distractor":[19],"directions":[20],"as":[22],"word":[23],"length,":[24],"which":[25],"is":[26,107],"efficiently":[27],"performed":[28],"with":[29,68,118],"linear":[30],"discriminant":[31],"analysis.":[32],"(2)":[33],"The":[34,98],"\"brain\"":[35],"intermediate-scale":[36],"structure":[37,101],"has":[38,112],"significant":[39],"spatial":[40,63],"modularity;":[41],"for":[42],"example,":[43],"math":[44],"code":[46],"features":[47],"form":[48],"a":[49,113],"\"lobe\"":[50],"akin":[51],"to":[52],"functional":[53],"lobes":[54,67],"seen":[55],"in":[56,121],"neural":[57],"fMRI":[58],"images.":[59],"quantify":[61,126],"locality":[64],"these":[66],"multiple":[69],"metrics":[70],"clusters":[74],"co-occurring":[76],"features,":[77],"at":[78],"coarse":[79],"enough":[80],"scale,":[81],"also":[82,125],"cluster":[83],"together":[84],"spatially":[85],"far":[86],"more":[87],"than":[88],"one":[89],"would":[90],"expect":[91],"if":[92],"feature":[93,104],"geometry":[94],"were":[95],"random.":[96],"(3)":[97],"\"galaxy\"-scale":[99],"large-scale":[100],"point":[105],"cloud":[106],"not":[108],"isotropic,":[109],"but":[110],"instead":[111],"power":[114],"law":[115],"eigenvalues":[117],"steepest":[119],"slope":[120],"middle":[122],"layers.":[123],"how":[127],"clustering":[129],"entropy":[130],"depends":[131],"on":[132],"layer.":[134]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":6}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}