{"id":"https://openalex.org/W7127324972","doi":"https://doi.org/10.48550/arxiv.2602.02224","title":"Spectral Superposition: A Theory of Feature Geometry","display_name":"Spectral Superposition: A Theory of Feature Geometry","publication_year":2026,"publication_date":"2026-02-02","ids":{"openalex":"https://openalex.org/W7127324972","doi":"https://doi.org/10.48550/arxiv.2602.02224"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2602.02224","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2602.02224","pdf_url":"https://arxiv.org/pdf/2602.02224","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2602.02224","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5124944808","display_name":"Georgi Ivanov","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ivanov, Georgi","raw_affiliation_strings":["Harvard University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Harvard University","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5119862711","display_name":"Narmeen Fatimah Oozeer","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Oozeer, Narmeen","raw_affiliation_strings":["Martian"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Martian","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070746960","display_name":"Shivam Raval","orcid":"https://orcid.org/0000-0002-3742-8209","observed_orcids":["https://orcid.org/0000-0002-3742-8209"]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Raval, Shivam","raw_affiliation_strings":["Harvard University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Harvard University","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5124906230","display_name":"Tasana Pejovic","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pejovic, Tasana","raw_affiliation_strings":["Martian"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Martian","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5124941144","display_name":"Shriyash Upadhyay","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Upadhyay, Shriyash","raw_affiliation_strings":["Martian"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Martian","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5124941064","display_name":"Amir Abdullah","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Abdullah, Amir","raw_affiliation_strings":["Martian","Thoughtworks"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Martian","institution_ids":[]},{"raw_affiliation_string":"Thoughtworks","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T10320","display_name":"Neural Networks and Applications","score":0.6166999936103821,"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":0.6166999936103821,"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/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.17419999837875366,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.07989999651908875,"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/spectral-superposition","display_name":"spectral superposition","score":0.890999972820282},{"id":"https://openalex.org/keywords/feature-geometry","display_name":"feature geometry","score":0.8690000176429749},{"id":"https://openalex.org/keywords/neural-network-interpretability","display_name":"neural network interpretability","score":0.7799999713897705},{"id":"https://openalex.org/keywords/frame-operator","display_name":"frame operator","score":0.8769999742507935},{"id":"https://openalex.org/keywords/eigenspaces","display_name":"eigenspaces","score":0.6669999957084656},{"id":"https://openalex.org/keywords/tight-frames","display_name":"tight frames","score":0.6990000009536743},{"id":"https://openalex.org/keywords/capacity-saturation","display_name":"capacity saturation","score":0.8330000042915344},{"id":"https://openalex.org/keywords/spectral-localization","display_name":"spectral localization","score":0.8299999833106995},{"id":"https://openalex.org/keywords/association-schemes","display_name":"association schemes","score":0.7799999713897705},{"id":"https://openalex.org/keywords/feature-interaction","display_name":"feature interaction","score":0.33000001311302185},{"id":"https://openalex.org/keywords/operator-theory","display_name":"operator theory","score":0.7409999966621399}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5541999936103821},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.5329999923706055},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.47350001335144043},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4648999869823456},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4528000056743622},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.4366999864578247},{"id":"https://openalex.org/C98306654","wikidata":"https://www.wikidata.org/wiki/Q2409122","display_name":"Spectral theory","level":3,"score":0.4350000023841858},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.42179998755455017},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4088999927043915},{"id":"https://openalex.org/C176641082","wikidata":"https://www.wikidata.org/wiki/Q2446767","display_name":"Spectral signature","level":2,"score":0.3594000041484833},{"id":"https://openalex.org/C73301696","wikidata":"https://www.wikidata.org/wiki/Q5469984","display_name":"Formalism (music)","level":3,"score":0.3578000068664551},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.3547999858856201},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.3537999987602234},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.35179999470710754},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.3458999991416931},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.3441999852657318},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.33250001072883606},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3273000121116638},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3224000036716461},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.2754000127315521},{"id":"https://openalex.org/C51364350","wikidata":"https://www.wikidata.org/wiki/Q3123529","display_name":"Spectral geometry","level":2,"score":0.26919999718666077},{"id":"https://openalex.org/C197115733","wikidata":"https://www.wikidata.org/wiki/Q1003136","display_name":"Forcing (mathematics)","level":2,"score":0.26919999718666077},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.2621000111103058},{"id":"https://openalex.org/C154968394","wikidata":"https://www.wikidata.org/wiki/Q5535474","display_name":"Geometric analysis","level":5,"score":0.2529999911785126},{"id":"https://openalex.org/C2778740170","wikidata":"https://www.wikidata.org/wiki/Q7575210","display_name":"Spectral space","level":2,"score":0.25279998779296875},{"id":"https://openalex.org/C32990609","wikidata":"https://www.wikidata.org/wiki/Q306542","display_name":"Transformation geometry","level":2,"score":0.25}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2602.02224","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2602.02224","pdf_url":"https://arxiv.org/pdf/2602.02224","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2602.02224","doi":"https://doi.org/10.48550/arxiv.2602.02224","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.02224","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2602.02224","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2602.02224","pdf_url":"https://arxiv.org/pdf/2602.02224","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7127324972.pdf","grobid_xml":"https://content.openalex.org/works/W7127324972.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7212009435","https://openalex.org/W7171735843","https://openalex.org/W7212194494","https://openalex.org/W7168278967","https://openalex.org/W7108726538","https://openalex.org/W7134947197","https://openalex.org/W7223221167","https://openalex.org/W7148398718","https://openalex.org/W4415086011","https://openalex.org/W4403785798"],"abstract_inverted_index":{"Neural":[0],"networks":[1],"represent":[2],"more":[3],"features":[4,12,24,39,95,114],"than":[5],"they":[6],"have":[7],"dimensions":[8],"via":[9,127],"superposition,":[10,101],"forcing":[11],"to":[13,106,144,166],"share":[14],"representational":[15],"space.":[16],"Current":[17],"methods":[18,87],"decompose":[19],"activations":[20],"into":[21,120],"sparse":[22],"linear":[23],"but":[25],"discard":[26],"geometric":[27,36],"structure.":[28],"We":[29],"develop":[30],"a":[31,64,160],"theory":[32,105,165],"for":[33],"studying":[34],"the":[35,42,55,81,89],"structre":[37],"of":[38,47,100,150],"by":[40],"analyzing":[41],"spectra":[43],"(eigenvalues,":[44],"eigenspaces,":[45,118],"etc.)":[46],"weight":[48,146],"derived":[49],"matrices.":[50],"In":[51,97],"particular,":[52],"we":[53,102],"introduce":[54],"frame":[56],"operator":[57,164],"$F":[58],"=":[59],"WW^\\top$,":[60],"which":[61],"gives":[62],"us":[63],"spectral":[65,86,112,140],"measure":[66,141],"that":[67,108],"describes":[68],"how":[69],"each":[70],"feature":[71,151],"allocates":[72],"norm":[73],"across":[74],"eigenspaces.":[75],"While":[76],"previous":[77],"tools":[78],"could":[79],"describe":[80],"pairwise":[82],"interactions":[83],"between":[84],"features,":[85],"capture":[88],"global":[90],"geometry":[91],"(``how":[92],"do":[93],"all":[94,131],"interact?'').":[96],"toy":[98,154],"models":[99],"use":[103],"this":[104],"prove":[107],"capacity":[109],"saturation":[110],"forces":[111],"localization:":[113],"collapse":[115],"onto":[116],"single":[117],"organize":[119],"tight":[121],"frames,":[122],"and":[123],"admit":[124],"discrete":[125],"classification":[126],"association":[128],"schemes,":[129],"classifying":[130],"geometries":[132],"from":[133],"prior":[134],"work":[135],"(simplices,":[136],"polygons,":[137],"antiprisms).":[138],"The":[139],"formalism":[142],"applies":[143],"arbitrary":[145],"matrices,":[147],"enabling":[148],"diagnosis":[149],"localization":[152],"beyond":[153],"settings.":[155],"These":[156],"results":[157],"point":[158],"toward":[159],"broader":[161],"program:":[162],"applying":[163],"interpretability.":[167]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-02-04T00:00:00"}