{"id":"https://openalex.org/W7204856728","doi":"https://doi.org/10.48550/arxiv.2608.28150","title":"Intrinsic Interaction Geometry Controls the Low-Rank Complexity of Softmax Attention","display_name":"Intrinsic Interaction Geometry Controls the Low-Rank Complexity of Softmax Attention","publication_year":2026,"publication_date":"2026-08-28","ids":{"openalex":"https://openalex.org/W7204856728","doi":"https://doi.org/10.48550/arxiv.2608.28150"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.28150","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28150","pdf_url":"https://arxiv.org/pdf/2608.28150","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","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.28150","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5147337357","display_name":"Yuhe Sui","orcid":"https://orcid.org/0009-0002-7456-0804","observed_orcids":["https://orcid.org/0009-0002-7456-0804"]},"institutions":[{"id":"https://openalex.org/I172675005","display_name":"Nanyang Technological University","ror":"https://ror.org/02e7b5302","country_code":"SG","type":"education","lineage":["https://openalex.org/I172675005"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Sui, Yuhe","raw_affiliation_strings":["Nanyang Technological University , Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Nanyang Technological University , Singapore","institution_ids":["https://openalex.org/I172675005"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5148793155","display_name":"Jianing Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhang, Jianing","raw_affiliation_strings":["Carnegie Mellon University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5061159129","display_name":"Yingzhi Tang","orcid":"https://orcid.org/0000-0002-5412-3634","observed_orcids":["https://orcid.org/0000-0002-5412-3634"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tang, Yingzhi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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/T10028","display_name":"Topic Modeling","score":0.35030001401901245,"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/T10028","display_name":"Topic Modeling","score":0.35030001401901245,"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":0.18150000274181366,"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/T10036","display_name":"Advanced Neural Network Applications","score":0.0697999969124794,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/softmax-attention","display_name":"softmax attention","score":0.9779999852180481},{"id":"https://openalex.org/keywords/low-rank-approximation","display_name":"low-rank approximation","score":0.7590000033378601},{"id":"https://openalex.org/keywords/interaction-matrix","display_name":"interaction matrix","score":0.48399999737739563},{"id":"https://openalex.org/keywords/gauge-invariance","display_name":"gauge invariance","score":0.7070000171661377},{"id":"https://openalex.org/keywords/bert","display_name":"BERT","score":0.6140000224113464},{"id":"https://openalex.org/keywords/gpt-2","display_name":"GPT-2","score":0.9409999847412109},{"id":"https://openalex.org/keywords/qwen2-5","display_name":"Qwen2.5","score":0.8980000019073486},{"id":"https://openalex.org/keywords/vision-transformer","display_name":"vision transformer","score":0.6790000200271606}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8317000269889832},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6360999941825867},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6243000030517578},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.6043999791145325},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5419999957084656},{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.42100000381469727},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.40869998931884766},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.3917999863624573},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.3765999972820282},{"id":"https://openalex.org/C188441871","wikidata":"https://www.wikidata.org/wiki/Q7554146","display_name":"Softmax function","level":3,"score":0.37310001254081726},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3693999946117401},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.36340001225471497},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.3587999939918518},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.3529999852180481},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.34360000491142273},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3292999863624573},{"id":"https://openalex.org/C2778158742","wikidata":"https://www.wikidata.org/wiki/Q1665115","display_name":"Rayleigh quotient","level":3,"score":0.3188999891281128},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.31540000438690186},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.3109000027179718},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.30489999055862427},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.2964000105857849},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.287200003862381},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.28290000557899475},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.2802000045776367},{"id":"https://openalex.org/C78397625","wikidata":"https://www.wikidata.org/wiki/Q192487","display_name":"Discriminant","level":2,"score":0.2750000059604645},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.2644999921321869},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.2590000033378601},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.2581000030040741},{"id":"https://openalex.org/C27142425","wikidata":"https://www.wikidata.org/wiki/Q1139334","display_name":"Almost everywhere","level":2,"score":0.25279998779296875},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.28150","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28150","pdf_url":"https://arxiv.org/pdf/2608.28150","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.2608.28150","doi":"https://doi.org/10.48550/arxiv.2608.28150","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.28150","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:2608.28150","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28150","pdf_url":"https://arxiv.org/pdf/2608.28150","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204856728.pdf","grobid_xml":"https://content.openalex.org/works/W7204856728.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W3166324261","https://openalex.org/W4327989522","https://openalex.org/W4386560149","https://openalex.org/W4400376897","https://openalex.org/W4401189978","https://openalex.org/W4415344216","https://openalex.org/W7151576043","https://openalex.org/W7160446104","https://openalex.org/W7160641250","https://openalex.org/W7171555168"],"related_works":["https://openalex.org/W7202130506","https://openalex.org/W7220474976","https://openalex.org/W7160446104","https://openalex.org/W7171555168","https://openalex.org/W7165467458","https://openalex.org/W7151576043","https://openalex.org/W7202355490","https://openalex.org/W7162102497","https://openalex.org/W4416068864","https://openalex.org/W7133196742"],"abstract_inverted_index":{"How":[0],"much":[1],"matrix":[2],"rank":[3,22,27],"is":[4,110,199],"required":[5],"to":[6],"preserve":[7],"every":[8,78],"bounded":[9,33],"value":[10],"output":[11],"of":[12,56],"normalized":[13],"softmax":[14,37],"attention?":[15],"We":[16,61,138],"study":[17,171],"the":[18,25,39,53,97,103,121,163,175,206,221],"unrestricted":[19],"maximum-row-\\(\\ell_1\\)":[20],"approximation":[21],"\\(r_\\varepsilon(A)\\),":[23],"exactly":[24],"least":[26],"achieving":[28],"uniform":[29],"error":[30],"over":[31],"all":[32],"vector-valued":[34],"values.":[35],"Row":[36],"exposes":[38],"intrinsic":[40,197],"interaction":[41,81],"\\(C=P_m(\\log":[42],"A)P_N\\),":[43],"whereas":[44],"invertible":[45],"\\(Q/K\\)":[46],"gauges":[47],"leave":[48],"\\(A\\)":[49],"fixed":[50,134],"while":[51],"changing":[52],"Euclidean":[54],"geometry":[55,178],"a":[57,67,127,191,216],"chosen":[58],"query/key":[59],"factorization.":[60],"replace":[62],"that":[63,174,203,220],"coordinate-dependent":[64],"description":[65],"by":[66],"projective":[68],"residual":[69],"\\(q(C-T)\\)":[70],"and":[71,125,136,154,208],"an":[72,211],"attained":[73],"factor-radius":[74],"size":[75,131],"\\(\u03ba(T)\\).":[76],"For":[77],"rank-\\(r\\)":[79],"retained":[80],"with":[82,96],"\\(\u03c4(T)<\\varepsilon\\),":[83],"we":[84],"prove":[85],"$$":[86,95],"r_\\varepsilon(A)\\le":[87],"\\min\\left\\{":[88],"N,\\;":[89],"C_r\\left(":[90],"1+\\frac{\u03ba(T)}":[91],"{(\\varepsilon-\u03c4(T))^2}":[92],"\\right)^{r/2}":[93],"\\right\\},":[94],"same":[98,122],"unknown":[99],"dimension":[100],"constant":[101],"as":[102],"underlying":[104],"weighted":[105],"Gibbs-row":[106],"cover.":[107],"The":[108,195],"profile":[109],"gauge":[111],"invariant,":[112],"termwise":[113],"no":[114,186],"worse":[115],"than":[116],"native":[117],"retained-subspace":[118],"bounds":[119],"at":[120,133],"declared":[123],"dimension,":[124],"has":[126],"worst-case":[128],"sharp":[129],"\\(r/2\\)":[130],"exponent":[132],"\\(r\\)":[135],"\\(\\varepsilon\\).":[137],"then":[139],"measure":[140],"\\(r_\\varepsilon(A)\\)":[141],"directly":[142],"on":[143],"learned":[144],"attention":[145],"using":[146],"9,978":[147],"certified":[148],"brackets":[149],"across":[150],"BERT,":[151],"GPT-2,":[152],"Qwen2.5,":[153],"two":[155],"ViT":[156],"checkpoints;":[157],"where":[158],"certificates":[159],"do":[160],"not":[161,200,225],"close,":[162],"optimum":[164],"remains":[165],"interval-valued.":[166],"A":[167],"pre-specified":[168],"2,302-cell":[169],"held-out":[170],"further":[172],"shows":[173],"historical":[176],"native-coordinate":[177],"block":[179],"contains":[180],"coarse,":[181],"mostly":[182],"head-level":[183],"information":[184,189],"but":[185],"detectable":[187],"incremental":[188],"beyond":[190],"strong":[192],"calibrated":[193],"baseline.":[194],"new":[196],"descriptor":[198],"evaluated":[201],"in":[202],"study.":[204],"Together,":[205],"theory":[207],"measurements":[209],"distinguish":[210],"operator-intrinsic":[212],"complexity":[213],"control":[214],"from":[215],"stronger":[217],"empirical":[218],"explanation":[219],"learned-head":[222],"evidence":[223],"does":[224],"support.":[226]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-01T00:00:00"}