{"id":"https://openalex.org/W7170037525","doi":"https://doi.org/10.48550/arxiv.2607.18584","title":"On the Diverse Dynamical Behaviors Arising in Deep Linear Transformers","display_name":"On the Diverse Dynamical Behaviors Arising in Deep Linear Transformers","publication_year":2026,"publication_date":"2026-07-20","ids":{"openalex":"https://openalex.org/W7170037525","doi":"https://doi.org/10.48550/arxiv.2607.18584"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2607.18584","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.18584","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,"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.2607.18584","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5143424218","display_name":"Sixu Li","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Sixu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5143454673","display_name":"Thomas Jacob Maranzatto","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Maranzatto, Thomas Jacob","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037001756","display_name":"Jan Peszek","orcid":"https://orcid.org/0000-0002-9363-0954"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Peszek, Jan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008286456","display_name":"Trevor Teolis","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Teolis, Trevor","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5114535869","display_name":"Semih Akko\u00e7","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Akkoc, Semih","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5088292594","display_name":"Konstantin Riedl","orcid":"https://orcid.org/0000-0002-2206-4334"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Riedl, Konstantin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5143424344","display_name":"Sennur Ulukus","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ulukus, Sennur","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5114761446","display_name":"Nicol\u00e1s Garc\u00eda Trillos","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Trillos, Nicol\u00e1s Garc\u00eda","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/T11804","display_name":"Quantum many-body systems","score":0.6836000084877014,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11804","display_name":"Quantum many-body systems","score":0.6836000084877014,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.05400000140070915,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.03830000013113022,"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/embedding","display_name":"Embedding","score":0.5526000261306763},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4399999976158142},{"id":"https://openalex.org/keywords/transformer","display_name":"Transformer","score":0.42089998722076416},{"id":"https://openalex.org/keywords/granularity","display_name":"Granularity","score":0.4154999852180481},{"id":"https://openalex.org/keywords/linear-dynamical-system","display_name":"Linear dynamical system","score":0.4108999967575073},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.3736000061035156},{"id":"https://openalex.org/keywords/hamiltonian","display_name":"Hamiltonian (control theory)","score":0.3610999882221222},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.35420000553131104}],"concepts":[{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5526000261306763},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5260000228881836},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5133000016212463},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4399999976158142},{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.42089998722076416},{"id":"https://openalex.org/C177774035","wikidata":"https://www.wikidata.org/wiki/Q1246948","display_name":"Granularity","level":2,"score":0.4154999852180481},{"id":"https://openalex.org/C114275822","wikidata":"https://www.wikidata.org/wiki/Q621512","display_name":"Linear dynamical system","level":3,"score":0.4108999967575073},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.3736000061035156},{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.3610999882221222},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.35420000553131104},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.3456000089645386},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3433000147342682},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.3179999887943268},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3068000078201294},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.29760000109672546},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.295199990272522},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.28929999470710754},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2822999954223633},{"id":"https://openalex.org/C77405623","wikidata":"https://www.wikidata.org/wiki/Q598451","display_name":"System dynamics","level":2,"score":0.27730000019073486},{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.27709999680519104},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.27709999680519104},{"id":"https://openalex.org/C145912823","wikidata":"https://www.wikidata.org/wiki/Q113558","display_name":"Dynamics (music)","level":2,"score":0.26489999890327454},{"id":"https://openalex.org/C121770821","wikidata":"https://www.wikidata.org/wiki/Q2072471","display_name":"Hamiltonian system","level":2,"score":0.2639999985694885},{"id":"https://openalex.org/C163175372","wikidata":"https://www.wikidata.org/wiki/Q3339222","display_name":"Linear model","level":2,"score":0.2639000117778778},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.25859999656677246}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2607.18584","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.18584","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,"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.2607.18584","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.18584","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,"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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,33],"study":[1],"the":[2,11,47,70,77,88,95,126,141,145,154,179],"inference-time":[3],"behavior":[4,148],"of":[5,13,76,83,102],"deep":[6],"linear":[7,30,106],"encoder-only":[8],"transformers":[9],"through":[10,28],"lens":[12],"interacting":[14],"particle":[15],"systems.":[16],"In":[17],"this":[18],"perspective,":[19],"tokens":[20],"are":[21,72,117],"modeled":[22],"as":[23,52,149],"particles":[24],"that":[25,35,94,137,178],"interact":[26],"dynamically":[27],"successive":[29],"self-attention":[31],"layers.":[32],"show":[34,93,136],"in":[36,105,125,160,171,183,188],"embedding":[37],"dimension":[38,161],"two,":[39,162],"for":[40,167],"any":[41],"key,":[42],"query,":[43],"and":[44,111,115],"value":[45],"matrices,":[46],"dynamics":[48,71,138],"can":[49],"be":[50],"reformulated":[51],"a":[53,81,99,121,130],"generalized":[54],"Kuramoto-type":[55],"model":[56],"with":[57,87],"pure":[58],"second-harmonic":[59],"coupling.":[60],"This":[61],"formulation":[62],"is":[63],"amenable":[64],"to":[65],"Watanabe--Strogatz":[66],"theory":[67,159],"which":[68],"reveals":[69],"intrinsically":[73],"low-dimensional":[74],"regardless":[75],"parameter":[78,96,169],"matrices.":[79],"For":[80],"class":[82],"token":[84],"initializations":[85],"associated":[86],"Ott--Antonsen":[89],"(OA)":[90],"manifold,":[91],"we":[92,134,163],"matrices":[97],"induce":[98],"diverse":[100],"variety":[101],"long-time":[103,147,180],"behaviors":[104,181],"transformers,":[107],"including":[108],"clustering,":[109],"oscillations,":[110],"bifurcations.":[112],"The":[113],"oscillations":[114],"bifurcations":[116],"characterized":[118,182],"by":[119,157],"uncovering":[120],"hidden":[122],"Hamiltonian":[123],"structure":[124],"dynamics.":[127],"By":[128],"establishing":[129],"structural":[131],"stability":[132],"result,":[133],"further":[135],"initialized":[139,151],"near":[140],"OA":[142],"manifold":[143],"exhibit":[144],"same":[146],"those":[150],"exactly":[152],"on":[153],"manifold.":[155],"Motivated":[156],"our":[158,184],"conduct":[164],"numerical":[165,175],"experiments":[166,176],"analogous":[168],"regimes":[170],"higher-dimensional":[172],"transformers.":[173],"Our":[174],"suggest":[177],"theoretical":[185],"results":[186],"persist":[187],"higher":[189],"dimensions.":[190]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-07-23T00:00:00"}
