{"id":"https://openalex.org/W7154723229","doi":"https://doi.org/10.48550/arxiv.2604.14727","title":"Expressivity of Transformers: A Tropical Geometry Perspective","display_name":"Expressivity of Transformers: A Tropical Geometry Perspective","publication_year":2026,"publication_date":"2026-04-16","ids":{"openalex":"https://openalex.org/W7154723229","doi":"https://doi.org/10.48550/arxiv.2604.14727"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.14727","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.14727","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.2604.14727","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5133867890","display_name":"Ye Su","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Su, Ye","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5133840237","display_name":"Yong Liu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liu, Yong","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/T11435","display_name":"Polynomial and algebraic computation","score":0.667900025844574,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11435","display_name":"Polynomial and algebraic computation","score":0.667900025844574,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.02459999918937683,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.018699999898672104,"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/tropical-geometry","display_name":"Tropical geometry","score":0.5436999797821045},{"id":"https://openalex.org/keywords/voronoi-diagram","display_name":"Voronoi diagram","score":0.5178999900817871},{"id":"https://openalex.org/keywords/differential-geometry","display_name":"Differential geometry","score":0.4309000074863434},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.39969998598098755},{"id":"https://openalex.org/keywords/bottleneck","display_name":"Bottleneck","score":0.3815000057220459},{"id":"https://openalex.org/keywords/minkowski-space","display_name":"Minkowski space","score":0.3714999854564667},{"id":"https://openalex.org/keywords/combinatorial-explosion","display_name":"Combinatorial explosion","score":0.3555999994277954},{"id":"https://openalex.org/keywords/polyhedron","display_name":"Polyhedron","score":0.3531999886035919},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.3528999984264374},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.3434000015258789}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6347000002861023},{"id":"https://openalex.org/C202652594","wikidata":"https://www.wikidata.org/wiki/Q2982669","display_name":"Tropical geometry","level":2,"score":0.5436999797821045},{"id":"https://openalex.org/C24881265","wikidata":"https://www.wikidata.org/wiki/Q757267","display_name":"Voronoi diagram","level":2,"score":0.5178999900817871},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.4309000074863434},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.39969998598098755},{"id":"https://openalex.org/C2780513914","wikidata":"https://www.wikidata.org/wiki/Q18210350","display_name":"Bottleneck","level":2,"score":0.3815000057220459},{"id":"https://openalex.org/C79464548","wikidata":"https://www.wikidata.org/wiki/Q464794","display_name":"Minkowski space","level":2,"score":0.3714999854564667},{"id":"https://openalex.org/C115908005","wikidata":"https://www.wikidata.org/wiki/Q2668364","display_name":"Combinatorial explosion","level":2,"score":0.3555999994277954},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.3531999886035919},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.3528999984264374},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3434000015258789},{"id":"https://openalex.org/C2874115","wikidata":"https://www.wikidata.org/wiki/Q17099562","display_name":"Persistent homology","level":2,"score":0.3416999876499176},{"id":"https://openalex.org/C4107886","wikidata":"https://www.wikidata.org/wiki/Q906377","display_name":"Discrete geometry","level":2,"score":0.32409998774528503},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.32190001010894775},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.32120001316070557},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.31690001487731934},{"id":"https://openalex.org/C183992945","wikidata":"https://www.wikidata.org/wiki/Q2495574","display_name":"Universality (dynamical systems)","level":2,"score":0.3147999942302704},{"id":"https://openalex.org/C12713177","wikidata":"https://www.wikidata.org/wiki/Q1900281","display_name":"Perspective (graphical)","level":2,"score":0.31439998745918274},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.3046000003814697},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.29679998755455017},{"id":"https://openalex.org/C110202963","wikidata":"https://www.wikidata.org/wiki/Q1783542","display_name":"Convex geometry","level":5,"score":0.2888999879360199},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.2849000096321106},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.28119999170303345},{"id":"https://openalex.org/C2779521785","wikidata":"https://www.wikidata.org/wiki/Q5535529","display_name":"Geometry processing","level":3,"score":0.2687999904155731},{"id":"https://openalex.org/C2776477805","wikidata":"https://www.wikidata.org/wiki/Q4460773","display_name":"Topological data analysis","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.2653999924659729},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2574999928474426},{"id":"https://openalex.org/C156135169","wikidata":"https://www.wikidata.org/wiki/Q5535522","display_name":"Geometry and topology","level":2,"score":0.2529999911785126},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.25290000438690186},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.2522999942302704},{"id":"https://openalex.org/C165263651","wikidata":"https://www.wikidata.org/wiki/Q2443460","display_name":"Topological conjugacy","level":2,"score":0.2517000138759613},{"id":"https://openalex.org/C29123130","wikidata":"https://www.wikidata.org/wiki/Q874709","display_name":"Computational geometry","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.14727","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.14727","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.2604.14727","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.14727","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":[{"id":"https://metadata.un.org/sdg/11","score":0.4731372594833374,"display_name":"Sustainable cities and communities"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"To":[0],"quantify":[1],"the":[2,40,57,66,72,85,91],"geometric":[3],"expressivity":[4],"of":[5,60,75,93],"transformers,":[6],"we":[7,29,47,83,118],"introduce":[8],"a":[9,24,35,49,100],"tropical":[10,26],"geometry":[11],"framework":[12],"to":[13,34,69,80],"characterize":[14],"their":[15],"exact":[16],"spatial":[17],"partitioning":[18],"capabilities.":[19],"By":[20],"modeling":[21],"self-attention":[22],"as":[23],"vector-valued":[25],"rational":[27],"map,":[28],"prove":[30],"it":[31],"evaluates":[32],"exactly":[33],"Power":[36],"Voronoi":[37],"Diagram":[38],"in":[39,96],"zero-temperature":[41],"limit.":[42],"Building":[43],"on":[44,90],"this":[45,79,121],"equivalence,":[46],"establish":[48],"combinatorial":[50,101],"rationale":[51],"for":[52],"Multi-Head":[53],"Self-Attention":[54],"(MHSA):":[55],"via":[56,135],"Minkowski":[58],"sum":[59],"Newton":[61],"polytopes,":[62],"multi-head":[63],"aggregation":[64],"expands":[65],"polyhedral":[67,123],"complexity":[68],"$\\mathcal{O}(N^H)$,":[70],"overcoming":[71],"$\\mathcal{O}(N)$":[73],"bottleneck":[74],"single":[76],"heads.":[77],"Extending":[78],"deep":[81],"architectures,":[82],"derive":[84],"first":[86],"tight":[87,137],"asymptotic":[88],"bounds":[89],"number":[92],"linear":[94],"regions":[95],"transformers":[97],"($\u0398(N^{d_{\\text{model}}L})$),":[98],"demonstrating":[99],"explosion":[102],"driven":[103],"intrinsically":[104],"by":[105],"sequence":[106],"length":[107],"$N$,":[108],"ambient":[109],"embedding":[110],"dimension":[111],"$d_{\\text{model}}$,":[112],"and":[113],"network":[114],"depth":[115],"$L$.":[116],"Importantly,":[117],"guarantee":[119],"that":[120],"idealized":[122],"skeleton":[124],"is":[125],"geometrically":[126],"stable:":[127],"finite-temperature":[128],"soft":[129],"attention":[130],"preserves":[131],"these":[132],"topological":[133],"partitions":[134],"exponentially":[136],"differential":[138],"approximation":[139],"bounds.":[140]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-04-18T00:00:00"}