{"id":"https://openalex.org/W7220651443","doi":"https://doi.org/10.48550/arxiv.2610.06653","title":"The Birkhoff Geometry of Manifold-Constrained Hyper-Connections: Two Channels, Vertex Viscosity, and Sinkhorn as a Retraction","display_name":"The Birkhoff Geometry of Manifold-Constrained Hyper-Connections: Two Channels, Vertex Viscosity, and Sinkhorn as a Retraction","publication_year":2026,"publication_date":"2026-10-05","ids":{"openalex":"https://openalex.org/W7220651443","doi":"https://doi.org/10.48550/arxiv.2610.06653"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.06653","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06653","pdf_url":"https://arxiv.org/pdf/2610.06653","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":null,"license_id":null,"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/2610.06653","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157706266","display_name":"Xiaoyu Li","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]},{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]},{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["AU","HK","US"],"is_corresponding":false,"raw_author_name":"Li, Xiaoyu","raw_affiliation_strings":["University of New South Wales University of Texas at Austin City University of Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of New South Wales University of Texas at Austin City University of Hong Kong","institution_ids":["https://openalex.org/I168719708","https://openalex.org/I31746571","https://openalex.org/I86519309"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110738653","display_name":"Zhizhou Sha","orcid":"https://orcid.org/0009-0004-2494-1945","observed_orcids":["https://orcid.org/0009-0004-2494-1945"]},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]},{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]},{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["AU","HK","US"],"is_corresponding":false,"raw_author_name":"Sha, Zhizhou","raw_affiliation_strings":["University of New South Wales University of Texas at Austin City University of Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of New South Wales University of Texas at Austin City University of Hong Kong","institution_ids":["https://openalex.org/I168719708","https://openalex.org/I31746571","https://openalex.org/I86519309"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5157716884","display_name":"Chiwun Yang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]},{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]},{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["AU","HK","US"],"is_corresponding":false,"raw_author_name":"Yang, Chiwun","raw_affiliation_strings":["University of New South Wales University of Texas at Austin City University of Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of New South Wales University of Texas at Austin City University of Hong Kong","institution_ids":["https://openalex.org/I168719708","https://openalex.org/I31746571","https://openalex.org/I86519309"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.6362000107765198,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.6362000107765198,"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/T10320","display_name":"Neural Networks and Applications","score":0.07119999825954437,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.054099999368190765,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/birkhoff-polytope","display_name":"Birkhoff polytope","score":0.9570000171661377},{"id":"https://openalex.org/keywords/doubly-stochastic-matrices","display_name":"doubly stochastic matrices","score":0.7889999747276306},{"id":"https://openalex.org/keywords/sinkhorn-algorithm","display_name":"Sinkhorn algorithm","score":0.8349999785423279},{"id":"https://openalex.org/keywords/fisher-information-metric","display_name":"Fisher information metric","score":0.7549999952316284},{"id":"https://openalex.org/keywords/gradient-flow","display_name":"gradient flow","score":0.7509999871253967},{"id":"https://openalex.org/keywords/convergence-rate","display_name":"convergence rate","score":0.37299999594688416},{"id":"https://openalex.org/keywords/permutation-matrix","display_name":"permutation matrices","score":0.4050000011920929},{"id":"https://openalex.org/keywords/fading-memory","display_name":"fading memory","score":0.37700000405311584}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8119999766349792},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49880000948905945},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.49129998683929443},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.45350000262260437},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.4519999921321869},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.4350000023841858},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.4187999963760376},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.4058000147342682},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.37940001487731934},{"id":"https://openalex.org/C140331021","wikidata":"https://www.wikidata.org/wiki/Q1868104","display_name":"Logit","level":2,"score":0.3666999936103821},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.3598000109195709},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.3573000133037567},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33230000734329224},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32749998569488525},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.3249000012874603},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.31369999051094055},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.29739999771118164},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.28450000286102295},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.27480000257492065},{"id":"https://openalex.org/C141042865","wikidata":"https://www.wikidata.org/wiki/Q200125","display_name":"Expected value","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.263700008392334},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2621000111103058},{"id":"https://openalex.org/C58041806","wikidata":"https://www.wikidata.org/wiki/Q1660484","display_name":"Imputation (statistics)","level":3,"score":0.2615000009536743},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.25679999589920044},{"id":"https://openalex.org/C194483076","wikidata":"https://www.wikidata.org/wiki/Q380382","display_name":"Valuation of options","level":2,"score":0.25119999051094055}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.06653","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06653","pdf_url":"https://arxiv.org/pdf/2610.06653","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.06653","doi":"https://doi.org/10.48550/arxiv.2610.06653","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.06653","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:2610.06653","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06653","pdf_url":"https://arxiv.org/pdf/2610.06653","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":null,"license_id":null,"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/W7220651443.pdf","grobid_xml":"https://content.openalex.org/works/W7220651443.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W250209774","https://openalex.org/W1665115054","https://openalex.org/W1966662421","https://openalex.org/W1970789124","https://openalex.org/W1982143037","https://openalex.org/W1990283121","https://openalex.org/W2003447360","https://openalex.org/W2016384870","https://openalex.org/W2045361520","https://openalex.org/W2064583534","https://openalex.org/W2085728653","https://openalex.org/W2086953401","https://openalex.org/W2165469059","https://openalex.org/W2302255633","https://openalex.org/W2522454188","https://openalex.org/W2786130756","https://openalex.org/W3039256091","https://openalex.org/W3173515307","https://openalex.org/W4323539420","https://openalex.org/W4403813222","https://openalex.org/W4407759118","https://openalex.org/W4416506257","https://openalex.org/W7117976675","https://openalex.org/W7123271459","https://openalex.org/W7126266340","https://openalex.org/W7127171306","https://openalex.org/W7131087313","https://openalex.org/W7138028481","https://openalex.org/W7140235693","https://openalex.org/W7148648387","https://openalex.org/W7162072752","https://openalex.org/W7163361253","https://openalex.org/W7163998015","https://openalex.org/W7165385035","https://openalex.org/W7171363439","https://openalex.org/W7202190163","https://openalex.org/W7204099838","https://openalex.org/W7207570375","https://openalex.org/W7211886651"],"related_works":["https://openalex.org/W7162072752","https://openalex.org/W4287869224","https://openalex.org/W7123271459","https://openalex.org/W7133351021","https://openalex.org/W7121244666","https://openalex.org/W7204636379","https://openalex.org/W7214561367","https://openalex.org/W7154992751","https://openalex.org/W7148648387","https://openalex.org/W7131087313"],"abstract_inverted_index":{"Hyper-connections":[0],"widen":[1],"the":[2,17,45,54,89,107,113,126,139,152,170,173,182,197,211,215],"residual":[3,66],"stream":[4,55],"of":[5,33,41,99,154,161,201],"a":[6,23,38,49,57,65,69,93,97,117,134,159,206],"Transformer":[7],"to":[8,172],"$n$":[9],"parallel":[10],"streams.":[11],"Their":[12],"manifold-constrained":[13],"version":[14],"(mHC)":[15],"mixes":[16],"streams":[18],"at":[19,158,162,185,192],"each":[20,73,155],"layer":[21,74],"with":[22,96],"doubly":[24,50],"stochastic":[25,51],"matrix,":[26],"which":[27,61,72],"it":[28],"computes":[29],"by":[30,76],"Sinkhorn":[31,202],"normalization":[32],"exponentiated":[34],"logits.":[35],"We":[36],"give":[37],"geometric":[39],"theory":[40],"this":[42],"design":[43],"on":[44,60],"Birkhoff":[46],"polytope.":[47],"First,":[48],"mixer":[52],"splits":[53],"into":[56],"mean":[58],"channel,":[59,71],"mHC":[62],"is":[63,92,116,124,142,169,203],"exactly":[64,125,143],"network,":[67],"and":[68,102,120,138,180],"difference":[70],"contracts":[75],"its":[77,121],"second":[78],"singular":[79],"value":[80],"$\u03c3_2":[81],"\\le":[82],"1":[83],"-":[84],"n":[85],"\\min_{ij}":[86],"H_{ij}$.":[87],"Thus":[88,129,176],"extra":[90],"width":[91],"fading":[94],"memory":[95],"horizon":[98],"$1/(1-\u03c3_2)$":[100],"layers,":[101],"among":[103],"nonnegative":[104],"mixers":[105],"only":[106,184],"permutations":[108],"do":[109],"not":[110],"collapse.":[111],"Second,":[112],"Sinkhorn-logit":[114],"map":[115],"global":[118],"chart,":[119],"logit":[122,130,149],"gradient":[123,131,150,177],"Fisher-Rao":[127,136],"gradient.":[128],"flow":[132,151,178],"follows":[133],"squared":[135],"metric,":[137],"straight-through":[140],"update":[141],"entropic":[144],"mirror":[145,189],"descent.":[146],"Third,":[147],"under":[148],"logarithm":[153],"entry":[156],"moves":[157,191],"rate":[160,186],"most":[163],"$4n^3\\|\\nabla":[164],"f\\|_\\infty":[165],"\\varepsilon$,":[166],"where":[167],"$\\varepsilon$":[168],"distance":[171],"nearest":[174],"permutation.":[175],"approaches":[179],"leaves":[181],"vertices":[183],"$1/t$,":[187],"but":[188],"descent":[190],"an":[193],"exponential":[194],"rate.":[195],"Fourth,":[196],"local":[198],"convergence":[199],"factor":[200],"$\u03c3_2^2$,":[204],"so":[205],"fixed":[207],"iteration":[208],"budget":[209],"limits":[210],"horizon.":[212],"Experiments":[213],"confirm":[214],"predicted":[216],"rates.":[217]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}