{"id":"https://openalex.org/W7224108229","doi":"https://doi.org/10.48550/arxiv.2610.36488","title":"Second-order Sinkhorn Geometry and $\\sqrt{n}$-Inference","display_name":"Second-order Sinkhorn Geometry and $\\sqrt{n}$-Inference","publication_year":2026,"publication_date":"2026-10-27","ids":{"openalex":"https://openalex.org/W7224108229","doi":"https://doi.org/10.48550/arxiv.2610.36488"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.36488","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.36488","pdf_url":"https://arxiv.org/pdf/2610.36488","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.36488","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5130714035","display_name":"Tristan Luca Saidi","orcid":"https://orcid.org/0009-0007-9577-5210","observed_orcids":["https://orcid.org/0009-0007-9577-5210"]},"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":"Saidi, Tristan Luca","raw_affiliation_strings":["Department of Statistics and Data Science , Carnegie Mellon University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics and Data Science , Carnegie Mellon University","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014488868","display_name":"Florian F. Gunsilius","orcid":"https://orcid.org/0000-0002-1698-6324","observed_orcids":["https://orcid.org/0000-0002-1698-6324"]},"institutions":[{"id":"https://openalex.org/I150468666","display_name":"Emory University","ror":"https://ror.org/03czfpz43","country_code":"US","type":"education","lineage":["https://openalex.org/I150468666"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Gunsilius, Florian","raw_affiliation_strings":["Department of Economics , Emory University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Economics , Emory University","institution_ids":["https://openalex.org/I150468666"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078598841","display_name":"Gonzalo E. Mena","orcid":"https://orcid.org/0000-0003-4432-9679","observed_orcids":["https://orcid.org/0000-0003-4432-9679"]},"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":"Mena, Gonzalo","raw_affiliation_strings":["Department of Statistics and Data Science , Carnegie Mellon University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics and Data Science , Carnegie Mellon University","institution_ids":["https://openalex.org/I74973139"]}]}],"institutions":[],"countries_distinct_count":1,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9976000189781189,"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"}},"topics":[{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9976000189781189,"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"}},{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.0003000000142492354,"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"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.0003000000142492354,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/sinkhorn-divergence","display_name":"Sinkhorn divergence","score":0.9729999899864197},{"id":"https://openalex.org/keywords/entropic-optimal-transport","display_name":"entropic optimal transport","score":0.8539999723434448},{"id":"https://openalex.org/keywords/metric-tensor","display_name":"metric tensor","score":0.7009999752044678},{"id":"https://openalex.org/keywords/hadamard-differentiability","display_name":"Hadamard differentiability","score":0.9679999947547913},{"id":"https://openalex.org/keywords/geodesics","display_name":"geodesics","score":0.9179999828338623},{"id":"https://openalex.org/keywords/parallel-transport","display_name":"parallel transport","score":0.9990000128746033},{"id":"https://openalex.org/keywords/reproducing-kernel-hilbert-space","display_name":"reproducing kernel Hilbert space","score":0.9509999752044678},{"id":"https://openalex.org/keywords/exponential-map","display_name":"exponential map","score":0.5130000114440918},{"id":"https://openalex.org/keywords/logarithmic-map","display_name":"logarithmic map","score":0.9549999833106995},{"id":"https://openalex.org/keywords/bootstrap-consistency","display_name":"bootstrap consistency","score":0.6769999861717224},{"id":"https://openalex.org/keywords/space-of-probability-measures","display_name":"space of probability measures","score":0.6050000190734863}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6482999920845032},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.5464000105857849},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.48570001125335693},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.42320001125335693},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4068000018596649},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.3894999921321869},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.35370001196861267},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.35030001401901245},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.3434999883174896},{"id":"https://openalex.org/C9136319","wikidata":"https://www.wikidata.org/wiki/Q362640","display_name":"Covariant transformation","level":2,"score":0.3393000066280365},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.3262999951839447},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.32120001316070557},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3125999867916107},{"id":"https://openalex.org/C2777634741","wikidata":"https://www.wikidata.org/wiki/Q768993","display_name":"Wasserstein metric","level":2,"score":0.3070000112056732},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3066999912261963},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.3028999865055084},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.2913999855518341},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2872999906539917},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.28380000591278076},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.2827000021934509},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.2815999984741211},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.27630001306533813},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.2718999981880188},{"id":"https://openalex.org/C108598597","wikidata":"https://www.wikidata.org/wiki/Q124255","display_name":"Conic section","level":2,"score":0.27129998803138733},{"id":"https://openalex.org/C21696900","wikidata":"https://www.wikidata.org/wiki/Q1333055","display_name":"Semiring","level":2,"score":0.2705000042915344},{"id":"https://openalex.org/C19492387","wikidata":"https://www.wikidata.org/wiki/Q2287715","display_name":"Covariant derivative","level":2,"score":0.2702000141143799},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.2632000148296356},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.2599000036716461},{"id":"https://openalex.org/C2011187","wikidata":"https://www.wikidata.org/wiki/Q383851","display_name":"Directional derivative","level":2,"score":0.2581000030040741}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.36488","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.36488","pdf_url":"https://arxiv.org/pdf/2610.36488","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.2610.36488","doi":"https://doi.org/10.48550/arxiv.2610.36488","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.36488","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.36488","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.36488","pdf_url":"https://arxiv.org/pdf/2610.36488","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7224108229.pdf","grobid_xml":"https://content.openalex.org/works/W7224108229.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4415435147","https://openalex.org/W7216086300","https://openalex.org/W7213327304","https://openalex.org/W4392404692","https://openalex.org/W4286230930","https://openalex.org/W3128439473","https://openalex.org/W3180551705","https://openalex.org/W2770187377","https://openalex.org/W2897105053","https://openalex.org/W3120509864"],"abstract_inverted_index":{"The":[0],"Sinkhorn":[1,31,95,124,225],"divergence,":[2],"a":[3,35,43,76,82,111,159,261],"debiased":[4],"form":[5],"of":[6,54,88,108,114,122,152,213,243,267],"entropic":[7],"optimal":[8,55],"transport,":[9],"has":[10],"been":[11],"widely":[12],"used":[13],"in":[14,134,260],"various":[15],"scientific":[16],"disciplines":[17],"due":[18],"to":[19,51,143,155,173,177],"its":[20],"computational":[21],"and":[22,37,81,127,147,180,222,252],"statistical":[23],"tractability.":[24],"Unlike":[25],"the":[26,30,86,94,123,130,140,145,175,178,210,214,217,223,231,241,248,265],"Wasserstein":[27],"distance,":[28],"however,":[29],"divergence":[32],"is":[33,172,234],"not":[34,40],"metric":[36,83,125],"it":[38],"does":[39],"naturally":[41],"admit":[42],"formal":[44,77],"Riemannian":[45,78],"structure.":[46],"These":[47],"properties":[48],"are":[49,247],"central":[50],"many":[52],"applications":[53],"transport":[56,64,149,182,221],"methods":[57],"that":[58,91,194,230],"utilize":[59],"interpolation,":[60],"barycentric":[61],"averaging,":[62],"parallel":[63,148,181,220],"or":[65],"other":[66],"geometric":[67,258],"operations.":[68],"Recent":[69],"work":[70],"addresses":[71],"these":[72,199,246],"shortcomings":[73],"by":[74],"constructing":[75],"tangent":[79],"space":[80,87,189,266],"tensor":[84,126],"on":[85,158,264],"probability":[89,268],"measures":[90],"quadratically":[92],"approximates":[93],"divergence.":[96],"In":[97],"this":[98,102,135],"work,":[99],"we":[100,118,128,137,228],"extend":[101],"framework":[103],"with":[104,166],"two":[105],"major":[106],"threads":[107],"contributions.":[109],"For":[110],"suitable":[112],"class":[113],"finite-smoothness":[115],"cost":[116],"functions,":[117],"prove":[119],"Hadamard":[120],"differentiability":[121],"derive":[129],"Levi-Civita":[131],"covariant":[132,141],"derivative":[133,142],"geometry;":[136],"then":[138],"use":[139],"obtain":[144],"geodesic":[146,179],"equations,":[150],"each":[151],"which":[153],"amount":[154],"an":[156],"ODE":[157,190],"fixed":[160],"Reproducing":[161],"Kernel":[162],"Hilbert":[163],"Space":[164],"(RKHS)":[165],"smooth":[167],"coefficients.":[168],"Our":[169],"second":[170],"contribution":[171],"linearize":[174],"solutions":[176],"equations":[183],"using":[184],"standard":[185,196],"tools":[186],"from":[187],"Banach":[188],"theory.":[191],"We":[192],"show":[193,229],"under":[195],"regularity":[197],"conditions":[198],"linearizations":[200],"imply":[201],"$\\sqrt{n}$-limiting":[202],"Gaussian":[203],"processes":[204],"centered":[205],"at":[206],"population":[207],"values":[208],"for":[209,236,256],"plugin":[211],"estimators":[212],"exponential":[215],"map,":[216,219],"logarithmic":[218],"induced":[224],"distance.":[226],"Moreover,":[227],"nonparametric":[232],"bootstrap":[233,253],"consistent":[235],"all":[237],"objects":[238],"mentioned.":[239],"To":[240],"best":[242],"our":[244],"knowledge,":[245],"first":[249],"$\\sqrt{n}$--limit":[250],"theorems":[251],"consistency":[254],"results":[255],"canonical":[257],"operations":[259],"transport-based":[262],"geometry":[263],"measures.":[269]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-29T00:00:00"}
