{"id":"https://openalex.org/W7124683740","doi":"https://doi.org/10.48550/arxiv.2601.10992","title":"Constant Metric Scaling in Riemannian Computation","display_name":"Constant Metric Scaling in Riemannian Computation","publication_year":2026,"publication_date":"2026-01-16","ids":{"openalex":"https://openalex.org/W7124683740","doi":"https://doi.org/10.48550/arxiv.2601.10992"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2601.10992","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"Article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5099087315","display_name":"Kisung You","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"You, Kisung","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5099087315"],"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/T12417","display_name":"Morphological variations and asymmetry","score":0.679099977016449,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12417","display_name":"Morphological variations and asymmetry","score":0.679099977016449,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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":0.17800000309944153,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.022099999710917473,"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/scaling","display_name":"Scaling","score":0.7437999844551086},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.7091000080108643},{"id":"https://openalex.org/keywords/statistical-manifold","display_name":"Statistical manifold","score":0.6977999806404114},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.675599992275238},{"id":"https://openalex.org/keywords/information-geometry","display_name":"Information geometry","score":0.5929999947547913},{"id":"https://openalex.org/keywords/riemannian-geometry","display_name":"Riemannian geometry","score":0.5882999897003174},{"id":"https://openalex.org/keywords/riemannian-manifold","display_name":"Riemannian manifold","score":0.5867000222206116},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5630000233650208}],"concepts":[{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.7437999844551086},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.7091000080108643},{"id":"https://openalex.org/C169391604","wikidata":"https://www.wikidata.org/wiki/Q7604402","display_name":"Statistical manifold","level":5,"score":0.6977999806404114},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.675599992275238},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6654000282287598},{"id":"https://openalex.org/C109546454","wikidata":"https://www.wikidata.org/wiki/Q3798604","display_name":"Information geometry","level":4,"score":0.5929999947547913},{"id":"https://openalex.org/C181104567","wikidata":"https://www.wikidata.org/wiki/Q761383","display_name":"Riemannian geometry","level":2,"score":0.5882999897003174},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.5867000222206116},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5630000233650208},{"id":"https://openalex.org/C4306028","wikidata":"https://www.wikidata.org/wiki/Q2185349","display_name":"Fundamental theorem of Riemannian geometry","level":4,"score":0.5157999992370605},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.5156999826431274},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.506600022315979},{"id":"https://openalex.org/C102224218","wikidata":"https://www.wikidata.org/wiki/Q25344559","display_name":"Exponential map (Riemannian geometry)","level":5,"score":0.4927000105381012},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.45579999685287476},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.42719998955726624},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.40529999136924744},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39070001244544983},{"id":"https://openalex.org/C127104897","wikidata":"https://www.wikidata.org/wiki/Q5454858","display_name":"Fisher information metric","level":4,"score":0.3346000015735626},{"id":"https://openalex.org/C121705324","wikidata":"https://www.wikidata.org/wiki/Q1510587","display_name":"Pseudo-Riemannian manifold","level":4,"score":0.3197000026702881},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.29989999532699585},{"id":"https://openalex.org/C129049296","wikidata":"https://www.wikidata.org/wiki/Q548675","display_name":"Levi-Civita connection","level":5,"score":0.2777000069618225},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.27559998631477356},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27079999446868896},{"id":"https://openalex.org/C144386022","wikidata":"https://www.wikidata.org/wiki/Q1332997","display_name":"Scale factor (cosmology)","level":5,"score":0.2590999901294708}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2601.10992","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"Article"},{"id":"doi:10.48550/arxiv.2601.10992","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.10992","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:doi:10.48550/arxiv.2601.10992","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"Article"},"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":{"Constant":[0],"rescaling":[1,123],"of":[2,60,124,131,137],"a":[3,13,56,76,121,129,149],"Riemannian":[4,66,110,158],"metric":[5,62,114,151],"appears":[6],"in":[7,36,44,157],"many":[8],"computational":[9],"settings,":[10],"often":[11,117],"through":[12],"global":[14,122,150],"scale":[15,152],"parameter":[16,153],"that":[17,72,89],"is":[18,27,140,144],"introduced":[19,156],"either":[20],"explicitly":[21],"or":[22,48],"implicitly.":[23],"Although":[24],"this":[25,52,138],"operation":[26],"elementary,":[28],"its":[29],"consequences":[30],"are":[31],"not":[32],"always":[33],"made":[34],"clear":[35],"practice":[37],"and":[38,83,86,99,102,143],"may":[39],"be":[40,118,155],"confused":[41],"with":[42],"changes":[43],"curvature,":[45],"manifold":[46],"structure,":[47],"coordinate":[49],"representation.":[50],"In":[51],"note":[53,139],"we":[54],"provide":[55],"short,":[57],"self-contained":[58],"account":[59],"constant":[61,113],"scaling":[63,115],"on":[64,165],"arbitrary":[65],"manifolds.":[67],"We":[68,105],"distinguish":[69],"between":[70],"quantities":[71],"change":[73],"under":[74],"such":[75,92],"scaling,":[77],"including":[78],"norms,":[79],"distances,":[80],"volume":[81],"elements,":[82],"gradient":[84],"magnitudes,":[85],"geometric":[87,163],"objects":[88],"remain":[90],"invariant,":[91],"as":[93,120],"the":[94,132,162],"Levi--Civita":[95],"connection,":[96],"geodesics,":[97],"exponential":[98],"logarithmic":[100],"maps,":[101],"parallel":[103],"transport.":[104],"also":[106],"discuss":[107],"implications":[108],"for":[109],"optimization,":[111],"where":[112],"can":[116,154],"interpreted":[119],"step":[125],"sizes":[126],"rather":[127],"than":[128],"modification":[130],"underlying":[133],"geometry.":[134],"The":[135],"goal":[136],"purely":[141],"expository":[142],"intended":[145],"to":[146],"clarify":[147],"how":[148],"computation":[159],"without":[160],"altering":[161],"structures":[164],"which":[166],"these":[167],"methods":[168],"rely.":[169]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2026-01-20T00:00:00"}