{"id":"https://openalex.org/W7161139062","doi":"https://doi.org/10.48550/arxiv.2605.13079","title":"Spectral Flattening Is All Muon Needs: How Orthogonalization Controls Learning Rate and Convergence","display_name":"Spectral Flattening Is All Muon Needs: How Orthogonalization Controls Learning Rate and Convergence","publication_year":2026,"publication_date":"2026-05-13","ids":{"openalex":"https://openalex.org/W7161139062","doi":"https://doi.org/10.48550/arxiv.2605.13079"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.13079","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.13079","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":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.13079","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005670374","display_name":"Tien-Phat Nguyen","orcid":"https://orcid.org/0000-0002-4204-842X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nguyen, Tien-Phat","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5136091387","display_name":"Truong N. Nguyen","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nguyen, Truong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5133718120","display_name":"Minh-Phuc Truong","orcid":"https://orcid.org/0009-0005-3834-4882"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Truong, Minh-Phuc","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5136128638","display_name":"Tuc Nguyen","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nguyen, Tuc","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5136145924","display_name":"James Bailey","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bailey, James","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5003751580","display_name":"Trung Le","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Le, Trung","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":6,"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/T12579","display_name":"Muon and positron interactions and applications","score":0.3375999927520752,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12579","display_name":"Muon and positron interactions and applications","score":0.3375999927520752,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10048","display_name":"Particle physics theoretical and experimental studies","score":0.15620000660419464,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy 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/T11044","display_name":"Particle Detector Development and Performance","score":0.10379999876022339,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/orthogonalization","display_name":"Orthogonalization","score":0.690500020980835},{"id":"https://openalex.org/keywords/muon","display_name":"Muon","score":0.652899980545044},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5400999784469604},{"id":"https://openalex.org/keywords/momentum","display_name":"Momentum (technical analysis)","score":0.5372999906539917},{"id":"https://openalex.org/keywords/flattening","display_name":"Flattening","score":0.5277000069618225},{"id":"https://openalex.org/keywords/spectrum","display_name":"Spectrum (functional analysis)","score":0.47269999980926514},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.4496000111103058}],"concepts":[{"id":"https://openalex.org/C47559304","wikidata":"https://www.wikidata.org/wiki/Q1702189","display_name":"Orthogonalization","level":2,"score":0.690500020980835},{"id":"https://openalex.org/C205334942","wikidata":"https://www.wikidata.org/wiki/Q3151","display_name":"Muon","level":2,"score":0.652899980545044},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5400999784469604},{"id":"https://openalex.org/C60718061","wikidata":"https://www.wikidata.org/wiki/Q1414747","display_name":"Momentum (technical analysis)","level":2,"score":0.5372999906539917},{"id":"https://openalex.org/C19444555","wikidata":"https://www.wikidata.org/wiki/Q212750","display_name":"Flattening","level":2,"score":0.5277000069618225},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5127000212669373},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.47269999980926514},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.4496000111103058},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.40630000829696655},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.4059999883174896},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.32409998774528503},{"id":"https://openalex.org/C136272165","wikidata":"https://www.wikidata.org/wiki/Q4048889","display_name":"Singular spectrum analysis","level":3,"score":0.30320000648498535},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.2833999991416931},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2831999957561493},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.2815000116825104},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.27570000290870667},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.266400009393692},{"id":"https://openalex.org/C175706884","wikidata":"https://www.wikidata.org/wiki/Q1130194","display_name":"Moving average","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.13079","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.13079","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":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.13079","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.13079","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":"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":{"Muon":[0,21,79,115],"orthogonalizes":[1],"the":[2,38,60,65,69,95,100,104,107,127],"momentum":[3],"buffer":[4],"before":[5],"each":[6],"update,":[7],"replacing":[8],"its":[9],"singular":[10,62],"values":[11],"with":[12,59,99],"ones":[13],"via":[14],"Newton-Schulz":[15],"iterations.":[16],"This":[17],"simple":[18],"change":[19],"lets":[20],"tolerate":[22],"far":[23],"larger":[24],"learning":[25,119],"rates":[26,120],"and":[27,43,85,131],"converge":[28],"faster":[29],"than":[30,68],"other":[31],"optimizers,":[32],"but":[33],"why?":[34],"We":[35],"show":[36],"that":[37,52,92,121],"mechanism":[39],"is":[40],"spectral":[41],"flattening,":[42],"develop":[44],"two":[45],"results":[46,146],"around":[47],"it.":[48],"First,":[49],"we":[50,77],"prove":[51],"Muon's":[53,153],"maximal":[54],"stable":[55,117],"step":[56,141],"size":[57],"scales":[58],"average":[61],"value":[63],"of":[64,106],"gradient":[66,74,83,108],"rather":[67],"largest,":[70],"which":[71],"bottlenecks":[72],"standard":[73],"descent.":[75],"Second,":[76],"recast":[78],"as":[80],"a":[81,88,148],"preconditioned":[82],"method":[84],"show,":[86],"under":[87],"Kronecker-factored":[89],"curvature":[90],"model,":[91],"it":[93],"improves":[94],"effective":[96],"convergence":[97],"factor,":[98],"improvement":[101],"controlled":[102],"by":[103],"spectrum":[105],"covariance.":[109],"Extensive":[110],"experiments":[111],"validate":[112],"both":[113],"results:":[114],"remains":[116],"at":[118,139],"cause":[122],"SGD":[123],"to":[124],"diverge":[125],"within":[126],"first":[128],"few":[129],"iterations,":[130],"reaches":[132],"accuracy":[133],"milestones":[134],"several":[135],"epochs":[136],"earlier":[137],"even":[138],"identical":[140],"sizes.":[142],"Taken":[143],"together,":[144],"our":[145],"offer":[147],"principled,":[149],"geometric":[150],"explanation":[151],"for":[152],"empirical":[154],"success.":[155]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-05-15T00:00:00"}
