{"id":"https://openalex.org/W7212902800","doi":"https://doi.org/10.48550/arxiv.2609.12123","title":"Rank-Efficient LoRA via Joint Tangent-Space Optimization under Isotropic Curvature","display_name":"Rank-Efficient LoRA via Joint Tangent-Space Optimization under Isotropic Curvature","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7212902800","doi":"https://doi.org/10.48550/arxiv.2609.12123"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.12123","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.12123","pdf_url":"https://arxiv.org/pdf/2609.12123","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/2609.12123","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151676914","display_name":"Zihan Zhu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhu, Zihan","raw_affiliation_strings":["University of Pennsylvania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Pennsylvania","institution_ids":["https://openalex.org/I79576946"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101264788","display_name":"Zhehang Du","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Du, Zhehang","raw_affiliation_strings":["University of Pennsylvania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Pennsylvania","institution_ids":["https://openalex.org/I79576946"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5151676475","display_name":"Xuyang Chen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Chen, Xuyang","raw_affiliation_strings":["University of Pennsylvania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Pennsylvania","institution_ids":["https://openalex.org/I79576946"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014387645","display_name":"Tim Tsz-Kit Lau","orcid":"https://orcid.org/0000-0002-5653-7232","observed_orcids":["https://orcid.org/0000-0002-5653-7232"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lau, Tim Tsz-Kit","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5151618514","display_name":"Jiayuan Wu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wu, Jiayuan","raw_affiliation_strings":["University of Pennsylvania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Pennsylvania","institution_ids":["https://openalex.org/I79576946"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5151634365","display_name":"X. Y. Han","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Han, X. Y.","raw_affiliation_strings":["University of Chicago"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Chicago","institution_ids":["https://openalex.org/I40347166"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113336765","display_name":"Qi Long","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Long, Qi","raw_affiliation_strings":["University of Pennsylvania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Pennsylvania","institution_ids":["https://openalex.org/I79576946"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5136521634","display_name":"Weijie Su","orcid":"https://orcid.org/0000-0003-1787-1219","observed_orcids":["https://orcid.org/0000-0003-1787-1219"]},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Su, Weijie","raw_affiliation_strings":["University of Pennsylvania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Pennsylvania","institution_ids":["https://openalex.org/I79576946"]}]}],"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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.8773999810218811,"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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.8773999810218811,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.08370000123977661,"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/T10028","display_name":"Topic Modeling","score":0.021299999207258224,"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/lora","display_name":"LoRa","score":0.859000027179718},{"id":"https://openalex.org/keywords/low-rank-adaptation","display_name":"low-rank adaptation","score":0.9819999933242798},{"id":"https://openalex.org/keywords/effective-rank","display_name":"effective rank","score":0.9210000038146973},{"id":"https://openalex.org/keywords/isotropic-curvature","display_name":"isotropic curvature","score":0.9169999957084656},{"id":"https://openalex.org/keywords/optimizer-design","display_name":"optimizer design","score":0.7379999756813049},{"id":"https://openalex.org/keywords/singular-spectrum","display_name":"singular spectrum","score":0.5460000038146973},{"id":"https://openalex.org/keywords/muon-optimizer","display_name":"Muon optimizer","score":0.7459999918937683},{"id":"https://openalex.org/keywords/adamw","display_name":"AdamW","score":0.9779999852180481},{"id":"https://openalex.org/keywords/parameter-efficient-fine-tuning","display_name":"parameter-efficient fine-tuning","score":0.7480000257492065},{"id":"https://openalex.org/keywords/language-model-adaptation","display_name":"language model adaptation","score":0.6909999847412109},{"id":"https://openalex.org/keywords/gpt-2","display_name":"GPT-2","score":0.8059999942779541}],"concepts":[{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.5656999945640564},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.515999972820282},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4961000084877014},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.4767000079154968},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.46709999442100525},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.3797000050544739},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3700999915599823},{"id":"https://openalex.org/C38935604","wikidata":"https://www.wikidata.org/wiki/Q4330363","display_name":"Stylized fact","level":2,"score":0.36239999532699585},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.3391999900341034},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.33399999141693115},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.32899999618530273},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3149999976158142},{"id":"https://openalex.org/C184050105","wikidata":"https://www.wikidata.org/wiki/Q273163","display_name":"Isotropy","level":2,"score":0.30059999227523804},{"id":"https://openalex.org/C2779863119","wikidata":"https://www.wikidata.org/wiki/Q6423048","display_name":"Knot (papermaking)","level":2,"score":0.29989999532699585},{"id":"https://openalex.org/C12302492","wikidata":"https://www.wikidata.org/wiki/Q1752097","display_name":"Setpoint","level":2,"score":0.2892000079154968},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.2888999879360199},{"id":"https://openalex.org/C186394612","wikidata":"https://www.wikidata.org/wiki/Q7098942","display_name":"Optimal design","level":2,"score":0.28839999437332153},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.2761000096797943},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.27239999175071716},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.2700999975204468},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.26820001006126404},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.26669999957084656},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.2619999945163727},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.2596000134944916},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.2590000033378601},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.25519999861717224},{"id":"https://openalex.org/C144171764","wikidata":"https://www.wikidata.org/wiki/Q48103","display_name":"Torque","level":2,"score":0.2508000135421753},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.2500999867916107}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.12123","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.12123","pdf_url":"https://arxiv.org/pdf/2609.12123","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.2609.12123","doi":"https://doi.org/10.48550/arxiv.2609.12123","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.12123","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:2609.12123","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.12123","pdf_url":"https://arxiv.org/pdf/2609.12123","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/W7212902800.pdf","grobid_xml":"https://content.openalex.org/works/W7212902800.grobid-xml"},"referenced_works_count":51,"referenced_works":["https://openalex.org/W2116837464","https://openalex.org/W2155894447","https://openalex.org/W2618017917","https://openalex.org/W2785678896","https://openalex.org/W2797328513","https://openalex.org/W2908510526","https://openalex.org/W2913946806","https://openalex.org/W2964121744","https://openalex.org/W3023528699","https://openalex.org/W3152956381","https://openalex.org/W3166140588","https://openalex.org/W3174770825","https://openalex.org/W3205270560","https://openalex.org/W4330338093","https://openalex.org/W4361229539","https://openalex.org/W4378509498","https://openalex.org/W4384263459","https://openalex.org/W4385572156","https://openalex.org/W4387800062","https://openalex.org/W4391988255","https://openalex.org/W4392575839","https://openalex.org/W4393138529","https://openalex.org/W4394029298","https://openalex.org/W4398192691","https://openalex.org/W4399911609","https://openalex.org/W4400104096","https://openalex.org/W4400611699","https://openalex.org/W4401306886","https://openalex.org/W4402671897","https://openalex.org/W4403815243","https://openalex.org/W4405172828","https://openalex.org/W4405655184","https://openalex.org/W4410061981","https://openalex.org/W4414802004","https://openalex.org/W4414846785","https://openalex.org/W4414968374","https://openalex.org/W4415037178","https://openalex.org/W4415428213","https://openalex.org/W4415797461","https://openalex.org/W4415938525","https://openalex.org/W4416046424","https://openalex.org/W4416113483","https://openalex.org/W7127566804","https://openalex.org/W7133233472","https://openalex.org/W7133235158","https://openalex.org/W7133241261","https://openalex.org/W7148703782","https://openalex.org/W7160599108","https://openalex.org/W7160889841","https://openalex.org/W7164564891","https://openalex.org/W7207708772"],"related_works":["https://openalex.org/W7214330414","https://openalex.org/W7169847197","https://openalex.org/W7207708772","https://openalex.org/W7163543289","https://openalex.org/W4416034389","https://openalex.org/W4402619439","https://openalex.org/W7154470314","https://openalex.org/W7169860126","https://openalex.org/W4415037178","https://openalex.org/W7161427964"],"abstract_inverted_index":{"Low-Rank":[0],"Adaptation":[1],"(LoRA)":[2],"is":[3,22,38,56],"an":[4,26,118,230],"effective":[5,96,171,199],"approach":[6],"for":[7],"adapting":[8],"large":[9],"pretrained":[10],"models":[11],"by":[12],"learning":[13],"low-rank":[14],"weight":[15,135],"updates.":[16,62],"In":[17,63],"practice,":[18],"the":[19,41,45,48,59,81,122,130,155,205],"LoRA":[20,112,123,156,210,222],"rank":[21,43,149,172,200,215],"used":[23,57],"to":[24],"control":[25],"adapter's":[27],"parameter":[28],"budget":[29],"and":[30,94,106,201,224],"representational":[31,46],"capacity.":[32],"We":[33,158,186],"show":[34],"that":[35,54,120,140,166,226],"this":[36,160,188],"view":[37],"incomplete:":[39],"while":[40,151],"nominal":[42,83],"determines":[44],"capacity,":[47],"optimizer":[49,77,119,227],"shapes":[50],"how":[51],"much":[52],"of":[53,67,104],"capacity":[55],"in":[58,134,221],"induced":[60,131],"weight-space":[61],"a":[64,74,101,178,182,218],"case":[65],"study":[66],"GPT-2":[68],"adaptation":[69,192],"with":[70,90,154,162,204],"LoRA,":[71],"we":[72],"observe":[73],"strong":[75],"rank-dependent":[76],"effect.":[78],"Despite":[79],"using":[80],"same":[82],"rank,":[84,97],"AdamW":[85],"often":[86],"produces":[87],"per-step":[88],"updates":[89,125,139],"concentrated":[91],"singular":[92,146],"spectra":[93],"low":[95],"whereas":[98],"Muon":[99],"uses":[100],"richer":[102],"set":[103],"directions":[105],"benefits":[107],"more":[108,143],"consistently":[109],"from":[110],"increasing":[111],"rank.":[113],"These":[114],"observations":[115],"motivate":[116],"ISO-LoRA,":[117],"couples":[121],"factor":[124,220],"through":[126,177],"spectral":[127],"descent":[128],"on":[129,190],"tangent":[132],"perturbation":[133],"space.":[136],"ISO-LoRA":[137,167,197],"promotes":[138],"distribute":[141],"energy":[142],"evenly":[144],"across":[145,193],"directions,":[147],"improving":[148],"utilization":[150,216],"preserving":[152],"compatibility":[153],"parameterization.":[157],"complement":[159],"design":[161,189,228],"theoretical":[163],"guarantees":[164],"showing":[165],"can":[168],"achieve":[169],"higher":[170],"than":[173],"standard":[174],"factor-wise":[175],"optimizers":[176],"one-step":[179],"analysis":[180],"under":[181],"stylized":[183],"spiked-gradient":[184],"model.":[185],"validate":[187],"language-model":[191],"0.1B-7B-parameter":[194],"models,":[195],"where":[196],"improves":[198],"downstream":[202],"performance,":[203],"strongest":[206],"gains":[207],"at":[208],"moderate-to-large":[209],"ranks.":[211],"Our":[212],"results":[213],"highlight":[214],"as":[217],"key":[219],"optimization":[223],"suggest":[225],"offers":[229],"important":[231],"path":[232],"toward":[233],"stronger":[234],"parameter-efficient":[235],"adaptation.":[236]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-15T00:00:00"}