{"id":"https://openalex.org/W4396628331","doi":"https://doi.org/10.1088/1742-5468/ae4bba","title":"Scaling and renormalization in high-dimensional regression","display_name":"Scaling and renormalization in high-dimensional regression","publication_year":2026,"publication_date":"2026-04-01","ids":{"openalex":"https://openalex.org/W4396628331","doi":"https://doi.org/10.1088/1742-5468/ae4bba"},"language":"en","primary_location":{"id":"doi:10.1088/1742-5468/ae4bba","doi":"https://doi.org/10.1088/1742-5468/ae4bba","is_oa":false,"landing_page_url":"https://doi.org/10.1088/1742-5468/ae4bba","pdf_url":null,"source":{"id":"https://openalex.org/S167943036","display_name":"Journal of Statistical Mechanics Theory and Experiment","issn_l":"1742-5468","issn":["1742-5468"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-c","jufo-1","ki-jl-1","norway-1","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310311669","host_organization_name":"Institute of Physics","host_organization_lineage":["https://openalex.org/P4310311669"],"host_organization_lineage_names":["Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Statistical Mechanics: Theory and Experiment","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2405.00592","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046607604","display_name":"Alexander Atanasov","orcid":"https://orcid.org/0000-0002-3338-0324","observed_orcids":["https://orcid.org/0000-0002-3338-0324"]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Alexander Atanasov","raw_affiliation_strings":["Center for Brain Science, Harvard University, Cambridge, MA, United States of America","Department of Physics, Harvard University, Cambridge, MA, United States of America"],"raw_orcid":"https://orcid.org/0000-0002-3338-0324","affiliations":[{"raw_affiliation_string":"Center for Brain Science, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Department of Physics, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5063822685","display_name":"Jacob A. Zavatone-Veth","orcid":"https://orcid.org/0000-0002-4060-1738","observed_orcids":["https://orcid.org/0000-0002-4060-1738"]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jacob A Zavatone-Veth","raw_affiliation_strings":["Center for Brain Science, Harvard University, Cambridge, MA, United States of America","Department of Physics, Harvard University, Cambridge, MA, United States of America","John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, United States of America","Society of Fellows, Harvard University, Cambridge, MA, United States of America"],"raw_orcid":"https://orcid.org/0000-0002-4060-1738","affiliations":[{"raw_affiliation_string":"Center for Brain Science, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Department of Physics, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Society of Fellows, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023195984","display_name":"Cengiz Pehlevan","orcid":"https://orcid.org/0000-0001-9767-6063","observed_orcids":["https://orcid.org/0000-0001-9767-6063"]},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Cengiz Pehlevan","raw_affiliation_strings":["Center for Brain Science, Harvard University, Cambridge, MA, United States of America","John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, United States of America","Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University, Cambridge, MA, United States of America"],"raw_orcid":"https://orcid.org/0000-0001-9767-6063","affiliations":[{"raw_affiliation_string":"Center for Brain Science, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University, Cambridge, MA, United States of America","institution_ids":["https://openalex.org/I136199984"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5023195984","https://openalex.org/A5046607604","https://openalex.org/A5063822685"],"corresponding_institution_ids":["https://openalex.org/I136199984"],"apc_list":{"value":3290,"currency":"USD","value_usd":3290},"apc_paid":null,"fwci":15.8622,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.98557505,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"2026","issue":"4","first_page":"043404","last_page":"043404"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.38040000200271606,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11716","display_name":"Random Matrices and Applications","score":0.38040000200271606,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.3296000063419342,"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/T10136","display_name":"Statistical Methods and Inference","score":0.0471000000834465,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/ridge-regression","display_name":"ridge regression","score":0.9800000190734863},{"id":"https://openalex.org/keywords/high-dimensional-regression","display_name":"high-dimensional regression","score":0.8769999742507935},{"id":"https://openalex.org/keywords/benign-overfitting","display_name":"benign overfitting","score":0.8579999804496765},{"id":"https://openalex.org/keywords/random-matrix-theory","display_name":"random matrix theory","score":0.7620000243186951},{"id":"https://openalex.org/keywords/free-probability","display_name":"free probability","score":0.9850000143051147},{"id":"https://openalex.org/keywords/s-transform","display_name":"S-transform","score":0.9660000205039978},{"id":"https://openalex.org/keywords/power-law-scaling","display_name":"power-law scaling","score":0.7289999723434448},{"id":"https://openalex.org/keywords/generalization-gap","display_name":"generalization gap","score":0.7279999852180481},{"id":"https://openalex.org/keywords/generalized-cross-validation","display_name":"generalized cross-validation","score":0.847000002861023},{"id":"https://openalex.org/keywords/bias-variance-decomposition","display_name":"bias-variance decomposition","score":0.9599999785423279},{"id":"https://openalex.org/keywords/random-feature-models","display_name":"random feature models","score":0.9990000128746033},{"id":"https://openalex.org/keywords/overparameterized-models","display_name":"overparameterized models","score":0.8519999980926514},{"id":"https://openalex.org/keywords/neural-scaling-laws","display_name":"neural scaling laws","score":0.7720000147819519}],"concepts":[{"id":"https://openalex.org/C166124518","wikidata":"https://www.wikidata.org/wiki/Q1047702","display_name":"Renormalization","level":2,"score":0.7565771341323853},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.7551790475845337},{"id":"https://openalex.org/C83546350","wikidata":"https://www.wikidata.org/wiki/Q1139051","display_name":"Regression","level":2,"score":0.5358755588531494},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5006554126739502},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4247283935546875},{"id":"https://openalex.org/C152877465","wikidata":"https://www.wikidata.org/wiki/Q208042","display_name":"Regression analysis","level":2,"score":0.41290855407714844},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3689839541912079},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.27252113819122314},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.2489551305770874},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.05116915702819824}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1088/1742-5468/ae4bba","doi":"https://doi.org/10.1088/1742-5468/ae4bba","is_oa":false,"landing_page_url":"https://doi.org/10.1088/1742-5468/ae4bba","pdf_url":null,"source":{"id":"https://openalex.org/S167943036","display_name":"Journal of Statistical Mechanics Theory and Experiment","issn_l":"1742-5468","issn":["1742-5468"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-c","jufo-1","ki-jl-1","norway-1","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310311669","host_organization_name":"Institute of Physics","host_organization_lineage":["https://openalex.org/P4310311669"],"host_organization_lineage_names":["Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Statistical Mechanics: Theory and Experiment","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2405.00592","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2405.00592","pdf_url":"https://arxiv.org/pdf/2405.00592","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.2405.00592","doi":"https://doi.org/10.48550/arxiv.2405.00592","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2405.00592","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:2405.00592","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2405.00592","pdf_url":"https://arxiv.org/pdf/2405.00592","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":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G3521671702","display_name":"A Theory of Learned Representations in Artificial and Natural Neural Networks","funder_award_id":"2134157","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G381467788","display_name":"CAREER: Developing Neural Network Theory for Uncovering How the Brain Learns","funder_award_id":"2239780","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7002119356","display_name":"Kavli Institute for Theoretical Physics","funder_award_id":"2309135","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8497063558","display_name":"Kavli Institute for Theoretical Physics","funder_award_id":"1748958","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320332349","display_name":"Society of Fellows, Harvard University","ror":null},{"id":"https://openalex.org/F4320337427","display_name":"Kavli Institute for Theoretical Physics, University of California, Santa Barbara","ror":"https://ror.org/02t274463"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4396628331.pdf","grobid_xml":"https://content.openalex.org/works/W4396628331.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4392822353","https://openalex.org/W4415796071","https://openalex.org/W4405602848","https://openalex.org/W4415976639","https://openalex.org/W4309045102","https://openalex.org/W7132859504","https://openalex.org/W2967536008","https://openalex.org/W3172995164","https://openalex.org/W4225681315","https://openalex.org/W4389073345"],"abstract_inverted_index":{"Abstract":[0],"From":[1,136],"benign":[2],"overfitting":[3],"in":[4,11,44,78,90,119,148,212,223,238],"overparameterized":[5,214,240],"models":[6,187,202,226,252],"to":[7,21,24,109,158,194,207,232],"rich":[8],"power-law":[9,146],"scalings":[10],"performance,":[12],"simple":[13],"ridge":[14,37,60,102],"regression":[15,38,61],"displays":[16],"surprising":[17],"behaviors":[18],"sometimes":[19],"thought":[20],"be":[22,95],"limited":[23],"deep":[25,81],"neural":[26,254],"networks.":[27],"This":[28,104,191],"balance":[29],"of":[30,42,66,100,123,129,133,145,167,184,253],"phenomenological":[31],"richness":[32],"with":[33,76,188],"analytical":[34],"tractability":[35],"makes":[36],"the":[39,63,85,101,114,127,130,154,159,168,204,208,213,239],"model":[40,149],"system":[41],"choice":[43],"high-dimensional":[45],"machine":[46],"learning.":[47,82],"In":[48,151],"this":[49],"paper,":[50],"we":[51,140,174],"present":[52],"a":[53,98,120,180,196,247],"unifying":[54,248],"perspective":[55,249],"on":[56,59,250],"recent":[57],"results":[58,243],"using":[62],"basic":[64],"tools":[65],"random":[67,185,200,224],"matrix":[68],"theory":[69],"and":[70,80,116,163,230,245],"free":[71,134],"probability,":[72],"aimed":[73],"at":[74],"readers":[75],"backgrounds":[77],"physics":[79],"We":[83,216],"highlight":[84],"fact":[86],"that":[87],"statistical":[88],"fluctuations":[89],"empirical":[91],"covariance":[92],"matrices":[93],"can":[94,141,227],"absorbed":[96],"into":[97],"renormalization":[99],"parameter.":[103],"\u201cdeterministic":[105],"equivalence\u201d":[106],"allows":[107,192],"us":[108,193],"obtain":[110],"analytic":[111],"formulas":[112],"for":[113,179,199,235],"training":[115],"generalization":[117,161],"errors":[118],"few":[121],"lines":[122],"algebra":[124],"by":[125],"leveraging":[126],"properties":[128],"S":[131,155],"-transform":[132,156],"probability.":[135],"these":[137,172],"precise":[138],"asymptotics,":[139],"easily":[142],"identify":[143],"sources":[144],"scaling":[147,197,255],"performance.":[150],"all":[152],"models,":[153],"corresponds":[157],"train-test":[160],"gap,":[162],"yields":[164],"an":[165],"analog":[166],"generalized-cross-validation":[169],"estimator.":[170],"Using":[171],"techniques,":[173],"derive":[175],"fine-grained":[176],"bias-variance":[177],"decompositions":[178],"very":[181],"general":[182],"class":[183],"feature":[186,201,225],"structured":[189],"covariates.":[190],"discover":[195],"regime":[198],"where":[203],"variance":[205],"due":[206],"features":[209],"limits":[210],"performance":[211,229],"setting.":[215,241],"also":[217],"demonstrate":[218],"how":[219],"anisotropic":[220],"weight":[221],"structure":[222],"limit":[228],"lead":[231],"nontrivial":[233],"exponents":[234],"finite-width":[236],"corrections":[237],"Our":[242],"extend":[244],"provide":[246],"earlier":[251],"laws.":[256]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":8}],"updated_date":"2026-10-20T09:52:01.089006","created_date":"2024-05-04T00:00:00"}