{"id":"https://openalex.org/W7220909375","doi":"https://doi.org/10.48550/arxiv.2610.07551","title":"Is $\\sqrt{d}$ Separation Necessary for Gradient EM to Learn Gaussian Mixtures in High Dimensions?","display_name":"Is $\\sqrt{d}$ Separation Necessary for Gradient EM to Learn Gaussian Mixtures in High Dimensions?","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220909375","doi":"https://doi.org/10.48550/arxiv.2610.07551"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.07551","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.07551","pdf_url":"https://arxiv.org/pdf/2610.07551","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/2610.07551","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157930635","display_name":"Yiran Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhang, Yiran","raw_affiliation_strings":["University of California , Berkeley"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of California , Berkeley","institution_ids":["https://openalex.org/I95457486"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5157930407","display_name":"Mo Zhou","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I201448701","display_name":"University of Washington","ror":"https://ror.org/00cvxb145","country_code":"US","type":"education","lineage":["https://openalex.org/I201448701"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhou, Mo","raw_affiliation_strings":["University of Washington"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Washington","institution_ids":["https://openalex.org/I201448701"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5157900077","display_name":"Weihang Xu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I201448701","display_name":"University of Washington","ror":"https://ror.org/00cvxb145","country_code":"US","type":"education","lineage":["https://openalex.org/I201448701"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xu, Weihang","raw_affiliation_strings":["University of Washington"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Washington","institution_ids":["https://openalex.org/I201448701"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102886973","display_name":"Maryam Fazel","orcid":"https://orcid.org/0000-0001-5329-4522","observed_orcids":["https://orcid.org/0000-0001-5329-4522"]},"institutions":[{"id":"https://openalex.org/I201448701","display_name":"University of Washington","ror":"https://ror.org/00cvxb145","country_code":"US","type":"education","lineage":["https://openalex.org/I201448701"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Fazel, Maryam","raw_affiliation_strings":["University of Washington Amazon, Inc"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Washington Amazon, Inc","institution_ids":["https://openalex.org/I201448701"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5033061754","display_name":"Simon S. Du","orcid":"https://orcid.org/0000-0003-0056-8299","observed_orcids":["https://orcid.org/0000-0003-0056-8299"]},"institutions":[{"id":"https://openalex.org/I201448701","display_name":"University of Washington","ror":"https://ror.org/00cvxb145","country_code":"US","type":"education","lineage":["https://openalex.org/I201448701"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Du, Simon S.","raw_affiliation_strings":["University of Washington"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Washington","institution_ids":["https://openalex.org/I201448701"]}]}],"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":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/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.9922000169754028,"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/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.9922000169754028,"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":0.002199999988079071,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.0007999999797903001,"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/gaussian-mixture-model","display_name":"Gaussian mixture model","score":0.9520000219345093},{"id":"https://openalex.org/keywords/expectation-maximization-algorithm","display_name":"expectation-maximization algorithm","score":0.7850000262260437},{"id":"https://openalex.org/keywords/global-convergence","display_name":"global convergence","score":0.6190000176429749},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.5600000023841858},{"id":"https://openalex.org/keywords/component-separation-technique","display_name":"component separation","score":0.6330000162124634},{"id":"https://openalex.org/keywords/high-dimensional-learning","display_name":"high-dimensional learning","score":0.7300000190734863},{"id":"https://openalex.org/keywords/random-initialization","display_name":"random initialization","score":0.7110000252723694}],"concepts":[{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.6460999846458435},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6360999941825867},{"id":"https://openalex.org/C138777275","wikidata":"https://www.wikidata.org/wiki/Q6884054","display_name":"Mixing (physics)","level":2,"score":0.6309999823570251},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6101999878883362},{"id":"https://openalex.org/C61224824","wikidata":"https://www.wikidata.org/wiki/Q2260434","display_name":"Mixture model","level":2,"score":0.5781999826431274},{"id":"https://openalex.org/C2776061190","wikidata":"https://www.wikidata.org/wiki/Q7451805","display_name":"Separation (statistics)","level":2,"score":0.5608000159263611},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.5430999994277954},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5281000137329102},{"id":"https://openalex.org/C168167062","wikidata":"https://www.wikidata.org/wiki/Q1117970","display_name":"Component (thermodynamics)","level":2,"score":0.4918000102043152},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.4442000091075897},{"id":"https://openalex.org/C51432778","wikidata":"https://www.wikidata.org/wiki/Q1259145","display_name":"Independent component analysis","level":2,"score":0.43309998512268066},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.42399999499320984},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4196000099182129},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.3889999985694885},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3837999999523163},{"id":"https://openalex.org/C120317606","wikidata":"https://www.wikidata.org/wiki/Q17105967","display_name":"Blind signal separation","level":3,"score":0.3747999966144562},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.3628000020980835},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.32440000772476196},{"id":"https://openalex.org/C182081679","wikidata":"https://www.wikidata.org/wiki/Q1275153","display_name":"Expectation\u2013maximization algorithm","level":3,"score":0.3188999891281128},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3034999966621399},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.2912999987602234},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C2780551164","wikidata":"https://www.wikidata.org/wiki/Q2306599","display_name":"Column (typography)","level":3,"score":0.2736999988555908},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.26440000534057617},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.25029999017715454}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.07551","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.07551","pdf_url":"https://arxiv.org/pdf/2610.07551","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.2610.07551","doi":"https://doi.org/10.48550/arxiv.2610.07551","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.07551","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:2610.07551","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.07551","pdf_url":"https://arxiv.org/pdf/2610.07551","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":[{"id":"https://openalex.org/G5250394633","display_name":"Collaborative Research: AF: Medium: Machine Learning Markets: Dynamics, Competition, and Interventions","funder_award_id":"2312775","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5522209294","display_name":"CAREER: Toward a Foundation of Over-Parameterization","funder_award_id":"2143493","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5882832730","display_name":"TRIPODS: Institute for Foundations of Data Science","funder_award_id":"2023166","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6151270557","display_name":"Collaborative Research: SCALE MoDL: Adaptivity of Deep Neural Networks","funder_award_id":"2134106","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6920479218","display_name":"Collaborative Research: CIF: Medium: MoDL:Toward a Mathematical Foundation of Deep Reinforcement Learning","funder_award_id":"2212261","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G826810440","display_name":"AI Institute for Societal Decision Making (AI-SDM)","funder_award_id":"2229881","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"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220909375.pdf","grobid_xml":"https://content.openalex.org/works/W7220909375.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4417119109","https://openalex.org/W7128638666"],"related_works":["https://openalex.org/W4417119109","https://openalex.org/W3120364812","https://openalex.org/W4400267488","https://openalex.org/W2794125141","https://openalex.org/W2769553966","https://openalex.org/W2617827719","https://openalex.org/W2146758737","https://openalex.org/W7099006363","https://openalex.org/W3003642205","https://openalex.org/W3036438643"],"abstract_inverted_index":{"Learning":[0],"Gaussian":[1,191],"mixture":[2],"models":[3],"(GMMs)":[4],"using":[5],"the":[6,35,39,46,59,69,77,92,121,129,140,146,172,185],"Expectation-Maximization":[7],"(EM)":[8],"algorithm":[9,114],"and":[10,124],"its":[11],"gradient-based":[12],"variants":[13],"is":[14,22,83,91,103,142,154],"a":[15,111,149],"fundamental":[16],"problem":[17],"in":[18,34,58,105,145,164,171,196],"machine":[19],"learning.":[20],"It":[21],"known":[23],"that":[24,44,68,99,115,138],"randomly":[25],"initialized":[26],"(gradient)":[27],"EM":[28,54,113,118,126,163,195],"fails":[29],"to":[30,85,156],"learn":[31],"multi-component":[32],"GMMs":[33],"exact-parameterized":[36],"setting,":[37,61],"where":[38,62,89],"number":[40],"of":[41,45,52,151,160],"components":[42,64,71,82],"matches":[43],"ground-truth":[47,70,81,186],"GMM.":[48],"Recently,":[49],"global":[50,158],"convergence":[51,159],"gradient":[53,125,162,194],"has":[55],"been":[56],"established":[57],"over-parameterized":[60,173],"more":[63],"are":[65,72],"used,":[66],"provided":[67],"well":[73],"separated.":[74],"In":[75,94],"particular,":[76],"minimum":[78],"separation":[79,150,187],"between":[80],"required":[84,188],"scale":[86],"as":[87],"$\u03a9(\\sqrt{d})$,":[88],"$d$":[90],"dimension.":[93],"this":[95,100],"paper,":[96],"we":[97,109,136],"show":[98],"dimensional":[101],"dependence":[102],"unavoidable":[104],"high-dimensional":[106],"settings.":[107],"Specifically,":[108],"consider":[110],"hybrid":[112],"uses":[116],"standard":[117],"updates":[119,127],"for":[120,128,189],"mixing":[122],"weights":[123],"component":[130],"means.":[131],"For":[132],"any":[133],"$\u03b5>":[134],"0$,":[135],"prove":[137],"when":[139],"dimension":[141],"sufficiently":[143],"large,":[144],"worst":[147],"case":[148],"order":[152],"$\u03a9(d^{0.5-\u03b5})$":[153],"insufficient":[155],"guarantee":[157],"population":[161],"sub-exponential":[165],"time":[166],"under":[167],"random":[168],"initialization,":[169],"even":[170],"regime.":[174],"Our":[175],"result":[176],"establishes":[177],"an":[178],"almost":[179],"optimal":[180],"worst-case":[181],"lower":[182],"bound":[183],"on":[184],"learning":[190],"mixtures":[192],"via":[193],"high":[197],"dimensions.":[198]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-09T00:00:00"}