{"id":"https://openalex.org/W7224870072","doi":"https://doi.org/10.48550/arxiv.2610.40566","title":"Self-regularization and structural adaptation of NPMLE for Gaussian mixture models in all dimensions","display_name":"Self-regularization and structural adaptation of NPMLE for Gaussian mixture models in all dimensions","publication_year":2026,"publication_date":"2026-10-29","ids":{"openalex":"https://openalex.org/W7224870072","doi":"https://doi.org/10.48550/arxiv.2610.40566"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.40566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.40566","pdf_url":"https://arxiv.org/pdf/2610.40566","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/2610.40566","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5160162817","display_name":"Yanjun Han","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I36672615","display_name":"Courant Institute of Mathematical Sciences","ror":"https://ror.org/037tm7f56","country_code":"US","type":"education","lineage":["https://openalex.org/I36672615","https://openalex.org/I57206974"]},{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Han, Yanjun","raw_affiliation_strings":["Courant Institute of Mathematical Sciences and Center for Data Science , New York University ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Courant Institute of Mathematical Sciences and Center for Data Science , New York University ,","institution_ids":["https://openalex.org/I36672615","https://openalex.org/I57206974"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5160191099","display_name":"Yandi Shen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Shen, Yandi","raw_affiliation_strings":["Department of Statistics and Data Science , Carnegie Mellon University ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics and Data Science , Carnegie Mellon University ,","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5160218052","display_name":"Yihong Wu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wu, Yihong","raw_affiliation_strings":["Department of Statistics and Data Science , Yale University ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics and Data Science , Yale University ,","institution_ids":["https://openalex.org/I32971472"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"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":0.9990000128746033,"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":0.9990000128746033,"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":0.00039999998989515007,"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/T10637","display_name":"Advanced Clustering Algorithms Research","score":9.999999747378752e-05,"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/nonparametric-maximum-likelihood-estimation","display_name":"nonparametric maximum likelihood estimation","score":0.9319999814033508},{"id":"https://openalex.org/keywords/gaussian-mixture-model","display_name":"Gaussian mixture model","score":0.9620000123977661},{"id":"https://openalex.org/keywords/self-regularization","display_name":"self-regularization","score":0.9110000133514404},{"id":"https://openalex.org/keywords/structural-adaptation","display_name":"structural adaptation","score":0.9480000138282776},{"id":"https://openalex.org/keywords/model-complexity","display_name":"model complexity","score":0.6840000152587891},{"id":"https://openalex.org/keywords/number-of-mixture-components","display_name":"number of mixture components","score":0.7300000190734863},{"id":"https://openalex.org/keywords/small-ball-probabilities","display_name":"small ball probabilities","score":0.7990000247955322},{"id":"https://openalex.org/keywords/hellinger-distance","display_name":"Hellinger distance","score":0.7680000066757202},{"id":"https://openalex.org/keywords/high-dimensional-statistics","display_name":"high-dimensional statistics","score":0.4690000116825104}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8514999747276306},{"id":"https://openalex.org/C61224824","wikidata":"https://www.wikidata.org/wiki/Q2260434","display_name":"Mixture model","level":2,"score":0.6841999888420105},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6000000238418579},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.5616000294685364},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5385000109672546},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5110999941825867},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.4472000002861023},{"id":"https://openalex.org/C50454189","wikidata":"https://www.wikidata.org/wiki/Q2075125","display_name":"Karush\u2013Kuhn\u2013Tucker conditions","level":2,"score":0.43779999017715454},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.43140000104904175},{"id":"https://openalex.org/C189508267","wikidata":"https://www.wikidata.org/wiki/Q17088227","display_name":"Density estimation","level":3,"score":0.421999990940094},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4187999963760376},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.3955000042915344},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.39169999957084656},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3781000077724457},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.36399999260902405},{"id":"https://openalex.org/C153024298","wikidata":"https://www.wikidata.org/wiki/Q3030678","display_name":"Hellinger distance","level":2,"score":0.34790000319480896},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.33559998869895935},{"id":"https://openalex.org/C102366305","wikidata":"https://www.wikidata.org/wiki/Q1097688","display_name":"Nonparametric statistics","level":2,"score":0.30550000071525574},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.2856999933719635},{"id":"https://openalex.org/C177384507","wikidata":"https://www.wikidata.org/wiki/Q1149000","display_name":"Multivariate normal distribution","level":3,"score":0.2700999975204468},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.2639000117778778},{"id":"https://openalex.org/C21596040","wikidata":"https://www.wikidata.org/wiki/Q2896771","display_name":"Unimodality","level":2,"score":0.2603999972343445},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.25920000672340393},{"id":"https://openalex.org/C167928553","wikidata":"https://www.wikidata.org/wiki/Q1376021","display_name":"Estimation theory","level":2,"score":0.257099986076355},{"id":"https://openalex.org/C20538924","wikidata":"https://www.wikidata.org/wiki/Q822139","display_name":"Characteristic function (probability theory)","level":3,"score":0.2526000142097473},{"id":"https://openalex.org/C7218915","wikidata":"https://www.wikidata.org/wiki/Q1054475","display_name":"Gaussian function","level":3,"score":0.2524000108242035},{"id":"https://openalex.org/C66690126","wikidata":"https://www.wikidata.org/wiki/Q2914302","display_name":"Concave function","level":3,"score":0.2506999969482422}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.40566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.40566","pdf_url":"https://arxiv.org/pdf/2610.40566","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.2610.40566","doi":"https://doi.org/10.48550/arxiv.2610.40566","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.40566","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:2610.40566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.40566","pdf_url":"https://arxiv.org/pdf/2610.40566","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7224870072.pdf","grobid_xml":"https://content.openalex.org/works/W7224870072.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3059391138","https://openalex.org/W7140273563","https://openalex.org/W7216052325","https://openalex.org/W7223252578","https://openalex.org/W7154460026","https://openalex.org/W7155220060","https://openalex.org/W7203695693","https://openalex.org/W2548889165","https://openalex.org/W2774003859","https://openalex.org/W7214030217"],"abstract_inverted_index":{"We":[0,21,119],"establish":[1,120],"structural":[2,59],"properties":[3],"of":[4,71,92,95],"the":[5,27,46,55,69,89,96,101,126,146,156],"nonparametric":[6],"maximum":[7],"likelihood":[8],"estimator":[9],"(NPMLE)":[10],"for":[11,145,155],"$d$-dimensional":[12],"Gaussian":[13],"mixtures":[14],"whose":[15],"means":[16],"lie":[17],"in":[18,76],"a":[19,37,58,66,83,135],"hypercube.":[20],"show":[22],"that,":[23],"with":[24,40,100,148],"high":[25],"probability,":[26],"NPMLE":[28,56],"based":[29],"on":[30,116],"$n$":[31],"independent":[32],"observations":[33],"self-regularizes":[34],"by":[35,124],"fitting":[36],"mixture":[38],"model":[39,48,52],"$O((\\frac{\\log":[41],"n}{\\log\\log":[42],"n})^d)$":[43],"components,":[44],"achieving":[45],"optimal":[47],"complexity":[49],"without":[50],"explicit":[51],"selection.":[53],"Furthermore,":[54],"exhibits":[57],"adaptation":[60],"property:":[61],"For":[62],"data":[63],"drawn":[64],"from":[65],"finite":[67],"mixture,":[68],"number":[70,91],"fitted":[72],"components":[73],"is":[74,108],"bounded":[75],"probability":[77,143],"as":[78],"$n\\to\\infty$.":[79],"Our":[80],"proof":[81],"applies":[82],"multivariate":[84],"Jensen-type":[85],"bound":[86,111,123],"to":[87,109],"control":[88],"typical":[90],"critical":[93],"points":[94],"KKT":[97,127],"function":[98,128],"associated":[99],"first-order":[102],"optimality":[103],"conditions.":[104],"The":[105],"key":[106],"step":[107],"lower":[110,122],"its":[112],"gradient":[113],"norm":[114],"uniformly":[115],"complex":[117],"spheres.":[118],"this":[121],"decomposing":[125],"into":[129],"an":[130],"empirical":[131],"fluctuation":[132],"term":[133],"and":[134,151],"relative":[136],"density":[137],"estimation":[138],"error":[139,153],"term,":[140],"combining":[141],"small-ball":[142],"estimates":[144],"former":[147],"polynomial":[149],"approximation":[150],"Hellinger":[152],"bounds":[154],"latter.":[157]},"counts_by_year":[],"updated_date":"2026-11-02T09:45:03.842371","created_date":"2026-10-31T00:00:00"}
