{"id":"https://openalex.org/W7161936803","doi":"https://doi.org/10.48550/arxiv.2605.20235","title":"Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine","display_name":"Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine","publication_year":2026,"publication_date":"2026-05-16","ids":{"openalex":"https://openalex.org/W7161936803","doi":"https://doi.org/10.48550/arxiv.2605.20235"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.20235","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.20235","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.20235","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5136667070","display_name":"Wei Huang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Huang, Wei","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031625303","display_name":"Andi Han","orcid":"https://orcid.org/0000-0003-4655-655X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Han, Andi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081238418","display_name":"Mingyuan Bai","orcid":"https://orcid.org/0000-0002-2454-4219"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bai, Mingyuan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5136721589","display_name":"Huanjian Zhou","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhou, Huanjian","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5136716098","display_name":"Qixin Zhang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhang, Qixin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5136628137","display_name":"Taiji Suzuki","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Suzuki, Taiji","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5012922872","display_name":"Kenji Fukumizu","orcid":"https://orcid.org/0000-0002-3488-2625"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fukumizu, Kenji","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"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/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.362199991941452,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.362199991941452,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.2727000117301941,"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/T13702","display_name":"Machine Learning in Healthcare","score":0.08009999990463257,"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/diffusion-map","display_name":"Diffusion map","score":0.6966999769210815},{"id":"https://openalex.org/keywords/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.6934999823570251},{"id":"https://openalex.org/keywords/nonlinear-dimensionality-reduction","display_name":"Nonlinear dimensionality reduction","score":0.6705999970436096},{"id":"https://openalex.org/keywords/noise-reduction","display_name":"Noise reduction","score":0.5394999980926514},{"id":"https://openalex.org/keywords/noise","display_name":"Noise (video)","score":0.5329999923706055},{"id":"https://openalex.org/keywords/dimensionality-reduction","display_name":"Dimensionality reduction","score":0.5189999938011169},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.5026999711990356},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4975999891757965},{"id":"https://openalex.org/keywords/heuristic","display_name":"Heuristic","score":0.48570001125335693},{"id":"https://openalex.org/keywords/singularity","display_name":"Singularity","score":0.4596000015735626}],"concepts":[{"id":"https://openalex.org/C55128770","wikidata":"https://www.wikidata.org/wiki/Q5275440","display_name":"Diffusion map","level":4,"score":0.6966999769210815},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.6934999823570251},{"id":"https://openalex.org/C151876577","wikidata":"https://www.wikidata.org/wiki/Q7049464","display_name":"Nonlinear dimensionality reduction","level":3,"score":0.6705999970436096},{"id":"https://openalex.org/C163294075","wikidata":"https://www.wikidata.org/wiki/Q581861","display_name":"Noise reduction","level":2,"score":0.5394999980926514},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.5329999923706055},{"id":"https://openalex.org/C70518039","wikidata":"https://www.wikidata.org/wiki/Q16000077","display_name":"Dimensionality reduction","level":2,"score":0.5189999938011169},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.5026999711990356},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4975999891757965},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4909999966621399},{"id":"https://openalex.org/C173801870","wikidata":"https://www.wikidata.org/wiki/Q201413","display_name":"Heuristic","level":2,"score":0.48570001125335693},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.475600004196167},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.4596000015735626},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4523000121116638},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4332999885082245},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4302000105381012},{"id":"https://openalex.org/C189508267","wikidata":"https://www.wikidata.org/wiki/Q17088227","display_name":"Density estimation","level":3,"score":0.42719998955726624},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.40860000252723694},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.40700000524520874},{"id":"https://openalex.org/C30732413","wikidata":"https://www.wikidata.org/wiki/Q17092636","display_name":"Intrinsic dimension","level":3,"score":0.4034999907016754},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.367900013923645},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.35850000381469727},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3303000032901764},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.3278000056743622},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.3057999908924103},{"id":"https://openalex.org/C21080849","wikidata":"https://www.wikidata.org/wiki/Q13611879","display_name":"Data point","level":2,"score":0.29750001430511475},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.29170000553131104},{"id":"https://openalex.org/C136389625","wikidata":"https://www.wikidata.org/wiki/Q334384","display_name":"Supervised learning","level":3,"score":0.290800005197525},{"id":"https://openalex.org/C68710425","wikidata":"https://www.wikidata.org/wiki/Q5275442","display_name":"Diffusion process","level":3,"score":0.2904999852180481},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.29030001163482666},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.28610000014305115},{"id":"https://openalex.org/C184509293","wikidata":"https://www.wikidata.org/wiki/Q5136711","display_name":"Clustering high-dimensional data","level":3,"score":0.2782999873161316},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.2727000117301941},{"id":"https://openalex.org/C65660741","wikidata":"https://www.wikidata.org/wiki/Q3952743","display_name":"Score","level":2,"score":0.2703000009059906},{"id":"https://openalex.org/C29265498","wikidata":"https://www.wikidata.org/wiki/Q7047719","display_name":"Noise measurement","level":3,"score":0.2603999972343445},{"id":"https://openalex.org/C160920958","wikidata":"https://www.wikidata.org/wiki/Q7662746","display_name":"Synthetic data","level":2,"score":0.2542000114917755},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.25209999084472656},{"id":"https://openalex.org/C2779530757","wikidata":"https://www.wikidata.org/wiki/Q1207505","display_name":"Quality (philosophy)","level":2,"score":0.2506999969482422}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.20235","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.20235","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.20235","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.20235","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"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":{"Diffusion":[0,91],"models":[1],"generate":[2],"high-dimensional":[3],"data":[4,23,68],"with":[5],"remarkable":[6],"quality,":[7],"yet":[8],"how":[9],"their":[10],"training":[11,75],"efficiently":[12],"learns":[13],"the":[14,18,39,42,50,54,62,67,77,81,113,125,131,136],"score":[15,43,55,109],"function,":[16],"bypassing":[17],"curse":[19],"of":[20,41,53,61,117],"dimensionality":[21],"when":[22],"is":[24],"supported":[25],"on":[26,80,130,140],"low-dimensional":[27],"manifolds,":[28],"remains":[29],"theoretically":[30],"unexplained.":[31],"We":[32,84,122],"identify":[33],"a":[34,57,93,106],"collapse-and-refine":[35],"mechanism":[36],"driven":[37],"by":[38],"geometry":[40],"function":[44],"itself:":[45],"at":[46,71],"small":[47],"noise":[48,73],"scales,":[49,74],"diverging":[51],"singularity":[52],"drives":[56],"rapid":[58],"dimensional":[59],"collapse":[60],"induced":[63],"denoising":[64,108],"map":[65],"onto":[66],"manifold":[69,99],"projection;":[70],"moderate":[72],"refines":[76],"intrinsic":[78,132],"density":[79,102],"learned":[82],"manifold.":[83],"instantiate":[85],"this":[86],"principle":[87],"as":[88],"Score-induced":[89],"Latent":[90],"(SiLD),":[92],"two-stage":[94],"framework":[95],"in":[96,157],"which":[97],"both":[98],"learning":[100],"and":[101,145,160],"estimation":[103],"emerge":[104],"from":[105],"single":[107],"matching":[110],"objective,":[111],"replacing":[112],"heuristic":[114],"KL":[115],"regularization":[116],"VAE-based":[118,155],"latent":[119],"diffusion":[120],"models.":[121],"prove":[123],"that":[124,150],"resulting":[126],"sample":[127],"complexity":[128],"depends":[129],"dimension":[133],"rather":[134],"than":[135],"ambient":[137],"dimension.":[138],"Experiments":[139],"Stacked":[141],"MNIST,":[142],"CelebA":[143],"variants,":[144],"molecular":[146],"generation":[147,158],"benchmarks":[148],"show":[149],"SiLD":[151],"matches":[152],"or":[153],"outperforms":[154],"LDMs":[156],"quality":[159],"consistently":[161],"improves":[162],"reconstruction,":[163],"validating":[164],"our":[165],"theoretical":[166],"predictions.":[167]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-05-22T00:00:00"}
