{"id":"https://openalex.org/W7165384385","doi":"https://doi.org/10.48550/arxiv.2606.19894","title":"Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures","display_name":"Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures","publication_year":2026,"publication_date":"2026-06-18","ids":{"openalex":"https://openalex.org/W7165384385","doi":"https://doi.org/10.48550/arxiv.2606.19894"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.19894","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.19894","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":"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"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.19894","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120425793","display_name":"Xinhe Mu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mu, Xinhe","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111714864","display_name":"Z.K. Shang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shang, Zaijiu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017842177","display_name":"Zheng Zhou","orcid":"https://orcid.org/0000-0002-9616-4215"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhou, Zhaoqi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138985812","display_name":"Chuan Zhou","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhou, Chuan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138995979","display_name":"Qi Meng","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Meng, Qi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5129842200","display_name":"Guiying Yan","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yan, Guiying","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5102998706","display_name":"Zhiming Ma","orcid":"https://orcid.org/0000-0002-7560-9043"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ma, Zhiming","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":0.8510000109672546,"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":0.8510000109672546,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.039000000804662704,"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/T12303","display_name":"Tensor decomposition and applications","score":0.022099999710917473,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"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/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.6355000138282776},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.6344000101089478},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.3822000026702881},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.3750999867916107},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.35910001397132874},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.3537999987602234},{"id":"https://openalex.org/keywords/approximation-theory","display_name":"Approximation theory","score":0.32120001316070557},{"id":"https://openalex.org/keywords/compact-space","display_name":"Compact space","score":0.31619998812675476}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7099999785423279},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6355000138282776},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6344000101089478},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4115000069141388},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3822000026702881},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3750999867916107},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.35910001397132874},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.3537999987602234},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.32600000500679016},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.32120001316070557},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.31619998812675476},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.313400000333786},{"id":"https://openalex.org/C167966045","wikidata":"https://www.wikidata.org/wiki/Q5532625","display_name":"Generative model","level":3,"score":0.305400013923645},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.2994999885559082},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2906999886035919},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.28540000319480896},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.28220000863075256},{"id":"https://openalex.org/C27142425","wikidata":"https://www.wikidata.org/wiki/Q1139334","display_name":"Almost everywhere","level":2,"score":0.27709999680519104},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26989999413490295},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.2694999873638153},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.2662999927997589},{"id":"https://openalex.org/C91873725","wikidata":"https://www.wikidata.org/wiki/Q3445816","display_name":"Function approximation","level":3,"score":0.26499998569488525},{"id":"https://openalex.org/C193290586","wikidata":"https://www.wikidata.org/wiki/Q17022030","display_name":"Heavy traffic approximation","level":2,"score":0.2637999951839447},{"id":"https://openalex.org/C68710425","wikidata":"https://www.wikidata.org/wiki/Q5275442","display_name":"Diffusion process","level":3,"score":0.26179999113082886},{"id":"https://openalex.org/C186603090","wikidata":"https://www.wikidata.org/wiki/Q55505926","display_name":"Distribution function","level":2,"score":0.2540000081062317}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.19894","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.19894","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.19894","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.19894","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":"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"},"sustainable_development_goals":[{"score":0.4032247066497803,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0],"remarkable":[1],"success":[2],"of":[3,71,106],"score-based":[4],"diffusion":[5,118,128],"models":[6,129],"has":[7],"spurred":[8],"significant":[9],"efforts":[10],"to":[11,49,132],"establish":[12],"their":[13,138],"theoretical":[14],"foundations.":[15],"However,":[16],"existing":[17,111],"complexity":[18,95],"bounds":[19],"for":[20,63,120],"score":[21,58,85],"approximation":[22,59],"rely":[23],"heavily":[24],"on":[25,67,113],"restrictive":[26],"assumptions":[27],"like":[28],"Lipschitz":[29],"continuous":[30],"densities":[31],"or":[32],"smooth":[33],"manifold":[34],"supports,":[35],"which":[36],"are":[37],"routinely":[38],"violated":[39],"by":[40],"the":[41,84,103,116],"singularities,":[42],"sharp":[43],"boundaries,":[44],"and":[45],"disjoint":[46],"clusters":[47],"inherent":[48],"real-world":[50,141],"perceptual":[51],"data.":[52],"This":[53],"work":[54,125],"establishes":[55],"a":[56,77,91],"universal":[57],"theorem":[60],"that":[61,83,127],"works":[62],"any":[64,68],"distribution":[65],"supported":[66],"compact":[69,122],"set":[70],"upper":[72],"Minkowski":[73],"dimension":[74],"$d$.":[75],"Using":[76],"novel":[78],"discrete-mixture":[79],"formulation,":[80],"we":[81],"prove":[82],"function":[86],"can":[87],"be":[88],"approximated":[89],"with":[90,99,110],"ReLU":[92],"network":[93],"whose":[94],"grows":[96],"exponentially":[97],"only":[98],"$d$,":[100],"thus":[101],"breaking":[102],"exponential":[104],"curse":[105],"ambient":[107],"dimensionality.":[108],"Combined":[109],"theories":[112],"accurately":[114],"solving":[115],"backward":[117],"SDE":[119],"arbitrary":[121],"distributions,":[123],"our":[124],"shows":[126],"readily":[130],"adapt":[131],"irregular,":[133],"non-smooth":[134],"data":[135],"structures,":[136],"explaining":[137],"competence":[139],"in":[140],"generative":[142],"tasks.":[143]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-20T00:00:00"}
