[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85402-en":3,"doc-seo-85402-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85402,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Low-dimensional adaptation of diffusion models: Convergence in total variation","This paper studies how diffusion generative models exploit (unknown) low-dimensional structure to speed up sampling. For two standard samplers—DDIM and DDPM—iteration complexity under exact score functions is shown to scale at most on the order of k/ε (up to logarithmic factors), where ε is the total-variation precision and k is an intrinsic dimension of the target distribution. The work extends results to learned scores, proving graceful degradation under score-estimation assumptions and establishing kernel-based estimators with finite-sample adaptation, without requiring smoothness or log-concavity.","arXiv :2501 . 12982v3 [ stat .ML] 12 Jul 2026  \nLow-dimensional adaptation of diffusion models: Convergence in total variation  \nJiadong Liang∗ Zhihan Huang∗ Yuxin Chen∗  \nJanuary, 2025; Revised: July 2026  \nAbstract  \nThis paper investigates how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling. Focusing on two mainstream samplers — the denoising diffusion implicit model (DDIM) and the denoising diffusion probabilistic model (DDPM), we prove that their iteration complexities under exact score functions are at most the order of k/ε (up to log factor), where ε is the precision in total variation distance and k is some intrinsic dimension of the target distribution. We further extend these convergence guarantees to the setting in which the score functions are learned from data rather than known exactly, showing that the convergence performance degrades gracefully under suitable score estimation assumptions. We then show that these assumptions are attainable via kernelbased score estimators with finite-sample guarantees that also adapt to the low-dimensional structure. Our results apply to a broad family of target distributions without requiring smoothness or log-concavity. Our findings provide the first rigorous evidence for the adaptivity of the DDIM-type samplers to unknown low-dimensional structure, and improve over the state-of-the-art DDPM theory regarding total variation convergence.  \nContents  \n1 Introduction 2  \n1.1 Score-based generative modeling: DDPM and DDIM ....................... 3  \n1.2 Harnessing low-dimensional structure? ............................... 3  \n1.3 This paper .............................................. 4  \n2 Preliminaries 5  \n3 Main results 8  \n3.1 Theoretical guarantees under exact scores ............................. 8  \n3.2 Theoretical guarantees under noisy score estimates ........................ 10  \n3.3 Other alternatives of coefficient design? .............................. 14  \n4 Related work 15  \n5 Discussion 16  \nA Interpretation from the lens of differential equations 17  \nB Technical lemmas 18  \n∗ Department of Statistics and Data Science, the Wharton School, University of Pennsylvania; email:{jdl97,zhihanh,[yuxinc](yuxinc}@wharton.upenn.edu)[}](yuxinc}@wharton.upenn.edu)[@wharton.upenn.edu](yuxinc}@wharton.upenn.edu).  \nAccepted for presentation at the Conference on Learning Theory (COLT) 2025 .  \nC Analysis for DDIM (proof of Theorem 2) 20  \nC.1 Main steps for proving Theorem 2 ................................. 21  \nC.2 Proof of Lemma 6 .......................................... 24  \nC.3 Proof of Lemma 7 .......................................... 25  \nC.4 Proof of Lemma 8 .......................................... 27  \nC.5 Proof of Lemma 9 .......................................... 28  \nC.6 Proof of Lemma 10 .......................................... 31  \nC.7 Proof of Lemma 11 .......................................... 32  \nD Analysis for DDPM (proof of Theorem 5) 33  \nD.1 Preparation .............................................. 33  \nD.2 Main steps for proving Theorem 5 ................................. 34  \nD.3 Proof of Lemma 13 .......................................... 40  \nD.4 Proof of Lemma 14 .......................................... 40  \nD.5 Proof of Lemma 15 .......................................... 40  \nD.6 Proof of Lemma 16 .......................................... 41  \nD.7 Proof of Lemma 17 .......................................... 41  \nD.8 Proof of Lemma 18 .......................................... 42  \nE Equivalence between relation (26) and Song et al. (2020, Eq. (12)) 43  \nF Proofs about reverse-time differential equations 44  \nF.1 Generalized reverse-time differential equations ........................... 44  \nF.2 Proof of Proposition 2 ........................................ 45  \nF.3 Proof of Proposition 1 ........................................ 47  \nG Proof of the lower bound in Theorem 7 47  \nH Auxiliary lem","cbCait0BnpaGBpSV","https://ap.wps.com/l/cbCait0BnpaGBpSV","pdf",932875,1,93,"English","en",105,"# Introduction\n## Score-based generative modeling: DDPM and DDIM\n## Harnessing low-dimensional structure?\n## This paper\n# Preliminaries\n# Main results\n## Theoretical guarantees under exact scores\n## Theoretical guarantees under noisy score estimates\n## Other alternatives of coefficient design?\n# Related work\n# Discussion","[{\"question\":\"What is the main goal of this paper on diffusion models?\",\"answer\":\"The paper aims to quantify how diffusion generative models can use unknown low-dimensional structure to accelerate sampling while guaranteeing convergence in total variation distance.\"},{\"question\":\"How do DDIM and DDPM converge under exact score functions?\",\"answer\":\"Under exact score functions, the paper proves that their iteration complexities are at most on the order of k/ε (up to logarithmic factors), linking convergence to an intrinsic dimension k and target precision ε.\"},{\"question\":\"What changes when the score functions are learned from data?\",\"answer\":\"When scores are learned rather than known exactly, the convergence guarantees extend by showing performance degrades gracefully under suitable score-estimation assumptions, supported by finite-sample results for kernel-based score estimators.\"}]",1784203151,234,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"low-dimensional-adaptation-of-diffusion-models-convergence-in-total-variation","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/low-dimensional-adaptation-of-diffusion-models-convergence-in-total-variation/85402/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of this paper on diffusion models?","Question",{"text":75,"@type":76},"The paper aims to quantify how diffusion generative models can use unknown low-dimensional structure to accelerate sampling while guaranteeing convergence in total variation distance.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do DDIM and DDPM converge under exact score functions?",{"text":80,"@type":76},"Under exact score functions, the paper proves that their iteration complexities are at most on the order of k/ε (up to logarithmic factors), linking convergence to an intrinsic dimension k and target precision ε.",{"name":82,"@type":73,"acceptedAnswer":83},"What changes when the score functions are learned from data?",{"text":84,"@type":76},"When scores are learned rather than known exactly, the convergence guarantees extend by showing performance degrades gracefully under suitable score-estimation assumptions, supported by finite-sample results for kernel-based score 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