[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85410-en":3,"doc-seo-85410-105":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},85410,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Riemannian Denoising Diffusion Probabilistic Models","Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) learn probability distributions on Euclidean submanifolds defined as level sets of smooth functions. Unlike manifold generative approaches that require geometric artifacts such as geodesics or Laplace–Beltrami eigenfunctions, RDDPMs use a projection-based scheme that only needs function values and first-order derivatives of the defining function. A continuous-time theoretical analysis clarifies the relationship to score-based generative models on manifolds, and demonstrations cover implicit complex manifolds with applications in statistical mechanics and molecular dynamics.","arXiv :2505 .04338v 3 [ cs .LG] 12 Jul 2026  \nRIEMANNIAN DENOISING DIFFUSION PROBABILISTIC MODELS  \nZICHEN LIU∗ , WEI ZHANG†, CHRISTOF SCH¨UTTE‡, AND TIEJUN LI§  \nAbstract. We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications. Existing methods for generative modeling on manifolds rely on substantial geometric information such as geodesic curves or eigenfunctions of the LaplaceBeltrami operator and, as a result, they are limited to manifolds where such information is available. In contrast, our method, built on a projection scheme, can be applied to more general manifolds, as it only requires being able to evaluate the value and the first order derivatives of the function that defines the submanifold. We provide a theoretical analysis of our method in the continuous-time limit, which elucidates the connection between our RDDPMs and score-based generative models on manifolds. The capability of our method is demonstrated on distributions over complex manifolds implicitly represented as level sets, with applications in statistical mechanics and molecular dynamics. Our code is available at: [https://github.com/ZichenLiu1999/RiemannianDDPM](https://github.com/ZichenLiu1999/RiemannianDDPM).  \nKeywords. generative modeling; diffusion probabilistic model; submanifold; projection scheme. AMS subject classifications. 58J65; 60J05; 60J60 .  \n1. Introduction  \nGenerative models have achieved remarkable success in learning data distributions across various fields. Among them, diffusion models stand out for their superior ability to generate high-quality samples that resemble the data distributions. Two prominent frameworks are Denoising Diffusion Probabilistic Models (DDPMs; [10]), which minimize a variational bound in variational inference, and Score-based Generative Models (SGMs;[30, 31]), which learn the score function [14] . Both frameworks have demonstrated significant success in learning data distributions in Euclidean spaces.  \nIn many scientific domains, data distributions are constrained to Riemannian manifolds rather than Euclidean spaces. For example, spheres are used in geographical sciences [23], while SE(3) and SO(3) are considered in studying protein structures [34] and robotic movements [27] . Other manifolds include SU(3) in lattice quantum chromodynamics [22], triangular meshes in 3D computer graphics [11], and the Poincar´e disk in cell development research [17] . These applications highlight the need for developing generative models that can handle distributions on manifolds.  \nSeveral recent works have extended diffusion-based models to Riemannian manifolds. The Riemannian Score-based Generative Model (RSGM; [7]) extends SGM to Riemannian manifolds by incorporating the heat kernel into the denoising score-matching objective. Since heat kernels on manifolds are generally intractable, RSGM approximates them using eigenfunction expansion or Varadhan’s approximation. Furthermore, RSGM leverages the exponential map to enable trajectory sampling on manifolds. The Riemannian Diffusion Model [13] adopts a variational diffusion model framework on Riemannian manifolds. It considers submanifolds embedded in an Euclidean space and utilizes a  \n∗ Equal contribution. Center for Data Science, Peking University, Beijing 100871, P.R. China ([zcliu@stu.pku.edu.cn](zcliu@stu.pku.edu.cn)).  \n†Equal contribution. Zuse Institute Berlin, Takustrasse 7, Berlin 14195, Germany (wei.zhang@fu[berlin.de](berlin.de)).  \n‡Institute of Mathematics, Freie Universit¨at Berlin and Zuse Institute Berlin, Takustrasse 7, Berlin 14195, Germany ([schuette@zib.de](schuette@zib.de)).  \n§ Corresponding author. LMAM and School of Mathematical Sciences, Center for Machine Learning Research and Center for Data Science, Peking University, Beijing 100871, P.R. China ([tieli@pku.edu.cn](tieli@pku.ed","cbCais9Y8a1nx3zs","https://ap.wps.com/l/cbCais9Y8a1nx3zs","pdf",1366402,1,29,"English","en",105,"# Introduction\n## Diffusion and score-based generative frameworks\n## Generative modeling on manifolds\n## Related diffusion-based manifold methods\n## Motivation for geometric-free approaches\n# Riemannian Denoising Diffusion Probabilistic Models (RDDPMs)\n## Projection scheme and Markov chains on submanifolds\n## Scope via level-set submanifolds\n## Advantages over existing manifold methods","[{\"question\":\"What problem do RDDPMs address in generative modeling?\",\"answer\":\"RDDPMs address learning distributions constrained to Riemannian submanifolds rather than unconstrained Euclidean spaces.\"},{\"question\":\"What key requirement distinguishes RDDPMs from prior manifold diffusion methods?\",\"answer\":\"RDDPMs avoid needing geodesic curves or heat kernels; they only require evaluating the defining function and its first-order derivatives.\"},{\"question\":\"How do RDDPMs relate to score-based generative models on manifolds?\",\"answer\":\"A continuous-time theoretical analysis explains the connection between RDDPMs and score-based generative models on manifolds.\"}]",1784203196,73,{"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},"riemannian-denoising-diffusion-probabilistic-models","",{"@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/riemannian-denoising-diffusion-probabilistic-models/85410/",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 problem do RDDPMs address in generative modeling?","Question",{"text":75,"@type":76},"RDDPMs address learning distributions constrained to Riemannian submanifolds rather than unconstrained Euclidean spaces.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What key requirement distinguishes RDDPMs from prior manifold diffusion methods?",{"text":80,"@type":76},"RDDPMs avoid needing geodesic curves or heat kernels; they only require evaluating the defining function and its first-order derivatives.",{"name":82,"@type":73,"acceptedAnswer":83},"How do RDDPMs relate to score-based generative models on manifolds?",{"text":84,"@type":76},"A continuous-time theoretical analysis explains the connection between RDDPMs and score-based generative models on manifolds.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]