[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83423-en":3,"doc-seo-83423-105":30,"detail-sidebar-cat-0-en-105":92},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83423,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Score Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling","Score matching minimizes average error under forward marginals, yet a discretized reverse-time sampler evaluates the learned score along its own sampling trajectory. The work shows that small forward-marginal score error does not ensure numerical stability. It constructs smooth score fields with arbitrarily small forward-marginal L2 error and controlled reverse-time behavior in path space, while Euler–Maruyama discretizations exhibit probability-level convergence but divergence of every positive moment and all Wasserstein distances Wp for p≥1. It extends the mechanism to fixed architectures and high dimensions, and proves a stability condition via projecting learned denoisers onto a bounded closed convex set containing the data support.","arXiv :2607 .08757v1 [ stat .ML] 9 Jul 2026  \nScore Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling  \nYiwei Zhou∗  \nAbstract  \nScore matching controls average error under the forward marginals, but a discretized reverse-time sampler evaluates the learned score along its own trajectory. We show that small forward-marginal error does not guarantee numerical stability. We construct a single smooth score field with arbitrarily small forward-marginal L2 error. The learned reverse-time process is nonexplosive, has moments of every order, and can be arbitrarily close to the exact reverse-time process in path-space total variation. Yet its Euler–Maruyama discretizations converge in probability while every positive moment diverges. Thus weak convergence can hold even though every Wasserstein distance Wp , p ≥ 1, diverges.  \nThe same failure can occur within one fixed finite neural architecture. We construct a family of bounded, globally Lipschitz denoisers for which both the forward-marginal error and the path-space total variation distance tend to zero, while their Euler–Maruyama endpoints diverge in every Wp. For compactly supported data, we also give a simple positive result. Projecting the learned denoiser onto a known bounded closed convex set containing the support preserves pointwise accuracy, gives grid-uniform moment bounds, and yields Wasserstein convergence under mild local regularity. Experiments with a small fixed DiT-style network show large growth along rare numerical trajectories and its suppression by denoiser projection, while overall trajectory errors remain small.  \nKeywords: score-based diffusion models; reverse-time sampling; numerical stability; Euler– Maruyama discretization; Wasserstein convergence; denoiser projection.  \nMSC 2020: Primary 65C30; Secondary 60H35, 60J60, 68T07 .  \n1 Introduction  \nIn score-based diffusion models, the exact reverse-time process uses the true score st in its drift. ReplEulear–cingMarsutybyamaa learnediscretdizaepdproximation sfor sampling bt gives the learned reverse-time process, which is then  \nA standard score-learning error has the form  \nEscore = Zδ T EXt ∼pt ∥sbt (Xt) − st (Xt)∥2 dt. (1)  \n∗ Email: [yiwei.zhou@utexas.edu](yiwei.zhou@utexas.edu)  \nWe call it on-path score error because it is measured under the forward marginals. A natural question is:  \n“Suppose both the exact and learned reverse-time processes are stable, and the learned process can be made arbitrarily close to the exact one in path-space total variation. Is an arbitrarily small on-path score error then enough to guarantee stability of the discretization?”  \nPerhaps surprisingly, the answer is no. We construct a single smooth score field with arbitrarily small on-path error. The learned reverse-time process is nonexplosive, has moments of every order, and can be arbitrarily close to the exact reverse-time process in path-space total variation. Yet its Euler–Maruyama endpoints converge in probability while every positive moment diverges. Weak convergence holds while the Wasserstein distance Wp diverges for every p ≥ 1. The same mechanism extends to arbitrary dimension and to probability-flow ODEs.  \nFurthermore, this error alone does not rank samplers by numerical stability. Two smooth score fields can have the same arbitrarily small on-path error while one Euler–Maruyama discretization has grid-uniform moments of every order and the other loses every positive moment. The same obstruction persists even within a fixed finite GELU architecture: bounded, globally Lipschitz denoisers can have arbitrarily small on-path error and path-space total variation distance while their Euler–Maruyama endpoints diverge in every Wp.  \nThe mechanism is simple. We place an inward superlinear perturbation in a remote region carrying very little mass under the forward marginals. The learned reverse-time process rarely reaches this region, so its path law remains close to that of the","cbCairfSrd73WDSs","https://ap.wps.com/l/cbCairfSrd73WDSs","pdf",506835,5,1,27,"English","en",105,"# Introduction\n## On-path score error and the stability question\n## Construction of counterexamples for Euler–Maruyama\n## Mechanism: rare trajectories in low-mass regions\n## Positive result via denoiser projection and Wasserstein convergence\n## Related work","[{\"question\":\"Why doesn’t small on-path (forward-marginal) score error guarantee numerical stability?\",\"answer\":\"Because Euler–Maruyama discretization follows rare numerical trajectories that can enter regions poorly controlled by the forward-marginal score error. The paper constructs examples where forward-marginal error is arbitrarily small and the continuous-time process stays stable, yet discretized endpoints lose all positive moments and Wasserstein stability.\"},{\"question\":\"What counterexample behavior is demonstrated for Euler–Maruyama discretizations?\",\"answer\":\"Weak convergence can still hold, while every positive moment diverges. As a result, all Wasserstein distances Wp for p≥1 diverge even though Euler–Maruyama endpoints converge in probability.\"},{\"question\":\"How does denoiser projection recover a stability guarantee?\",\"answer\":\"When data support lies in a known bounded closed convex set, projecting the learned denoiser onto that set preserves pointwise accuracy and yields grid-uniform moment bounds. Under mild local regularity, weak numerical convergence then implies Wasserstein convergence; for image-box data this corresponds to clipping predicted clean samples.\"}]",1784187511,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"score-accuracy-along-the-forward-diffusion-does-not-certify-numerical-stability-in-diffusion-sampling","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/score-accuracy-along-the-forward-diffusion-does-not-certify-numerical-stability-in-diffusion-sampling/83423/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why doesn’t small on-path (forward-marginal) score error guarantee numerical stability?","Question",{"text":76,"@type":77},"Because Euler–Maruyama discretization follows rare numerical trajectories that can enter regions poorly controlled by the forward-marginal score error. The paper constructs examples where forward-marginal error is arbitrarily small and the continuous-time process stays stable, yet discretized endpoints lose all positive moments and Wasserstein stability.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What counterexample behavior is demonstrated for Euler–Maruyama discretizations?",{"text":81,"@type":77},"Weak convergence can still hold, while every positive moment diverges. As a result, all Wasserstein distances Wp for p≥1 diverge even though Euler–Maruyama endpoints converge in probability.",{"name":83,"@type":74,"acceptedAnswer":84},"How does denoiser projection recover a stability guarantee?",{"text":85,"@type":77},"When data support lies in a known bounded closed convex set, projecting the learned denoiser onto that set preserves pointwise accuracy and yields grid-uniform moment bounds. Under mild local regularity, weak numerical convergence then implies Wasserstein convergence; for image-box data this corresponds to clipping predicted clean samples.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]