[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83776-en":3,"doc-seo-83776-105":30,"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":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},83776,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Asymptotic-Preserving A Posteriori Analysis of Diffusion and Flow-Matching Samplers","Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale to a small terminal floor σmin where the score becomes stiff and forms a boundary layer. Treating σmin as a singular perturbation, the work characterizes fixed-step schemes that are asymptotic-preserving: stable and uniformly accurate as σmin→0. Criteria are expressed as a posteriori audit residuals with σmin-uniform coefficients computable on a pretrained checkpoint, without ground-truth scores or exact trajectories. Terminal-layer rigidity identifies unique layer-exact discretizations for specific clocks, and analyzes where log(1/σmin) cost arises.","arXiv :2607 .04 1 13v 1 [ cs .LG] 5 Jul 2026  \nASYMPTOTIC-PRESERVING A POSTERIORI ANALYSIS OF DIFFUSION AND FLOW-MATCHING SAMPLERS  \nShiheng Zhang  \nUniversity of Washington [shzhang3@uw.edu](shzhang3@uw.edu)  \nABSTRACT  \nDiffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale down to a small terminal floor σmin , at which the score is stiff and the flow develops a boundary layer. We treat σmin as a singularperturbation parameter and determine which fixed-step samplers are asymptoticpreserving (AP)—stable and uniformly accurate as σmin → 0—casting the criteria as an a posteriori audit: residual functionals with σmin-uniform coefficients, computable on a pretrained checkpoint without ground-truth scores or exact trajectories. On the terminal layer, Euler in the σ-clock, the deterministic DDIM update, is the unique layer-exact discretization up to affine reparameterization, with rectified flow its flow-matching counterpart; the λ-clock is stable only for steps h ≤ h⋆ = 1 + W(1/e), and the uniform-σ2 heat clock stalls a σmin-independent distance from the data. On two solvable models (rank-deficient Gaussian, symmetric two-point mixture), deterministic samplers remain first-order uniformly accurate with no log(1/σmin) factor, even across a symmetric posterior-switching interface whose distributional budget is a universal constant; the logarithm is charged entirely to the It term of stochastic samplers, whose path-KL scales as Λ2 /N against the ODE’s O(Λ2 /N 2 ) budget, Λ = log(σmax /σmin) . On the EDMCIFAR-10 checkpoint, spectra measured once predict held-out residual budgets across step count, schedule, and noise level against pre-specified gates with no per-configuration refitting, and calibrate the It coefficient at M1 = 1 .00 ± 0.01.  \nThe clock decides stability; the noise, not the geometry, charges the logarithm.  \n1 INTRODUCTION  \nDiffusion and flow-matching models sample by integrating a learned field from noise down to data, stopping at a small but nonzero terminal noise scale: EDM (Karras et al., 2022) uses σmin = 0 .002. The stop is structural: as σmin → 0 the score becomes stiff, its Lipschitz constant growing like σ−m2in (2 .5 × 105 at the EDM floor) . The floor is therefore a genuine singular parameter of the sampling problem, yet it is typically tuned by hand and excluded from the analysis, which either assumes a time-uniformly Lipschitz score or studies an early-stopped process whose iteration count grows like log(1/σmin) (Chen et al., 2022; Benton et al., 2024; De Bortoli, 2022) .  \nThis paper analyzes the floor directly. Near the data the probability-flow ODE develops a boundary layer, the structure that asymptotic-preserving (AP) numerics was built to handle (Roos et al., 2008; Jin, 2022), and the appropriate criterion for a fixed-step sampler is stability and accuracy uniform in σmin. Written in the denoiser D and its normalized residual η, the samplers in use (deterministic, stochastic, and flow-matching) are a single flow traversed under different choices of clock, gauge, and noise level (§2), so the AP criterion reduces to a classification of traversal choices. Under this classification, empirical design wisdom on schedules, parameterizations, and stochasticity becomes a set of rigidity theorems, packaged as an a posteriori audit whose functionals are computable on apretrained checkpoint (§3) .  \nStability on the pure terminal layer is characterized first (§4) . A frozen-field Euler step is layerexact if and only if its clock is affine in σ, which makes the σ-clock step, precisely DDIM (Songet al., 2020a), the unique fitted operator up to affine reparameterization, with rectified flow its flowmatching counterpart; the λ-clock is stable only below the Lambert-W threshold h⋆ = 1+W(1/e);  \nand the uniform-σ2 heat clock cannot resolve the layer at all, stalling a σmin-independent distance from the data.  \nAccuracy beyond the linear regime follows. On the rank-defici","cbCaim9DTCh6Udpm","https://ap.wps.com/l/cbCaim9DTCh6Udpm","pdf",639513,3,1,21,"English","en",105,"# Introduction\n# Preliminaries: One Flow, Many Samplers\n# Asymptotic-Preserving Terminal-Layer Analysis\n# Accuracy Across Interfaces and Logarithmic Costs\n# Predictive Calibration on EDM CIFAR-10","[{\"question\":\"What does “asymptotic-preserving” mean for diffusion and flow-matching samplers?\",\"answer\":\"It means a fixed-step sampler remains stable and uniformly accurate as the terminal noise floor σmin approaches 0. The analysis treats σmin as a singular perturbation and checks uniform behavior using audit functionals.\"},{\"question\":\"How are asymptotic-preserving criteria evaluated without ground-truth scores or exact trajectories?\",\"answer\":\"The paper reformulates the AP requirement as an a posteriori audit using residual functionals with σmin-uniform coefficients. These residuals are computable from a pretrained checkpoint.\"},{\"question\":\"Where does the log(1/σmin) cost arise according to the paper?\",\"answer\":\"The logarithmic cost is charged entirely to the It term of stochastic samplers. Deterministic samplers avoid this logarithmic factor on the studied models, including across a symmetric posterior-switching interface.\"}]",1784190345,53,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"asymptotic-preserving-a-posteriori-analysis-of-diffusion-and-flow-matching-samplers","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/asymptotic-preserving-a-posteriori-analysis-of-diffusion-and-flow-matching-samplers/83776/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","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 does “asymptotic-preserving” mean for diffusion and flow-matching samplers?","Question",{"text":75,"@type":76},"It means a fixed-step sampler remains stable and uniformly accurate as the terminal noise floor σmin approaches 0. The analysis treats σmin as a singular perturbation and checks uniform behavior using audit functionals.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are asymptotic-preserving criteria evaluated without ground-truth scores or exact trajectories?",{"text":80,"@type":76},"The paper reformulates the AP requirement as an a posteriori audit using residual functionals with σmin-uniform coefficients. These residuals are computable from a pretrained checkpoint.",{"name":82,"@type":73,"acceptedAnswer":83},"Where does the log(1/σmin) cost arise according to the paper?",{"text":84,"@type":76},"The logarithmic cost is charged entirely to the It term of stochastic samplers. Deterministic samplers avoid this logarithmic factor on the studied models, including across a symmetric posterior-switching interface.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"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":106,"slug":138},19,"General","general"]