[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84259-en":3,"doc-seo-84259-105":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},84259,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance","Classifier-free guidance (CFG) strengthens class conditioning in diffusion and flow-matching samplers but, at high guidance, oversaturates and destabilizes, motivating step inflation or limited-interval schedules. The work analyzes CFG via asymptotic-preserving numerical analysis, showing that guidance re-stiffens discriminative subspaces to an anomalous terminal exponent 1+w, breaking DDIM’s fitted-operator protection. A one-coefficient, terminal-exact repair modifies only the guidance-direction term, removing σmin-divergent blow-up and improving stability and residual behavior on tested models.","arXiv :2607 .07665v 1 [ cs .LG] 8 Jul 2026  \nGUIDANCE BREAKS THE FITTED OPERATOR: A TERMINAL-FITTED REPAIR FOR CLASSIFIER-FREE GUIDANCE  \nShiheng Zhang  \nUniversity of Washington [shzhang3@uw.edu](shzhang3@uw.edu)  \nABSTRACT  \nClassifier-free guidance (CFG) is the standard way to strengthen classconditioning in diffusion and flow-matching samplers, yet at large guidance it oversaturates and destabilizes—symptoms practitioners suppress with more steps or limited-interval schedules. We analyze CFG through an asymptotic-preserving, numerical-analysis lens. Building on a recent result [12] that the deterministic DDIM step is the unique fitted operator for the unguided terminal layer—exact on the final, small-σ stretch of sampling—we show that guidance re-stiffens exactly the discriminative subspace to an anomalous exponent 1 + w (guided coordinates contract like σ1+w rather than σ) . DDIM is therefore no longer fitted there, and on coarse meshes its guided residual diverges as σmin → 0. We prove a guided clock barrier with three ordered step-size thresholds, and read one-step oversaturation as its endpoint—a solver artifact on the calibration model rather than the continuous guided law. The same analysis yields a one-coefficient, zero-extra-NFE repair:  \nreplace CFG’s w(r − 1) by r 1+w − r on the guidance direction. This coefficient is the unique spectrum-free terminal-exact one, and it preserves the sign of every analyzed coordinate. On the calibration model’s discriminative crossover it removes CFG’s σmin-divergent blow-up and is first-order accurate against the exact guided flow as σmin → 0—asymptotic-preserving in σmin , though not uniform in w. On learned CIFAR-10 checkpoints—and, as a cross-domain smoke test, on Stable Diffusion 1.5 DDIM—it acts as a high-guidance stabilizer at no extra cost rather than a universal quality knob: it cuts residual amplification and saturation and gives 9/9 point-FID wins over CFG on the tested grid, while in the hard-cell blocks its classifier-proxy target accuracy stays close to CFG and terminal guidance shutdown loses much more. We report the limits alongside: it is not a universal image-quality win (KID can favor CFG; an interval can win FID in some cells), and against a dense vanilla-CFG reference it is not a uniformly better integrator of that field.  \n1 INTRODUCTION  \nDiffusion and flow-matching samplers integrate a probability-flow ODE whose velocity field stiffensas the noise scale σ → σmin: on a coordinate normal to the data manifold the exact flow contracts at exponent one, by σn+1/σn per step. A recent fitted-operator analysis of unguided samplers [12] shows that this terminal layer admits a unique fitted (layer-exact) one-step operator: a frozen-field Euler step—Euler with the velocity held at its start-of-step value—reproduces that exact contraction if and only if its integration variable is affine in σ . This singles out the σ-clock step, algebraically identical to deterministic DDIM [11], and makes it asymptotic-preserving (AP) on the exactly solvable calibration models studied there [12]—uniformly accurate as σmin → 0. This paper asks one question of that lens: what survives of DDIM’s fitted-operator protection once we turn on guidance?  \nGuidance re-stiffens the discriminative subspace. Classifier-free guidance (CFG) [7] runs the sampler on an extrapolated denoiser Dw = (1 + w)Dc − wDu , that is, on a second velocity field ηw = ηc + w(ηc − ηu) (the η’s are noise-prediction fields; Section 2) with guidance scale g =  \nµw (σ)  \nλ = −log σ (terminal →)  \nFigure 1: The guided terminal exponent µw (σ) by layer type on the commuting model (schematic; noise decreases to the right) . Guidance leaves the shared-normal exponent at one and freezes tangential directions, but re-stiffens the discriminative directions to 1 + w—the subspace on which DDIM+CFG exits the class of fitted operators. Here a, b are the class and marginal variances along the direction (Section 2) .  \n1 + w. This is ","cbCail1IS9ds3NbX","https://ap.wps.com/l/cbCail1IS9ds3NbX","pdf",389937,4,1,14,"English","en",105,"# Abstract\n# Introduction\n## Guidance re-stiffens discriminative directions\n## Two orthogonal sources of guidance pathologies\n## One-coefficient terminal repair","[{\"question\":\"Why does classifier-free guidance become unstable at large guidance scales?\",\"answer\":\"At large guidance, CFG oversaturates and destabilizes the sampling dynamics, manifesting as terminal oversaturation and norm inflation. The paper links part of this behavior to solver discretization breaking the fitted-operator property.\"},{\"question\":\"What does the paper mean by DDIM’s “fitted operator” protection?\",\"answer\":\"For unguided sampling, the deterministic DDIM step is identified as the unique fitted one-step operator for the terminal layer, preserving exact contraction under the right affine σ-clock condition. This protection holds asymptotically-preserving on solvable calibration models.\"},{\"question\":\"What is the proposed terminal-fitted repair and what problem does it address?\",\"answer\":\"The repair replaces the CFG guidance-direction coefficient w(r−1) with a modified one-coefficient term that corrects the anomalous terminal exponent 1+w. It removes the σmin-divergent residual blow-up on coarse meshes while preserving sign structure and improving stability without extra function evaluations per step.\"}]",1784194424,35,{"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},"guidance-breaks-the-fitted-operator-a-terminal-fitted-repair-for-classifier-free-guidance","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/guidance-breaks-the-fitted-operator-a-terminal-fitted-repair-for-classifier-free-guidance/84259/",{"url":52,"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","2026-07-27","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},"Why does classifier-free guidance become unstable at large guidance scales?","Question",{"text":75,"@type":76},"At large guidance, CFG oversaturates and destabilizes the sampling dynamics, manifesting as terminal oversaturation and norm inflation. The paper links part of this behavior to solver discretization breaking the fitted-operator property.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the paper mean by DDIM’s “fitted operator” protection?",{"text":80,"@type":76},"For unguided sampling, the deterministic DDIM step is identified as the unique fitted one-step operator for the terminal layer, preserving exact contraction under the right affine σ-clock condition. This protection holds asymptotically-preserving on solvable calibration models.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the proposed terminal-fitted repair and what problem does it address?",{"text":84,"@type":76},"The repair replaces the CFG guidance-direction coefficient w(r−1) with a modified one-coefficient term that corrects the anomalous terminal exponent 1+w. It removes the σmin-divergent residual blow-up on coarse meshes while preserving sign structure and improving stability without extra function evaluations per step.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"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":20,"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"]