Probability-Conserving Flow Guidance
Junha Hyung∗ KAIST
Amirhossein Dadashzadeh University of Bristol
Jaegul Choo KAIST
CFG
Majid Mirmehdi University of Bristol
Ours (AdaMaG)
arXiv:2605.20079v1 [cs.CV] 19 May 2026
Parsa Esmati∗ University of Bristol
Figure 1: AdaMaG vs. CFG. CFG (top) vs AdaMaG (bottom), same prompts and seeds. CFG shows saturation and hallucinated artefacts that grow with guidance scale; AdaMaG restores probability conservation along trajectories, yielding clean, on-manifold generations with no inference overhead.
Abstract Diffusion and flow-based generative models dominate visual synthesis, with guidance aligning samples to user input and improving perceptual quality. However, Classifier-Free Guidance (CFG) and extrapolation-based methods are heuristic linear combinations of velocities/scores that ignore the generative manifold geometry, breaking probability conservation and driving samples off the learned manifold under strong guidance. We analyse guidance through the continuity equation and show its effect decomposes into a divergence term and a score-parallel term defined invariantly across parameterisations. We prove the divergence term blows up structurally as sampling approaches the data manifold, motivating a time-dependent schedule alongside score-parallel attenuation. The resulting plug-and-play rule, Adaptive Manifold Guidance (AdaMaG), bounds both terms at no additional inference cost. Finally, we show that most empirical heuristics for reducing saturation or improving generation quality correspond directly to the two terms in our decomposition. Across image generation benchmarks, AdaMaG improves realism, reduces hallucinations, and induces controlled desaturation in high-guidance regimes. ∗ Equal contribution.
Preprint.
1
Introduction
We consider the problem of guided sampling from high-dimensional data distributions with diffusion and flow-matching models, where the 𝑀!"$ 𝑀!"$ generative process is realised by integrating 𝑔! 𝑔! learned ordinary differential equations (ODEs) 𝑔!% or stochastic differential equations (SDEs) de- 𝑀!"# 𝑀!"# 𝑣$! 𝑣! 𝑣! 𝑣$! fined by a time-dependent velocity (or score) field [Song et al., 2021, Lipman et al., 2022, 𝑀! Rombach et al., 2022, Albergo and Vanden- 𝑀! (a) CFG Guidance (b) AdaMaG (Ours) Eijnden, 2023, Esser et al., 2024, Wan et al., 2025]. Figure 2: Conceptual overview. Unlike CFG (a), Classifier-free guidance (CFG) [Ho and Sali- which extrapolates from the unconditional field mans, 2021] and its variants have become the and drifts off the manifold Mt , AdaMaG (b) atde facto standard for steering samples toward tenuates the score-parallel component of guidance the high support conditional regions by linearly and applies a time-dependent schedule, keeping extrapolating the conditional prediction away trajectories on-manifold at no extra cost. from the unconditional using a guidance scale [Ho and Salimans, 2021, Chung et al., 2024]. Such extrapolation has been shown to improve perceptual quality, producing sharper and more detailed samples with stronger conditional alignment. Yet, applying CFG, and related autoguidance methods [Hong et al., 2023, Karras et al., 2024, Hong, 2024, Hyung et al., 2025, Ifriqi et al., 2025], violates the learnt probability conservation during sampling, leading to artifacts and oversaturation that grow more severe as the guidance scale increases. We first revisit guidance from the perspective of conservation of probability and the model’s learned generative manifold. In the ideal continuous-time formulation, the unconditional and conditional velocity fields are trained to satisfy a continuity equation that approximately preserves probability mass along trajectories, but the linear combinations used in current guidance mechanisms break this structure at sampling time, introducing unconstrained divergences and off-manifold drifts. We formalise the contribution of a guidance term to the continuity equation and show that it decomposes into two components: (i) a divergence term that locally creates or removes probability mass, and (ii) a score-parallel flux that transports mass across iso-density surfaces. When these contributions cancel, guidance is conservative and offers little fine-grained control; when their sum grows unchecked, trajectories depart the model’s manifold. Motivated by this analysis and to keep the model bounded to the conservation of probability, we introduce Adaptive Manifold Guidance (AdaMaG), a plug-and-play modification to the sampler that treats the conditional velocity as a primary, approximately probability-preserving flow, and restricts guidance to act predominantly along its orthogonal directions. At each step, we decompose the guidance term into components parallel and orthogonal to the primary score direction, preserve the orthogonal updates, and modulate tangential guided corrections. This construction strengthens conditional alignment while regularising the geometry of the update and mitigating drift away from the model’s learned manifold without additional function evaluations. We further prove that this late-time divergence spike is structurally inevitable, blowing up as sampling approaches the data manifold at a rate determined by the conditional/unconditional posterior covariance gap. To characterise the divergence empirically along the generative path, we measure it for the primary manifold velocities and the residual guidance field. We observe that, across most denoising steps, the primary velocity dominates by orders of magnitude, with guidance divergence only becoming comparable in the final iterations as samples approach the data manifold. Consequently, we introduce a time-dependent guidance schedule that preserves strong early conditioning while attenuating guidance near the endpoint to suppress this late-time divergence spike. In summary, our contributions are: (i) Conservation-based view of guidance. We analyse guidance as a source term in the continuity equation and identify two components, a divergence and a scoreparallel flux, that violate probability conservation, and show that prior manifold-preserving methods fall under this framework. (ii) Adaptive Manifold Guidance. A plug-and-play sampler that attenuates the score-parallel term and schedules guidance to suppress late-stage divergence, with no extra function evaluations. (iii) Empirical evaluation. On SD3, SD3.5, and Flux, we conduct 2
comprehensive quantitative and qualitative evaluation and show AdaMaG consistently improves FID, IS, and saturation across guidance scales, with ablations isolating each component’s contribution.
2
Related works
We review guidance methods in diffusion and flow models, and then consider recent manifold-aware sampling rules most relevant to our approach. Guidance in diffusion and flows. Diffusion and flow-matching models generate samples by integrating a learned time-dependent score or velocity field from noise to data [Sohl-Dickstein et al., 2015, Ho et al., 2020, Song et al., 2021, Lipman et al., 2022, Albergo and Vanden-Eijnden, 2023]. To steer these models toward desired conditions, guidance terms were added at sampling time. Early work used classifier-based guidance, augmenting the score with the gradient of a separately trained classifier [Dhariwal and Nichol, 2021, Nichol et al., 2021]. CFG [Ho and Salimans, 2021] replaced the external classifier with conditional and unconditional predictions and has become the de facto standard in text-to-image and text-to-video frameworks [Ho and Salimans, 2021, Rombach et al., 2022, Balaji et al., 2022, Esser et al., 2024, Wan et al., 2025]. Beyond classifier-based and CFG-style methods, guidance can also be derived from CLIP, energy, or reward models, as well as training-free surrogates that approximate CFG-like behaviour without dedicated conditional/unconditional training [Nichol et al., 2021, Yu et al., 2023, Song et al., 2023, Lu et al., 2023, Sadat et al., 2025, Jang et al., 2025]. However, CFG and its variants induce large mismatches between the unconditional and conditional fields, increasing the curvature of the sampling trajectory [Chung et al., 2024, Hyung et al., 2025] and pushing samples off the model’s learned manifold, which in turn reduces diversity and yields distorted or oversaturated images [Sadat et al., 2024]. Manifold preservation. Most recently, direct manifold preservation during sampling has drawn attention. Manifold-Preserving Gradient Descent (MPGD) [He et al., 2024] for instance enforces manifold consistency by applying guidance on the denoised clean estimate and restricting the update to the autoencoder’s image manifold. Their method is however developed for training-free, loss-based conditioning at inference time, and thus does not directly address extrapolation based guidance methods. CFG++ [Chung et al., 2024] instead targets classifier-free guidance in text-conditional diffusion models, and mitigates off-manifold drift by a minimal sampling-rule change. It specifically forms the guided denoised estimate, but keeps the renoising/noise term unconditional, effectively favoring interpolation over extrapolative CFG at high scales. Although the interpolation view keeps updates bounded, it is fundamentally tied to diffusion samplers where the update decomposes into denoising and renoising terms. In the same spirit, Rectified-CFG++ [Saini et al., 2025] extends this idea to rectified-flow models via a predictor–corrector scheme at the cost of roughly doubling the number of function evaluations. Characteristic [Zheng and Lan, 2023] gives a geometric view of the score-induced Fokker-Planck dynamics and shows that linearly combining conditional and unconditional scores generally violates this non-linear PDE; they therefore add a corrective term. However, this term is obtained through an iterative approach which raises computational cost. APG [Sadat et al., 2024] further reports, empirically, that the tangential component of guidance (defined with respect to the conditional approximate posterior) is a primary driver of saturation and artifacts, and proposes both tangential downscaling and an update bound to curb overshooting. In contrast, we view the generative manifold as the geometry learned by the model under the continuity equation, and show that guidance decomposes into two conservation-violating components whose bounding keeps trajectories on-manifold. Prior methods, including APG as a special case, fall under this decomposition.
3
Method
We aim to improve generation quality and mitigate saturation by reducing off-manifold effects of guidance in conditional normalizing flow (CNF) models. We start by preliminaries, then analyse guidance through the lens of probability conservation, and finally introduce the AdaMaG update rule that follows from this analysis. 3
3.1
Preliminaries
Rectified flow. Let xt ∈ RD denote the latent state at time t ∈ [0, 1], and let y be the condition (with ∅ denoting the empty prompt). We model the probability-flow dynamics with a velocity field vt (·, y) parameterised by a neural network vθ (x, t, y), and define the unconditional and conditional fields as vtu (x) := vθ (x, t, ∅), and vtc (x) := vθ (x, t, y) respectively. Samples are then obtained by solving the probability-flow ODE. dxt = vθ (xt , t, y), t ∈ [0, 1], (1) dt initialized from x0 ∼ p0 . Following standard flow convention [Lipman et al., 2022] , we parameterize the marginal xt as a linear mixture of a source sample x0 and a target (or data) sample x1 , such that xt = αt x1 + σt x0 ,
(2)
where (αt , σt ) is a scalar schedule with α0 = 0, σ0 = 1 and α1 = 1, σ1 = 0. These conditions can be reversed to follow the convention used by large-scale models such as SD3 [Esser et al., 2024], and WAN [Wan et al., 2025]. Classifier-free guidance. Classifier-free guidance forms the guided velocity by linearly combining vtc (x) and vtu (x) as vtcfg (x) = vtu (x) + ω vtc (x) − vtu (x) , (3) where ω > 1 is the guidance scale. Larger ω typically improves generation quality and text alignment, but also amplifies approximation errors and drives trajectories into artifact-prone regions, often accompanied by increased saturation. 3.2
Characterising off-manifold flows
Our goal is to strengthen semantic guidance while avoiding off-manifold drifts, i.e., perturbations that significantly distort the model’s learned density pt . We view such drifts through the lens of probability conservation and demonstrate that off-manifold flows can be decomposed into two contributions. Consequently, we derive a simple constraint on the guidance field gt that minimizes these contributions while guiding the generation. 3.3
Guidance and probability conservation
We now characterise when a guidance field perturbs the model’s density and when it does not. The analysis applies identically to the conditional and unconditional flows; we therefore drop superscripts (u, c) until the guidance formulation is fixed in Sec. 3.5. Let pt denote the density of xt induced by a primary velocity field vt . Probability conservation is expressed by the continuity equation [Chen et al., 2018] ∂t pt + ∇·(pt vt ) = 0.
(4)
Adding a guidance field gt to form ṽt := vt + gt preserves pt as a solution of (4) if and only if ∇·(pt gt ) = 0. Expanding this constraint with the identity ∇pt = pt ∇ log pt and using pt > 0 on the support yields the following equivalent condition. Proposition 3.1 (Conservation under guidance). The guided velocity ṽt = vt + gt preserves pt under the continuity equation (4) if and only if (see Appendix A) ∇·gt (x) + gt (x)⊤ ∇ log pt (x) = 0. | {z } {z } |
(i) divergence
(5)
(ii) score-parallel flux
The two terms have distinct geometric meaning: (i) is the local volume change induced by gt , and (ii) is the flux of probability mass across level sets of pt . Only their sum is constrained – either may be non-zero individually, and both must vanish jointly for gt to be conservative. A computable surrogate. Score-based models provide direct access to st (x) := ∇ log pt (x), but the divergence ∇·gt is intractable for high-dimensional neural fields at inference time. Our empirical analysis (Sec. 3.4, Fig. 3) shows that |∇·gt | is negligible relative to the divergence of the learned 4
Optimal scale SD3
High-guidance scale
SD3.5
Flux
SD3 (ω=15)
SD3.5 (ω=15)
Flux (ω=3)
Method
FID
IS
SAT FID
IS
SAT FID
IS
SAT FID
IS
SAT FID
IS
SAT FID
IS
SAT
CFG Rect-CFG++ TAG APG Ours
32.4 32.4 32.4 39.0 30.4
33.2 30.2 32.8 25.7 32.9
0.53 0.55 0.53 0.59 0.51
28.8 26.0 29.7 20.3 30.3
0.53 0.55 0.53 0.58 0.48
35.2 34.9 34.8 35.4 36.6
0.38 0.37 0.38 0.35 0.34
24.9 25.1 25.9 26.1 29.4
0.63 0.70 0.64 0.62 0.59
18.3 18.1 18.6 20.2 21.5
0.70 0.69 0.71 0.71 0.66
31.4 34.1 31.3 34.2 34.4
0.42 0.37 0.41 0.38 0.35
35.8 39.6 35.2 54.9 32.1
36.1 35.7 36.1 34.3 34.8
42.6 43.5 42.2 39.7 35.6
62.7 63.4 61.7 54.9 54.6
37.8 35.4 37.4 36.0 36.9
Table 1: Robustness under optimal and strong guidance. AdaMaG dominates four baselines at the optimal (left) and high-stress (right) guidance scales, with the lead growing as ω increases. Arrows: FID ↓, IS ↑, SAT ↓. Bold: best; underlined: second-best. The optimal ω values per model are reported in Sec. 4. For optimal scales and implementations see Figure 5, and Appendix F. velocities for the bulk of the trajectory and grows sharply only as t → 1. We therefore enforce a relaxed condition and differ the study of divergence term to Section 3.4, gt (x)⊤ ∇ log pt (x) ≈ 0,
(6)
which constrains gt to remain orthogonal to the score and confines samples to iso-density surfaces of pt . The omitted divergence is addressed separately via a time-dependent guidance schedule (Sec. 3.4). Relating score and rectified-flow velocity. For rectified flow parameterisations of the form described in subsection 3.1, corresponding probability-flow velocity vt admits a closed-form representation in terms of the score α̇t σ̇t σt αt − α̇t σt2 vt (x) = x− ∇x log pt (x), (7) αt αt where dots denote derivatives with respect to t (derivation in Appendix B). Defining scalar coefficients at :=
α̇t , αt
bt :=
σ̇t σt αt − α̇t σt2 , αt
(8)
we can rewrite (7) as vt (x) = at x − bt ∇x log pt (x). Consequently, the score-parallel term demonstrated in (5) can be obtained from 1 gt (x)⊤ ∇x log pt (x) = gt (x)⊤ at x − vt (x) . bt {z } |
(9)
(10)
=: nt (x)
Thus, in rectified flows, we find that guidance that is orthogonal to at x − vt (x), keeps samples primarily within iso-density surfaces, approximately satisfying the continuity equation. 3.4
Divergence and guidance schedule
To investigate the significance of the intractable divergence term in (5), we approximate ∇x · gt (x) using Hutchinson’s unbiased estimator. This approach utilises random probe vectors ξ satisfying E[ξξ ⊤ ] = I and is implemented via Jacobian–vector products. While this procedure is too memoryintensive for deployment during sampling, it serves as a critical analytical tool. As detailed in Figure 3, we observe that the divergence of the guidance field remains negligible throughout the majority of the trajectory but exhibits a sharp increase near the end of sampling. Consequently, we rely on the orthogonality approximation gt (x)⊤ ∇x log pt (x) ≈ 0 for earlier timesteps, while introducing a power-law decay schedule to attenuate guidance intensity during later stages, where divergence is non-trivial and continuity equation does not hold. Starting from a reference scale ωref , we modulate the guidance as t → 1, bounded by a minimum strength ωmin , ω(t) = max ωmin , ωref (1 − t)γ , t ∈ [0, 1]. (11) Here, ωref corresponds to the standard CFG guidance strength, while ωmin and γ govern the rate of late-time attenuation. Systematic ablation studies confirm that this schedule consistently improves generation quality across a wide range of hyperparameters. 5
Model Method Color ↑ Shape ↑ Texture ↑ Spatial ↑
Model Method FID ↓ IR ↑ PS ↑ HPSv2 ↑ SAT ↓
SD3
CFG + Ours
0.7374 0.8134
0.5789 0.5811
0.7131 0.7790
0.3211 0.3241
SD3
CFG +Ours
23.89 0.98 0.441 22.34 1.043 0.557
0.275 0.288
0.51 0.48
SD3.5
CFG + Ours
0.7415 0.8026
0.5743 0.5997
0.7514 0.7821
0.2851 0.2970
SD3.5
CFG +Ours
20.29 1.04 0.492 19.18 1.118 0.594
0.281 0.293
0.49 0.46
Flux
CFG + Ours
0.5943 0.6118
0.4114 0.4231
0.5428 0.6429
0.2409 0.2376
Flux
CFG +Ours
21.47 1.08 0.512 20.83 1.14 0.601
0.285 0.29
0.37 0.34
Table 2: T2I-CompBench results. AdaMaG im- Table 3: High-resolution evaluation at proves over CFG across attribute-binding categories 1024×1024. IR: ImageReward, PS: PickScore, and most spatial settings. Bold: best per row pair. SAT: Saturation. 3.5
Manifold-aware guidance update
We now formalise the AdaMaG algorithm. We adopt the standard CFG framework (3), where the unconditional velocity vtu (x) defines the base flow and conditioning is introduced via a guidance field gt (x). Our goal is to suppress the score-parallel flux. We first calculate the score proportional nt (x) as nt (x) = at x − vtc (x) (Eq.(10)). We utilise the conditional flow vtc for this approximation rather than vtu because we empirically find it provides consistently larger gains, particularly at low guidance scales. We then decompose the raw guidance field gt (x) into components parallel and orthogonal to this normal direction nt (x). The parallel component is given by projecting gt onto nt : ⟨gt (x), nt (x)⟩ nt (x), ∥nt (x)∥2
10−1
(12)
yielding orthogonal component gt⊥ (x) := ∥ gt (x) − gt (x). Finally, combining these components with the schedule ω(t) from (11), we define the refined guidance field as: ∥ g˜t (x) := ω(t) gt⊥ (x) + βgt (x) , (13)
∇⋅v u , ∇⋅v c
β = 0.1 β = 0.5 β = 1.0 (CFG)
|∇ ⋅ g|
∥
gt (x) :=
100
10−2
10−3
0
5
10
15
Steps
20
25
30
Figure 3: Divergence magnitude (normalised by where β ∈ (0, 1) is a fixed scalar that dampens dimensionality) along the sampling trajectory. the score-parallel (orthogonal to iso-densities) Black curves show the divergence of the condicomponent. By prioritising gt⊥ , this update con- tional and unconditional velocities as references; fines the trajectory to approximate iso-density coloured curves show the guidance residual under surfaces while retaining a controlled score- varying score-parallel damping β (with β = 1.0 orthogonal component gt⊥ to maintain condi- recovering CFG). tioning. We apply stronger guidance early in the process to leverage the regime where probability conservation still holds and guidance is most effective. As sampling progresses (t → 1), we attenuate the intensity; this prevents the saturation artifacts caused by the late-stage explosion in guidance divergence which leads to violation of probability conservation. Although the analysis in Section 3.3 is exact, the design choice to address those are built on two assumptions: (i) generality of late stage divergence blow up, and (ii) independence of the two terms in Eq. (5). For these we provide detailed theoretical and empirical evidence in Appendix C.
4
Experiments
We evaluate our method on text-to-image generation against standard classifier-free guidance (CFG) and recent baselines that explicitly target manifold preservation and saturation. We include RectCFG++ [Saini et al., 2025], TAG [Cho et al., 2025], and APG [Sadat et al., 2025]. Implementation details and hyperparameter settings for all baselines can be found in the Appendix F. Experiments are conducted on three large-scale models: Stable Diffusion 3 (SD3) [Esser et al., 2024], Stable Diffusion 3.5 (SD3.5), and Flux. SD3 and SD3.5 use standard classifier-free guidance, 6
Figure 4: Qualitative comparison between AdaMaG, and other baselines at their optimal setting. which our method directly targets, while Flux applies CFG through negative prompting, testing the same manifold-aware principles in a guidance-distilled setting. Following the standard textto-image evaluation protocol, we generate 5,000 images from COCO validation-set prompts at 256×256, and 1024×1024, reporting Fréchet Inception Distance (FID), Inception Score (IS), ImageReward [Xu et al., 2023], PickScore [Kirstain et al., 2023], HPSv2 [Wu et al., 2023], and a saturation metric capturing the desaturation behaviour characteristic of strong guidance. Saturation here is implemented following APG [Sadat et al., 2024]. We further evaluate compositional alignment on T2I-CompBench [Huang et al., 2023] across color, shape, texture, and spatial attributes. Qualitative results use 150 custom prompts at 1024×1024 (listed in Appendix G). All images are generated with Euler integration and 30 solver steps on a single H100 GPU. Divergence study We quantify the divergence term of Eq. 5 along the sampling trajectory using the procedure of Subsection 3.4, focusing on its magnitude in early steps versus its behaviour as t → 1. Figure 3 reports the divergence magnitude (normalised by dimensionality) for the conditional, unconditional, and guidance velocities; the learned velocities serve as a natural manifold baseline since the model is trained to approximately satisfy the continuity equation. Quantitative comparisons To benchmark AdaMaG against CFG and related baselines, Table 1 reports FID , IS, and saturation at both moderate and high guidance scales. Since the SD family and guidance-distilled Flux exhibit different guidance behaviour, we use model-specific scales as indicated in the table. Across all models, AdaMaG consistently improves generation quality and reduces saturation. Particularly on SD3 and SD3.5, AdaMaG achieves an average 8.19% reduction in FID and a 6.03% reduction in saturation while also improving Inception Score, consistently outperforming other baselines. A further observation is that AdaMaG remains competitive without retuning hyperparameters. In contrast, APG, the second strongest baseline in high guidance scale ranges, is sensitive to its momentum coefficient and radial bound, which can introduce instabilities, particularly at low guidance scales. We study parameter sensitivity for AdaMaG in Section 5. 7
Table 4: Score-parallel damping ablation. Effect of varying β on generation quality.
Table 5: Guidance schedule exponent ablation. Effect of varying γ at fixed β = 0.1.
Setting
FID ↓
IS ↑
SAT ↓
Recall ↑
Setting
FID ↓
IS ↑
SAT ↓
Recall ↑
CFG
32.64
33.04 ± 1.44
0.52
0.68
Ours β = 0.1
31.39
33.17 ± 1.74
0.50
0.70
β = 0.0 β = 0.1 β = 1.0 β = 5.0
31.47 31.39 31.55 32.10
32.67 ± 2.41 33.17 ± 1.74 33.12 ± 1.05 33.13 ± 1.24
0.50 0.50 0.50 0.51
0.70 0.70 0.69 0.70
γ = 0.1 γ = 1.0 γ = 2.0 γ = 4.0
31.15 29.97 29.55 29.29
32.79 ± 1.32 33.14 ± 0.88 33.32 ± 1.55 33.06 ± 1.69
0.497 0.493 0.491 0.489
0.71 0.71 0.71 0.72
Beyond standard quality metrics, AdaMaG also strengthens compositional alignment and preferencealigned quality at high resolution (Tables 2, 3). On T2I-CompBench, AdaMaG consistently improves over CFG on attribute-binding categories (Color, Shape, Texture) across all three models, with smaller but generally positive gains on Spatial layout. At 1024×1024, AdaMaG yields concurrent gains in FID, ImageReward, PickScore, HPSv2, and saturation across SD3, SD3.5, and Flux, indicating the manifold-preservation gains transfer cleanly to high-resolution and human aligned outputs. SD3 60
FID ↓
50
SD3.5 CFG TAG APG Rect-CFG++ Ours
40
30
Inception Score ↑
Qualitative comparisons. Fig. 4 presents qualitative comparisons across the considered baselines, highlighting generation quality and artifact frequency. All samples are generated using each method’s optimal guidance scale as selected from Fig. 5. Overall, AdaMaG produces higher-quality generations with noticeably fewer artifacts. Since Fig. 4 uses scales optimised jointly for FID and saturation (Fig. 5), Fig. 7 provides a focused comparison under heavy guidance between CFG, APG (a desaturationfocused method), and AdaMaG.
32.5 30.0 27.5 25.0 22.5 20.0
Saturation
0.65 To complement the automated metrics, we con0.60 ducted a pairwise preference study with 30 eval0.55 uators on 30 prompts using SD3, with each 0.50 method run at its optimal guidance scale. For ev0.45 0.40 ery prompt, evaluators chose between AdaMaG 4 6 8 10 12 14 4 6 8 10 12 14 Guidance Scale (ω) Guidance Scale (ω) and a baseline along three axes (text alignment, image quality, and overall preference) or marked Figure 5: Guidance-scale sweeps for FID, IS, and the pair as a tie. Figure 6 reports the aggregated saturation across methods for SD3 (left column) preferences. AdaMaG is preferred over CFG, and SD3.5 (right column). TAG, and APG across all three criteria, with the strongest margins against APG (63.3% vs. 13.3% on text alignment) and CFG (60.0% vs. 10.0% on overall preference). The lead over TAG is narrower but consistent across all three axes (46.7–53.3% in our favour vs. 13.3–20.0%), indicating that the manifold-preservation gains translate into perceptually meaningful improvements beyond FID and IS.
5
Ablations
Score-parallel damping. We first ablate the score-parallel term attenuation in AdaMaG by varying the score-parallel scaling factor β, which controls the magnitude of the score-parallel component in our update. Table 4 reports the resulting trade-off in fidelity and artifact suppression, isolating the effect of minimizing the second term in Eq. 5. This table shows that score-parallel component damping consistently improves fidelity over CFG, reducing FID, and marginally saturation while improving IS. Recall remains essentially unchanged across β, suggesting that scaling down the score-parallel component does not harm the coverage. Guidance schedule. The first term in (5) captures the divergence of the guidance field, which we mitigate using the power-law guidance schedule introduced in Section 3.4. Table 5 reports an ablation over the exponent γ, showing how the schedule strength affects FID, IS, and saturation. Presented results indicate that increasing γ consistently reduces both FID and saturation, while IS exhibits an 8
Ours wins
Tie
Preference (%)
Text Align 60 40 20 0
Baseline wins
Quality
63.3
56.7
30.0 13.3
vs. CFG
60.0
53.3
50.0 36.7
33.3
vs. TAG
13.3
vs. APG
60.0
33.3 20.0
13.3
vs. CFG
vs. TAG
26.7 13.3
vs. APG
56.7
53.3
46.7
23.3 13.3
Overall
30.0 10.0
vs. CFG
30.0 16.7
vs. TAG
26.7 16.7
vs. APG
Figure 6: Human evaluation. AdaMaG is preferred over CFG, TAG, and APG on text alignment, image quality, and overall preference. optimum at an intermediate setting. Importantly, the schedule never underperforms the no-schedule baseline, and although larger γ continues to improve fidelity and desaturation, we use γ = 4.0 throughout the study to maintain a competitive IS.
6
Discussion
We provide an additional interpretation of our framework next and defer the discussion of the theoretical connection to other works to Appendix E. Practical Interpretation. From the decomposition in Eq. 5, guidance splits into an scoreparallel component and a divergence component. Both terms violate probability conservation, but they play qualitatively different roles. The divergence term introduces the source/sink mechanism that provides the actual conditioning signal, whereas the score-parallel term induces additional off-manifold drift, which manifests as hallucination and oversaturation. Consequently, using a larger guidance scale early in sampling strengthens the source/sink effect and helps establish global semantics and structure under stronger conditioning. As t → 1, we damp guidance to avoid excessive late-stage forcing, which can otherwise produce overly sharp, cartoonish details. In practice, this can be achieved either by reducing the guidance scale ω(t) or by bounding the magnitude of the guidance residual via a radial norm constraint. Since gt scales with Figure 7: Qualitative saturation comparison at ω (e.g., gt = ω (vtc − vtu )), both mechanisms ef- ω = 15. fectively limit the guidance strength over time.
7
Conclusion
We introduced AdaMaG, a guidance update rule that leverages probability conservation as a proxy for the model’s generative manifold, and explicitly limits deviations from this constraint during sampling. Our analysis identified two primary sources of violation, (a) the score-parallel component of the guidance update, and (b) divergence of the guided field. We addressed these effects via score-parallel attenuation and a simple time-dependent guidance schedule, respectively. Through text-guided image generation experiments, we showed that AdaMaG consistently improves upon standard CFG and other baselines, with performance that is largely insensitive to the choice of hyperparameters. Limitations. A primary limitation of our framework is that it does not explicitly calculate the divergence term during the sampling process due to computational and memory constraints. Addressing this requires future research into generation with cheap, instance-specific likelihood evaluation.
9
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Tero Karras, Miika Aittala, Tuomas Kynkäänniemi, Jaakko Lehtinen, Timo Aila, and Samuli Laine. Guiding a diffusion model with a bad version of itself. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. Yuval Kirstain, Adam Polyak, Uriel Singer, Shahbuland Matiana, Joe Penna, and Omer Levy. Picka-pic: An open dataset of user preferences for text-to-image generation. Advances in neural information processing systems, 36:36652–36663, 2023. Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747, 2022. Cheng Lu, Huayu Chen, Jianfei Chen, Hang Su, Chongxuan Li, and Jun Zhu. Contrastive energy prediction for exact energy-guided diffusion sampling in offline reinforcement learning. In International Conference on Machine Learning, pages 22825–22855. PMLR, 2023. Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen. Glide: Towards photorealistic image generation and editing with text-guided diffusion models. arXiv preprint arXiv:2112.10741, 2021. Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. Highresolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 10684–10695, June 2022. Seyedmorteza Sadat, Otmar Hilliges, and Romann M Weber. Eliminating oversaturation and artifacts of high guidance scales in diffusion models. In The Thirteenth International Conference on Learning Representations, 2024. Seyedmorteza Sadat, Manuel Kansy, Otmar Hilliges, and Romann M. Weber. No training, no problem: Rethinking classifier-free guidance for diffusion models. In The Thirteenth International Conference on Learning Representations, 2025. Shreshth Saini, Shashank Gupta, and Alan C Bovik. Rectified-cfg++ for flow based models. arXiv preprint arXiv:2510.07631, 2025. Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In Francis Bach and David Blei, editors, Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pages 2256–2265, Lille, France, 07–09 Jul 2015. PMLR. Jiaming Song, Qinsheng Zhang, Hongxu Yin, Morteza Mardani, Ming-Yu Liu, Jan Kautz, Yongxin Chen, and Arash Vahdat. Loss-guided diffusion models for plug-and-play controllable generation. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett, editors, Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 32483–32498. PMLR, 23–29 Jul 2023. Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021. Team Wan, Ang Wang, Baole Ai, Bin Wen, Chaojie Mao, Chen-Wei Xie, Di Chen, Feiwu Yu, Haiming Zhao, Jianxiao Yang, et al. Wan: Open and advanced large-scale video generative models. arXiv preprint arXiv:2503.20314, 2025. Xiaoshi Wu, Yiming Hao, Keqiang Sun, Yixiong Chen, Feng Zhu, Rui Zhao, and Hongsheng Li. Human preference score v2: A solid benchmark for evaluating human preferences of text-to-image synthesis. arXiv preprint arXiv:2306.09341, 2023. Jiazheng Xu, Xiao Liu, Yuchen Wu, Yuxuan Tong, Qinkai Li, Ming Ding, Jie Tang, and Yuxiao Dong. Imagereward: Learning and evaluating human preferences for text-to-image generation. Advances in Neural Information Processing Systems, 36:15903–15935, 2023. 11
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A
Proof of Proposition 3.1
Proposition (Conservation under guidance). Let pt ∈ C 1 (RD ; R>0 ) satisfy ∂t pt + ∇·(pt vt ) = 0 for vt ∈ C 1 (RD ; RD ), and let gt ∈ C 1 (RD ; RD ). Then pt satisfies the continuity equation under the guided velocity ṽt := vt + gt if and only if ∇·gt + g⊤ t ∇ log pt = 0
on supp(pt ).
(14)
Proof. By linearity of the divergence, ∇·(pt ṽt ) = ∇·(pt vt ) + ∇·(pt gt ), so ∂t pt + ∇·(pt ṽt ) = 0 holds if and only if ∇·(pt gt ) = 0. The product rule gives ⊤ ∇·(pt gt ) = pt ∇·gt + g⊤ (15) t ∇pt = pt ∇·gt + gt ∇ log pt , where the second equality uses ∇ log pt = ∇pt /pt , valid since pt > 0. As pt does not vanish on its support, ∇·(pt gt ) = 0 on supp(pt ) is equivalent to (14).
B
Velocity-score conversion
We provide a self-contained derivation of the conversion between the marginal score st (x) := ∇x log pt (x) and the marginal velocity field vt (x) for the Gaussian probability path t ∈ [0, 1],
Xt = αt X1 + σt X0 ,
(16)
where X0 ∼ N (0, I) is independent of X1 . Let pt denote the marginal density of Xt , and let pt|1 (x | x1 ) denote the conditional density of Xt given X1 = x1 . Step 1: Conditional score. From (16), we have pt|1 (x | x1 ) =
N x | αt x1 , σt2 I
1 2 ∝ exp − 2 ∥x − αt x1 ∥2 . 2σt
(17)
Taking logs and collecting all terms independent of x into a constant C(t) yields log pt|1 (x | x1 ) = C(t) −
1 ∥x − αt x1 ∥22 . 2σt2
(18)
Differentiating (18) with respect to x gives the conditional score ∇x log pt|1 (x | x1 ) = −
1 (x − αt x1 ). σt2
(19)
R Step 2: Fisher’s identity and the marginal score. Using pt (x) = pt|1 (x | x1 ) q(x1 ) dx1 and differentiating under the integral sign, Z ∇x pt (x) = ∇x pt|1 (x | x1 ) q(x1 ) dx1 Z = pt|1 (x | x1 ) ∇x log pt|1 (x | x1 ) q(x1 ) dx1 . (20) p
(x|x )q(x )
1 1 Dividing by pt (x) and recognising the posterior p(x1 | x) = t|1 pt (x) yields Fisher’s identity, ∇x log pt (x) = E ∇x log pt|1 (Xt | X1 ) Xt = x . (21)
Substituting (19) into (21) gives 1 st (x) = E − 2 Xt − αt X1 Xt = x . σt 12
(22)
By (16), we have the identity Xt − αt X1 = σt X0 , hence st (x) = −
1 1 E[σt X0 | Xt = x] = − E[X0 | Xt = x]. σt2 σt
(23)
Defining x0|t (x) := E[X0 | Xt = x], we obtain the score–x0 conversion x0|t (x) = −σt st (x).
(24)
Step 3: Velocity in terms of conditional expectations. We define the marginal velocity field vt (x) := E[Ẋt | Xt = x].
(25)
Differentiating (16) with respect to t yields Ẋt = α̇t X1 + σ̇t X0 , and therefore vt (x) = α̇t E[X1 | Xt = x] + σ̇t E[X0 | Xt = x] = α̇t x1|t (x) + σ̇t x0|t (x),
(26)
where x1|t (x) := E[X1 | Xt = x]. Step 4: Eliminating x1|t and expressing vt via the score. Rearranging (16) gives X1 = αt−1 (Xt − σt X0 ) (for αt ̸= 0), hence taking conditional expectations yields 1 σt x− x0|t (x). αt αt
(27)
α̇t α̇t x0|t (x). x + σ̇t − σt αt αt
(28)
x1|t (x) = Substituting (27) into (26) gives vt (x) =
Finally, using (24) to eliminate x0|t (x) yields the score–velocity conversion vt (x) =
C
α̇t α̇t x − σt σ̇t − σt2 st (x). αt αt
(29)
Trajectory dynamics of guidance
This appendix studies the structural dynamics of the guidance field along the sampling trajectory and provides formal justification for the two design choices in AdaMaG: the time-dependent guidance schedule and the independent tunability of the score-parallel damping parameter β. We establish two findings: • Late-stage divergence is structurally inevitable C.1. Under the standard flow parameterisation, |∇x · gt | blows up as t → 1 at a rate determined entirely by the conditional/unconditional posterior covariance gap. The blow-up is not an artefact of training or sampling; it is a direct consequence of the manifold-supported nature of conditional and unconditional priors. • Guidance dynamics exhibit two distinct C.2. The decomposition of gt into score-parallel and score-orthogonal components reveals a regime transition along the trajectory: early steps are parallel-dominant, while late steps are orthogonal-dominant and host the divergence blow-up. This regime structure is not predicted by the proposition’s dimensional argument but is verified empirically across the trajectory, and explains why β-damping and the schedule ω(t) address structurally distinct contributions to conservation violation. All proofs are deferred to Appendix D. C.1
Late-stage divergence spike.
The empirical spike in |∇x · gt | as t → 1 (Figure 3) admits a clean structural explanation. The flow parameterisation gives an exact identity for the divergence, and the spike emerges from a posterior-covariance gap of the clean data failing to vanish at a specific dimensional rate. 13
Proposition C.1 (Late-stage divergence behaviour). Let xt = αt x1 + σt x0 under the Lipman linear schedule with x0 ∼ N (0, I) and x1 ∼ pdata . Let gt := vtc − vtu denote the guidance residual, and define the trace gap ∆t (x) := tr Covu [X1 | Xt = x] − tr Covc [X1 | Xt = x], between the unconditional and conditional posterior covariances of the clean data X1 given the noisy state Xt = x. Then for every t ∈ (0, 1), ∇x · gt (x) =
αt ∆t (x) . σt3
(30)
In particular, |∇x · gt (x)| → ∞ as t → 1 whenever |∆t (x)| vanishes strictly slower than σt3 . Proof in Appendix D. The sharp rise of |∇x · gt | as t → 1 in Figure 3 confirms that |∆t (x)| decays slower than σt3 for real conditional data, consistent with the manifold hypothesis under which conditional and unconditional priors concentrate on distinct submanifolds and the posterior covariance gap fails to vanish (formal treatment in Appendix D). C.2
Decoupling of the two controls.
AdaMaG introduces two hyperparameters: β damping the score-parallel component, and (γ, ωmin ) shaping the guidance schedule ω(t). The flux contribution scales exactly with β since gt⊥ is orthogonal to the score, g̃t⊤ ∇x log pt = ω(t) β gt⊤ ∇x log pt . (31) The following proposition shows that modifying β leaves the divergence term approximately unchanged. ∥ Proposition C.2 (Divergence insensitivity to score-parallel damping). Let g˜t = ω(t) gt⊥ + β gt be the AdaMaG guidance field (Eq. 13). Under the assumption that the Jacobian Jgt has eigenvalues of comparable order across directions, the sensitivity of the divergence to β is suppressed by the ambient dimensionality: ∥ ∇x · g t 1 = O , (32) D ∇x · g t where D is the latent dimensionality. Consequently (proof in Appendix D), 1 − β ∇x · g̃t = ω(t) ∇x · gt · 1 + O . D
(33)
∥
The argument relies on a structural asymmetry: gt is rank-1 along the score normal n̂t , so its Jacobian contributes to the trace along a single direction, whereas gt⊥ spans the (D−1)-dimensional orthogonal subspace. For typical latent dimensions (D=4×32×32=4,096 at 256×256 resolution), the predicted cross-talk between β and the divergence is negligible. Direct empirical verification. Because the comparable-eigenvalue assumption may fail in regions where gt has anisotropic structure, we directly measure all three quantities in Eq. 32 along the ∥ sampling trajectory across β ∈ [0.1, 20]. Figure 8 reports |∇ · gt |, |∇ · gt |, and |∇ · gt⊥ | at every denoising step. The total divergence |∇ · gt | remains essentially flat across two orders of magnitude in β at every step, directly confirming the operational claim of Proposition C.2: β can be tuned with negligible effect on probability conservation. Two regimes along the trajectory. Beyond verifying the decoupling, the per-step decomposition reveals an interesting structural feature of the guidance field. The dominance between the parallel and orthogonal divergence components shifts systematically along the trajectory: ∥
∥
• Parallel-dominant regime (early steps, t near 0): |∇ · gt | > |∇ · gt⊥ |, with |∇ · gt | ≈ |∇ · gt | in the earliest steps. Guidance acts predominantly along the score direction. 14
|∇⋅g ∥ |
divergence
step 30
step 28
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0.004
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0.4
|∇⋅g ⟂ |
100
β
101
0.001 10−1
100
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101
10−1
100
β
101
∥
Figure 8: Divergence components vs. β along the trajectory. Each panel shows |∇·gt |, |∇·gt |, and |∇ · gt⊥ | as a function of β ∈ [0.1, 20] at a fixed denoising step. The total divergence |∇ · gt | (green) remains flat across two orders of magnitude in β at every step, directly verifying Proposition C.2. ∥ Early steps exhibit a parallel-dominant regime (|∇ · gt | > |∇ · gt⊥ |), while late steps exhibit an orthogonal-dominant regime where the divergence spike of Proposition C.1 concentrates in gt⊥ . • Orthogonal-dominant regime (late steps, t near 1): |∇ · gt⊥ | dominates and exhibits the late-stage ∥ spike of Proposition C.1, while |∇ · gt | remains small. This regime structure has a concrete design implication: because the late-stage divergence spike lives ∥ in gt⊥ , β-damping (which acts on gt ) cannot suppress it. The separate schedule mechanism ω(t) is therefore not redundant with β but addresses a structurally distinct contribution to conservation violation. The factorial ablation over (β, γ) in Section 5 confirms that their improvements to FID and saturation are approximately additive.
D
Proofs
Here we provide complete proofs of the results stated in Section C. D.1
Laplacian of the log-density via Tweedie
First we obtain an expression of the laplacian log density required for proposition C.1. We take Tweedie’s formula as the starting point: for the Gaussian convolution model Xt = αt X1 + σt X0 with X0 ∼ N (0, I) independent of X1 ∼ pk (k ∈ {c, u}), the marginal density pkt satisfies, at every x with pkt (x) > 0, ∇x log pkt (x) =
αt Ek [X1 | Xt = x] − x . σt2 15
(34)
Lemma D.1 (Laplacian of the log-density). Under the setup above, α2 tr Covk [X1 | Xt = x] − Dσt2 ∆x log pkt (x) = t . σt4
(35)
Proof. Differentiating (34) with respect to x gives the Hessian of the log-marginal: αt ∇x Ek [X1 | Xt = x] − I ∇2x log pkt (x) = . σt2 From Tweedie’s equation, it follows that αt ∇x Ek [X1 | Xt = x] = 2 Covk [X1 | Xt = x], σt so substituting yields the matrix identity α2 Covk [X1 | Xt = x] I ∇2x log pkt (x) = t − 2. σt4 σt Taking the trace of both sides yields (35) where D is the dimension, and ∆x is the Laplacian. D.2
Proof of Proposition C.1
Proposition C.1 (Late-stage divergence behaviour). Let Xt = αt X1 + σt X0 with X0 ∼ N (0, I) independent of X1 ∼ pdata , under the Lipman linear schedule αt = t, σt = 1 − t. Let gt := vtc − vtu denote the guidance residual, and define the trace gap ∆t (x) := tr Covu [X1 | Xt = x] − tr Covc [X1 | Xt = x], where the conditional and unconditional posterior covariances are taken under pdata (· | y) and pdata respectively. Then for every t ∈ (0, 1) and every x in the joint support of pct and put , αt ∇x · gt (x) = 3 ∆t (x) . σt In particular, |∇x · gt (x)| → ∞ as t → 1 whenever |∆t (x)| vanishes strictly slower than σt3 . Proof. The proof has three steps: reduce the divergence to a Laplacian gap of log-densities (Step 1), apply Lemma D.1 to express this gap in terms of posterior covariances (Step 2), and combine with the schedule-specific value of bt (Step 3). Step 1: Reduce divergence to a Laplacian gap. From the velocity–score relation (9), vtk (x) = at x − bt ∇x log pkt (x) for k ∈ {c, u}. The at x terms are identical and cancel under subtraction, giving gt (x) = −bt ∇x log pct (x) − ∇x log put (x) . Since bt depends only on t, the divergence commutes with the scalar factor: (36) ∇x · gt (x) = −bt ∆x log pct (x) − ∆x log put (x) . Step 2: Express the Laplacian gap via posterior covariances. Applying Lemma D.1 to both k = c and k = u gives α2 tr Covk [X1 | Xt = x] − Dσt2 ∆x log pkt (x) = t . σt4 The dimensional terms −Dσt2 /σt4 are identical for c and u and cancel under subtraction: α2 (37) ∆x log pct (x) − ∆x log put (x) = − 4t ∆t (x). σt Step 3: Compute bt and combine. Under the linear Lipman schedule αt = t, σt = 1 − t, a direct computation gives σ̇t σt αt − α̇t σt2 σt bt = = − . αt αt Substituting (37) and this value of bt into (36), σ α2 αt t ∇x · gt (x) = − − · − 4t ∆t (x) = − 3 ∆t (x). αt σt σt Taking absolute values yields the claimed identity. 16
Late-stage blow-up. The final claim follows directly from the identity above. As t → 1, αt → 1 and σt → 0, so αt /σt3 → ∞ at rate Θ(1/(1 − t)3 ). If |∆t (x)| vanishes strictly slower than σt3 , then |∇x · gt (x)| diverges. Empirically from 3 and across seeds we observe the same pattern as demonstrated in proposition. D.3
Proof of Proposition C.2
Proposition C.2 (Divergence insensitivity to score-parallel damping). Let g̃t denote the AdaMaGmodified guidance defined in Eq. (13), and let gt be the raw guidance with orthogonal decomposition ∥ gt = gt +gt⊥ relative to the score-normal direction nt . Under the assumption that Jgt has eigenvalues of comparable order across directions and that the score direction n̂ = nt /∥nt ∥ varies smoothly, ∥
∇x · g t ∇x · g t
h i , ∇x · g̃t = ω(t) ∇x · gt 1 + O 1−β D
1 = O D ,
where D is the latent dimensionality. ∥
Proof. By linearity of the divergence operator and the identity gt = gt + gt⊥ , ∥ ∇x · g̃t = ω(t) ∇x · gt − (1 − β) ∇x · gt , ∥
(38) ∥
so ∂(∇x ·g̃t )/∂β = ω(t) ∇x ·gt and the sensitivity to β is controlled by ∇x ·gt alone. ∥
Writing the parallel component as gt (x) = λ(x) n̂(x) with λ(x) = ⟨gt (x), nt (x)⟩/∥nt (x)∥ and n̂ = nt /∥nt ∥, the product rule yields Jg∥ = n̂ (∇x λ)⊤ + λ Jn̂ ,
(39)
t
whose trace is the divergence ∥
∇x · gt = n̂⊤ ∇x λ + λ ∇x · n̂.
(40)
The first term is a single directional derivative along n̂. The second term, λ ∇x · n̂, can in principle aggregate contributions from all D coordinate directions; under the stated regularity, however, no single direction (in particular n̂) carries disproportionate weight in Jgt , so the trace of the rank-one part n̂(∇x λ)⊤ contributes O(1) to a sum ∇x ·gt = tr(Jgt ) of D comparably-sized eigenvalues. This gives ∥ |∇x · gt | = O(1/D). |∇x · gt | Substituting into (38) and factoring out ∇x ·gt yields the stated approximation. For latent dimensions typical of flow-based generative models (D = 4,096 at 256×256; D = 16,384 at 512×512), this ratio is negligible, confirming that the divergence is effectively invariant to changes in β.
E
Connections to related works
Connection to APG. APG [Sadat et al., 2024] empirically observed that attenuating the tangential component of the CFG update reduces saturation, with effects most visible under x0 -prediction and a decomposition defined relative to x̂0cond . We make three explicit advances over this empirical observation. First, we identify the underlying geometric principle. APG’s tangential attenuation is the parameterisation-specific manifestation of a broader conservation argument (Eq. 5) in which the score-parallel flux across iso-density surfaces governs off-manifold drift. Where APG operates on a heuristic decomposition, our framework derives the same construction from probability conservation. Second, our formulation is parameterisation-invariant. Working directly with the score st = ∇x log pt rather than x̂0 or ε, the relevant projection is uniquely determined. APG arises as a special case of Eq. 10: under x0 -prediction with at = 1/t, the scalar at factors out and the projection reduces 17
to APG’s decomposition relative to x̂0cond . In other parameterisations, this scalar reduction does not occur and the score-line decomposition diverges from a clean-image decomposition. Third, our framework predicts where APG’s effect weakens. In ε-prediction, the score relates to the noise predictor by sθ = −εθ /σt , so the geometric object that should be attenuated is rescaled by σt and the empirical benefit of tangential attenuation is masked. In DDPM/VP-style models, our framework predicts that working in score-space recovers the equivalent geometric attenuation that APG cannot directly access in ε-space. Beyond these advances, AdaMaG addresses a second contribution to conservation violation, the divergence term in Eq. 5, which APG does not consider. The schedule ω(t) targets this term independently of the score-parallel attenuation, making AdaMaG more complete in its coverage of conservation violations. Connection to Rectified-CFG++. Rectified-CFG++[Saini et al., 2025] adopts a predictor-corrector sampling strategy in which each guided (predictor) update is followed by a corrective step that re-applies the base generative dynamics. Interpreted through our lens, this correction acts as an implicit manifold update inserted between successive guided updates. Consequently, the method approximately enforces our deviation constraint in an alternating manner such that after each guided step that may introduce off-manifold drift, the subsequent corrector step reduces the accumulated deviation by steering the iterate back toward the model’s learned trajectory. In effect, this achieves nt (x) = 0 every other step.
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Baselines
Rect-CFG++. Rect-CFG++ is implemented as a predictor–corrector scheme and therefore requires two network evaluations per integration update (one prediction and one correction), i.e., roughly 2× NFE compared to single-evaluation baselines. To keep comparisons compute-matched, we fix the total NFE budget across methods and split it evenly between predictor and corrector steps. Concretely, with a 30-NFE budget we run 15 predictor updates and 15 corrector updates, yielding the same total number of model calls as the 30-step Euler baselines. APG adaptation to rectified flows. APG was originally introduced for diffusion models with an explicit x0 (clean-image) prediction. We adapt APG to rectified flows by first forming a one-step estimate of x0 from the current state xt and the model velocity at the same noise level. We then apply the standard APG update in x0 -space using the original coefficients from the diffusion formulation. Finally, we map the updated x̂0 back to an equivalent rectified-flow update by taking a single Euler step that returns to the same noise level t, and define the corresponding velocity that realizes this update.
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Custom prompts Prompt list (In order of appearance) P1: A ballerina mid-pirouette in a flowing white tutu with motion blur on the fabric and sharp focus on her concentrated expression P2: Monk meditating in a serene temple with incense smoke curling and golden Buddha statue behind P3: Tennis player serving with ball tossed high and body arched in powerful motion P4: A master sushi chef slicing fresh tuna with a long knife at a traditional counter while perfect nigiri pieces are arranged on a wooden board with wasabi and pickled ginger P5: Steam rising from a copper Turkish coffee pot on an ornate brass tray with ceramic cups painted in cobalt blue patterns P6: A white horse galloping through shallow surf at sunrise with spray catching pink and gold light and mane flowing
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P7: A cormorant drying its wings on a wooden post with water droplets and soft morning backlight P8: A vintage sailboat with weathered wooden hull and cream canvas sails gliding across turquoise Mediterranean waters at golden hour P9: A woman with freckles and copper hair laughing in golden hour light with wind catching loose strands P10: A woman in a white linen shirt reading on a sun-dappled balcony with coffee steaming beside her P11: A ballerina tying her pointe shoe ribbons with focus and determination visible in her expression P12: A florist arranging peonies with green-stained fingers and apron covered in petals P13: Close-up of a woman applying red lipstick in a vintage compact mirror P14: A barista creating latte art with intense focus and steam rising around her P15: A woman wrapped in cashmere reading by firelight with snow falling outside the window P16: A dancer stretching at a barre with morning sun streaming through tall windows P17: A solitary cypress tree on a Tuscan hilltop at golden hour with rolling green hills and distant farmhouses P18: An elderly craftsman’s hands shaping wet clay on a pottery wheel with earth tones and soft window light P19: A woman in white linen walking barefoot through shallow tide pools with reflections of clouds and blue sky P20: A gondolier in striped shirt steering through a narrow Venice canal with laundry hanging above and golden walls P21: A woman hiking at sunrise with wind in her hair and mountains behind her P22: A woman floating in a turquoise cenote with white dress billowing underwater P23: Close-up of hands holding a worn leather journal and fountain pen P24: An archer drawing her bow with intense focus and forest background P25: A woman running through a sunflower field with arms outstretched and joy on her face P26: A traditional wooden rowboat half submerged in clear lake water with autumn trees reflected around it P27: A woman with wind-blown dark hair on a cliff overlooking a turbulent sea in a camel wool coat P28: A woman with auburn hair in a rust-colored sweater holding a steaming mug on a foggy morning porch P29: A sommelier in crisp white shirt decanting wine by candlelight P30: A woman running through a sunflower field with arms outstretched and joy on her face P31: Antique maps spread on a table with a brass compass magnifying glass and leather case P32: A barn owl in flight at dusk with wings spread and soft feathers and golden field below P33: A falconer with hawk landing on her gloved hand against stormy sky P34: A toucan with massive bill in orange yellow and black tossing a berry in the air against rainforest backdrop P35: A Wilson’s bird-of-paradise displaying with turquoise crown curled tail feathers and cape of iridescent green P36: A Bengal cat with marbled coat lounging on a velvet cushion in deep teal with dust motes in sunlight
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P37: A hyacinth macaw pair preening each other with cobalt blue feathers and bright yellow eye rings against jungle green P38: A stop sign saying all way underneath it P39: A white horse galloping through shallow surf at splashing water as running P40: A giraffe drinking water by the lake
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Additional results
We provide further qualitative comparisons between AdaMaG and standard CFG at their respective optimal guidance scales. Each pair is generated under the same prompt and seed, illustrating how AdaMaG affects visual fidelity and artifact frequency across a larger set of examples.
Figure 9: Additional qualitative comparisons between AdaMaG and CFG at their respective optimal guidance scales (part 1/4).
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Figure 10: Additional qualitative comparisons between AdaMaG and CFG at their respective optimal guidance scales (part 2/4).
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Figure 11: Additional qualitative comparisons between AdaMaG and CFG at their respective optimal guidance scales (part 3/4).
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Figure 12: Additional qualitative comparisons between AdaMaG and CFG at their respective optimal guidance scales (part 4/4).
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