Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Authors: Cheng Jin
Zhenyu Xiao
Chutao Liu
Yuantao Gu
Venue: ICML 2025 poster (2025)
Keywords: Diffusion Model
Classifier Free Guidance
Text-to-Image Generation
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: A diffusion guidance method by enhancing in the angular domain, improving text-image alignment and image quality.

Abstract: Classifier-free guidance (CFG) has emerged as a pivotal advancement in text-to-image latent diffusion models, establishing itself as a cornerstone technique for achieving high-quality image synthesis. However, under high guidance weights, where text-image alignment is significantly enhanced, CFG also leads to pronounced color distortions in the generated images. We identify that these distortions stem from the amplification of sample norms in the latent space. We present a theoretical framework that elucidates the mechanisms of norm amplification and anomalous diffusion phenomena induced by classifier-free guidance. Leveraging our theoretical insights and the latent space structure, we propose an Angle Domain Guidance (ADG) algorithm. ADG constrains magnitude variations while optimizing angular alignment, thereby mitigating color distortions while preserving the enhanced text-image alignment achieved at higher guidance weights. Experimental results demonstrate that ADG significantly outperforms existing methods, generating images that not only maintain superior text alignment but also exhibit improved color fidelity and better alignment with human perceptual preferences.

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Cheng Jin'
Zhenyu Xiao 1
Chutao Liu
Yuantao Gu
Abstract
2024). These models operate by performing diffusion in the latent space, which is then decoded into the image space, ef- fectively transforming textual descriptions into high-quality images with intricate details and compelling visual effects. A key component of this process is the Classifier-Fres Guid- ance (CFG) mechanism (Ho & Salimans. 2021), which enhances generation quality through línear extrapolation of the conditional score function with guidance weight.
Classifier-free -ree guidance (CFG) has emerged as a pivotal advancement in text-to-image latent diffu- sion models, establishing itself as cornerstone technique for achieving high-quality image syn- thesis. However, under high guidance weights, where text-image alignment significantly en- hanced, CFG also leads to pronounced color dis- tortions in the generated images. We identify that these distortions stem from the amplifica- tion of sample norms in the latent space. We present a theoretical framework that elucidates the mechanisms of norm amplification and anoma- lous diffusion phenomena induced by classifier- free guidance. Leveraging our theoretical insights and the latent space structure, we propose an An- gle Domain Guidance (ADG) algorithm. ADG constrains magnitude variations while optimizing angular alignment, thereby mitigating color dis- tortions while preserving the enhanced text-image alignment achieved at higher guidance weights. Experimental results demonstrate that ADG sig- nificantly outperforms existing methods, generat- ing images that not only maintain superior text alignment but also exhibit improved color fidelity and better alignment with human perceptual pref- erences.
Despite its effectivenesss, CFG introduces several challenges. Research has shown that increasing the guidance weight im- proves text alignment but at the cost of reduced sample diver- sity (Ho & Salimans, 2021). More critically, when the guid- ance weight surpasses certain threshold, the quality the generated images deteriorates significantly, leading loss of detail and overall visual degradation. Theoretically, this phenomenon may be attributed to the non non-commutativity between the tilting process, driven by the conditional like- lihood, and the noise addition process (Wu et al., 2024; Chidambaram et al., 2024). This non-commutativity results in a deviation of the CFG-generated samples from the tar- get distribution, manifesting as performance degradation, particularly at high guidance weights.
Several approaches have been proposed to mitigate these issues. Techniques such as variable weighting schemes and additional Langevin sampling iterations have been intro- duced to counteract the decline in image quality (Chung et al., 2025; Xia al., 2024; Bradley & Nakkiran, 2024; Sadat et al., 2025). While these methods have shown some efficacy in reducing quality degradation for specific tasks, they remain constrained by their reliance on linear combi- nation of conditional score functions. This linear framework limits their ability to maintain generation quality, highlight- ing the need for a more comprehensive solution.
1. Introduction
The Stable Diffusion series (Rombach al., 2022; Esser al., 2024), along with other latent-diffusionnbassse genera- tive models (Ramesh et al., 2022; Labs, 2024), has revolu- tionized text-to-image generation and related downstream tasks (Zhang et al., 2023; Zhao et al., 2024; Zhang et al.,
In this paper, we first observe that higher guidance weights are associated with larger sample norms in latent space, leading to high image saturation and color distortion. We then provide a theoretical analysis of the CFG mechanism, demonstrating that the linear extrapolation of the score func- tion inevitably causes norm amplification and anomalous diffusion behavior. Building on this analysis, we propose an Angle Domain Guidance (ADG) algorithm, inspired by the high-dimensional Gaussian assumption in latent space
'Equal contribution 'Department of Electronic Engineering, Tsinghua University, Beijing, China "Zhili College, Tsinghua University, Beijing, China. Correspondence to: Yuantao Gu <gyt@tsinghua.cduu
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
ADG (with lows guidanes weight)
ADG (with higha guidance weight)
ADG bwith high guiclance weight)
CFG (with low guidance weight)
CFG (with high guidance weight)
CBG (with high guidunnce weight)
Figure 1. Comparison of ADG and CFG using the same random seed with the prompt "AA bathroom with toilet and bathtub." At low guidance weights, ADG accurately aligns with the prompt, while CFG fails do so. At high guidance weights, ADG remains stable, whereas CFG suffers from abnormal saturation and color distortion.
inherent in variational autoencoders. The ADG algorithm transforms the score function into an expectation over time-dependent distribution, which then enhanced in the angular domain before being re-mapped back into the score function for diffusion. Experimental results on the COCO dataset demonstrate that the ADG algorithm outperforms the baseline, particularly in generating images that better aligned with textual descriptions, underscoring its potential to overcome the limitations of existing methods. The imple- mentation is available at github.oom/jnec441//ACCC
where wo denotes the Wiener process, and Po(-Ic) represents the conditional target distribution, with denoting con- dition (such as text in text-to-image models). Specifically, po(-()) represents the unconditional distribution.
In this work, we adopt the widely used Variance Preserving (VP) SDE (Song al., 2021b) for the forward process:
B(t) dæ, 2
x, + / 2o 1o~po0 po(-lc). (2)
where the dynamics ensure that pr approximates standard Gaussian distribution for sufficiently large T. The transition distribution of æ, conditioned on Wp is given by:
2. Background
2.1. Conditional Diffusion Model
2; 20g N (ay B,I),
Diffusion models composed of two fundamental pro- cesses: forward process that gradually transforms the target distribution into Gaussian noise, and a reverse pro- cess that reconstructs the target distribution from the noise distribution (Song et al., 2021b). The forward process is governed by the following stochastic differential equation (SDE):
where ā, = exp (o. - (T) and B, = 1- ō
The reverse process, which reconstructs the target distribu- tion, corresponds to the time-reversed SDE:
(((( a. 3(t) V log (x.,c)) dt ((t)dw. dē,
(4)
where w represents the reverse Wiener process, pa is the distribution of æ; in (2), and log Pt is referred as the
dæ, = f(a,, t)dt g(t)der ao ~ ~po(lc),
2

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
score function. Alternatively, the sampling process can be described by the following reverse-time ordinary differential equation (ODE):
and the gradient of the log-probability becomes:
V log Po, (x)) (1 w) V log po( + wV logpo(#llc.
8(1) Tt
3(1) 2
dæ
Va, logp:( dt, (5)
Inspired by this formulation, the Classifier-Free Guidance (CFG) method replaces the score function Væ, log pr (a c) with a weighted combination of the unconditional and con- ditional score functions, denoted Setg,w (it 1, e), defined
2
which eliminates the stochastic noise sampling during the process, making it more commonly used in practice.
According to classical results from probability theory (An- derson, 1982), if the reverse-time process-
SDE or ODE-is initialized with şr = pr. then the marginal dis- tributions match at all times, i.e., = Pt for all t E [o, T]. Consequently, if it is possible to sample from pr and ob- tain the score function v log Pt (2 c) for different times t, one can generate samples from the target distribution po by discretizing either (4) or (5).
Seig (T,, = (1 W) a log Pi (
+ w a, log Pt T2 ()),
(8)
in the reverse process. For the ODE sampler, this process can be expressed as:
dæ. = I-2: 8efg, (De, i, c)]
(9)
Since the score function V ( c) is generally in- tractable and pr is approximated as standard Gaussian distribution, practical first-order discretization methods for (4) and (5) correspond to the DDPM sampler (Ho et al., 2020) and DDIM sampler (Song et al., 2021a), respectively:
In the above equation, w, referred to as the guidance weight, quantifies the strength of the tilting toward the conditional distribution which is always bigger than 1. However, due to the non-commutativity of the tilting process with the forward process, for any >>> 0,
(10) where (x)) is the distribution of o, in (2) with po(-lc) replaced by Pr..- This discrepancy implies that CFG does not strictly correspond sampling process targeting Pa.w as the true target distribution.
V log (t, &
αt-At At
+ (1-
Õt âi-sc)
so(2,,t,c)
āt-Ai 17,
01-A
04 At
0g + 2
(1
at
80 s0( t, c),
3. Revisiting Classifter-Free Guidance
3.1. Norm Amplification in CFG
where (a c) is a neural network trained to approxi- mate the score function, 77 ~ (O, I), and æy ~ (0, (0,1). I).
While Classifier-Free Guidance (CFG) effectively addresses the limitations of vanilla conditional diffusion models, introduces notable side effects. Specifically, increasing the guidance weight has a significant impact the generated samples: they exhibit larger fo-norms, and the correspond- ing decoded images show higher saturation levels, often accompanied by distortion. Figure illustrates this phe- nomenon, depicting how the 42-nrrm of wo varies with changes in the guidance weight, and the relationship be- tween the latent variable norm and image saturation.
To enhance the stability of neural network training, in prac- tice, the network is trained to approximate -V3.V V log by cg. rather than directly approximating V log.p by 88- This approach mitigates numerical instabilities and im- proves convergence during training.
2.2. Classifier-Free Guidance
While the theoretical foundations of vanilla conditional sam- pling based on diffusion models are well-established, di- rectly using (4) or (5) for generation often results in low- quality outputs (Ho & Salimans, 2021). To address this limitation, the Classifier-Free Guidance (CFG) technique was introduced. CFG aims to sample from modified dis- tribution po,,(2)) x Po (o))) (c(z), which represents the unconditional data distribution tilted by the conditional like- lihood. Using Bayes' theorem, this distribution can be re- formulated as:
This phenomenon can be understood through the interplay between the conditional and unconditional distributions. When CFG operates on conditional probability distribution that aligns with the "surface" of the unconditional distribu- tion, the generated samples tend to deviate further from the overall distribution, resulting in larger norms. To formalize this, we consider the case of a Gaussian mixture model:
Po poxx)=
Σ (aN aml
(11)
(x)) x pb 1-- (x)) (x)c).
3

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
where me denotes the mixing coefficient satisfying Ee Te = 1, and the conditional distribution is given by:
move toward the outer regions of the distribution compared to samples generated by standard conditional diffusion (5). This tendency is captured in Theorem 3.2, which quantifies the norm amplification effect.
Po(2]c) = (1(2) I).
Definition 3.1 (surface class), A class ct is surface class if there exists hyperplane defined by w x + b = 0 such that w M.. + 6 = 0, and for all c + c", (w n. + b) < O, indicating that μe lies strictly on the same side of the hyperplane and w points to the outer side of the uncoditional distribution.
Theorem 3.2 (Norm amplification). For the same initial point æT, let de denote the sample obtained using the CFG sampler (9), and vg denote the sample obtained using the ODE sampler (5). Then:
w
xo w
To.
(12)
indicating that CFG-sampled points align more strongly with the outer direction defined by w.
We focus on a surface class c², defined as class whose mean Mee- lies at a vertex of the polytope formed by C [pe)- -1- This concept is formalized in Definition 3.1.
The result highlights how CFG encourages samples move toward outer regions, with their f2-norms growing as the guidance weight increases. Figure 3 visualizes this behavior, showing how samples deviate from the data distribution under different guidance weights.
225
ADC
C3
2.00
More
1.75
3.2. Anomalous Diffusion in CFG
1.50
Building on Theorem 3.2, we further investigate the geomet- ric and probabilistic properties of CFG sampling in the latent space. Specifically, we define family of time-dependent latent manifolds MM as:
Latert
1.00
0.75
M, = far s s. (2,, (x, t, V (x)c")) - 0}. (13)
0.50
0.5
10
Regions defined by M, exhibit anomalous diffusion, where the update direction of sampled points tends toward areas with lower probability density.
Guidance Weights
2.a Latent Variable Norm with Different Guidance Weights
250
Theorem 3.3 (Anomalous diffusion). For the unit normal vector w of the hyperplane in Definition 3.1, there exists constant C > O, dependent on po. w, and I, such that:
Samoles
Regression
200
aette + kw MMs E V
(14)
Furthermore, the constant C1 increases as the guidance weight w become larger.
Theorem 3.3 establishes that CFG sampling induces anoma- lous diffusion phenomena within the neighborhood extend- ing beyond the boundaries of the conditional distribution. As the guidance weight increases, the anomalous regions defined by Mr expand, amplifying the deviation of CFG- generated samples from the original conditional distribution.
50
Latent Vartable Norm
2.b Relationship Between Latent Variable Norm and Image Saturation
To provide theoretical insight while maintaining concise- ness, we outline the proof of Theorem 3.3. First, we charac- terize the score function via Lemma 3.4.
Figure 2. Average latent variable norm (&2) and image saturation under different guidance weights using CFG with SD3.5 (d=38). Prompts are randomly selected from 100 samples in the COCO dataset. As the guidance weight increases, the norm of the hidden space samples generated by the CFG algorithm increases, which leads oversaturation of the corresponding images.
Lemma 3.4 (Lemma of (Huang et al., 2023)). For any E (0, T). the score function is given by:
Vā (c)
V (o/p)
(15)
Samples generated by CFG diffusion (9) are more likely to
B

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
3.a =
3.b
3.c 5
Figure 3. Generated sample distributions of four-component Gaussian mixture model under varying guidance weights. True component means are marked, and samples corresponding to surface classes move further from the distribution as w increases.
where
reinforces To linearly, leading to generated samples with excessively high norms. While CFG effectively captures features aligned with the target distribution, it also inflates sample magnitudes, resulting in oversaturation, artifacts, and unrealistic images at high guidance weights. To miti- gate these issues, we propose the Angle- Guidance Sampling (ADG) method, which focuses on the directional alignment of samples with the target while limiting changes in their magnitudes. A simplified version of ADG is pro- vided in Appendix E as Algorithm 8.
(c) 1
Exp x. [oo)
|| Vā,wol 1
ac,t, (Zo) (o(ooll) exp
23
Lemma 3.4 shows that the score function can be character- ized by the expectation of the conditional distribution tilted by a time-dependent Gaussian distribution. Using this result, Sefg.w can be rewritten as:
Or (#)
(c) (0)
4.1. Motivation for ADG
Sofg,w (w 1)
Be
The primary motivation underlying ADG is that the direc- tional information encoded in drg is often sufficient guide samples toward the target distribution. Latent space dif- fusion models are typically based on the assumption of high-dimensional isotropic Gaussian inherent in variational autoencoders (Kingma, 2013), which concentrates around spherical shell with fixed radius (Wainwright, 2019). Although due to limitations in network capacity, training data, and other factors, the latent space distribution cannot be fully modeled by high-dimensional Gaussian, this as- sumption still offers valuable insights. Specifically, CFG can be seen as enhancing do in the linear domain, while emphasizing the differences in both the angle and magni- tude between (c) and ( While norm adjustments may marginally improve semantic alignment, excessive norm amplification at high guidance weights becomes detrimen- tal, resulting in oversaturation and distortion. Therefore, ADG shifts the focus differences in the angular domain, while constraining variations in the magnitude domain.
x
(16)
3
The distinction between Selg,w and V log logpe(xxlc") lies in the first term of (16). For a point a kw with sufficiently small k. we find:
w (())
(1)) e,
(17)
where e > 0. Further, noting that Væ log Pr (x)):(()e))) e*)) ) -kw, we have:
de. T (x,t, t, c"),,,""VVV" log (o < ck + 22
(18)
thereby proving the original proposition.
From the analysis, it can be seen that the CFG method is equivalent to enhancing & to xx)()) + (w (w)1)(# 20
While this enhancement effectively amplifies features that align with the target distribution, it unintentionally leads to excessively large norms, causing oversaturation and distor- tion artifacts when the guidance weight is high.
4.2. Methodology
The proposed ADG algorithm comprehensively described in Algorithm 1. Figure 5 illustrates the distinct behaviors of ão under ADG compared CFG, emphasizing the en- hanced stability and directional control achieved by ADG.
4. Angle-Domain Guidance Sampling (ADG)
Building upon the insights presented in Section 3, we iden- tify Classifier-Free Guidance (CFG) as an operation that
5

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Figure 4. Comparison of generated images using different guidance strategies for the prompt "Part of a small bathroom in need of repair." The rows correspond to different guidance algorithms: ADG (top), CFG (middle), and CFG++ (bottom). columns represent increasing guidance weights: 2, 4, 6, 8, and 10 (from left to right).
Unlike CFG, which often leads to exaggerated magnitudes Φp. ADG introduces a refined mechanism that prioritizes alignment with direction. Moreover, Proposition 4.1 formal- izes the constraints imposed by ADG the magnitude of 2b. These constraints ensure balanced trade-off between directional guidance and structural integrity, resulting in improved sample quality.
in Algorithm I satisfy the following property:
|0 2l2 S v21l@! l1.
(19)
ADG demonstrates remarkable flexibility, making it adapt- able to a wide range of sampling frameworks. It compat- ible with advanced deterministic samplers, such as DPM- Solver (Lu et al., 2022a;b), as well as stochastic samplers like DDPM. This versatility arises from the fact that all sam- pling algorithms of Dt. Lp. and Gaussian noise. By replacing the original to with To derived from ADG, the method seamlessly generalizes across various diffusion-based approaches. Ad- ditionally, ADG extends naturally to flow flow-matching genera- tive models (Lipman et al., 2023), which are mathematically equivalent to diffusion models with some assumptions et al., 2024). Detailed derivations of ADG's extension to flow fow-matching models provided in Appendix F.
Algorithm Angle
Angle-Domain Domain Guidance Sampling (ADG)
be expressed weighted combinations
Require: Zr (Q I),1 < W E I. Decoder D
for / T to do
a (x:
vI areo t, c))/
a = (x:
vI (xco t, 0))/ Vai
(z())) a/e)
7 arccos
! [|2| & (00)
Tu threshold(
1)7, T/3)
Co.w
= cos(Ti) )a0g + sin(tw)
(c)
sin(?)
(#)
(e) proj (2)
20
(c)
P:-1 vār- 1000 + VI O:-1 V1-ac
a. 50
5. Experiment
end for
I D(xn)
5.1. Text-To- To-t-TT-Immge Task
To evaluate the effectiveness of our proposed method, we conduct experiments using Stable Diffusion v3.5 (d=38)
Proposition 4.1. For any E (0, T), a and ( defined
6

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Table Results of 10 NEE generation with SD v3.5 (d=38) COCO1OK, hest performance includes finer grained data.
4
METHOD
CLIP 1
IR
FID 4
CLIP
IR f
FID
CLIP
IR f
FID +
ADG (PROPOSED)
0.315
0.727
17.4
0.319
0.928
17.1
0.321
0.964
16.9
CFG
0.316
0.711
17.5
0.318
0.835
16.9
0.317
0.586
17.1
CFG++ w/12.5)
0.315
0.574
17.1
0.306
-0.148
18.0
0.282
-0,980
19.8
APG
0.315
0.651
17.5
0.317
0.910
17.2
0.319
0.964
16.9
10
15
BEST PERFORMANCE
CLIP f
IR 1
FID 4
CLIP
IR 1
FID
CLIP
IR 1
FID 4
ADG (PROPOSED)
0.322
0.970
16.6
0.324
0.940
16.8
0.324
0.970
16.6
CFG
0.304
-0.142
17.7
0.290
-0.659
20.3
0.318
0.843
16.6
CFG++(A w/12.5)
0.257
-1.509
21.4
0.246
-1.680
22.6
0.315
0.574
17.1
APG
0.318
0.935
16.4
0.316
0.691
16.1
0.319
0.966
16.1
weights, maintaining stable performance under condi- tions where baselines exhibit significant color distortions and quality degradation. Second, ADG consistently outper- forms baselines across all guidance weights in text-image consistency metrics (CLIP and IR) while remaining compet- itive FID. Notably, ADG achieves substantial improve- ments in IR, indicating better alignment with human pref- erences. Finally, ADG exhibits wide operational range, delivering superior generation performance across broad spectrum of guidance weights. In contrast, baseline methods experience rapid performance decay beyond their optimal operating points, highlighting the limitations of these meth- ods comparison ADG.
2CRC
26
8
Figure 5. Comparison of angular domain guidance (ADG) and classifier-free guidance (CFG). The left diagram illustrates the ADG update, which preserves the latent variable norm by focusing on angular adjustments, while right diagram shows the CFG, which amplities the norm due to direct linear adjustments. The black line represents the potential manifold structure.
5.2. Ablation Study
We investigate two variants of the proposed ADG algorithm, described Appendix as Algorithm 6 and Algorithm The first variant removes the maximum turning angle con- straint, while the second introduces normalization to the norm of to Table 2 presents the performance metrics for these variants under = 8.
on the COCO dataset([.in in et al., 2014), comparing against CFG (Ho Salimans, 2021), CFG++ (Chung et al., 2025), and APG (Sadat al., 2025). For CFG++ and APG, we adopt the recommended parameter setting from the origi- nal papers. We assess sample quality using three metries: FID (Heusel et al., 2017), CLIP score (Radford et al., 2021), and Reward (Xu et al., 2024). ImageReward is de- signed to align with human preferences, evaluating images based on text alignment, aesthetic appeal, and safety con- siderations. Quantitative results are summarized in Table 1, while Figure 4 provides a qualitative comparison of gen- erated images. To ensure fair comparison, fine- ane-grained experimental results for the baseline methods are included in the appendix. The "Best performance" column in Table aggregates these results.
Table 2. Results of 10 NFE gemeration with SD v3.5 (d=38) COC010k under w = 8.
Method
CLIP t
IR 1
FID J
ADG (Proposed)
0.322
0.970
16.7
ADG w/o angle constraint
0.275
-0.782
28.6
ADG normalization
0.322
0.958
16.6
The experimental results indicate that normalization has limited impact ADG's performance. The comparable performance between the normalized ADG and the original framework demonstrates that the ADG algorithm effectively controls the norm of generated samples. This finding further confirms that angular domain adjustment, rather than norm adjustment, serves as the primary mechanism for enhancing
Our experiments reveal several key advantages of ADG. First, ADG demonstrates robustness high guidance

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
6.a ADG
6.b ADG without angle constraint
6.c ADG with normalization
Figure 6. Comparison of different ADG variants. ADG without angle constraint exhibits significant degradation, while ADG with normalization remains stablc.
text-image alignment.
tion. ADG addresses critical limitations of CFG, including norm amplification and oversaturation under high guidance weights, by leveraging angular-domain updates in the latent space. This heuristic approach stabilizes the sampling pro- cess and consistently improves semantic alignment, image fidelity, and diversity metrics.
Furthermore, the necessity of the maximum turning angle constraint is evident. Removing this constraint leads to a substantial decline in performance across all evaluation met- rics. As illustrated in Figure 6, the unconstrained variant exhibits severe instability, particularly when the angular de- viation (w 1)7 urpasses the critical threshold of x, leading to pathological guidance directions and ultimately catas- trophic generation failures.
The theoretical analysis presented in this work primarily focuses on the limitations of CFG. Specifically, demon- strate how Linear-domal amplification in CFG leads to deviations from the target distribution, resulting in undesir- able artifacts such as oversaturation. While these insights informed the design of ADG, it important to note that, like CFG, ADG remains heuristic approach without formal theoretical guarantees.
5.3. Compatibility with High order Samplers
To demonstrate the compatibility of our proposed Angular Domain Guidance (ADG) algorithm with different mod- els and samplers, we conducted experiments with SD v2.1 using DPM-Solver sampler(Lu et al., 2022a). As shown in Table 3, ADG consistently outperforms CFG across evaluation metries, achieving higher CLIP scores and Im- ageReward while maintaining competitive FID performance. These results highlight ADG's seamless integration with high-order samplers and its ability to enhance performance across multiple dimensions.
In comparison to existing methods, ADG introduces a novel perspective by shifting the focus from linear to angular- domain adjustments. This shift not only mitigates the iden- tified issues with CFG but also ensures seamless integra- tion with current samplers and latent diffusion frameworks. However, ADG is specifically designed for latent diffusion models, which dominate modern T2I generation due to their computational efficiency. If future advancements explore diffusion processes in the image domain, ADG may require significant adaptations to remain effective.
Table 3, Results of 25 NFE generation with SD v on COCO using DPM-Solver under w 8.
The ADG framework is highly extensible and can adapted to other generative tasks, such as video synthesis(L.in et al., 2024), 3D content generation(Lin impainting(Sun al., 2025). Future research could also explore integrating ADG with techniques aimed at reducing sampling iterations(Li & Cai, 2024; Lu et al., 2022a), to enhance its efficiency for real-time applications. These extensions would further solidify ADG's role as a versatile and robust solution for advancing generative modeling.
METHOD
CLIP
IR t
FID |
al., 2023) depth
ADG (PROPOSED)
0.322
0.467
16.6
CFG
0.313
0.399
16.6
6. Diseussion
Angle Domain Guidance (ADG) is introduced as an al- ternative to the widely adopted Classifier-Fas Guidance (CFG) latentdiifusion--alt text-to-image (T2I) genera-
8

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
7. Conclusion
CFG++: Manifold-constrainedd classifier free guidance for diffusion models. In The Thirteenth International Conference on Learning Representations, 2025.
We present Angular Domain Guidance (ADG), a novel ap- proach designed to address the limitations of Classifier Free Guidance (CFG) in diffusson-basse text-to-image genera- tion. By leveraging angular-domain updates in the latent space, ADG enable stable and high-quality image genera- tion. Experimental results demonstrate that ADG consis- tently outperforms existing methods across multiple evalua- tion metrics, including semantic alignment, image fidelity, and diversity. These findings underscore ADG's potential as robust and effective alternative for advancing text-to- image generation.
Esser, P., Kulal, S., Blattmann, A., Entezari, R., Müller, J., Saini, H., Levi, Y., Lorenz, D., Sauer, A., Boesel, F., et al. Scaling rectified flow transformers for high nigh-resolution image synthesis. In Forty-first International Conference on Machine Learning, 2024.
Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., and Hochreiter, GANs trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems, 30, 2017.
Acknowledgements
Ho, J. and Salimans, T. Classifier-free diffusion guidance. NeurlPS 2021 Workshop on Deep Generative Models and Downstream Applications, 2021.
This work was supported by NSAF (Grant No. U2230201) and a Grant from the Guoqiang Institute, Tsinghua Uni- versity. The authors are affiliated with the Department of Electronic Engineering and the Beijing National Research Center for Information Science and Technology Tsinghua University.
Ho, J., Jain, A., and Abbeel, P. Denoising diffusion proba- bilistic models. Advances in neural information process- ing systems, 33:6840-6851, 2020.
Huang, X., Dong, H., Yifan, H., Ma, Y., and Zhang, T. Re- verse diffusion monte carlo. In The Twelfth International Conference on Learning Representations, 2023.
Impact Statement
This work re-examines the foundational mechanisms of guidance in latent diffusion models and presents a princi- pled alternative to the widely used Classifier-Free Guidance (CFG). By conducting a theoretical analysis of CFG's fail- ure modes-aaeee particulaall norm amplification and anomalous diffusion under high guidance weights-thh the paper identifies key limitations in -omain extrapolation. The pro- posed Angle Domain Guidance (ADG) algorithm addresses these issues through angular-domain updates, offering conceptually novel and practically effective framework. By shifting guidance from algebraic extrapolation to geomet- ric rotation, ADG provides a new lens for understanding and designing structure-preservinn sampling strategies in generative modeling.
Kingma, D. P. Auto-encoding variational bayes. arXiv preprint arXiv: 2013.
Labs, B. F.
Flux. https: / l.igghh...oom com/ 2024.
Li, G. and Cai, C. Provable acceleration for diffusion models under minimal assumptions. arXiv preprint arXiv:2410.23285, 2024.
Lin, C.-H., Gao, J., Tang, I.., Takikawa, T., Zeng, X., Huang, X., Kreis, K., Fidler, S., Liu, M.-Y., and Lin, T.-Y. Magic3d: High-resolution text-to-3d content cre- ation. In Proceedings of the IEFF/CVF conference on computer vision and pattern recognition, pp. 300-309, 2023.
References
Lin, T.-Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ra- manan, D.,, Dollár, P., and Zimick, C. L. Microsoft coco: Common objects in context. In Computer vision-ECCV 2014: 13th European conference, zurich, Switzerland, September 6-12, 2014, proceedings, part v 13, pp. 740- 755. Springer, 2014.
Anderson, B. D. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313- 326, 1982.
Bradley, A. and Nakkiran, P. Classifier-frea guidance is a predictor-corrector,, arXiv preprint arXiv:2408.0000 2024.
Lipman, Y., Chen, R. T., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. In 11th International Conference on Learning Representations, ICLR 2023, 2023.
Chidambaram, M., Gatmiry, K., Chen, S., Lee, H., and Lu, What does guidance do? a fine-grained analysis a simple setting. In The Thirty-eighth Annual Conference Neural Information Processing Systems, 2024.
Liu, Y., Zhang, K., Li, Y., Yan, Z., Gao, C., Chen, R., Yuan, Z., Huang, Y., Sun, H., Gao, J., al. Sora: A review on
Chung, H., Kim, J., Park, G. Y., Nam, H., and Ye, C.

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
background, technology, limitations, and opportunities of large vision models. arXiv preprint arXiv:2402.17177, 2024.
Wu, Y., Chen, M., Li, Z., Wang. M., and Wei, Y. Theoretical insights for diffusion guidance: case study for gaussian mixture models. In Forty-first International Conference on Machine Learning. 2024.
Lu, C., Zhou, Y., Bao, F., Chen, J., Li, C., and Zhu, J. DPM- solver: A fast ODE solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems, 35:5775- 5787,, 2022a.
Xia, M., Xue, N., Shen, Y., Yi, R., Gong, T., and Liu, Y.-J. Rectified diffusion guidance for conditional generation. arXiv preprint arXiv:2410. 18737, 2024.
Lu, C., Zhou, Y., Bao, F., Chen, J., Li, C., and Zhu, J. DPM- solver++: Fast solver for guided sampling of diffusion probabilistic models. arXiv preprint arXiv:221 1.01095, 2022b.
Xu, J., Liu, X., Wu, Y., Tong, Y., Li, Q., Ding, M., Tang. J., and Dong, Y. ImageReward: Learning and evaluating hu- man preferences for text-to-image generation. Advances in Neural Information Processing Systems. 36, 2024.
Zhang. J., Zhuang, J., Jin, C., Li, G., and Gu, Y. Unleashing the denoising capability of diffusion prior for solving in- verse problems. The Thirty-eighth Annual Conference Neural Information Processing Systems, 2024.
Patel, Z., DeLoye, J., and Mathias, L. Exploring diffu- sion and flow matching under generator matching. arXiv preprint arXiv:2412.1100 110244 2024.
Radford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp. 8748-8763, PMLR, 2021.
Zhang, L., Rao, A., and Agrawala, M. Adding conditional control to text-to-image diffusion models. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 3836-38477 2023.
Zhao, S., Chen, D., Chen, Y-C.,, Bao, J., Hao, S., Yuan, L., and Wong, K.-Y. Uni-controlnet: All-in-one control text-to-image diffusion models. Advances in Neural Information Processing Systems, 36, 2024.
Ramesh, A., Dhariwal, P., Nichol, A., Chu, C., and Chen, M. Hierarchical text-conditional image generation with clip latents. arXiv preprint arXiv:2204.06125 1(2):3, 2022.
Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent diffuusion models. In Proceedings of the IEEE/CVF con- ference on computer vision and pattern recognition, pp. 10684-10695, 2022.
Sadat, S., Hilliges, O., and Weber, R. M. Eliminating over- saturation and artifacts of high guidance scales in diffu- sion models. In The Thirteenth International Conference on Learning Representations, 2025.
Song, J., Meng, C., and Ermon, S. Denoising diffusion implicit models. In International Conference on Learning Representations, 2021a.
Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A.. Er- mon, S., and Poole, B. Score- based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021b.
Sun, T., Hu, D., Dai, Y., and Wang, G. Diffusion-based depth inpainting for transparent and reflective objects. IEEE Transactions Circuits and Systems for Video Technology, 35(1):394-404, 2025, doi: 10.1109/TCSVT. 2024.3434740.
Wainwright, M. J. High-dimensional statistics: A non- asymptotic viewpoint, volume 48. Cambridge university press, 2019.
10

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
A. Related Work
A.I. Theoretical Analyses of Classifier-Free Guidance
To the best of our knowledge, four key studies have provided theoretical analyses of Classifier-Free Guidance (CFG). Below, we elucidate the connections and distinctions between our theoretical contributions and these prior works.
Comparison to Chidambaram et ahhhhaaamaa et al., 2024).
This work studies the performance of the CFG-based ODE sampling method in two scenarios: (1) one-dimensional compactly supported distributions with two components, and (2) one-dimensions Gaussian mixture distributions with two components. They conclude that as the guidance weight increases, the sampling process tends to concentrate on the edges of the conditional distribution. Additionally, even a small nonzero error in score estimation can cause sampling results to deviate significantly from the target distribution's 's support under sufficiently large guidance weights.
The key difference in our theoretical analysis lies in its broader applicability to nigh-dimensional settings with multiple components. We demonstrate that the phenomenon of sampling concentrating on the edges only occurs for surface classes, distinction that is absent in (Chidambaram et al., 2024) due to the nature of their one-dimensional two-component scenarios, where all classes inherently behave as surface classes. By incorporating this insight, our work provides more nuanced understanding of CFG's behavior in complex, high-dimensional generative tasks, offering theoretical explanations for phenomena that remain unaddressed in their framework.
Comparison Wu et al. et al., 2024).
This work investigates the impact of the guidance weight on sampling performance under Gaussian Mixture Models (GMMs). The authors evaluate two key metrics: "classification confidence," characterized by the conditional likelihood of the output distribution, and "distribution diversity." quantified by the differential entropy of the output distribution. They theoretically prove that CFG sampling not only increases classification confidence but also reduces distribution diversity.
Our work differs from theirs in two significant aspects. First, we operate under more relaxed assumptions; while (Wu et al., 2024) requires the components of the Gaussian mixture to be approximately orthogonal, our analysis accommodates more general configurations of GMMs. Second, we argue that high classification confidence is not always desirable. For example, in one-dimensional GMM with two components centered at I and -1, respectively, and sharing the same variance 1, sampling point located at 1000 exhibit extremely high classification confidence but would not constitute meaningful sample from the target distribution.
Comparison to Bradley and Naradey & Nakkiran, 2024).
This work analyzes CFG under the assumption that both the conditional and unconditional distributions follow pero-mean meedimensionall Gaussian distributions. Under this assumption, they derive a closed-form solution for the output distribution, showing that it does not correspond to the expected gamma-weighted distribution. Additionally, they conduct numerical studies on one-dimensional two-component Gaussian mixture models, finding that the output distribution similarly fails to match the expected gamma-weighte distribution.
The authors further reinterpret CFG as a predictor-correcct method, alternating between a denoising predictor (based on the ODE of the conditional distribution) and sharpening corrector (employing Langevin dynamics). This perspective provides new insights into the iterative nature of CFG sampling.
In contrast, work extends beyond the limitations of one-dimensional Gaussian assumptions. We study CFG in high- dimensional settings involving multi-component distributions and introduce the notion of surface classes. Our analysis reveals that phenomena such norm amplification are closely linked to the geometry of these surface classes, providing a more general and nuanced understanding of CFG behavior in complex generative settings.
Comparison to Xia et (Xi et al., 2024).
This work builds on the findings of (Bradley & Nakkiran, 2024) and extends the data assumptions to high-dimensional isotropic Gaussian distributions with different parameters for the conditional and unconditional distributions. Under these assumptions, the authors derive closed-form solution for the CFG output distribution and further confirm its inconsistency with the gaiga-weighted distribution. Additionally, they propose relaxing the constraints in the gamma- weighted distribution by introducing more flexible guidance coefficients, allowing the corrected distribution to better align with diffusion theory.

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
In contrast, abandon the Gaussian distribution assumption gain deeper insights into the behavior of CFG outpu distributions in more general scenarios. Our analysis covers high-dimensional settings with multiple components and highlights the geometric and probabilistic distinctions that emerge in such cases. Specifically, we introduce the concept of surface classes, demonstrating that the phenomena of norm amplification and edge sampling predominantly occur for surface classes. By broadening the theoretical scope and proposing practical solutions such as angular-domain adjustments, our work addresses the limitations of CFG in real-world generative tasks with greater generality and flexibility.
A.2. Improvements to Classifier-Free Guidance
Comparison to CFG++(Chung et al., 2025).
This work, inspired by diffusion model-based inverse problem solvers, proposes performing denoising under the guidance of a weight smaller than followed by re-noising guided by the unconditional distribution. However, this method is
Virai-1
CFG++ remains confined equivalent to using a time-varying guidance weight in CFG, defined as wi => > Vir0e-1- P4-
to the framework of linear-domain guidance and still exhibits color distortions under high A. Additionally, our experiments reveal that its optimal guidance weight is highly sensitive to the inference step size, further limiting its applicability (see Appendix G.5).
Algorithm 2 CFG++
Require: ør ~ ((0,,),0<
for T to do
[c]
X) (eg(
#0
(ic)
(at v1 o((E,w)/ V0t
(c)
Ft-1 vā (c)
vI (0,
end for
Comparison PCG(Brad & Nakkiran, 2024).
This work decomposes each iteration of the DDPM-style CFG sampler into denoising step, corresponding DDIM, and a sharpening step, corresponding to one step of Langevin dynamics. The proposed PCG algorithm performs one denoising step followed by N sharpening steps within each iteration, aiming to enhance the output. However, this enhancement is also carried out within the linear domain. Due to the N-fold increase in the number of function evaluations (NFE) compared to standard CFG algorithms, its practicality is significantly reduced. Furthermore, the authors explicitly state, "we present PCG primarily as a tool understand CFG." As such, we did not compare the proposed algorithm in this work with our method.
step of
Algorithm 3 PCG
Require: DT ~ (N(0,I),1 <EER
for t to 1 do
-3,co (Ite -Bico(Ze;; c)
((() = (x: VI-
0reo (zz t, c,c))//v
Ft-1 va
VI 0,
81
- m
vi
K = [ &r
āc-1
for k = to N do
1 ~ I0.
c- = (1 - w) t 1,0) + weo - c,
(e)
It- = xt-1
f-1x
8, + Vin
end for end for
12

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Comparison ReCFG(Xia et al., 2024).
This work relaxes the constraint of the weighting coefficients summing to one in traditional CFG by introducing precomputed lookup table f. Strict implementation of ReCFG requires precomputing this lookup table for all conditions, which is impractical for open-condition models like T2I. For datasets like ImageNet, where images have explicit category labels, the authors propose precomputing the table based on categories rather than text conditions. However, in real-world scenarios, text prompts do not always correspond to specific categories, making this method unsuitable for general T21 tasks. Consequently, we do not compare their algorithm with ours.
Algorithm 4 ReCFG
Require: ær ~ (V( I).1 < w E I, trained lookup table f.
for t T to do
A f(c)
1, = X(1 - w))
[c]
xr, 0) + weo(2,
60 = (It v1 (() \/V
Ft-1 = vāc- + VI 16,16
end for
Comparison APG(Sadat al., 2025)
This work attributes image oversaturation and degradation to the parallel component of the difference vector A a -=(") with respect to ( denoted as o Building on this observation, they propose reducing the influence of the parallel component in CFG by replacing AF A@o.j| + A with A 7/ A where v to mitigate its adverse effect image quality. In addition, this work introduces extra negative momentum mechanism and imposes a constraint on the norm of Aão. While removing the parallel component slows the norm growth, it does not fully resolve the issue of norm amplification. However, its algorithm does not directly constrain the norm of A and thus fails to address the issue of excessive latent norm under high guidance weights. As a result, image degradation still occurs when the guidance weight is large.
Algorithm 5 APG
Require: Ør ~ (O, I),1 w WER <0, <, R,, < <
Adoo =
for t T do
i (I: vI âies xz, t, c))/
20
(0) (I, -11 -
(x, t, (0)0)////
(c)
(0)
Adg 20 an
A&l (Aáo 1/ (())
(a) A
AĀT = Ado Ad
AZo A Il
Aāo
Ado Ado min 1, |
Aeg,history
Aão AA Dihittoryy
No.w 31 + (w - 1) A0
2t-1 10 + VI 22 Go
V1-āi
end for
B. Proof of Theorem 3.2
We now consider a strengthened proposition of Theorem 3.2:
Theorem B. Assume the data follows the model (11) and ot the surface class of the distribution. Consider two state
13

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
variables wa and Z₁ governed by the following ODEs:
dæ;
{(t)
l-x Væ, logpi
(20)
df
dz:
3(()
1-2 2t +1+ logre(z.lc") w., log (z:)]=
(21)
dt
If the initial conditions satisfy TT S 2 then for any 1 E [0, T) the following inequality holds:
T; Me < 2 pe
Proof. For Gaussian distributions and Gaussian mixture distributions, the score functions admit closed-form expressions. For the conditional score function:
Va logp((xlc") V (- log(2T)
| 0z 11
2 aift
(22)
and for the marginal score function:
V De To((ol(mueo)) logpe(x))
E (o((o(
-x- á
2 (
(23)
where (x) Σタ J (2) [J)
Ta 1
Substituting (22) and (23) into (20) and (21) yields:
dæ B(t)
(24)
dt
dz
B(t))
- wāi Ho Σ nhe (1)) )
(25)
dt
By taking the projection along wi T Mo. we have:
dw T x,
{(t)
[ā T
Mo+]
(26)
dt 2
dw 2,
3(t)
w,w" Me wā, Σ [σ(2)) (w pe w uc)
(27)
From Definition 3.1, it follows that:
w T
Mo. w Ho < <0, Vc
which implies:
dw 21
dw T Et VI E (0,T).
dl
dl
Using the ODE comparison theorem and the initial condition conclude that:
T u S xTT.. (notice that the ODE is time-reversed), we
x He < 2, Hot, E (O,
14

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
C. proof of Theorem 3.3 Using Lemma 3.4, we express the score functions as:
V logp:( Vir, mo -ot a.m [oo] - x
(28)
B
ValL 98.,0, [2o] - 2
log))(x)(0)
(29)
Bg
where the conditional distributions get,t,m and 1 given by:
| - v 00 113
le (20) x ( Mor exp
(30)
2/3
(-11t
v / 11
(00)
WoN WaN(aro exp
(31)
23g
For aet,1 ,a, we calculate:
| (-11e
v 0020l
|| V 04.00 112
N( Me I) exp
x exp
-1 142 112
(32)
233
23,
(-11+
at
v V
x exp
1
26 wo +
Ma.
+
200
(33)
31
3g
Since 3, -- and Cr =e 22- the expectation is:
E Emo [ro] = Bept. + Voga.
(34)
Substituting this back, we find:
V logp)( Vor(Bi + +vaaa
(35)
Br
=Voitt - x.
(36)
Thus, for x Vortlee + knw, we have:
Va log kir.
(37)
similarly, we have
Σ waNiaro: Me, exp
(-
[_1x
v ar
W. exp -1|200 -Mel12
x
v
20 11
(38)
2,,
2,3r
α
2 o=l
w
(1) N
xo Beo: + Vasit
1
(39)
3r
where
Wa w. exp (110 - voruell
(1)
(40)
Let
WeN(x;; vasMo. 1)
w.
Di=1 wiN(x;; Iarmy.1)
15

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
We then have:
Bete, + vāna, 1 )
f0 (Du)
Σ
To:
To:
(41)
BBa
From this, the expectation of 2o with respect to 16,r. becomes:
I3 20 18 [ro] = Σ (3(3) (Bitle + vara)
= Vvr + Be Σ w
(42)
Now consider the difference between the expectations under 9ct ,ta and 90,1.=
E Ez
[2o)
E Ex
[20]]
3e
Σ
w (1r. 140
Be
w (3)
e, (M Mc),
(43)
exce
where the second equality uses the fact that for = E", the term cancels out.
Next, we calculate the dot product of Sefaw with V (oolp())
Sefguw Vlogpi(x)c')= (w- 1) v ae [Bx 1.
[oo] E -uu s. [xo))
Vlogp((</("))
v llgg((<))c))
B
=-k(w- 1)
(()
(M.. pc))
w + w w.
(44)
Bg
c/c
where the second term Po e*) contributes w T w and the projection difference in expectations contributes the first term.
Without loss of generality, assume is unit vector. From Definition 3.1, we know that p1.- w 14. for + c². Define the following terms for clarity:
8 min (1a 14.) Tw 0,
(45)
cgc
R² max 111..12
(46)
min We
(47)
ogfe We-
Now consider the dot product of 8efg.w and V Vlogp((<r((")) For Va, R, we have:
ofg. V c -kiw - 1) Vā
w 8 + 12. c.m
(48)
Be
cafe
Substituting the expression for ( we get:
WaN(xx: v I
8 8//9 v Voggp((x.c))
Skk k(w 1)
(49)
31
cc - wiN(x; Varp4.
16

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Using the decomposition of the weights, write:
Σ w.N(x; Varka =1 Wo ,
(50)
cffe Ei Ei (x; I
Ei= wN;(2; vorlu.
Substitute the Gaussian density terms and simplify:
(a::N(x;; Vorlter
(51)
Eis (a: (osp,,)
Ma 1 E+400 we Mo
Using the exponential decay property of the Gaussian distribution:
N(:; I)
(- IIVan + kw varu
N( va.p... $ exp
(52)
2
Combining terms:
We-N((;; Vorltet Dinuu II
(53)
1 (C exp
na.
Substitute back into the original inequality:
Vot 5
Befg. V log c))
sk2 k(W 1)
1
(54)
Br
1+ (C- 1)xexp
9. R2
For min (VTtR.
(VodR.(W- (w- 11) yā= (
(1)-120111111- 1-(C-1)Aexp/
we conclude:
3t
8ef Vlogpr(zx,c)
D. Proof of Proposition 4.1
For convenience, we define the following notation:
sin(?) (a))) - proj 20 (a) ))
Dasist
It follows that:
1 = 12.000
We now compute the norm of ioo.:
(B Vsin² ((w 1)y) | 11 2si ((w 1)7) cos ((w 1)r) (a a60)) + cos ((w 1)7)
(c) + (2( - -1)7) (Rasist, (())
= | ||2
x
2
[ |
v1 + (2 1)7)
< 2 1
This completes the proof.
17

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
E. Variants of ADG
The implementation details of the two variant algorithms mentioned in Section are outlined here, Algorithm removes the constraint on the maximum turning angle, allowing for more flexible updates. On the other hand, Algorithm normalizes the corrected øy ensure consistency in its magnitude.
Algorithm ADG w/o angle constraint
Require: øT ~ (0, WEER
for t = T to do
20 (a: v1 āgco (x, t, c))c/v// 20
(0) (xz v1 āeco (x, t, 0)) v
(
7 arcoos
|106
11b ||2| 2 l2
Th (w 1)7
F. = cos(11 Yw 0
+ sin(?w) (e) proj (0 (z/)))
sin(?) (00g
It-1 Vā 10 + VI 0,- m1 0,.0
V1-
end for
Algorithm ADG with Normalization
Require: ør ~ (O WER
for f = T to 1 do
ab (x: /1 ârea (x, 1, c.c))/ Va.
(0)) (x: VI aeea (2, t, 0)) V
(m a/e)
7 arccos
|
~ threshold( (w 1)7, /33
C = cos(T1 (Yo + sin(Tw) (e)
sin(?) ( proj [ (2)
Co.w = To |
Ft-1 vac-120,4 + VI a1-1 VI-
21 a 0,00.
end for
Moreover, considering that ADG introduces relatively substantial modifications compared to CFG, which may hinder its ease of adoption in existing frameworks, we further propose a simplified variant that can be more seamlessly integrated into current pipelines. This simplified method is also derived from our analysis of CFG, and directly mitigates the degradation issue by explicitly constraining the norm of ft..
Algorithm 8 Simplified ADG
Require: 2T ~ (0, I),1 < W E
for t T to do
(c) (#: vI (xt 1, cc)c///r
a = (x: /I area (x, 1.0))/var
fo In + (w 12 -
(c)
(c)
| ||
Ao Do,CFG io,.
Ft-1 = 120.A1100AAAG + VI /x-1
ADC
VI-ai
end for
18

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Table 4. Results of 10 NFE generation with SD v3.5 (d=38) on COCO1Ok (CPG).
VALUE
2
w 3.5
4
4.5
CLIPt
0.316
0.317
0.318
0.318
0.318
IRT
0.711
0.831
0.843
0.835
0.792
FID4
17.5
17.2
17.0
16.9
16.6
VALUE
5
8
10
BEST
CLIPT
0.318
0.317
0.311
0.304
0.318
IRT
0.735
0.586
0.219
-0.142
0.843
FID
16.6
17.1
17.2
17.7
16.6
F. Extension of Flow Matching Model for ADG Algorithm
The generative model based on Flow Matching introduces timedeepenent vector field v. dæs, which is used to learn the continuous transformation path from Gaussian distribution Po to target distribution p1 and subsequently perform sampling. Notably, this notation differs from that in diffusion models, where the target distribution is denoted as p1- A key concept in the Flow Matching-based model is the conditional probability paths, which are defined as:
de
p (a,, Ue oz) (x))).
The conditional probability paths most used in generative tasks are expressed as:
p( = Na tan - Tmin 21)
Under this setting, the ideal reference flow is given by:
Eg lan [ 0 min ) )x:
0 0min) )
Thus, the Flow Matching model's Adaptive Directional Guidance (ADG) can be written as:
$ (c) (1 (1 min )tlv (c) (xx) + (1 Tmin)
(55)
i 1 = (1 (1 m v/v (0) (ar:) + (1 Tmin
(0)
(56)
(z) ) x
Y arccos
(57)
||00|00 || | (c)
||2
% threshold(
1)7, π /3),
(58)
1ww cos(Tw)af
sin (w))
sin(7) (z -proj (o(()) )),
(59)
V, = 11 (1 -mmin) )x,
(60)
(1 Jmin
U++ = v,AL + Fz.
(61)
G. Futher Experiment
G.I. Fine Grained Experimental Results
We present the fine-grained experimental results of CFG and CFG+ here. Around the maximum value of the coarse boarse-grained results, the step size is reduced to identify better-performing w values for the reference algorithms. Notably, ADG without fine-grained hyperparameter tuning consistently outperforms the reference algorithms across the metrics and maintains its superiority over wide range of w. This is attributed to its emphasis on angular domain guidance.
19

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Table 5. Results 10 NFE generation with SD v3.5 (d=38) on COCO1Ok (CFG+, X w/1..))
VALUE
1.5
2.5
CLIP
0.296
0.311
0.315
0.315
0.314
IRT -0.161
0.456
0.574
0.477
0.300
FIDL
18.1
17.3
17.1
17.2
17.2
VALUE
w
BEST
CLIP
0.306
0.282
0.268
0.257
0.315
IRT
-0.148
-0.980
-1.290
-1.509
0.574
FIDL
18.0
19.8
20.6
21.4
17.1
G.2. Evaluation of Generated Image Quality with LLM
In this section, we leverages GPT-4o to assess the quality of generated images from multiple perspectives. Our evaluation simultaneously considers three crucial aspects: image quality and authenticity, text-image alignment, and semantic consis- tency. As demonstrated Table 6, the evaluation results reveal that ADG exhibits overwhelming advantages over CFG, particularly under large guidance weights. The superior performance of ADG can be attributed its ability to maintain high-fidelity image generation while ensuring precise alignment with textual descriptions, even when subjected strong guidance conditions. This multi-faceted evaluation approach provides robust and reliable assessment of the generated images.
Table 6. Evaluation of Generated Image Quality with GPT-4o.
2
10
ADG better
228(83.21 %)
1284 %)
4744(95.44%) 179(4.56%)
6276(97.96% 131(2.04%)
8001 %)
CFG better
46(16.79%)
221((4.69%% 8495
81(100%%
Similar
9726
5077
3593
1918
G.3. Evaluation with Complex Text Prompts
To evaluate ADG under complex prompts, curated 500 complex text prompts using GPT. Our experiments show that ADG significantly outperforms CFG under these conditions. For guidance weight = 8, ADG achieves CLIP score of 0.355 and an IR score of 1.566 while CFG scores lower with a CLIP score of 0.338 and an IR score of 0.766. Examples of the generated outputs are partially illustrated Figure 7.
G.4. Qualitative Results at Extreme Guidance Weight
To further evaluate the robustness of ADG under extreme guidance conditions, we conduct additional experiments using a guidance weight of w = 20. Figure presents representative samples generated by both CFG and ADG.
As shown, CFG exhibits severe artifacts, including oversaturation, unnatural textures, and significant semantic drift from the input prompts. In contrast, ADG consistently produces visually coherent and semantically faithful images, even under such high guidance weight. These results further validate the stability and effectiveness of ADG in extreme settings.
G.S. Results with Different Guidance Weight and NFE
We show the methods with different guidance weights and NFEs as in Figures 9 11. Notably, the ADG and CFG algorithms maintain relative stability across various NFEs, while the CFG++ algorithm does not. For instance, with guidance weight of 0.2, the CFG++ algorithm exhibits relatively normal saturation NFE=10. However, at NFE=40, it shows noticeable oversaturation, and similar trends can be observed with guidance weights of 0.3, 0.4, and 0.5.
This behavior may be due to the CFG++ algorithm being equivalent to a time-varying guidance weight CFG algorithm. The time-varying guidance weight, influenced by the discrete step size, likely causes its instability relative to both the CFG and ADG algorithms.
20

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
7.a ADG
7.b CFG
Figure Comparison of ADG and CFG with complex text prompts under While CFG shows notable degradation, ADG with normalization maintains stable performance.
Another noteworthy point is that, for different NFE values, ADG consistently generates images that align well with the textual prompt while maintaining relative stability in image content under the same random seed. In contrast, CFG and CFG++ exhibit significant variations in image content across different NFE values. This further highlights the relative stability of our proposed algorithm.
Figure 9. Generated images using ADG for the prompt "The church is up on the hill in the country" Rows correspond to different numbers function evaluations (NFE): 40 (top). (middle), and (bottom). Columns represent increasing guidance weights: 2, 4, 8, and 10 (from left to right). The same random seed is used across all generations.
21

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
8.a ADG
8.b CFG
Figure 8 Comparison of ADG and CFG with complex prompts at w 20. ADG maintains stability, while CFG produces distorted and misaligned results. The prompt room blue walls white sink and door".
Figure 10. Generated images using CFG for the prompt "The church is up on the hill in the country" Rows correspond to diffcrent
numbers of function evaluations (NFE): 40 (top), 20 (middle), and 10 (bottom). Columns represent increasing guidance weights: 2, 4, 6, 8, and 10 (from left to right). The same random seed is used across all generations.
22

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Figure 11. Generated images using CFG++ for the prompt "The church is up on the hill in the country" Rows correspond to different numbers of function evaluations (NFE); 40 (top), 20 (middle), and 10 (bottom). Columns represent increasing guidance weights: 0.1, 0.2, 0.3, 0.4, and 0.5 (from left to right). The same random seed is used across all generations.
G.6. More Results COCO datasets
In this section, provide more images generated by ADG and reference algorithms for comparison as in Figures 12 to 14.
23

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Figure 12, conditional COCO samples using ADG (w = 6, NFE=10).
24

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
MTM
Figure 13, conditional COCO samples using CFG (w 6. NFE=10).
25

---

Angle Domain Guidance: Latent Diffusion Requires Rotation Rather Than Extrapolation
Figure 14, conditional COCO samples using CFG++ (A = 0.3, NFE=10).
26


Reviews:
[{"review_id": "gCjQtmVfB6", "reviewer": "Reviewer_6TQ2", "summary": "This paper attempts to analyze the distributions of conditional generation vs. unconditional generation, and claims that in some occasions the direction of classifier-free guidance may be \"abnormal\", i.e., leads to low-probability density areas, and proposes \"Angle-Domain Guidance Sampling\" (ADG) as a remedy.\n\nUPDATE after author response:\nThe author response makes me understand the paper and some statements better. In particular, I appreciate the authors provide extra comparative examples with APG. I'd like to raise the rating to weak accept.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their thoughtful comments. While the overall evaluation was critical, your constructive feedback is highly valuable and will help us improve both the clarity and impact of our work. Below, we provide detailed responses to your key concerns.\n\n### 1. Comparison with Adaptive Projected Guidance  \n**Reviewer Concern:**  \nThe reviewer notes the absence of discussion on Adaptive Projected Guidance (APG) [ICLR 2025], suggests high similarity with our proposed ADG method, and questions whether ADG offers substantive novelty.\n﻿\n\n**Response:**  \nWe appreciate the reviewer for pointing out this important and timely work. APG appeared on arXiv in October 2024, shortly before our submission, and was therefore not included in our original analysis. We will include appropriate citations and a dedicated comparison section in the final version.\n\nTo clarify the distinctions between ADG and APG, we summarize the differences below:\n﻿\n1. **Attribution of Image Degradation:**  \n- APG attributes image oversaturation and degradation to the **parallel component** of the difference vector $\\Delta \\hat x_0$ relative to $\\hat x_{0, c}$, denoted as $\\Delta \\hat x_0^\\parallel$.  \n- In contrast, we attribute oversaturation and distortion to the **large norm** of $\\hat x_{0, CFG}$:  \n$$\n\\hat x_{0, CFG} = \\hat x_{0, c} + (\\omega - 1) \\Delta \\hat x_0.\n$$\nRemoving the parallel component slows norm growth but **fails to address norm amplification**. Additional experiments show that normalizing $\\hat x_{0, CFG}$ mitigates image degradation more effectively.  \n[Visual example](https://files.catbox.moe/2msq2b.png)\n﻿\n2. **Algorithmic Difference:**  \n- APG modifies $\\Delta \\hat x_0$ in CFG:  \n$$\n\\Delta \\hat x_{0, APG} = \\eta \\Delta \\hat x_0^\\parallel + \\Delta \\hat x_0^\\perp, \\quad \\hat x_{0, APG} = \\hat x_{0, c} + \\omega \\Delta \\hat x_{0, APG},\n$$\nwhere $\\eta < 1$. However, APG retains **linear enhancement**, leading to norm growth at high guidance weights.  \n- ADG, in contrast, performs angular-domain updates, effectively constraining norm growth (Proposition 4.1).  \n[ADG vs. APG Visual](https://files.catbox.moe/1nnve4.png)\n﻿\n3. **Empirical Performance and Stability:**  \n\nUnder high guidance weights, **ADG successfully mitigates oversaturation and artifacts**, whereas **APG-generated images exhibit significant oversaturation**, which reduces text-image alignment. [visual examples](https://files.catbox.moe/c76emo.png)  \n   ADG outperforms CFG and CFG++ in both alignment (CLIP) and human preference metrics (ImageReward), especially under high guidance weights.  [quantitative results](https://files.catbox.moe/cjzl1o.png)  \n\n﻿\n### 2. Clarification on Theoretical Derivations and Algorithm Motivation  \n**Reviewer Concern:**  \nThe reviewer seeks clarification on how the theoretical derivation leads to the conclusion that \"CFG leads to excessively large norms of features.\" Additionally, they express confusion about the connection between the theoretical derivations and the proposed method. The reviewer also suggests illustrating the mathematical derivations with concrete generation examples where possible.  \n﻿\n**Response:**  \nTheorem 3.2 shows that **linear extrapolation in CFG** shifts samples toward the outer regions of the unconditional distribution, increasing the norms of the latent variables (features).  \n﻿\nThe paper's logical flow is:  \n1. **CFG Limitation:** Norm amplification and anomalous diffusion for surface-class samples (Theorem 3.2 and Theorem 3.3).  \n2. **Source of Norm Amplification:** Linear enhancement of $\\hat{x}_0$ leads to larger norms.  \n3. **ADG Motivation:** Angular-domain updates mitigate norm amplification (Proposition 4.1).  \n\nFig. 3 illustrates this phenomenon with a Gaussian mixture model. The green, blue, and orange clusters represent surface classes, while the red cluster denotes non-surface classes. Higher guidance weights push surface-class samples away from the unconditional distribution, amplifying norms.  \n﻿\n---\n﻿\n### 3. Clarification on “Magnitude Differences Are Secondary” Statement  \n**Reviewer Concern:**  \nThe reviewer finds the statement *\"The focus on magnitude differences is secondary, ...\"* ambiguous.\n﻿\n\n**Response:**  \nWe appreciate the reviewer's feedback. The intended insight is:  \n﻿\n- **Linear enhancement** methods (e.g., CFG) modify both the magnitude and direction of $\\hat{x}_0$.  \n- While **magnitude enhancement** may improve semantic alignment slightly, it becomes detrimental at high guidance weights due to excessive norm amplification, resulting in image degradation.\n#### Revised Statement:\n\"Linear enhancement methods simultaneously modify the norm and direction of $\\hat{x}_0$. While norm adjustments may marginally improve semantic alignment, excessive norm amplification at high guidance weights becomes detrimental, resulting in oversaturation and distortion.\"\n\nWe appreciate your feedback and will revise the paper to clarify contributions and address concerns.", "claims_and_evidence": "1. After two hours of reading, I feel very confused about the theoretical derivations, especially I don't understand why they lead to the \"Angle-Domain Guidance Sampling\" (ADG). In particular,\n    1) The theorems and lemmas in section 3.2 don't seem to lead to the claim that \"CFG leads to excessively large norms of features\". \n    2) More confusingly, when the authors present motivations of ADG, they said \"The focus on magnitude differences is secondary, and in the case of high guidance weights, it can even be detrimental\", so this means that large norms are not important, but angles are important? Then why in the abstract, \"these distortions stem from the amplification of sample norms in the latent space\"?", "methods_and_evaluation_criteria": "1. As listed in \"Claims And Evidence\", the derivations are confusing and I don't understand why ADG is a logical consequence of the theorems.\n2. The empirical evaluation seems to be fine.", "theoretical_claims": "Same as those listed in \"Claims And Evidence\".", "experimental_designs_or_analyses": "The empirical evaluation seems to be fine. However, perhapse the most important baseline is adaptive projected guidance (APG) [a], which is not mentioned or compared with.\n\n[a] Eliminating Oversaturation and Artifacts of High Guidance Scales in Diffusion Models. ICLR 2025.", "relation_to_broader_scientific_literature": "ADG seems to be highly similar to adaptive projected guidance (APG) [a], although ADG is obfuscated by some non-essential transformations in Algorithm 1 (arccos, discount then cos). Whereas APG is not cited (It's not a concurrent work since APG has been on arxiv since Oct 2024).\n\n[a] Eliminating Oversaturation and Artifacts of High Guidance Scales in Diffusion Models. ICLR 2025.", "essential_references_not_discussed": "[a] Eliminating Oversaturation and Artifacts of High Guidance Scales in Diffusion Models. ICLR 2025.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "1. Please center your derivations around your main claims/methods.\n2. Preferablly, illustrate math derivations with actual examples of generation.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Uu4KqDDjys", "reviewer": "Reviewer_pHtg", "summary": "The paper presents angle domain guidance (ADG), an alternative to classifier-free guidance (CFG) for conditional diffusion models. The key observation is that CFG leads to excessively large sample norms, causing oversaturated colors in the generated images. The paper claims that this is a result of CFG's linear extrapolation scheme in the latent space, presents an analysis on norm magnitude in a simplified setting where the target distribution is a Gaussian mixture model, and proposes to instead extrapolate in the angular domain to align the directions of latents, thereby effectively controlling norm magnitude. The experimental results demonstrate that ADG outperforms CFG given large guidance weights.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive comments. We provide our responses below.\n\n### **1. Plot norm against guidance weight for ADG**\n\n**Reviewer Comment:**  \n> To further strengthen this claim, I encourage the authors to plot norm against guidance weight (similar to Figure 2a) to find out whether ADG can empirically control sample norm as suggested by the theoretical result.\n\n**Response:**  \nWe appreciate this suggestion. We have added a new plot to **Figure 2a**, showing the norm against guidance weight for ADG.  \n[View Figure 2a](https://files.catbox.moe/mtxaye.png)  \n\nThe experimental results indicate that **ADG effectively controls sample norms**, preventing norm amplification even at high guidance weights. \n\n\n### **2. Experimental results for flow-based models**\n\n**Reviewer Comment:**  \n> However, no experimental results are provided for flow-based models.\n\n**Response:**  \nThank you for highlighting this point. We would like to clarify that our primary results were obtained on **SDv3.5**, and Stable Diffusion series starting from version 3 employs flow-based models[a].  \n\nTo avoid any ambiguity, we will emphasize this point more explicitly in the camera-ready version.\n\n\n### **3. CFG with normalization as a potential baseline**\n\n**Reviewer Comment:**  \n> One potential baseline could be CFG with normalization. That is, performing CFG at each sampling step, followed by normalizing and re-scaling the estimated x_0's, similar to Algorithm 6. This could be a simpler remedy to the norm amplification issue compared to ADG.\n\n**Response:**  \nThis is a good suggestion. Since our motivation stems from the observation that **angular information in the x_0 domain better aligns with semantics**, while **norm amplification** under high CFG weights degrades sample quality, we designed ADG to enhance angular consistency. The method proposed by the reviewer is a simple way to control sample norms.  \n\nWe implemented and tested this potential baseline (CFG and normalization in the x_0 domain). Experimental results show that although this method performs slightly worse than ADG, it significantly outperforms the original CFG. Our analysis suggests that the angle between the predicted x_0 and the unconditional prediction under this baseline is smaller than the angle between the conditional prediction and the unconditional prediction in ADG, limiting semantic alignment despite effective norm control.  \n[View comparison](https://files.catbox.moe/e0scpg.png)\n\n### **4. Explanation of non-commutativity and Equation 10**\n\n**Reviewer Comment:**  \n> I encourage the authors to expand on what they mean by \"non-commutativity of the tilting process with the forward process\" (L126-127) and briefly explain why Equation 10 holds.\n\n**Response:**  \nDue to space constraints, we could not elaborate on this point in the main text. However, as noted in [c], **non-commutativity** means that the weighted gradient used in CFG does not align with the true gradient of the tilted target distribution for t>0.  \n\nTo illustrate this, consider a simple example where:  \n- The unconditional distribution is modeled as a Gaussian with zero mean and variance 10.  \n- The conditional distribution is another Gaussian with zero mean and unit variance.  \n\nWhen applying classifier-free guidance (CFG) with a guidance weight of 2, the tilted distribution becomes a Gaussian with:  \n- Zero mean.  \n- A variance of 10/19.  \n\nAfter applying the forward process, which introduces Gaussian noise over time (considering a Variance Exploding SDE, i.e., VE-SDE), the resulting distribution is convolved with a Gaussian kernel whose variance increases with time.  \nTo simplify the computation, assume that the time-dependent Gaussian kernel at a particular time t* is modeled as a Gaussian with variance 1.  \n\nIn this setting:  \n- The gradient of the CFG-modified distribution is a weighted sum of the gradients of the conditional and unconditional distributions, which results in a gradient proportional to **-10/11 \\* x**.  \n- However, this gradient does **not** match the gradient of the tilted distribution obtained after applying the forward process, which is proportional to **-19/29 \\* x**, highlighting the **non-commutativity** between the tilting process and the forward process.  \n\nOnce again, thank you for your constructive feedback and for considering our paper for acceptance. We will revise our paper accordingly.\n\n## **References**\n\n- [a] [Stable Diffusion 3](https://stability.ai/news/stable-diffusion-3)  \n- [b] Exploring diffusion and flow matching under generator matching. arXiv\n- [c] What does guidance do? A fine-grained analysis in a simple setting. NeurIPS 2024.", "claims_and_evidence": "- The paper claims that CFG amplifies sample norm, which is indicative of poor sample quality. This claim is substantiated by empirical evidence (Figure 2) showing that norm magnitude is proportional to guidance weight, and color values are positively correlated with norm magnitude. The paper further presents a theoretical analysis (Section 3) which echoes the empirical findings. The analysis is performed in a simplified setting, where the target distribution is assumed to be a mixture of Gaussians. Nevertheless, this analysis sheds light on the challenge and informs the design of the new guidance algorithm.\n\n- The paper claims that ADG mitigates norm amplification, thereby enhancing sample quality. This is backed by theoretical (Proposition 4.1) and experimental results (Section 5). To further strengthen this claim, I encourage the authors to plot norm against guidance weight (similar to Figure 2a) to find out whether ADG can empirically control sample norm as suggested by the theoretical result.\n\n- The paper claims that ADG generalizes across samplers and can work with flow-based models. This is supported by experimental results with the DPM sampler (Table 3) and an extension of ADG that fits the flow-matching formulation (Appendix F). However, no experimental results are provided for flow-based models.", "methods_and_evaluation_criteria": "- The proposed method aims to address a limitation of CFG, namely poor sample quality under large guidance weights. The method is motivated by a theoretical analysis and is both simple and effective.\n\n- The benchmarks cover multiple aspects of sample quality, namely image quality (FID), image-text alignment (CLIP score), and human preference (IR). The same set of metrics has been used by the community for the evaluation of text-to-image diffusion models. Additional qualitative results further highlight the strength of the proposed method.", "theoretical_claims": "The paper provides a theoretical analysis on the impact of CFG on sample norms. The analysis is performed in a simplified setting, where the target distribution is assumed to be a mixture of Gaussians. I did not carefully check the correctness of the proofs, yet the claims make intuitive sense and are partially justified by the experimental results.", "experimental_designs_or_analyses": "- The experiments compare ADG against two baselines, namely CFG and CFG++, under varying guidance weights. An ablation study is performed to understand what design choices impact the performance of ADG. I do not have major concerns about the experimental setting, although I encourage the authors to showcase qualitative results on more diverse text prompts.\n\n- One potential baseline could be CFG with normalization. That is, performing CFG at each sampling step, followed by normalizing and re-scaling the estimated x_0's, similar to Algorithm 6. This could be a simpler remedy to the norm amplification issue compared to ADG.", "relation_to_broader_scientific_literature": "The proposed method fits in the literature of conditional diffusion models and provides an alternative to the widely used CFG for improving sample quality and condition following.", "essential_references_not_discussed": "I am not familiar with recent literature on the analysis of diffusion model sampling. My impression is that the paper has adequately covered the most related works, given the quite extensive discussion in the supplementary material.", "other_strengths_and_weaknesses": "- Overall, the paper is well motivated and clearly written.\n\n- I encourage the authors to expand on what they mean by \"non-commutativity of the tilting process with the\nforward process\" (L126-127) and briefly explain why Equation 10 holds.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qtvIMzUp1x", "reviewer": "Reviewer_FRgS", "summary": "This paper focuses on the problem of color distortions in the generated images when classifier-free guidance is set to a high value. This paper identifies that these distortions come from the amplification of sample norms in the latent space. To address this problem, this paper proposes Angle Domain Guidance (ADG) algorithm. ADG constrains magnitude variations while optimizing angular alignment, thereby mitigating color distortions while preserving the enhanced text-image alignment achieved at higher guidance weights. Experimental results demonstrate the effectiveness of ADG.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive comments. We provide our responses below.\n\n### **1. Performance of Higher CFG Values**\n\n**Reviewer Comment:**  \n> I am curious about the performance of a higher CFG, e.g., larger than 20. This would further show the effectiveness of ADG.\n\n**Response:**  \nWe appreciate this insightful suggestion. We have conducted additional experiments with higher guidance weights, specifically at weight = 20. The results demonstrate that ADG maintains stable performance even under extreme guidance conditions, whereas CFG suffers from severe color distortions and semantic misalignment.These results highlight the robustness of ADG at high guidance levels. We will include qualitative visualizations at weight = 20 in the supplementary material.\n[View Images](https://files.catbox.moe/lnh69r.png)  \n\n\n### **2. Evaluation with Complex Text Prompts**\n\n**Reviewer Comment:**  \n> Since sdv3.5 is pretrained on text-to-image generation, it's better to show more results containing complex text prompts as guidance (instead of simple prompts that describe a single object). This would further show how text alignment becomes when we increase the guidance scale.\n\n**Response:**  \nThank you for this helpful suggestion. To evaluate ADG under more complex prompts, we curated 500 complex text prompts using GPT.\nOur experiments show that ADG significantly outperforms CFG under these conditions. For guidance weight = 8:\n- ADG achieves a CLIP score of 0.355 and an IR score of 1.566\n- CFG scores lower with a CLIP score of 0.338 and an IR score of 0.766\n\nThese results confirm that ADG preserves semantic alignment and image quality even with complex instructions. We will include detailed metrics and visual comparisons in the supplementary material. [View Images](https://files.catbox.moe/p5ayb9.png)  \n\n\n### **3. Experiments on State-of-the-Art Models (e.g., Flux)**\n\n**Reviewer Comment:**  \n> More experiments on state-of-the-art models (e.g., Flux) would make the claim more solid.\n\n**Response:**  \nThank you for raising this important point. We initially considered incorporating results from state-of-the-art open-source models, including **Stable Diffusion 3.5 large** and **Flux.1 [dev]**. However, during our experiments, we discovered that Flux.1 [dev] applies guidance weight as a **model input** rather than through the conventional CFG mechanism.  \n\nUpon further investigation of the Flux architecture and related blog posts, we learned that Flux.1 [dev] is derived via guidance distillation from Flux.1 [pro], which is unfortunately **not open-source**. Since access to Flux.1 [pro] is required to modify the guidance mechanism, it was impossible for us to conduct experiments on the Flux family.  \n\nOnce again, thank you for your constructive feedback and for considering our paper for acceptance. We will revise our paper accordingly.", "claims_and_evidence": "The main claim of this paper is to address the color distortion problem when classifier-free guidance is set to a high value. The experiment results successfully validate the claim.", "methods_and_evaluation_criteria": "The method and evaluation are pretty complete, but I still have some questions:\n- I am curious about the performance of a higher CFG, e.g.,  larger than 20. This would further show the effectiveness of ADG.\n- Since sdv3.5 is pretrained on text-to-image generation, it's better to show more results containing complex text prompts as guidance (instead of simple prompts that describe a single object). This would further show how text alignment becomes when we increase the guidance scale.\n- More experiments on state-of-the-art models (e.g., Flux) would make the claim more solid.", "theoretical_claims": "The theoretical claims are correct.", "experimental_designs_or_analyses": "See \"Methods And Evaluation Criteria\".", "relation_to_broader_scientific_literature": "The contribution of this paper can be further applied to broader generation models such as motion generation and video generation, which would have a broader impact on the research field.", "essential_references_not_discussed": "The references are complete and comprehensive.", "other_strengths_and_weaknesses": "This paper addresses an important issue in this field and can lead to a broader impact on more general generation methods. The paper writing of this paper is clear and easy to follow.", "other_comments_or_suggestions": "See above.", "questions_for_authors": "See above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BX0MamCiXA", "reviewer": "Reviewer_QHkd", "summary": "This paper introduces Angle Domain Guidance (ADG), a simple and effective sampling algorithm designed to improve the performance of text-to-image latent diffusion models, particularly under high guidance weights. The authors focus on the shortcomings of Classifier-Free Guidance (CFG), specifically its tendency to cause norm amplification in the latent space, which leads to color distortions and oversaturation in generated images. The paper provides a comprehensive theoretical analysis showing that CFG’s linear extrapolation mechanism results in sample norm inflation and anomalous diffusion phenomena. Based on this insight, ADG is proposed as an alternative that focuses on angular alignment rather than magnitude extrapolation in the latent space, ensuring better text-image alignment without sacrificing image fidelity. Experimental results on the COCO dataset demonstrate that ADG achieves superior performance in terms of CLIP Score, ImageReward, and FID across a wide range of guidance weights, outperforming both CFG and CFG++.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive comments. We provide our responses below.\n### 1. **Clarification on Computational Costs of ADG**\n**Reviewer Concern:** \nThe reviewer mentions that computational costs associated with ADG, particularly in high-dimensional spaces, are not discussed in detail.\n\n\n**Response:** \nWe appreciate the reviewer’s observation. ADG introduces negligible computational overhead compared to standard sampling procedures. It does **not require additional neural network evaluations**; instead, its overhead consists solely of basic vector operations (e.g., normalization and angle clipping).\n\nTo quantify this:\n- On **SD3.5-large** (latent dimensionality ≈ $2 \\times 10^5$), the additional operations per inner loop cost around $1 \\times 10^6$ FLOPs.\n- In contrast, one full sampling step requires approximately $9 \\times 10^{13}$ FLOPs (measured using the `thop` package).\n- Empirical measurements over 100 generations on an A100 GPU show:\n  - **CFG**: Avg. generation time = **6.74s**\n  - **ADG**: Avg. generation time = **6.72s**\n\nThis confirms that **ADG does not increase runtime**, and minor variation may be attributed to system noise.\n\n### 2. Behavior of CFG in Pixel-Space Diffusion Models\n**Reviewer Concern:** \nThe reviewer asks whether similar trends of norm amplification and image degradation occur in **pixel-space diffusion models**.\n\n\n**Response:** \nYes, similar degradation patterns are observed in pixel-space diffusion models under high guidance weights. As in latent models, CFG-induced norm amplification pushes pixel values toward extreme ranges, leading to oversaturation and unnatural contrast. [visual example](https://files.catbox.moe/dqymkj.png)\n\nThese observations reinforce our hypothesis that norm amplification is a core issue, not limited to latent spaces.\n\n### 3. Justification for Assuming Spherical Gaussian Latent Space \n**Reviewer Concern:** \nThe reviewer requests further elaboration or visualization to validate the assumption that the latent space follows a spherical Gaussian distribution. \n\n\n**Response:** \nDuring the pretraining of VAEs used in latent diffusion models, the latent space is regularized to approximate a standard multivariate Gaussian distribution through a Kullback-Leibler (KL) divergence loss term. We acknowledge that Gaussian priors in VAEs are often idealized, and real-world latent spaces may deviate from this assumption due to model imperfections or data complexity. However, this Gaussian prior remains a practical and effective approximation that helps explain why directional information carries more semantic meaning than magnitude information. \n\nSamples drawn from a high-dimensional Gaussian distribution tend to concentrate around a thin spherical shell, where the angular relationship between latent variables retains critical semantic information. Consequently, emphasizing angular alignment, as done in Angle-Domain Guidance (ADG), effectively preserves text-image consistency while mitigating the undesirable effects of norm amplification. This is also one of the reasons why we emphasize that, although ADG is theoretically inspired and has some theoretical guarantees, it remains a heuristic algorithm.\n### 4. Choice of ADG over Spherical Linear Interpolation (SLERP)\n**Reviewer Concern:** \nThe reviewer suggests that using **spherical linear interpolation (SLERP)** could be a natural alternative to ADG for performing angular guidance and inquires why it was not considered.\n\n\n**Response:** \nWhile **SLERP** is an elegant mathematical alternative, it is not suitable for high guidance weights in our setting. The reasons are as follows:\n- SLERP inherently performs interpolation, not extrapolation. In cases where the guidance weight $\\omega > 1$, SLERP **extrapolates** beyond the two endpoints, leading to uncontrolled norm growth. \n- Our proposed ADG method, by contrast, **constrains the norm** of the generated sample under high guidance weights, as demonstrated by **Proposition 4.1**. \n- Consider a simple example where: \n$$\n\\hat x_{0, c} = [1, 0]^\\top, \\quad \\hat x_{0, \\emptyset} = [0.5, 0.01]^\\top.\n$$ \nUsing SLERP with $\\omega = 5$, the result is: \n$$\n\\text{slerp}(\\hat x_{0, \\emptyset}, \\hat x_{0, c}, 5) \\approx [15.9, -1.2]^\\top,\n$$ \nwhich exhibits extreme sensitivity when the vectors are nearly aligned. \nIn contrast, ADG mitigates this instability by ensuring angular alignment while maintaining controlled magnitude growth.\nWe chose ADG over SLERP due to its norm control under high guidance weights, which is crucial for maintaining image fidelity.\n\n\nOnce again, thank you for your constructive feedback and for considering our paper for acceptance. We will revise our paper accordingly.", "claims_and_evidence": "The paper claims that (1) CFG leads to significant color distortions at high guidance weights due to norm amplification in latent space, (2) ADG effectively mitigates these issues by controlling magnitude variation while enhancing angular alignment, and (3) ADG offers better text-image alignment, improved color fidelity, and superior perceptual quality. The evidence is compelling: both theoretical analyses and extensive empirical experiments support these claims. Figures and quantitative metrics (CLIP, ImageReward, FID) consistently demonstrate ADG’s robustness and effectiveness at high guidance weights, where CFG and CFG++ degrade. The ablation studies further substantiate the role of angular constraints in preventing catastrophic failures, solidifying the evidence behind the paper’s key claims.", "methods_and_evaluation_criteria": "The proposed ADG method is well-motivated, deriving from a sound analysis of the pitfalls of existing CFG. The authors introduce a geometrically inspired approach by emphasizing angular guidance in the latent space, consistent with the assumption that latent representations follow high-dimensional spherical Gaussian distributions. Evaluation criteria include standard and accepted metrics in text-to-image generation: CLIP Score (semantic alignment), ImageReward (human preference alignment), and FID (distributional similarity). \nThe authors run experiments on the COCO dataset using Stable Diffusion v3.5 and validate compatibility with advanced samplers such as DPM-Solver. \nThe evaluation is adequate, though it relies primarily on automated metrics without additional human evaluations, which are often necessary for perceptual alignment validation in generative modeling.", "theoretical_claims": "The paper provides a theoretical framework analyzing the shortcomings of CFG. Specifically, Theorem 3.2 introduces the norm amplification effect of CFG, and Theorem 3.3 shows anomalous diffusion, which offers insight into how CFG might induce undesirable latent space behaviors at high guidance scale.\nThe authors extend beyond prior analyses by introducing the concept of surface classes and offering proofs in high-dimensional settings with multiple components. \nHowever, while the theoretical arguments are sound, the practical extension of these results to complex real-world latent spaces (beyond Gaussian mixtures) could be discussed in more depth.", "experimental_designs_or_analyses": "The experimental design is sound and systematic. \nThe authors test ADG across varying guidance weights ($\\omega=2$  to $\\omega=10$), compare against CFG and CFG++, and include ablation studies to validate the contribution of angular constraints and normalization. Experiments on both Stable Diffusion v3.5 (COCO dataset) and Stable Diffusion v2.1 with DPM-Solver highlight ADG’s generality. Key metrics are reported clearly, with ADG consistently outperforming baselines in ImageReward (often used as a proxy for human preference) and CLIP Score. Ablation studies demonstrate the necessity of angular constraints to avoid instability.", "relation_to_broader_scientific_literature": "The proposed paper provides a method to improve image generation, while the method could be further investigated to diffusion models on other data, e.g., protein, text, etc., as the conditional generation with diffusion model plays a crucial role in various machine learning tasks.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper’s primary strength lies in its novel conceptual shift from linear extrapolation to angular guidance in latent diffusion sampling. ADG is a theoretically motivated and practically effective approach that mitigates common issues with CFG, especially at high guidance weights. The theoretical analysis is comprehensive and extends prior literature in meaningful ways. Additionally, the method is shown to be compatible with various samplers and diffusion frameworks, highlighting its flexibility.\n\nOn the weakness side, the empirical validation is narrowly focused on latent diffusion models (i.e., Stable Diffusion). While the authors acknowledge ADG’s heuristic nature and its dependence on the latent space structure of variational autoencoders, a deeper analysis of potential limitations when applied to non-latent diffusion models (e.g., pixel-space diffusion) would be helpful. Moreover, computational costs associated with ADG, particularly in high-dimensional spaces, are not discussed in detail.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. Does the similar trends of CFG also resides in pixel diffusion models? \n2. The paper assumes that the latent space follows spherical Gaussian distribution, could the author elaborate or provide additional visualization to validate this claim?\n3. While original CFG use linear interpolation, one natural way is to use spherical linear interpoliation (slerp) to perform angular guidance. Is there any reason why did not tried slerp as the design choice for angular guidance?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes Angle-Domain Guidance Sampling (ADG), a new guidance method focusing on angular alignment rather than magnitude extrapolation in the latent space. 

Strengths

* Clear motivation from both theoretical and empirical analysis of existing method (i.e., CFG)

* Effective method derived from analysis

* Strong empirical evaluation

Weaknesses

* Limited to latent diffusion models

The authors have addressed the reviewers’ concerns in the rebuttal, and I believe this paper presents ADG as a compelling guidance method, backed by comprehensive experiments and clear presentation, making it an impactful contribution to the field.
Decision: Accept (poster)