Zigzag Diffusion Sampling: Diffusion Models Can Self-Improve via Self-Reflection

We appreciate the reviewer 9XhW for the thorough feedback, and are delighted that our work has received your recognition. We concur with reviewer's suggestion to compare with more methods and reference/discuss them in the paper. Below are the experimental results.

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Q1: Would the author compares Z-sampling with other training-free enhancement techniques?

A1: Thanks for the suggestion. Yes, we would like to. We present the results compared across multiple baselines below, and we report more details in Table 9 and Table 10 of Appendix D.1. It further supports that Z-Sampling can still outperform more recent baselines.

• SAG[1]/SEG[2]

  We note that SAG[1] employs blur guidance and intermediate self-attention maps to achieve higher quality samples. Building upon this, SEG[2] further optimizes this method from the energy landscape perspective. We cite both of them in revision. Since SEG is a more recent powerful method, we report the comparison results with SEG in Pick-a-Pic Dataset given the rebuttal time.

  In SEG, its characteristic is that it can generate well under unconditional conditions without specifying a default cfg guidance scale. Therefore, we adopt the default setting in our paper (i.e., cfg guidance scale \gamma_{1} = 5.5). We found that Z-Sampling outperforms SDXL and SEG.

  (Pick-a-Pic)	HPS v2↑	AES↑	PickScore↑	IR↑	(DrawBench)	HPS v2↑	AES↑	PickScore↑	IR↑
  SDXL	0.2989	6.0870	21.6353	0.5865	SDXL	0.2881	5.5595	22.3086	0.6075
  SEG	0.3053	6.1231	21.4186	0.6157	SEG	0.2960	5.6596	22.1453	0.6042
  Ours	0.3128	6.1302	21.8477	0.7922	Ours	0.3050	5.6739	22.4581	0.7997

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• CFG++[3]

  As for CFG++[3], since it restricts the cfg scale to the range [0,1], while the classic Z-Sampling is larger, such as the range [3,10], a fair comparison is not possible. Given this, we use $\omega = 0.5$ in CFG++, corresponding to a cfg scale of 5.5 in Z-Sampling. (Refer to Table 1 in [3]).

  It is worth noting that in the official implementation of CFG++, the VAE encoder uses "madebyollin/sdxl-vae-fp16-fix" checkpoint. For fair comparison, we follow this setting, so the results reported for SDXL and Z-Sampling are slightly different from the previous results.

  (Pick-a-Pic)	HPS v2↑	AES↑	PickScore↑	IR↑	(DrawBench)	HPS v2↑	AES↑	PickScore↑	IR↑
  SDXL	0.3004	6.1121	21.8053	0.6007	SDXL	0.2885	5.6245	22.4213	0.6761
  CFG++	0.3028	6.0989	21.8337	0.6730	CFG++	0.2865	5.6174	22.3797	0.6266
  Ours	0.3124	6.1170	21.8444	0.7855	Ours	0.3035	5.6594	22.4392	0.7911

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Q2: Would the author compares Z-sampling with the training mathods "Text-to-Image Rectified Flow as Plug-and-Play Priors"[4]?

A2: Thanks for the reviewer's suggestion. Due to RFDS's need for a pretrained flow model and latent space training, comparing it with Z-Sampling, is challenging. Specifically, we note that

• RFDS

  A method utilizing fixed prior weights $\phi$ to assist in the training of learnable parameters $\theta$.

• IRFDS

  A method utilizing fixed prior weights $\phi$ to assist in the training of learnable noise $\epsilon$.

• RFDS-REV

  Combining RFSD and iRFSD, i.e., optimizing noise $\epsilon$ before updating parameters $\theta$, to enhance generation quality.

RFSD and RFDS-REV improve the training process, and iRFSD enhances image editing performance (Table 2 of [4]). We believe that iRFSD, as a method for optimizing noise based on training, can reduce the approximation error of inversion in Z-Sampling. Thanks to the reviewer's suggestion, we also discuss this point in Appendix B.

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We hope that these additional experiments and explanations resolve the reviewer’s concerns and questions. We are more than happy to discuss if the reviewer 9XhW has further questions or comments.

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[1] Hong, Susung, et al. "Improving sample quality of diffusion models using self-attention guidance." Proceedings of the IEEE/CVF International Conference on Computer Vision. 2023.

[2] Hong, Susung. "Smoothed Energy Guidance: Guiding Diffusion Models with Reduced Energy Curvature of Attention." The Thirty-eighth Annual Conference on Neural Information Processing Systems.

[3] Chung, Hyungjin, et al. "CFG++: Manifold-constrained Classifier Free Guidance for Diffusion Models." arXiv preprint arXiv:2406.08070 (2024).

[4] Yang, Xiaofeng, et al. "Text-to-Image Rectified Flow as Plug-and-Play Priors." arXiv preprint arXiv:2406.03293 (2024).

We appreciate the reviewer BMZe for the thoughtful feedback. We are glad to hear that the reviewer appreciates the usability and general applicability of Z-Sampling. We hope these discussions can further clarify and elaborate on the contributions of Z-Sampling.

We duly address the reviewer BMZe’s all concerns below.

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Q1: The authors’ insight—that the semantic information contained in the latent affects generation results—is quite evident.

A1: We first thank the reviewer for acknowledging the point that "the semantic information contained in the latent affects generation results," which means our empirical observations (see Sec 3.1) have been reasonably recognized.

It is important to emphasize that, prior to our work, no research had identified that each step of the zigzag operation can accumulate the information gain between strong guidance and weak guidance. And we are the first to explicitly articulate this concept, offer a credible theoretical explanation for it, and substantiate it with extensive empirical evidence.

Here, we want to outline the insights from our work andhope to clarify our contributions again:

• What signifies a better latent variable in the Diffusion Model? (Sec 3.1)

  Observing the latent's generation results under weak conditions to judge its performance under specific strong conditions.

• How to improve the properties of an initial latent variable to enhance generation quality? (Sec 3.2)

  Demonstrating that the gap between cfg scales is a key component, controlling this gap can determine the direction and intensity of optimization.

• Why end-to-end methods is inferior to step-by-step approaches? (Sec 3.3)

  Conducting theoretical and empirical analysis to reveal the impact of semantic term $\tau _{1}$ and error terms $\tau _{2}$, further improving the algorithm.

We understand the reviewer's concern. While it seems that insight #1 is a reasonable observation, we believe that deriving subsequent conclusions from this observation is inspiring and interesting work, including investigating the classifier guidance gap, capturing semantics based on this observation, or deriving inversion errors.

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Q2: Z-Sampling and the time-travel approach seem somewhat similar.

A2: We respectfully note that we have introduced this method (termed as Re-sampling) in related work section and compared it with our approach. This method was first introduced in Repaint [1], where it was termed Resampling. Subsequently, Freedom [2] adopted this approach, referring to it as Time-Travel. In essence, both Resampling and Time-Travel are the same algorithm.

Here, we emphasize the differences between Z-Sampling and Resampling.

• Empirically, we provide a extensive comparison between Z-Sampling and Resampling in Table 1 and Table 2. Z-Sampling consistently outperforms Resampling across nearly all metrics (such as HPS v2, PickScore, etc.) on all benchmarks.

• Theoretically, Z-Sampling is supported by a clear theoretical foundation, which involves continuously accumulating semantic information from the gap between strong guidance and weak guidance. This allows us to flexibly control the direction and intensity of optimization, as well as provide guidance for further algorithm optimization. In contrast, Resampling has been mostly treated as a minor performance enhancement module in previous works, such as...

  • In Repaint [1], they provided a hypothetical description of less than a page (Sec 4.2 in [1]) for the specific task of repaint. There were no detailed experiments and theoretical analysis to explain why Resampling works; and it was merely treated as a performance-enhancing trick.

  • In Time Travel [2], they also allocated less than one page to introduce Time Travel (Sec 4.2 in [2]), merely stating that "... it has empirically shown to inhibit the generation of disharmonious results...".

  Why it works theoretically, how to develop better algorithms, and what controls the effect (e.g., guidance gap) — these aspects have received scant attention in previous work.

Specifically, we believe in the importance of uncovering the underlying mechanisms—we earnestly hope that Reviewer BMZe also agrees with the significance of this endeavor. And we organize the relationships with other works in Appendix B, hoping this could further clarify the contributions of our work.

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[1] Lugmayr, Andreas, et al. "Repaint: Inpainting using denoising diffusion probabilistic models." 2022.

[2] Yu, Jiwen, et al. "Freedom: Training-free energy-guided conditional diffusion model." 2023.

Q8: In Figure 10, how is the x-axis varied?

A(8): We apologize for the inadequate description of Figure 10. It is because we use various sampling steps in this experiment, so we can oberserve the signficant improvement of Z-Sampling across various sampling steps/inference time costs .

Here is a more detailed explanation:

We note that in Diffusion Models, we can alter the total inference time step T to control the time cost. For example, in SDXL, the default T is 50. When we set T to 10, the time to generate one image becomes 1/5 of the original. Therefore, by selecting different values of T, the time cost to generate one image, represented by the horizontal axis in Figure 10, also changes accordingly.

In standard sampling, the total number of denoising predictions, denoted as $T_{sd}$, is equal to T. While in Z-Sampling, this number, denoted as $T_{z}$, is equal to $T + 2*\lambda$. To achieve a fair comparison in Figure 10, we set T=50 in standard sampling, meaning $T_{sd}$=50. And we set T=25 and $\lambda = \frac{T}{2}$ in Z-Sampling, mean $T_{z} = T + 2*\lambda = 25 + \frac{25}{2} * 2 = 50$. In this scenario, the time cost to generate one image is nearly the same for Z-Sampling and standard sampling. The results in Figure 10 show that Z-Sampling consistently outperforms standard sampling.

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Q9: How might Z-Sampling be integrated into the training phase?

A9: Thanks for the constructive suggestion. As we focused on sampling phase, the training phase is beyond the scope of this work. It will be very interesting to include the effect of Z-Sampling into the training phase. In future, we plan to explore training/distillating diffusion models along the path of Z-Sampling. This could leverage the improvement of Z-Sampling with no extra inference cost.

Thanks for the constructive suggestion. As we focused on sampling phase, the training phase is beyond the scope of this work. It will be very interesting to include the effect of Z-Sampling into the training phase. In future, we plan to explore training/distillating diffusion models along the path of Z-Sampling. This could leverage the improvement of Z-Sampling with no extra inference cost. (copy xie sir.)

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Q10: The paper appears to lack analysis of the unconditional branch.

A10: The response here is the same as in A5.

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Finally, we sincerely thanks Reviewer A4t2 for the hard work and comments again. We respectfully hope that the reviewer could reevaluate our work given the reponses addresing your main concerns. We appreciate it very much in advance.

Q5: Our work lacks analysis of the role of the unconditional branch.

A5: We respectfully argue that our analysis also sheds light on $u_{\theta}(x_{t},c,t)-u_{\theta}(x_{t},\varnothing,t)$, rather than just $u_{\theta}(x_{t},c,t)$ or $u_{\theta}(x_{t},\varnothing,t)$, as indicated in Theorem 3. Following the suggestion, we further discuss the the unconditional branch deeply in Appendix F of the revision. It shows that the role of the unconditional branch is to provide a base signal and decouple useful semantic information from the conditional branch (in the form of a differential). Notably, this differential itself is independent of $\gamma_{1}$ or $\gamma_{2}$ (as evident from the input to $u_{\theta}$), and the main role of $\gamma_{1}-\gamma_{2}$ is to amplify or attenuate the magnitude of this differential.

In the case of neglecting the approximation error $\tau_{2}$, the effect of Z-Sampling can be considered as

$\delta_{Z-sampling} = \sum_{t=1}^{T} \alpha_{t}h_{t}^{2}\left((\gamma_{1}-\gamma_{2})\left(u_{\theta}(x_{t},c,t)-u_{\theta}(x_{t},\varnothing,t)\right)\right)^{2}.$

In this formula, even if $\gamma_{2}$ is set to zero, the difference $\delta_{\gamma} = \gamma_{1} - \gamma_{2}$ is not zero, which means that the uncondtional branch $u_{\theta}(x_{t},\varnothing,t)$ still plays a role in this game, and $u_{\theta}(x_{t},\varnothing,t)$ is provided by the denoising process rather than the inversion process in this scenario. By controlling the magnitude and sign of $\delta_{\gamma}$, we determine the effect of the difference between $u_{\theta}(x_{t},c,t)$ and $u_{\theta}(x_{t},\varnothing,t)$. It is noted that in the ideal case, i.e., neglecting the inversion approximation error, even if $\gamma_{1}$ and $\gamma_{2}$ change, this difference $u_{\theta}(x_{t},c,t)-u_{\theta}(x_{t},\varnothing,t)$ remains unchanged.

Additionally, it is important to note that our experiments is not limited to the setting where $\gamma_{2}=0$. In Figure 7/8, we conducted experiments with different inversion guidance $\gamma_{2}$. For example, when $\gamma_{2}>0$, the unconditional branch participates in the process, and $\delta_{\gamma}$ becomes smaller, reducing the gains of Z-Sampling. We note $\gamma_{2}=0$ is just a hyperparameter for main experiments after considering multiple metrics (such as HPS v2, AES, etc.), and it is not a special value.

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Q6: There is a typo error in the algorithm 1.

A6: Thanks for the reviewer's feedback. However, after a thorough check, we think there are no typographical errors in Algorithm 1. We are afarid that the reviewer may misundertand the method a little bit.

In Algorithm 1, at each time step $t$, a normal denoising operation is first performed, followed by an optional (inversion and then denoising) operation based on the value of $\lambda$. Following this logic, the notation in Algorithm 1 is correct. Here we provide a more detailed explanation of Algorithm 1.

We note that $\Phi^{t}$ is denoising operation at timestep $t$ , and $\Psi^{t}$ is inversion operation. Assuming the current step is $t$, our goal is to obtain $x_{t-1}$ from $x_{t}$.

• In each iteration, we first denoise $x_{t}$ under high guidance $\gamma_{1}$ to obtain a temporary $x_{t-1}$. (Line 5)

• Determine whether zigzag operation is needed. If not, skip optimization; otherwise, perform zigzag operation. (Line 6)

• Backtrack the temporary $x_{t-1}$ to obtain the inversion result $\tilde{x}_{t}$ under low guidance $\gamma _{2}$. (Line 7)

• Re-denoise $\tilde{x} _{t}$ under high guidance $\gamma _{1}$ to obtain the optimized latent $x _{t-1}$. (Line 8)

• End the current iteration and proceed to the next denoising step. (Line 9)

We hope this further clarifies the symbols and logical meaning in the Algorithm 1.

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Q7: How is the guidance scale $\gamma_{1}$ set in Table 1/2?

A(7): Due to our oversight, we only mentioned $\gamma _{1}=5.5$ in Sec 3.2 (line 235), and forgot to reiterate it in Sec 4.1. In our settings, for SDXL and SD2.1, we use $\gamma _{1}=5.5$. For DreamShaper-xl-turbo-v2, we use $\gamma _{1}=3.5$. For Hunyuan DiT, we use $\gamma _{1}=3.5$. We note that these settings are the recommended values from the official sources.

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Q3: The theoretical analysis of Z-Sampling is insufficient, particularly lacking in analysis related to training.

A3: We believe that the true value of a theoretical analysis depends on its novelty and the insight it provides for subsequent research.

• In terms of novelty，we note that before us, there is no research realized that the gap between strong guidance and weak guidance could be used to study how this accumulated gain affects the generation results. We are the first to explicitly articulate this concept, offer a theoretical explanation, and substantiate it with extensive experiments.

• In terms of insights for subsequent research, we thoroughly explored the impact of Z-Sampling's components on generation effects. Based on our theory, future work can be optimized in multiple directions, such as designing better accumulation methods for the semantic term $\tau_{1}$, exploring approaches to reduce the inversion error term, and applying this semantic accumulation gain to zero-shot discriminative tasks (such as image classification).

Additionally, since our research focuses on the diffusion sampling algorithm, training is beyond the scope of this paper. As the reviewers noted, Z-Sampling holds significant potential in the training phase, which we will explore in future work.

Finally, we clarify the theoretical contributions of our work:

• Theorems 1 and 2 show that the effect of Z-Sampling can be decomposed into the semantic term $\tau_{1}$ and the error term $\tau_{2}$.

• Theorem 3 unveils the pivotal aspect of this iterative inverse-and-denoise approach, emphasizing that the CFG guidance gap is indeed significant. Specifically, positive optimization can only be achieved when $\gamma_{1} - \gamma_{2}>0$, a perspective that has not been sufficiently explored.

We hope the reviewer could understand the efforts we have made, which aim to bring more new thinking to the community through Z-Sampling and its core guidance gap mechanism. And we will continue this work in future researche, especially theoretical the analysis on how latent spaces possess prior semantic information.

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Q4: Limited technical innovation of Z-Sampling.

A4: We thank the reviewer A4t2 for mentioning this work [2], and we will reference and discuss it in the revision. We note that Z-Sampling is the first work that has conducted a detailed analysis of the zigzag mechanism, including cfg guidance, error terms, semantic accumulation, and extensive ablation experiments to validate the correctness of the theory. In contrast, as a broad inference paradigm, most research simply treats it as a plug-and-play sub-module to improve performance**, for example:

• In Tune-a-video [2], only a quarter of a page is devoted to discussing this iterative paradigm (Sec 3.3 in [2]). They just say "well, this seems to help stabilize the structural integrity, which is enough". In fact, Sec 3.3 in [2] just employs an end-to-end optimization approach (which has been proven suboptimal, see Appendix E.2 in the revision), and doesn't consider that further reducing the guidance scale during inversion might yield better performance.

• In Time Travel [3], they also allocate less than a page to introduce time travel algorithm (Sec 4.2 in [3]), merely stating that "... it has empirically shown to inhibit the generation of disharmonious results...".

We highly value this feedback, which has helped us clarify our contributions and refine the positioning of our work. We hope that Z-Sampling, can provide a extensive study of this iterative paradigm and inspire the community with further optimization directions (such as better semantic accumulation methods or smaller inversion errors). And we also hope the reviewer can understand this point and look forward to further discussions on this topic.

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[1] Ho, Jonathan, and Tim Salimans. "Classifier-Free Diffusion Guidance." NeurIPS 2021.

[2] Wu, Jay Zhangjie, et al. "Tune-a-video: One-shot tuning of image diffusion models for text-to-video generation." ICCV 2023.

[3] Yu, Jiwen, et al. "Freedom: Training-free energy-guided conditional diffusion model." ICCV 2023.

We appreciate the reviewer A4t2 for the thoughtful feedback, which has deepened our reflection on our work. Below we attempt to address each of the reviewer A4t2 's concerns.

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Q 1: The visual results reported in the paper appear to be oversaturated.

A 1: We respectfully argue that the oversaturation is mild and is not responsible for the improvement in quantitative metrics for Z-Sampling. We note that Z-Sampling could find a superior trade-off between image quality and prompt adherence.

First, we conduct experiments at an extremely high CFG scale (equivalent to three times the recommended guidance scale). In this scenario, standard sampling performs unquestionably poorly, exhibiting severe artifacts/oversaturation. However, using Z-Sampling, we are able to mitigate these issues to some extent (Please see Figure 24 in the revision). For example, if saturation is too low, Z-Sampling increases it (as pointed out by the reviewer A4t2); if saturation is too high (typically occurring at excessively high CFG scale), Z-Sampling decreases it.

Second, in Table 2, we report the performance of Z-Sampling on GenEval, a complex benchmark that includes metrics such as counting, position, and attribute binding, which are unrelated to saturation. We note that Z-Sampling still achieves better performance.

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Q 2: Z-Sampling seems to have similar effects to high CFG, sharing the same drawbacks.

A 2: We respectfully note that the effects of Z-Sampling are actually different from high CFG scales. Figure 24 in the revision reveals this point. To further validate our claim, we also present the quantitative results under different CFG values in the table below (also shown in Table 13 in the revision). Due to deadline constraints, we use DreamShaper-xl-turbo-v2 as the base model.

The standard sampling performs best at $\gamma_{1}=3.5$, which is also the official recommended guidance sclae. While $\gamma_{1}>3.5$, the performance of standard sampling begins to decline. However, Z-Sampling consistently yields signficant performance gains, indicating that our method can still work effectively under high guidance scales. Specifically, in Figure 23, we note that even at extremely high CFG scale (e.g., $\tau_{1}=11.5$), the winning rate remains above 80%.

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We respectfully note that your final concern is not a weakness, but a feature, which does not harm the qualitative or quantitative performance of Z-Sampling. We kindly argue that your evaluation concern on the qualitative results is highly subjective, limited to very few cases, and very inconsistent with more observations, popular metrics, other three reviewers, and even previous works. In contrast, the effectiveness of Z-Sampling are widely supported and verified in terms of rich dimensions.

From a qualitative perspective, in many previous works, such as Fig 3 of Diffusion-DPO [1] and Fig 2 of FreeU [2], similar oversaturation are observed. Such saturation enhancement has been widely presented and mentioned in visualization cases as an addition to some users' preference.

Particularly, We note that rating the effectiveness of a method according to one user's preference to very few cases is very subjective and unreasonable. The human preference models, including HPS v2, PickScore, and Image Reward are trained with the preference of a large group of users, which reflect preference of more users, support the effectiveness of Z-Sampling under all studied CFG scales. And, even you agreed that, Figures 24 and 27 are convincing and can mitigate saturation. Table 18 even makes the advantage of Z-Sampling more convincing under various CFG scales.

The dimensions of evaluation metrics include:

• Higher HPS v2/PickScore/Image Reward: Better human preferences of three different large groups of users.

• Higher AES: More aesthetic.

• Higher GenEval performance: Better prompt adherence.

• Lower FID: Better image quality/consistency with real-world images.

To the best of our knowledge, no sampling method can achieve such impressive and comprehensive improvement compared to ours. If you have some candidate, please feel free to let us know. Moreover, our theoretical analysis can even further help understanding diffusion sampling and enhance our contribution beyond previous work.

We gratefully thank you in advance for reconsidering the rating from a more objective perspective.

Your effort will be a valuable addition to our community!

Reference:

[1] Wallace, Bram, et al. "Diffusion model alignment using direct preference optimization." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024.

[2] Si, Chenyang, et al. "Freeu: Free lunch in diffusion u-net." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024.

Q3: The paper lacks a deeper analysis of the inversion error term

A3: We greatly appreciate the reviewer's suggestion. We note that the theoretical analysis of the information gain term is novel and very helpful for us to design Z-Sampling. We also agreed that a deeper analysis of the error term can further provide the interesting insights and advantages of Z-Sampling beyond previous studies. We present more discussion on the error term here and in Appendix E.2/E.3 of the revision.

To only study the error term $\tau _{2}$, we eliminated the semantic term $\tau _{1}$ by setting $\delta _{\gamma} = 0$, Equation 10 is equivalently transformed as

$(x_{T}-\tilde{x}_{T})^{2}=\sum _{t=1} ^{T} \alpha _{t} h _{t} ^{2} ( \underbrace{\epsilon _{\theta} ^{t}(x _{t})-\epsilon _{\theta} ^{t}(\tilde{x} _{t-1})} _{\tau _{2}(t)})$.

Under this setting, we performed end-to-end and step-by-step experiments with SDXL.

It can be observed that without semantic gain (e.g., $\delta _{\gamma}=0$), the cumulative error by end-to-end inversion significantly degrades image quality. In contrast, under the step-by-step setting, image quality only slightly decreases, indicating a minimal impact from $\tau _{2}$.

To further illustrate, we added an additional Gaussian error term $s*error _{gs}$ as

$(x_{T}-\tilde{x}_{T})^{2}=\sum _{t=1} ^{T} \alpha _{t} h _{t} ^{2} ( \underbrace{\epsilon _{\theta} ^{t}(x _{t})-\epsilon _{\theta} ^{t}(\tilde{x} _{t-1})} _{\tau _{2}(t)}+s* \frac{norm(\epsilon _{\theta} ^{t}(x _{t}))}{norm(error _{gs})} error _{gs})$

We find that as $s$ increases, Z-Sampling performs worse. Therefore, based on the above experiments, we note that:

• An increase in the error term $\tau_{2}$ degrades the sampling effect.

• The step-by-step approach helps reduce the error term $\tau_{2}$, mitigating this negative gain.

We provide further detailed discussions in Appendix E.2 and E.3.

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Q4: Can Z-Sampling be extended to more generative tasks?

A4: Thanks for the suggestion from reviewer uffk, which opens up more general potential for our method. The table below demonstrates that the mechanism of Z-Sampling remains effective in video generation.

We choose AnimateDiff [1] as the baseline and test it on Chronomagic-Bench-150 [2], and we set $\gamma_{1}=7.5$ and $\gamma_{2}=0$ in Z-Sampling. We note that Z-Sampling outperforms both AnimateDiff and another train-free sampling method FreeInit [3] in UMT-FVD, UMT-Score, GPT4o-Score. More details can be found in Table 11 in Appendix D.1.

Due to the deadline constraints, we will subsequently report the results on the 3D generation task.

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Finally, we sincerely thank the reviewer uffk’s feedback. It definitely inspires us to further improve our work and clarification for more readers.

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[1] Guo, Yuwei, et al. "AnimateDiff: Animate Your Personalized Text-to-Image Diffusion Models without Specific Tuning." ICLR, 2024.

[2] Yuan, Shenghai, et al. "ChronoMagic-Bench: A Benchmark for Metamorphic Evaluation of Text-to-Time-lapse Video Generation." NeurIPS, 2023.

[3] Wu, Tianxing, et al. "Freeinit: Bridging initialization gap in video diffusion models." ECCV, 2024.

We appreciate the reviewer uffk for the thorough feedback and the recognition of our contribution in both theoretical and experimental aspects. Below we address each of the reviewer uffk's concerns.

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Q1: Z-Sampling incurs additional computational overhead.

A1: We frankly admit that Z-Sampling incurs additional computational costs. We will further explore how to do Z-Sampling more efficiently in future, such as including zigzag into the training phase. However, we also note the time consumption is controllable and often very limited compared with the performance improvement. Moreover, even under the same computation overhead with less denoising steps, Z-Sampling can achieve significant improvements, according to Figure 10. This is already a powerful advantage, as the reviewer uffk also agreed.

Moreover, the absolute time consumption can be often very limited in practice.

• Figure 9 demonstrates that even with a small number of steps (e.g., the first 10 steps in SDXL) employing the zigzag operation, a nearly 70% increase in winning rates can be achieved. We note that this incurs less than a fifth of the additional time consumption.

• In Table 1, we validate that Z-Sampling can effectively bring benefits with distillation diffusion model. Notably, the official recommended denoising steps of DreamShaper-xl-turbo-v2 is 4, which means the additional inference steps $T_{z} - T_{sd}$ brought by Z-Sampling is at most 6, a very small cost, but it can increase the winning rate to 80% (Please see Figure 6).

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Q2: Is Z-Sampling exclusively employed within deterministic sampling schedulers?

A2: Yes, we observe that Z-Sampling does not perform well with stochastic samplers like Euler (A), as detailed in Appendix E.1. This is primarily due to their inability to provide precise inversion operations.

We note that Z-Sampling exhibits favorable performance with mainstream ODE schedulers (such as DDIM, DPM-Solver). We respectfully contend that this underscores the versatility of our method, given that most community endeavors (such as generation acceleration[1], image editing[2]) are predicated on ODE schedulers.

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[1] Lu, Cheng, et al. "DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models." arXiv e-prints (2022): arXiv-2211.

[2] Hertz, Amir, et al. "Prompt-to-Prompt Image Editing with Cross-Attention Control." The Eleventh International Conference on Learning Representations.