Inner Classifier-Free Guidance and Its Taylor Expansion for Diffusion Models

Q3: Extra computation cost.

In comparison, CFG requires 2 forward passes, meaning our method incurs a 50% increase in computation. To further minimize this additional computation, we propose a method to selectively apply 2nd order ICFG only during key timesteps. Upon examining the insights from the second figure in Figure 1, it is apparent that the guidance is more significant during the middle approximately 40% of the time. Capitalizing on this observation, our plan is to implement the second ICFG exclusively during these specific time steps. This strategic approach is expected to result in a substantial reduction, decreasing the extra inference time from 50% to approximately 20%.

Our gratitude extends to Reviewer UdAr for acknowledging the novelty in our approach to combining guidance strength and condition. Their assistance in correcting definitions, perfecting proofs and the formulation of the theorem, along with their guidance in including additional empirical results and speeding up our method, has been invaluable.

Q1: Concerns about the theoretical analysis in Section 3.

Thank you very much for your valuable suggestions. We acknowledge that our theoretical analysis had some shortcomings. To address these issues, we have made several improvements in the proof, and we would like to highlight the key changes:

• We have explicitly written out the normalization constant.
• We provide a clearer formula for typical training scenarios.
• We have revised Theorem 3.1, retaining only the sufficient conditions and acknowledging that we will prove their necessity for typical training scenarios later.

We apologize for any confusion caused by the symbol system. The enhanced intermediate distribution is consistently denoted as $\bar{q}$, but when explicitly expressing the enhancement strength $\beta$, we can omit the symbol $\bar{q}$. The correct formula is as follows:

This equation holds under the cone assumption, allowing the neural network to capture them together. The Corollary 3.1.1 is now more clear in this context.

We also apologize for the lack of interpretation. In Eq. 12, the term $\beta(x_t)$ is a general energy function, chosen arbitrarily to modify the enhanced intermediate distribution, and it is unrelated to the guidance strength $\beta$.

Thank you once again for your insightful questions. We believe that these improvements significantly enhance the theoretical foundation of the manuscript.

Q2-1: The improvement of second-order ICFG is not clear.

We apologize for any confusion in our experiment descriptions. While maintaining the same training policy as CFG, the continuous feature of the condition space enables second-order ICFG to consistently outperform CFG in both FID and CLIP scores across various CFG strengths ($w$).

For a clearer understanding of our advancements in generation quality, consider the comparison between second-order ICFG with $w = 2.0, v = 0.25, C=C_{all}$ and CFG with $w = 3.0$ (depicted as the second orange spot and the third blue spot in Figure 2). Despite nearly identical CLIP scores (26.12 and 26.11), we observed a notable improvement of $(16.68 - 15.28)/16.68 =$ 8.3% in FID. This implies that with increased control strength, the generated quality is significantly enhanced.

It is noteworthy that the CLIP Score improvement achieved by our second-order ICFG is nearly half of the overall improvements observed from Stable Diffusion v1.1 to v1.5 (comparison chart). This indicates a substantial improvement in performance.

Due to GPU limitations, we were unable to train SD from scratch. However, we are currently training another framework, U-ViT, with a resolution of 256x256 on the COCO dataset from scratch to fully explore the capabilities of our ICFG. The preliminary results are listed below, and we plan to update them upon completion of the training.

We plan to report the final FID and CLIP scores after completing the entire training process. As of now, we have observed a notable 3% improvement in FID with ICFG training algorithm and CFG sampling algorithm compared with original training policy.

Q2-2: Other datasets and carefully hyperparameter tuning.

Thank you very much for the suggestions. We will consider use more datasets to tune the $v$ and middle point in a more data driven way. In fact we just choose the $v$ and middle point under the following consideration:

1. Because the U-Net doesn't trained to realize the cone structure, middle point should be near the original condition point.
2. Due to the reminder term $R_2{(\beta)}$ will be larger with larger $\beta$ (Because $R_n{(\beta)}\sim o((\beta-1)^n)$), $\sqrt{v}$ is an upper bound of the 2nd order "step size" and should be small.

Thank you sincerely for your prompt and valuable feedback. We also acknowledge and apologize for any confusion resulting from our previous incomplete statement in the rebuttal.

Q1: If there is any problem in this study by reducing the theorem to the sufficient condition of the original Theorem 3.1.

There is no issue since the sufficient condition is enough to demonstrate that our ICFG does not exhibit inconsistency in the enhanced intermediate distributions.

Q2: Experiments on other datasets.

Due to limitations of time, we haven't extensively tuned hyperparameters on larger datasets like Laion. However, to demonstrate the stability and universal applicability of the existing improvements, we have taken the following two measures:

1. For FID and CLIP Score, we conducted five calculations to present mean and standard deviation results for key hyperparameters, specifically setting $middle = 1.1$ and $v = 0.25$. Although we have completed only three points so far, we plan to extend this analysis further. Analyzing our existing experimental results, it's noteworthy that the standard deviation is significantly smaller than the observed improvement. This pattern suggests a high degree of stability in our achieved enhancements.

CLIP Score (higher values indicate better performance):

FID (lower values indicate better performance):

2. We have also included several visual comparisons on anything-V4 (https://huggingface.co/xyn-ai/anything-v4.0). It is evident from the images that second-order ICFG outperforms CFG, producing images with better-rendered hands and closer alignment to the provided prompts. The results are visualized in Figure 5 of Appendix F. It shows the universal applicability of our ICFG.

We are downloading the Laion dataset and we will further tune the hyperparameters for the future work.

Q3: Experiments about the speedup.

We conducted our experiments on an NVIDIA GeForce RTX 3090, using a batch size of 4. We performed 100 samplings to calculate the timings and utilized 10,000 images for the computation of FID and CLIP scores.

• Due to the text encoder and VAE decoder, the real-time consumption of the 2nd-order ICFG is less than the estimated extra computation of the U-Net.
• Through a preliminary selection of key timesteps (0.2-0.8) for applying the 2nd-order ICFG, we achieve nearly full FID benefits and a 74% improvement in CLIP scores, with a reduced extra inference time of 26.27%. We anticipate further enhancements in extra inference time by refining the selection of key timesteps.

Dear reviewer UdAr,

Thank you sincerely for appreciating our previous rebuttal. We understand your concerns, and to further alleviate them, we have included additional information regarding the results on other datasets. We also provide more information to make it easier for you to review and validate our proof.

Experiments on other datasets.

1. We want to clarify the previous response to Q2. The visual comparisons in Figure 5 of Appendix F were conducted on the anything-V4 dataset (https://huggingface.co/xyn-ai/anything-v4.0) with the same hyperparameters $middle = 1.1$ and $v = 0.25$. It is important to note that this model was trained on other datasets, specifically in the anime domain.
2. Additionally, to address the concerns raised by reviewer UdAr, we performed CLIP Score comparisons on the pokemon-blip-captions dataset (https://huggingface.co/datasets/lambdalabs/pokemon-blip-captions). The results involve sampling algorithms ICFG and CFG, along with both Stable Diffusion v1.5 and U-Net fine-tuned models. We used the script available at https://github.com/huggingface/diffusers/blob/main/examples/text_to_image/train_text_to_image_lora.py. The reported CLIP Score varies with different $v$. It's important to note that FID is not reported due to the dataset's small size, making it impractical to calculate FID with the most general settings (as referenced in https://github.com/bioinf-jku/TTUR/issues/4).

For the original Stable Diffusion v1.5, with $w = 2.0$:

For the fine-tuned Stable Diffusion v1.5 on the pokemon-blip-captions dataset, with $w=2.0$:

Despite the untuned middle points and consequently suboptimal results, 2nd-order ICFG consistently outperforms CFG. Notably, with larger $v$ in the range from 0 to 1, it exhibits superior CLIP Score performance, aligning with the findings in COCO. We consider to incorporate additional results from larger datasets in the final version.

Concern on the proof.

We appreciate your concern, and we would like to highlight key points for validation of the proof.

• When $w \ne 0$, there exists an inconsistency between the enhanced intermediate distributions. One approach involves obtaining the conditional $x_0$ and then disturbing it, while the other involves acquiring the disturbed $x_t$ and subsequently applying the condition.
• However, when $w = 0$, there is no such inconsistency.
• Due to our approach of placing the condition enhancement in the condition space, our ICFG consistently exhibits $w=0$ in the score domain, effectively circumventing any potential inconsistency.

We extend our sincere gratitude to you once again.

We are deeply thankful for Reviewer jYKn's recognition of our novel formulation and well-studied hyper-parameters. We sincerely value their guidance in recommending the inclusion of additional empirical results, particularly in the form of visual comparisons.

Q1: Need more experiment results.

We added more experiments here, and after discussion, we will add all the experiments to the final manuscript.

1. For the training algorithm, due to GPU limitations, we were unable to train SD from scratch. However, we are currently training another framework, U-ViT, with a resolution of 256x256 on the COCO dataset from scratch to fully explore the capabilities of our ICFG. The preliminary results are listed below, and we plan to update them upon completion of the training.

We plan to report the final FID and CLIP scores after completing the entire training process. As of now, we have observed a notable 3% improvement in FID with ICFG training algorithm and CFG sampling algorithm compared with original training policy.

2. For the sampling algorithm, We incorporated CLIP Scores on the fine-tuned anything-v4.0 using prompts from the test dataset available at a prompt dataset.

We observe that our sampling algorithm is applicable to other fine-tuned Stable Diffusion (SD)-based models.

3. We have also included several visual comparisons on anything-V4 (https://huggingface.co/xyn-ai/anything-v4.0). It is evident from the images that second-order ICFG outperforms CFG, producing images with better-rendered hands and closer alignment to the provided prompts. The results are visualized in Figure 5 of Appendix F.

Furthermore, we plan to include more additional qualitative samples, contemplating the addition of more visual comparisons to vividly demonstrate the effectiveness of our methods.

Q2: Is the few-shot training of ICFG on a small dataset generalizable to large-scale setting?

We apologize for any confusion. For the training algorithm, we encourage the application of the training algorithm to the entire dataset to more comprehensively capture the guidance strength in the U-Net during training. However, the primary improvements highlighted in Table 1 do not require fine-tuning; the ICFG training algorithm is not necessary. We believe that the effectiveness of the second-order term is attributed to the continuity of the condition space. Consequently, we consider the improvements demonstrated by the second-order ICFG to be generalizable to large-scale settings.

Q3: It would be good to show the effectiveness of ICFG on other fine-tuned variations of Stable Diffusion such as anything-V4 (https://huggingface.co/xyn-ai/anything-v4.0), etc.

As explained in Q1, the second-order ICFG can be applied immediately and gain improvements. Furthermore, we plan to include more additional qualitative samples, contemplating the addition of more visual comparisons to vividly demonstrate the effectiveness of our methods.

As explained in response to Q1, the second-order ICFG can be applied immediately to achieve improvements. We have included several visual comparisons on anything-V4 (https://huggingface.co/xyn-ai/anything-v4.0). Additionally, we are in the process of including more qualitative samples. We are considering the addition of more visual comparisons to vividly demonstrate the effectiveness of our methods.

Dear reviewer jYKn,

Thank you for your invaluable review. We have thoroughly considered your concerns, particularly those pertaining to further qualitative sample comparison experiments of fine-tuned variations of Stable Diffusion, and the universal applicability of our training and sampling algorithm. In our response, we have diligently addressed these issues to the best of our ability.

As the discussion stage approaches its conclusion, we would like to inquire if you have any additional questions or suggestions. We would be delighted to engage in further discussion with you.

W2-1: The differences in the transition kernels.

The computation of transition kernels may pose challenges, but calculating the two distinct enhanced intermediate distributions resulting from differences in transition kernels is more straightforward.

We provide a theoretical analysis of the ratio between these two types of enhanced intermediate distributions. One approach involves obtaining the conditional $x_0$ and then disturbing it, while the other involves acquiring the disturbed $x_t$ and subsequently applying the condition.

In common scenarios, where we have $N$ paired $(\mathbf{x}_0^i, \mathbf{c}^i), i = 1, ..., N$ training data, given $\mathbf{c}=\mathbf{c}^1$, the ratio of the probabilities of the two enhanced intermediate distributions at $\mathbf{x}_t$ is expressed as:

where $Z_t$ and $Z_0$ are constants. Additional calculation details can be found in Appendix A.1.

In a common scenario, when $\mathbf{x_t}$ is approximately equidistant between $\alpha_t \mathbf{x}_0^1$ and $\alpha_t \mathbf{x}_0^2$ with $w\ne 0$, the ratio is evidently not 1 and becomes highly sensitive to changes in $\mathbf{x_t}$.

W2-2: How much difference how it affect the generation quality?

To ensure a fair evaluation of generation quality, we compare FIDs at the same CLIP Score. Our training-free second-order ICFG demonstrates a maximum improvement of 8.3% in FID. Further details can be found in the response to W1-2.

W3: The assumptions (Assumptions 3.1 and 4.1) made to achieve the proposed method do not seem realistic and were not verified in the experiments.

We appreciate your understanding, and we would like to clarify Assumptions 3.1 and 4.1 for better comprehension.

In practice, we utilize the CLIP embedding space as the condition space, which is 77x768-dimensional. To construct a cone structure, we use a finite number of outputs from CLIP as the basis, extending it by multiplying it with any positive number.

The true assumption, in essence, is that there are no identical directions among the outputs of CLIP for different texts, which is more realistic. Importantly, Assumption 3.1 aligns with CFG. While CFG operates in the score domain (output of U-Net), ICFG applies the cone assumption within the condition space.

The challenge posed by Assumption 3.1 lies in adapting the score predictor (U-Net for Stable Diffusion) because, during training, the U-Net has never encountered the extended cone structure. However, owing to the continuity of the extended condition space, the unadapted U-Net performs well within a small range of the cone structure. Consequently, the approximated second-order ICFG with the unadapted U-Net still demonstrates a notable improvement compared to CFG.

While Assumption 4.1 may not always hold for uniformly bounded $M_n$, it serves as a conceptual aid. In reality, the bounds need not be uniform and can be extended, for instance, to $M_{n+1} \sim o\left(\sqrt{n+1}\left[\frac{n+1}{eB}\right]^{n+1}\right)$, where values like $M_{n+1}= 10^n$ are acceptable. This extension provides a more practical perspective. Additional details can be found in Appendix B.

Q1: What is the main reason we should use ICFG instead of CFG?

Referring to W1-3.

Q2: How much does the enhanced transition kernel deviate from the original transition kernel (as in Theorem 3.1?) How much does the deviation affect the generation quality?

Referring to W2.

We thank reviewer cPdv very much for recognizing the theoretical contributions and novelty of our idea. Your assistance in refining assumptions and conducting a deeper analysis of the gap between enhanced intermediate distributions has been particularly insightful.

W1-1: It is not clear which additional information the second-order term provides.

We apologize for any confusion in my previous wording. The second-order ICFG provides more consistent enhanced intermediate distributions at various times, leading to improved overall performance. To elaborate further, the enhanced intermediate distributions at different time steps, denoted as $\bar{q}(\mathbf{x_{t_1} | c})$ and $\bar{q}(\mathbf{x_{t_2} | c})$ ($t_1 < t_2$), align more closely with the diffusion process ${\rm d}\mathbf{x} = \mathbf{f}(\mathbf{x},t) {\rm d} t + g(t) {\rm d} \mathbf{w}$. In this way, the original diffusion reverse process can be more easily accommodated for the conditional reverse diffusion process due to the reduced mismatch of the enhanced intermediate distributions.

W1-2: The improvement of second-order ICFG is not clear.

We apologize for any confusion in our experiment descriptions. While maintaining the same training policy as CFG, the continuous feature of the condition space enables second-order ICFG to consistently outperform CFG in both FID and CLIP scores across various CFG strengths ($w$).

For a clearer understanding of our advancements in generation quality, consider the comparison between second-order ICFG with $w = 2.0, v = 0.25, C=C_{all}$ and CFG with $w = 3.0$ (depicted as the second orange spot and the third blue spot in Figure 2). Despite nearly identical CLIP scores (26.12 and 26.11), we observed a notable improvement of $(16.68 - 15.28)/16.68 =$ 8.3% in FID. This implies that with increased control strength, the generated quality is significantly enhanced.

It is noteworthy that the CLIP Score improvement achieved by our second-order ICFG is nearly half of the overall improvements observed from Stable Diffusion v1.1 to v1.5 (comparison chart). This indicates a substantial improvement in performance.

Due to GPU limitations, we were unable to train SD from scratch. However, we are currently training another framework, U-ViT, with a resolution of 256x256 on the COCO dataset from scratch to fully explore the capabilities of our ICFG. The preliminary results are listed below, and we plan to update them upon completion of the training.

We plan to report the final FID and CLIP scores after completing the entire training process. As of now, we have observed a notable 3% improvement in FID with ICFG training algorithm and CFG sampling algorithm compared with the original training policy.

W1-3: The main reason we should use ICFG instead of CFG.

We apologize for any confusion, and we want to clarify that we have made significant updates to our manuscript to enhance clarity regarding the assumptions and calculation of the mismatch. Following a thorough rebuttal, we will incorporate all additional experiments into the manuscript.

While CFG has achieved notable success in various impactful projects, our focus is on elucidating potential theoretical inconsistencies. This inconsistency serves as the motivation behind our ICFG, and we present both theoretical and experimental analyses of the problem.

Theoretically, we provide a detailed analysis and present policies for addressing the issue on both the training and sample sides. Our proposed ICFG in the condition space effectively mitigates the distribution mismatch problem.

Experimentally, for integration into existing Stable Diffusion, we introduce a practical approach, the second-order ICFG. This modification requires only a minor adjustment involving the addition of a second term, without the need for additional training. Notably, this approach consistently improves both FID and CLIP scores. In scenarios with high CLIP scores, we observe an FID improvement of approximately 8%. Additionally, we have undertaken efforts to optimize the speed of our approach, aiming to make it more widely applicable and accepted within the community.

Thank you sincerely for your valuable feedback. We also acknowledge and apologize for any confusion resulting from Table 1. We hope the following explanation will be helpful.

Q1: About Table 1 Experiments.

For a clearer understanding of our improvements, we summarize key points here, and trust that the subsequent explanation will provide further clarity.

• When $v=0$, it indicates CFG, and when $v = 0.25$, it indicates ICFG settings.
• For FID, lower values indicate better performance.
• For CLIP Score, larger values indicate better performance.
• We conducted 5 times of sampling to ensure stable results for our ICFG.

Q1-1: Compare with same $w$.

In comparing the first two columns, with same $w=2$, FID of ICFG (15.25) is better than CFG(15.42); CLIP Score of ICFG(26.13) is better than CFG(25.80).

Similarly, in comparing the last two columns, with same $w=3$, FID of ICFG (16.58) is better than CFG(16.68); CLIP Score of ICFG(26.36) is better than CFG(26.12).

In summary of the Table 1 experiments, for all $w=1,2,3,4,5$, the FID and CLIP Score of ICFG consistently outperform those of CFG.

While we acknowledge that the improvement may appear marginal when comparing the same $w$, we attribute this to enhancements in both text-image alignment (CLIP Score) and generated quality (FID).

To provide a more comprehensive comparison, we introduce the following analysis. The motivation behind CFG and ICFG is to achieve enhanced control strength, as evaluated by the CLIP Score, which measures the alignment between the control condition text and the generated image. Our aim is to assess generation quality under the constraint of the same text-image alignment. Therefore, we conduct a comparison with the same CLIP Score to fairly evaluate the improvement of generation quality itself.

In summary, when referring to the same $w$, we are essentially comparing the desired control strength in the input. However, the evaluation of whether the control strength is sufficient being determined by the CLIP Score. This is the reason why we conduct the following experiments.

Q1-2: Compare with same CLIP Score.

Comparing the middle two columns, with almost the same CLIP Score (26.11), the decrease in our FID is $(16.68 - 15.28)/16.68 = 8.3$%, which is substantial. We apologize for any subjective expressions and leave it to the reader to assess the improvement.

Q1-3: More experiments.

We conduct our experiments on another fine-tuned model, anything-V4 (https://huggingface.co/xyn-ai/anything-v4.0). It is evident from the images that second-order ICFG outperforms CFG, producing images with better-rendered hands and closer alignment to the provided prompts, and the results are visualized in Figure 5 of Appendix F.

Q2: The response of W3 does not justify applying the cone assumption for realistic setting.

Our previous response emphasizes that the cone is typically constructed by extending the original conditional space. The crucial factor is whether the score predictor can adapt to it. If the space is continuous, it usually works within a small range. From our perspective, if the space can be extended to form a cone, the ICFG should be adaptable within a limited range.

To assess the U-Net's ability to capture the cone structure, we employ two complementary approaches. Firstly, we directly apply the guidance strength on the condition space to observe the generated results, as demonstrated in Appendix E. Secondly, we calculate the FID and CLIP Score of the samples as metrics for evaluation.

We thank you very much for the invaluable reviews and suggestions.

• The improvement is stable.

• We will include more validations on the comparison with same CLIP Score.

• We can check them by FIDs of same CLIP Score.

Dear reviewer cPdv,

Thank you for your insightful review. We have thoroughly considered your concerns, particularly those related to additional experiments, the advantages of ICFG, refining assumptions, and conducting a more in-depth analysis of the gap between enhanced intermediate distributions. In our response, we have endeavored to address these issues to the best of our ability.

As the discussion stage nears its conclusion, we welcome any further questions or suggestions you may have. We are eager to engage in further discussion and address any remaining points of consideration.

We extend our appreciation to Reviewer LfUU for acknowledging both our theoretical analysis and empirical results. Their guidance in accelerating our method and perfecting the non-strict 2nd order term reasons has been tremendously helpful.

Q1: Cone assumption.

We appreciate your understanding, and we would like to clarify Assumption 3.1 for better comprehension. In practice, we utilize the CLIP embedding space as the condition space, which is 77x768-dimensional. To construct a cone structure, we use a finite number of outputs from CLIP as the basis, extending it by multiplying it with any positive number.

The true assumption, in essence, is that there are no identical directions among the outputs of CLIP for different texts, which is more realistic. Importantly, Assumption 3.1 aligns with CFG. While CFG operates in the score domain (output of U-Net), ICFG applies the cone assumption within the condition space.

The challenge posed by Assumption 3.1 lies in adapting the score predictor (U-Net for Stable Diffusion) because, during training, the U-Net has never encountered the extended cone structure. However, owing to the continuity of the extended condition space, the unadapted U-Net performs well within a small range of the cone structure. Consequently, the approximated second-order ICFG with the unadapted U-Net still demonstrates a notable improvement compared to CFG.

In summary, the cone is typically constructed by extending the original conditional space. The crucial factor is whether the score predictor can adapt to it. If the space is continuous, it usually works within a small range. From our perspective, if the space can be extended to form a cone, the ICFG should be adaptable within a limited range.

To assess the U-Net's ability to capture the cone structure, we employ two complementary approaches. Firstly, we directly apply the guidance strength on the condition space to observe the generated results, as demonstrated in Appendix E. Secondly, we calculate the FID and CLIP Score of the samples as metrics for evaluation.

Q2: Algorithm 3 relaxing 2nd order term.

Thank you very much for your question, which prompted significant revisions to our manuscript. In the introduction of the non-strict approach, the parameter $v$ does make logical sense.

Given the order of approximation, the remainder term $R_n{(\beta)}$ becomes larger with increasing $\beta$ (since $R_n{(\beta)}\sim o((\beta-1)^n)$), leading to potentially worse results. Therefore, a strict 2nd-order algorithm is more effective for smaller $\beta$. Additionally, we considered whether limiting $(\beta - 1)^2$ with an upper bound $v$ would result in consistent improvements for larger $\beta$ values.

Experiments confirmed our hypotheses. Viewing $v$ as a tuning parameter is also meaningful: $\sqrt{v}$ serves as an upper bound for the 2nd order "step size." Therefore, from an experimental standpoint, the non-strict 2nd-order algorithm appears to be more suitable.

Q3: Forward passes for naive 2nd order ICFG implementation.

The calculation is a crucial aspect of our concern. The nth-order term itself requires n+1 points for estimation. In this computation, we need $\frac{(n+1)(n+3)}{2}$ forward passes to obtain all the terms. However, the nth order term can reuse n points to estimate the (n-1)th order term, reducing the actual forward passes to 3 for 2nd order ICFG.

In comparison, CFG requires 2 forward passes, meaning our method incurs only a 50% increase in computation. To further minimize this additional computation, we propose a method to selectively apply 2nd order ICFG only during key timesteps. Upon examining the insights from the second figure in Figure 1, it is apparent that the guidance is more significant during the middle approximately 40% of the time. Capitalizing on this observation, our plan is to implement the second ICFG exclusively during these specific time steps. This strategic approach is expected to result in a substantial reduction, decreasing the extra inference time from 50% to approximately 20%.

Dear reviewer LfUU,

We express our heartfelt appreciation for your valuable review and advice. We firmly believe that with all your guidance, our manuscript has undergone significant improvement. We are committed to incorporating all discussions into the final version, and we are also considering the inclusion of some heuristics to estimate a suitable choice for $v$.

Once again, we extend our sincere gratitude to reviewer LfUU.