Your Absorbing Discrete Diffusion Secretly Models the Conditional Distributions of Clean Data

Thank you for your thoughtful feedback and for recognizing the contributions and insights of our work.

W1: Related work

Thanks for pointing out the related work, we will add the following discussions in the revision.

W1.1: Is RADD's time-independent parameterization unique?

We clarify that our time-independent parameterization is a unique theoretical contribution of our work. Specifically, the parameterization in [1*] remains time-dependent, as explicitly shown in their Equation (11). In contrast, our approach simplifies the expression to its most fundamental form, completely removing the explicit dependency on $t$. For further details, please refer to the response to Common Concern 3.

Additionally, [2*] lacks a theoretical understanding and rigorous analysis for removing the time condition and the caching trick.

For a more detailed comparison and discussion with [1*] and [2*], please see Lines 486 to 507 of our submission.

W1.2: Comparison to prior works concerning equivalence discussion

We acknowledge that [3*] establishes the equivalence between ARMs and the ELBO of the absorbing diffusion models, corresponding to the $\mathcal{L}_ {\text{DSE}}^T(\boldsymbol{x}_ 0)$ and $\mathcal{L}_ {\text{AO}}(\boldsymbol{x}_ 0)$ terms in Eq. (3.7) of our submission. However, our approach offers unique contributions by:

• Leveraging the time-independent properties under the reparameterization formulation, which provides an alternative proof of this equivalence.
• Extending the analysis to demonstrate the equivalence of four losses including $\mathcal{L}_ {\text{DSE}}^T(\boldsymbol{x}_ 0)$ and $\mathcal{L}_ {\text{AO}}(\boldsymbol{x}_ 0)$ in Eq. (3.7), deepening our understanding of absorbing discrete diffusion.
• Conducting a comprehensive study of these losses, presented in Table 1, which has not been previously explored before to our knowledge.

Please see more details in the common concern 2.

Q1:Parametrisation and Algorithms of AO-ARM

Yes, the denoising model trained with diffusion loss can be reparametrised as an AO-ARM. We have included a seesion in the Appendix of the revision to discuss the parametrisation and algorithms of AO-ARM for better understanding.

[1*] Shi, Jiaxin, et al. "Simplified and Generalized Masked Diffusion for Discrete Data." arXiv preprint arXiv:2406.04329 (2024).

[2*] Sahoo, Subham Sekhar, et al. "Simple and Effective Masked Diffusion Language Models." arXiv preprint arXiv:2406.07524 (2024).

[3*] Hoogeboom, Emiel, et al. "Autoregressive diffusion models." arXiv preprint arXiv:2110.02037 (2021).

We sincerely appreciate reviewer 4Udp's thorough and timely discussion and apologize for our earlier misunderstanding. We agree that previous works also attempt to remove $t$, and our main contribution lies in providing theoretical and experimental support for why removing $t$ potentially yields better outcomes.

Thank you again for your constructive feedback and acknowledgment of our contributions.

Thank you for the detailed review and acknowledgment of our contributions. Below we address specific questions.

W1:Actual sampling efficiency under cache strategy

Thank you for your feedback. Your analysis is correct: in static-shape compiled programs, such as those implemented in JAX, our caching strategy does not reduce the workload due to the lack of dynamic batching support. However, PyTorch, which is widely used in both research and industry, supports dynamic shapes. This allows our caching strategy to effectively accelerate sampling, as shown in our results. For futher discussion on mini-batch sampling, please refer to our response to W1.1 for reviewer fLbR.

W2 & Q1 : Is RADD's time-independent parameterization unique?

We clarify that our time-independent parameterization is a unique contribution. In particular, the parameterization in [1*] remains time-dependent, as explicitly shown in their Equation (11). In contrast, our approach simplifies the expression to its most fundamental form, completely removing explicit dependency on $t$. For further details, please refer to the common concern 3.

We will make it clearer in the revision.

[1*] Shi et al. Simplified and generalized masked diffusion for discrete data, 2024.

Thank you for your extremely thorough review and constructive feedback on our paper. Below, we address your concerns and suggestions.

W1: Mimi-batch issue and comparison with Chen et al. 2024

W1.1: Efficiency of our caching strategy with mini-batch

Our caching strategy enables efficient batch sampling by dynamically adjusting neural network computations to process only the samples that change at each timestep. To illustrate this efficiency, we conducted experiments generating 64 samples under various batch sizes on an NVIDIA 4090 GPU (24GB VRAM, maximum batch size = 8).

The results, summarized in the table below, demonstrate that the average generation time per sample (in seconds) with our caching strategy consistently outperforms SEDD across different batch sizes and timestep configurations:

SEDD-medium:

RADD-medium:

We provide a detailed analysis below:

• Batch Size and GPU Utilization: For a fixed model, increasing the batch size leads to improved GPU utilization before reaching the maximum batch size, reducing the average sampling time per sample. In the case of RADD, sampling time decreases significantly as the batch size increases from 1 to 4. However, the reduction in sampling time becomes minimal beyond a batch size of 4, indicating that GPU utilization has nearly reached its maximum capacity at this point. This demonstrates the scalability of our approach within the limits of the hardware's parallel processing capabilities.
• SEDD vs RADD: Under identical batch sizes and timesteps, RADD consistently outperforms SEDD in sampling speed. This highlights the efficiency of RADD’s design, where the caching mechanism reduces redundant computations and achieves faster generation, especially for larger batch sizes and higher timesteps.

We will add the results in the revision, which improves the quality of the paper.

W1.2: Comparison with Chen et al. 2024

Our paper and your comment focus on continuous-time setting. As detailed in Common Concern 1, directly applying the continuous-time samplers from [1*] to SEDD results in $\text{NFEs} = d$ (equivalent to ARMs). Moreover, Chen et al. [1*] did not explore methods to reduce NFEs below $d$ for continuous-time models. In contrast, RADD can directly employ its caching strategy in continuous-time settings, allowing NFEs to be reduced below $d$, as the input to the model remains independent of time.

We will include the discussion in the revised version.

W2&Q2: Related work

We sincerely thank the reviewer for pointing out the related works. Below, we provide our comparisons and clarifications:

• Comparison with Austin et al., 2021 Our work provides a rigorous proof of the equivalence, as discussed in detail in the response to Common Concern 2.
• Comparison with Hoogeboom et al., 2021 We acknowledge that Hoogeboom et al., 2021, establishes the equivalence between ARMs and the ELBO of absorbing diffusion models. However, our approach offers unique contributions, including: an alternative proof of this equivalence leveraging time-independent properties, extending the analysis to demonstrate the equivalence of four losses in Eq. (3.7), and conducting a comprehensive study of these losses, presented in Table 1. For more details, please refer to Common Concern 2.
• Comparison with Campbell et al., 2024 Campbell et al., 2024, is a relevant work on discrete diffusion. However, we did not find a discussion or connection to AO-ARMs in their work.
• Comparison with Chen et al., 2023 Regarding the connection between caching and derandomized sampling in Chen et al., 2023, please see our detailed response to W1.2.

Further, while our work uses a time-independent cross-entropy loss based on the reparameterization, we do not claim to propose a new loss function.

We are happy to include discussions of these related works in our revised manuscript. These additions will enhance the paper’s quality while leaving our main contributions unchanged.

Q1: Experiments

Q1.1: Which model is used for evaluating the generation quality?

For the evaluation of generative perplexity in Section G.4, we also used the GPT-2-Large model. The higher generative perplexity values compared to SEDD [2*] are primarily due to differences in precision during categorical sampling. We explain the precision choices below:

• Previous works, including SEDD, employed fp32 precision for categorical sampling. This introduced precision errors that effectively acted as a form of annealing, resulting in deceptively lower perplexity values (see Appendix G.3).
• In contrast, we utilized fp64 precision to minimize precision errors. While this approach provides more accurate sampling, it inherently results in higher perplexity values due to the lack of annealing effects.

To address your concern, we add new results using fp32 precision, where SEDD’s generative perplexity matches the values reported in their original paper. The results are as follows:

RADD-$\lambda$-DCE (medium):

SEDD (medium):

These results demonstrate that under fp32 precision, the perplexity of RADD-$\lambda$-DCE (medium) and SEDD (medium) are similar when using the same number of sampling steps. However, RADD's caching strategy enables it to achieve comparable perplexity with fewer NFEs (Number of Function Evaluations), highlighting its advantage in sampling efficiency. Namley, the conclusion in fp32 precision holds the same as in fp64 precision.

Q1.2: Training Ablation Experiments

Thank you for the suggestion. We agree that an ablation study would provide valuable insights. Unfortunately, due to time constraints, we are unable to complete this experiment within the rebuttal period. However, we are currently running the experiment and will include the results in the final version of the manuscript.

Nevertheless, we believe that the 400K results, conducted following the settings of SEDD, are sufficient to demonstrate the effectiveness of our approach.

[1*] Chen, Zixiang, Huizhuo Yuan, Yongqian Li, Yiwen Kou, Junkai Zhang, and Quanquan Gu. Fast Sampling via Discrete Non-Markov Diffusion Models, 2024.

[2*] Aaron Lou, Chenlin Meng, and Stefano Ermon. Discrete diffusion modeling by estimating the ratios of the data distribution, 2024.

[3*] Austin, Jacob, Daniel D. Johnson, Jonathan Ho, Daniel Tarlow, and Rianne Van Den Berg. "Structured denoising diffusion models in discrete state-spaces." Advances in Neural Information Processing Systems 34 (2021): 17981-17993.

[4*] Hoogeboom, Emiel, Alexey A. Gritsenko, Jasmijn Bastings, Ben Poole, Rianne van den Berg, and Tim Salimans. "Autoregressive diffusion models." arXiv preprint arXiv:2110.02037 (2021).

[5*] Campbell, Andrew, Jason Yim, Regina Barzilay, Tom Rainforth, and Tommi Jaakkola. "Generative flows on discrete state-spaces: Enabling multimodal flows with applications to protein co-design." arXiv preprint arXiv:2402.04997 (2024).

Dear Reviewer fLbR,

Thank you once again for your constructive feedback. We would like to kindly remind you that we have now included additional experiments to address your concerns:

• Validation of the acceleration performance of our cache strategy under varying batch sizes.
• Generative perplexity results under FP32 precision, aligned with SEDD.

Additionally, we have also added a detailed comparison with all related works including Chen et al. 2024, Austin et al., 2021, Hoogeboom et al., 2021, and Campbell et al., 2024.

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As the discussion period is coming to a close in two days, we look forward to your response and would be happy to address any further comments or questions you may have.

Best,

The Authors

Dear Reviewer fLbR,

Thank you once again for your valuable comments. We believe that our responses have addressed your concerns comprehensively. Could you kindly share whether our replies meet your expectations and if you would consider reevaluating the paper?

Best regards,

The Authors

Thank you for your timely feedback and acknowledgment of our theory. Regarding the caching mechanism, we clarify that the NFE mentioned in our paper refers to a single sample, but it applies to minibatch cases, as detailed in our response to your W1.1.

In particular, we agree that within a minibatch, some samples may remain unchanged while others may evolve. In our implementation, we account for this by employing a dynamic batch-size approach. Only the samples that have changed are passed through the neural network for computation. While this still involves a "batch-level NFE" (your NEF according to our understanding), the overall computational load is reduced, leading to faster sampling. In the response to your W1.1, we included a comparison of sampling speeds for batch sizes of 1, 2, 4, and 8 (the maximum batch size per GPU). The results show that caching consistently accelerates sampling.

In comparison, concurrent work [Sahoo et al., 2024] does not consider dynamic batch size in their implementation, so their practical acceleration falls short of the theoretical potential.

We sincerely appreciate your feedback, which helps make our paper clearer. We have already presented the accelerated sampling results for minibatch scenarios in Appendix J.4 of our revision. We will further incorporate the above discussion to provide additional clarity. Please feel free to reach out with any further questions.

Thank you very much for your response.

The clarification of generative perplexity evaluation gap makes sense to me, given the differences between the fp32 and fp64 implementations. My primary concern was in the presentation of the paper's claims. I agree with Reviewer 4Udp that the main novelty of the work lies in establishing the theoretical connection between the parameterization of score entropy (ratio) and clean data prediction. But the practical contribution is less pronounced, as the performance are not very different under one form or the other, especially in terms of generation quality. Additionally, the theory surrounding time-independent clean data reconstruction is relatively well-known and not novel. Therefore, I would recommend placing greater emphasis on the theoretical connection in the reparameterization framework and less on Section 3.3.

Regarding caching, my concern is more about the number of NFEs required as the batch size increases. In such cases, it is highly likely that at least one sample in the batch will require a transition from a mask to a token. This introduces a limit to caching efficiency as batch size grows, as observed in concurrent work [Sahoo et al., 2024].

I have updated my evaluation score and look forward to seeing the revised version address the points I raised, particularly those related to the theoretical presentation. [1] Sahoo, Subham Sekhar, Marianne Arriola, Yair Schiff, Aaron Gokaslan, Edgar Marroquin, Justin T. Chiu, Alexander Rush, and Volodymyr Kuleshov. "Simple and Effective Masked Diffusion Language Models." arXiv preprint arXiv:2406.07524 (2024).

We sincerely appreciate your constructive feedback and the score update. We would like to clarify that the concurrent work [Sahoo et al., 2024] does not claim omitting $t$ is beneficial for training, nor does it explain why this is the case.

To provide some context, we first reference the concurrent work [Sahoo et al., 2024] as described in their Sec 3.2.2:

"Therefore, we introduce a model $x_\theta(z_t, t): V \times [0, 1] \to \Delta^K$ that approximates $x$ with a neural network. We can also omit explicit dependence of $x_\theta$ on time $t$, which simplifies sampling, yielding a 2x inference speed-up”

It is evident that [Sahoo et al., 2024] approximates the time-dependent function $x_\theta(z_t, t)$ with a time-independent $\text{NN}(z_t)$ solely for sampling, without providing any explanation as to why this omission does not impact training.

Moreover, in Appendix E.5 of [Sahoo et al., 2024], Table 12 empirically demonstrates that removing $t$ results in slightly better PPL on a single task. However, they do not claim and explain the improvement at all, stating only the following without further analysis:

"We observe that time-conditioning has minimal impact on perplexity."

In conclusion, we believe it is fair to assert that the concurrent work [Sahoo et al., 2024] does not claim that omitting $t$ is beneficial for training, nor do they provide any theoretical justification for it.

In contrast, our work provides a rigorous theoretical foundation demonstrating that transition probability can be parameterized in a manner that eliminates dependence on $t$ in both the score parameterization (Theorem 1) and the mean parameterization (Eqn. E1 in Appendix E of our submission) as used in [Sahoo et al., 2024]. This theoretical insight clarifies why our reparameterization on omitting $t$ simplifies the training target and, consequently, benefits the training process.

Such a theory provides a significant practical improvement. In particular, the SEDD-Scale and RADD-DSE models in our submission differ only in whether $t$ is removed or retained, which naturally serves as an ablation study on $t$ and a validation of our theoretical insights (see more details in response to Q2 from Reviewer rVfV). The results in Tables 1 and 2 of our submission demonstrate that RADD-DSE consistently and substantially outperforms SEDD-Scale, providing strong empirical support for our contributions, as also acknowledged by Reviewer 4Udp.

Once again, we deeply appreciate your valuable feedback. We will revise Section 3.1 as soon as possible as the rebuttal PDF submission deadline is closing soon.

Q4: Additional experiments on sampling

Thank you for raising this point. Regarding the $\tau$-leaping sampler mentioned in the question, we are unsure whether you are referring to $\tau$-leaping sampler for discrete diffusion from [4*] or the Tweedie $\tau$-leaping sampler proposed in [3*]. To clarify:

• $\tau$-leaping sampler [4*] is an earlier sampling method.
• Tweedie $\tau$-leaping sampler [3*], which is a more recent and efficient approach, has been adopted in our work.

If the question refers to the Tweedie $\tau$-leaping sampler, we present the following perplexity results for RADD-$\lambda$-DCE medium:

Except for the last column (corresponding to the AO-ARM results under the random row in Table 4), all values were measured using Tweedie $\tau$-leaping. The results demonstrate that with increasing steps, the performance of Tweedie $\tau$-leaping converges and aligns closely with the AO-ARM results. All evaluations were conducted on the same RADD-$\lambda$-medium model to ensure a fair comparison.

We will add the new results and dicussion in the revision. If you are referring to the $\tau$-leaping sampler from [4*], please let us know, and we will be happy to perform the necessary experiments.

Q5:Training Tokens and Iterations:

Thank you for your question. The training tokens for SEDD and RADD are the same when using the same number of iterations. Specifically, for both models:

• 400K iterations correspond to approximately 105 billion tokens.
• 1000K iterations correspond to approximately 262 billion tokens. It is worth noting that the entire OpenWebText dataset contains only 9 billion tokens, meaning both models cycled through the dataset multiple times during training.

In the main paper, we report results for models trained for 400K iterations to align with the primary baseline, SEDD [3*], which was also trained for 400K iterations. This alignment ensures a fair comparison. Training for 1000K iterations was conducted to observe convergence behavior and evaluate the performance trends of our RADD model more comprehensively.

We will clarify it in the revision.

[1*] Aaron Gokaslan and Vanya Cohen. Openwebtext corpus. http://Skylion007.github.io/ OpenWebTextCorpus, 2019

[2*] Gulrajani, I. and Hashimoto, T. Likelihood-based diffusion language models. In Advances in Neural Information Processing Systems, 2023

[3*] Aaron Lou, Chenlin Meng, and Stefano Ermon. Discrete diffusion modeling by estimating the ratios of the data distribution, 2024.

[4*] Andrew Campbell, Joe Benton, Valentin De Bortoli, Tom Rainforth, George Deligiannidis, and Arnaud Doucet. A Continuous Time Framework for Discrete Denoising Models, 2022.

Thank you for your detailed review and insightful comments. Below, we address your concerns and suggestions.

W1: Limitations of experiments

We utilized OpenWebText and the zero-shot likelihood metric due to their representativeness and widespread use in benchmarking language models [1*, 2*, 3*], ensuring comparability with prior work. Additionally, this choice aligns with SEDD, our primary competitor, to maintain fairness. We are open to conducting additional experiments based on specific recommendations and will include the results in the final version given the time limits during rebuttal.

W2 & Q3: Differences in zero-shot likelihoods in Tables 1, 2, and 5

Thank you for your constructive feedback. The equivalence of the objectives we refer to is in terms of their expected values. Specifically, in an ideal scenario with infinite data, the two loss functions would converge to the same solution. However, for models trained on finite datasets using gradient-based optimization methods, the gradient estimation for each loss function differs. Combined with the non-convex nature of neural networks, these factors lead the models to converge to different local optima, which explains the discrepancies in zero-shot likelihoods observed in Tables 1, 2, and 5.

We also appreciate your observation that one RADD model underperforms SEDD on LAMBADA while another outperforms it. This inconsistency likely stems from the inherent stochasticity and non-convexity of the optimization process. Nonetheless, across the five datasets, the overall performance of all RADD loss functions surpasses that of SEDD.

We will clarify it in the final version.

W3 & Q1 & Q2 : Architectures and ablation

Part 1: Architecutural comparison between RADD and SEDD

We would like to clarify a potential misunderstanding regarding the architectural differences between RADD and SEDD. The theoretical foundation of RADD demonstrates that modeling the concrete score function requires both the removal of time conditioning and the inclusion of a scaling term to achieve optimal performance. The architectural differences between these models are summarized below:

Part 2: Ablation of the time condition

It can be seen that adding the time condition back into RADD would essentially revert it to the same model as SEDD-Scale. In our submission, we trained a version of RADD with the same DSE loss as SEDD, referred to as "RADD-DSE". These models have similar parameter counts, with the main difference being the inclusion or exclusion of time conditioning. Therefore, the results of SEDD-Scale and RADD-DSE in Tables 1 and 2 of our submission serve as an ablation study on the impact of time conditioning. Notably, RADD-DSE significantly outperforms SEDD-Scale in these comparisons, underscoring the effectiveness of RADD’s architectural design.

We will make it clearer in the revision.

W4: AR sampling speed

Thanks for the constructive comments. As highlighted in Fig. 1 of SEDD [3*], SEDD surpasses AR sampling in terms of speed. Moreover, RADD further improves upon SEDD in sampling speed, as demonstrated in Fig. 1b and Table 3 of our submission, indicating that RADD also outperforms AR sampling.

We will include this discussion in the revised version. If the reviewer is interested, we are open to including a direct comparison with ARMs in the final version, as the time constraints of the rebuttal phase limit our ability to provide this analysis now.

Thank you for your timely response. We appreciate your recognition of our clarifications and are glad to address the remaining points:

1. Test Perplexity on OWT: OpenWebText [1*] does not include a predefined test set. In line with the methodology followed in SEDD, we did not partition the OpenWebText into a separate test set for evaluation. Consequently, neither SEDD nor RADD could report test perplexity on OWT.

2. Mauve Scores of Conditionally Generated Text: We acknowledge the importance of evaluating conditional generation quality using metrics such as Mauve. While we have been actively working to obtain Mauve scores for comparison with Table 5 of the SEDD paper, based on the official code of SEDD, we have encountered challenges replicating SEDD's reported Mauve results. Notably, other works [5*] have also reported similar difficulties as discussed in Appendix A.4 . We are continuing to resolve these issues and hope to report the results soon.

3. Variance of the Objectives: Yes, different objectives have distinct variances. Namely, the gradients of the losses are different when estimated on finite (i.e. a mini-batch of) samples, influencing training dynamics and convergence speed. This could further lead to discrepancies in zero-shot performance even when training for the same number of steps. To the best of our knowledge, quantifying the variances of such objectives is challenging due to the presence of the neural network. We will add a discussion in the revision.

If you have further questions or requirements, we would be happy to continue the discussion. Thank you again for your thoughtful feedback!

[1*] Aaron Gokaslan and Vanya Cohen. Openwebtext corpus. http://Skylion007.github.io/ OpenWebTextCorpus, 2019

[5*] J. Deschenaux and C. Gulcehre, “Promises, outlooks and challenges of diffusion language modeling,” 2024, https://arxiv.org/abs/2406.11473

Thank you for your extremely thorough review and constructive feedback on our paper. Below, we address your concerns and suggestions.

W1: Can the RADD methodology be applied to other discrete diffusion models?

We appreciate your thoughtful question. RADD is specifically designed for absorbing discrete diffusion, which has demonstrated SOTA performance among discrete diffusion models [1*,2*]. The key intuition behind RADD lies in factoring concrete score into a time-independent term and a time-dependent term. To the best of my knowledge, this specific decomposition is not directly applicable to other models like multinomial diffusion, which have fundamentally different score formulations. We will clarify this in the manuscript.

W2: Comparison with Chen et al. 2024

Thank you for your insightful question. Below, we summarize the key differences, with additional details provided in Common Concern 1. The samplers proposed in Chen et al. (2024) [3*] and in this work are applicable in different scenarios, as discussed in the response to W2.2.

W2.1: Similar NFEs estimation with Chen et al. 2024

Although the samplers in this paper and in Chen et al. (2024) originate from different formulations (detailed in Common Concern 1), the results of Theorem D.1 [3*] for the discrete-time sampler align with those of our Eq. (3.4). However, as discussed later, the two samplers are applicable in different scenarios. If the reviewer has additional insights, we would welcome further discussion.

W2.2: If using methods in Chen et al. 2024 for sampling, does RADD still have any advantage over SEDD?

SEDD is a continuous-time model. As outlined in Common Concern 1, directly applying the continuous-time samplers from [3*] to SEDD results in $\text{NFEs} = d$ (equivalent to ARMs). Moreover, Chen et al. [3*] did not explore methods to reduce NFEs below $d$ for continuous-time models like SEDD. In contrast, RADD can directly employ its caching strategy in continuous-time settings, allowing NFEs to be reduced below $d$, as the input to the model remains independent of time.

Additionally, RADD not only simplifies but also improves the performance of SEDD (see Table 1 of our submission), providing advantages that extend beyond fast sampling.

We will include all related discussions in the revised manuscript.

[1*] Austin, Jacob, Daniel D. Johnson, Jonathan Ho, Daniel Tarlow, and Rianne Van Den Berg. "Structured denoising diffusion models in discrete state-spaces." Advances in Neural Information Processing Systems 34 (2021): 17981-17993.

[2*] Aaron Lou, Chenlin Meng, and Stefano Ermon. Discrete diffusion modeling by estimating the ratios of the data distribution, 2024.

[3*] Chen, Zixiang, Huizhuo Yuan, Yongqian Li, Yiwen Kou, Junkai Zhang, and Quanquan Gu. "Fast Sampling via Discrete Non-Markov Diffusion Models.", 2024

Dear Reviewer u3V7,

Thank you once again for your constructive feedback. We would like to kindly remind you that we have clarified the applicable scenarios of RADD and added a detailed comparison with Chen et al. 2024.

As the discussion period is coming to a close in two days, we look forward to your response and would be happy to address any further comments or questions you may have.

Best,

The Authors

Dear Reviewer u3V7,

Thank you very much for your time and valuable comments. We regret that we have not received your response during the rebuttal period. As the rebuttal phase is set to conclude in one day, we believe that our responses have sufficiently addressed your concerns.

Best regards,

The Authors