The Emergence of Reproducibility and Consistency in Diffusion Models

Q4: “Have the authors investigated larger datasets, such as CoCo, ImageNet?”

A4: We thank the reviewer for the valuable suggestions to study the phenomenon more comprehensively. During the revision, we conducted more experiments considering two extra cases: (i) the Conditional Diffusion Model on the ImageNet Dataset [1], and (ii) Stable Diffusion [2] on the LAION-5B dataset [3]. The results can be found in Appendix A.1 and A.2 of the revised paper. As such, our new experiments not only covered larger datasets in (i, ii), but also text-to-image diffusion models in (ii). Specifically, we find the following:

1. Conditional Diffusion Model on the ImageNet Dataset: We evaluated the reproducibility of EDM [4] and ADM [5], as shown in the following table:

   $\text{RP} \text{ Score}$	EDM	ADM
   EDM	1.0	0.81
   ADM	0.81	1.0

   Detailed experimental settings and visual results are presented in Appendix A.1. We find that the model reproducibility still exists on ImageNet.

2. Stable Diffusion on the LAION-5B Dataset: We evaluated the reproducibility of stable diffusion models with versions v1-1 to v1-4, with their respective reproducibility scores shown in the table below:

   $\text{RP} \text{ Score}$	V1-1	V1-2	V1-3	V1-4
   V1-1	1.0	0.10	0.03	0.03
   V1-2	0.10	1.0	0.21	0.20
   V1-3	0.03	0.21	1.0	0.63
   V1-4	0.03	0.20	0.63	1.0

   The relationship between V1-1 to V1-4 could be summarized as

   • Versions v1-1, v1-2, and v1-3 each are trained on different subsets of the LAION-5B dataset.
   • Versions v1-3 and v1-4 share the same training subset from LAION-5B.
   • Version v1-2 is resumed from v1-1, while v1-3 and v1-4 are resumed from v1-2.

   Notably, only versions v1-3 and v1-4 were trained on the exact same dataset, resulting in the highest reproducibility scores (0.63). Lesser but noticeable reproducibility scores (below 0.21) are observed among v1-1, v1-2, and v1-3, this can be attributed to their sequential training and overlapping datasets. There is no model reproducibility between v1-1 and others, because the training datasets are nonoverlapping and different. Further details on the experimental setup and visual results can be found in Appendix A.2.

In conclusion, our findings suggest that this phenomenon persists for larger diffusion models trained on large datasets.

[1]Deng, Jia, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. "Imagenet: A large-scale hierarchical image database." In 2009 IEEE conference on computer vision and pattern recognition, pp. 248-255. Ieee, 2009.

[2]Rombach, Robin, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. "High-resolution image synthesis with latent diffusion models." In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10684-10695. 2022.

[3] Schuhmann, Christoph, Romain Beaumont, Richard Vencu, Cade Gordon, Ross Wightman, Mehdi Cherti, Theo Coombes et al. "Laion-5b: An open large-scale dataset for training next generation image-text models." Advances in Neural Information Processing Systems 35 (2022): 25278-25294.

[4] Karras, Tero, Miika Aittala, Timo Aila, and Samuli Laine. "Elucidating the design space of diffusion-based generative models." Advances in Neural Information Processing Systems 35 (2022): 26565-26577.

[5] Dhariwal, Prafulla, and Alexander Nichol. "Diffusion models beat gans on image synthesis." Advances in neural information processing systems 34 (2021): 8780-8794.

Q3: “How does the ODE solver and the NFE influence the image?”

A3:

We thank the reviewer for the question. We want to clarify that the model reproducibility is regarding the content or shape of the image, not image quality. Starting from the same noise, we find that different diffusion models generate the same content, but the image quality can be different depending on the model.

To the reviewer’s question, for the ODE solver, we have covered DPM-Solver [1], Heun-Solver [2], and DDIM [3] for unconditional reproducibility. We find that different ODE solvers do not affect model reproducibility, although they may affect the image quality. The experiment settings are in Table 1, Appendix B. The reproducibility scores are shown in Figure 14.

For NFE, we added more experiments in Appendix A.4 during the revision. While a lower NFE tends to degrade the generated image quality, our observations reveal that the content of the generated images remains remarkably consistent across varying NFE levels; see Figure 12.

[1] Lu, Cheng, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. "Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps." Advances in Neural Information Processing Systems 35 (2022): 5775-5787.

[2] Karras, Tero, Miika Aittala, Timo Aila, and Samuli Laine. "Elucidating the design space of diffusion-based generative models." Advances in Neural Information Processing Systems 35 (2022): 26565-26577.

[3] Song, Jiaming, Chenlin Meng, and Stefano Ermon. "Denoising diffusion implicit models." arXiv preprint arXiv:2010.02502 (2020).

We thank the reviewer for acknowledging the quality of our presentation, experiments, and the timeliness of our results. We value the constructive feedback that the reviewer provided. In the following, we carefully respond to each of the concerns and suggestions that the reviewer raised.

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Q1: “But it seems that there is no further improvement in the training of the current diffusion model.”

A1:

We agree with the reviewer that improving the training of diffusion models is not the main focus of this work. However, we believe that research progress can be driven by two distinct avenues: (i) technical enhancements and (ii) the pursuit of deeper insights. Both aspects hold equal significance. In our investigation, we directed our efforts towards gaining a deeper understanding of diffusion models, unveiling several intriguing phenomena, which we would like to highlight as follows.

• The first systematical and quantitative study of the model reproducibility phenomenon on the unconditional diffusion model.

• Moreover, we discovered that model reproducibility occurs in two distinct regimes, the “memorization regime” and “generalizability regime”, and model reproducibility has a strong correlation with model generalizability. Moreover, we have theoretically and experimentally justified the reproducibility in the memorization regime.

• We generalized the study of model reproducibility in a variety of settings, including conditional diffusion models, diffusion models for inverse problems, and fine-tuning diffusion models. For these different settings, we showed that the model reproducibility manifests in different but principled ways.

Our findings could have a variety of potential applications that we discuss in Q2.

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Q2: “How can we get a better training strategy from the current conclusion?”

A2:

We thank the reviewer for this important question, and we believe that our understanding opens many interesting application ideas beyond improving training efficiency. One major insight gained from this study is that the mapping from the noise space to the image space is a unique one-to-one mapping, regardless of network architectures or training procedures.

As we discussed in Section 1, this observation implies that the noise space contains valuable information that can be utilized for controlled and efficient data-generative processes using diffusion models. For instance,

• In text-driven image generation, insights into this question could help to guide the content generation (e.g., adversarial attacking [1], robust defending [2], copyright protection [3]) using the same prompt but varying noise inputs.

• Moreover, in the context of solving inverse problems, an answer to this question could guide us to select the input noise for improving the sampling efficiency, and reducing the uncertainty and variance in our signal reconstruction [4, 5, 6].

• Theoretically, understanding the question will shed light on how the mapping function is learned and constructed between the noise and data distributions, which is crucial for understanding the generation process through the distribution transformation or identifiable encoding [7, 8].

[1] Zou, Andy, Zifan Wang, J. Zico Kolter, and Matt Fredrikson. "Universal and transferable adversarial attacks on aligned language models." arXiv preprint arXiv:2307.15043 (2023).

[2] Zhu, Kaijie, Jindong Wang, Jiaheng Zhou, Zichen Wang, Hao Chen, Yidong Wang, Linyi Yang et al. "PromptBench: Towards Evaluating the Robustness of Large Language Models on Adversarial Prompts." arXiv preprint arXiv:2306.04528 (2023).

[3] Somepalli, Gowthami, Vasu Singla, Micah Goldblum, Jonas Geiping, and Tom Goldstein. "Understanding and Mitigating Copying in Diffusion Models." arXiv preprint arXiv:2305.20086 (2023).

[4] Jalal, Ajil, Marius Arvinte, Giannis Daras, Eric Price, Alexandros G. Dimakis, and Jon Tamir. "Robust compressed sensing mri with deep generative priors." Advances in Neural Information Processing Systems 34 (2021): 14938-14954.

[5] Chung, Hyungjin, and Jong Chul Ye. "Score-based diffusion models for accelerated MRI." Medical image analysis 80 (2022): 102479.

[6] Luo, Guanxiong, Moritz Blumenthal, Martin Heide, and Martin Uecker. "Bayesian MRI reconstruction with joint uncertainty estimation using diffusion models." Magnetic Resonance in Medicine 90, no. 1 (2023): 295-311.

[7] Roeder, Geoffrey, Luke Metz, and Durk Kingma. "On linear identifiability of learned representations." In International Conference on Machine Learning, pp. 9030-9039. PMLR, 2021.

[8] Khemakhem, Ilyes, Diederik Kingma, Ricardo Monti, and Aapo Hyvarinen. "Variational autoencoders and nonlinear ica: A unifying framework." In International Conference on Artificial Intelligence and Statistics, pp. 2207-2217. PMLR, 2020.

Dear Reviewer uEsZ:

I would like to begin by expressing my sincere gratitude for the time and effort you have dedicated to reviewing our work. Your insights and feedback are invaluable to us. We hope our responses addressed your questions. If you have any further concerns, we are appreciated and more than willing to provide any further information or clarification.

If our responses have addressed your concerns, we would appreciate it a lot if you could improve the score. Thank you once again for your valuable contribution to improving our work. Looking forward to your feedback.

Q2: Why are the reproducibility scores between transformer-based models and unet-based models low as shown in Figure 7(b)? According to Theorem 1, the reproducibility scores should be high. Could you please provide more explanation?

A2: Thanks the reviewer for this very good question. When solving inverse problems with diffusion models, our experiments in Figure 7 showed that the model reproducibility between different type of network architectures are very low. We use Diffusion Posterior Sampling (DPS) [1] to solve the inpainting problem, with details provided in Appendix F. We conjecture that the lack of reproducibility across network architectures is due to the following reasons:

1. Here, DPS introduces the gradient term $\dfrac{\partial \epsilon_{\theta} ( x_t, t)}{\partial x_t}$ during the sampling, and this extra term might break the reproducibility for different type of architectures. In contrast, for the unconditional diffusion model, the reproducibility holds as long as the denoiser function $\epsilon_{\theta} ( x_t, t)$ is reproducible.
2. Second, for image inpainting or inverse problem solving in general, the data $x_t$ passed into the denoiser $\epsilon_{\theta}( x_t, t)$ is out-of-distribution. As such, the reproducibility between different types of architectures might not hold for out-of-distribution data generation.

We will incorporate this discussion into the revised paper in Section 4 (when the extra page is provided).

[1] Chung, Hyungjin, Jeongsol Kim, Michael T. Mccann, Marc L. Klasky, and Jong Chul Ye. "Diffusion posterior sampling for general noisy inverse problems." arXiv preprint arXiv:2209.14687 (2022).

We thank the reviewer for appreciating the quality of our presentation, experiments, and the timeliness of our results. We value the constructive feedback that the reviewer provided. In the following, we will carefully respond to each of the concerns and suggestions that the reviewer raised.

---

Q1: Lack of discussion about text-to-image diffusion models.

A1: We thank the reviewer for the valuable suggestions to study the phenomenon more comprehensively. During the revision, we conducted more experiments considering two extra cases: (i) the Conditional Diffusion Model on the ImageNet Dataset [1], and (ii) Stable Diffusion [2] on the LAION-5B dataset [3]. The results can be found in Appendix A.1 and A.2 of the revised paper. As such, our new experiments not only covered larger datasets in (i, ii), but also text-to-image diffusion models in (ii). Specifically, we find the following:

1. Conditional Diffusion Model on the ImageNet Dataset: We evaluated the reproducibility of EDM [4] and ADM [5], as shown in the following table:

   $\text{RP} \text{ Score}$	EDM	ADM
   EDM	1.0	0.81
   ADM	0.81	1.0

   Detailed experimental settings and visual results are presented in Appendix A.1. We find that the model reproducibility still exists on ImageNet.

2. Stable Diffusion on the LAION-5B Dataset: We evaluated the reproducibility of stable diffusion models with versions v1-1 to v1-4, with their respective reproducibility scores shown in the table below:

   $\text{RP} \text{ Score}$	V1-1	V1-2	V1-3	V1-4
   V1-1	1.0	0.10	0.03	0.03
   V1-2	0.10	1.0	0.21	0.20
   V1-3	0.03	0.21	1.0	0.63
   V1-4	0.03	0.20	0.63	1.0

   The relationship between V1-1 to V1-4 could be summarized as

   • Versions v1-1, v1-2, and v1-3 each are trained on different subsets of the LAION-5B dataset.
   • Versions v1-3 and v1-4 share the same training subset from LAION-5B.
   • Version v1-2 is resumed from v1-1, while v1-3 and v1-4 are resumed from v1-2.

   Notably, only versions v1-3 and v1-4 were trained on the exact same dataset, resulting in the highest reproducibility scores (0.63). Lesser but noticeable reproducibility scores (below 0.21) are observed among v1-1, v1-2, and v1-3, this can be attributed to their sequential training and overlapping datasets. There is no model reproducibility between v1-1 and others, because the training datasets are nonoverlapping and different. Further details on the experimental setup and visual results can be found in Appendix A.2.

In conclusion, our findings suggest that this phenomenon persists for larger diffusion models trained on large datasets.

[1]Deng, Jia, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. "Imagenet: A large-scale hierarchical image database." In 2009 IEEE conference on computer vision and pattern recognition, pp. 248-255. Ieee, 2009.

[2]Rombach, Robin, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. "High-resolution image synthesis with latent diffusion models." In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10684-10695. 2022.

[3] Schuhmann, Christoph, Romain Beaumont, Richard Vencu, Cade Gordon, Ross Wightman, Mehdi Cherti, Theo Coombes et al. "Laion-5b: An open large-scale dataset for training next generation image-text models." Advances in Neural Information Processing Systems 35 (2022): 25278-25294.

[4] Karras, Tero, Miika Aittala, Timo Aila, and Samuli Laine. "Elucidating the design space of diffusion-based generative models." Advances in Neural Information Processing Systems 35 (2022): 26565-26577.

[5] Dhariwal, Prafulla, and Alexander Nichol. "Diffusion models beat gans on image synthesis." Advances in neural information processing systems 34 (2021): 8780-8794.

Dear Reviewer 6ysm:

I would like to begin by expressing my sincere gratitude for the time and effort you have dedicated to reviewing our work. Your insights and feedback are invaluable to us. We hope our responses addressed your questions. If you have any further concerns, we are appreciated and more than willing to provide any further information or clarification.

If our responses have addressed your concerns, we would appreciate it a lot if you could improve the score. Thank you once again for your valuable contribution to improving our work. Looking forward to your feedback.

Dear Reviewer 6ysm,

Thanks for your further questions and concerns.

Q3 ``Also, I agree with Reviewer LTJ7 that Theorem 1 might be rather trivial. The model in Theorem 1 is too simple and too ideal: Theorem 1 only considers the optimal score function, which is impossible to obtain in practical training. The result in Eq.(2), which can be derived by computing the minimizer of a quadratic function, is trivial due to the simple modeling.''

A3: We agree with the reviewer that Theorem 1 is simple, but the result is nontrivial. This is corroborated by our experiments in Figure 5, where we showed that practical diffusion models do obtain the analytical results in practice when the training data size is small. As such, it well explains the reproducibility phenomenon in the memorization regime.

Moreover, we want to highlight our results in Figure 2, where our work has identified an interesting generalization regime where reproducibility and generalizability co-emerge. The phenomenon in the generalization regimes is very different from that of the memorization regime, which cannot be simply explained by Reviewer LTJ7’s claim.

Q4 ``Yet the explanation for the low scores between diffusion models with different architectures is not convincing. According to theorem 1, the score function also follows the pattern even given out-of-domain data. Why low reproducibility scores between transformer-based and unet-based models might need further exploration.''

A4: We thank the reviewer for raising the concerns, but we believe that there are some misunderstandings on the phenomenon between the memorization regime and generalization regime (see Figure 2) that we discovered in our paper.

For inverse problem solving using the diffusion model, the experiment settings are in the generalization regime, which is out-of-score of Theorem 1. Our claim on Theorem 1 is only to explain the phenomenon in the memorization regime, as we discussed in A1. We have highlighted this in the figures of our paper.

Moreover, as pointed out by Reviewer vqyB, the low reproducibility between transformer-based and unet-based models for inverse problems in the generalization regime is counter-intuitive and also refutes the claim by Reviewer LTJ7 that this phenomenon is well-known or well-studied. We have provided potential explanations of this in our previous response A2 that are worth further investigation. Again, this opens a new theoretical question for the diffusion model community.

We hope our responses addressed your questions. If you have any further concerns, we are more than willing to provide any further clarification. Thank you once again for your valuable feedback on improving our work.

We very much appreciate the reviewer for acknowledging the quality of our presentation, experiments, and the timeliness of our results. We also thank the reviewer for carefully reading our paper and detailed comments. We value the constructive feedback and encouragement that the reviewer provided. In the following, we carefully respond to each of the concerns and suggestions that the reviewer raised.

---

Q1: “The present study is mainly on CIFAR-10, and it would be interesting to see how do the results transfer to larger data sets, like ImageNet. Especially when moving to the very large data sets used for text-conditional generation.”

A1: We thank the reviewer for the valuable suggestions to study the phenomenon more comprehensively. During the revision, we conducted more experiments considering two extra cases: (i) the Conditional Diffusion Model on the ImageNet Dataset [1], and (ii) Stable Diffusion [2] on the LAION-5B dataset [3]. The results can be found in Appendix A.1 and A.2 of the revised paper. As such, our new experiments not only covered larger datasets in (i, ii), but also text-to-image diffusion models in (ii). Specifically, we find the following:

1. Conditional Diffusion Model on the ImageNet Dataset: We evaluated the reproducibility of EDM [4] and ADM [5], as shown in the following table:

   $\text{RP} \text{Score}$	EDM	ADM
   EDM	1.0	0.81
   ADM	0.81	1.0

   Detailed experimental settings and visual results are presented in Appendix A.1. We find that the model reproducibility still exists on ImageNet.

2. Stable Diffusion on the LAION-5B Dataset: We evaluated the reproducibility of stable diffusion models with versions v1-1 to v1-4, with their respective reproducibility scores shown in the table below:

   $\text{RP} \text{ Score}$	V1-1	V1-2	V1-3	V1-4
   V1-1	1.0	0.10	0.03	0.03
   V1-2	0.10	1.0	0.21	0.20
   V1-3	0.03	0.21	1.0	0.63
   V1-4	0.03	0.20	0.63	1.0

   The relationship between V1-1 to V1-4 could be summarized as

   • Versions v1-1, v1-2, and v1-3 each are trained on different subsets of the LAION-5B dataset.
   • Versions v1-3 and v1-4 share the same training subset from LAION-5B.
   • Version v1-2 is resumed from v1-1, while v1-3 and v1-4 are resumed from v1-2.

   Notably, only versions v1-3 and v1-4 were trained on the exact same dataset, resulting in the highest reproducibility scores (0.63). Lesser but noticeable reproducibility scores (below 0.21) are observed among v1-1, v1-2, and v1-3, this can be attributed to their sequential training and overlapping datasets. There is no model reproducibility between v1-1 and others, because the training datasets are nonoverlapping and different. Further details on the experimental setup and visual results can be found in Appendix A.2.

In conclusion, our findings suggest that this phenomenon persists for larger diffusion models trained on large datasets.

[1]Deng, Jia, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. "Imagenet: A large-scale hierarchical image database." In 2009 IEEE conference on computer vision and pattern recognition, pp. 248-255. Ieee, 2009.

[2]Rombach, Robin, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. "High-resolution image synthesis with latent diffusion models." In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10684-10695. 2022.

[3] Schuhmann, Christoph, Romain Beaumont, Richard Vencu, Cade Gordon, Ross Wightman, Mehdi Cherti, Theo Coombes et al. "Laion-5b: An open large-scale dataset for training next generation image-text models." Advances in Neural Information Processing Systems 35 (2022): 25278-25294.

[4] Karras, Tero, Miika Aittala, Timo Aila, and Samuli Laine. "Elucidating the design space of diffusion-based generative models." Advances in Neural Information Processing Systems 35 (2022): 26565-26577.

[5] Dhariwal, Prafulla, and Alexander Nichol. "Diffusion models beat gans on image synthesis." Advances in neural information processing systems 34 (2021): 8780-8794.

Q2: ”When studying the encoding from the noise hyperplane to the image manifold, would it be more appropriate to interpolate the noises using spherical linear interpolation instead of direct linear interpolation?”

A2: Thanks for your very insightful suggestion. Given the spherical linear interpolation is confined to a single dimension and time limit, we opted not to visualize the manifold reproducibility in a figure. Instead, we developed a manifold reproducibility score to measure the manifold reproducibility. The metric is defined as follows:

The metric begins by selecting two initial noise vectors, $(\mathbf{\epsilon}^{(0)}, \mathbf{\epsilon}^{(1)})$, from the noise space $\mathcal E$. These vectors are then processed through two distinct diffusion model architectures, resulting in pairs of clear images: ($\mathbf x^{(0)}_1$, $\mathbf x^{(1)}_1$) from the first model and ($\mathbf x^{(0)}_2$, $\mathbf x^{(1)}_2$) from the second. The manifold reproducibility score for the two unconditional diffusion models is defined as:

$RP_{manifold} Score := \mathbb{P}( \mathcal M_{SSCD} ( x^{(\alpha)}_1, x^{(\alpha)}_2)>0.6 \mid \alpha \in [0,1] )$

where $\mathbf x^{(\alpha)}_i$ are generated from spherical linear interpolation based on ($\mathbf x^{(0)}_i$, $\mathbf x^{(1)}_i$):

$ & \mathbf x^{(\alpha)}_i = \frac{\text{sin}((1 - \alpha) \theta)}{\text{sin}(\theta) } \mathbf x^{(0)}_i + \frac{\text{sin}(\alpha \theta)}{\theta} \mathbf x^{(1)}_i, \ \ i \in \{1, 2\} \\ & \theta = \text{arccos} \left(\frac{ \left(x^{(0)}_i\right )^T x^{(1)}_i}{||x^{(0)}_i|| ||x^{(1)}_i||} \right) $

We applied this score to compare the manifold reproducibility across various unconditional diffusion models. Results are shown below. Detailed descriptions of the experimental settings can be found in Appendix A.3. In this case, our observation of model reproducibility is still consistent with our current findings.

We thank the reviewer for acknowledging the quality of our presentation, experiments, and the timeliness of our results. We value the constructive feedback that the reviewer provided. Although we agree that the some preliminary observations on the reproducibility phenomenon were reported by Song et al (2021) [1] in a very empirical way, our investigation delves deeper and provides a more systematic study about the fundamentals for this phenomenon for generic diffusion models which haven’t been done in previous works to our best knowledge, offering a much more thorough analysis. In particular, the key contributions/findings of our paper that differ from prior works include:

• we introduced quantitative measures to study reproducibility more systematically;

• we demonstrate that reproducibility manifests in two unique regimes and is closely tied to the generalizability of diffusion models, as depicted in Figure 2 of our study—an interesting relationship that has never been explored previously.

• we also studied model reproducibility in conditional diffusion models, diffusion models for inverse problems, and fine-tuning, which indicates how this phenomenon could be useful in these different applications.

Reviewer vqyB has also noted these significant findings in his comments. In the following, we carefully respond to each of the concerns that the reviewer raised.

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Q1: “I believe the main finding of this paper is already known.” “why their main finding is different compared to prior work.”

A1: We agree with the reviewer that Song et al. (2021) presented initial findings on the reproducibility phenomenon with a brief description of this observation. Please note that we've also acknowledged this point at the bottom of Page 3 of our paper. However, we respectfully disagree that our main discovery is well-known or extensively researched. This has also been pointed out by Reviewer vqyB.

1. First, in Section 2, we introduced reproducibility metrics to study this phenomenon more systematically and quantitatively. Our findings that reproducibility appears across different noise corruption processes have not been shown in prior works and the phenomenon is nontrivial, which we will discuss in detail in A2 for Q2.

2. Second, in Section 3, our work showed that model reproducibility occurs in two distinct regimes, the “memorization regime” and “generalizability regime”, and model reproducibility has a strong correlation with model generalizability (see Figure 2). This phenomenon has never been discovered nor studied in previous works, and we believe this is very important for understanding the generalizability properties of diffusion models. Moreover, we have theoretically and experimentally justified the reproducibility in the memorization regime, as illustrated in Figure 5.

3. Third, in Section 4 we studied the model reproducibility of diffusion models under different settings, such as conditional diffusion models, diffusion models for inverse problems, and fine-tuning diffusion models. For each of these different settings, the model reproducibility manifests in different but principled ways. These have never been studied in previous works.

[1] Song, Yang, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. "Score-based generative modeling through stochastic differential equations." arXiv preprint arXiv:2011.13456 (2020).

Q3: “The claim, that models trained with different corruption processes will have this property, doesn’t make much sense to me. This is because for VE SDE and VP SDE for example, even the terminal distribution is different, so how do we even start from the same noise? What distribution does this noise follow?”

A3: Although using different perturbation kernels will result in different noise distributions, the distribution difference only lies in the scaling of the variance for the Gaussian. Therefore, for a fair comparison, we can always get rid of the scaling issues by normalizing the variance and then sampling from the noise. We have discussed this in detail in “Initial Noise Consistency” of Appendix B of our work.

For example, for VP and subVP noise perturbation kernels in [1], we define the noise space as $\mathcal E = \mathcal{N}(\mathbf 0, \mathbf I)$, whereas the VE noise perturbation kernel [1] introduces a distinct noise space with $\mathcal E = \mathcal{N}(\mathbf 0, \sigma^2_{\text{max}} \cdot \mathbf I)$, where $\sigma_{\text{max}}$ is predefined. So during the experiment, we sample 10K initial noise $\mathbf \epsilon_{\text{vp, subvp}} \sim \mathcal{N}(\mathbf 0, \mathbf I)$ for the generation with VP and subVP noise perturbation kernel. For VE noise perturbation kernel, the initial noise is scaled as $\mathbf \epsilon_{\text{ve}} = \sigma_{\text{max}} \mathbf \epsilon_{\text{vp, subvp}}$.

[1] Song, Yang, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. "Score-based generative modeling through stochastic differential equations." ICLR, 2021.

Q2: "The Probability Flow ODE depends on the functions f, g, and the score. For given functions f, g, the score function is unique! There is a unique score function because there is a unique likelihood function induced by corrupting the data distribution. All diffusion models are trained to estimate the exact same score function — even if they have different architectures and are trained separately — the target is always the same. Hence, if the diffusion models are trained perfectly, then it is expected that they will all map the same noise to the same output.”

A2: First of all, we appreciate the reviewer's thoughtful remarks regarding the model reproducibility. However, to the best of our knowledge, except for the preliminary findings reported by Song et al. in 2021 [1], we have not come across any empirical or theoretical evidence substantiating the above claims that (i) “all diffusion models (in terms of all different settings and cases) are designed to estimate exactly the same score function in practice”, and (ii) “all diffusion models (in terms of all different settings and cases) can always be perfectly trained for score estimation”. We would appreciate it if the reviewer could share the references that can support these claims other than Song et al. 2021.

On the other hand, to partially support/refute the reviewer’s claims, our work has quantitatively conducted extensive experiments to study this problem in both memorization and generalization regimes under various settings, and we provided a theoretical study in our work in the memorization regime.

1. First, in the memorization regime when the training data is limited, we demonstrate that the diffusion model can only reproduce or “memorize” training data. And the reviewer’s Claims (i) and (ii) are corroborated by both our theoretical analysis and experimental findings. Our research demonstrates the existence of an optimal denoiser and scoring function, as established in our Theorem 1. Furthermore, our work showed that the practical diffusion models converge towards this optimal denoiser, which is unique given f and g, as evidenced in Figure 5. Therefore, in the memorization regime, this observation is in agreement with the reviewer's statement.

2. In the generalization regime, where the model reproducibility and generalizability co-exist, whether and how the diffusion model has been trained perfectly is not clear. As depicted in Figure 13 of our revised paper, it is evident that in the generalization regime, the denoiser obtains a relatively high training loss, but still has good generations, good generalizability, and also has reproducibility. This requires additional investigation before we can assert this claim, and our empirical results have raised numerous intriguing theoretical questions that merit further exploration.

3. Moreover, our results surprisingly show that model reproducibility still exists in the generalization regime, even if we use different functions $f$,$g$ by using different perturbation kernels (we address the noise distribution question in Q3). This observation may diverge from the assumption of the reviewer’s claim.

[1] Song, Yang, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. "Score-based generative modeling through stochastic differential equations." ICLR, 2021.

Dear Reviewer LTJ7:

I would like to begin by expressing my sincere gratitude for the time and effort you have dedicated to reviewing our work. Your insights and feedback are invaluable to us. We hope our responses addressed your questions. If you have any further concerns, we are appreciated and more than willing to provide any further information or clarification.

If our responses have addressed your concerns, we would appreciate it a lot if you could improve the score. Thank you once again for your valuable contribution to improving our work. Looking forward to your feedback.

I would like to briefly here also raise discussion on how obvious is the result that all the different models produce similar images. From what I can see, it is true that the goal in diffusion models is to estimate the $\mathbb{E}[x_0|x_t]$, and if all the models accurately estimate this (and thus the ground-truth score function for the data set), it seems clear to me that all models behave similarly: They reproduce the original data set. This is also pointed out in Theorem 1 of the paper, regarding the "memorization regime". I may be mistaken, but I do not see a similar result holding in the "generalization regime", where none of the models recover the ground-truth score, but instead make "errors" and thus generalize. Intuitively, it seems that all models make highly similar errors for some reason or the other. This seems quite non-obvious to me, and in that sense an interesting direction for research.

Additionally, from what I can see (but again may be wrong), the paper contains new aspects of the phenomenon that have not been noticed in previous work. For instance, 1) The fact that there's a point between the "generalization" and "memorization" regimes where the models do not produce similar results. 2) The paper points out a case where reproducibility does not hold (UViT/DiT compared to DDPM models in inpainting).

I wonder if the others would agree on these points?

Dear AC and reviewers:

We really appreciate it that AC initiated such a discussion, and thank Reviewer vqyB for helping us to clarify the contributions of our work. We thank Reviewer LTJ7, 6ysm, and uEsZ for their valuable feedback for improving our work. We also want to thank the AC for pointing us to valuable references, which we will cite and carefully discuss in our revised paper.

In the following, we want to use this opportunity to further clarify the contributions of our work.

1. We showed model reproducibility in two distinct “memorization’’ and “generalization” regimes, as highlighted in Figure 2. This finding in Figure 2 and Section 3 is significant because it demonstrates that reproducibility in diffusion models is not just a byproduct of a deterministic sampling process. Instead, it arises from various factors like dataset size, model capacity, and generalization ability. We found that model reproducibility occurs either in the memorization regime when the dataset is small enough for the model to memorize training data (as seen in the left part of Figure 2(a)(b)), or in the generalization regime when the dataset is large enough for the model to generate new samples (right part of Figure 2(a)(b)). Moreover, Figure 2 shows a co-emergence of reproducibility and generalizability, which we believe is quite intriguing. Prior research often mixed up these two scenarios without distinguishing them. Our work uniquely identifies the root of this phenomenon, showing that reproducibility stems from the model either memorizing data or learning the underlying data distribution (generalization), a topic not thoroughly examined in previous studies.

2. Diffusion models estimate different distributions in these two distinct regimes while maintaining reproducibility. The AC made an insightful observation that the primary objective in diffusion models is to estimate $\mathbb E[x_0|x_t]$. However, the practical methodologies by which diffusion models are trained for this estimation have not been thoroughly explored in previous studies. Our research offers a comprehensive investigation of this aspect.

   • In the memorization regime, we demonstrate both theoretically and experimentally that the diffusion model has the capacity to effectively reduce the training loss as defined in equation (1), which can be depicted in the left portion of Figure 13. This process leads to convergence towards the optimal denoiser in equation (2) of Theorem 1, as illustrated in the left portion of Figure 5. The mathematical equivalence of this optimal denoiser to $\mathbb E_{x_0 \sim p_{\text{training data}}}[x_0|x_t]$ is established in [1], where $p_{\text{training data}}$ represents the distribution of the training data.
   • In the context of generalization, the findings presented in Figure 13 indicate that the diffusion model does not appear to estimate $\mathbb E_{x_0 \sim p_{\text{training data}}}[x_0|x_t]$. This is obvious from the substantial increase in the training loss (equation (1)) as the size of the training data increases. We posit that, in the generalization regime, the diffusion model may instead be learning to estimate $\mathbb E_{x_0 \sim p_{\text{true data}}}[x_0|x_t]$, where $p_{\text{true data}}$ represents the distribution of the true underlying data. This hypothesis can be supported by large generalization score in the right part of Figure 2(b), worth further theoretical investigation.

3. Systematic yet counter-intuitive findings in the generalization regime for further investigation. We highlight these findings below:

   • Model reproducibility holds even under different perturbation processes. This implies that in the generalization regime, the diffusion model has the ability to consistently converge and learn the actual data distribution, denoted as $p_{\text{training data}}$, regardless of the specific noise perturbation process applied.
   • Model reproducibility diminishes between different types of network architectures for inverse problems. Specifically, in the context of inverse problem solving, we observe that reproducibility does not persist when comparing U-Net-based and Transformer-based architectures in the generalization regime. This implies that by adding constraints or conditions, the posterior distribution may change in a way that it does not inherit the reproducibility from unconditional data distribution. These findings may be potentially quite insightful to guide how we utilize the distribution prior learned from diffusion models for solving inverse problems, which is unexplored and worth further investigation.
   • In the transition between memorization and generalization regimes, there is no reproducibility between different diffusion models. There is an interesting phase transition phenomenon to be well understood.

[1] Yi, Mingyang, Jiacheng Sun, and Zhenguo Li. "On the Generalization of Diffusion Model." arXiv preprint arXiv:2305.14712 (2023).

4. Model reproducibility beyond unconditional diffusion models. Our study has revealed a more structured pattern of reproducibility when examining conditional diffusion models in the generalization regime. This applies to scenarios like diffusion models for inverse problems and fine-tuning diffusion models. Although these settings have distinct characteristics compared to the unconditional diffusion model, they exhibit a systematic and well-defined behavior. As illustrated in Figure 6, there is a distinctive reproducibility pattern that remains consistent between conditional and unconditional diffusion models. To our knowledge, these findings represent novel and inspiring contributions to the existing body of research.

In summary, while the idea may appear intuitive initially, the reproducibility of diffusion models remains a largely uncharted territory that worth a more comprehensive investigation by the research community, rather than relying solely on intuition. Our study represents one of the first efforts in this area. We are confident that our result is new and inspiring, particularly in the generalization regime as illustrated in Figure 2.

Moreover, we believe that our paper opened many interesting theoretical problems to the field on ``why diffusion model generalizes’’, that worth further investigation, and our result sheds insights for more interpretable and controllable data generation in practice.