Diffusion Models as Cartoonists: The Curious Case of High Density Regions

We thank the Reviewer for their scrutiny of our work. We address the raised concerns below.

1. To use the assumption that the data is gaussian in order to use remark 1 results in incorrect results.

In Remark 1 we show that HP-ODE generates the mode in the case of Gaussian data. For non-gaussian data we are no longer guaranteed to find the mode using HP-ODE, but in section 4.3 we empirically show that it generates points with higher likelihoods than regular samples.

We do not use the Gaussian assumption for real-world data experiments and we explicitly emphasize it in lines 349-351: "Even though equation (18) finds the mode of $p(x_0|x_t)$ only in the Gaussian case [...], we will empirically show that even for non-Gaussian data it finds points with much higher likelihoods than regular samples."

2. Specifically this ODE doesn't sample from the correct target distribution, and therefore we observe those cartoon like images.

Indeed, HP-ODE does not generate samples from the target distribution, which is evident is Figures 7 and 9. This was precisely the point of "High probability" seeking ODE: To find samples with larger likelihoods than those of regular samples.

This was the question we were exploring (see e.g. lines 028 or 290-291): What happens if we explicitly bias the sampler towards higher likelihood regions? Deviation from the data distribution was expected.

3. Additionally when comparing the likelihood of images sampled normally or from the HP-ODE the likelihoods were measured using different methods. In section 4.3 it is explained how they compute the likelihood of. Empirically they show that it results in higher likelihood than , however since these likelihoods are being evaluated in different ways this comparison doesn't make sense.

Indeed, we explain in section 4.3 that $p(y_0|x_t)$ is evaluated differently from $p(x_0|x_t)$. The sole reason for this is computational. We mention this briefly in a footnote (line 377), but are happy to elaborate on it.

The goal is to generate many samples $x_0 \sim p(x_0|x_t)$ and for each $x_0$ compare $\log p(x_0|x_t)$ with $\log p(y_0|x_t)$. To generate diverse samples $x_0$ conditioned on the same $x_t$, the sampling needs to be done stochastically (deterministic PF-ODE sampling would always generate the same sample). As we show in Equation (20), the main computational challenge is estimating $\log p_0(x_0) - \log p_t(x_t)$. Since $\log p_0(x_0) - \log p_t(x_t)=-\int_0^t d \log p_s(x_s)$, we can estimate it "for free" using Theorem 1.

4. If done correctly the sample would be evaluated using the same way as, so we would evaluate the likelihood of the same distribution and results would not be as reported.

That said, we can estimate both $\log p(y_0|x_t)$ and $\log p(x_0|x_t)$ using the augmented probability flow ODE as is standard practice (albeit more expensive than our proposed approach).

Motivated by your concerns we now repeated the experiment in Section 4.3 (Figure 7), but estimated both $\log p(x_0|x_t)$ and $\log p(y_0|x_t)$ using the same standard method, i.e. PF-ODE (equation 3). Specifically, we use PF-ODE to esimate $\log p_0(x_0)$, $\log p_0(y_0)$ and $\log p_t(x_t)$ and use equation (20) to estimate $\log p(y_0|x_t)$ and $\log p(x_0|x_t)$.

The results are almost identical to the ones that we reported in the paper. The correlation between two different methods of estimation is at 0.998. We again find that for all samples $x_0 \sim p(x_0|x_t)$ we have $\log p(x_0|x_t) < \log p(y_0|x_t)$. This is to be expected, because (as we have shown in section 3.2) the difference between $\log p$ estimation using our novel method (Theorem 1) and the standard approach (Equation 3) is very small. However, we chose to use our novel method, because it does not incur any additional cost unlike the standard approach.

5. Unfortunately the final conclusion of the paper is just incorrect. It is not true that diffusion models sample away from the high likelihood regions.

We hope that the above explanations convinced you that our analysis in the paper does in fact demonstrate that regular diffusion model samples have lower likelihoods than 1) samples generated by HP-ODE (Figures 7 and 9); and blurry images (Figure 10).

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Many thanks for the opportunity to clarify some key aspects of HP-ODE. We hope we have resolved all your concerns and would be grateful if this could be reflected in an improved score. If you have any further questions or concerns or suggestions, we are more than happy to address them.

I thank the authors for the insightful comments. I would like to apologize for the initially bad review, I think some of the points discussed here are perhaps under/miss-explained in the main paper. I agree that in high dimensions we might "miss" the highest likelihood point, however there is an important distinction between high likelihood points and high probability regions. In many parts of the paper we keep talking about sampling high probability (some examples are line 107, the whole high probability seeking ODE section and the conclusion of the paper in line 533).

As it might be true that some deep high likelihood regions are avoided, these are low probability. While other regions with a more "standard" likelihood (meaning that is neither low or the highest (the set up considered here)) are actually high probability. So diffusion models still sample from high probability regions, while avoiding the low probability regions. This can be seen for instance in the GMM case you provided, if we make a large enough ball around the heavy mode then many samples would be in this region.

I think it is important to have these explained correctly in order to avoid these confusions.

Could you please provide a full table detailing $\mathbb{E}_{x}[ -\log(p_0(x)))]$ where $-\log(p_0(x))$ is measured using the PF ode for the following 3 cases:

1. x is sampled using the traditional PF ODE
2. x is sampled using the SDE
3. x is sampled using your HP-ODE (it would be nice if we get several values of t, but at least one would be nice) Additionally it would be good to add what happens when the measurement is done using theorem 1. It would also be nice to have the variances. This experiment is very important to have, because in the current set up we are only getting to see a single sample. (edit: for this table bits per dimension is also good, i.e. NLL/dim)

For instance:

• Figure 7 just makes a claim without detailing any of the values and the result only apply for the 3 images there presented
• Figure 8 doesn't provide any details of how many samples were used to compute these estimates. I suspect the results are correct but this information is missing
• In Figure 9 we only get to see the samples, but no information about their likelihoods or how they compare with other samples is provided

Intuitively the results here presented can make sense, but no quantitative evidence is provided and this is a big downside of this work. The theoretical insights are very interesting, however since assumptions were made we need to properly show that these resulted in the claimed results. If these two are resolved I would be willing to significantly increase my score.

We thank the Reviewer for their engagement in the discussion and valuable feedback!

1. there is an important distinction between high likelihood points and high probability regions

We agree. Let us formalize this: For a random variable $X$ with probability density of a point $x$, $p_X(x)$, the probability of a set $A$ is $P_X(A)=\int_A p_X(x)d\mu(x)$, where $\mu$ is the Lebesgue measure on $\mathbb{R}^D$.

We believe we have reached a consensus that there can exist a typical set $T$ and a high-density set $H$ such that $P_X(T) \approx 1$, $P_X(H)\approx 0$ and that $\sup\lbrace\log p_X(x)|x\in H\rbrace > \sup\lbrace\log p_X(x)|x\in T\rbrace$.

In other words $H$ is low probability, but it contains points of larger density than all elements of $T$. What we study in the paper are exactly such sets $H$ (which we call high-density regions). The regular samples from the model belong to the typical set, which does not contain the highest density points, but is of course high-probability.

Thank you for this comment as it helps to improve the clarity of the paper. We will change the wording and use "high-density regions" throughout the paper and change the name of Algorithm 1 to "high-density (HD) sampling".

2. Could you please provide a full table detailing $\mathbb{E} [-\log p_0(x_0)]$

Below we provide the table with the values of $\mathbb{E} [-\log p_0(x_0)]$ (in bits-per-dim) for different models and sampling strategies. In all cases $\log p_0(x_0)$ was estimated using the PF-ODE. The values are mean ± one standard deviation. We see that HD sampling generates samples with higher density than regular samples accross different models and values of the threshold parameter $t$.

where:

• CIFAR10-ML - maximum likelihood training, i.e. unweighed ELBO
• CIFAR10-SQ - (Sample Quality) weighted ELBO training as recommended in [1]
• Rev-SDE(Th 1) - $\log p$ evaluated with Theorem 1. Note that it can only be used for Rev-SDE sampling.
• "Original" sampling - same sampling procedure as in the corresponding paper, i.e. Predictor-Corrector sampling for FFHQ256 and CHURCH256 [2]; and stochastic Heun sampler for IMAGENET64 [3]. We chose original samplers to faithfully reproduce the "regular samples". Note that neither of these samplers are the Reverse SDE, so we could not use Theorem 1 to evaluate $\log p_0$.
• For CIFAR10 models we used 1024 samples in each experiment. For the remaining models we used 192 samples and only two values of $t$ for each model due to computational budget constraints.
• Values of $t$ is HD sampling are in different ranges for different models, because they use different SDEs and different noise schedules. CIFAR10 uses VP SDE with linear log-SNR schedule, [2] uses VE SDE and exponential schedule and [3] uses VE SDE with $\sigma=t$.

3. Figure 8 doesn't provide any details of how many samples were used

To generate Figure 8, we used 192 samples and for each created 7 versions by applying different strengths of blur ($\sigma \in \lbrace 0, 1, 2, 5, 10, 20, 50\rbrace$) so in total 192 * 7=1344 samples were used. For every image, $\log p_0$ was estimated with PF-ODE.

The distributions of $\log p_0$ for different blur levels are on the right of Figure 10.

4. In Figure 9 we only get to see the samples, but no information about their likelihoods or how they compare with other samples is provided

On the right-hand side of Figure 9 we show the plots of distributions of $\log p_0$ for all models, for regular samples and for HD for the two presented values of $t$ for each model. These density plots were based on $\log p_0$ evaluated on 192 samples. The means and standard deviations are as in the table above.

We thank the Reviewer again for your thorough review, and hope that our clarifications and these additional details have convincingly addressed your concerns. We will appreciate if the same can be reflected in your revised score. Many thanks!

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[1] Kingma et al. "Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation" (NeurIPS 2024)

[2] Song et al. "Score-Based Generative Modeling through Stochastic Differential Equations" (ICLR 2021)

[3] Karras et al. "Elucidating the Design Space of Diffusion-Based Generative Models" (NeurIPS 2022)

We thank the Reviewer for acknowledging the strengths of the paper and insightful questions. We address the raised concerns below.

1. It isn't clear whether analysis from Section 5 can lead to the creation of better stochastic samplers or improve the quality of image generation. Overall practical implications of this work are quite poor [...] Have the authors thought about how the proposed theory can be used to improve the quality of image generation?

This is a valid point and we address it under "How can this study lead to improved sampling strategies?" in the Global Response.

2. There is no intuition behind observations from Figure 4. More precisely, what does it mean that the model optimized for sample quality yields a smaller difference between $p_0^{\text{SDE}}$ and $p_0^{\text{ODE}}$?

Thank you for the opportunity to elaborate. As we mention in Lines 181-182, the SDE model and the ODE models are not equivalent in practice (because the score function is not learned perfectly). Therefore, for a learned score function $\mathbf{s}(t, x) \approx \nabla_x \log p_t(x)$, we have two different probability distributions $p_0^{\text{SDE}}$ and $p_0^{\text{ODE}}$.

As we know from [1], if the score function was learned perfectly, i.e. $\mathbf{s}(t, x) = \nabla_x \log p_t(x)$, then we would have $p_0^{\text{SDE}} \equiv p_0^{\text{ODE}}$. Therefore, the discrepancy between $p_0^{\text{SDE}}$ and $p_0^{\text{ODE}}$ can be interpreted as a measure of self-consistency of the diffusion model.

In section 3, we derive a tractable upper bound on this discrepancy as measured by $KL[p_0^{\text{SDE}} || p_0^{\text{ODE}}]$, i.e. $\mathcal{R}^U(\mathbf{s})$. Empirically, we found that $\mathcal{R}^U(\mathbf{s})$ is smaller for a model optimized for sample quality than for a model optimized with maximum likelihood training. This was a surprising discovery because it suggests that, the principled learning objective, i.e. maximum likelihood, leads to a model, which is worse in terms of the self-consistency as defined above. Conversely, the model optimized for sample quality, i.e. trained with a weighed ELBO, with the weighting function chosen empirically [3], leads to a model, which is more self-consistent.

Furthermore, we believe that $\mathcal{R}^U(\mathbf{s})$ might become a useful tool for evaluating and comparing diffusion models. Especially, since FID is expensive to compute in practice and is sensitive to the choice of a random seed [4].

3. It would greatly improve the scope of the work if the experiments were performed on the frontier models

Thank you for this suggestion. Based on this, we now extended the study with Stable Diffusion v2.1. Please see "Lack of analysis of more sophisticated models." in the Global Response for more details.

4. Can we use the proposed estimation for the log probability to evaluate model quality the same way as [1] (Table 2)? Will there be any difference between SDE and ODE estimation? Which one is better to compare different models?

Very good question!

First, our main novelty in likelihood estimation in stochastic SDEs are theorems 1 and 3. They show that during stochastic sampling, one can get an estimate of $\log p_0^{\text{SDE}}(x_0)$. However, this only holds, when the model is generating the sample and we can keep track of the likelihood along the stochastic trajectory. If we are presented with a sample $x_0$ (e.g. from a test set like in Table 2 in [1]), theorems [1] and [3] do not apply, because the model did not generate the sample.

In that case, we would need to apply the augmented forward stochastic dynamics, which apply for any given input $x_0$ (in particular, it can be from some test set). However, in that case, we show (In Equation (13)) that this ends up being equivalent to the known formula for the lower bound on $\log p_0^{\text{SDE}}(x_0)$ [2].

Finally, in that case, you can use both and the results most likely will be different for two reasons: 1) For $\log p_0^{\text{SDE}}(x_0)$ we only have a bound, whereas $\log p_0^{\text{ODE}}(x_0)$ we can estimate exactly; and 2) as we mention in the paper, $\log p_0^{\text{SDE}}$ and $\log p_0^{\text{ODE}}$ correspond to two different models, which are not equivalent in practice! In section 3 we even propose a tractable way to quantify the Kullback-Leibler divergence between them.

Thank you again for a thorough review, we hope that we addressed all concerns and would kindly ask you to reconsider the score given the new evidence and clarifications.

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[1] Song et al. "Score-based generative modeling through stochastic differential equations". (ICLR 2021)

[2] Song et al. "Maximum Likelihood Training of Score-Based Diffusion Models" (NeurIPS 2021)

[3] Kingma et al. "Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation" (NeurIPS 2024)

[4] Karras et al. "Elucidating the Design Space of Diffusion-Based Generative Models" (NeurIPS 2022)

We thank the Reviewer for their thorough review and support of this work! We address the raised concerns below.

1. The proposed mode-tracking approach has a very high computational cost and this is not clearly discussed. The discussion on how this mode-tracking approach scale and its computational limitations should be discussed and quantified

This is a good point, thank you for bringing our attention to this. In the revised version of the paper, we added Appendix K: "Cost of mode-tracking", where we discuss in detail how the mode-tracking ODE scales and why it is prohibitively expensive, especially in larger dimensions. For example, on CIFAR10 dataset, mode tracking ODE is at least 6000x times more expensive to evaluate than the PF-ODE (or HD-ODE). On 256x256 images, it becomes 400000x more expensive than PF-ODE (or HD-ODE).

2. The high-likelihood samples discovered by the proposed analysis does not seem to have a real practical advantage.

This is a good point that we address in the Global Response under "How can this study lead to improved sampling strategies?". We argue, that while the high-density sampler does not produce high-quality samples, our findings together with [1] suggest that the highest-quality images may have likelihoods in the lower end of "moderate" likelihood values. We leave this exploration for future work.

3. The limitations of the work are not discussed at all, a limitations section covering all the main potential limitations should be added.

Thank you for pointing this out. We have now added the Limitations section in Appendix R: "Limitations" and refer to it in the conclusion. The main limitations are the inherent limitations of stochastic sampling as compared to deterministic carry over to likelihood estimation; and the impracticality of the exact mode-tracking ODE for general, non-Gaussian distributions. However, as our experiments suggest, we can dispense with the computationally expensive term to get significantly higher-likelihood samples than regular ones.

4. The work does not report any implementation details, code and implementation are not submitted in the supplementary materials. It would be important to openly release the contribution upon acceptance to ensure reproducibility and improve transparency.

In the revised version of the paper, we added appendices M: "CIFAR models hyperparameters" and N: "Quantitative analysis of likelihoods of samples generated with algorithm 1", where we provide all the hyperparameters we used for our CIFAR10 models and other baselines used. Specifically, the CIFAR models we trained from scratch, whereas for the other baselines: ImageNet64, FFHQ256, Churches256 and now Stable Diffusion, we used the original implementations with default sampling parameters. We implemented high-density sampling on top of each of them. We are committed to releasing the code of our experiments upon acceptance.

5. The analysis of high-density regions is done on small and simple diffusion models trained on restricted datasets with limited variability. Moreover the selected models and study is done on uncoditional sampling without any guidance from text. In my opinion it would be valuable to explore real-world and more complex diffusion models (such as SD, Flux etc), especially focusing on the impact of text.

Thank you for this suggestion. We address this in the Global Response under "Lack of analysis of more sophisticated models.". Based on your suggestion, we added an experiment with Stable Diffusion v2.1 and confirmed that even text-guided, latent diffusion models exhibit similar behavior, i.e. high-density samples correspond to blurry and cartoon-like images.

6. The related work does not take into consideration a very relevant paper "Null-text Guidance in Diffusion Models is Secretly a Cartoon-style Creator", Zhao et al which is not discussed. In particular it would be relevant to highlight some connection and insights with this previous related work.

Thank you for pointing this out and we apologize for this omission. We have now included a discussion on the comparison with the paper you provided in Appendix Q "Cartoon generation" and we point to it in the related works section. The main difference between is in the motivation. Unlike existing models, our aim was not to build the best possible cartoon generator. We studied the high-density regions and discovered that they comprise of cartoon-like and blurry images.

We would like to thank the Reviewer again for their thoughtful and constructive review. We hope that we have addressed all concerns and you will consider an even stronger support for this work. For a full summary of the updates in the manuscript, please refer to the Global Response. We are committed to engaging further in case you have any further concerns, questions, or suggestions.

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[1] Karras et al. "Guiding a diffusion model with a bad version of itself" (NeurIPS 2024)

Dear Authors,

Thanks a lot for your answers, I particularly appreciate the effort made with running additional experiments with SD2.1. I think this work is a nice analysis, toward a better understanding of the probability landscape of diffusion models.

We thank the Reviewer for their insightful review! We address the raised concerns below.

1. The paper describes the emergence of cartoon-like images in high-likelihood samples but provides limited exploration into why or how this phenomenon occurs.

Thank you for raising this point. Based on your question, we have now revised the manuscript and added Appendix P: "Why do cartoons and blurry images occupy high-density regions?", where we provide an explanation for why this behavior occurs. We argue that the diffusion model's estimate of the (negative) loglikelihood can be interpreted as a notion of the amount of detail in an image. Therefore highest-density regions will be occupied by images with the least amount of detail, such as cartoons or blurry images.

2. Although the paper sheds light on high-density regions and likelihood estimation, it lacks a clear discussion on how these findings could practically inform the design or improvement of diffusion models in applied settings. Without recommendations on balancing high-likelihood sampling with image quality, it’s challenging to draw valuable insights from the findings, particularly for practitioners focused on real-world applications

This is a good point. We address this in the Global Response under "How can this study lead to improved sampling strategies". We propose that, while the high-density sampler does not produce high-quality samples, our findings, alongside [1], suggest that the highest-quality images may correspond to the lower range of "moderate" likelihood values. We leave a deeper investigation of this for future work.

3. The experiments are primarily conducted on well-known datasets like CIFAR-10 and FFHQ-256. Expanding the study to other types of diffusion models or additional data domains would strengthen the generalizability of the findings.

Another good point! Based on your suggestion we extended the study to include the Stable Diffusion v2.1 model. Please see "Lack of analysis of more sophisticated models." section in the Global Response.

4. Additionally, clarifying any specific architectural limitations of the proposed high-probability sampler could benefit those aiming to extend the approach.

Good question. We have not encountered any architectural limitations. The qualitative analysis shows similar phenomena for different types of SDEs and noise schedules in Section 5. Furthermore, the results now extend to the Stable Diffusion model, which not only is text-guided but also operates in the latent space as opposed to the pixel space like the other baselines.

5. While the paper emphasizes the likelihood of generated samples, it does not address diversity within these high-likelihood outputs.

Thank you for drawing attention to this. We provide some intuition regarding this in Figure 6, where we show that the threshold parameter $t$ in the high-density sampler allows for controlling the diversity/likelihood tradeoff. In terms of real-world data, a similar observation can be made. The higher the $t$, the lower the diversity of the generated outputs. This is caused by the fact that the highest-density samples have the least degrees of freedom (we discuss this in more detail in Appendix P) of the revised manuscript.

Considering the two extremes: when $t \approx 0$, then HD sampling coincides with regular sampling (whether it's PF-ODE or Rev SDE). Then the variability is the same as of the base diffusion model. On the other hand, when $t \approx T$, then as can be seen in the bottom row of Figure 1, the outputs are monochromatic patches, which have significantly less variation than natural images.

6. For applications that prioritize both high likelihood and visual quality, what adjustments to the high-probability sampler could mitigate the production of low-quality images?

This is a very interesting question. As we show in Section 5 and now elaborate on in the Global Response under "How can this study lead to improved sampling strategies?", we argue that it might be impossible to have both. This is caused by the fact that the highest likelihoods are occupied by images with the least detail. We leave the exploration of optimization of perceptual quality from the perspective of likelihood for future work.

We thank the Reviewer again for their suggestions and questions. We hope that we have addressed all concerns and that you will consider increasing your score. For a full summary of updates in the manuscript, please refer to the Global Response. We remain dedicated to addressing any further concerns, questions, or suggestions you may have.

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[1] Karras et al. "Guiding a diffusion model with a bad version of itself" (NeurIPS 2024)

Thank you for your thoughtful comments and for emphasizing this important point. We appreciate the opportunity to clarify our findings and their implications.

The main takeaway from Section 5 of our paper is that, across different diffusion SDEs and models — now also including the latent diffusion model (Stable Diffusion v2.1) as per your suggestion — we consistently observe that high-density regions contain unrealistic images. This finding is particularly surprising in light of [1], which attributes the success of diffusion models to their ability to avoid low-density regions.

Combining these observations — the insights from [1], our findings, and now also the explanation of this phenomenon inspired by your question (Appendix P in the revised manuscript) — we hypothesize that the highest-quality samples likely occupy regions of moderate likelihood (neither too low nor too high). Thus, samplers explicitly targeting these regions may produce the best results. However, we do not propose a specific solution, as addressing this challenge is nontrivial and requires an additional in-depth investigation that we leave for future work.

While the high-density sampler does not directly lead to high-quality sampling, it was crucial in making these observations, which we believe will ultimately inform the design of better samplers.

Finally, we wish to emphasize that the high-density sampler and the empirical findings in Section 5 are only part of this work's contributions. Our theoretical developments in Sections 2 and 3, along with the practical computational tool for estimating likelihood in stochastic diffusion models, offer broader potential applications. For instance, as noted in lines 156–157, they could enable importance sampling without additional computational cost.

We hope that this addresses your remaining concern and that you will consider stronger support for this work. Thank you again for your feedback and for the opportunity to expand on this aspect of our study.

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[1] Karras et al. "Guiding a diffusion model with a bad version of itself" (NeurIPS 2024)

6. Additional experiments using Stable Diffusion - We added Appendix O: "Stable Diffusion Samples", where we describe an experiment conducted with stable diffusion, which extends the score of the study to 1) latent diffusion models and 2) text-guided diffusion models.
   • We found that even though the SDE governs the evolution of the latent representation of an image, we observe similar behavior as in pixel-space diffusion models. Specifically, we found that high-density samples decode to images, which are either blurry or exhibit cartoon-like features
7. Explanation of cartoons and blurry images occupying the high-density regions - We added Appendix P: "Why do cartoons and blurry images occupy high-density regions?", where we note the connection between local intrinsic dimension and diffusion model's likelihood estimation.
   • We argue that the diffusion model's estimate of the (negative) log-likelihood can be interpreted as the amount of detail in an image. Therefore, since blurry and cartoon-like images have much less detail than regular images, they lie in high-density regions.
8. Addition of cartoon generation methods to related work - We added Appendix Q: "Cartoon generation", where we discuss the connection between our results and other cartoon generation methods.
   • The main difference is the motivation. Unlike the existing cartoon-generation method, our aim was not to build the best possible cartoon generator. Rather, we set out to conduct a theoretical analysis of the high-density regions and made a surprising discovery that they consist mainly of cartoons and blurry images.
9. Updated conclusion - We added to the first part of the conclusion: "While [1] argued that avoiding low-density regions is crucial for the success of diffusion models, our analysis reveals that high-density regions should also be avoided in high-quality image generation."
   • We added this to emphasize the takeaway from section 5.
10. Limitations - We added Appendix R, where we discuss the limitations of our work, which are split into two groups:
    • stochastic likelihood estimation - the inherent limitations of stochastic sampling as compared to deterministic carry over to likelihood estimation;
    • exact-mode tracking ODE holds only for linear SDEs and is too expensive to be used in practice.

We thank the Reviewers again for their constructive feedback!

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[1] Karras et al. "Guiding a diffusion model with a bad version of itself" (NeurIPS 2024)

[2] Rombach et al. "High-resolution image synthesis with latent diffusion models" (CVPR 2022)