Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure

Q1. First, most of section 3 is rather obvious for people with a signal/image processing background; e.g., Theorem 1 is well known.

We sincerely appreciate the review for pointing us to the Wiener filter. We agree that Theorem 1 is well studied (we will add citations on the Wiener filter to acknowledge this), but we believe many findings of our work are not obvious as the reviewer suggested. Please see A.2. in the global response.

Q2: A second observation that the authors did not make is that any linear denoiser leads to Gaussian generated images. Further, the Gaussian denoiser produces samples from the Gaussian distribution with the same mean and covariance as the training set, which is thus a very simple model for which samples can easily be produced without any diffusion procedure.

Firstly, in this work, we are interested in the mapping from the noise space z to the image space x. An arbitrary linear denoiser does not produce similar images as the actual diffusion models therefore it is worth no interest. Furthermore, we cannot compare diffusion mapping with the Gaussian mapping if we directly sample from the Gaussian distribution without the reverse diffusion procedure.

Q3. It is not surprising that good diffusion models are capable of capturing the covariance of their training data

Again, this point is not well understood as well. First of all, we not only show that diffusion models capture the covariance of their training data but more importantly, we show that the obtained diffusion denoisers share similar function mapping with the Gaussian denoisers. The first observation doesn’t explicitly imply the second. Since even the optimal denoisers under the multi-delta distribution assumption captures the covariances, after all, they are able to reproduce the whole training dataset. On the contrary, it is not well understood how gradient descent on deep networks arrive at a denosier that is close to the Gaussian denoisers. To understand this requires rigorous analysis on the gradient dynamics of the deep network and to our best knowledge, no such analysis exists in the current literature. Furthermore, in the current literature on diffusion models, this problem is circumvented by either assuming an infinite number of training data or directly restricting the architectures of the diffusion models.

Q4. The similarity breaks down for more complex datasets. The inductive biases of network denoisers cannot be reduced to the Gaussian structure only, as they are capable of learning much more complex structures despite the curse of dimensionality.

Our results in Figure 15 in the appendices and Figure 2 (b) of our newly uploaded PDF demonstrate that this similarity also persists in more complex datasets (e.g., LSUN-Churches, AFHQ datasets and Cifar-10 dataset). This similarity indicates that the first and second order statistics of the finite training dataset is heavily utilized by diffusion models. Otherwise, we should not be able to observe the similarity. However, we do agree the generation power cannot be reduced to the Gaussian structure only, but it indeed plays a critical role.

Q5. The surprisingly high cosine similarities reported in Figure 1 look suspicious to me.

We conduct experiments to measure the linearity of the score functions as the reviewer suggested. The results are shown in Figure 3 of our newly uploaded PDF, where we see that measuring linearity of score is not very meaningful. This is because the noise magnitude can be much higher compare to the denoising output (in the range of [-1,1]), therefore subtracting the denoising outputs from the noisy image does not change the noisy image significantly except in the low noise variance regime. For this reason, we always see a high linearity of the scores for most of the noise variances. In the figure, denoised_img_1 and dneoised_img_2 correspond to $D(x_1|\sigma_t)$ and $D(x_2|\sigma_t)$, denoised_img_1+denoised_img_2 corresponds to $\frac{1}{\sqrt{2}} D(x_1|\sigma_t)+\\frac{1}{\sqrt{2}}D(x_1|\sigma_t)$ and denoised_image_additive corresponds to $D(\frac{1}{\sqrt{2}}x_1|\sigma_t)+D(\frac{1}{\sqrt{2}}x_2|\sigma_t)$. $x_1$ and $x_2$ are two randomly sampled noisy image.

Q5. Two observations relative to memorization and generalization made by the authors are straightforward to me...

First of all, this point is not well perceived by the current literature. Current theorems normally assume you can directly sample from the ground truth training data distribution instead of just a finite number of training data. In this case, exactly minimizing the score denoising matching loss indeed results in the ground truth score function. However, less work focuses on the finite training setting, in which minimizing the score denoising matching loss results in overfitting. As you mentioned, the trained network is suboptimal in a specific way and enables them to generalize correctly. Our results indicate that this specific way is to be close to the Gaussian denoiser and why this happens is not well understood.

Secondly, our discussion on strong generalizability focuses on the actual diffusion model rather than the Gaussian model. Yes, a Gaussian model will strongly generalize in the sense of [15] when the two training sets have similar first and second-order statistics. We use this fact to explain why diffusion models exhibit strong generalizability. Since the diffusion models learn similar function mappings as their corresponding Gaussian denoisers in the generalization regime, strong generalization of the Gaussian models leads to the strong generalization of the actual diffusion models. This is supported by the fact that we can direct the model towards strong generalization by either early stopping or decreasing the model scale. Remember these two actions prompts the emergence of the Gaussian structure, which further highlights the necessity of the Gaussian structure in diffusion model’s generalizability.

1. I do not understand why the scores appear linear except for very small noise levels... This would indicate than there is non-negligible non-linearity in the score, which are the objects that appear in the reverse SDE/ODE.

Yes, the linearity of the score for noise larger than 1 is not meaningful since the pixel range of our images lie in [-1,1]. In fact, we've already implicitly measure the linearity of the diffusion denoisers, as shown in Figure 2 (left). Notice that the difference between the actual diffusion denoisers and the corresponding linear denoisers are the largest for $\sigma$ in the range of [0.4, 10]. This aligns with our original measure of linearity shown in Figure 1, where we see the most nonlinear part lies in the range of [0.4,10] as well. Therefore, we believe our original linearity measure is good enough. Let's then go back to the reviewer's original question: "Given the points above, the surprisingly high cosine similarities reported in Figure 1 look suspicious to me. If denoisers were truly linear, then they would generate Gaussian images, and we know that they learn much better models than this. " Please notice that we never claim the denoisers are exactly linear. However, we do see the trend that the denoisers become increasingly linear as it transitions from memorization to generalization. In fact, as we've shown in the Figure 2 (a) of our newly uploaded PDF, the denoising outputs of the actual diffusion denoisers are highly similar to those from the Gaussian denoisers. Because of this similarity, the linear models obtained through distillation match the Gaussian models. Please let us know if you have further doubt about the emergence of linearity observed in our paper.

2. I agree that capturing the Gaussian/linear structure of the score is the first-order necessary condition for generalization, and it goes a relatively long way in producing images that are correlated with the images generated by a generalizing network. However, I maintain that pointing this out does not teach us a lot about how diffusion models generalize. When I said that this was well understood, I meant that very simple and classical approaches lead to generative models that capture the linear/Gaussian structure and generalize in the sense of [15]. The mystery lies in how non-Gaussian/linear structure can still be estimated from limited samples despite the curse of dimensionality (by diffusion models or other approaches).

Could the reviewer provides us with some references on "very simple and classical approaches lead to generative models that capture the linear/Gaussian structure and generalize in the sense of [15]?" We are sincerely eager to learn more about this. Furthermore, we want to emphasize that our findings don't just show the diffusion models capture the first and second order statistics, but more importantly, diffusion denoisers are quite close to the Gaussian denoisers. We want to emphasize that capture first and second order statistics doesn't mean the latter, since even in the case of memorization, the model still capture the second and first order statistics, but the diffusion denoisers in this case are not close to Gaussian.

Furthermore, for other generative models such as VAE and GAN, they can also be interpreted as denoisers since they generate images by mapping noise space to the image space. Moreover, they also capture the first and second order statistics of the training dataset since they have a strong generation power. However, their function mappings don't share any similarity with the Gaussian denoisers. For this reason, we believe our results are intriguing and meaningful since we demonstrate that the function mapping of the diffusion models share high similarity with the Gaussian models. To be more specific, our results demonstrate that the best linear approximation of the nonlinear diffusion models are nearly identical to the Gaussian models.

Please share your thoughts about this, we are open to any criticize, in the hope that we can make the paper better.

Q1. Although the paper's focus is on generalization in diffusion models, it lacks any quantitative measure of generalization within the experiments presented.

A1: The memorization and generalization can be clearly observed in our experimental results by comparing the generated image with the nearest neighbor (NN) in the training dataset (see Figure 6 to Figure 9). To measure the memorization and generalization quantitatively, we can empirically define the generalization score as follows

$

\text{GL Score} := \frac{1}{k}\sum_{i=1}^k\frac{||x_i-\text{NN}_Y(x_i)||_2}{||x_i||_2}
%\mathcal{L}(\mb x_k,\text{NN}_Y(\mb x_i)),

$

where $[x_1, x_2, ..., x_k]$ are sampled images from the diffusion models, and $Y:=[y_1, y_2, ..., y_N]$ denote the training dataset. We used this metric to assess generalization versus memorization, with the results presented in Figure 1 of our global response. As shown in Figure 1 (d) to (g), diffusion models start to exhibit memorization behavior when the number of training images becomes smaller than 8750 (GL Score around 0.5). Therefore we choose 0.6 as the threshold to distinguish between generalization and memorization. As shown in Figure 1(a) and (b), the diffusion denosiers exhibit increasing linearity as the diffusion models shift from memorization to generalization.

Q2. The work primarily investigates the linearity of the learned score functions across various noise levels. However, the connection between these experiments and the necessity of linearity for generalization in diffusion models is unclear and not well substantiated.

A2. Please refer to A.3. of our global response.

Q3. The term "inductive bias" is frequently used throughout the paper, but it is never clearly defined.

A3: Here we give a precise definition of the inductive bias in our work: “When the model size is relatively small compared to the training dataset size, training diffusion models with the score matching loss (Equation (3)) results in diffusion denoisers that behave similarly to the linear Gaussian denoisers (though with certain amount of difference especially in the intermediate noise variance region). Furthermore, even when the model is overparameterized, such similarity emerges in the early training epochs." Our finding is consistent across various architectures including VE, VP and ADM (see Figure (13)).

Though in the paper, we mainly test on EDM configuration, which specifies specially designed forward process, and denosier parameterization (Equation (9)), we expect the inductive bias will manifest for other forward process and parameterization as well since recent work [18] shows that diffusion models trained with different forward process and parameterization generate almost identical images starting from the same random seed, which suggests that even in those cases, the diffusion denoiser share similar function mappings.

Q4. The paper lacks detailed information on the experimental setup, including training procedures and the hyperparameters of the architectures used.

A4: In the revision, we will include more experimental details (e.g., the model architectures, hyperparameters, training procedures) in our paper for reproducibility of our results.

Q5. Can the authors explain how from their set of experiments we can conclude that the emergence of Gaussian structure leads to strong generalizability?

A5: Firstly, based upon our studies in Section 3 and 4 we find strong correlation between generalization and Gaussian structures. Furthermore, in section 5 we show that decreasing model scale and early stopping are able to prompts strong generalization. Remember those two activations prompts the emergence of the Gaussian structure.

Q6. Could the authors better discuss the comparison with [15] in more detail? I fail to understand how learning "common structural features" differs from learning the density.

A6: Our work offers complementary explanations based on the common Gaussian structures from non-overlapping datasets and the inductive bias towards these structures. In contrast, [15] hypothesized that the models learn the same underlying distribution. Notice that learning the ground truth distribution is only a sufficient condition not a necessity for the strong generalization. Just because two models produce the same images doesn’t mean they learn the underlying distribution. It only implies certain common structures of the two datasets are captured but such a common structure might not be the optimal. Our experiment shows that such a common structure is highly related to the Gaussian structure.

Q7. The paper shows how the linear (distilled) model resembles the optimal denoiser for a multivariate Gaussian distribution. Would the same phenomena be observed if we would linearize the trained model instead of distilling it?

A7: Yes, our experiment results show that directly training a linear diffusion model based on the denoising score matching loss results in the Gaussian denoisers. However, this is not the main point of the paper since what is interesting is that diffusion models behave closely to the Gaussian denoisers even without this explicit linear constriant.

Q8. The authors refer to edm-ve, edm-vp, and edm-adm as different "model architectures". Could they clarify this terminology?

A8: The EDM paper [4] proposes a novel diffusion configuration (a specially designed forward process, time schedule, network parameterization). This configuration can be adapted to various network architectures. EDM-VE [3], EDM-VP [2] and EDM_ADM [23] are all trained with the EDM configuration but with different network architectures. Here VE stands for the architecture proposed in the paper [3], VP stands for the architecture proposed in the paper [2] and ADM corresponds to the architecture proposed in [23]. We refer the reviewer to the EDM [4] paper for more detail.

Dear Reviewer 6yNJ,

We have tried our best to address your concerns in our response. Since the deadline for the discussion period is approaching, we'd like to know if you have further questions so that we can do our best to respond further. Please feel free to raise any questions. Thanks for your insightful feedback.

Due to the space limitation of the rebuttal, we could not add the experimental setup of our paper. Here, we include it below for your reference:

Here we provide a more detailed description of our experiment setup:

• Section 3: we train linear models to distillate the actual diffusion models (including EDM-VE, EDM-VP, EDM-ADM). The actual diffusion models are trained on the FFHQ dataset (70000 images in total) for around 1000 epochs. The details of training of linear models are in the appendix B. We use the default hyperparameters (including learning rate, detailed network parameters) provided in the EDM code base.

• Section 4: we study the impact of (i) dataset size (ii) model scale and (iii) training time on diffusion models’s generalizability. For figure 6, we train the same models on datasets with various sizes. The datasets are randomly sampled from the FFHQ dataset. The diffusion model is trained with EDM-VE configuration. All models are trained for around 2 days to ensure convergence. For figure 7, we fix the dataset size as 1094. We still use the same EDM-VE training configuration but vary the model scales [4,8,16,32,64,128,128]. For Figure 8, we train a diffusion model with EDM-VE (scale 128)] configuration on FFHQ dataset with 1094 images. We early stop the model at various epochs specified in the paper.

• Section 5: we study the strong generalizability. We randomly split FFHQ dataset into non overlapping datasets with size 35000 and 1094. All the models are trained with EDM-VE configurations.

In the revision, we will make sure to include those experimental details (e.g., the model architectures, hyperparameters, training procedures) in our paper for the reproducibility of our results. We will make our code public upon publication to ensure reproducibility.

Best,

Authors

Thanks for your reply, we are happy to further address your concerns.

Q1. Inductive bias

As we've stated in our last response (please see A.3), a more precise interpretation of the inductive bias in this paper should be:

When the model size is relatively small compared to the training dataset size, training diffusion models with the score matching loss (Equation (3)) results in diffusion denoisers that behave similarly to the linear Gaussian denoisers (though with certain amount of difference especially in the intermediate noise variance region). Furthermore, even when the model is overparameterized, such similarity emerges in the early training epochs.

Please notice that our findings don't just show the diffusion models capture the first and second order statistics, but more importantly, when diffusion models transition from generalization to memorization, the corresponding diffusion denoisers progressively get closer to the Gaussian denoisers. We want to emphasize that capturing first and second order statistics doesn't mean the latter. Consider diffusion models in the memorization regime, they also capture the first and second order statistics of the training dataset since they can perfectly reproduce the training data, however, in this case the diffusion denoisers are not close to the Gaussian denoisers.

Q2. Why this finding is surprising and considered as a bias ?

Remember that we are training diffusion models on finite number of training data. The optimal solution to the training objective (equation 3) in this scenario has the form of equation 8, which has no generalizability. On the contrary, our findings show that in practice, when the model capacity is relatively small compared to the training dataset size, training diffusion models with the score matching objective (equation 3) leads to diffusion denoisers that share high similarity with the Gaussian denoisers. Furthermore, we also show that even when the model capacity is sufficiently large, such similarity emerges in the early training epochs. We considered this behavior as a bias since the diffusion models in generalization regime bias towards learning similar diffusion denoisers as the Gaussian models. Our findings are interesting since they are not well understood in the current literature.

We encourage the reviewer to take a look at our discussion with reviewer N1bc, who asked questions similar to Q1. and Q2. We hope the discussion there can help the reviewer understand our paper better.

Q3. The use of RMSE.

We use RMSE because it is able to accurately characterize the difference between the Guassian denoisers and the actual diffusion denoisers. To address your concerns on normalization, we have conducted experiments using normalized MSE, where the trend in Figure (2,6,7,8) remains the same and does not change our conclusions. This is because the pixel values of the outputs of the diffusion denoisers for different time steps consistently lie in the range [-1,1].

In the final version, we will definitely include these results and clarification as suggested.

Q4. Novelty in Theorem 1.

Although we agree that Theorem 1 is well studied (we will add citations also suggested by Reviewer N1bC), we state the result mainly to establish the connection between the linear denoisers and the Gaussian denoisers which is not our main contribution. Our main novelty and contribution mainly instead amounts to:

(i) Establishing the inductive bias that diffusion models exhibit emerging linearity as they transition from memorization to generalization, and the corresponding diffusion denoisers become progressively closer to the Gaussian denoisers (section 3).

(ii) Showing such inductive bias is governed by the relative model capacity compared to the dataset size. Furthermore, in the overparameterized setting, such inductive emerges in early training epochs.

(Iii) Showing that generalization of diffusion models can happen with a small training dataset size.

Q4. Gaussian Structure and Strong Generalizability

First of all, we agree with the reviewer that generating high quality images is important. However, we believe that our experiment results which show that diffusion models trained on non-overlapping dataset generate nearly identical images (figure 9 (c)) is also generalization. As we’ve discussed in our last response to you (please see Q.6), strong generalization happens when the diffusion models capture certain common information between two non-overlapping datasets. Such common information might be the underlying ground truth image distribution, might be something else. We don’t know that yet in the current literature. However, please notice that the generated images of diffusion models trained on 1094 (figure 9 (c)) and 35000 (figure 9 (a) bottom) images are highly similar. This high similarity indicates that much of (more than first and second order statistics) of the information in the large dataset, which is essential for generalization, is already there in the smaller dataset. Our experiments in section 5 are meaningful since we demonstrate we can exploit this important structural information in a small dataset by either using a small model or applying early stopping. As we’ve demonstrated in section 4, by applying these two actions, we are prompting the diffusion models to learn similar function mappings as the corresponding Gaussian models, which indicates that the Gaussian inductive bias is important for the emergence of strong generalizability.

From a function mapping perspective, in the strong generalization regime, the high similarity between generated images of diffusion models trained on 1094 and 35000 images indicates that their diffusion denoisers share high similarity. But what are these function mappings? According to our experiments in section 4, the function mappings of these models are also close to the Gaussian denoisers. This again implies the Gaussian inductive bias plays an important role in the strong generalization of diffusion models. Notice that the Gaussian denoisers generate images that share similar structure as those generated by actual diffusion models in the strong generalization regime.

However, we agree with the reviewer that larger dataset contains more information which is essential for generating images with higher quality. The differences among the diffusion denoisers (1094, 35000) and the Gaussian denoisers play an important role in generating finer image structures. We really appreciate the reviewer for pointing this out and we will emphasize this in our revision. Nevertheless, our main message in section 5 is to show that we can prompt generalization on small datasets by steering the diffusion denoisers to get closer to the Gaussian denoisers.

Q5. Model Linearization.

The reference provided by the author is performing Taylor expansion, which can only approximate a function locally. For this reason, it doesn't serve our purpose since in this work we aim to find a linear model that can approximate the nonlinear deep network globally, i.e., we want to well approximate the input output mapping of the deep network for inputs that are widely spread. We will cite the paper and carefully discuss it.

Q1. The study is only limited to a few generative models and a face dataset. The observation might be different on different datasets. The face dataset has a more obvious Gaussian distribution. For example, the mean of all faces is a face, and correlations between pixels are relatively consistent. Therefore, a Gaussian distribution is somewhat reasonable for such a dataset. It is unknown whether a Gaussian distribution is still a reasonable choice for a set of diverse images. In such a case, does the diffusion model still need to mimic the Gaussian model?

A1: Although the Gaussian structure is the most evident on face images, the same linear phenomenon persists in other datasets as well. As shown in Figure 15 in the appendix, for more complicated dataset such as AFHQ and LSUN-Churches, there is still a clear similarity between the generated images from the diffusion models and those from the Gaussian model. Furthermore, we conducted extra experiments on Cifar-10 during the rebuttal, with results in Figure 2 (b) of our newly uploaded PDF. We can observe that the generated images for Cifar-10 dataset exhibits similarity with Gaussian models.

Q2. The current analysis still could not explain all observations. For example, the behavior in noise levels between 20 and 80 is not well explained in Figure 6. These observations might be worth more in-depth discussion.

A2: For the noise variance region between 20 and 80, the behavior of the learned diffusion denoisers are relatively stable. From Figure 6, we see that the difference between the diffusion denoisers and the Gaussian denoisers in the high noise region does not change as much compared to the intermediate noise variance region when varying the dataset size. Furthermore, from Figure 7 we see that the difference between the diffusion denoisers and the Gaussian denoisers seems to decrease consistently as the model scale increases. These observations indicate that with sufficient training, the diffusion denoisers for the high noise variance region converge to the Gaussian denoisers without overfitting. This observation might be attributed to the fact that for the high noise variance region, the Gaussian denoisers are approximately the global minimizer of the denoising score matching objective under finite training dataset assumption (see Equation (5)); this is proved in [19]. Therefore, we should expect that with a large enough model capacity, deep networks converge to such a minimizer. Since it is a global minimizer, we observe less overfitting.

We will include this discussion in our revised paper and make those points clear. Nevertheless, this region doesn’t influence the final generated images as much as the intermediate noise regions, which is why we mainly focus on the discussion of the intermediate noise region.

Q3. Have you considered using standard Gaussian distribution as the convergent distribution of the diffusion process?

A3: We start the sampling process from a standard Gaussian distribution with std = 80. Since the image lies in the range of [-1,1], the noise magnitude is much larger, which means the convergent distribution is roughly a standard Gaussian distribution with std equals to 80.

We thank the review for the insightful comments, please let us know if you have further questions.