Paper: Generalization in diffusion models arises from geometry-adaptive harmonic representations (ICLR-2024)

Reviewer comment (Experiments):
further validation on a more realistic data might be beneficial to strengthen the main points of the paper

Author response:
Thank you for your review. 
- In addition to the LSUN bedroom dataset (in the appendix), we’re continuing to test our method on other datasets, which will be included in the final version of the paper.

Reviewer comment (Writing):
lack of any sort of description of more "general DNN GAHBs"

Author response:
Thank you for your review. 
- We acknowledge (in the Discussion) that we don’t (yet) have a mathematically precise definition of GAHBs, which is an interesting open question.

Reviewer comment (Novelty):
The paper's results were obtained using a straightforward CNN architecture, but prior studies have indicated that CNN architectures can effectively learn and utilize harmonic bases (https://arxiv.org/abs/1810.12136). The paper's only weakness lies in not acknowledging this perspective that has been previously established in the literature.

Author response:
Thank you for your review. The paper you mention is indeed interesting, and explains the importance of phase correlations within successive layers of a DNN, and their relevance to image representation.  However, it does not define orthonormal bases and it is not so related to the GAHB bases of the diffusion models, although we do agree that it is worth citing.

Reviewer comment (Novelty):
Would you be able to add some of this line of literature to your discussion?

Author response:
Thank you for your review. The paper you mention is indeed interesting, and explains the importance of phase correlations within successive layers of a DNN, and their relevance to image representation.  However, it does not define orthonormal bases and it is not so related to the GAHB bases of the diffusion models, although we do agree that it is worth citing.

Reviewer comment (Novelty):
In Section 3, subsection S3.1 seems to be largely based on (Mohan et al., 2020). If so, please state this clearly at the onset, or make it clear where the new contributions begin.

Author response:
Thank you for your very positive review, and constructive suggestions, which we will incorporate in the revision.

Regarding S3.1: The use of a bias-free network in equation 6, and the Jacobian analysis of locally-linear behavior, is taken from Mohan et al 2020.  Both citations in the beginning of section 3.1 refer to those contributions. But the rest of section 3 (equation 7 onward) is new and independent of contributions made in Mohan et al 2020.

Reviewer comment (Theory):
It seems that the theoretical framework posited in section 3 is specific to the particular denoiser architectures being studied in the paper (BF-CNN). For example, PixelNet++-type denoisers used in modern diffusion models do not seem to be amenable to a decomposition like equation (6), because they involve attention layers and positional embeddings (presumably with affine components) to implement different conditioning operations. It would be good if the authors could comment on this issue, and how they see the theory extending to this modern setting.

Author response:
- Architecture:
a) We constrained the architecture to have a zero net bias (Eq 6), which allows the eigen-analysis in section 3.1.  We think this choice is unlikely to affect the results.  In particular, note that Mohan et al 2019 (from which we borrowed the architecture) showed the effective bias of the network tends to zero within the training range of noise levels (in our paper, denoisers are trained on a large range of noise levels). 
b) We have explored the effect of the number of parameters in our architecture (its depth and width). Larger networks with larger images require more training samples to generalize. We will include these results in the appendix for the final revision. 
c) Other major changes in the architecture, however, could potentially result in different inductive biases and  generalization regimes (or even a failure of generalization). An exhaustive exploration over architectures (such as PixelNet++) is an open empirical question which we’ve not addressed. Our method for testing generalization, however, can be used in future to test and detect memorization for any architecture/training scheme. We will make this point clearer in our revision of the discussion.

Reviewer comment (Experiments):
How robust the main empirical insights are to the specific training setup and model architecture being studied in the experiments? For example, what if one used a modern noise-conditional diffusion model instead of training a single unconditioned model on multiple noise scales; what if one considered larger-scale training (eg on datasets of higher-resolution, photorealistic images, as one considers for modern diffusion models); what if one performed generation with an ODE/SDE-type sampling procedure, rather than Algorithm 1 used in the paper? It would be helpful to know how the authors see Figures 1 and 2 translating into these settings, whether they would expect changes, etc.

Author response:
1) Robustness of our results to choices of architecture, dataset characteristics, and the details of the synthesis procedure: 
 - Architecture:
a) We constrained the architecture to have a zero net bias (Eq 6), which allows the eigen-analysis in section 3.1.  We think this choice is unlikely to affect the results.  In particular, note that Mohan et al 2019 (from which we borrowed the architecture) showed the effective bias of the network tends to zero within the training range of noise levels (in our paper, denoisers are trained on a large range of noise levels). 
b) We have explored the effect of the number of parameters in our architecture (its depth and width). Larger networks with larger images require more training samples to generalize. We will include these results in the appendix for the final revision. 
c) Other major changes in the architecture, however, could potentially result in different inductive biases and  generalization regimes (or even a failure of generalization). An exhaustive exploration over architectures (such as PixelNet++) is an open empirical question which we’ve not addressed. Our method for testing generalization, however, can be used in future to test and detect memorization for any architecture/training scheme. We will make this point clearer in our revision of the discussion. 

- Image resolution/size: Increasing the image dimensionality will surely require a larger number of samples to transition to generalization (Figs 1  and 2).  It thus remains open whether state-of-the-art diffusion models have reached the generalization regime or not. We will comment on this in the discussion. 
- Synthesis procedure: we believe that the transition from memorization to generalization is robust to the details of the synthesis algorithm, including the noise-conditional vs. unconditional synthesis. However, this should be verified empirically. We will comment on this in the discussion.

Reviewer comment (Theory):
Could you clarify behind the thinking behind the claim in the discussion that "deep networks are more adapted towards high-dimensional structures [than low-dimensional structures]"? Although I think I follow at a high-level — eg in Figure 5, the adaptive basis at the noisy image is not exactly equal to the optimal 5-dimensional basis for the low-dimensional tangent space — it still seems to me that there is a bias in what has been learned towards low-dimensional structure, because the correct basis for the true tangent space seems to be contained in the actual adaptive basis, and the eigenvalues of the Jacobian decay at a reasonable rate (making the spectrum overall "compressed", if not sparse/low-dimensional). Hence when I try to interrogate the claim at a lower level of detail, I cannot exactly 'compile' it.

Author response:
2) The sentence "deep networks are more adapted towards high-dimensional structures [than low-dimensional structures]" was indeed cryptic - apologies for that. There is a common belief that deep networks generalize due to the presence of low-dimensional structure in the training data (the “manifold” hypothesis). If networks were indeed inductively biased towards low-dimensional manifolds, we would expect them to be optimal on such distributions (e.g., the disk images, or the single face or sinusoid images in the appendix), and deviate from optimality on higher-dimensional distributions (e.g., the C-alpha family). We observe that the opposite happens, which is indicative of a different inductive bias.