Probing the Latent Hierarchical Structure of Data via Diffusion Models

We thank the reviewer for finding our work clearly written, our analysis nice, and our experiments well-designed. We answer to their concerns below.

1. On the implication of a peak in the susceptibility

In this work, we demonstrate that very popular models such as Gaussian data or Gaussian random fields fail to exhibit such a peak, whereas the Random Hierarchy Model, text and images do. This work thus emphasizes the limit of modeling data as Gaussian, and points toward the need for richer structures. That said, the reviewer is correct that the observation of a peak in susceptibility does not imply that the data must be exactly tree-like. As discussed in our reply to other reviewers, we believe that mild context dependence will not affect qualitatively our conclusions. Yet, a full classification of which graphical models may capture this phenomenon is a question for the future, as now emphasized in the text (highlighted in blue).

2. Mean-field approximation

The reviewer is correct: the epsilon process can be seen as a mean-field approximation of multinomial discrete diffusion, where the uncertainty in the reconstruction is uniformly spread onto the sequence of tokens. We then use a mean-field approach, averaging over the possible rule realizations of the RHM, to compute the correlations.

3. Diffusion as spectral autoregression

We assume the reviewer is referring to the analysis presented in this blog. Spectral autoregression aligns with the behavior observed in our Gaussian random field model, discussed in Section 3 and Appendix B, where higher frequencies are noised first in the diffusion process. Crucially, this perspective, as described in the blog, does not incorporate assumptions about or analyze the conditional probabilities between different scales (or frequencies). By contrast, these correlations among scales are central to the analyses of Marchand et al., 2022 and Kadkhodaie et al., 2023, who explore their structure and implications. Thus, we view these two lines of work as distinct.

Questions

1. Does the susceptibility divergence tell us anything about how many levels of hierarchy are likely present? Or just that there is at least one level?

In the Random Hierarchy Model, the phase transition and the associated length scale divergence hold in the limit of large depth. In practice, at finite depth, one observes a smooth crossover with an associated finite susceptibility peak, which increases with increasing depth (fixing the parameters). Conversely, the RHM with just a single level does not exhibit this phenomenology. In the case of real data, the susceptibility peak highlights the presence of structured dependencies consistent with a multi-level hierarchy but does not provide specific information on the depth.

2. The RHM is discrete, and discrete vs continuous diffusion are rather different; can you justify why RHM should be a good model for continuous data/diffusion as well?

While images are inherently continuous, they can be described at an abstract, semantic level using discrete hierarchies. As discussed in the related work section, these hierarchical representations have been formalized in pattern theory (Stoyan, 1997), where the decomposition is inspired by parsing methods used in linguistics. In this framework, visual scenes are decomposed hierarchically into objects, parts, and primitives, leading to practical algorithms for semantic segmentation and scene understanding. We refer to the paragraph "Hierarchical models of images and text" in the Related Work section for further references.

3. Do the MDLM and ImageNet expts actually confirm that a phase transition occurs? Or do we just observe the susceptibility peak and infer a phase transition by analogy to RHM? In particular, it seems that for ImageNet it might actually be possible to run a classifier to determine whether the class changed.

The susceptibility peak in real data strongly suggests an underlying hierarchical structure. In the case of images, as suggested by the reviewer, we ran a convolutional classifier (a state-of-the-art ConvNeXt pre-trained on ImageNet) to determine when the class changes (note that Sclocchi et al., 2024, performed equivalent experiments). We reported the results in Figure 12 in the updated manuscript. Clearly, in correspondence to the susceptibility peak, the class of the generated images displays a sharp transition.

We thank the reviewer for their feedback and address their specific concerns below.

1. Presentation

We appreciate the reviewer's feedback on improving the readability of our paper. Our work follows a structured approach: we first present our theoretical framework using the synthetic hierarchical model of data, the RHM. Then, we validate these theoretical results through numerical experiments. Finally, we test our predictions in real-world scenarios involving state-of-the-art diffusion models for both images and text. In response to your suggestion, we have taken the following steps to enhance the clarity of our presentation:

• We have explicitly highlighted the structure of the paper by adding a paragraph after the introduction.
• We have separated and presented the main definitions in a more structured, standalone format, making them easier to follow.
• We have moved the detailed computations of the dynamical correlation length analysis (originally in Subsection 3.1.1) to the Appendix. In the main text, we have provided a more concise and accessible description of the results.

We believe these revisions (highlighted in blue in the updated pdf) have significantly improved the clarity and readability of our work. We welcome any further feedback or suggestions on how we might continue to refine our presentation.

2. BP vs. neural network for denoising

Score-based generative models generate samples by reversing a forward diffusion process that progressively adds noise to data. In practice, these models employ a neural network to approximate the score function, i.e., the gradient of the log density of the data. The score enables running the backward dynamics. The score function is implicitly related to the conditional expectation $\mathbb{E}[x_0|x_t]$, where $x_t$ is a noisy observation at time $t$. Given this conditional expectation, new samples can be generated by running a time-discretized backward dynamics.

For tree-like models such as the RHM, the conditional expectation $\mathbb{E}[x_0|x_t]$ can be computed exactly via message-passing algorithms like Belief Propagation (BP). As noted in L157, this corresponds to having access to a neural network that has learned the exact population score, which can be used to run the backward dynamics. However, BP also allows for the sampling of new data directly from the exact posterior $p(x_0|x_t)$ without running the reverse process. In the limit of an infinite number of diffusion steps, the two sampling processes are equivalent. Thus, our predictions with the RHM are directly comparable to real-world data.

We have incorporated these clarifications into the updated manuscript (highlighted in blue).

3. Practical applications

See our reply to reviewer ohKX. We want to clarify further that the primary goal of our research is to provide a fundamental analysis of the hierarchical structure of data belonging to different modalities and how diffusion models capture and utilize this structure. Our findings contribute to the growing body of work on the interpretability of latent representations in neural networks, offering a fresh perspective by examining correlated changes observed in forward-backward experiments with diffusion models. On the one side, our work supports the hypothesis that hierarchical latent structures are universal properties underlying natural data as diverse as images and language. On the other side, it opens new possibilities for future work on the interpretation of these correlated changes, e.g., in terms of the syntactic structure of a language. These further works will be data-specific, whereas the present work emphasizes the universal connection between theory and observations in very diverse settings.

We clarified the text to make that point (highlighted in blue).

Thanks for the clarifications and additional experiments. I now understand that the authors are analyzing the entire progressive sampling process, which resembles the forward-backward experiment. Moreover, they use mean-field theory for simplification, without which, understanding the sampling process could be challenging for unstructured data distributions.

I think the paper is accomplished, but I am still not very confident about how realistic the RHM is. I can increase my ratings and lower my confidence score if the authors would like.

We thank the reviewer for appreciating our work, recognizing that it is well-written and that our theoretical claims are supported by experiments on natural data. Below, we address the reviewer's specific concerns.

1. Practical applications

Fundamental science contributes to practical problems by inspiring follow-up research on different time scales. In this specific case, the theoretical and experimental results we provide have quite direct potential for practical applications. The most important one concerns the interpretability of deep networks - a central issue of the field. The hierarchical representation they build is believed to reflect the combinatorial structure of data. Tools to study the latter are scarce. Here, we show that the effect of changing latent variables at different depths can be studied by monitoring the noise level in forward-backward experiments, opening a new avenue to characterize data structure. Finally, the presence of a transition at a certain noise level suggests that it may be especially useful to train diffusion models particularly around this noise value - an idea that practitioners just started to explore [1].

As suggested by the reviewer, we added these two points in our conclusions in the revised pdf document (highlighted in blue).

[1] Barceló, R. et al. Avoiding mode collapse in diffusion models fine-tuned with reinforcement learning. arXiv preprint (2024).

2. Context-free vs. context-sensitive models

The question raised by the reviewer is both interesting and subtle. On the one hand, some apparently non-tree-like graphs of latent variables can be made tree-like, if one allows for complex latent variables that encode more information. On the other hand, this is not always the case, and for example, context-dependence can be present. More broadly speaking, controlling analytically diffusion models for a general graph of latent variables is untractable - in fact, it can take a time exponential in the number of latent variables just to sample from the model.

In practice, it is known that context-free grammars are not expressive enough to capture all phenomena in the syntax of natural languages, requiring mildly context-sensitive models [2]. Our observations on WikiText thus support that our conclusion holds beyond context-free grammars, at least for mildly context-dependent structures. In the future, building models that depart gradually from a context-free (tree-like) structure may give a handle to progress on this difficult question.

In the revised manuscript, we have added a discussion in the conclusion section to address the limitations of the tree model and the potential for incorporating context dependencies in future work (highlighted in blue).

[2] Jäger, G., and James, R. Formal language theory: refining the Chomsky hierarchy. Philosophical Transactions of the Royal Society B: Biological Sciences 367, no. 1598 (2012): 1956-1970.

We thank the reviewer for recognizing the novelty of our contribution and acknowledging the rigor of our analysis, along with the solid experimental validation provided. Below, we address the reviewer's specific concerns.

1. Presentation

We have made several improvements to the presentation and structure of our paper in response to feedback from multiple reviewers, including:

• We have added a new paragraph at the end of the introduction section that explicitly outlines the structure of the paper, providing a clearer roadmap for the reader.
• We have isolated the main definitions from the core text, presenting them in a more structured and standalone manner to improve readability.
• We have moved the detailed computations of the dynamical correlation length analysis (originally in Subsection 3.1.1) to the Appendix. In the main text, we have provided a more concise and accessible description of the results.

We welcome any further suggestions for enhancing the clarity of our work and are open to elaborating on specific points that may require additional explanation.

2. Binary vs. continuous spin variables

In Equation 3 (Equation 2 of the updated manuscript), our choice to use binary spin variables $\sigma_i$ stems from the discrete nature of the model under consideration in that section, where features are equidistant and represented categorically. However, we agree that a continuous measure is indeed more suitable for continuous data types, such as images. Specifically, for image data, we account for the continuity of variations by measuring the L2 distance between patch embeddings before and after the forward-backward procedure (see Equation 7). To clarify this point, we have added a sentence when introducing the discrete spin variables, noting that we will extend them to accommodate continuous data in the subsequent sections (highlighted in blue).

3. Content dependence of spatial correlation structure

We acknowledge that spatial correlations between changes in a single image are influenced by the content. Specifically, variations in camera distance or object placement can alter the spatial correlation structure of an individual image. Nevertheless, our analysis in Figure 6 reports average spatial correlations across a large set of images. These average correlations are robust to individual content variations as long as the data distribution of the initial images remains consistent. In other words, while the spatial correlation for any individual image is content-dependent, the trends in Figure 6 represent an aggregate measure, capturing the statistical properties of the dataset.

We hope our responses provide sufficient clarification and would be happy to elaborate further or address additional concerns as needed.