Learning Discrete Concepts in Latent Hierarchical Models

Thank you for your thoughtful review and dedicating your valuable time to our work! We address your concerns as follows.

W1: “From Sec. 2, given the description that the continuous latent variables c seem to control a lower level of features of data, while in Figure 1. a, it seems to be independent to concept factor d at the same level of the hierarchical structure. Could you elaborate on the relation and difference between c and d?”

Sure, we are happy to elaborate! As you correctly observed in Figure 1.a, the continuous variable $\mathbf{c}$ directly influences the observed variable (e.g., images) $\mathbf{x}$ and is independent of the discrete variables $\mathbf{d}$ (please also see Equation 1).

Intuitively, $\mathbf{d}$ contains all the discrete concepts/information in the image distribution, for instance, object classes or categories of shapes, whereas $\mathbf{c}$ captures the information, such as illumination, angles. These two sources of information are complementary and together fully describe the image information.

Please note that the independence is not restrictive because even if $\mathbf{c}$ depends on $\mathbf{d}$ we can reduce it to the independent case by replacing $\mathbf{c}$ with its the exogenous variable, which is independent of $\mathbf{d}$.

Please let us know if we have cleared your question – thank you!

W2: “For LD experiments, at the early steps of the diffusion process (i.e., T), Figure 5 presents controlling of bread and species (high-level) features, while the low-level (e.g., object angle, background) can remain the same. But, in Figure A8, such capability does not hold, especially for the shoes. Can the author explain the reason behind this?”

Thank you for the interesting question! We consider the object angle as a continuous variable (i.e., part of $\mathbf{c}$ in Figure 1.a) due to its continuous nature, so it does not belong to the discrete hierarchical structure. We speculate the consistent camera angles in Figure 5 result from the high probability of this specific angle in the data distribution, since most of the close-up face photos are captured from this angle in the training dataset.

Please let us know if you have remaining issues -- we are more than happy to engage!

Thank you for recognizing our theoretical contribution as “interesting, important, and novel” and we highly appreciate your detailed, constructive suggestions on the writing. In light of your suggestion, we have put a lot of effort into revising the paper to explain all the involved definitions clearly, as we will detail below. We hope that this contribution should be visible to the community sooner. Therefore, we’d like to follow the tradition of publishing the paper at NeurIPS. We’d appreciate your feedback – thank you in advance!

W1: Writing suggestions.

We are grateful for your thoughtful suggestions. In light of your feedback, we have made the following modifications to improve the readability of Sec. 3.3:

1. Trek-separation and non-negative ranks: we have moved definitions in the appendix to Sec. 3.3 (original lines 234) and explained their distinctions with d-separation and ranks.
2. Minimal-graph operator: we have added references to Figure A1 (a)(b) in the main text and moved its definition to the original line 288.
3. Condition A3.15 and the skeleton operator: we have moved Theorem 3.9 to the appendix and keep the reference in lines 300-30. Consequently, we keep Condition A3.15 and the skeleton operator at the current location in the appendix.
4. Algorithm 1: as our theory only involves simple changes of the original algorithms in [20], we tentatively leave the algorithm at its original location. Instead, we have added the following line in line 299 to highlight the modifications “our modifications consist of changing the original rank to the non-negative rank over probability tables and setting discovered covers as discrete variables as mentioned above”.
5. Adaptive sparsity: we have moved its description (original lines 1013-1015) to Sec. 6.2.
6. Literature review: we feel that the introduction has covered key literature, so for now we leave it in the appendix and will move it to the main text if given one additional page for the final version.

To compensate for the space, we have moved large chunks of discussion in Sec. 5 to the appendix, along with Theorem 3.9 and its discussion. We’d like to note that all these edits are limited to Sec. 3.3 and Sec. 5 locally and the global structure of the paper remains.

We’d love to hear your feedback!

W2: “Diffusion model experiments.”

Thank you for the comment. In light of your comments, a portion of Sec. 5 has been moved to the appendix, while the main paper highlights its connection to our main results and findings. In our humble opinion, the hierarchical model is a valuable framework for reasoning about diffusion generation processes. We believe this could provide inspiration for the community to advance towards more controllable and interpretable generative models.

W3.1: Estimator definitions.

Many thanks! We have added the following to line 132: “where we assume access to the full population $p(\mathbf{x})$”.

W3.2: Table 1, 2 references,

Thanks for the reminder! We’ve added references in lines 314, and 318 respectively.

W3.3: notion of splitting variables.

Thanks a lot! We’ve replaced “splitting” as “(i.e., turning a latent variable $z_ {i}$ into $\tilde{z} _{i,1}$ and $\tilde{z} _{i,2}$ with identical neighbors and matched cardinalities $|\Omega^{z} _{i} | = | \tilde{\Omega}^{z} _{i,1} | + | \tilde{\Omega}^{z} _{i,2} |$ ).”

W3.4: Random variable support.

Great point! We have modified “open” to “closed”, as this doesn’t affect our main proof.

W3.5: definition phrasing.

Thanks! We’ve edited it to “We introduce … [43].”.

W3.5: t-separation notations.

Thanks! We’ve edited it as “ a partition $(\mathbf{L} _{1}, \mathbf{L} _{2})$ t-separates…” in Theorem 3.5.

W3.6: small text.

Thanks for pointing it out! We have updated the fonts.

W3.7: pure children clarification.

You’re totally right! We’ve moved Definition A3.8 to line 119.

W3.8: typos.

Thanks! We have corrected it to $v_{1}$.

Q1: Condition 3.1.

Great question. We acknowledge that this condition may be strong and we wished to avoid it. Unfortunately, it seems technically necessary without additional assumptions. For instance, its necessity is discussed in [24] for one-layer mixture models and we believe this is also the case for hierarchical models. Since science is established step by step, we hope our results could serve as the basis for more relaxed conditions in the community.

In the distribution of dog images, suppose “head” has only two children “eyes” and “nose”. This condition means for each type of “head”, all combinations of shapes of “eyes” and “nose” should have non-zero probabilities. In reality, some combinations may be rare but can still appear at extremely small probabilities. That said, we agree that there are certainly cases where this can be violated, e.g., deterministic relations.

Yes, you’re totally right about $\text{ne}(v)$. We’ve included “$\text{ne}(v):= \text{Pa}(v) \cup \text{Ch}(v)$” in line 119 - thanks!

Q2: Discussion on sparsity conditions.

Great suggestion! We’ve added the following discussion to the original line 184: “Condition 3.3-iii is related to the notation of sparsity in disentanglement literature. Brady et al. [b] divide the latent representation into blocks and assume no shared children among blocks, which can be stringent if one aims to identify fine blocks. Moran [c] assumes pure observed children for each discrete variable, which is strictly stronger than Condition 3.3-iii. The structural sparsity in Zheng et al. [d] implies Condition 3.3-iii. By contrapositive, if latent variable $z_{0}$’s children form a subset of a distinct variable $z_{1}$’s children, then we cannot find a subset of observed variables whose parent is $z_{0}$ alone.”

Q3: Definition 3.6 clarification.

It’s the collection of all states of variables in the cover. We’ve defined this notation in line 236 – thanks!

Please let us know if you have further concerns and we are more than happy to discuss more!

Thank you so much for your feedback. We're really glad to hear that you think our changes can improve the manuscript by quite a bit. Your suggestions have been incredibly helpful -- thank you again!

Thank you for your encouraging words and thoughtful comments! We address your concerns as follows.

W1: “The identification conditions (Condition 3.3 and 3.7) may be too restrictive… for complex high-dimensional data.”

Thank you for your comments. Maybe counterintuitively, the high dimensionality of image data actually makes Conditions 3.3 and 3.7 more likely to hold.

Specifically, for the invertibility condition, note that since $\mathbf{x}$ is generated by $\mathbf{z}$ ($\mathbf{x} := g(\mathbf{z})$), $\mathbf{x}$ cannot contain more information than $\mathbf{z}$. Thus, the non-invertibility problem arises only when $\mathbf{x}$ fails to preserve the information of $\mathbf{z}$. High dimensionality gives $\mathbf{x}$ sufficient capacity to contain such rich information and facilitates invertibility. This assumption is also well-accepted in the community on image data [32-34]. This also applies to Condition 3.3-iii and Condition 3.7-ii: with a large dimension of the observed variable $\mathbf{x}$, the children of each latent variable $d$ are less likely to overlap (Condition 3.3-iii), and latent variables are more likely to have unique, observed descendants (Condition 3.7-ii).

At the same time, we acknowledge that there do exist situations where some assumptions are violated. However, since science is constructed step by step, we hope our results can shed light on further development of this field.

W2: “...rank tests (Theorem 3.5), which can be computationally expensive and numerically unstable…”

Thank you for raising this concern. We agree that rank tests are not easy (as noted in lines 415-417). However, we hope and believe that as these tests become increasingly important [19,20,41,42], more stable and efficient methods will be developed shortly. Therefore, we hope the current limitations of rank tests do not diminish the value of this contribution.

W3: “... overlapping relationships that may not fit neatly into a DAG structure”.

By "overlapping relationships," we were wondering if you referred to cyclic relations among latent variables. (Please let us know if we've misunderstood this.) We agree that this paper focuses on the DAG structure. If non-DAG is necessary to address specific issues, we believe the ideas articulated in this paper can still provide valuable insights, although many practical issues should be considered.

W4: “... how to handle noise or uncertainty in the observed data..”

Thank you for raising the issue of noise or uncertainty in the observed data, which is a significant and challenging problem in causal discovery and causal representation learning, as noted in recent contributions [a]. Addressing these problems is highly nontrivial, so we believe it should and will be tackled after solutions to the basic settings are clear, which is what we aim to achieve in this paper.

[a] Causal Discovery with Linear Non-Gaussian Models under Measurement Error: Structural Identifiability Results. Zhang et al. UAI 2018.

W5: Concrete mechanism for improving diffusion models.

Thank you for the comment. In our initial attempts, we applied basic ideas like sparsity to improve concept extraction techniques from the diffusion model, as shown in our experiments (Section 6.2 & Figure A6). We are actively working on building more principled generative models using our theoretical insights. Additionally, we hope the connection to hierarchical causal models presented in this work can inspire the community to develop more controllable and interpretable generative models.

Q1: robustness to invertibility violation and possible relaxations.

This is a great question. We have been working on this for a long time. Recently, it seems hopeful to solve this problem by greatly extending [b]. While they consider a single latent variable in the probabilistic setting, extending to multiple latent variables seems highly possible but requires substantial investigation.

[b] Instrumental variable treatment of nonclassical measurement error models. Hu et al. Econometrica, 2008

Q2: “... to design a diffusion process that explicitly learns and respects a given hierarchical concept structure?”

Great question! We believe this correspondence can benefit the controllability of current diffusion models, which remains challenging for existing methods [c,d]. Unlike standard diffusion models, our framework suggests injecting different concepts at various diffusion steps, ensuring that concepts at different levels are properly rendered in the image [d]. We see more structured language and image interaction as a promising direction where our work could provide valuable insights.

[c] Compositional Text-to-Image Generation with Dense Blob Representations. Nie et al. ICML 2024.

[d] ELLA: Equip Diffusion Models with LLM for Enhanced Semantic Alignment. Hu et al. ArXiv 2024.

Q3: The discrepancy between discrete variables and continuous diffusion latent space; possible extensions to continuous latent variables.

Thank you for the question. We view the continuous latent space of diffusion models as an ensemble of embedding vectors for discrete concepts, similar to word embeddings in NLP (see lines 330-335). Park et al. [52] and our experiments (lines 388-399) support this interpretation, showing that the continuous space can be decomposed into a finite set of basis vectors, each carrying distinct semantic information.

Regarding potential extensions, our theoretical analysis is connected to identification results on continuous latent variable models [19,20] (see lines 223-233) and can be adapted accordingly. We should note that for continuous models, strong assumptions like linearity are still required [19,20]. We hope to develop a completely non-parametric framework to handle such models in the near future.

Please let us know if you have further questions -- thank you so much!

Thank you for your careful assessment and thoughtful comments on our work! We address your concerns point to point as follows.

M1. "Practicality of Recovering Hierarchical Graphs".

Thank you for the great question! In light of your suggestion, we’ve included in our revision the following experiment, where we extract concepts and their relationships from Stable Diffusion through our hierarchical model interpretation. Please find the results in the submitted PDF file in the global response.

Our recovery involves two stages: determining the concept level and identifying causal links. We add a textual concept, like "dog", into the prompt and identify the latest diffusion step that would render this concept properly. If "dog" appears in the image only when added at step 0 and "eyes" appears when added from step 5, it indicates that "dog" is a higher-level concept than "eyes". After determining the levels of concepts, we intervene in a high-level concept and observe changes in low-level ones. No significant changes indicate no direct causal relationship.

As shown in the submitted PDF, we explore the relationships among the concepts "dog", "tree", "eyes", "ears", "branch", and "leaf". We provide the final recovered graph and some intermediate results of both stages. Although the current experiment scale (number of involved concepts) is relatively small, we hope these results demonstrate how to leverage the hierarchical causal model in our work to investigate black-box diffusion models.

M2: "Empirical Evidence for Real-world Data".

Thank you for the interesting question. Given your question, we spent quite some time attempting to develop a quantitative relation between the attention sparsity and the graphical structure. Unfortunately, the gap seems nontrivial given that these two quantities belong to quite different objects (causal graphs, deep-learning layers) -- the same attention sparsity may correspond to a large family of causal graphs. For now, we think this metric may better serve as a qualitative indicator of the graphical connectivity, as in the manuscript.

M3: "Realism of Condition 3.3".

Thank you for pointing this out. Given your suggestions, we’ve included the following discussion in our revised manuscript (original line 184).

Condition 3.3 i: “In practice, the continuous variable $\mathbf{c}$ often controls extents/degrees of specific attributes (e.g., sizes, lighting, and angles) and takes values from intervals (which are connected spaces). For instance, the variable related to “lightning” ranges from the lowest to the highest intensity continuously.”

Condition 3.3 ii: “For images, the invertibility condition assumes that images preserve the semantic information from the latent variables. Thanks to their high dimensionality, images often have adequate capacity to contain rich information to meet this condition. For instance, the image of a dog contains a detailed description of the dog’s breed, shape, color, lighting intensity, and angles, all of which are decodable from the image.”

Condition 3.3 iii: “Practically, this condition indicates that lowest-level concepts influence diverse parts of the image. Lowest-level concepts are often atomic such as a dog’s ear, eyes, or even finder. As we can see, these atomic features often don’t overlap in the image (e.g., ears and eyes are separate).”

m1: "Partial Literature Review".

Thank you for the helpful suggestion! Thanks to your pointer, we have included the following paragraph in our revision. Please let us know if we have overlooked important works and we would be happy to include them.

“A plethora of work has been dedicated to extracting interpretable concepts from high-dimensional data such as images. Concept-bottleneck [12] first predicts a set of human-annotated concepts as an intermediate stage and then predicts the task labels from these intermediate concepts. This paradigm has attracted a large amount of follow-up work [13][a-e]. A recent surge of pre-trained multimodal models (e.g., CLIP [17]) can explain the image concepts through text directly [14-16]. In contrast with these successes, our work focuses on the formulation of concept learning and theoretical guarantees.”

[a] Post-hoc Concept Bottleneck Models. Yuksekgonul et al. ICLR 2023.

[b] Probabilistic Concept Bottleneck Models. Kim et al. ICML 2023.

[c] Addressing Leakage in Concept Bottleneck Models. Havasi et al. NeurIPS 2022.

[d] Incremental Residual Concept Bottleneck Models. Shang et al. CVPR 2024.

[e] Interactive concept bottleneck models. Chauhan et al. AAAI 2023.

m2: "Clarity of Theoretical Explanations".

Thank you for your suggestion! Besides the examples for Condition 3.3 as quoted above, we have included the following for Condition 3.7.

"Intuitively, Condition 3.7-ii requires that each discrete variable has sufficiently many children and neighbors to preserve its influence while avoiding problematic triangle structures to ensure the uniqueness of its influence. Condition 3.7-iii requires sufficient side information (large $|\mathbf{A}|$) to identify collider $\mathbf{C}$."

Q: "applicability of this framework in a supervised setting".

Thank you for the intriguing question! We believe this is possible. In the supervised setting, the prediction target is often a high-level concept. To make correct predictions, the model may need to leverage low-level concepts, like ears and fur for classifying cats. Thus, one may impose sparsity constraints on the representation to view which features are engaged for predicting a specific class and infer certain concept structure therein. For example, if predicting cats and dogs calls for latent variables $(z_{1}, z_{2})$ and $(z_{2}, z_{3})$ respectively, one may infer that "cat", as a high-level variable, is a parent to $(z_{1}, z_{2})$, and "dog" is a parent to $(z_{2}, z_{3})$.

Please let us know if it is unclear and we would be happy to discuss further!