Paper: Revealing Hidden Causal Variables and Latent Factors from Multiple Distributions (ICLR-2024)

Reviewer comment (Experiments):
While the bulk of the paper establishes fundamental results for causal inference, this seems to have little impact on the design or guarantees of machine learning algorithms. Though there is a Simulations section in which the authors apply their theory, there seem to be many details and analysis missing. For instance, it is not clear to me how the data is generated, and the results shown in Figures 2 and 3 have little explanation. Hence it is quite challenging to understand what exactly the authors are trying to demonstrate, and how well their procedure performs.

Author response:
Thanks for this insightful comment. The theoretical results have two major impacts on the design of machine learning algorithms: (1) learning  $(\hat{g}, \hat{f},p_{\hat{Z}})$ to achieve Eq. (2), and (2) applying sparsity constraint on the edges of Markov network over $\hat{Z}$. We will make this clear in the revised manuscript.

Reviewer comment (Theory):
The technical side seems reasonable. However, I didn't understand the significance of the linearly independent vectors (see, e.g., bullet point 2 of Thm 1). Please explain its importance.

Author response:
The linearly independent vectors are crucial to the proof of the theoretical result, and is commonly used in the literature [1]. Specifically, it allows us to obtain Eqs. (11), (12), and (13) from the equation above them that involves linear combinations of the corresponding terms. Intuitively speaking, it requires the distribution to vary sufficiently across different domains. We will explain this in the revised manuscript.

Reviewer comment (Writing):
On page 3, should it read $Z_j \to X_i$ if and only if $Z_j \in PA(X_i)$?

Author response:
Thanks for spotting this. It should read $Z_j\rightarrow Z_i$ if and only if $Z_j\in\textrm{PA}(Z_i)$, and we have fixed this typo in the revision.

Reviewer comment (Reproducibility):
How should we imagine that $(\hat{g}, \hat{f}, p_{\hat{Z}})$ is learned to achieve Eq. (2)?

Author response:
Thanks for this insightful question. We estimate a model $(\hat{g}, \hat{f},p_{\hat{Z}})$ that assumes the same data generating process as in Eq. (1) and matches the true distribution of $X$. We did not specify exactly how to achieve it, and leave the door open for different approaches to be used, such as normalizing flow or variational approaches. For example, we adopt a variational approach in Section 5. We have included this discussion in the updated manuscript.

Reviewer comment (Presentation):
Unclear how well structure is recovered, just from looking at Figure 2. What do these scatter plots represent? Same comment for Figure 3.

Author response:
Thanks for your question. We assume that there are 4 components in $z$. In each subfigure, there are $x\times 4$ graphs, graph $i,j$ denotes the scatter plot between the true component $z_i$ and the estimated $z_j$. Firstly, we observe that there is one-to-one correpondence between the estimated components and the true components, supporting our component-wise identifiability theory. On the other hand, for the Y-structure, we have $z_1\rightarrow z3 \leftarrow z2$ and $z_3 \rightarrow z_4$. As shown in the Fig. 3(a)(b), the estimated $\hat{z}_1$ recovers $z_2$ and $\hat{z}_1$ is independent from $z_1$; the estimated $\hat{z}_2$ recovers $z_1$ and $\hat{z}_2$ is independent from $z_2$. Therefore, our estimated $\hat{z}_1, \hat{z}_2$ recovers the $z_2, z_1$ and their independence relationship. As for $\hat{z3}$, it is dependent on $\hat{z}_1$ and $\hat{z}_2$. Therefore, we have $\hat{z}_1\rightarrow \hat{z}_3 \leftarrow \hat{z}_2 \Leftrightarrow z_2 \rightarrow z_3 \leftarrow z_1$. As for $\hat{z}_4$, it is dependent on all previous variables, i.e., $\hat{z}_1, \hat{z}_2, \hat{z}_3$. Since we have the sparisty regularization, we can remove the redundant link from $\hat{z}_1$ and $\hat{z}_2$ to $\hat{z}_4$. Therefore, we are able to recover the true latent cause structure. In addition, in eq 6, we have an adjacency matrix to learn, the learned adjacency matrix is identical to the true adjacency matrix, which further supports our theory.

Reviewer comment (Theory):
Is the condition (sufficient change) of theorem 1 too strict? What are some practical use cases for this?

Author response:
Thanks for this question. The requirement of a sufficient number of environments is actually standard in the literature (see e.g. a recent survey [1]). Various real-world experimental results (e.g., visual disentanglement [2], domain adaptation [3], video analysis [4], and image-to-image translation [5]) in this line of research indicate that it is likely to hold in practice. Meanwhile, for many tasks, the number of environments is actually accessible, just like the number of domains in transfer learning and the number of time indices in time series data. We have updated the manuscript to include this discussion.

Reviewer comment (Theory):
The number of latents needs to be known in advance.

Author response:
Thanks for this comment. Although it is a rather standard assumption in causal representation learning to assume that the number of latents is known in advance (e.g., [1, 6]), let us discuss how to deal with number of latent variables in practice. Some possible approaches are (1) selecting the number of latent variables via model selection, or (2) use a mask to automatically select the number of latent variables, similar to [7]. We will include this discussion in the revised manuscript.

Reviewer comment (Theory):
I am a little confused about one of the main objectives of the paper: What does identifying potential shifts even mean? Even if I can learn $\theta$ up to component-wise invertible transformations? What does knowing $\theta$ tell me? And I don’t think theorem 5 is verified by experiments either.

Author response:
$\theta$ denotes the latent (changing) factor (or effective parameters) associated with each latent structural equation model, which governs the changes in  causal mechanisms. Knowing $\theta$ tells us whether a particular causal mechanism changes across certain domains.

Reviewer comment (Theory):
Can you explain how the last line of the proof of theorem 5 leads to the conclusion? Also where in the proof do you need the modular change assumption?

Author response:
Thanks a lot for this question. The statement $\frac{\partial \theta_l}{\partial \hat{\theta}_i} \frac{\partial \theta_l}{\partial \hat{\theta}_j} = 0$ implies that each $\theta_l$ is a function of at most one of $\hat{\theta}_1,\dots,\hat{\theta}_n$. By considering the inverse of Jacobian matrix, we also have $\frac{\partial \hat{\theta}_l}{\partial \theta_i} \frac{\partial \hat{\theta}_l}{\partial \theta_j} = 0$, indicating that each $\hat{\theta}_l$ is a function of at most one of $\theta_1,\dots,\theta_n$. Combining both of them, $\theta_i$ are identifiable up to component-wise invertible transformation.

Furthermore, the modular change assumption is needed to obtain the equation below Eq. (16). We have modified the manuscript to make these clear.

Reviewer comment (Theory):
The sparsity condition is not made explicit in theorem 2? Can you state it formally?

Author response:
The sparsity constraint implies the minimal number of edges of Markov network $\mathcal{M}_{\hat{Z}}$ over $\hat{Z}$. We have upadated the manuscript to make this explicit.

Reviewer comment (Theory):
Not sure how to interpret the results of theorem 1? It is not intuitive? If the main purpose of theorem 1 is to prove later theorem, maybe it should be a lemma?

Author response:
Thanks for the constructive suggestion. We have modified Theorem 1 to be a lemma in the revised manuscript.

Reviewer comment (Theory):
The proof of theorem 1 uses assumption 3 which is not assumed by theorem 1?

Author response:
Thanks for your careful reading and spotting this. We have fixed the typo in the proof of Theorem 1. That is, the proof does not involve Assumption A3, which instead should be Assumption A2.

Reviewer comment (Theory):
What's the relation between modular change and ICM? Why do you use the "not related via equality constraint"? What if $\theta_i$ and $\theta_j$ are related by some inequalities? Does your result still hold?

Author response:
Thanks for asking this question. One can consider the modular change condition as an instantiation of ICM. We will discuss this conenction in the revised manuscript to explain clearly the relation between modular changes and ICM.

In our case, we have not been aware the situation where $\theta_i$ and $\theta_j$ are related by some inequalities. Intuitively, if it is possible for them to be related by inequality cosntraints, then in this case, one should include inequality constraints into Assumption A2.

Reviewer comment (Novelty):
How is your multi-distribution setting different from soft interventions?

Author response:
If soft intervention is defined this way as in [8], then it is completely general, as long as there is change in the conditional distribution. If so, then we assume soft intervention for all latent causal variables. We will discuss this in the revised manuscript.

Reviewer comment (Theory):
The main "sufficient changes" assumption is not discussed. It is not explained when we can expect it to hold, nor is it evaluated in any actual scenarios. This is the main issue.

Author response:
Thanks a lot for your great suggestions. In light of these, we have added more discussions in the updated manuscript, including more real-world examples. At the same time, we would like to mention that this assumption is common in the literature. For example, [4, 5] require $2n+1$ environments in order for the system to be identifiable, where $n$ is the number of variables. Some real-world experiments (e.g., visual disentanglement [6], domain adaptation [7], video analysis [8], and image-to-image translation [9]) also suggests that this assumption is likely to hold at least in certain real-world scenarios.

Reviewer comment (Presentation):
The presentation is misleading as it positions the paper as "treating the case of multiple environments, which is not treated in the literature". This gives the impression that the approach is a generalisation of a single environment case. However, it is not. The proposed methods cannot be used for a single environment. The method exploits the differences in environments, and there must be many of them.

Author response:
Thanks for your question. We fully agree with you that our method exploits the distributional change to identify latent causal variables and the structure among them. Actually, we thought it is exactly the considered task and didn’t treated it as a generalisation of the single environment case. In fact, the single environment case is way more difficult and most previous works in causal representation learning assume a sufficient number of different environments (e.g., [1,2]). Therefore, the multi-environments case is not a generalization of a single environment case. We have modified the introduction in order to avoid any potential confusion.

Reviewer comment (Writing):
Some critical definitions are not given in the main text. Specifically, "Modular Changes" of Theorem 5 is not defined, and "sparsity constraint" of Theorem 3 is not defined.

Author response:
Thank you for the reminder, and apologize for the potential confusion. The term “modular changes” is defined in A3 of Theorem 5 (“$\theta_i$ across different $i$ are not related via equality constraints”). The sparsity constraint implies the minimal number of edges of Markov network $\mathcal{M}_{\hat{Z}}$ over $\hat{Z}$.  We have updated the theorem statement to make this explicit.

Reviewer comment (Novelty):
There is a significant amount of research on multiple environments via causality that is not mentioned in the paper. The neural network architectures presented in the experiments are small modifications of known architectures (normalising flows for causal discovery). Proper references should be given.

Author response:
Thanks for the suggestion. As detailed in A1, our theoretical results take the existence of multiple environments as an assumption of the identifiability theory for causal representation learning, which differs from previous works trying to address the challenge of multiple environments (e.g., CD-NOD [3]). And one can consider causal representation learning as an extension of causal discovery, which aims at identifying both the latent causal variables and hidden structures. We have incorporated the discussion and futher highlighted the differences in the manuscript as well as the recommended references.

Reviewer comment (Theory):
The setting (1) is restrictive in two ways: first, the observations \(X\) are a deterministic function (i.e., not noisy observations). Second, and more importantly, the observation map \(g\) is assumed to be invertible. For at least somewhat smooth maps this means the dimension of \(X\) and \(Z\) must be the same, which is unrealistic. How important is invertibility?

Author response:
We appreciate the insightful questions. Let us apologize for the typo in the paragraph before Eq. 9. The map is actually injective but not invertible, which is a standard assumption in the previous work in causal representation learning [2,4]. Moreover, since most identifiability theorems in the literature stems from the theory of nonlinear ICA, many works follows the setting of it, where the observation $X$ is a deterministic (nonlinear) function of latent variables $Z$. Our work lies in this line of research and thus follows the previous setting. We will provide additional discussion on it to make the context clearer, according to your constructive suggestions.

Reviewer comment (Theory):
The fundamental assumption of this paper, the Sufficient Changes assumption, appears very strong. It requires independence for every point \(z\). The number of environments must be at least as large as the number of variables (if the same set of environments is good for all \(z\)). This condition must be discussed. When does it hold? Examples? It is suggested to remove the Experiments section and discuss this condition in detail.

Author response:
Thanks a lot for your great suggestions. In light of these, we have added more discussions in the updated manuscript, including more real-world examples. At the same time, we would like to mention that this assumption is common in the literature. For example, [4, 5] require $2n+1$ environments in order for the system to be identifiable, where $n$ is the number of variables. Some real-world experiments (e.g., visual disentanglement [6], domain adaptation [7], video analysis [8], and image-to-image translation [9]) also suggests that this assumption is likely to hold at least in certain real-world scenarios.

Reviewer comment (Theory):
In the experiments section, regarding the phrase at the end of Section 5 about Gaussian noise making it more challenging and aligning with theory (e.g., Theorem 3), it is not clear how this result aligns with the theory, and in particular with Theorem 3. Theorem 3 does not appear to have assumptions that distinguish between Gaussianity and non-Gaussianity. This should be explained.

Author response:
Thanks for raising this point. Actually, Theorem 3 perserves some degree of mixture after our estimation, and that is why we stated that the observation that some components are mixed together align with Theorem 3. Intuitively, we conjecture that non-Gaussianity might be helpful for further decreasing the indeterminacy, just as that in the linear ICA case. Thus, we also tested our model on dataset with non-Gaussianity, and our empirical results indeed show that introducing non-Gaussianity is helpful. However, this is just an empirical validation of our intuition, so we only mentioned it in experiments.

Reviewer comment (Novelty):
It seems like the identification technique with second-order derivatives of the log-density from Lin (1997) is a standard technique for identifying structural causal models (SCMs), e.g., used in [1, 2]. Also, I don’t understand the connections between existing identifiable SCMs, like additive noise models and post-non-linear models, and the statements of Theorems 1 and 3, as those SCMs must be special cases of the proposed Theorems.

Author response:
Q1: Connections to previous works.

A1: Thanks for the question. We fully agree with you that the connection between independence and second-order derivatives of the log-density was the key to the identifiability of some functional causal models. At the same time, the task we are trying to address is different from causal discovery, since we need to first identify the latent causal variables and then discover the hidden structure, all from some general nonlinear mixtures of these causal variables, i.e., causal representation learning. To the best of our knowledge, our proof technique has never been introduced in this literature. 
Furthermore, as you correctly mentioned, we are dealing with a setting that might be more general than existing identifiable SCMs, since we do not have similar constraints on the functional relations and we allow both the causal variables and structures to not be observed. At the same time, these two settings are different since we are incorporating additional information such as sufficient changes in the distribution.

Reviewer comment (Writing):
I found it hard to understand the claims of the Theorems, as some notation was not introduced properly or sufficiently. For example, how are $\tilde{Z}$, $h(\cdot)$, and $q(\cdot)$ defined? What is $m$ in Theorem 1, and which sparsity constraints Theorem 2 refers to? Are $\theta$ one-dimensional factors (parameters)? What is $u$ in Eq. 3? This is problematic, as, e.g., I failed to understand the proof of Theorem 1.

Author response:
Q2: Clarification of theorems and notations.

A2: Thanks for your suggestions and sorry for the potential confusion and typos. We have incorporated the clarification into the updated manuscript. Specifically, we have made the following clarifications/corrections:

- (1) $\tilde{Z}$ should be $\hat{Z}$, which denotes the estimated latent variables.
- (2) $h(\cdot)$ denotes the mapping between the estimated latent variables $\hat{Z}$ and the true variables $Z$. This is also the case for $q(\cdot)$. For consistency, we have changed them to $h$ or $h^{-1}$.
- (3) $m$ should be $u$, which denotes the auxiliary variable.
- (4) $\theta$ represents the latent factor that govern the change of hidden causal mechanisms. $u$ in Eq. 3 denotes the auxiliary variable.
- (5) For Theorem 1, we do not need to assume that $g$ is invertible as long as the function between estimated latent variables $\hat{Z}$ and the true variables $Z$ (i.e., $h$) is invertible, since the change of variable formula only applies between $p(\hat{Z};\theta)$ and $p(Z;\theta)$. Similar to previous work (e.g., [1,2]), $g$ only needs to be injective. We have corrected it in the manuscript.
  
Apologies again for the potential confusion. We sincerely appreciate your detailed suggestions, which are very helpful in improving the clarity of our manuscript.

Reviewer comment (Theory):
Some statements might be even wrong, given the current version of the paper. For example, I do not understand, how the dimensions of $X$ and $Z$ can be different and $g(\cdot)$ is an invertible transformation, and at the same time, the determinant of Jacobian is well-defined (App. A). In the case of the dimensions mismatch, an invertible transformation would be non-continuous (see Netto's theorem). On the other hand, if we assume a same-dimensional manifold embedded in higher-dimensional space, the change of variables formula used in App. A would be different, e.g., see [3].

Author response:
Thank you so much for your prompt response. We are very grateful for your further feedback, which has helped us identify some typos that led to the confusion. We have made the following changes for clarification:

1. In the current manuscript, $m$ has been changed to $u$.

2. Regarding the confusion about the dimensionality of $X$ and $Z$, we realized it was actually caused by a typo. The dimension of $Z$ should be $n$ and that of $X$ should be $d$. So the dimension of $Z$ is actually smaller or equal to that of $X$. We are very sorry about this and have corrected it in the current manuscript. Specifically, the updated part of the manuscript is as follows:

> Let $X=(X_1,\dots,X_d)$ be a $d$-dimensional random vector that represents the observations (e.g., images). As is standard in literature [1,2, 3], we assume that they are generated by $n$ hidden causal variables $Z=(Z_1,\dots,Z_n)$ via a nonlinear injective mixing function $g:\mathbb{R}^n\rightarrow\mathbb{R}^d$ ($d \geq n$), which is also a $\mathcal{C}^2$ diffeomorphism.

Note that this is a standard assumption in related literature, see e.g. Sec. 3.1 in [1], Sec  4.2 in [2], and Assumption 1 in [3]. Since our proof is based on the compound of mixing and demixing functions, we do not have to introduce $g$ in the proof, which is also the case in previous work.

We apologize again for the potential confusion caused by these typos. We hope our clarification and further updates resolve the remaining issues.

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References:

[1] Khemakhem, Ilyes, et al. "Variational autoencoders and nonlinear ICA: A unifying framework." ATSTATS 2020

[2] Hälvä, Hermanni, et al. "Disentangling identifiable features from noisy data with structured nonlinear ICA." NeurIPS 2021

[3] Buchholz, Simon, et al. "Learning linear causal representations from interventions under general nonlinear mixing." NeurIPS 2023

Reviewer comment (Theory):
The major discussion is missing on whether the assumptions in Theorems 1 and 5 are realistic. For example, how could one assume the dimensionality of $Z$ only given $X$? Or even when assuming it, how can we verify, given the data from several environments, whether the number of environments is sufficient? I would love to see a real-world application or a case—study, where we could make those assumptions, or at least speculate about them.

Author response:
Q3: Realisticness of the assumption.

A3: Thanks for your question. As mentioned in A2(5), we do not assume that $Z$ and $X$ are of the same dimension, similar to many previous works in causal representation learning (e.g., [1, 2]). And the determination of the dimension in practice is a typical model selection problem. In practice, one may extend model selection method such as cross-validation or sparsity regularization on a mask to determine the number of dimension of $Z$ [9]. The requirement of a sufficient number of environments is also standard in the literature (see e.g. a recent survey [3]). Various real-world experimental results (e.g., visual disentanglement [5], domain adaptation [6], video analysis [7], and image-to-image translation [8]) in this line of research indicate that it is likely to hold in practice. Meanwhile, for many tasks, the number of environments is actually accessible, just like the number of domains in transfer learning and the number of time indices in time series data. We have updated the manuscript to hightlight the related discussion.

Reviewer comment (Reproducibility):
The provided experiments aim to support the claims of Theorem 3, and I did not find any regarding Theorem 5 for identifying latent factors. The provided experiments still contain very little implementation and evaluation details, and there is no source code available. For example, it is not clear, how the synthetic data is sampled, e.g., what are “noises” and “biases” in Sec. 5.3? Additionally, I don’t understand, how the provided synthetic datasets satisfy the assumptions of Theorem 3, e.g., what is the sufficient number of environments, what is the dimensionality of $X$, or how we ensure that the MLP is an invertible function. On the other hand, the evaluation itself is ambiguous, as it is unclear, how the scatter plots in Figures 2 and 3 support the claim that “the hidden structure is recovered”.

Author response:
Q4: Experiments and evaluations.

A4: Thanks for pointing this out. We assume that the influences from the domain and its parents are location and scaling transformations. Specificically, for each component $z_i$ in each domain, suppose the parents are $PA(i)$, the $z_i=\sum_{j\in PA(i)} s_{j,i}z_j+b_i$, where $s_{j,i}$ are scalars sampled from $Unif[0.5, 2]$ and $b_i\in Unif[-2,2]$. We use 100 domains such that the number of environments is sufficient. As for the nonlinear mixing function, we follow the standard nonlinear ICA literature and construct an MLP whose output dimension is the same as the input dimension. Each layer of MLP consists of an orthogonal matrix weight and a leaky-relu function. Since the weights are orthogonal and the activation is invertible, the MLP is invertible too. For the Y-structure, we have $z_1\rightarrow z_3 \leftarrow z_2$ and $z_3 \rightarrow z_4$. As shown in the Fig. 3(a)(b), the estimated $\hat{z}_1$ recovers $z_2$ and $\hat{z}_1$ is independent from $z_1$; the estimated $\hat{z}_2$ recovers $z_1$ and $\hat{z}_2$ is independent from $z_2$. Therefore, our estimated $\hat{z}_1, \hat{z}_2$ recovers the $z_2, z_1$ and their independence relationship. As for $\hat{z}_3$, it is dependent on $\hat{z}_1$ and $\hat{z}_2$. Therefore, we have $\hat{z}_1\rightarrow \hat{z}_3 \leftarrow \hat{z}_2 \Leftrightarrow z_2 \rightarrow z_3 \leftarrow z_1$. As for $\hat{z}_4$, it is dependent on all previous variables, i.e., $\hat{z}_1, \hat{z}2, \hat{z}_3$. Since we have the sparsity regularization, we can remove the redundant link from $\hat{z}_1$ and $\hat{z}_2$ to $\hat{z}_4$. Therefore, we are able to recover the true latent cause structure. In addition, in eq 6, we have an adjacency matrix to learn, the learned adjacency matrix is identical to the true adjacency matrix, which further supports our theory.